“Is that to say we are against Free Trade? No, we are for Free Trade, because by Free Trade all economical laws, with their most astounding contradictions, will act upon a larger scale, upon the territory of the whole earth; and because from the uniting of all these contradictions in a single group, where they will stand face to face, will result the struggle which will itself eventuate in the emancipation of the proletariat.”
Karl Heinrich Marx · Marx-Engels Collected Works, Vol. VI, p. 290
Originally published on September 10, 2021 Revised and expanded edition · October 29, 2026
· · ·
Abstract
When does a discrepancy between observed and expected frequencies constitute evidence against a hypothesis? Answering this question requires constructing a measure of discrepancy and understanding its behavior under the proposed model. This essay reconstructs that connection, from Pearson’s criterion to quadratic forms in the multivariate normal distribution, ellipsoidal level sets, their transformation into spheres, and the radial integration that yields the chi-squared distribution. It distinguishes exact results from asymptotic approximations, develops the original examples, and provides reproducible R code.
A difference between what is observed and what is expected does not speak for itself. Its meaning depends on the variability of the process, the constraints on the data, and the sampling model. The chi-squared distribution connects these dimensions: it translates certain quadratic discrepancies into probabilities that we can interpret under explicit assumptions.
1. Two frequency problems
Snedecor and Cochran present two situations that capture the problem. A geneticist expects a particular tomato cross to produce red flesh and yellow flesh in a 3:1 ratio. Examining 400 fruits yields 310 red and 90 yellow. If the model were correct, the expected counts would be 300 and 100. Is the difference compatible with sampling fluctuation? [1]
In another example, 480 men and 420 women are observed among 900 hospitalized cases. Under the hypothesis of equal proportions, 450 cases would be expected in each category. How much weight should we attach to a deviation of thirty cases from the expected count?
Both problems require separating two operations: measuring discrepancy and calibrating its magnitude probabilistically. A positive number is not inferentially large or small without a reference distribution and degrees of freedom.
2. Pearson and the historical problem
Karl Pearson, a foundational figure in mathematical statistics and a communist, formulated a criterion for assessing whether a system of discrepancies could reasonably be attributed to random sampling. The central reference for this essay is his 1900 paper in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, volume 50, pages 157–175. This date refers to the cited publication; a reconstruction of its antecedents must distinguish it from other stages of development. [2]
The contribution connects a measure of deviation with a probability calculated under a hypothesis. It links the empirical problem, its mathematical representation, and the distribution of the statistic. The historical interest of the argument lies precisely in that transition.
Pearson’s political characterization is retained in the sense used in the original essay. His scientific contribution and the consistency of all his social positions are separate questions. The following exposition reconstructs the statistical argument; it does not assume that his ideological trajectory was uniform.
The 1900 text uses notation that is unfamiliar today. We therefore present a modern reconstruction and its connections with determinants, cofactors, and principal axes. The multivariate normal distribution provides the starting point for the geometry; the multinomial model will subsequently bring us back to frequencies.
3. Constructing a measure of discrepancy
Let Oi be the observed count and Ei the expected count in category i. With k categories, Pearson’s statistic adds their individual contributions:
XP2 = ∑i=1k(Oi − Ei)2Ei
(01)
Conceptual reading. Subtraction measures disagreement within each category. Squaring prevents positive and negative deviations from canceling and gives greater weight to large discrepancies. Dividing by the expected count weights the deviation according to the scale of the count. Finally, summation combines the individual discrepancies. Each included term requires Ei > 0.
The statistic is a finite sum, not an infinite series. The lower limit indicates where the sequence of terms begins and the upper limit where it ends. When all observed frequencies match their expected values, the statistic is zero. If at least one differs, it is positive.
Weighting by expected counts has a more precise justification than simply “accounting for sample size.” In count models, variability depends on the expected frequency. The multinomial structure introduces dependencies and a joint constraint; we will explain why this sum nevertheless acquires an approximate chi-squared law.
Ei = Npi, ∑i=1k pi = 1
(02)
Conceptual reading. N is the total number of observations and pi the probability of the category under the hypothesis. Multiplying them gives the average count that would arise under repeated sampling. An expected count may be fractional even though every observed count is an integer.
XP2 = N ∑i=1k(p̂i − pi)2pi, p̂i = Oi/N
(03)
Conceptual reading. Written in terms of proportions, the same formula shows that a fixed relative discrepancy receives more weight as N increases. More data can detect smaller deviations; statistical significance and substantive magnitude are therefore different concepts.
4. Variables, dimensions, and covariances
Before introducing matrices, distinguish three concepts. A deviation is the difference between a value and its reference. The standard deviation measures the dispersion of a variable. The standard error of a mean measures the dispersion of that mean across repeated samples.
SE(X̄) = σ/√N
(04)
Conceptual reading. If N observations are independent and have the same finite variance σ², the mean fluctuates less than an individual observation: its standard deviation is σ/√N. Under dependence, this expression must be modified to incorporate covariances. Here N counts observations, not dimensions.
In the geometric development, d denotes the number of components of a vector, N the sample size, and k the number of categories. A vector with d components lies in ℝd. Its effective degrees of freedom may be smaller when constraints apply.
Σij = Cov(Xi, Xj), Σii = σi2
(05)
Conceptual reading. Σ, uppercase sigma, is a d × d matrix recording the variability of each component and the covariation between pairs. Its diagonal contains variances. Other entries can be positive, negative, or zero. The symbol Σ is not the summation operator ∑.
A symmetric matrix for d components contains d variances and d(d−1)/2 distinct off-diagonal covariances. Counting those parameters does not change the dimension of the space containing each observation. Nor does it produce a space of dimension d(d−1).
Σ = DRD, D = diag(σ1, …, σd)
(06)
Conceptual reading. R is the correlation matrix, and D contains the standard deviations, assumed positive. R describes associations without units; multiplying by D on both sides restores the original scales. This identity distinguishes a formulation using standardized variables from one using variables in their original units.
5. The quadratic form and the inverse
Suppose X has a multivariate normal distribution with population mean μ and positive definite population covariance matrix Σ. Its joint density can be written as follows. [3]
fX(x) = 1(2π)d/2|Σ|1/2 exp[−½(x − μ)⊤Σ−1(x − μ)]
(07)
Conceptual reading. The density describes probabilistic concentration around x; integrating it over a region gives that region’s probability. The initial factor normalizes the total probability to one. |Σ| is the determinant, which adjusts the volume scale. The exponential decreases as the quadratic distance from the center increases. This is not a cumulative distribution function.
Q = (X − μ)⊤Σ−1(X − μ)
(08)
Conceptual reading. X−μ expresses displacement from the center. Σ−1 weights that displacement for dispersion and correlations. Transposition, denoted by ⊤, converts the first column vector into a row so that the product returns a scalar. Q is the squared Mahalanobis distance and has no units.
The exponent −1 denotes a matrix inverse: multiplying Σ by its inverse gives the identity matrix, the operation that leaves every vector unchanged. It does not represent a power of −1, subtraction, or alternating terms. The entries are not inverted separately either.
d = 1: Q = (X − μ)2/σ2
(09)
Conceptual reading. A deviation of ten units yields Q=25 when the standard deviation is two, and Q=0.25 when it is twenty. The same absolute displacement can be exceptional or ordinary depending on dispersion. In multiple dimensions, we must also adjust for the information shared by correlated variables.
Q = z12 − 2ρz1z2 + z221 − ρ2, |ρ| < 1
(10)
Conceptual reading. z1 and z2 are deviations divided by their standard deviations; ρ is their correlation. The cross term adjusts for the joint direction of displacement. If ρ is positive, moving in the same direction in both variables is less unusual than moving in opposite directions. If ρ=0, Q reduces to the sum of the two squares. The negative sign in this expansion comes from matrix inversion, not an alternating series.
6. From historical to matrix notation
Pearson’s argument combines deviations, scales, and correlations. Its structure is now recognizable as a quadratic form. For standardized variables z, the relevant matrix is R, the correlation matrix; for unstandardized deviations, it is Σ. [2][4]
Conceptual reading. Diagonal terms weight squared components. Cross terms incorporate relationships between components. Since the inverse matrix is symmetric, each pair appears twice in the full sum: aijzizj and ajizjzi. Combining them explains the factor 2. The condition i<j counts each pair once.
Mij = det(R with row i and column j removed), Cij = (−1)i+jMij
(12)
Conceptual reading. Deleting one row and one column produces a submatrix; its determinant is the minor Mij. The cofactor Cij introduces a positive sign when i+j is even and a negative sign when it is odd. Here an alternation of signs does occur, arising from the determinant’s structure. It must not be confused with the inverse exponent −1.
(R−1)ij = Cji/det(R)
(13)
Conceptual reading. Transposing the cofactor matrix forms the classical adjugate and explains the interchange of indices. Dividing by a nonzero determinant completes the inversion. Minors and cofactors are not defined through elementary row operations, although those operations can help calculate determinants and inverses.
This translation preserves the algebraic content of the historical exposition while replacing less explicit summation symbols with defined limits. It also distinguishes an observed realization q from the random variable Q: fixing a value of the quadratic form defines a surface; allowing the vector to vary generates a distribution of values.
7. Ellipsoids, spheres, and topology
(x − μ)⊤Σ−1(x − μ) = c, c > 0
(14)
Conceptual reading. All points on this surface have the same squared Mahalanobis distance and the same normal density. Replacing equality with ≤ describes the interior together with its boundary. In two dimensions, the boundary is an ellipse; in three, it is an ellipsoidal surface. The bell-shaped density graph is a different object: it adds a further coordinate representing height.
Principal axes show the natural directions of dispersion. A noncircular ellipse has perpendicular major and minor axes centered at μ. In d dimensions, we choose d orthogonal directions. If some eigenvalues coincide, axes within the corresponding subspace are not unique, but an orthogonal choice is possible.
Σ = UΛU⊤, Λ = diag(λ1, …, λd), U⊤U = I
(15)
Conceptual reading. The columns of U are unit eigenvectors defining the principal directions. The positive eigenvalues λj are variances along those directions. At level c, each semiaxis has length √(cλj). The identity matrix I has ones on its diagonal and zeros elsewhere: it represents leaving a vector unchanged.
Y = Λ−1/2U⊤(X − μ), Q = Y⊤Y = ∑j=1d Yj2
(16)
Conceptual reading. First subtract the center. U⊤ expresses the displacement in principal-axis coordinates. Λ−1/2 divides each coordinate by √λj, giving unit variance. Ellipsoids become spheres. Under normality, the coordinates of Y are independent standard normals: lack of correlation implies independence here because of the normality assumption.
Figure 1. A two-dimensional schematic of whitening. The contours are levels of a quadratic form, not a density graph. The drawing illustrates the procedure; the scales in the two panels are conventional.
dx = |Σ|1/2 dy
(17)
Conceptual reading. The inverse transformation is x=μ+UΛ1/2y. Translation does not change volumes; an orthogonal matrix has a determinant with absolute value one; and the product of the factors √λj is |Σ|1/2. This factor cancels the determinant term in the density normalizer. Probabilities are preserved by jointly transforming the region, density, and volume element.
8. Radial integration and solid angles
In whitened coordinates, the density depends only on distance r from the origin. The probability of falling beyond a given radius is obtained by integrating over the entire exterior of a sphere, not over a single surface. Distinguishing radius from squared radius will be decisive.
r = ‖y‖ = √(y⊤y), q = r2
(18)
Conceptual reading. r measures distance and q measures squared distance. The radius follows a chi distribution; its square follows a chi-squared distribution. We use r as the radial integration variable and q as the variable of the final distribution to keep their densities distinct.
dy = rd−1 dr dΩ, ∫Sd−1 dΩ = 2πd/2Γ(d/2)
(19)
Conceptual reading. A thin spherical shell has a volume approximately equal to its surface area times its thickness dr. Scaling the radius by a factor r scales its d−1 tangential directions by that factor, producing rd−1. dΩ captures directions on the unit sphere Sd−1. For d=2, the element is r dr dθ; for d=3, it is r² dr dΩ. In d=1 there are two directions, one on each half-line.
In three dimensions, a solid angle measures the area of a region’s radial projection onto the unit sphere; its unit is the steradian. The whole sphere spans 4π steradians. A face of a cube centered at the origin subtends 2π/3: its six faces divide spatial directions symmetrically. This is the geometric construction illustrated in the original document. In other dimensions, we speak of angular measure on the unit sphere. [6]
Figure 2. Schematic of radial projection, inspired by the cube construction cited by Weisstein. By symmetry, each face of a cube centered at the origin subtends one sixth of the total solid angle. The lines are schematic and do not represent areas to scale.
P(Q ≥ q0) = ∫√q₀∞ e−r²/2rd−1 dr∫0∞ e−r²/2rd−1 dr
(20)
Conceptual reading. The numerator accumulates mass from the observed radius to infinity; the denominator normalizes by total radial mass. The angular integral and common constants cancel because all directions are included in both cases. This cancellation follows from radial symmetry and the same angular region, not from matching asymptotic behavior of eigenvectors.
These are volume integrals reduced using polar coordinates. They do not assign positive probability to a single surface of fixed radius: in this continuous model, that surface has probability zero. Probability mass occupies regions with radial thickness.
q = r2, dr = dq2√q, rd−1dr = ½qd/2−1dq
(21)
Conceptual reading. Changing the variable also changes the width of each interval. The factor 1/(2√q) adjusts for that change. Multiplying it by rd−1 produces the exponent d/2−1. This “minus one” comes from the Jacobian and the combination of powers; it does not indicate an alternating series either.
Γ(a) = ∫0∞ ta−1e−t dt, a > 0
(22)
Conceptual reading. Γ is the function that supplies the density normalizer. It extends the factorial: for positive integers m, Γ(m)=(m−1)!. Substituting t=r²/2 in the total radial integral gives 2d/2−1Γ(d/2). The preceding ratio thus becomes a calculable, normalized probability.
9. The chi-squared law and its properties
fQ(q) = qd/2−1e−q/22d/2Γ(d/2), q > 0
(23)
Conceptual reading. The power term captures shell geometry and the change of variable; the exponential captures the decline of the normal density with distance. The denominator makes the integral over all positive values equal to one. This is a one-dimensional density, even though it arises from a d-dimensional problem.
We write Q ∼ χ²d: ∼ means “has distribution,” whereas = states equality between objects or expressions. In this derivation, d is a positive integer because it counts independent normal coordinates. The gamma family defines χ² distributions for any positive real number of degrees of freedom. [7]
Conceptual reading. Each squared standard normal contributes mean one and variance two. Summing d independent terms adds their means and variances. The gamma equivalence specifies scale two; under the rate parameterization, the rate is 1/2. Confusing rate and scale changes the model.
The support is [0,∞), although the density formula is stated for q>0. The density can diverge as q approaches zero when the degrees of freedom are less than two, without creating a point mass at zero. For d=2 it is exponential; increasing d makes it less asymmetric. A density can exceed one: it is integrated probability that must lie between zero and one.
Figure 3. Curves calculated from the χ² density. The vertical scale extends to 0.66; the ν=1 curve continues toward infinity as q approaches zero. The curves are densities, not cumulative probabilities.
p = PH₀(Q ≥ qobs) = ∫qobs∞ fQ(q)dq
(25)
Conceptual reading. The p-value is the probability, under H₀ and the model assumptions, of a statistic at least as discrepant as the observed one. Including or excluding the endpoint does not change this continuous integral. It is neither the probability that H₀ is true nor the probability that the result is “due to chance” in an absolute sense.
10. Why it works for counts
The preceding normal result is exact when its assumptions hold and population μ and Σ are used. Category counts are not exactly normal. Connecting them with χ² requires the multinomial model and a limiting argument.
(O1, …, Ok) ∼ Multinomial(N; p1, …, pk)
(26)
Conceptual reading. Each independent observation belongs to exactly one category, the categories are exhaustive, and their probabilities remain constant across observations. The counts sum to N. Although individual observations are independent, counts in different categories are not: when one increases with the total fixed, less room remains for the others.
Var(Oi) = Npi(1−pi), Cov(Oi,Oj) = −Npipj (i≠j)
(27)
Conceptual reading. The fixed total induces negative dependence. This explains why it would be incomplete to justify Pearson’s statistic by assuming that each count has variance exactly equal to its expectation and that all counts are independent. That idea is natural under an independent Poisson model, but the multinomial requires incorporating the constraint on the total.
∑i=1k(Oi−Npi) = 0
(28)
Conceptual reading. Once k−1 deviations are known, the total determines the remaining one. There are therefore only k−1 free directions, and the joint covariance matrix is singular in the full space. The relevant geometry lies in a subspace of dimension k−1.
Wi = Oi−Npi√(Npi), Cov(W) = I−ss⊤, s = (√p1, …, √pk)⊤
(29)
Conceptual reading. W denotes the vector of Pearson residuals. Dividing by √(Npi) produces dimensionless residuals. Because the probabilities sum to one, s has unit length. The matrix I−ss⊤ removes direction s and leaves the k−1 directions orthogonal to it unchanged. It is a projection of rank k−1; the covariance identity is exact under the multinomial model.
With k fixed, fixed positive probabilities, and increasing N, the multinomial central limit theorem approximates these residuals by a normal vector in the free subspace. Choosing an orthonormal basis in that subspace gives k−1 independent standard normal coordinates. The sum of their squares yields the limit:
XP2 = ∑i=1kWi2 ⟶distribution χ²k−1
(30)
Conceptual reading. The arrow denotes convergence in distribution as N increases, not exact equality for every sample. This is the bridge between normal geometry and counts. Subtracting one reflects the constraint on the total rather than an arbitrary rule.
ν = k−1−m
(31)
Conceptual reading. In regular goodness-of-fit problems, estimating m identifiable parameters can consume m additional directions. This formula requires appropriate estimation and regularity conditions, such as maximum likelihood based on the grouped counts. Mechanically subtracting parameters is insufficient: estimating them from ungrouped data before grouping can alter calibration. Structurally impossible categories also need specific treatment. [9]
11. The original examples, worked through
11.1. Tomatoes: a 3:1 ratio
The hypothesis specifies probabilities of 0.75 and 0.25. The 400 fruits are treated as independent observations under the model; clustering by plant or batch would require reassessing that assumption.
Table 1. Decomposing discrepancy in the genetic example.
Conceptual reading. The absolute deviations are equal, but ten fruits carry more weight in a category with an expected count of one hundred than in one with an expected count of three hundred. With one degree of freedom, the approximate right-tail probability is 0.2482. At α=0.05, the 3:1 ratio is not rejected; this does not prove it is exactly true.
11.2. Hospital cases: a 1:1 ratio
Table 2. Decomposing the hospital example under equal proportions.
Category
O
E
O − E
(O − E)²/E
Men
480
450
30
2
Women
420
450
−30
2
Total
900
900
0
4
XP2 = 900/450 + 900/450 = 4; ν = 1; p ≈ 0.045500
(33)
Conceptual reading. Under equal proportions, both categories contribute equally. The value 4 is not interpreted in isolation: with one degree of freedom, its right-tail probability is approximately 0.0455. At α=0.05, equal proportions are rejected within the model; at α=0.01, they are not. The significance level should be chosen before examining the result.
Figure 4. Shading represents the right tail in the hospital example. The plot ends at q=12; the stated probability integrates to infinity. The curve’s height is not the p-value.
In the second example, the observed proportion of men is 480/900≈0.5333: about 3.33 percentage points above 0.5. This illustrates how a relatively small discrepancy can cross a conventional threshold. Its clinical or social relevance requires a substantive question and suitable data.
12. Applications, conditions, and limitations
Goodness of fit
This compares counts with specified probabilities. Categories must follow a clear definition; when continuous data are grouped, interval choices affect the result. Each interval’s probability must be calculated under the corresponding model. If parameters are estimated, the reference distribution requires justification.
Independence and homogeneity
In a table with r rows and c columns, independence states that joint probabilities factor into marginal probabilities. Homogeneity compares categorical distributions across populations or groups. Both can produce the same statistic, although their sampling designs differ.
Eij = Oi+O+jN, ν = (r−1)(c−1)
(34)
Conceptual reading. Oi+ is the total for row i and O+j the total for column j. Their proportions estimate marginal probabilities under independence. The degrees-of-freedom formula accounts for the relevant constraints in an ordinary full table without structural zeros or other restrictions.
The count table contains observed values; expected values are calculated under the hypothesis. In R, xtabs() constructs the observed table, while chisq.test(... )$expected provides expected counts. Independence of observations does not follow from the independence that the test evaluates: it is a design assumption.
V = √XP2N min(r−1,c−1)
(35)
Conceptual reading. V summarizes the magnitude of association in a contingency table using the uncorrected Pearson statistic. It indicates neither direction nor causality; its interpretation depends on table structure and may benefit from bias corrections and intervals. In a 2 × 2 table, it equals the absolute value of φ. Residuals and proportions show which cells account for the association.
Variance of a normal population
The same distributional family appears in another problem with different assumptions: inference about a normal population variance.
T = (N−1)S2σ2 ∼ χ²N−1
(36)
Conceptual reading. For an independent sample from a normal population, S² uses N−1 in its denominator. Estimating the mean consumes one direction. Under H₀: σ²=σ₀², σ₀² is used in T. Depending on the alternative, the test can use the left tail, the right tail, or both; not every test based on χ² is therefore a right-tailed test. [10]
Diagnostics before calculation
Observational unit: identify what is being counted and whether dependence, pairing, clustering, or repeated measurements are present.
Counts: use frequencies rather than percentages without their sample size. Weighted frequencies from complex surveys may require design adjustments such as Rao–Scott corrections.
Categories: they must be mutually exclusive and exhaustive under the model; structural zeros are not ordinary sampling zeros.
Expected counts: inspect their magnitudes and distribution across cells. “All at least five” is a conservative rule of thumb, not a universal theorem or a condition on observed counts.
Sparse data: consider an exact procedure, simulation consistent with the hypothesis, or scientifically justified pooling. Do not pool categories to obtain a desired result.
Reporting: accompany the statistic with degrees of freedom, p-value, design, diagnostics, proportions, and a suitable measure of magnitude.
In 2 × 2 tables, Yates’s continuity correction modifies the approximation. R applies it by default in that case through chisq.test(), but it is not a universal remedy for sparse data. Fisher’s exact test and Monte Carlo methods use specific reference distributions. For tables, simulation in chisq.test() conditions on both margins, requires positive margins, and omits Yates’s correction; it does not automatically reproduce every sampling design. [11]
13. Reproduction and exploration in R
The code uses base R. RStudio can provide the working environment, but the functions belong to R. No external data files are required. The numerical outputs stated above are deterministic approximations; simulation experiments illustrate variability and depend on their replicates.
13.1. Manual calculation and the built-in test
# Original examples: probabilities fixed by H0.
# No parameters of the category probabilities have been estimated.
pearson_fixed <- function(observed, probabilities) {
stopifnot(is.numeric(observed), is.numeric(probabilities),
length(observed) == length(probabilities),
length(observed) >= 2L,
all(is.finite(observed)), all(observed >= 0),
all(observed == floor(observed)), sum(observed) > 0,
all(is.finite(probabilities)), all(probabilities > 0),
abs(sum(probabilities) - 1) < 1e-10)
expected <- sum(observed) * probabilities
contributions <- (observed - expected)^2 / expected
statistic <- sum(contributions)
df <- length(observed) - 1L
# Direct upper tail avoids subtracting a probability near 1 from 1.
p_value <- pchisq(statistic, df = df, lower.tail = FALSE)
automatic <- chisq.test(observed, p = probabilities)
stopifnot(isTRUE(all.equal(statistic,
unname(automatic$statistic))),
isTRUE(all.equal(p_value, automatic$p.value)))
list(table = data.frame(observed, expected, contributions),
X2 = statistic, df = df, p = p_value,
test = automatic)
}
tomatoes <- pearson_fixed(c(red = 310, yellow = 90), c(0.75, 0.25))
hospital <- pearson_fixed(c(men = 480, women = 420), c(0.5, 0.5))
tomatoes$table
c(X2 = tomatoes$X2, df = tomatoes$df, p = tomatoes$p)
# X2 = 1.333333333; df = 1; p = 0.248213079
hospital$table
c(X2 = hospital$X2, df = hospital$df, p = hospital$p)
# X2 = 4; df = 1; p = 0.045500264
# Exact binomial alternatives: these are not Pearson approximations.
binom.test(310, 400, p = 0.75)
binom.test(480, 900, p = 0.5)
pchisq() returns cumulative probabilities; lower.tail = FALSE requests the upper tail. Given a vector of counts and probabilities, chisq.test() performs a goodness-of-fit test. The correct argument concerns 2 × 2 tables and does not change these vector-based tests. The checks compare manual summation with the built-in function.
13.2. An illustrative contingency table
The following table is an added example constructed to demonstrate the procedure; it is not part of the original data and does not represent an empirical study.
tab <- matrix(c(40, 20, 30, 50), nrow = 2, byrow = TRUE,
dimnames = list(group = c("A", "B"),
response = c("Yes", "No")))
test <- chisq.test(tab, correct = FALSE)
test
test$observed
test$expected
test$residuals # (O - E) / sqrt(E)
test$stdres # Adjust for the variance of each residual.
X2_manual <- sum((tab - test$expected)^2 / test$expected)
df_manual <- (nrow(tab) - 1) * (ncol(tab) - 1)
p_manual <- pchisq(X2_manual, df = df_manual, lower.tail = FALSE)
V <- sqrt(X2_manual /
(sum(tab) * min(nrow(tab) - 1, ncol(tab) - 1)))
c(X2 = X2_manual, df = df_manual, p = p_manual, V = V)
stopifnot(isTRUE(all.equal(p_manual, test$p.value)))
# Compare procedures: explicitly report which one is used.
chisq.test(tab, correct = TRUE) # Yates, only for 2 x 2 tables.
fisher.test(tab) # Fisher's exact test.
set.seed(20261029)
chisq.test(tab, simulate.p.value = TRUE, B = 99999)
With B=99,999 replicates, the smallest attainable simulated p-value is 1/(B+1)=0.00001. This resolution is not precision: the estimate retains Monte Carlo error. Standardized residuals help locate discrepancies, but assessing many cells requires considering multiplicity and dependence. [11]
13.3. From the normal distribution to χ²: checking the geometry
set.seed(20261029)
B <- 50000L
d <- 3L
Sigma <- matrix(c(4, 1, 0.5,
1, 2, 0.2,
0.5, 0.2, 1), nrow = d, byrow = TRUE)
# chol returns upper-triangular C with t(C) %*% C = Sigma.
C <- chol(Sigma)
Z <- matrix(rnorm(B * d), nrow = B, ncol = d)
X <- Z %*% C # Each row has mean zero and covariance Sigma.
Q <- rowSums((X %*% solve(Sigma)) * X)
Q_standard <- rowSums(Z^2)
stopifnot(max(abs(Q - Q_standard)) < 1e-8)
c(empirical_mean = mean(Q), theoretical_mean = d,
empirical_variance = var(Q), theoretical_variance = 2 * d)
hist(Q, breaks = 70, probability = TRUE,
col = "#f5f0e8", border = "#d4c8b0",
main = "Normal quadratic form and chi-squared law",
xlab = "q")
curve(dchisq(x, df = d), add = TRUE, col = "#8b4513", lwd = 2)
# solve(Sigma) makes the inverse explicit for teaching purposes.
# For large problems, prefer factorizations and linear solves.
The identity between the two calculations of Q holds to numerical precision. By contrast, empirical means and variances only approximate their theoretical values because the simulation has finitely many replicates.
13.4. From counts to the asymptotic limit
set.seed(20261029)
B <- 30000L
prob <- c(0.2, 0.3, 0.5)
df <- length(prob) - 1L
threshold <- qchisq(0.95, df = df)
explore <- function(N) {
O <- rmultinom(B, size = N, prob = prob) # One sample per column.
E <- N * prob
residuals <- sweep(O, 1, E, "-")
contributions <- sweep(residuals^2, 1, E, "/")
X2 <- colSums(contributions)
c(N = N, minimum_expected = min(E),
mean = mean(X2), rejection_005 = mean(X2 > threshold),
percentile_95 = unname(quantile(X2, 0.95)))
}
t(vapply(c(20L, 100L, 1000L), explore, numeric(5)))
qchisq(0.95, df = df)
# The mean of X2 is k - 1 even at finite N under this model.
# That does not make the chi-squared distribution exact.
# The percentile and rejection rate examine the approximation.
# Density, CDF, quantile, and random generation are distinct operations:
dchisq(4, df = 1) # Density height.
pchisq(4, df = 1, lower.tail = FALSE) # Tail probability.
qchisq(0.95, df = 1) # 95th-percentile threshold.
rchisq(5, df = 1) # Five simulated values.
Simulation evaluates the procedure under a particular model; it does not guarantee validity for other designs. An approximation can improve with N while still exhibiting Monte Carlo fluctuation. The discreteness of counts also persists in a finite sample.
14. Synthesis and glossary
Discrepancy becomes interpretable evidence when its scale, constraints, and calibrating distribution are made explicit.
The journey began with observed and expected frequencies. The multivariate normal distribution represented discrepancy through a quadratic form with ellipsoidal levels. Centering, rotating, and rescaling produced standard normal coordinates; integrating over radial shells and squaring the radius yielded the χ² law. The multinomial model brought the argument back to counts and explained the asymptotic character of Pearson’s test.
This connection does not turn every quadratic deviation into a chi-squared variable, every geometric relationship into probabilistic equivalence, or every significant result into a causal conclusion. Its strength lies in showing how assumptions enable the transition between observation, mathematical structure, and inference.
Table 3. Notation and concepts to consult while reading.
Symbol or concept
Meaning in this essay
N
Sample size or total count.
d
Dimension of the nondegenerate normal vector.
k
Number of categories in a goodness-of-fit test.
ν
Degrees of freedom of the reference distribution.
Σ / ∑
Covariance matrix / summation operator.
Σ⁻¹
Matrix inverse, existing under nonsingularity.
R / W
Correlation matrix / vector of Pearson residuals.
Q / q
Random quadratic variable / a value of that variable.
X²P
Pearson’s statistic for counts.
χ²ν
Family of distributions indexed by degrees of freedom.
r / q=r²
Radius / squared radius.
Density
Concentration whose integral over a region gives probability.
Homoscedasticity
Constancy or equality of variances within a model.
α / p-value
Nominal level of the procedure / tail probability under the hypothesis.
Diffeomorphism
A smooth bijection with a smooth inverse.
15. References and the original paper
The original references are retained as part of the intellectual development. Added sources document the modern formulation and applications. The revised mathematical exposition is an explanatory reconstruction, not a verbatim transcription of the 1900 paper. The linked original essay is in Spanish; this English edition presents the revised argument in full.
Snedecor, G. W., and Cochran, W. G. (1967). Statistical Methods, 6th ed. Iowa State University Press. See the introductory examples on pp. 20–21 cited in the original paper.
Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling.Philosophical Magazine, series 5, 50(302), 157–175. DOI and publisher’s record.
Penn State, STAT 505. Multivariate Normal Distribution. Multivariate normality, Mahalanobis distance, and ellipsoidal contours. Teaching materials.
Boyd, S. (2007). Symmetric matrices, quadratic forms, matrix norm, and SVD. EE263, Stanford University. Teaching reference cited in the original. 2007–2008 course archive. See also Lall, S., Symmetric matrices and quadratic forms, EE263 notes.
Gómez Julián, J. M. (2021; originally published under the name Isadore Nabi). Hacia una interpretación dialéctica-materialista de la topología general: génesis histórica-teórica de la topología desde la geometría y la teoría de conjuntos [Toward a dialectical-materialist interpretation of general topology: its historical and theoretical genesis from geometry and set theory]. The author’s essay, in Spanish. It contextualizes reflection on structural equivalences; probabilistic conditions are justified in the present text.
Weisstein, E. W. Solid Angle. MathWorld, Wolfram Research. Consulted for the original essay in 2021. Definition and cube example.
Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung [Foundations of the Theory of Probability]. Springer. Historical reference added to contextualize the axiomatization of probability.
Wikipedia. Elipse [Ellipse]. Geometric source cited in the original and consulted there in 2021. Reference entry, in Spanish. Retained for traceability; the algebraic reconstruction relies on the teaching sources listed above.
The essay archive
The original version preserves the 2021 formulation and presentation in Spanish. This English page develops its examples, clarifies its notation, and expands the geometric and statistical foundations.
Mathematical statistics · Theory and practice in R
The Quantile Function in R
Theoretical foundations, probabilistic interpretation, and applications
JOSÉ MAURICIO GÓMEZ JULIÁN
Originally published on September 5, 2021 Revised and expanded edition · October 29, 2026 English translation of the revised Spanish edition
Knowing a probability and knowing the value associated with it are related but distinct problems. We may ask what proportion of a distribution lies below a particular score; we may also ask which score marks a proportion chosen in advance. The cumulative distribution function answers the first question. The quantile function allows us to answer the second.
This article develops that relationship from its mathematical definition to its implementation in R. The running example retains a normal model for test scores with mean 72 and standard deviation 15.2, showing why a probability of 0.7850824 yields approximately 84 for the lower tail and 60 for the upper tail. It then examines discrete quantiles, sample quantiles, and the role of critical values in inference.
1. Two different questions
Suppose a test score is represented by a random variable X. If we fix a score x, we can calculate the probability that X is less than or equal to it. If we fix a probability p, we can seek the score that reaches that cumulative level. In one case, the threshold is given; in the other, the probability level is given.
From value to probability and from probability to value
Question
Normal-distribution function
Returns
What probability lies to the left of x?
pnorm(x, ...)
A probability
Which value corresponds to cumulative probability p?
qnorm(p, ...)
A threshold
What is the density at x?
dnorm(x, ...)
A density height
How can observations be generated from this model?
rnorm(n, ...)
n pseudorandom realizations
The prefixes d, p, q, and r organize many of the probability-distribution families available in R. For discrete variables, d…() functions return probability masses; for continuous variables, they return densities. A density height is not the probability of observing exactly that value: in the continuous case, probabilities are obtained over intervals. See references [1] and [2].
2. The cumulative distribution function
For every real-valued random variable X, the cumulative distribution function is defined as:
This function is nondecreasing and right-continuous; its limits are 0 as x tends to −∞ and 1 as x tends to +∞. It may be continuous, have jumps, or combine both features. Its existence does not depend on our knowing an elementary expression that can be evaluated by hand.
For a standard normal distribution, the cumulative distribution function is denoted by Φ:
This integral is well defined, although it cannot be expressed using elementary functions. In R, pnorm() evaluates it numerically and qnorm() provides its quantiles. Quantiles are useful because of the question they answer, not because the cumulative distribution function is somehow absent.
The cumulative distribution function starts with a value and measures probability; the quantile starts with a probability and locates a value.
3. The quantile function as a generalized inverse
To include distributions with jumps or constant stretches, we define the lower quantile as:
The symbol inf denotes the infimum: the greatest lower bound of the set of values whose cumulative probability reaches or exceeds p. For a real-valued random variable and a level strictly between 0 and 1, this definition selects the first threshold satisfying the condition.
If F is continuous and strictly increasing, the quantile function coincides with its ordinary inverse and F(Q(p)) = p. With jumps, that equality may fail. The relevant general relationship is:
Here, F(q−) = P(X < q) is the left-hand limit, whereas F(q) = P(X ≤ q) includes any probability mass at the threshold. Thus, in general, a quantile does not leave exactly a proportion p below it: it ensures that the cumulative level is reached or exceeded.
Properties that aid interpretation
Order: as the level p increases, the quantile does not decrease.
Units: if X measures scores, incomes, or times, its quantiles retain those units.
Position:Q(0.5) is a choice of median; Q(0.25) and Q(0.75) are the quantiles associated with the first and third quartiles.
Convention: when a distribution has several medians, the definition above selects one of them. The convention used should be stated.
The terms percent-point function and inverse cumulative distribution function are also used. A percentile expresses the level p on a scale of one hundred: the 95th percentile corresponds to p = 0.95. For finite samples, its calculation also requires specifying a method.
4. The test-score example: one probability and two tails
Suppose test scores follow, as a working model, a normal distribution with mean μ = 72 and standard deviation σ = 15.2:
The second parameter in this notation is the variance, σ². By contrast, the R argument sd takes the standard deviation directly. For this family:
This identity shows how a standard normal quantile is transformed to the original scale by multiplying it by σ and adding μ. The expression qnorm(p) uses μ = 0 and σ = 1 by default; supplying other parameters specifies a different normal distribution rather than the standard normal distribution.
4.1. Lower tail: the threshold is approximately 84
We seek a score whose lower-tail cumulative probability is 0.7850824. The call and its verification are:
R · Lower tail and verification
target_p <- 0.7850824
score_mean <- 72
score_sd <- 15.2
lower_threshold <- qnorm(target_p, mean = score_mean,
sd = score_sd)
lower_threshold
# Approximately 84.00000012
pnorm(lower_threshold, mean = score_mean, sd = score_sd)
# Approximately 0.7850824
Under the model, approximately 78.50824% of scores are less than or equal to 84. The returned value is very close to 84, but it is not identical in exact arithmetic: the supplied decimal probability has been rounded.
4.2. Upper tail: the threshold is approximately 60
We now seek a value above which that same probability remains. The tail changes while the requested probability level stays fixed:
R · Upper tail and verification
upper_threshold <- qnorm(target_p, mean = score_mean,
sd = score_sd, lower.tail = FALSE)
upper_threshold
# Approximately 59.99999988
pnorm(upper_threshold, mean = score_mean,
sd = score_sd, lower.tail = FALSE)
# Approximately 0.7850824
pnorm(upper_threshold, mean = score_mean, sd = score_sd)
# Approximately 0.2149176
The interpretation is that approximately 78.50824% of the distribution lies above 60. Its lower-tail cumulative probability is approximately 0.2149176. For this continuous distribution, including or excluding the exact threshold does not change the probability, because every individual point has zero probability mass.
Figure 1. The shaded area represents approximately 78.50824% in both panels. The distribution stays the same; the tail and the threshold change. The curve is shown over a finite range for readability.
For a normal distribution, symmetry explains why both thresholds are equally far from the mean: 84 − 72 = 72 − 60 = 12, up to numerical rounding. This symmetry is a property of this model and should not be assumed for every distribution.
The same probability p = 0.7850824
Requested tail
Threshold
Probability under the model
Lower
≈ 84
P(X ≤ 84) ≈ 0.7850824
Upper
≈ 60
P(X > 60) ≈ 0.7850824
Lower tail at the second threshold
≈ 60
P(X ≤ 60) ≈ 0.2149176
Figure 2.pnorm() follows the relationship from value to probability; qnorm() allows it to be followed from probability to value.Figure 3. The quantile curve takes probabilities on the horizontal axis and returns scores on the vertical axis. For a nondegenerate normal distribution, its limits are −∞ and +∞ as the probability approaches 0 and 1.
4.3. Is a normal model appropriate for test scores?
The normal distribution has support on the entire real line. If scores are bounded between 0 and 100, the model assigns some probability to impossible values. With these parameters, the probability above 100 is approximately 3.27%: a discrepancy worth examining, especially when high percentiles are of interest.
Its use requires examining the shape of the data and the purpose of the analysis. Depending on the situation, empirical quantiles, a truncated distribution, or a model with bounded support may be more appropriate. The example illustrates the mechanism of inversion; it does not automatically validate normally distributed scores.
5. Syntax, arguments, and numerical precision in R
A valid call with all the main arguments specified explicitly is:
R · Complete call
qnorm(p = 0.95, mean = 0, sd = 1,
lower.tail = TRUE, log.p = FALSE)
# Approximately 1.644854
Interpreting the arguments
Argument
Meaning
Main consideration
p
Probability level
Use 0.95, not 95, for 95%
mean
Model mean
If using an estimate, state how it was obtained
sd
Model standard deviation
Neither variance nor standard error
lower.tail
Selects the lower or upper tail
The supplied probability refers to the selected tail
log.p
Indicates whether the input is the natural logarithm of the probability
Does not request the logarithm of the returned quantile
An explanatory phrase such as “TRUE or FALSE” should not be copied literally as an argument: a specific logical value must be chosen in each call. It is preferable to spell out TRUE and FALSE. The abbreviations T and F can be reassigned and may become ambiguous during a working session.
5.1. Complementary tails
For a normal distribution and an interior probability level p, the upper-tail threshold corresponds to the lower-tail quantile at level 1 − p. However, changing lower.tail returns a different value, not the complementary probability as its output. In extreme tails, it is preferable to supply the probability of the tail of interest directly.
R · Logarithms and extreme tails
qnorm(log(0.95), log.p = TRUE)
# The same quantile as qnorm(0.95)
qnorm(1 - 1e-20)
# Inf: the subtraction rounds to 1
qnorm(1e-20, lower.tail = FALSE)
# Approximately 9.26234
qnorm(-1000, lower.tail = FALSE, log.p = TRUE)
# Finite quantile for upper-tail probability exp(-1000)
Subtracting a tiny probability from 1 can produce exactly 1 in floating-point representation. Thus, qnorm(1 - 1e-20) may return Inf, whereas the upper-tail call retains a finite threshold. With log.p = TRUE, a probability can be represented by its logarithm even when the ordinary probability is too small to be stored with useful precision.
For the nondegenerate normal interpretation, we assume sd > 0. R also handles sd = 0 as a point mass and flags negative values of sd as invalid. A probability level outside [0, 1] is also invalid when log.p = FALSE. Options and numerical details are documented in [2].
6. When the distribution has jumps
Consider the number of successes in four independent Bernoulli trials, each with success probability 0.5. For X ~ Binomial(4, 0.5), we obtain:
No value has a cumulative probability exactly equal to 0.5. Up to 1 success, the accumulated probability is 0.3125; at 2 successes, it reaches 0.6875. The quantile at level 0.5 is therefore 2: it is the smallest value that reaches or exceeds the requested level. This is the operational definition of qbinom(), described in [3].
Figure 4. The first value that reaches or exceeds 0.5 is 2. Filled points indicate the value included in each step; open points indicate the excluded endpoint. No outcome has a cumulative probability exactly equal to 0.5.
In this example, P(X < 2) = 0.3125, P(X ≤ 2) = 0.6875, P(X > 2) = 0.3125, and P(X ≥ 2) = 0.6875. Including the threshold changes the probability because P(X = 2) = 0.375.
7. Theoretical quantiles and sample quantiles
qnorm() calculates quantiles of a specified normal distribution. quantile() works with a sample and a computational method. Even if the data appear normal, these operations involve different constructions: one depends on the model and its parameters; the other depends on the ordered data and the chosen convention.
Consider six illustrative scores, created solely to demonstrate the computational methods:
The empirical cumulative distribution function assigns to each threshold the observed proportion of data less than or equal to it:
The expression 𝟙 denotes an indicator function: it equals 1 when the condition holds and 0 otherwise. For 0 < p ≤ 1, the generalized inverse of this function corresponds to type = 1. Writing the ordered observations as x(1), …, x(n), it returns x(⌈np⌉). For example, with six scores and p = 0.5, the index is 3 and the result is 70.
7.2. Default interpolation
The default R method is type = 7. For a finite sample of size at least two, it places the level p at position h = 1 + (n − 1)p and interpolates linearly between adjacent observations when that position is not an integer. Thus, p = 0.25 gives h = 2.25 and the result is 65 + 0.25(70 − 65) = 66.25.
At level 0.5, interpolation falls between 70 and 72 and returns 71. Both methods follow legitimate conventions; they are not resolving an arithmetic disagreement. The interpolated result need not coincide with an observed score.
R provides nine methods for calculating sample quantiles. It is unnecessary to develop all their algorithms here, but the chosen method should be stated, especially when comparing results across programs. The official documentation presents the nine definitions and their relationship to order statistics; see [4].
Removing missing values allows the calculation to proceed, but does not explain why they are missing. A study should report the sample size used, the policy for handling missing data, the requested probability level, and the quantile method. If sampling weights or a complex design are involved, the procedure must account for them as well.
8. Quantiles, critical values, and confidence intervals
Quantile functions appear in many inferential procedures because they locate critical values of a reference distribution. However, calling qnorm() or qt() does not, by itself, construct a confidence interval. One must specify the parameter being estimated, the statistic being used, and the assumptions under which its distribution is known or approximated.
8.1. An interval for the mean
If X1, …, Xn are independent observations from the same normal distribution with mean μ and unknown positive variance, the pivot:
follows a Student t distribution with n − 1 degrees of freedom. Inverting the probability inequality for this pivot gives the two-sided confidence interval:
In this formula, the sample mean is the estimator; S/√n is its estimated standard error; and the quantile t1−α/2,n−1 is the critical value calculated by qt(). See [5] and [6].
For these six illustrative scores, the 95% interval is approximately [60.50649, 84.82684]. The calculation demonstrates the procedure; a sample this small requires careful examination of the plausibility of the normality assumption. For nonnormal populations, approximate use of this interval needs justification and can be problematic with pronounced skewness, heavy tails, dependence, or influential outliers.
8.2. Three objects that should be distinguished
The object of interest determines the procedure
Object
Question
Example
Central interval of a distribution
Between which values does a proportion of the model lie?
Two normal quantiles at levels 0.025 and 0.975
Confidence interval
How can a parameter be estimated with specified coverage?
Student t interval for the mean
Prediction interval
How can a future observation be bounded with specified coverage?
A procedure incorporating individual variation and estimation uncertainty
R · Central interval of the normal model
qnorm(c(0.025, 0.975), mean = 72, sd = 15.2)
# Approximately: 42.20855 101.79145
# Central interval of the model, not an interval for its mean
For a known normal distribution with these parameters, the central 95% interval is approximately [42.20855, 101.79145]. It is not a confidence interval for μ. The upper limit also illustrates why caution is necessary when using a normal distribution for scores whose actual maximum is 100.
8.3. Critical values and p-values
In a two-sided test concerning a mean, the critical value is obtained by setting a level α before evaluating the statistic. The p-value is calculated from the observed statistic and its distribution under the null hypothesis. The letter p in the arguments of qnorm() or qt() denotes an input probability; it does not automatically mean “the p-value of a test.”
In this example, both decisions agree: μ = 72 is not rejected at level 0.05. This does not prove that the null hypothesis is true or that a substantive difference is impossible. The calculation uses a symmetric reference distribution; other tests require the appropriate tail, statistic, and distribution to be established.
9. Simulation through the quantile function
The quantile function also allows realizations from a distribution to be constructed from a uniform variable. If U is uniformly distributed on (0, 1), then QX(U) has cumulative distribution function FX. Defining the quantile through a generalized inverse extends this principle to discrete distributions as well.
R · Inverse transform
set.seed(2021)
uniform_values <- runif(1000)
simulated_scores <- qnorm(uniform_values, mean = 72, sd = 15.2)
mean(simulated_scores)
sd(simulated_scores)
# Sample approximations to 72 and 15.2
# Normal generation using the dedicated function:
set.seed(2021)
direct_scores <- rnorm(1000, mean = 72, sd = 15.2)
# Element-by-element equality between the two methods is not expected
This is the inverse transform principle. A seed makes the sequence reproducible within a compatible random-number-generator configuration; it does not eliminate sampling variation or make the observed mean equal to the exact parameter. From the user’s perspective, rnorm() provides direct normal generation, but this example does not describe every internal algorithm through which R can generate it.
This application makes the article’s central relationship visible: a collection of probability levels is transformed into values of a variable. Beyond simulation, the same relationship appears in descriptive percentiles, decision thresholds, and diagnostics comparing empirical quantiles with theoretical quantiles.
10. Reproducing the plots in R
The plots embedded in this article represent the mathematical functions used in the examples. The following block produces their equivalents using base R tools: two density areas, a cumulative distribution curve, a quantile curve, and a discrete cumulative distribution. No additional packages need to be installed.
R · Plots with base R
plot_quantiles <- function() {
previous_parameters <- par(no.readonly = TRUE)
on.exit(par(previous_parameters))
paper <- "#fffdf7"
ink <- "#2c2418"
copper <- "#8b4513"
area <- "#e4d1bb"
mu <- 72
sigma <- 15.2
p <- 0.7850824
q_left <- qnorm(p, mean = mu, sd = sigma)
q_right <- qnorm(p, mean = mu, sd = sigma,
lower.tail = FALSE)
grid <- seq(12, 132, length.out = 1201)
density_values <- dnorm(grid, mean = mu, sd = sigma)
par(mfrow = c(2, 1), mar = c(4, 4, 3, 1),
bg = paper, fg = ink, col.axis = ink,
col.lab = ink, col.main = ink)
for (tail_choice in c("lower", "upper")) {
cutoff <- if (tail_choice == "lower") q_left else q_right
bounds <- if (tail_choice == "lower") {
c(12, cutoff)
} else {
c(cutoff, 132)
}
xs <- seq(bounds[1], bounds[2], length.out = 600)
ys <- dnorm(xs, mean = mu, sd = sigma)
plot(grid, density_values, type = "n",
xlab = "Score", ylab = "Density",
main = paste("Tail", tail_choice, "- probability", p))
polygon(c(xs[1], xs, tail(xs, 1)), c(0, ys, 0),
col = area, border = NA)
lines(grid, density_values, col = copper, lwd = 2)
segments(cutoff, 0, cutoff,
dnorm(cutoff, mean = mu, sd = sigma), col = copper)
abline(v = mu, col = "#c4956a", lty = 3)
mtext(sprintf("Threshold: %.2f", cutoff), side = 3,
line = 0.2, cex = 0.8)
}
par(mfrow = c(1, 1))
plot(grid, pnorm(grid, mean = mu, sd = sigma),
type = "l", col = copper, lwd = 2, ylim = c(0, 1),
xlab = "Score", ylab = "Cumulative probability",
main = "From p to threshold: cumulative distribution")
segments(12, p, q_left, p, lty = 2, col = copper)
segments(q_left, 0, q_left, p, lty = 2, col = copper)
points(q_left, p, pch = 19, col = copper)
plot_levels <- seq(0.005, 0.995, length.out = 1000)
plot(plot_levels,
qnorm(plot_levels, mean = mu, sd = sigma),
type = "l", col = copper, lwd = 2,
xlab = "Cumulative probability p", ylab = "Quantile",
main = "Normal quantile function")
points(c(0.5, p), c(mu, q_left), pch = 19, col = copper)
plot(NA_real_, NA_real_, type = "n", xlim = c(-0.5, 4.5),
ylim = c(0, 1), xlab = "Number of successes",
ylab = "Cumulative probability",
main = "Binomial(4, 0.5): cumulative distribution with jumps")
abline(h = 0.5, lty = 2, col = "#c4956a")
segments(-0.5, 0, 0, 0, col = copper, lwd = 2)
for (k in 0:4) {
cumulative <- pbinom(k, size = 4, prob = 0.5)
endpoint <- min(k + 1, 4.5)
segments(k, cumulative, endpoint, cumulative,
col = copper, lwd = 2)
points(k, cumulative, pch = 19, col = copper)
if (k < 4) {
points(k + 1, cumulative, pch = 21,
col = copper, bg = paper)
}
}
segments(2, 0, 2, pbinom(2, 4, 0.5), lty = 2, col = copper)
invisible(NULL)
}
plot_quantiles()
# In RStudio, use the Plots pane arrows to view the plots.
# To save all pages in a PDF from R:
# pdf("quantile_plots.pdf", width = 8, height = 7)
# plot_quantiles()
# dev.off()
Shaded regions under the densities correspond to probabilities; the heights are densities. In the cumulative distribution plot, height represents cumulative probability. In the quantile curve, the roles of the axes change: the probability level becomes the input and the variable’s value becomes the output.
11. Common mistakes
A guide to interpretation and review
Misconception
Correct interpretation
qnorm() returns a probability
It returns a threshold associated with a probability
sd means standard error
It is the standard deviation of the normal model
The quantile function is used because the cumulative distribution does not exist
The cumulative distribution exists; inversion answers a different question
Every quantile satisfies exactly F(q) = p
Jumps can make the attained level exceed the requested one
The upper tail is simply the same threshold with a different explanation
Keeping p fixed while changing the tail generally changes the threshold
A quantile lists the observations below it
It locates a value; identifying observations requires the data
A model proportion holds exactly in every sample
Observed frequencies vary across samples
quantile() and qnorm() are interchangeable
They work with a sample and a specified model, respectively
A pair of quantiles is always a confidence interval
Inferential coverage depends on a procedure and its assumptions
x <- rnorm generates scores
It stores a function; the function must be called with arguments
Interpreting any result requires recording four elements: the variable and its units, the distribution or sample used, the requested probability level, and the computational convention. If the result is used for inference, the parameter, statistic, and assumptions supporting the procedure must also be made explicit.
· · ·
A quantile turns a proportion into a position within a distribution. Its interpretation depends on which distribution is being studied and how that position is defined.
12. Complete script and checks
The expandable section collects the examples and plots in execution order. It can be copied to an .R file and run in a fresh session. The six test scores are illustrative and do not come from an empirical study. The last printed decimal places may vary with the version and platform.
Expand the complete R script
R · Complete file
# THE QUANTILE FUNCTION IN R
# JOSÉ MAURICIO GÓMEZ JULIÁN
# Illustrative data; base R and stats functions.
# LOWER TAIL AND VERIFICATION
target_p <- 0.7850824
score_mean <- 72
score_sd <- 15.2
lower_threshold <- qnorm(target_p, mean = score_mean,
sd = score_sd)
lower_threshold
# Approximately 84.00000012
pnorm(lower_threshold, mean = score_mean, sd = score_sd)
# Approximately 0.7850824
# UPPER TAIL AND VERIFICATION
upper_threshold <- qnorm(target_p, mean = score_mean,
sd = score_sd, lower.tail = FALSE)
upper_threshold
# Approximately 59.99999988
pnorm(upper_threshold, mean = score_mean,
sd = score_sd, lower.tail = FALSE)
# Approximately 0.7850824
pnorm(upper_threshold, mean = score_mean, sd = score_sd)
# Approximately 0.2149176
# PROBABILITY OUTSIDE THE 0–100 SCALE
p_below_0 <- pnorm(0, mean = score_mean, sd = score_sd)
p_above_100 <- pnorm(100, mean = score_mean,
sd = score_sd, lower.tail = FALSE)
c(p_below_0 = p_below_0, p_above_100 = p_above_100)
# COMPLETE CALL
qnorm(p = 0.95, mean = 0, sd = 1,
lower.tail = TRUE, log.p = FALSE)
# Approximately 1.644854
# LOGARITHMS AND EXTREME TAILS
qnorm(log(0.95), log.p = TRUE)
# The same quantile as qnorm(0.95)
qnorm(1 - 1e-20)
# Inf: the subtraction rounds to 1
qnorm(1e-20, lower.tail = FALSE)
# Approximately 9.26234
qnorm(-1000, lower.tail = FALSE, log.p = TRUE)
# Finite quantile for upper-tail probability exp(-1000)
# MULTIPLE LEVELS AND ENDPOINTS
qnorm(c(0.025, 0.5, 0.975), mean = 72, sd = 15.2)
# Approximately: 42.20855 72.00000 101.79145
qnorm(c(0, 1))
# -Inf Inf
# BINOMIAL QUANTILE AND ATTAINED PROBABILITIES
discrete_quantile <- qbinom(0.5, size = 4, prob = 0.5)
discrete_quantile
# 2
pbinom(discrete_quantile - 1, size = 4, prob = 0.5)
# 0.3125
pbinom(discrete_quantile, size = 4, prob = 0.5)
# 0.6875
qbinom(0.5, size = 4, prob = 0.5, lower.tail = FALSE)
# 2
pbinom(2, size = 4, prob = 0.5, lower.tail = FALSE)
# 0.3125
# COMPARING TWO SAMPLE-QUANTILE METHODS
scores <- c(58, 65, 70, 72, 80, 91)
levels <- c(0.25, 0.5, 0.75)
quantile(scores, probs = levels, type = 1)
# 25% 50% 75%
# 65 70 80
quantile(scores, probs = levels, type = 7)
# 25% 50% 75%
# 66.25 71.00 78.00
empirical_cdf <- ecdf(scores)
empirical_cdf(70)
# 0.5
# MISSING VALUES
scores_with_na <- c(scores, NA_real_)
sum(is.na(scores_with_na))
# 1
quantile(scores_with_na, probs = levels,
type = 7, na.rm = TRUE)
# 66.25 71.00 78.00
# STUDENT T INTERVAL FOR THE MEAN
scores <- c(58, 65, 70, 72, 80, 91)
n <- length(scores)
confidence <- 0.95
alpha <- 1 - confidence
sample_mean <- mean(scores)
sample_sd <- sd(scores)
standard_error <- sample_sd / sqrt(n)
critical_t <- qt(1 - alpha / 2, df = n - 1)
mean_ci <- sample_mean + c(-1, 1) *
critical_t * standard_error
mean_ci
# Approximately: 60.50649 84.82684
automatic_ci <- t.test(scores, conf.level = confidence)$conf.int
unname(automatic_ci)
stopifnot(isTRUE(all.equal(
as.numeric(mean_ci), as.numeric(automatic_ci),
tolerance = 1e-10
)))
# CENTRAL INTERVAL OF THE NORMAL MODEL
qnorm(c(0.025, 0.975), mean = 72, sd = 15.2)
# Approximately: 42.20855 101.79145
# Central interval of the model, not an interval for its mean
# CRITICAL QUANTILE AND TWO-SIDED P-VALUE
null_mean <- 72
observed_t <- (sample_mean - null_mean) / standard_error
two_sided_p <- 2 * pt(abs(observed_t), df = n - 1,
lower.tail = FALSE)
c(observed_t = observed_t,
critical_t = critical_t, two_sided_p = two_sided_p)
abs(observed_t) > critical_t
# FALSE
two_sided_p < alpha
# FALSE
# INVERSE TRANSFORM
set.seed(2021)
uniform_values <- runif(1000)
simulated_scores <- qnorm(uniform_values, mean = 72, sd = 15.2)
mean(simulated_scores)
sd(simulated_scores)
# Sample approximations to 72 and 15.2
# Normal generation using the dedicated function:
set.seed(2021)
direct_scores <- rnorm(1000, mean = 72, sd = 15.2)
# Element-by-element equality between the two methods is not expected
# PLOTS WITH BASE R
plot_quantiles <- function() {
previous_parameters <- par(no.readonly = TRUE)
on.exit(par(previous_parameters))
paper <- "#fffdf7"
ink <- "#2c2418"
copper <- "#8b4513"
area <- "#e4d1bb"
mu <- 72
sigma <- 15.2
p <- 0.7850824
q_left <- qnorm(p, mean = mu, sd = sigma)
q_right <- qnorm(p, mean = mu, sd = sigma,
lower.tail = FALSE)
grid <- seq(12, 132, length.out = 1201)
density_values <- dnorm(grid, mean = mu, sd = sigma)
par(mfrow = c(2, 1), mar = c(4, 4, 3, 1),
bg = paper, fg = ink, col.axis = ink,
col.lab = ink, col.main = ink)
for (tail_choice in c("lower", "upper")) {
cutoff <- if (tail_choice == "lower") q_left else q_right
bounds <- if (tail_choice == "lower") {
c(12, cutoff)
} else {
c(cutoff, 132)
}
xs <- seq(bounds[1], bounds[2], length.out = 600)
ys <- dnorm(xs, mean = mu, sd = sigma)
plot(grid, density_values, type = "n",
xlab = "Score", ylab = "Density",
main = paste("Tail", tail_choice, "- probability", p))
polygon(c(xs[1], xs, tail(xs, 1)), c(0, ys, 0),
col = area, border = NA)
lines(grid, density_values, col = copper, lwd = 2)
segments(cutoff, 0, cutoff,
dnorm(cutoff, mean = mu, sd = sigma), col = copper)
abline(v = mu, col = "#c4956a", lty = 3)
mtext(sprintf("Threshold: %.2f", cutoff), side = 3,
line = 0.2, cex = 0.8)
}
par(mfrow = c(1, 1))
plot(grid, pnorm(grid, mean = mu, sd = sigma),
type = "l", col = copper, lwd = 2, ylim = c(0, 1),
xlab = "Score", ylab = "Cumulative probability",
main = "From p to threshold: cumulative distribution")
segments(12, p, q_left, p, lty = 2, col = copper)
segments(q_left, 0, q_left, p, lty = 2, col = copper)
points(q_left, p, pch = 19, col = copper)
plot_levels <- seq(0.005, 0.995, length.out = 1000)
plot(plot_levels,
qnorm(plot_levels, mean = mu, sd = sigma),
type = "l", col = copper, lwd = 2,
xlab = "Cumulative probability p", ylab = "Quantile",
main = "Normal quantile function")
points(c(0.5, p), c(mu, q_left), pch = 19, col = copper)
plot(NA_real_, NA_real_, type = "n", xlim = c(-0.5, 4.5),
ylim = c(0, 1), xlab = "Number of successes",
ylab = "Cumulative probability",
main = "Binomial(4, 0.5): cumulative distribution with jumps")
abline(h = 0.5, lty = 2, col = "#c4956a")
segments(-0.5, 0, 0, 0, col = copper, lwd = 2)
for (k in 0:4) {
cumulative <- pbinom(k, size = 4, prob = 0.5)
endpoint <- min(k + 1, 4.5)
segments(k, cumulative, endpoint, cumulative,
col = copper, lwd = 2)
points(k, cumulative, pch = 19, col = copper)
if (k < 4) {
points(k + 1, cumulative, pch = 21,
col = copper, bg = paper)
}
}
segments(2, 0, 2, pbinom(2, 4, 0.5), lty = 2, col = copper)
invisible(NULL)
}
plot_quantiles()
# In RStudio, use the Plots pane arrows to view the plots.
# To save all pages in a PDF from R:
# pdf("quantile_plots.pdf", width = 8, height = 7)
# plot_quantiles()
# dev.off()
# Numerical checks with tolerance
stopifnot(
abs(lower_threshold - 84) < 1e-5,
abs(upper_threshold - 60) < 1e-5,
abs(pnorm(lower_threshold, score_mean, score_sd) -
target_p) < 1e-10,
abs(pnorm(upper_threshold, score_mean, score_sd,
lower.tail = FALSE) - target_p) < 1e-10,
discrete_quantile == 2,
pbinom(discrete_quantile - 1, 4, 0.5) < 0.5,
pbinom(discrete_quantile, 4, 0.5) >= 0.5,
isTRUE(all.equal(
as.numeric(quantile(scores, levels, type = 1)),
c(65, 70, 80))),
isTRUE(all.equal(
as.numeric(quantile(scores, levels, type = 7)),
c(66.25, 71, 78)))
)
sessionInfo()
13. References and documentation
The following sources document the functions, conventions, and procedures used here. The mathematical explanations and examples are developed in the article itself; these links provide access to implementation details.
R Core Team. Sample Quantiles. Official R manual. quantile and its nine methods. The documentation cites Hyndman, R. J. and Fan, Y. (1996), Sample Quantiles in Statistical Packages, as a reference for this classification.
R Core Team. The Student t Distribution. Official R manual. dt, pt, qt, and rt.
Foundations of Estimation and Hypothesis Testing in R
Theory, simulation, annotated code, and interpretation: from the distribution of the data to the design of a study.
JOSÉ MAURICIO GÓMEZ JULIÁN
Originally published September 6, 2021 Comprehensive revision: October 29, 2026
· · ·
A hypothesis test does not begin with a function call. It begins by specifying the phenomenon under investigation, how the data were obtained, and which statement about the population can be examined using those data. R performs the calculations; understanding their foundations makes it possible to determine when those calculations address the research question and which conclusions they support.
This notebook connects descriptive statistics, simulation, estimation, and inference through a sequence of examples: a sample of stores, computationally generated normal populations, comparisons of means and variances, paired sleep measurements, and power calculations. Each procedure is developed through its question, mathematical foundations, implementation, and interpretation. R is the language and computing environment; RStudio is an integrated development environment that can be used to work with it.
01 / Foundations
From a scientific question to a statistical decision
The population is the set of units or the process to which the investigation refers; the sample is the collection of available observations. A parameter, such as the population mean μ or population variance σ², characterizes the population. An estimator, such as the sample mean X̄ or sample variance S², is a function of the sample. Its value calculated from the observed data is an estimate.
The distinction is essential: the data are observed; the parameter is usually unknown; the estimator varies from sample to sample. Inference studies that variation under a model and a data-collection design. A large sample does not, by itself, correct selection bias or make dependent observations independent.
X̄ = (1/n) ∑ᵢ₌₁ⁿ XᵢS² = [1/(n − 1)] ∑ᵢ₌₁ⁿ (Xᵢ − X̄)²Estimated SE of X̄ = S / √n
The standard deviation summarizes the dispersion of the observations. The standard error describes the dispersion of an estimator across samples. For independent, identically distributed observations with finite variance, the standard error of the mean is σ/√n; when σ is unknown, it is estimated by S/√n. This expression requires adjustment when there is dependence, clustering, weighting, or a complex sampling design.
Hypotheses, the test statistic, and the p-value
A null hypothesis H₀ specifies the reference model. For a mean, it may be stated as H₀: μ = μ₀. The two-sided alternative is H₁: μ ≠ μ₀; a one-sided alternative specifies μ > μ₀ or μ < μ₀. The direction must follow the scientific question and be established before using the observed result to choose it.
The test statistic summarizes the discrepancy between the data and H₀ on a scale whose distribution can be studied. The p-value is the probability, under H₀ and the model’s other assumptions, of obtaining a statistic at least as extreme as the observed one, as defined by the chosen alternative. It is not the probability that H₀ is true or the probability that the result is due to chance. Statistical significance does not measure the size or importance of an effect.[1]
Four possible outcomes of a testing rule
Decision
H₀ is true
A specific alternative is true
Reject H₀
Type I error; probability α for an exact test of size α.
Detection; probability 1 − β.
Do not reject H₀
A decision compatible with H₀.
Type II error; probability β for that effect.
The level α sets an upper bound on the testing rule’s Type I error; for conservative procedures, the actual probability may be lower. β depends on the true effect, sample size, variability, and test. Failing to reject H₀ may reflect compatibility with it, limited precision, or insufficient ability to detect a relevant effect. It does not establish equality.
A statistical result acquires meaning when it is related back to the phenomenon, the unit of measurement, and the conditions under which the data were produced.
02 / Preparation
Data, importing, and structure
The original example considers a random sample of 21 stores from different parts of the country. It records a cleanliness indicator, an index of products repaired and marketed, and units sold during the previous month. This edition retains that scenario as an exercise; the values below are illustrative. They do not support inferences about an actual population.
Variable dictionary for the exercise
Variable
Meaning
Caution
menor16
A cleanliness indicator; higher values represent greater cleanliness.
The original name is retained. The original scale must be known to decide whether its intervals are comparable.
ipc
The percentage of defective products that were repaired and subsequently marketed.
This is not a consumer price index. The operational definition of the denominator must be checked.
ventas
The quantity sold during the previous month.
This is a count; a continuous approximation requires attention to its distribution and the purpose of the analysis.
Within an RStudio project, it is useful to work with paths relative to the project directory. getwd() reports the current working directory; setwd("actual/path") changes it. The original string "(...)" is a placeholder, not an executable path. The following importing option assumes that the file has no header and contains three columns separated by spaces or tabs:
getwd()
# setwd("actual/path/to/project") # only if you need to change it
# If you have the original file, without a header:
# estudios <- read.table("estudios.txt", header = FALSE,
# col.names = c("menor16", "ipc", "ventas"))
read.table() reads the file; col.names assigns names to the columns of the data frame. It does not create the physical file. If the file already contains column names, use header = TRUE; if it uses a different separator or decimal mark, adjust sep and dec. Reading it twice, first without assignment and then with assignment, is unnecessary.
To run the article without that file, use the following complete dataset:
The commands names(), nrow(), ncol(), dim(), and str() check names, row count, column count, dimensions, and structure. Here they should identify 21 rows and three numeric columns. Original variable and object identifiers are retained in the English edition to preserve correspondence with the Spanish code; comments, explanations, and graphic labels are translated.
stopifnot(all(c("menor16", "ipc", "ventas") %in% names(estudios)),
all(vapply(estudios, is.numeric, logical(1))))
print(names(estudios)); print(nrow(estudios)); print(ncol(estudios))
print(dim(estudios)); str(estudios)
print(colSums(is.na(estudios)))
if (any(!is.finite(as.matrix(estudios)))) {
stop("There are missing or non-finite values: document their treatment before analysis.")
}
stopifnot(nrow(estudios) >= 2L)
Explicit access and the historical use of attach()
attach(estudios) adds the variables to the search path, allowing their names to be written without the data-frame prefix; detach(estudios) removes that entry. Understanding these commands is useful because they appear in the original, but an identically named object may mask another, and subsequent changes to the data frame are not necessarily reflected as expected. The revised code uses estudios$ventas, with(estudios, ...), or the argument data = estudios. The assignment ventas <- estudios$ventas creates an explicit vector and must be updated if the data frame is subsequently modified.
03 / Exploration
Means, dispersion, quantiles, and histograms
Description precedes testing. The mean and standard deviation convey different aspects of the data; quantiles and graphical displays can reveal asymmetry, extreme observations, or clusters that a single summary may conceal.
apply(estudios, 2, mean) applies the function to each column; MARGIN = 1 would apply it by row. There is no need to execute mean() or quantile() beforehand. With a mixed-type data frame, apply() may convert everything into a character matrix; this is why the columns are checked to be numeric here. vapply() and lapply() can work column by column without that global conversion.[2]
What quantiles express
Quantiles locate values of the variable from cumulative proportions. By default, quantile(ventas) returns the minimum, quartiles, and maximum; probs = c(0.025, 0.975) requests the 2.5th and 97.5th percentiles. R provides several algorithms; the default, type = 7, uses interpolation.[3]
In the illustrative records, the mean is 187.667, the standard deviation is 23.258, and these two percentiles are 148 and 225. The values 141 and 243 mentioned in the original are not reproduced here because they belong to a different data file. With small samples, tail quantiles are especially sensitive to a few observations.
abline(v = qv) draws vertical lines at the quantiles; col changes their color and lty their line type. A numeric breaks value suggests a number of classes, but R may choose convenient boundaries and return a different number. To require precise boundaries, provide an explicit vector of breakpoints. When classes have different widths, distinguishing frequencies from densities is important.
Figure 01 Dashed lines: the 2.5th and 97.5th percentiles (148 and 225). These describe the data; they are not a confidence interval for their mean.
04 / Simulation
Normal populations and user-defined functions
We adopt the notation X ∼ N(μ, σ²), with variance as the second parameter. In R, however, rnorm(n, mean, sd) takes the standard deviation. Thus, rnorm(1000, 35, 2) generates 1,000 observations from N(35, 4). In this call, 35 and 2 are parameters of the generating distribution, not the mean and standard deviation actually observed in the sample.[4]
set.seed(20210906)
y <- rnorm(1000, mean = 35, sd = 2)
qy <- quantile(y, probs = c(0.025, 0.975))
hist(y, probability = TRUE, col = beige, border = cobre,
main = "N(35, 4): estimated and theoretical densities", xlab = "Value")
abline(v = qy, col = cobre, lty = 2)
lines(density(y), col = cobre, lwd = 2)
curve(dnorm(x, mean = 35, sd = 2), add = TRUE, lty = 3, lwd = 2)
legend("topright", c("Kernel estimate", "Theoretical normal"),
col = c(cobre, tinta), lty = c(1, 3), bty = "n", cex = 0.7)
The seed supplied by set.seed() allows the pseudorandom sequence to be repeated in a reproducible environment. R versions and random-number-generator settings also matter; sessionInfo() documents the session. A histogram with probability = TRUE displays densities and has unit total area, allowing comparison with density curves.
density(y) estimates a curve from the data using kernels and a bandwidth. It does not know the normal density that generated the sample. Use dnorm() to display that theoretical density. Approximate agreement between the two is a simulation result, not an identity.[5]
A function that simulates, plots, and returns results
The plot.m() function retains the original example’s purpose, adds a line for the mean, checks its inputs, and returns the data and summaries. invisible() avoids automatically printing a lengthy object while still allowing the result to be saved.
plot.m <- function(n, m, s) {
stopifnot(length(n) == 1L, is.finite(n), n >= 2, n == as.integer(n),
length(m) == 1L, is.finite(m), length(s) == 1L, is.finite(s), s > 0)
y <- rnorm(n, mean = m, sd = s)
qy <- quantile(y, c(0.025, 0.975))
hist(y, probability = TRUE, col = beige, border = cobre,
main = paste("Normal sample: n =", n), xlab = "Value")
abline(v = qy, col = cobre, lty = 2)
abline(v = mean(y), col = tinta, lwd = 2)
lines(density(y), col = cobre)
invisible(list(media = mean(y), sd = sd(y), cuantiles = qy, datos = y))
}
muestra_10 <- plot.m(10, 100, 15)
muestra_10000 <- plot.m(10000, 100, 15)
print(muestra_10[c("media", "sd", "cuantiles")])
print(muestra_10000[c("media", "sd", "cuantiles")])
In the original version, a heading announced n = 10, whereas the call used 10,000. Both sizes are shown here, under the same N(100, 15²) model, as distinct experiments. A larger sample usually gives a more stable description of the population; it does not reduce the population dispersion of the individual observations.
The four families of distribution functions should be distinguished: d* evaluates a density or probability mass, p* a cumulative probability, q* a quantile, and r* generates realizations. Thus, dnorm(), pnorm(), qnorm(), and rnorm() perform different operations on the same distribution family.
05 / Variation across samples
The sampling distribution and standard error of the mean
A population, a sample from it, and a distribution of sample means are different objects. To study the third, repeat the same procedure: draw n observations, calculate their mean, save the result, and start again with another independent sample.
If X₁, …, Xₙ are independent and Xᵢ ∼ N(μ, σ²):X̄ ∼ N(μ, σ²/n)E(X̄) = μ · SD(X̄) = σ/√n
Here the normal distribution of the mean is exact for every n. No asymptotic approximation is required to obtain it. For a non-normal population, the central limit theorem establishes an approximation under its conditions; no universal sample-size threshold guarantees a good approximation for every phenomenon.
n_sim <- 10000L; mu_sim <- 100; sigma_sim <- 15; B <- 1000L
medias <- numeric(B)
for (i in seq_len(B)) {
sam <- rnorm(n_sim, mean = mu_sim, sd = sigma_sim)
medias[i] <- mean(sam)
}
print(c(media_teorica = mu_sim, media_simulada = mean(medias),
ee_teorico = sigma_sim / sqrt(n_sim), ee_simulado = sd(medias)))
qm <- quantile(medias, c(0.025, 0.975))
hist(medias, probability = TRUE, col = beige, border = cobre,
main = "Distribution of means", xlab = "Sample mean")
abline(v = qm, col = cobre, lty = 2); lines(density(medias), col = cobre)
With σ = 15 and n = 10,000, the theoretical standard error is 0.15. sd(medias) approximates it through simulation; exact agreement is not required because the exercise uses 1,000 repetitions, not infinitely many. The mean of those 1,000 means should be close to 100. Its theoretical Monte Carlo standard error is 0.15/√1000, approximately 0.00474.
numeric(B) preallocates space for B results. Within the for loop, sam contains the complete sample for the current repetition; medias[i] stores only its mean. A loop repeats instructions: it is not itself a function, although it can be written inside one. The analogy with computational orbits may help represent iteration, but it does not imply that the procedure is a dynamical model of the phenomenon being investigated.
Figure 02 Theoretical distributions under normality and independence. The horizontal scales differ: the dispersion of the means is one hundred times smaller.
Extension: an initially asymmetric population
An exponential distribution with rate 1 has mean 1 and variance 1. Comparing means from samples of sizes 1, 5, and 30 shows how their shape changes and how their standard deviation approaches 1/√n. This exercise illustrates an approximation process distinct from the exact normality of the previous example.
replicate() repeats an expression; lapply() organizes the results for several sizes. The simulation mathematically represents repeated sampling. It does not replace a real experiment or, by itself, establish that a model adequately describes observations from a laboratory, stores, or patients.
06 / Estimation
Confidence intervals for a mean
For an independent sample from a normal population with unknown σ, the statistic (X̄ − μ)/(S/√n) follows a t distribution with n − 1 degrees of freedom. Inverting its central region gives the two-sided interval:
CI₁₋α(μ) = X̄ ± t₁₋α/₂, ₙ₋₁ · S/√nCenter: estimate · Half-width: critical value × standard error
The t distribution arises by combining a standardized normal mean with an independent estimate of its variance. Its tails are heavier than those of the standard normal distribution for finite degrees of freedom; as these increase, it approaches the normal distribution. qt(0.975, n - 1) returns the quantile with 97.5% probability to its left; for a two-sided 95% interval, 2.5% remains in each tail.[6]
The vector c(-1, 1) constructs the two endpoints simultaneously. The following function generalizes the calculation to confidence level p; this argument is not a test’s p-value. It allows explicit removal of missing values, requires at least two observations, and stops if there is no variation.
ic <- function(x, p = 0.95, na.rm = FALSE) {
stopifnot(is.numeric(x), length(p) == 1L, is.finite(p), p > 0, p < 1,
is.logical(na.rm), length(na.rm) == 1L, !is.na(na.rm))
if (na.rm) x <- x[!is.na(x)]
if (any(!is.finite(x))) stop("Missing or non-finite data.")
n_x <- length(x)
if (n_x < 2L) stop("At least two observations are required.")
if (sd(x) == 0) stop("There is no sample variation.")
ee_x <- sd(x) / sqrt(n_x)
critico <- qt((1 + p) / 2, df = n_x - 1)
setNames(mean(x) + c(-1, 1) * critico * ee_x, c("lower", "upper"))
}
print(ic(ventas, 0.95)); print(ic(ventas, 0.99))
print(ic(estudios$ipc, 0.95)); print(ic(estudios$menor16, 0.95))
stopifnot(isTRUE(all.equal(unname(ic(ventas)),
as.numeric(t.test(ventas)$conf.int))))
For the illustrative data, the 95% CI for mean sales is [177.080, 198.254], whereas the 99% CI is [173.226, 202.108]. Holding the data fixed while increasing confidence raises the critical value and widens the interval. This does not mean that the estimator becomes more robust to extreme observations or model misspecification.
The frequentist interpretation of 95% concerns the procedure: if sampling and interval construction were repeated under the same conditions, 95% of the intervals would contain μ. Once the data have been observed, the parameter is fixed and the particular interval either contains it or does not. Nor does the interval for μ claim that 95% of stores have sales within it.
Figure 03 Increasing confidence widens the interval for the same data. Both intervals contain 180.
07 / One parameter
The one-sample t-test for a mean
The exercise asks whether population mean sales differ from 180 units per month. We adopt H₀: μ = 180, H₁: μ ≠ 180, and α = 0.05. The reference value needs an external justification—for example, a standard or a prior hypothesis—and should not be selected to obtain a desired result.
tobs = (x̄ − μ₀)/(s/√n)Under H₀ and independent normal sampling: T ∼ tₙ₋₁p = P(|T| ≥ |tobs|) = 2 P(T ≤ −|tobs|)
alpha <- 0.05; mu0 <- 180
n_ventas <- length(ventas); gl <- n_ventas - 1L
ee_ventas <- sd(ventas) / sqrt(n_ventas)
t_obs <- (mean(ventas) - mu0) / ee_ventas
p_manual <- 2 * pt(-abs(t_obs), df = gl)
prueba_una <- t.test(ventas, mu = mu0, alternative = "two.sided",
conf.level = 1 - alpha)
print(prueba_una)
print(c(t_manual = t_obs, p_manual = p_manual))
stopifnot(isTRUE(all.equal(p_manual, prueba_una$p.value)))
# Choose a one-sided alternative before examining the results.
prueba_mayor <- t.test(ventas, mu = mu0, alternative = "greater")
The expression using abs() works for either sign of the statistic. pt() evaluates the t cumulative distribution function, not a density. Doubling is justified by that distribution’s symmetry and the two-sided alternative. Using pt(-abs(t_obs), ...) also avoids the subtraction 1 - pt(...), which may lose precision in a very small tail.[6]
Result · Illustrative data
Mean = 187.667; t = 1.511; df = 20; p = 0.1465; 95% CI = [177.080, 198.254].
At the 0.05 level, H₀ is not rejected. Under this test and its assumptions, the data do not provide sufficient evidence against μ = 180. This does not establish that μ is exactly 180 or that a relevant difference is impossible.
Figure 04 Total area in the two tails: p ≈ 0.1465. Calculated under H₀, the specified model, and the two-sided alternative.
One-sided tests and duality with confidence intervals
alternative = "greater" tests for a mean greater than μ₀; "less" tests for a smaller mean. The rejection region and the interval returned by R change accordingly. A tail must not be selected after observing the effect’s sign. For corresponding two-sided tests and two-sided intervals, at the same level and under the same procedure, excluding μ₀ is equivalent to rejecting H₀, apart from the convention at the boundary.
The test is not performed “inside” an interval as an independent operation: testing and interval estimation are two related ways of summarizing the same information under the model. The p-value does not replace the interval or the effect’s magnitude in its original units.
08 / Comparison
Constructing groups without losing sight of the original variable
The exercise divides stores into ipc < 17 and ipc ≥ 17. The threshold consistent with the original code is 17; the mention of 117 was an inconsistency in its heading. Creating two categories does not turn the study into an experiment or make an association causal.
ifelse() expresses the classification; factor() assigns levels and labels. A grouping variable classifies observations, but is not synonymous with a variable in general. tapply() applies a function to groups defined by factor levels; the technical reference to a ragged array does not mean that a data frame has rows of different lengths.[7]
In the constructed records, the ipc < 17 group contains 10 stores with mean 168.8; the ipc ≥ 17 group contains 11 stores with mean 204.818. The order of the levels fixes the direction of the difference: first minus second.
Boxplots and means with intervals
A boxplot summarizes the median and quartiles; its whiskers depend on the convention used and are not confidence intervals for the mean. Points with confidence bars represent a different object: mean estimates and their uncertainty. gplots::plotmeans() provides this second display and is retained as an option. A diamond-shaped marker does not turn every interval plot into a “diamond plot” with a universal statistical meaning.
plot(ipc ~ ipc2, data = estudios, col = beige, border = cobre,
xlab = "Group", ylab = "IPC", main = "Classification by IPC")
abline(h = umbral, col = cobre, lty = 2)
boxplot(ventas ~ ipc2, data = estudios, col = beige, border = cobre,
ylab = "Sales", xlab = "Group", main = "Sales by group")
ic_grupos <- sapply(split(estudios$ventas, estudios$ipc2), ic)
plot(seq_along(med_grupos), med_grupos, pch = 18, xaxt = "n",
xlim = c(0.5, 2.5), ylim = range(ic_grupos), xlab = "Group", ylab = "Sales",
main = "Means and 95% confidence intervals")
axis(1, at = 1:2, labels = names(med_grupos))
arrows(1:2, ic_grupos[1, ], 1:2, ic_grupos[2, ],
angle = 90, code = 3, length = 0.05, col = cobre)
# Optional resource from the original article:
if (requireNamespace("gplots", quietly = TRUE)) {
gplots::plotmeans(ventas ~ ipc2, data = estudios, connect = FALSE,
col = cobre, barcol = cobre)
}
Overlap or non-overlap between two individual 95% intervals does not replace a test of the difference between means. That question requires the standard error of the difference and its own interval. A useful display can accompany the test but does not mechanically replace it.
09 / Two means
Student’s pooled-variance test and Welch’s test
We test H₀: μ₁ − μ₂ = 0 against a two-sided alternative. Both versions require independence between units; they differ in their treatment of group variability. The subtraction order must be preserved when interpreting the sign.
Equal population variances
If σ₁² = σ₂² is justified, the pooled variance weights the sample variances by their degrees of freedom. It is not a simple arithmetic mean, nor is it unconditionally “more precise”: its justification depends on the common-variance model.
Welch estimates the variances separately and uses Welch–Satterthwaite degrees of freedom. These need not be integers. This is the default behavior of t.test() for two independent samples; var.equal = TRUE explicitly requests the preceding version.[8]
The difference “ipc < 17 minus ipc ≥ 17” is −36.018 units. Welch gives t ≈ −5.658, df ≈ 18.712, and p ≈ 0.0000198. The pooled-variance version gives p ≈ 0.0000184.
Both rules reject equality of means in this constructed example. The result describes a difference between the defined groups; it does not demonstrate that increasing the ipc indicator causes higher sales. The effect’s magnitude and direction should be reported alongside the interval returned by the test.
The interval for the difference is constructed as (x̄₁ − x̄₂) ± critical value × standard error, using the appropriate degrees of freedom. If the corresponding two-sided interval contains zero, equality is not rejected at the same level. This does not establish equivalence: that question requires substantive margins and a procedure designed to test them.
A preliminary F-test should not serve as an automatic switch between Student and Welch. Failing to reject equality of variances does not establish that they are equal; the full selection strategy changes the procedure’s properties. In this notebook, Welch is the main option, while the pooled version illustrates the common-variance case.
10 / Two variances
The F-test and an interval for a ratio
Comparing dispersion is a different question from comparing means. The general hypothesis is H₀: σ₁²/σ₂² = r₀, with r₀ > 0. For equality, set r₀ = 1. This does not require σ₁² = σ₂² = 1, only that the two variances be equal.
Fobs = (s₁²/s₂²)/r₀Under H₀, with independent normal samples:F ∼ F(n₁ − 1, n₂ − 1)
The F distribution is positive and asymmetric. For the equal-tail two-sided test implemented here, the p-value is twice the smaller tail probability, capped at one. Always doubling pf(F_obs, ...) may produce an incorrect probability, potentially exceeding one.[9]
If r̂ = s₁²/s₂², then:CI₁₋α(σ₁²/σ₂²) =[ r̂/F₁₋α/₂;ν₁,ν₂ , r̂/Fα/₂;ν₁,ν₂ ]
The interval estimates the population ratio; it is not symmetric around r̂ on the original scale. If it contains one, equality is not rejected by the corresponding two-sided test. When testing a different ratio, the comparison is with r₀, not necessarily one.
In the illustrative data, the sample ratio is approximately 1.049 and p ≈ 0.9338. Equality of variances is not rejected. This establishes neither their identity nor equality of the means.
11 / Within-pair dependence
Two sleep treatments in the same people
The historical sleep dataset contains 20 observations on ten patients, with two soporific treatments. extra is the increase in sleep hours relative to control; group identifies the treatment and ID the patient. These are not twenty independent people.[11]
Pairing should be constructed by identifier, rather than assuming that two vectors retain the same order. In sleep$group == "1", $ accesses the group column; “1” selects a treatment level and does not mean “column number one.” Selected rows and columns are distinguished by the comma in [rows, columns].
Define Dᵢ = Aᵢ − Bᵢ. The paired test of H₀: μD = 0 is a one-sample t-test applied to the differences:
tpaired = d̄/(sD/√npairs)Degrees of freedom = npairs − 1Var(A − B) = Var(A) + Var(B) − 2 Cov(A, B)
The covariance explains why paired analysis can have different precision from analysis that ignores the pairs. Its advantage depends on the dependence structure. Differences from distinct patients must be independent; for the exact t-test, normality concerns the differences, rather than requiring two separately normal marginal distributions.
dif_sueno <- pares$extra_A - pares$extra_B
prueba_pareada <- t.test(pares$extra_A, pares$extra_B, paired = TRUE)
prueba_diferencias <- t.test(dif_sueno, mu = 0)
prueba_independiente <- t.test(pares$extra_A, pares$extra_B)
print(prueba_pareada); print(prueba_independiente)
print(c(dz = mean(dif_sueno) / sd(dif_sueno)))
stopifnot(isTRUE(all.equal(prueba_pareada$p.value, prueba_diferencias$p.value)))
matplot(1:2, t(as.matrix(pares[c("extra_A", "extra_B")])),
type = "b", pch = c(1, 16), lty = 1, col = cobre,
xaxt = "n", xlab = "Treatment", ylab = "Increase in sleep (hours)",
main = "Paired measurements by patient")
axis(1, at = 1:2, labels = c("A", "B"))
# Patient-index graph retained from the original example:
plot(seq_len(nrow(pares)), pares$extra_A, type = "b", pch = 1,
col = cobre, ylim = range(pares[c("extra_A", "extra_B")]),
xlab = "Patient", ylab = "Increase in hours", main = "A and B by patient")
lines(seq_len(nrow(pares)), pares$extra_B, type = "b", pch = 16, lty = 2)
legend("topleft", c("A", "B"), col = c(cobre, tinta), pch = c(1, 16),
lty = c(1, 2), bty = "n")
Result · Historical sleep dataset
Mean difference A − B = −1.58 hours; t = −4.062; df = 9; p = 0.002833; 95% CI = [−2.460, −0.700] hours.
Treatment B has a greater mean increase in sleep than A in these records. The estimated difference is 1.58 hours in B’s favor. Clinical and causal inference requires considering the historical study’s design and conditions in addition to the calculation.
Figure 05 Each connection identifies the same person. The analysis uses ten differences, not twenty independent patients.
The call without paired = TRUE, retained as a teaching comparison, gives p ≈ 0.07939 because it treats the vectors as independent samples. For this design, that call ignores information supplied by the pairing. Analysis is not selected according to which result crosses 0.05: it is selected according to the data’s structure.
The paired standardized difference dz = d̄/sD is approximately −1.285. This definition differs from dividing an independent-groups difference by a pooled standard deviation; when reporting a standardized effect, identify its denominator.
The original graph comparing A and B along a patient index is also retained. That axis represents identifiers, not a time series. A display with one line per patient between treatments makes the pairing clearer. The marginal intervals from plotmeans(extra ~ group) are not the confidence interval for the paired difference.
12 / Research design
Power, detectable differences, and sample size
Power is the probability of rejecting H₀ when a specific alternative is true. If β is the Type II error probability for that effect, power is 1 − β. There is no single power independent of the true effect.
The example’s “resolution” is formalized as a difference that the study should be able to detect with specified power and significance level α. A minimum relevant difference must be justified by the phenomenon: it is not the smallest difference that a test could ever detect, nor a boundary making detection of smaller effects impossible.
One sample: d = δ/σTwo groups with a common variance: d = (μ₁ − μ₂)/σPaired data: dz = μD/σDPower(δ) = 1 − β(δ)
For planning, σ represents assumed variability or an estimate from relevant prior information. Using a pilot estimate introduces uncertainty; several scenarios should be explored. Conventional effect-size values do not replace a substantive justification for δ.
One sample: detecting three units
The original values are retained: δ = 3, σ = 10.2, target power 0.90, and a two-sided alternative. Here α = 0.05 is written explicitly. The effect size is d = 3/10.2 ≈ 0.29412. pwr is an additional package, not a native R database.
# Optional installation, once:
# install.packages("pwr")
# If installed, it can be attached with library(pwr).
if (requireNamespace("pwr", quietly = TRUE)) {
plan_pwr <- pwr::pwr.t.test(
n = NULL, d = 3 / 10.2, sig.level = 0.05,
power = 0.90, type = "one.sample", alternative = "two.sided"
)
print(plan_pwr)
print(ceiling(plan_pwr$n))
}
The result is n ≈ 123.403, requiring 124 complete observations under the adopted model and assumptions. With n = 124, calculated power is approximately 0.90139. Rounding to the nearest integer could leave power below the target; use ceiling().
The power.t.test() function in base R solves the same exercise without installing packages. Its delta argument uses original units, while sd supplies the standard deviation. strict = TRUE includes both rejection regions in the two-sided test and matches the two-sided calculation in pwr.[12]
With n = 40, which difference corresponds to power 0.90?
Fixing n = 40 and solving for δ, with the same σ and α, gives δ ≈ 5.3615 units. In the pwr parameterization, solve for d first, then multiply by σ. Calling a result “optimal” requires a cost function or another criterion; this exercise solves a power constraint, not a general optimization problem.
plan_40 <- power.t.test(n = 40, delta = NULL, sd = sigma_plan,
sig.level = alpha, power = potencia_objetivo,
type = "one.sample", strict = TRUE)
print(plan_40$delta)
# Equivalent with pwr: solve for d = NULL and multiply by sigma_plan.
A planning curve
A sample size is calculated for each difference from two to six units. Holding α, σ, and power fixed, detecting a smaller difference requires more observations. The curve may be displayed continuously before rounding; the final implementation uses integer sizes and therefore has a stepwise form.
deltas <- seq(2, 6, length.out = 30)
n_plan <- numeric(length(deltas)); d_grid <- deltas / sigma_plan
for (i in seq_along(deltas)) {
n_plan[i] <- power.t.test(delta = deltas[i], sd = sigma_plan,
sig.level = alpha, power = potencia_objetivo,
type = "one.sample", strict = TRUE)$n
}
plot(deltas, ceiling(n_plan), type = "s", col = cobre, lwd = 2,
xlab = "Difference to detect", ylab = "Sample size (rounded up)",
main = "90% power; alpha = 0.05")
Figure 06 Integer sample sizes, with σ = 10.2, α = 0.05, target power 0.90, and a two-sided one-mean test. No allowance for loss to follow-up is included.
Amylase: two independent groups
The second example retains a reference pooled variance of 290.9 and an effect of interest of 15 units/ml. The standard deviation is √290.9 ≈ 17.056, and d ≈ 0.87947. With α = 0.05 and power 0.90, the two-sided calculation for two equally sized groups gives n ≈ 28.166 per group: round up to 29 per group, 58 in total.
This calculation corresponds to the model of two independent groups with a common variance; it is not a general Welch power calculation with arbitrary variances and sample sizes. In paired samples, n counts pairs and the relevant standard deviation is that of the differences. Justified allowances for losses or incomplete records should be added; with an expected retention fraction r, an approximate recruitment target is ceiling(complete n/r), subject to the study design.
Calculating β: the noncentral t distribution
For a one-mean test under normality, with true effect δ = μ − μ₀ and σ specified as a planning assumption, T follows a noncentral t distribution with ν = n − 1 and noncentrality parameter λ = δ√n/σ. The test still estimates its standard error using S; knowing σ as a power-calculation assumption does not turn it into a z-test.
c = t₁₋α/₂,ν · λ = δ√n/σβ(δ) = Pν,λ(−c ≤ T ≤ c)Power = Pν,λ(T < −c) + Pν,λ(T > c)
With n = 124 and the preceding parameters, β ≈ 0.09861 and power ≈ 0.90139. Agreement with the planning function checks the calculation’s foundations. The script verifies this equality numerically.
13 / Scientific judgment
Assumptions, limitations, and reporting results
A function may return a number even when the question is poorly formulated. Analysis quality depends on the relationship among design, model, measurement, and phenomenon. The following criteria organize that assessment without turning it into a mechanical sequence of preliminary tests.
What to assess before interpreting a t-test
Aspect
Review question
Unit and independence
Does each row represent a distinct unit? Are there repeated patients, branches of the same chain, clusters, or temporal dependence?
Design
Was there random selection, random assignment, or convenience sampling? To which population can the findings be generalized?
Distribution
Are there strong asymmetries, extreme values, or very small samples? For pairs, what do the differences look like?
Variances
Is equality justified by the model? Is Welch preferable without imposing it?
Missing data
Which records were excluded, why, and with what possible bias?
Analytical decisions
Were hypotheses, tails, thresholds, and primary outcomes defined before searching for significance?
A quantile–quantile plot and inspection of the data can help assess distributional shape. Failing to reject a normality test does not establish normality, especially with small samples. Nor is every minor deviation a reason to abandon all parametric methods: what matters is how sensitive the particular inference is to that deviation.
If the tools change, preserve the question or declare its modification. A rank test is not automatically a test of equality of means. Bootstrap resampling and permutation methods require their own conditions; permuting labels needs a justification of exchangeability and does not, by itself, repair a dependent or biased design.
Significance, geometry, and magnitude
Observations can be represented geometrically as vectors, and means as projections onto the constant direction. In that language, a difference between estimates has magnitude and direction. Statistical significance additionally requires a reference distribution, an extremeness rule, and sampling assumptions. It is not defined solely as a visible change in a geometric pattern.
A small effect may be significant when estimated precisely; an important effect may have a wide interval. A binary decision does not exhaust the evidence. When many tests are performed, define the family of questions and consider appropriate multiplicity control; for example, p.adjust(valores_p, method = "holm") applies Holm’s adjustment to a specified family. There is no need to indiscriminately adjust every calculation presented for teaching purposes.
A way to report the result
A report should state the question and population, design, sample sizes actually analyzed, estimate in original units, interval, method and assumptions, statistic, degrees of freedom, and p-value, together with the limitations. For the sleep example: “The mean difference A − B was −1.58 hours (95% CI: −2.460 to −0.700); paired t-test, t(9) = −4.062, p = 0.002833, n = 10 pairs.” The report should then explain the meaning of that magnitude and how far the design permits generalization.
A very small p-value should be reported with sufficient digits or scientific notation, not as p = 0. A nonsignificant result should include its estimate and uncertainty; it should not be reframed as absence of an effect. A claim of equivalence requires margins and a specific analysis.
14 / Quick reference
A syntax guide and reproducible documents
Understanding the R resources used here
Resource
Operation
<-, =
Assignment in the contexts used here; <- is used for assignments and = for named arguments.
c(), numeric()
Construct a vector; preallocate a numeric vector.
$, [ , ]
Access a column by name; select rows and columns.
~
Express a formula. In these examples, the response is on the left and grouping on the right; its interpretation depends on the function receiving it.
factor(), table()
Represent categories; count observations by category.
apply(), tapply()
Apply over matrix margins; summarize groups defined by factors.
for, seq_len()
Repeat instructions; construct a safe sequence of indices.
pt(), qt()
Cumulative probability and quantile of a t distribution.
pf(), qf()
Cumulative probability and quantile of an F distribution.
library(), package::function()
Attach a package; use a function without attaching the entire package.
TRUE, FALSE
Logical constants. Prefer these to T and F, whose names can be reassigned.
Inspecting a test object with names(prueba_una) identifies its components. prueba_una$p.value, $statistic, $parameter, $conf.int, and $estimate extract results without manually copying them from the console. Each component’s name and interpretation should be checked in the function’s documentation.
R Markdown: results within prose
In an .Rmd file, an R expression enclosed in backticks and preceded by r is evaluated when the document is rendered. It is not an expression that WordPress executes or an instruction to paste unchanged into the R console.
The mean increase in sleep is `r mean(sleep$extra)` hours.
Hay `r nrow(sleep)` registros y `r length(unique(sleep$ID))` patients.
This distinguishes twenty records from ten patients in sleep. The mean of sleep$extra combines the two treatments; a comparison requires treatment-specific means or, in this design, patient-level differences. A reproducible document should also record transformations, exclusion criteria, and software versions.
15 / Executable appendix
Complete reproducible script
The following script assembles every step in order, includes the 21 illustrative records, produces the graphics, and compares manual calculations with R functions through numerical checks. Run it in a clean session. The main sections work with base R; if pwr and gplots are installed, the optional variants also run. Packages are not installed automatically.
To use the original file when it becomes available, change usar_archivo_original to TRUE and adapt importing to its actual format. The printed store results will change; the historical sleep results and planning scenarios do not depend on those records.
# FOUNDATIONS OF ESTIMATION AND HYPOTHESIS TESTING IN R
# JOSÉ MAURICIO GÓMEZ JULIÁN
# Revised edition. The 21 records are ILLUSTRATIVE, not recovered original data.
# Run in a clean session. Packages are not installed automatically.
# 1. Data and checks
usar_archivo_original <- FALSE
if (usar_archivo_original) {
estudios <- read.table("estudios.txt", header = FALSE,
col.names = c("menor16", "ipc", "ventas"))
} else {
estudios <- data.frame(
menor16 = c(12,15,11,18,14,16,13,19,17,12,20,15,18,14,16,19,13,17,11,20,18),
ipc = c(11,12,14,16,13,15,10,16,14,12,18,20,17,22,19,24,18,21,23,17,20),
ventas = c(145,158,172,183,167,192,151,176,164,180,188,205,194,217,201,229,182,212,221,196,208)
)
}
stopifnot(all(c("menor16", "ipc", "ventas") %in% names(estudios)),
all(vapply(estudios, is.numeric, logical(1))))
print(names(estudios)); print(nrow(estudios)); print(ncol(estudios))
print(dim(estudios)); str(estudios)
print(colSums(is.na(estudios)))
if (any(!is.finite(as.matrix(estudios)))) {
stop("There are missing or non-finite values: document their treatment before analysis.")
}
stopifnot(nrow(estudios) >= 2L)
ventas <- estudios$ventas
print(c(media = mean(ventas), sd = sd(ventas), varianza = var(ventas)))
print(apply(estudios, 2, mean)); print(apply(estudios, 2, sd))
print(quantile(ventas)); print(apply(estudios, 2, quantile))
qv <- quantile(ventas, probs = c(0.025, 0.975))
print(qv)
# Colors and base graphics: no additional packages required.
tinta <- "#2c2418"; cobre <- "#8b4513"; beige <- "#d4c8b0"
par(col = tinta, col.axis = tinta, col.lab = tinta, col.main = tinta,
bg = "#fffdf7", las = 1)
hist(ventas, breaks = 7, col = beige, border = cobre,
xlab = "Sales", ylab = "Frequency", main = "Sales: illustrative data")
abline(v = qv, col = cobre, lty = 2)
# 2. Simulation: population parameters, quantiles, and density
set.seed(20210906)
y <- rnorm(1000, mean = 35, sd = 2)
qy <- quantile(y, probs = c(0.025, 0.975))
hist(y, probability = TRUE, col = beige, border = cobre,
main = "N(35, 4): estimated and theoretical densities", xlab = "Value")
abline(v = qy, col = cobre, lty = 2)
lines(density(y), col = cobre, lwd = 2)
curve(dnorm(x, mean = 35, sd = 2), add = TRUE, lty = 3, lwd = 2)
legend("topright", c("Kernel estimate", "Theoretical normal"),
col = c(cobre, tinta), lty = c(1, 3), bty = "n", cex = 0.7)
plot.m <- function(n, m, s) {
stopifnot(length(n) == 1L, is.finite(n), n >= 2, n == as.integer(n),
length(m) == 1L, is.finite(m), length(s) == 1L, is.finite(s), s > 0)
y <- rnorm(n, mean = m, sd = s)
qy <- quantile(y, c(0.025, 0.975))
hist(y, probability = TRUE, col = beige, border = cobre,
main = paste("Normal sample: n =", n), xlab = "Value")
abline(v = qy, col = cobre, lty = 2)
abline(v = mean(y), col = tinta, lwd = 2)
lines(density(y), col = cobre)
invisible(list(media = mean(y), sd = sd(y), cuantiles = qy, datos = y))
}
muestra_10 <- plot.m(10, 100, 15)
muestra_10000 <- plot.m(10000, 100, 15)
print(muestra_10[c("media", "sd", "cuantiles")])
print(muestra_10000[c("media", "sd", "cuantiles")])
# 3. Sampling distribution of the mean: loop and theoretical equivalence
n_sim <- 10000L; mu_sim <- 100; sigma_sim <- 15; B <- 1000L
medias <- numeric(B)
for (i in seq_len(B)) {
sam <- rnorm(n_sim, mean = mu_sim, sd = sigma_sim)
medias[i] <- mean(sam)
}
print(c(media_teorica = mu_sim, media_simulada = mean(medias),
ee_teorico = sigma_sim / sqrt(n_sim), ee_simulado = sd(medias)))
qm <- quantile(medias, c(0.025, 0.975))
hist(medias, probability = TRUE, col = beige, border = cobre,
main = "Distribution of means", xlab = "Sample mean")
abline(v = qm, col = cobre, lty = 2); lines(density(medias), col = cobre)
# Extension: an exponential population with mean 1 and standard deviation 1.
set.seed(20210907)
medias_exp <- lapply(c(1L, 5L, 30L), function(nn) {
replicate(4000L, mean(rexp(nn, rate = 1)))
})
names(medias_exp) <- c("n1", "n5", "n30")
print(vapply(medias_exp, mean, numeric(1)))
print(vapply(medias_exp, sd, numeric(1)))
# 4. Confidence intervals: a function with input checks
ic <- function(x, p = 0.95, na.rm = FALSE) {
stopifnot(is.numeric(x), length(p) == 1L, is.finite(p), p > 0, p < 1,
is.logical(na.rm), length(na.rm) == 1L, !is.na(na.rm))
if (na.rm) x <- x[!is.na(x)]
if (any(!is.finite(x))) stop("Missing or non-finite data.")
n_x <- length(x)
if (n_x < 2L) stop("At least two observations are required.")
if (sd(x) == 0) stop("There is no sample variation.")
ee_x <- sd(x) / sqrt(n_x)
critico <- qt((1 + p) / 2, df = n_x - 1)
setNames(mean(x) + c(-1, 1) * critico * ee_x, c("lower", "upper"))
}
print(ic(ventas, 0.95)); print(ic(ventas, 0.99))
print(ic(estudios$ipc, 0.95)); print(ic(estudios$menor16, 0.95))
stopifnot(isTRUE(all.equal(unname(ic(ventas)),
as.numeric(t.test(ventas)$conf.int))))
# 5. One-sample t-test: manual and automatic calculations
alpha <- 0.05; mu0 <- 180
n_ventas <- length(ventas); gl <- n_ventas - 1L
ee_ventas <- sd(ventas) / sqrt(n_ventas)
t_obs <- (mean(ventas) - mu0) / ee_ventas
p_manual <- 2 * pt(-abs(t_obs), df = gl)
prueba_una <- t.test(ventas, mu = mu0, alternative = "two.sided",
conf.level = 1 - alpha)
print(prueba_una)
print(c(t_manual = t_obs, p_manual = p_manual))
stopifnot(isTRUE(all.equal(p_manual, prueba_una$p.value)))
# Choose a one-sided alternative before examining the results.
prueba_mayor <- t.test(ventas, mu = mu0, alternative = "greater")
# 6. Grouping and summaries. Illustrative threshold, not optimized.
umbral <- 17
estudios$ipc1 <- ifelse(estudios$ipc < umbral, 1L, 2L)
estudios$ipc2 <- factor(estudios$ipc1, levels = c(1, 2),
labels = c("ipc < 17", "ipc >= 17"))
plot(ipc ~ ipc2, data = estudios, col = beige, border = cobre,
xlab = "Group", ylab = "IPC", main = "Classification by IPC")
abline(h = umbral, col = cobre, lty = 2)
boxplot(ventas ~ ipc2, data = estudios, col = beige, border = cobre,
ylab = "Sales", xlab = "Group", main = "Sales by group")
med_grupos <- tapply(estudios$ventas, estudios$ipc2, mean)
sd_grupos <- tapply(estudios$ventas, estudios$ipc2, sd)
var_grupos <- tapply(estudios$ventas, estudios$ipc2, var)
n_grupos <- table(estudios$ipc2)
stopifnot(length(n_grupos) == 2L, all(n_grupos >= 2L), all(var_grupos > 0))
print(cbind(n = as.numeric(n_grupos), media = med_grupos,
sd = sd_grupos, varianza = var_grupos))
ic_grupos <- sapply(split(estudios$ventas, estudios$ipc2), ic)
plot(seq_along(med_grupos), med_grupos, pch = 18, xaxt = "n",
xlim = c(0.5, 2.5), ylim = range(ic_grupos), xlab = "Group", ylab = "Sales",
main = "Means and 95% confidence intervals")
axis(1, at = 1:2, labels = names(med_grupos))
arrows(1:2, ic_grupos[1, ], 1:2, ic_grupos[2, ],
angle = 90, code = 3, length = 0.05, col = cobre)
# Optional resource from the original article:
if (requireNamespace("gplots", quietly = TRUE)) {
gplots::plotmeans(ventas ~ ipc2, data = estudios, connect = FALSE,
col = cobre, barcol = cobre)
}
# 7. Independent means: pooled variance and Welch
dif_medias <- unname(med_grupos[1] - med_grupos[2])
sp2 <- sum((as.numeric(n_grupos) - 1) * var_grupos) / (sum(n_grupos) - 2)
ee_pool <- sqrt(sp2 * sum(1 / as.numeric(n_grupos)))
t_pool <- dif_medias / ee_pool
gl_pool <- sum(n_grupos) - 2
p_pool <- 2 * pt(-abs(t_pool), df = gl_pool)
prueba_pool <- t.test(ventas ~ ipc2, data = estudios, var.equal = TRUE)
a_w <- var_grupos / as.numeric(n_grupos)
ee_welch <- sqrt(sum(a_w))
gl_welch <- sum(a_w)^2 / sum(a_w^2 / (as.numeric(n_grupos) - 1))
t_welch <- dif_medias / ee_welch
p_welch <- 2 * pt(-abs(t_welch), df = gl_welch)
prueba_welch <- t.test(ventas ~ ipc2, data = estudios, var.equal = FALSE)
print(prueba_pool); print(prueba_welch)
stopifnot(isTRUE(all.equal(unname(p_pool), prueba_pool$p.value)),
isTRUE(all.equal(unname(p_welch), prueba_welch$p.value)),
isTRUE(all.equal(unname(t.test(ventas ~ ipc2, data = estudios)$p.value),
prueba_welch$p.value)))
# 8. F-test and variance-ratio interval
razon0 <- 1
razon_muestral <- unname(var_grupos[1] / var_grupos[2])
f_obs <- razon_muestral / razon0
gl1 <- as.numeric(n_grupos[1]) - 1; gl2 <- as.numeric(n_grupos[2]) - 1
p_f <- min(1, 2 * min(pf(f_obs, gl1, gl2),
pf(f_obs, gl1, gl2, lower.tail = FALSE)))
ic_razon <- razon_muestral / qf(c(1 - alpha/2, alpha/2), gl1, gl2)
prueba_f <- var.test(ventas ~ ipc2, data = estudios, ratio = razon0)
print(prueba_f); print(ic_razon)
stopifnot(isTRUE(all.equal(p_f, prueba_f$p.value)),
isTRUE(all.equal(ic_razon, as.numeric(prueba_f$conf.int))))
# 9. Paired samples: align by ID, rather than assuming row order
data("sleep", package = "datasets")
A <- sleep[sleep$group == "1", c("ID", "extra")]
B_sleep <- sleep[sleep$group == "2", c("ID", "extra")]
stopifnot(!anyDuplicated(A$ID), !anyDuplicated(B_sleep$ID),
setequal(A$ID, B_sleep$ID))
pares <- merge(A, B_sleep, by = "ID", suffixes = c("_A", "_B"), sort = TRUE)
stopifnot(nrow(pares) == 10L, all(complete.cases(pares)))
dif_sueno <- pares$extra_A - pares$extra_B
prueba_pareada <- t.test(pares$extra_A, pares$extra_B, paired = TRUE)
prueba_diferencias <- t.test(dif_sueno, mu = 0)
prueba_independiente <- t.test(pares$extra_A, pares$extra_B)
print(prueba_pareada); print(prueba_independiente)
print(c(dz = mean(dif_sueno) / sd(dif_sueno)))
stopifnot(isTRUE(all.equal(prueba_pareada$p.value, prueba_diferencias$p.value)))
matplot(1:2, t(as.matrix(pares[c("extra_A", "extra_B")])),
type = "b", pch = c(1, 16), lty = 1, col = cobre,
xaxt = "n", xlab = "Treatment", ylab = "Increase in sleep (hours)",
main = "Paired measurements by patient")
axis(1, at = 1:2, labels = c("A", "B"))
# Patient-index graph retained from the original example:
plot(seq_len(nrow(pares)), pares$extra_A, type = "b", pch = 1,
col = cobre, ylim = range(pares[c("extra_A", "extra_B")]),
xlab = "Patient", ylab = "Increase in hours", main = "A and B by patient")
lines(seq_len(nrow(pares)), pares$extra_B, type = "b", pch = 16, lty = 2)
legend("topleft", c("A", "B"), col = c(cobre, tinta), pch = c(1, 16),
lty = c(1, 2), bty = "n")
# 10. Power and sample size using base R
delta <- 3; sigma_plan <- 10.2; potencia_objetivo <- 0.9
d_plan <- delta / sigma_plan
plan_una <- power.t.test(n = NULL, delta = delta, sd = sigma_plan,
sig.level = alpha, power = potencia_objetivo,
type = "one.sample", alternative = "two.sided", strict = TRUE)
n_requerido <- ceiling(plan_una$n)
potencia_redondeada <- power.t.test(n = n_requerido, delta = delta,
sd = sigma_plan, sig.level = alpha,
type = "one.sample", strict = TRUE)$power
plan_40 <- power.t.test(n = 40, delta = NULL, sd = sigma_plan,
sig.level = alpha, power = potencia_objetivo,
type = "one.sample", strict = TRUE)
deltas <- seq(2, 6, length.out = 30)
n_plan <- numeric(length(deltas)); d_grid <- deltas / sigma_plan
for (i in seq_along(deltas)) {
n_plan[i] <- power.t.test(delta = deltas[i], sd = sigma_plan,
sig.level = alpha, power = potencia_objetivo,
type = "one.sample", strict = TRUE)$n
}
plot(deltas, ceiling(n_plan), type = "s", col = cobre, lwd = 2,
xlab = "Difference to detect", ylab = "Sample size (rounded up)",
main = "90% power; alpha = 0.05")
# Amylase: two independent groups, equal sizes, and a common variance.
s2p <- 290.9; sp <- sqrt(s2p); delta_amilasa <- 15
d_amilasa <- delta_amilasa / sp
plan_amilasa <- power.t.test(delta = delta_amilasa, sd = sp,
sig.level = alpha, power = potencia_objetivo,
type = "two.sample", strict = TRUE)
print(c(d = d_plan, n_continuo = plan_una$n, n_requerido = n_requerido,
potencia_redondeada = potencia_redondeada,
delta_con_n40 = plan_40$delta, d_amilasa = d_amilasa,
n_amilasa_por_grupo = plan_amilasa$n,
n_amilasa_redondeado = ceiling(plan_amilasa$n)))
# Original version using the optional pwr package:
# install.packages(c("pwr", "gplots")) # run only if you wish to install these packages
if (requireNamespace("pwr", quietly = TRUE)) {
plan_pwr <- pwr::pwr.t.test(n = NULL, d = d_plan, sig.level = alpha,
power = potencia_objetivo, type = "one.sample",
alternative = "two.sided")
plan_pwr40 <- pwr::pwr.t.test(n = 40, d = NULL, sig.level = alpha,
power = potencia_objetivo, type = "one.sample")
plan_pwr_amilasa <- pwr::pwr.t.test(n = NULL, d = d_amilasa,
sig.level = alpha, power = potencia_objetivo,
type = "two.sample")
print(plan_pwr); print(plan_pwr40$d * sigma_plan); print(plan_pwr_amilasa)
stopifnot(abs(plan_pwr$n - plan_una$n) < 0.001,
abs(plan_pwr_amilasa$n - plan_amilasa$n) < 0.001)
}
# 11. Exact power for a mean under normality: noncentral t
gl_plan <- n_requerido - 1
crit_plan <- qt(1 - alpha/2, gl_plan)
ncp_plan <- delta * sqrt(n_requerido) / sigma_plan
beta_plan <- pt(crit_plan, gl_plan, ncp = ncp_plan) -
pt(-crit_plan, gl_plan, ncp = ncp_plan)
print(c(beta = beta_plan, potencia = 1 - beta_plan))
stopifnot(abs(1 - beta_plan - potencia_redondeada) < 1e-10)
# R Markdown: write this inside an .Rmd, not as standalone R code:
# The mean increase in sleep is `r mean(sleep$extra)` hours.
# The number of records is `r nrow(sleep)`; there are `r length(unique(sleep$ID))` patients.
# 12. Verifiable summary and session information
cat("\nVERIFIED_SUMMARY\n")
print(list(
media_ventas = mean(ventas), sd_ventas = sd(ventas), cuantiles_ventas = qv,
ic95_ventas = ic(ventas), ic99_ventas = ic(ventas, 0.99),
una_muestra = c(t = unname(prueba_una$statistic), gl = gl, p = prueba_una$p.value),
grupos = c(diferencia = dif_medias, t_welch = unname(t_welch),
gl_welch = unname(gl_welch), p_welch = p_welch, p_pool = p_pool,
f = f_obs, p_f = p_f),
sueno = c(diferencia = mean(dif_sueno), sd_diferencias = sd(dif_sueno),
t = unname(prueba_pareada$statistic), p = prueba_pareada$p.value,
ic_inf = prueba_pareada$conf.int[1], ic_sup = prueba_pareada$conf.int[2],
p_independiente = prueba_independiente$p.value),
planificacion = c(n_una = plan_una$n, delta40 = plan_40$delta,
n_amilasa = plan_amilasa$n)))
sessionInfo()
16 / Sources
References and documentation
The linked documentation allows the arguments, distributions, and assumptions to be checked. Functions are cited where they are used; reference links do not replace the explanations developed in the text.
Wasserstein, R. L., and Lazar, N. A. (2016). The ASA’s Statement on p-Values: Context, Process, and Purpose. The American Statistician, 70(2), 129–133. doi:10.1080/00031305.2016.1154108.
R Core Team. Apply Functions Over Array Margins: apply().
R Core Team. Student’s Sleep Data: sleep. The documentation refers to Cushny and Peebles (1905) and Student (1908).
R Core Team. Power Calculations for One and Two Sample t Tests: power.t.test().
Champely, S., and contributors. pwr: Basic Functions for Power Analysis, version 1.3-0. Package manual. Calculations use Cohen’s (1988) effect-size parameterization.
The thematic connections to the author’s Spanish-language texts Análisis teórico de la función cuantil en R Studio [Theoretical Analysis of the Quantile Function in RStudio] and Modelos lineales generalizados [Generalized Linear Models] are also retained. The original publication mentions Part I on statistical theory; this notebook develops the foundations used here without attributing to that document content that has not been explicitly incorporated.
In Capitalist Political-Economic Systems in Particular and Economic Systems in General
JOSÉ MAURICIO GÓMEZ JULIÁN
Original study · 2021 / Complete revised and expanded edition · 2026
Abstract
An investigation of living labour, the means of production and technological devaluation. The essay examines controversies over values and prices, competition and profit rates, the temporality of models and vertical integration. It considers their historical and statistical foundations and explores the relations between value, energy and information. It includes the United States empirical appendix, all the graphic materials in the original document, and a reproducible methodological supplement.
Opening perspective
Labour and the transformation of its own conditions
Labour produces commodities and transforms the conditions under which those commodities can be produced again. This relationship raises a decisive question: how can an activity that creates value also contribute, through technological development, to the devaluation of the means of production and products that it has itself made possible?
This essay examines that question by bringing together Marxian political economy, the history of economic thought, the epistemology of models, statistics and input–output analysis. Its starting point is the distinction between the creation of new value by living labour and the transfer of the value of the means of production. Its dynamic problem concerns changes in the social conditions of reproduction: a machine can remain physically operational yet lose value when a cheaper or more productive alternative appears.
The physical persistence of a means of production does not guarantee the persistence of its economic value.
The guiding proposition of this edition
The debate requires a distinction between the material conditions of the labour process and the social forms of production. Every society must reproduce its material existence; generalised commodity production, wage labour and the valorisation of capital belong to specific historical configurations. Extending the analysis to economic systems in general means investigating continuities and differences without automatically attributing every category of capitalism to every society.
The exposition retains the complete trajectory of the 2021 study: the scenarios concerning the contribution of productive factors; the debate over value–price correlations; competition and profitability; transformation and temporality; vertical integration; depreciation; and the problem of information. It also includes the United States empirical methodology, its graphic materials, references and endnotes. The clarifications distinguish original results, revised inferences and proposed developments. The historical tables retain their figures and are not presented as new estimates.
Concepts and units
What Is Created, Transferred and Lost
Economic vocabulary used in the exposition
Category
Operational definition
Necessary distinction
Use-value
The utility of a good or service under specified conditions.
Utility and material abundance do not, by themselves, constitute a measure of value.
Value
In Marxian theory, the objectification of socially necessary abstract labour.
The time individually expended does not automatically determine the time socially recognised.
Exchange-value
The quantitative relation in which one commodity exchanges for another; a form of expression of value.
It is not simply identical to the substance or magnitude of value.
Price
A monetary expression whose formation requires specification of the market, productive conditions and theoretical framework.
A price change does not, by itself, identify the creation or destruction of value.
Value of labour-power
The labour socially necessary for its reproduction, with historical and social determinations.
It differs from the value created through the exercise of that capacity.
Surplus-value
The part of new value that exceeds the equivalent of the value of labour-power.
It is not synonymous with all new value or with any physical surplus.
Constant / variable capital
A distinction based on the function performed in valorisation.
It does not coincide with fixed / circulating capital, a distinction concerning turnover and transfer.
Material / technological depreciation
Physical wear and the loss of value associated with cheaper or improved conditions of reproduction.
The latter can occur before the physical life of the asset is exhausted.
A minimal conceptual accounting framework
In a simplified scheme, the value of the product is divided into the value transferred by the means of production consumed, the equivalent of labour-power, and surplus-value:
W = c + v + s · New value = v + s
Here c is the value of the means of production actually consumed during the period: the entire stock of fixed capital does not enter the product at once. Concrete labour preserves and transfers the value of the means used; labour considered abstractly adds new value. The part of that new value which replaces the wage is still newly created value. The production of surplus-value additionally requires the working day to exceed the time necessary to reproduce the equivalent of labour-power.
For example, if 40 units of value are transferred and living labour adds 60, of which 30 are equivalent to the value of labour-power, the product contains 100 and surplus-value is 30. All 60 units added constitute new value. This is a teaching example, not a statistical observation from the original study.
Units must be declared. If labour magnitudes are expressed in hours and monetary magnitudes in monetary units, a monetary expression of labour time—MELT—has the dimensions of money per hour. Multiplying hours by that coefficient expresses a labour magnitude in money under a particular theoretical definition; it does not prove that every individual price is proportional to labour or eliminate disputes over the determination of the MELT.
mt : money/hour · mtLt : money
For the rate of profit, it is necessary to distinguish capital advanced, its turnover periods and the annual flow of costs. The expression r = s/(C + V) requires compatible definitions and temporalities for numerator and denominator. Indiscriminately substituting costs consumed for the stock advanced mixes different magnitudes. The rate of surplus-value, by contrast, relates s to v.
Textual foundations for these distinctions: Marx, Capital, Volume I, the chapters on the purchase and sale of labour-power, the valorisation process, and constant and variable capital. Verifiable supplementary references are collected at the end.
I. Historical Background
There is a debate within Marxism, particularly between the New Interpretation (NI), associated with Foley and Duménil, and the Temporal Single System Interpretation (TSSI), associated with Alan Freeman and Andrew Kliman. Their disagreements concern several questions; of particular interest here is the discussion of the conservation or destruction of value.
Classical mechanics allows a limited analogy: physical work describes a transfer of energy associated with forces and displacements. Its general expression is the integral of force along a path. This relation neither identifies physical work with abstract labour nor turns economic theory into an application of mechanics. The historical influence of the natural sciences on economists must be investigated through documentary evidence; a formal resemblance to wave–particle duality likewise does not demonstrate an equivalence between value and price.
The first characteristic is the same definition of work, i.e., a force is said to perform a work when there is a displacement of the center of mass of the body over which the force is applied, in the direction of that force, while the second characteristic is that As by definition work is an energy transit (…) it is never referred to as an increase in work, nor is it symbolized as ΔW, as verified in (Wikipedia, 2021). The first characteristic implies (considering it not at a physical level but reflecting it philosophically) that the work has a direction (as a direction of a trajectory), a reason to be for that direction and an alteration of the location or state of the body on which it is applied; a synthetic way of seeing it is to think of physical work as the application of a force along a displacement (Wikipedia, 2021), which captures the notion that it consists in the application of a force that opposes the location of the object with sufficient intensity to displace it from its location within the system, regardless of the system in question. The second characteristic means that, by definition, since transitar is defined as Going or passing from one point to another via public roads or places (Real Academia Española, 2021), the physical work applied on an object transfers its energy to that object.
In the capitalist production process, labour is applied to means of production—raw materials, intermediate products, auxiliary materials and instruments of labour—and produces new commodities. The decisive Marxian distinction is twofold: concrete labour preserves and transfers the value of the means actually consumed, while abstract labour creates new value. The value of labour-power is determined by the social conditions of its reproduction, whereas its use can create a greater value. The production of surplus-value rests on that difference. Classical references to Ricardo and Marx must distinguish their theories of prices, money and international exchange; no energy analogy replaces historical and social argument.
II. Factor Contributions to the Production Process
A historical, epistemological, theoretical and econometric review
II.I. First scenario: no productive factor creates value
Suppose that neither labour nor the means of production create new value. Within the labour-theory framework, the origin of the increase in value and its relation to the surplus would then have to be explained. It does not follow that physical growth is impossible: different theories can conceptualise value and wealth differently. The scenario defines the problem addressed by the essay; it does not refute every alternative theory at once.
A critique of marginalist, neoclassical, Austrian, Keynesian and post-Keynesian economics requires recognition of their internal differences and objects of study. This investigation retains a Marxian perspective and questions explanations that separate valorisation from the social relations of production. A systematic refutation of each school exceeds this particular argument; their differences cannot be resolved by treating them as identical variants of subjectivism.
II.II. Second scenario: only the means of production create value
The hypothesis that only the means of production create value must be confronted with the function of labour in social reproduction. Ancient, slave and feudal societies depended on human labour, although their forms of appropriating the surplus differed. The necessity of labour for producing goods does not, by itself, provide a complete demonstration of the labour theory; it does prevent the material contribution of equipment from being treated as a socially self-sufficient explanation.
Automation transforms the distribution of labour across tasks, sectors and periods. It can reduce direct labour per unit while expanding the indirect requirements associated with equipment, energy, maintenance and knowledge. This transformation is compatible with the distinction between physical productivity and value creation, but it must be studied using data on the entire chain, without treating a machine as independent of the labour required for its production and operation.
The study of primitive accumulation, developed by Marx in Volume I of Capital, makes it possible to examine the expropriation of direct producers, the formation of labour markets, slavery and colonial expansion. European industrialisation and long factory working days belong to a sequence later than the European arrival in the Americas; this corrects the chronological inversion in the draft. Early societies used natural resources and tools of varying complexity: the development of means produced through labour must be analysed historically, without assuming a complete absence of instruments in every early society.
In (Sánchez & Ferràndez, Valores, prices of production and market prices based on data from the Spanish economy, 2010) a study is carried out on the relationship between direct prices (proportional to value), prices of production (the primus of capitalist industrial production, are the prices that form industrially) and market prices or sales prices (the final prices set in the market by the interaction of supply and demand – in Marx they are called sales prices because they are set in the sphere of commercial capital, not productive capital-), obtaining results that verify the previously made claims. This relationship is modeled by using regressions with log−log regression models of the form log(Yi) = β0 + β1 log(Xi) + ui, as shown below.
Plate 01 · original document, p. 6
Value–price regressions: results for Spain. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Ferràndez, Valores, precios de producción y precios de mercado a partir de los datos de la economía española, 2010, p. 93).
Plate 02 · original document, p. 7
Further comparisons of values and prices in Spain. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Ferràndez, Valores, precios de producción y precios de mercado a partir de los datos de la economía española, 2010, p. 94).
There are some questions to be asked. As is widely known, there exists in (Kliman, The law of value and laws of statistics: sectoral values and prices in the US economy, 1977-97, 2002) a critique of the experimental results obtained by Marxist econometrists in the last 15-20 years of research related to the statistical correlation between values and prices obtained from the input–output matrices of different countries across the globe, arguing about them that the correlation was spurious in a seemingly robust way and performing as an empirical demonstration of their argumentation a simulation in terms of statistically implying a non-spurious correlation between values and prices (and verified at the end of the simulation), by allocating terms to the specific perturbation rate; these disturbances in the same way express a mathematically variable value that exists in a system of political economy as a result of the existence of a system of constantly varying costs and gains, which is usually expressed in terms of the value of a given asset, such as the value of a given asset and the value of a given asset, and the value of a given asset, as a result of the average gains in a given system as a given value of money, as a constantly multiplied by the value of a given value of a given value. The reason why the average rate of profit (in its stochastic version) is theoretically proposed as a variable whose distribution is normal is explained by two reasons. 1
The first reason invoked by the draft for adopting normality concerned statistical limit theory. Its scope must be specified: a central limit theorem establishes, under particular assumptions, convergence in distribution of appropriately centred and normalised sums or means. It does not establish that the distribution of the original observations becomes normal as more years are observed. Nor is there a general division according to which the CLT is a topologically local law and the law of large numbers a global law: they are different results concerning different objects and modes of convergence.
The law of large numbers concerns the stabilisation of averages; ergodic theorems relate time averages to properties of a process under explicit assumptions. Results exist for dependent observations and for non-identical distributions. The relevance of any particular result requires checking its conditions: the dependence structure, moments, stationarity where appropriate, and normalisation. The economic interconnection of sectors makes it essential to specify this problem, but it does not demonstrate that every theorem for dependence applies to every dataset.
Temporal dependence, dependence between units and multicollinearity among regressors must also be distinguished. Linear regression does not require explanatory variables to be stochastically independent; the absence of perfect collinearity permits coefficient identification, while high collinearity can increase uncertainty. A non-zero inner product is not, by itself, equivalent to stochastic dependence. Variance inflation factors diagnose a regression-design problem; they neither certify the assumptions of a CLT for dependent series nor replace specification tests.
What happens is that, like light and sound when Nature leaves the sky with electric fury, practice travels faster than theory, so the deficiencies of the paradigm (in the sense of Thomas Kuhn) are first evident in the applications and it is after that that that theories are made about the new findings and the new paradigm is consolidated, since the paradigms are only one or more past scientific achievements, achievements that some particular scientific community recognizes, for a certain time, as a foundation for its later practice. 33). Thus, in research (Dedecker & Prieur, 2007) a central limit theorem for an empirical process under dependence conditions is studied, in research (Andrews, 1991) the existence of the CLT is demonstrated for empirical processes indicated by soft functions (in which the underlying stochastic variables can be temporarily dependent and distributed non-identally), in research (KO, RYU, KIM, & CHOI, 2007) a general version of the CLT is proposed for weighted sums of random variables with linear negative dependence (the central idea of this research is to weigh the elements of a non-negative triangular dependent matrix, whose elements have a measure, by weighing the elements of this matrix with a linear-negative dependencies), in research (KO, RYU, KIM, & CHOI, 2007) a general version of the CLT is proposed for weighted sums of random variables with linear negative dependence (the central idea of this research is to weigh the elements of a non-negative dependent matrix, whose elements have a measure, underlying the elements of a linear-negative dependent parameters with a linear-negative parameters), where under the parameters of a mathematical parameters are demonstrated in a mathematical framework (theoretically dependent parameters for theoretical parameters of the CLT, under theoretical parameters of the M−a−a−a−a−a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −a −b −b −b −b −b −a −b −b −b −b −a −b −b −a −a −b −b −b −b −b −b −b −b In multilinear stochastic processes, in research (Godwin & Zaremba, 1961) CLT verification is proposed when variables, although dependent on each other, are only partially dependent (the reason the authors use partly dependent instead of partially dependent is because, according to (Alan, 2011), parly” is used when an object is part of a tangible whole and partially when it is not). Thus, the state of the art on the subject points not only towards an empirical validation of the above, but also that the available results extend the field of inference under specific hypotheses, without proving by themselves its applicability to any economic series.
The second reason in the draft connected the long-run horizon of economic theory with a normal distribution of the rate of profit. This connection must be formulated as a testable hypothesis, rather than a necessary consequence of Marxism or of increasing sample size. There are three different objects: the distribution of profitability across sectors in a given year, the temporal evolution of an aggregate rate, and the sampling distribution of an estimator of that rate. A gamma distribution of sectoral profitability can coexist with asymptotic normality results for certain averages. Difficulties of estimation do not justify imposing normality.
The following two charts retain the original exploration of the United States, 1963–2008, prepared using Marquetti and Foley’s data. The first describes the series; the second shows diagnostics of a normal fit. The underlying data have not been re-estimated here. An updated examination would require declaring the definition of profitability, studying temporal dependence and possible structural changes, assessing the fit, and comparing alternative distributions.
Long-run evolution of the average rate of profit. United States, 1963–2008
Plate 03 · original document, p. 11
United States average rate of profit, 1963–2008. The original study material is retained; the surrounding discussion explains its interpretation and limitations.
Source: Author’s calculations in RStudio, based on (Marquetti & Foley, 2021).
Normal probability distribution fitted to the average rate of profit. United States, 1963–2008
Plate 04 · original document, p. 12
Graphical diagnostics of the normal fit: United States sample. The original study material is retained; the surrounding discussion explains its interpretation and limitations.
Source: Author’s calculations in RStudio, based on (Marquetti & Foley, 2021).
The graphical diagnostics document the normal fit examined in the original study. They do not, by themselves, establish the normality of the data-generating process or a universal law governing the distribution of profitability. Their scope is exploratory and specific to the sample and definition of the rate used.
In Cockshott & Cottrell’s research (Robust correlations between prices and labour values, 2005) a strong refutation of the criticism of the American Marxist economist Kliman is made, on which its essential aspects will be exposed. Computer scientists argue that Kliman’s point of spurious correlation is based on a misconception of what spurious correlation is. Thus, they point out that The danger of a false correlation is quite real. Take, for example, a study of the association between alcohol consumption and violent crime. Suppose a researcher makes a regression with the number of violent crimes as a dependent variable and the amount of alcohol consumed as an independent variable, for a sample of cities of very variable sizes. We should expect to find a significant positive coefficient on alcohol consumption, but this would have no scientific interest: simply, we would expect the larger cities to show both more crime and more alcohol consumption. The obvious correction here is to scale both variables of interest by expressing them per capita, dividing by the population of the city. If there is still a significant positive association, this could be of sociological interest. However, correlations in which the units of observation are of different “sizes” are not necessarily false. (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 310). 4
Subsequently, they propose a modified version of the previous example, modified in the sense that it poses the scenario of a researcher who has the hypothesis that population is the main factor that governs the size of cities (measured this size by their land area). Thus, they point out that One way to evaluate this statement would be to make a regression of the city’s territorial area over population and see if the relationship between these variables is close to proportional. In this case, one is well aware that both land area and population are measures of the “size” of the city, and the object of the exercise is to see how closely they are related. (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 310), to conclude with Suppose now that someone objected to this hypothetical study as follows: “This is a case of spurious correlation. Of course, larger cities will, in general, occupy more land area and have larger populations. To overcome this problem, you’ll have to deflate the land area and the population with an appropriate measure of the size of the city, say the number of residential units. The objection is out of place. In the former case, the size of the city (population) was an independent ‘third factor’ that could plausibly induce an apparent correlation between crime and alcohol consumption, while in the latter case there is no independent third factor at stake. (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 310)4. What follows is an epistemological questioning (valid, by the way) by the authors cited in relation to the analysis of unit values and prices.
Cockshott and Cottrell defend the relevance of studying aggregate sectoral magnitudes. That choice addresses a question about the social structure of production; it does not mean that political economy has no unit-level or disaggregated objects. Investigating totalities also requires studying relations among their parts. The comparison of aggregates, unit magnitudes and relative prices must be justified by the hypothesis under investigation, not merely by a philosophical preference for “the big picture”.
We have two answers to this suggestion. First, for many commodities there is no single natural ‘unit’ other than the total production of the industry concerned. The scope of commodities can be divided into two sub-sets: ‘accounting’ products and ‘massive’ products. Cars are a paradigmatic example of an accounting commodity: the natural unit is the individual car. You cannot buy half a car and there is no relevant natural unit larger than an individual car (e.g. you don’t buy cars per six pack). Electricity is a good example of a mass consumer good. What is the natural unit of electricity? The kilowatt-hour can be used for billing purposes, but is not entitled to special status as a natural unit. Liquids of all kinds (cooking oil, gasoline, milk, beer) are also mass consumption products, as are flour, cheese, etc. These products are “divided” into units of various sizes for the convenience of the consumer, but the units are not intended to be unique. Given the presence (and importance) of mass consumer products, there is simply no way to produce a non-arbitrary vector of industrial sizes in natural units. In other words, the idea of ‘unit price’ is generally not well defined and therefore cannot serve as a basis for a ‘correct’ price-value correlation measure. Secondly, even if we focus exclusively on commodities where there is an unequivocal natural unit (ignoring all commodities in mass), it can be argued that the effect of moving to unit prices and values, by dividing the aggregate output of industry by the size of industry into natural units, would not reduce the price-value correlation found at the aggregate level of industry. (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 310-311). The mathematical foundation of his second argument is presented below, although in the research where saw light appears in the appendices, it will be presented here immediately in order to achieve exponential continuity, hoping thereby to achieve greater clarity and depth at the analytical level.
They point out (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 316) that It might seem that, if all industries produced ‘accounted’ raw materials with an unequivocal natural unit of production, we could measure the ‘correct’ correlation between price and value if we first divided the price and value vectors of industry by an industry size vector, expressed in the appropriate physical units for each industry., to continue to argue that But note that there is no reason to expect this ‘physical size’ vector to be correlated with the price and value vectors of industry. The fact that the baking industry produces more bread than the automotive industry does not tell us anything about the price or relative value of the production of these industries. So what is the probable effect of a scale of the industry price and value vectors by the industry size vector? This is easier to see if we think in logarithmic terms. P and V are the vectors of industry prices and aggregate values, respectively, and p and v are the vectors of logarithmic prices and logarithmic values (industry aggregates). S is the size vector of the industry in physical units and the logarithmic size vector.
To express the scaling correctly, write p = log P, v = log V and s = log S. The correlation of p and v is compared with that of p − s and v − s; the second expression must contain V/S, rather than a second copy of P/S.
ρ(p,v) = Cov(p,v) / √[Var(p) Var(v)]
ρ(p−s,v−s) = Cov(p−s,v−s) / √[Var(p−s) Var(v−s)]
If Cov(p,s) = Cov(v,s) = 0, it is sufficient to speak of an absence of linear correlation with the scaling factor; zero covariance does not establish stochastic independence. Under those conditions, the correct identity is:
The denominator contains sums of variances. If Cov(p,v)=1 and Var(p)=Var(v)=2, the initial correlation is 1/2; adding a scaling variance of 1 gives 2/3, and adding 2 gives 3/4. This retains Cockshott and Cottrell’s numerical example while correcting the algebraic transcription.
In this example, scaling increases the correlation. Monotonicity must not be generalised without conditions to every combination of variances and covariances. The methodological lesson is that dividing both variables by a common factor can change their association: this change requires economic and statistical interpretation. Neither an increase nor the disappearance of correlation resolves the causal question by itself. Note 5 retains the discussion of the scaling factor, subject to the clarification given here.
Continuing their refutation of Kliman’s criticisms, the authors propose a process of selecting the scale variable, epistemologically justifying their proposal. They explain that:
There is a valid argument for taking industry size into account when measuring price-value correlations, but it is quite different from the ideas considered above. This point can be addressed by the concept of ‘specific price’ proposed by Farjoun and Machover (1983), i.e. the market price obtained per unit of labour embedded. Farjoun and Machover suggest that we think in terms of the probability distribution of a specific price. If prices everywhere were exactly proportional to the content of labour, this distribution would be degenerate, the whole mass contained in a single peak. We don’t expect to see that; the question is, how scattered is the distribution of the specific price, compared to other economic variables of interest such as interest rates or the organic composition of capital? When constructing an empirical price-specific probability distribution from industry data, it is necessary to weigh industries according to the added value of their production (see Cockshott and Cottrell, 1998). Each industry receives a weight equal to the proportion of the total value of production represented by the industry in question. This is because the starting point of the mind experiment is not “selecting a random commodity”, but “selecting a randomly built-in unit of working time”. If one thinks in these terms, there is a good case for applying work time weights of this type when calculating price-value correlations between industries, and also for using weighted minimum squares with wi weights when estimating price-value regressions. (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 311).
The above only corresponds to the criticism of the authors of Kliman’s claim (questioned by those authors) that it is necessary to deflate prices and values of industries in the context of aggregate economic analysis, not even to Kliman’s simulation as such (which is based on the above, disputed idea). Despite this, Cockshott and Cottrell also do an analysis of the simulation as such. The simulation carried out by Kliman is performed conditioning it beforehand so that the correlation is true, selecting amplitudes (of the term stochastic perturbation) such that at the end of the simulation correlations between price and cost, as well as between cost and value appear. The logic behind Kliman’s experiment is that if the correlations exist (as is the case with the simulation) then they should survive the deflationary process he conducted (i.e., divide by the size of the industry -which measures by the added costs of each industry-). As noted (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 312), This simulation seems convincing at first glance, but it overlooks a key feature of the actual empirical procedure. In Kliman’s simulated data, sector values are the given amount, while costs and prices are each obtained by taking a random step of the value vector. The corollary is that price and cost vectors are two random steps separated from each other, with the consequence that price and cost are less correlated than price and value. Kliman sets out the characteristics of his simulated data in Table 2 (2002, p. 304).”
Table 2 of the Kliman study to which reference is made is presented below in this investigation.
Plate 05 · original document, p. 19
Characteristics of Kliman’s simulation. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Kliman, The law of value and laws of statistics: sectoral values and prices in the US economy, 1977-97, 2002, p. 304).
Thus, continuing with the Cockshott and Cottrell exposition, It is not reported that the price-value correlation is higher than the price-cost correlation (…) With cost and price “farther away” than price and value, cost deflation reduces but does not destroy the price-value correlation.
Let us suppose that if we had independent empirical data on the value of industrial production, the results of deflation by industrial costs (unnecessarily) would be similar to those found in the Kliman simulation, reducing the price-value correlation but not destroying it. Unfortunately, however, we do not have independent data on securities. This would require a fully disaggregated input–output table together with data on labour input in hours. When we calculate industrial values in practice, our primary data are only industrial costs: we infer values from these costs, with an inevitable error. (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 312), and the above even though this way of deflating variables (using costs per industry) is not shared at an epistemological level by Cockshott and Cottrell, since, as has been seen before, the authors do not consider deflating necessary in such aggregate studies and, failing that, they would appear to find more good in the deflationary method proposed by Farjoun and Machover in 1983, which in the opinion of the author of this research is true insofar as the workforce, here in the form of units of work, is omnipresent at a methodological level throughout theoretical and applied analysis, which makes it a statistical practice at the level of what in economic theory is known as a commodity, the essence of which in philosophical terms is equivalent to a single characteristic of the dynamic system, which can be thought of as a common feature of the dynamic system in terms of radial length. 6
Finally, (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 312-313) point out that there are three empirical sources of divergence between the underlying real values and the values estimated from industrial cost data:
1. Values are defined in terms of hours of socially necessary working time, but in most empirical work direct working hours are represented by the wage mass. 2. The assumption that the rate of surplus-value is uniform in all sectors: most, if not all, empirical work on value uses this assumption, but it is unlikely to be strictly true. 3. The question of aggregation: each of the sectors or industries used in an empirical study represents the aggregation of a wide range of production processes. Aggregation is, of course, carried out in terms of monetary values. Therefore, the weightings taken by each subsector in the formation of figures for any given sector depend on market prices, which of course show some degree of divergence in values.
Table 1 presents the results of 500 replications of the simulation that Cockshott and Cottrell replicated from Kliman.
Plate 06 · original document, p. 21
Cockshott and Cottrell’s simulation replications. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 314).
As can be seen, the correlation between values and prices was effectively destroyed under deflation by the costs of industry (…)
Cockshott and Cottrell’s reply emphasises that a simulation must represent the procedure by which values are actually estimated. When these estimates are inferred from monetary costs, dividing by those same costs changes the source information itself. The following simplified identity retains the study’s algebraic argument:
Vi = ci + vi + s′vi = Ci[1+s′/(1+Oi)]
Ci = ci+vi · Oi = ci/vi · Vi/Ci = 1+s′/(1+Oi)
Here s′ represents an assumed rate of surplus-value and O a cost composition, under homogeneous definitions. The identity neither eliminates all information about labour nor proves that the original correlation is causal: it shows which scaling component is cancelled. Assessing the normalisation requires considering how each variable was estimated and which hypothesis is being tested.
Of course, the debate did not end there, appearing in (Kliman, Reply to Cockshott and Cottrell, 2005) the answer to the research of Marxist computer scientists. In it Kliman points to an aspect that has indeed been overlooked in Cockshott and Cottrell’s research, saying that: &C&C challenges one aspect of my demonstration, my simulation exercise, but the results of my simulation simply confirmed the results I had already tested deductively (Kliman, 2002, p. 303). For some reason, C&C simply ignores this test. (Kliman, Reply to Cockshott and Cottrell, 2005, p. Therefore, if the labour theory of relative prices were true, ln(A) would still be equal to A. However, the price-value correlation will necessarily be lower. (Kliman, The law of value and laws of statistics: sectoral values and prices in the US economy, 1977-97, 2002, p. 303) as referred to by Kliman in his 2005 investigation.
Although Kliman’s arguments are true, this does not necessarily imply that the results of regression are epistemologically invalid (for the reasons set out by Cockshott and Cottrell). This is because the logic that led C&C’s research is based on the idea that there is a correlation between values and significant prices (which Kliman claims is spurious and precisely to prove such an assertion is that Kliman poses the simulation in terms such that it assumes the correlation as non-spurious), it is not intended to verify that the econometric approaches (and their possible theoretical implications) made by some authors of the NI are or are not completely compatible with what was theorized by Marx and Engels. Thus, one thing is the statistical explanatory power of values in relation to prices (of production and market) and another thing is whether or not the theoretical underpinning of that explanatory power is epistemologically valid, although they are clearly related aspects, but not therefore equivalent. To what extent are both aspects of these economic investigations (purely statistical and theoretical-economic) linked and to what extent are they separate? 7
The discussion of the so-called labour theory of relative prices must be distinguished from the proof of the aggregate identities: sum of prices and values, sum of gains and surpluses, and consistency of profitability under compatible definitions and units. The basic document interprets Kliman’s criticism as affecting the reconstruction of those relations, while Cockshott and Cottrell’s defence focuses on the empirical procedure. This difference in questions does not allow a complete refutation to be awarded by a single correlation result.
Kliman belongs to the TSSI and maintains differences with other interpretations, but his personal intention is not presumed. The New Interpretation and the set of empirical value-price works are not identical sets. The consistency of a transformation must also be distinguished from the hypothesis of a trend fall in profitability, which requires its own dynamic analysis.
Proof of the above statement is found in (Cockshott, Cottrell, & Zachariah, Against the Kliman theory, 2019, p. 17). There, the authors point out that Kliman follows the error of Ricardo and Marx in assuming that the rate of profit can be considered the same in all capital. He puts all his effort into trying to construct a consistent theory with equal profit rates. But there is no evidence of a trend to equalize the rate of profit since real profit rates decrease with the intensity of capital! This question will then be addressed.
Competition, gravitation and persistent differences
The proposition that there is no tendency for equalisation must be confronted with the theoretical and empirical literature. This research defends the relevance of competition and capital mobility to explain the gravity of profitability, but the defence must distinguish a conditioned trend from a universal equality. Flaschel and Semmler, Glick and Ehrbar, Fröhlich and Kuroki are examined; among these works are theoretical modelling, simulations and data analysis, and their results do not have the same probative status.
The first research to be analysed is (Flaschel & Semmler, 1985). This research aims to be a reproduction of the process of capitalist competition from the dynamic perspective of the theory of classical economists, as its authors point out in the first paragraph of their research. This research also raises on page 2 the assumptions of the von Neumann model with regard to workers’ consumption and savings. Neumann’s model assumes, on the one hand, that Any product can be added. It is not necessary to distinguish between products and primary factors; even final demand is absorbed in demand for inputs for the next production period and labour is assimilated to a product generated by an input: the vital energy consumed at work (Mora Osejo, 1992, p. 219). In the same vein, Neumann assumes that commodity consumption takes place only within the production process, in the words of Neumann, quoted by (Mora Osejo, 1992, p. 219): The essential phenomenon we want to capture is this: commodities are produced by each other and we must determine: 1) what processes are used; 2) what is the relative rate of increase of all goods produced; 3) what prices will be generated; 4) what will be the interest rate; and, to completely isolate this phenomenon, we must include the additional assumption that commodity consumption takes place only through production processes, including consumption needs for the lives of employees and workers.
Von Neumann’s formulation can be put into dialogue with traditions of reproduction and surplus production, including Marxian. However, the exclusive belonging of a model does not follow from a later work by Sraffa or a post-Keynesian formulation appearing later. The chronology of publications does not prove an intention to hide influences nor does it allow to attribute to an author an attribution that it does not document. The historical question of the relationship between models is preserved, drawing inferences about personal motives.
Mora Osejo comments on the model and alludes to conceptual flexibility and its formal contributions. That judgment requires an assessment of the specific cases. On the savings side, Gloria-Palermo (2010, p. 154) points to the assumption that workers do not save. This restriction limits the reproduction represented by the model; it does not amount to a universal description of household behaviour.
In the light of the above, the research (Flaschel & Semmler, 1985) includes both working capital and fixed capital in its model and, if this was not sufficient for C&C, makes its estimates using simultaneous equations. After making a theoretical justification of how the proposed mathematical model is true to the classical theory (fundamentally retracing Adam Smith and his natural prices – which in Marx are the prices of production) comprised from page 3 to page 10, the authors perform several simulations, one of which is of interest in this particular research. The dynamic models of simultaneous equations are called cross-dual process models (cross-dual process, in English) and the computational-statistical simulation, which emulates such models, is configured by establishing that capital migrates from one productive branch to another in search of maximum possible profitability. The results of the simulation show that the dynamic process thus proposed will generate stable fluctuations around the stationary state (which is the neoclassical terminology – popularized by Robert Solow in A Contribution to the Theory of Economic Growth- to refer to the point of economic equilibrium) for input–output models that produce a single good and consider the existence of fixed capital.
The stability results of Flaschel and Semmler are propositions conditioned on the structure and adjustment rules of their model. The study relates changes in returns and capital mobility, and contrasts local and global stability properties. It is not appropriate to turn the result into an affirmation about every dynamic process imaginable. Gödel’s theorems, which refer to formal systems with certain properties, do not replace the examination of these economic assumptions nor do they prove that mathematics in general is not self-evident. The simulation reproduced below consists of 200 iterations; these steps are units of calculation and only represent economic periods if the model defines them as such.
Plate 07 · original document, p. 27
Simulated competition, prices and profitability. The original study material is retained; the surrounding discussion explains its interpretation and limitations.
Source: es (Flaschel & Semmler, 1985, p. 22).
Plate 08 · original document, p. 27
Additional simulation conditions and results. The original study material is retained; the surrounding discussion explains its interpretation and limitations.
Source: (Flaschel & Semmler, 1985, p. 23).
Plate 09 · original document, p. 28
Adjustment trajectories in Flaschel and Semmler’s model. The original study material is retained; the surrounding discussion explains its interpretation and limitations.
Source: (Flaschel & Semmler, 1985, p. 25).
Plate 10 · original document, p. 28
Comparison of results in the fixed-capital model. The original study material is retained; the surrounding discussion explains its interpretation and limitations.
Source: (Flaschel & Semmler, 1985, p. 25).
As is verified (Flaschel & Semmler, 1985, p. 27), the above results, belonging to a basic model consisting of two activities for producing product 1 and one activity for producing product 2, are generalized by Flaschel and Semmler, adding to the analysis the criterion of additional investment by firms in the different branches of production.
Plate 11 · original document, p. 29
Extensions of the adjustment results. The original study material is retained; the surrounding discussion explains its interpretation and limitations.
Source: (Flaschel & Semmler, 1985, p. 30).
Plate 12 · original document, p. 29
Additional results under the investment rule. The original study material is retained; the surrounding discussion explains its interpretation and limitations.
Source: (Flaschel & Semmler, 1985, p. 30).
The second research to be considered is (Glick & Ehrbar, 1988). This research, carried out in 1988, sought to use the then new data relating to long-term measurements of industrial profit rates to directly and exclusively compare the size and the theoretical significance of the differences observed in profit rates in industries in the United States and Europe.
The authors begin by laying out the theoretical foundations of their econometric analysis, in which they state that The idea that market prices should gravitate around ‘natural prices’ or ‘prices of production’ corresponding to industry’s equal profit rates has been a cornerstone of the classical analysis of competition. This view was shared by Adam Smith (Smith, 1965, chap. 7), David Ricardo (Ricardo, 1981, chap. 4), and also by Karl Marx (Marx, 1981, chap. 10). In the classical view of competition, we find a description of market prices that ‘gravitate’ around long-term natural prices or prices of production. These ‘gravity centres’ are prices corresponding to a state of uniform rates of profit between industries and are the result of the competitive process. In competition, according to the classical analysis, a double mechanism is introduced which should result in an equal rate of profit in the long term. (Glick & Ehrbar, 1988, p. 179).
With regard to the exodus of capital from one productive branch to another, the authors point out that On the one hand, companies that maximize profits move investment between industries seeking higher profit rate opportunities and thus expand supply in those industries with higher than average returns. On the other hand, this increased supply will face the penalty of a negative demand curve and prices and profits will be forced to fall. The same process can also work in reverse, as capital will come out of a sector with a low rate of profit, a move that reduces supply and therefore increases prices and profits. The perfect adjustment is strictly described for the long term, and while this imbalance process is underway, the industry’s profit rates are expected to be uneven. Furthermore, the existence of constant disturbances in the economy will transform this process into one of gravity. Instead of a uniform rate of profit actually expected in the economy, the expectation is that when competition prevails, industry rates of profit tend to fluctuate around equal average levels. Recently there has been a resurgence of interest in the study of classical competition analysis. There have been several attempts to formalize this process and analyze its mathematical properties (Dumenil / Levy, 1983, 1984; Boggio, 1984; Flaschel / Semmler, 1984; see also Steedman, 1984). Unfortunately, this renewed theoretical interest contrasts with a poverty of empirical analysis on this subject. The study of profitability differentials has rarely been directly considered. On the contrary, the analysis of competition has been dominated by consideration of the impact of market structures on profit rates. This was the result of the conviction of neoclassical and Marxist economists that price-fixing procedures in the American economy should be described as monopolistic. (Glick & Ehrbar, 1988, p. 179-180).
The authors consider that, in order to scientifically analyse the average rate of profit, its study must be carried out in isolation from the effect that the market structure has on it, in order to understand its natural behaviour, its behaviour in general, beyond the distortions of its dynamics that may arise as a result of a particular market structure. In this respect, they point out, together with others who are keen to point out the shortcomings which they believe exist in previous studies, that The basic assumption permeating all existing empirical literature on this subject is that profitability differentials exist and that the only relevant question is to provide an explanation of these differences. In our opinion, the evidence for this claim is not convincing for several reasons. First, the strongest criticism that can be made of attempts to measure profitability differentials between industries in the US is that the The United States It is only considered a short period of time (often only one year and a maximum of ten years). This was partly due to the failure to obtain consistent data at industry level over a long period, and partly was an effect of the neoclassical framework used. In the classical perspective, only the long-term estimate is relevant. Secondly, the question of how the rate of profit should be defined was not discussed. For example, the profit margin (i.e. profit on sales) is often mistakenly used to represent the profit rate. Finally, the levels of aggregation of enterprises and sectors are confused. The fact that individual firms have persistent differences in the rate of profit above the average is a different matter as to whether a particular market structure raises the average rate of profit for an entire industry. (Glick & Ehrbar, 1988, p. 181).
With regard to the drawbacks raised by the authors referred to on page 181 of the research, they propose the solution to them in the following terms: Strict adherence to the classical perspective provides guidelines for solving these problems: 1. Earnings rates are not equal, but gravitate around equal centers. Therefore, only a long-term calculation can produce the expected result. Data from the Department of Commerce have been available since 1983 to make this analysis possible. 2. From the classical point of view, the appropriate definition of the rate of profit must be that which guides capital movements between industries. In practice, two problems arise. First, the study of real investment behaviour is dominated by non-classical inspiration and is not well developed from a classical perspective. Secondly, there is no data available to accurately distinguish between the different profitability ratios. However, it is clear that the range of appropriate profitability measures should be restricted to ratios that consider a narrow definition of profit (e.g. net tax) and a broad definition of capital (e.g. gross inventory). It is clear that the profit margin is an inadequate investment guideline from a classical perspective. 3. Although less explicit in Smith and Ricardo, in Marx’s discussion of the equalisation of the rate of profit, prices of production correspond to the equalized rates of profit of industry. Within each industry, a hierarchy of performance rates reflects the existence of different technologies and organizations of enterprises (Marx, 1981, p. 281). Research should therefore be carried out on the equalisation of the rate of profit at industry level. (Glick & Ehrbar, 1988, p. 181-182).
As can be seen, the theoretical justification for econometric analysis is well founded, so it is time to proceed to analyse the quality of the data in terms of its processing and its sources.
Glick and Ehrbar’s research is very detailed in explaining the data they use because they were novel data at the time (no studies of the average rate of profit had been carried out for more than 10 years) and the authors were inexorably confronted with the need to demonstrate categorically that their data processing, definitions and measurement methodology in general did not present the shortcomings they pointed out of the methodologies with which the average rate of profit behaviour had previously been studied.
As noted in (Glick & Ehrbar, 1988, p. 182-183), the study used two primary data sources, the Eurostat database for Europe and the national income and product accounts corresponding to capital stock measurements for the United States.
Eurostat’s database is an effort by the OECD group of economists to develop a consistent set of definitions on industry for seven European countries (making it possible to analyse at industrial level) on a scale of 13 manufacturing and mining industries (with the characteristic of country-to-country comparability). This information allowed me to initiate a wave of studies on the spread of the rate of profit in the most industrially advanced capitalist countries. Eurostat provided the authors with data on the gross capital stock by industry (measured by its replacement cost in this case) for four countries (Federal Republic of Germany, France, Italy and the United Kingdom), so the study of European economies carried out by the author in his research is limited to these four countries. Finally, on the basis of Eurostat data, the authors point out that Data on the capital stock are provided for the period 1960-1981. Eurostat provides a selection of income data from the respective national income and product accounts of the different countries. Profit can be constructed from gross value added at market prices (available for the years 1960-81) and wages and salaries (available only from 1970 to 1981). As a result of these data constraints, we are restricted to a broad definition of profit that is constructed before taxes and depreciation, and includes net interest, as well as the period 1970 to 1981. (Glick & Ehrbar, 1988, p. 183).
For the US national accounts, the authors note that U.S. industry data is available for a wider range of industries and a longer period of time. The data also allow for a wider spectrum of definitions of the rate of profit. Our social capital data come from the most recent revision of the 4-digit SIC classification, 1948-79 (by Ken Rogers of BEA, 1984). This capital is constructed using the perpetual inventory method (…) (Glick & Ehrbar, 1988, p. 183). 8
Thus, they continue to expose that inventInventory data are available from the same source, but only beginning in 1958. We have supplemented the inventory series with the Annual Manufacturing Survey for the years 1949-57. Therefore, these early years do not fit the changes in the definition of the industry, as is the case with the series after 1958, but we find small changes in its pattern before and after 1958. Income data for profit-building comes from the US National Income and Product Accounts. (Glick & Ehrbar, 1988, p. 183). 9
In the same vein, they note that The accounts are broken down in the SIC 2-digit industry classification for the whole economy in GNP and components (14) by industry, 1948-79. (Glick & Ehrbar, 1988, p. 183), which in turn explains that Calculations with unweighted averages yield almost the same results. (Glick & Ehrbar, 1988, p. 199).
Our study of profitability differentials will be limited to a group of rather aggregated industries. In fact, it would be more desirable to use a more economically appropriate definition of ‘industry’. But this would require further breakdown. Unfortunately, the BEA only has a 2-digit industry definition for a large number of years, and Eurostat is also restricted to 13 industries. (Glick & Ehrbar, 1988, p. 183).
Thus, it is possible to move on to study the econometric model in its general form. According to (Glick & Ehrbar, 1988, p. 184), “Our study of industry profit rate differentials is based on an econometric model of profit rate adjustment. We do not explicitly model the mechanism of this adjustment, but we assume that profit rates are adjusted according to the three guidelines mentioned above. The rate of profit, rtj, in industry j in year t, is the sum of three components (…), which are those belonging to the expression:
rtj = αt + βj + utj
In this breakdown of Glick and Ehrbar, αt is the common component of the year; βj is the industry’s persistent differential from that component; and utj represents the unexplained adjustment. The time subindex of the common component shall be t. If the sectoral differentials are different from zero, the process may gravitate around persistently different levels.
The authors allow for heterogeneity of variance between industries and model temporal dependence through a first-order self-regressive process. Homocedasticity means constant conditional variance with respect to the domain concerned; it is not defined exclusively as temporary invariance. In this application, the sectoral dimension and the temporal dimension must be clearly distinguished. Following Wooldridge’s econometric explanation, a general AR(1) can include intercept; the error parameterization considered here has a mean of zero:
utj = ρjut−1,j + εtj, |ρj|<1
E(εtj)=0, Var(εtj)=τj²
The word innovation here refers to the new statistical shock of the process; not an observed technological innovation. Nor does a standardization used to write or calculate a covariance matrix prove that the economic variance of technological innovation is one. The scales, the Gaussian hypotheses of the estimate and the differentiation between disruption and statistical innovation should be preserved.
Estimation is carried out by maximum likelihood. To identify the levels of the components, Glick and Ehrbar normalise the mean of the sectoral differentials, weighted by capital stock, to zero: adding a constant to every annual effect and subtracting it from the sectoral effects would not change the fit. The differentials are then interpreted as distances from the weighted average profitability.
Their numerical procedure searches for the vector of common effects and optimises the persistence, differential and variance parameters for each industry. The article describes calculating persistence through a polynomial root, obtaining other parameters conditionally by generalised least squares, and a procedure combining elements of the Davidon–Fletcher–Powell and Gauss–Newton algorithms. This computational detail is retained because it helps distinguish the model specification from the method used to estimate it. The account continues with the initialisation and updating of the matrix employed by the algorithm.
For initialisation, the authors describe using an approximation to the inverse Hessian instead of starting with the identity matrix and subsequently updating it. They therefore characterise their procedure as a cross between DFP and Gauss–Newton (Glick and Ehrbar, 1988, p. 186). The industries, results and comparisons presented in the study are retained below.
Plate 13 · original document, p. 38
Industries and definitions in Glick and Ehrbar’s study. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Glick & Ehrbar, 1988, pp. 186-187).
Thus, defining the rate of profit as In the previous section, the rate of profit was a ratio of gross value added minus employees’ compensation on gross capital stock to replacement cost. In this section we will use a new rate of profit, where indirect taxes, net interest and an estimate of equivalent non-corporate salary are deducted from profit, while inventories are added to the capital stock. Profits and capital gains remain gross depreciation, and capital gains continue to be accounted for on the basis of their replacement cost. Although the industry-level data available to the United States allows for a greater variety of options than the European data, we are still subject to important limitations. The main limitation of the new definition of the rate of profit, in our opinion, is that profit is gross from corporate taxes and depreciation. (Glick & Ehrbar, 1988, p. 192).
Thus, the results for the United States case of the period
Plate 14 · original document, p. 39
United States industrial profitability results. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Glick & Ehrbar, 1988, p. 193).
Plate 15 · original document, p. 40
Comparison of profitability differentials. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Glick & Ehrbar, 1988, p. 196).
Finally, the authors conclude that Although the notion of an equal rate of profit has been a key concept in classical analysis, modern empirical work on this topic has been inadequate. The work mainly carried out by economists from industrial organisations, who were concerned with assessing the impact of market structure, has not directly investigated the question of whether profitability differentials actually exist. In addition, the methodology in this literature was based on neoclassical theory. Most studies have only considered profit rates for a very short period of time; inappropriate definitions of profitability are often used; and industry and enterprise levels are confused. This work has attempted to overcome these shortcomings and take advantage of the publication of new data of higher scope (comparability between different countries and longer time series). We have reported three main results:
1. In Section 2, we use a first definition of the rate of profit which we consider to be less than adequate, but which allows the comparison of four European countries and the United States (for the years 1970-79). Statistically significant differences in the rate of profit were found in all the countries considered. The size of the total dispersion is comparable with the exception of Italy. Furthermore, the profitability pattern produced in each country appeared similar in many respects. These observations suggest that these differences are the result of imperfect measurements or that they represent real values Economic processes. By imperfect measurement we mean inconsistent accounting procedures between industries. By real economic processes we mean such things as the structure of the market, the impact of international competition or other competitive mechanisms that can influence the formation of prices of production.
2. In the first part of section 3, we find strong support for the hypothesis that the definition of the rate of profit has an impact on the magnitude of the rate of profit differentials. By replacing a more appropriate definition of the rate of profit (for the United States), we got much smaller differentials.
3. In the second part of section 3, we tested the possibility of obtaining a higher result over a longer period of time. The maximum number of years for which data are available is 31 years. The profit rates measured during this period were somewhat less scattered, but the confidence intervals were widened, suggesting that industry-specific components of the profit rate may not be constant over such a long period. With an adequate rate of profit and a substantial number of years, it can be argued that market prices are not far from gravitating around prices of production. Although this result has only been proven for the United States, our analysis in the first section suggests that, with higher data, the same findings could be extended to other advanced capitalist countries. With regard to the remaining differentials, we are again faced with the dilemma described above. Are these differences the result of additional measurement biases or the expression of real economic processes? Further research should not exclude any of these possibilities. (Glick & Ehrbar, 1988, p. 197-198).
The third research to be studied is that of (Fröhlich, 2013). In it, Fröhlich begins by stating that During recent years, several empirical studies have found that deviations from labour values to market prices are quite small. However, most of these articles do not provide a detailed reason for this result. This article presents two theoretical justifications for the labour value theory together with some data on labour value, prices of production and market prices, based on German input–output tables from 2000 and 2004. In addition, the statistical characteristics of the profit are analysed by the rates. Both theoretical arguments are very much in line with empirical observations, because there is only a slight tendency to transformation and, at the same time, profit rates and capital intensity are negatively correlated. In addition, during the period under observation, the German economy appears to be in a statistical equilibrium. (Fröhlich, 2013, p. 1107). It is therefore necessary to proceed to analyse not the executive summary of the investigation but its actual content.
The initial theoretical specification requires a distinction between interpretative frameworks: In unconventional economic theory, there are generally two ways of explaining market prices. First, there is the labour value theory, which states that prices are driven by vertically integrated labour time (labour values). This approach, originally used by Karl Marx in Capital I, evoked the famous transformation problem, because the balance of the rate of profit is only possible in the case of a uniform capital intensity or zero profits. (Fröhlich, 2013, p. 1107). Fröhlich adopts a framework for the formalisation of the transformation problem which must be distinguished from the New Interpretation and from temporary readings. Discussion of whether that framework adequately reconstructed Marx requires comparing his assumptions. Interpretative disagreement does not automatically invalidate your measurements or authorize you to attribute a school to it without examining its definitions.
On the other hand, there is a growing body of empirical studies stating that deviations from values to prices are quite small (Shaikh, 1984; Petrović, 1987; Ochoa, 1989; Cockshott and Cottrell, 1997, 1998, 2003; Tsoulfidis and Maniatis, 2002; Zachariah, 2006; Tsoulfidis and Mariolis, 2007; Tsoulfidis, 2008). These authors found that correlation coefficients and determination coefficients R2 were considerably higher than 0.9. Labour values could therefore be as good at explaining market prices as neo-Ricardian prices of production. Although these results are rarely associated with theoretical debates, they pose a serious challenge to traditional approaches to classical economics. As expected, there has been fundamental criticism of these results (see Kliman, 2002, 2005; Díaz and Osuna, 2005-06, 2007, 2009). The purpose of this article is to connect theoretical and empirical arguments. (Fröhlich, 2013, p. 1108).
Then from the middle of page 1108 to page 1112 of his research. Fröhlich develops what he calls the theoretical framework of the labour-value theory and, as it could not be surprising after his declaration of intentions in the executive summary of the research, such a framework presents a formalisation whose relationship to the Marxian and Sarafian traditions must be specified. The substantive question is what categories and restrictions are retained, not the automatic attribution of validity or invalidity by school affiliation. Then, resuming the 1983 research of Farjoun and Machover, he introduces the concept of probabilistic political economy. In this regard, he comments very clearly that:
In the probabilistic approach developed by Farjoun and Machover (1983), all magnitudes, such as prices, labour values, earnings rates, etc., are random variables. Instead of analyzing a deterministic system with mechanical equilibrium properties, as traditional Marxist or neo-Ricardian theorists do, they examine the elements of an economic system in a similar way to how ideal gas molecules enclosed in a container described by statistical mechanics do (Farjoun and Machover, 1983, p. 39-56). In their view, the problem of transformation arises from the use of an inadequate concept of equilibrium, namely the adoption of a uniform rate of profit (Farjoun and Machover, 1983, p. 28–38). Instead, they assume that profit rates are described by a gamma distribution and replace the assumption of equalizing profit rates with the more sophisticated principle that for a given country in a state of equilibrium, the probability density function (PDF) of profit rates is virtually independent of time (Farjoun and Machover, 1983, p. 64-6). This concept is even compatible with large differences between sectoral earnings rates, whereas traditional approaches must assume that the distribution of empirical earnings rates must be quite narrow, which is almost not the case. (Fröhlich, 2013, p. 1112).
The profitability distributions discussed by Sánchez and Ferràndez, Zachariah and Farjoun and Machover should be compared by specifying unit of observation and horizon. A cross-cutting of enterprises or sectors is not the same distribution as an aggregate time series. Neither a sectoral gamma distribution is refuted for having a long series, nor does the CLT show that that distribution should converge to a normal one. Marxian long-term relevance requires research into competitive dynamics, trends and mechanisms; it does not impose a single probabilistic family.
Arghiri Emmanuel, in the first passages of his great work (already cited here), poses a sad reality that occurs in Political Economy. Contrary to what this research has shown, Emmanuel’s criticism considers that it applies to all economists since Quesnay, undermining the achievements of the physiocrats, classics and Marx; he obviously does not seem to be right in this. However, if we substitute A from for After and Quesnay for Marx, the sentence ends up being true (at least on a general level, there are clearly exceptions):
After Marx, economists work on models of noble logic and behave as if the world did not exist. They reason in the name of the spirit, but they cannot claim their wisdom to be like that of a veterinarian or an electronics technician. Since then, profans no longer need to keep silent and listen to them, as they usually do in front of those who know. Political economy is no longer a respectable science. (Emmanuel, 1972, p. 13).
The case described above by Emmanuel is precisely what happens to Farjoun and Machover (hereafter, F&M) in their research, since it is up to page 173 of the same that they present an empirical example with data taken from reality. Thus, F&M points out that &Consider the following diagram, which gives the empirical distribution of profit rates in the British non-oil manufacturing industries in 1972. In the same diagram, the theoretical distribution of Chapter 11 is plotted (Farjoun & Machover, 1983, p. 173). As mentioned above, Farjoun and Machover’s research is not long-term and therefore does not alone allow a long-term annual trajectory to be contrasted, because they operate, as explained in this research, through mechanisms and time scales that must be identified; long-term, sample size and mathematical limit are not interchangeable concepts. Thus, 172 pages of pure theory and a cross-sectional empirical analysis are present, i.e. it consists only of a temporary period.
The dependency structure of the probabilistic approach also requires discussion. Furthermore, they point out that beyond questions about the behaviour, the overall rate of profit of the economy lies in the problem of its behaviour for the sub-sectors, or even for individual firms. The distribution of the rate of profit is the result of a huge number of independent destinations of individual companies. The general distribution is far from arbitrary, as can be seen in empirical evidence and theoretical consideration. (Farjoun & Machover, 1983, p. 176). The dialectics of a process implies, inexorably, the interconnection of its parts, how could the competing firms be independent of each other’s behaviour? Rather than justifying this point by arguing with the same theoretical and applied research done in Statistics (as has been done here), assuming independence without justifying its scope is a constraint to be examined. Economic dependence does not exclude any probabilistic approximation; it requires precise units, conditionality and mechanisms
The examination of Farjoun and Machover must distinguish between economic interdependence and probabilistic assumptions. The specification AR(1), the sectoral variations and the definitions of Glick and Ehrbar are decisions of a particular study, not ideaideal assumptions universally binding for any experiment. The cross-sectional results of an investigation may be relevant for sectoral distributions without answering a question on time adjustment on their own. Fröhlich’s assessment must take into account that difference of objects.
Plate 16 · original document, p. 47
Empirical distribution of profitability in the probabilistic approach. The original study material is retained; the surrounding discussion explains its interpretation and limitations.
Source: (Farjoun & Machover, 1983, p. 181).
The position that Fröhlich reconstitutes generates an interpretative disagreement. The author expresses Thus, it is typically regarded as the state of the art and even prominent Marxist authors claimed that labour values play no role in the discussion of exchange and price (Roemer, 1981, p. 200). Different viewpoints that the transformation problem is likely insignificant did not prevail. (Fröhlich, 2013, p. 1107-1108).
Although the quote refutes itself, it is worth recalling, with some observations to be made later, that Many economists situated in the context of Marxism or in its vicinity consider that there are many important things in Marx’s thinking that must be preserved, such as his theory of exploitation, class struggle or historical materialism, his approach to social conflicts, his historical perspective, his interdisciplinary or socioeconomic sensibility, his concern for the institutional, etc. But they don’t realize that defending each of these elements, together or separately, is perfectly compatible with maintaining neoclassical economics as a theoretical skeleton. In fact, if all this is defended, but the labour theory of value is rejected, the essence of Marx’s economic thought is betrayed, and the resulting product must therefore be considered an impossible reading. 107). First, it’s not about loyalty or betrayal here, it’s about what’s intuitive, what’s logical and what’s credible in satisfactory proportions given the evidence available. Second, the distributive argument requires a distinction between Euler’s theorem, the marginal remuneration hypotheses and Marxian theory of exploitation. The following precision develops that distinction.
Criticism retains its substantive question: how labour theory of value, exploitation, classes and historical materialism relate. These relations require a conceptual reconstruction; social conflicts cannot be proved to disappear by adopting a particular production function. The discussion of what deserves to be called a Marxist must distinguish between interpretative fidelity and validity of the argument.
The temporal scope of Fröhlich’s study requires a correction: the German tables correspond to 2000 and 2004, two cross-sections separated by four years, not four annual observations. The author works with sectoral information and adapts its aggregation to make it compatible with capital data. This structure allows for the study of inter-sectoral relationships in the years considered; it is not sufficient to estimate an annual long-term adjustment path. This limitation should be assessed in relation to the question actually addressed in the study, without itself disqualifying the cross-sectional analysis. The editorial sheet of the article places its publication in 2013 and its acceptance in 2012; citations are regularised with the year of publication.
Finally, on Fröhlich’s research, it must be said that it has the merit of showing O&D that Now, the angle α between p̅ and i is a suitable way of measuring price-to-value deviations, because it is only based on add-on numbers. It does not measure the length but the direction of the relevant vectors, i.e. it is based on relative prices. Therefore, α must be independent of any physical unit and cannot be affected by the arbitrary choice of numériare (…) Both the independence of physical units and the independence of the choice of numériare are important. Although Diaz and Osuna (2009, p. 435) mention the second point, their criticism does not cover the first. Instead, they present an example in which physical units are defined in such a way that wλ = i (Díaz and Osuna, 2009, p. 437-8). As a result, their results depend on this arbitrary measurement method. But this procedure is not in line with the approach described above: in equation (37) all physical units are cancelled. Therefore, it is impossible for physical dimensions to have any influence on α when using the equation (37). (Fröhlich, 2013, p. 1116). As you can see, Fröhlich makes a brilliant refutation of O&D from the logic of simultaneous equations.
The fourth and last theoretical research to be studied is the research of (Kuroki, 1985). The theoretical model assumptions are detailed on page 35, which are presented below:
1) Production technology is linear (located within the framework of linear production models) and leads to constant returns at scale. 2) Workers spend all their wages on purchasing consumer goods and the real wage rate is constant. Underemployment may be the predominant state in the labour market (this is consistent with what Engels called the industrial reserve army).
3) Capitalists invest all their capital, including profits. 4) Capital moves constantly in search of the highest rates of profit. Thus, cross-sectoral capital movements will occur when profit rates are different from each other (which occurs in the process of adjustment or convergence to balance). 5) neither fixed capital nor joint production (where sectors are interrelated to generate output combining inputs from these sectors) is considered. 6) The production period is the same in all sectors and constant over time. 7) Price fluctuations depend on excess market demand (calculated as the level of demand that exceeds the level that would emptied the market – which would exhaust supply, which is the quantity of products produced on the market – during the analysis period), with a clear relationship between higher demand and higher price.
The assumptions listed limit the scope of the Kuroki model. Constant returns at scale refer to the response of the product when inputs vary proportionally with a given technology; technological change can displace that technology without logically invalidating this assumption. It is not right to infer rising returns on a scale from the mere fact that innovation exists.
The absence of labour savings and the assumption of full reinvestment of the surplus simplify the distribution closure. The relationship between savings and investment requires declaring whether an accounting identity, a closed economy, implemented variables or planned decisions are being discussed. Aggregate equality does not demonstrate that the mechanisms for saving and investment are identical or irrelevant. Marx’s relationship with later developments of Kalecki, Keynes and Joan Robinson deserves a specific historical reconstruction; it is not used here as a substitute proof of the modeled mechanism.
Capitalist consumption also affects the composition and realization of the product and may be relevant to accumulation. Failure to do so is a modelling decision whose effect must be examined. The same applies to excluding fixed capital and joint production, equalizing productive periods or keeping them unchanged. They are useful abstractions for certain questions and limiting for others. Its acceptability depends on the purpose and sensitivity of the results, not merely its frequency in the literature.
Despite all the theoretical shortcomings raised, including the use of simultaneous equations (which will be discussed later), the authors find results that significantly go in the same direction as the others previously presented: As a result of our analysis using closely the dynamic adjustment mechanism of competitive markets, it would be shown that the equalisation of sectoral profit rates through competition depends on industrial structure, i.e. the degree of technical interdependence between sectors. When the organic composition of the capital goods sector (…) is greater than that of the consumer goods sector (…) and the closer the economic structure is to the decomposable state), the higher the rates of profit which have a divergent trend and the gravity towards the uniform rate of profit does not work. We could then conclude that only the system in which both sectors use relatively much of each other’s production as their own input and therefore the degree of technical interdependence is large, has a stable uniform rate of profit (i.e. for example, the economy where much wheat is needed to make iron and at the same time much iron is used to produce wheat). (Kuroki, 1985, p. 48-49). In the methodological Annex IV.I, a complementary investigation of temporary associations between production and inputs is presented. That association does not automatically amount to the technical interdependence condition of the Kuroki model. In the same place, the methodology used (built to describe partnerships in the chosen period) is presented gradually (step by step), as well as providing all the statistical tables used, as well as the data manipulation code in the R Studio programme.
Finally, as is also widely known in the Marxist community (as pointed out by Sánchez and Ferràndez, henceforth S&F), the research (Díaz & Osuna, 2007) criticizes this type of research because of the indeterminacy between values and unit prices, however, the research (Sánchez & Ferràndez, Valores, prices of production and market prices based on data from the Spanish economy, 2010) focuses on values and aggregate prices, as well as all macroeconomic research of a Marxian nature. Nevertheless, it is appropriate to address the general nature of this criticism and to relate it to the above.
Differences between Marxist economists include historical, philosophical and methodological problems. In this controversy, Kliman answers Diaz and Osuna, who question the possibility of awarding an empirical victory through the negotiated normalisation procedures. The interest of the answer is to reconstruct what conditions would enable a relevant association to be identified and what evidence would be insufficient.
The conclusions of the D&O research are that Over the course of this work, we have shown the existence of an inescapable indeterminacy problem, which invalidates all the statistical results obtained from the cross-sectional regression models used in the Ricardian hypothesis test. The selection of units of measurement is the cause of the indeterminacy. We have shown in this article, analytically and empirically, that there is no escape from this trap. Indeed, all we can say is that any correlation measure, even with completely disaggregated data, is misleading, but not “strong” or “false”. Consequently, we cannot assess whether direct prices are (or are not) good predictors of prices of production. Therefore, any attempt to empirically prove the “Richard’s 93% value theory” by using correlation measures is pointless. (Díaz & Osuna, 2007, p. 398).
So, (Kliman, What is spurious correlation? A reply to Díaz and Osuna, 2008, p. 346) begins by stating in his reply to D&O that However, the finding of a spurious correlation has been questioned by Emilio Díaz and Rubén Osuna in a recent article in this journal. They correctly state that if other indicators of sector size are used instead of total costs [,]. . . the results obtained are different (20056, p. 356). Then they infer from this that the measures of correlation between prices and values are completely indeterminate, because they go from zero to one depending on the (arbitrary) choice of the size index of the industry (ibid., p. 356, emphasis on the original). Therefore, the dispute between the proponents of ‘the ‘strong correlation hypothesis’ and the ‘spurious correlation hypothesis’ cannot be resolved empirically’ (ibid., p. 356).”
Thus, his answer, which he understands as a challenge to his findings, is presented in general terms as follows: First, I maintain that the indictment of indeterminacy stems from Diaz and Osuna’s misunderstanding of the concept of spurious correlation. The results are equally ‘undetermined’ in many other cases of spurious correlation, including those that are obvious. Secondly, I show that cost deflation makes the value-price correlation disappear, not because of the effect of this procedure on the variability of the returned variables, but because the original correlation is spurious. (Kliman, What is spurious correlation? A reply to Díaz and Osuna, 2008, p. 346)
His reply begins by suggesting that Imagine that cross-sectional studies consistently support the claim that adherence to Christianity is the dominant determinant of income. For example, U.S. state-level data in 2000 indicate that the r2 between the number of Christians and income is 0.95, and that an additional Christian in the state increases state income by nearly $68,000. However, one researcher shows that the correlation is false; income and adherence to Christianity are highly correlated only because total income and total number of Christians are large in large states and small in small states. Once we control the differences in the size of the states, when we calculate the correlation between Christians as a proportion of the total population and income per capita, the correlation between Christianity and income fades away. (Kliman, What is spurious correlation? A reply to Díaz and Osuna, 2008, p. 346).
And he goes on to state that But the two critics point out that if we use other indices of the size of the states instead of the total population [,]. . . the results obtained are different. The area is not a less valid indicator of the size of the state than the population, but if we control the differences in the size of the state by dividing the number of Christians and incomes by the areas of the states, the original correlation between Christianity and income is almost completely preserved. Thus, critics conclude, “measures of the correlation between income and the number of Christians are completely indeterminate, because they range from zero to one depending on the (arbitrary) choice of the size index of the state”. The dispute between the proponents of “the ‘strong correlation hypothesis’ and the ‘spurious correlation hypothesis’ cannot be resolved empirically”. I doubt anyone, including Diaz and Osuna, will take seriously the critics’ accusation of indecision. However, what critics argue is strictly analogous to what they have argued. The statements of the critics in the last paragraph are the exact words of Diaz and Osuna, quoted in the second paragraph of this reply, except that I have replaced the terms sector and industry with adostate, cost with population, “Prices” with “income” and “values” with “number of Christians”. (Kliman, What is spurious correlation? A reply to Díaz and Osuna, 2014, p. 348).
The Kliman analogy illustrates the problem of a common factor of size; its economic relevance requires examining whether the comparative standardisation fulfils the same function in both examples. The reason for this is simple, it’s comparing, as my mother would say, mangos with avocados, which unlike peras with apples denotes a certain similarity in their physical form, but which a minimally careful eye can see that it doesn’t happen, if it’s just a faint similarity in physical form. Kliman explains the substitution of terms from his example by pointing out that The statements of the critics in the last paragraph are the exact words of Díaz and Osuna, quoted in the second paragraph of this answer, except that I have replaced the terms sector and industriaindustry by state, cost by population, “Prices” by “income” and “values” by “number of Christians” (…) (Kliman, What is spurious correlation? A reply to Díaz and Osuna, 2008, p. 348). This is, in short, because for him it is so common sense that empirical estimates inspired by the models of simultaneous equations cast spurious correlation as the inadequate example he used to prove his point and, furthermore, while properly pointing out to D&O that statistical laws should not be mixed with economic laws [This conclusion is a hasty generalization of a set of data. It is neither based nor supported by statistical theory. In fact, the correlation coefficient, which is defined as the covariance between two variables divided by the product of their standard deviations, has nothing to do with the relative sizes of the standard deviations. A reply to Díaz and Osuna, 2008, p. 349)] implicitly states on the other hand that the epistemological validity of the thesis that God provides is as robust as the vast set of statistical research in scientific research journals (with blind peer criteria and other issues) conducted by professionals who have devoted years of study to the subject. Don’t confuse these words, one issue is that statistical estimates (and conclusions derived from them) are wrong and another very different one is that their cognitive value is epistemologically equivalent to the God will provide that implies the hypothetical example of Kliman. This will be discussed further below.
The review allows three conclusions to be drawn with explicit scope. First, competition offers a mechanism of gravity for profitability, but its operation depends on capital mobility, production structure, barriers, risks and measurement conventions; the studies examined do not prove universal equality. Second, documented value-price associations deserve robust explanation and contrast, without the correlation alone identifying causation or closing the debate on sector size. Thirdly, the results favourable to the explanatory power of labour support the research of labour theory, but they alone do not constitute an exhaustive proof of causal exclusivity.
The decreasing trend of the rate of profit and cross-sectoral gravity are distinct hypotheses: one describes the evolution of an aggregate and another describes differences and adjustments between units. The previously cited work on unitary roots and non-stationaryness should be considered in your specific question. Here equalisation and gravitation have economic meaning; the lexicographic definition of equalisation in signal reproduction cannot replace that content.
II.II.1. Epistemological considerations on simultaneous equations
This is, undoubtedly, the most important section of this research, as it will establish the epistemological foundations required to reconcile theoretically all the aspects which, in the light of the evidence analysed, proved to be true, specifically the compatibility between the approach regarding the temporality of the transformation of values into prices of production supported by the TSSI and the empirical results found by the NI and Shaikh.
To do this, it is necessary to start by talking about the process of transforming values into prices of production from a historical perspective. As he rightly points out (Emmanuel, 1972, p. 427), the concept of value is rooted in predominantly pre-capitalist and largely non-commercial modes of production. However, going back the course of the evolution of capitalism is a fixed capital that is becoming less important and more and more personalized, the differences in organic compositions are reduced so that at the limit it is legitimate, at least from the methodological point of view and abstracting contingent impurities, to speak of a situation in which capital is practically nonexistent, or either equal or proportionate to wages. In order to move from this situation to that of developed capitalism, where the inequality of organic compositions becomes significant, it is necessary to accumulate quantitative changes which give rise to a given moment of qualitative change by which (…) value is transformed into prices of production.
Emmanuel, despite adopting a historical perspective, also supports the use of simultaneous equations. He argues: “As Bortkiewicz has shown, the transformation must be global or not take place at all. This transformation cannot be performed on output without simultaneously performing it on inputs, especially constant and variable capital” (Emmanuel, 1972, p. 428). His historical reflection nevertheless deserves consideration.
The relevant epistemological problem is not that algebraic simultaneity is incompatible, by definition, with history. It is what relationships the system represents, when inputs are valued, what changes between periods, and what assumptions allow for a transition from a description of reproduction to an explanation of transformation. A historical theory can use simultaneous relationships for a period; a dynamic theory also needs transitional laws and an explicit temporal interpretation.
The transition between commodity production and capitalism must be investigated historically. Its duration is not deduced from the convergence of an algorithm, nor does any long-term research require an invariable average. Simple commodity production can designate real relationships present in various forms; making it a universal and dominant stage requires specific historical evidence. The interest of the essay in the social genesis of the categories is preserved, specifying the scope of each statement.
Walras’ work is a central reference in the formalisation of general equilibrium by simultaneous equations. Its French title is Éléments d’économie politique pure; the translation by William Jaffé published in 1954 is entitled Elements of Pure Economics, or the Theory of Social Wealth. The terminological difference deserves historical attention, but it does not prove by itself an intention to conceal. The following facsimiles are preserved as bibliographic documentation of the editions cited.
Plate 17 · original document, p. 61
Bibliographic facsimile of the cited Walras edition. Documentary facsimile. The main discussion and its clarifications explain its role in the argument.
Source: Author’s scan, taken from (Walras, 1954, p. 29).
Plate 18 · original document, p. 62
Documentary page from the English edition of Walras. Documentary facsimile. The main discussion and its clarifications explain its role in the argument.
Source: Author’s scan, taken from (Walras, 1954, p. 3).
The content of the work must be read in full: its index includes theory of production, production equations, capital formation and circulation. The statement in the draft that there were no such sections is corrected. The relevant Marxian disagreement concerns how he conceptualizes production, ownership and surplus. Lesson 5 defines exchange value, commodity and market value. In this respect, it defines exchange value as follows: cambioExchange value is a property, which possesses certain things, not to be freely given or taken, but to be bought and sold, that is, to be received and transmitted in exchange for other things in certain quantitative proportions. 83). The definition poses a conceptual problem. First, it speaks of ‘certain things’, but what are the common properties inherent in them that allow them to be interrelated? The answer would seem to be ‘that they are not given, but are bought and sold, received and delivered, in defined quantitative proportions’, without that definition, considered in isolation, explaining the social relations of production and distribution. Finally, a profoundly elementary question arises: how are these quantitative ratios defined, based on what? This definition of exchange value aroused criticism even among other marginalists such as Wilfredo Pareto. In this respect, the first footnote of that chapter, made by the editor, reads that (…) cf. V. Pareto, Manuel d’économie politique, Paris, 1909, reprinted Paris, Giard, 1947, p. It is also important to note that this is not the case in the United States. 194-195. Pareto attacked Walras’ definition of “value in exchange” for lack of meaning, because it defines an unknown “value in exchange”, in terms of an equally unknown “property”. He finds it useless to refer to this metaphysical entity, when its concrete manifestation, the price, that is, the exchange relation, is all that is needed in the subsequent theoretical development. This is not entirely true, since, as Cournot pointed out, while value in exchange necessarily implies the idea of a relationship between two terms… a change achieved in the relationship is a relative effect, which can and should be explained by absolute variations in the terms of reason. 24. But apart from the question of the fairness of the criticism, the acrimony of Pareto’s comments directed against Walras’ definition seems utterly unnecessary. Walras was not responsible for introducing the term into the economic literature, but he conclusively demonstrated, as seen in §§ 101-102, that the only meaning that could have value in exchange was that of a term in a proportion with no autonomous existence outside that proportion. To this Pareto added nothing with his criticism. (Walras, 1954, p. 498).
The controversy between Pareto, Walras and the commentators raises the status of the categories and their formal relationships. Levins (1993) questions the claim to base science exclusively on logical analysis. The essay takes up that criticism to argue that a formal definition must be articulated with history, institutions and experience, without inferring that any abstraction lacks scientific content.
In political economy, mathematical relationships must explain what processes and practices they represent, how their variables are measured, and under what conditions they can be contrasted. A relational formulation can be useful without exhausting the explanation of production, exchange and appropriation. Marxian criticism questions these mediations; it does not need to state that all marginalist research is formally empty or that an equation is invalid because it shares structure with another scientific field.
Walras also defines the act of buying and selling as follows: The buyer of a thing is the seller of what he gives in return. The seller of a thing is the buyer of what he receives in return. That is to say, any exchange of one thing for another consists of a double purchase and a double sale. 83). Here the key is (and I thank my mother for noting it) that what Walras is implicitly raising (bringing to light that by the historical time of writing and publishing the work money as the general equivalent was already fully developed – and because the dimensional approach of marginalists and neomarginalists, i.e., the unit of measurement they employ of economic variables, is precisely money) that, for example, if the reader and the author of this research exchange, for example, a cell phone (pertaining to the reader) for USD$200 (pertaining to the author), the reader is the buyer of the money (and the seller of the cell phone) and the author is the
money seller and phone buyer. The reciprocity of a transaction can be described this way, but it is not sufficient to explain the origin of money nor does it require a specific theory of surplus on its own. The examination shall distinguish between the act of exchange and its formal representation of historical conditions of production and circulation.
Thus, the above exposition ends when Walras argues that:
Things that are valuable and interchangeable are also known as commodities. The market is a place where commodities are exchanged. So the phenomenon of exchange value manifests itself in the market, and we have to go to the market and study exchange value. Value in exchange, when left alone, arises spontaneously in the market as a result of competition. As buyers, merchants make their demands by overwhelming each other. As sellers, traders do it by bidding on each other. The union of buyers and sellers then results in giving goods certain values in exchange, sometimes rising, sometimes falling, sometimes stationary. The more perfectly competition works, the more rigorous the way to get value in exchange. (Walras, 1954, p. 83). Mr. Walras returned to Political Economy in the Stone Age, because even Aristotle or even more, even Jenofonte (more temporarily distant from the stagirite), had more sophisticated conceptions about use value, exchange value, and exchange in general; or, more accurately: he sent Political Economy to a state in which it never existed, because the economic world that Walras defines, inspired by Robinson Crusoe’s work (in fact, his example of exchange economy is the economics of Robinson and Friday, which, considering Friday was a slave, would seem like a whole statement of intentions).
The analytical priority that Walras gives to exchange differs from the Marxian reconstruction of production relations. This distinction does not authorize you to claim that your work lacks any theory of production or that the use of simultaneous equations logically eliminates any ethical question. The first lessons discuss definitions of political economy and distinguish between science, art and ethics; a rigorous critique must address that architecture.
Walras defines social wealth through material or intangible things that he considers scarce (1954, p. 65). The Marxian perspective asks how these things are produced, appropriated and distributed and what social relationships make their commodification possible. This substantive discrepancy is maintained: a definition of scarcity does not, by itself, reconstruct the history of property, wage labour or the appropriation of surplus. Nor can it be inferred from it, without mediation, that any conceivable commodity is ethically acceptable.
The simultaneous determination of prices and quantities expresses compatibility relationships within a model. It does not prove that the whole economy occurs in an instant or that there are no models with simultaneous relationships in each period and transitions between periods. Historical criticism should point out which mechanisms, institutions or dates omit a particular model, rather than deriving its invalidity from the algebraic instrument used.
In Marx, the general circuit of valuation can be deployed as money destined for means of production and labour-power, productive process and sale of product:
D — M {MP, FT} … P … M′ — D′
The circuit retains the notation of the Spanish source: D denotes money, M commodities, MP means of production, and FT labour-power.
Average output P between the purchase of productive elements and the sale of valued goods. D′ is increased money, not a commodity. The abbreviated formula DMD′ expresses the movement of capital but does not locate production simply between D and M. Criticism of the draft to the draft of those mediations is preserved, correcting its representation of the circuit.
The dimensional problems examined by Sánchez (2013) should be addressed by indicating units, stocks, flows and periods of each expression. Disagreement with general equilibrium does not prove that every system of simultaneous equations has these problems. Transformation research requires comparing economic specifications, not attributing a single school to a mathematical technique.
The problem of transformation was discussed at the beginning of the 20th century, among others, by Ladislaus von Bortkiewicz, a St. Petersburg-born mathematician and statistician active in Germany. It does not correspond to identifying him with the Austrian school. Its simultaneous reconstruction and the tradition that developed from it raise the question of how reprices of production, values, profitability and aggregate standardisation relate. The extent of the incompatibilities depends on the assumptions and the closure of the system.
1. The sum of values is equal to the sum of prices. 2. The sum of the gains is equal to the sum of the surpluses. 3. The rate of profit obtained is the same if it is calculated in terms of monetary units as in terms of physical units.
As can be seen, criticism is of a macroeconomic nature, but as prices of production are a microeconomic category, criticism also takes on a microeconomic dimension. Later on, it will return to the foundation of the historical approach and the critique of the use of simultaneous systems of equations (whose explanation of widespread use is also historical). Next, a critique will be presented, within the same framework of the simultaneous equations, which refutes Bortkiewicz’s critique.
As he points out (Valle Baeza, 1978), the mistakes made by Bortkiewicz are:
1. It does not consider fixed capital, on the contrary, it assumes that only circulating capital exists (i.e. that capital is consumed in the same period, which in accounting terms is understood as short-term).
2. No extended reproduction is employed, only simple reproduction (and simple reproduction has its roots in the genesis of capitalism, presumably requiring additional assumptions and additional information). 3. From the logic of simultaneous equation models, it is not significantly relevant that the rate of profit is quantitatively equivalent if calculated in monetary units or in physical units. For further information on this aspect, see (Valle Baeza, 1978, p. 200). 4. It also points out (Emmanuel, 1972, p. 431) that in the simultaneous transformation carried out by Bortkiewicz that the Russian economist of the Austrian school (…) makes the simultaneous transformation of inputs and products and its arrival scheme is a balance scheme that respects the difficulties and consistencies that are inherent in it, but the determinations by the quantities of work have disappeared from it. In his arrival scheme all trace of value as long as the chronometric quantity of work has disappeared.
Of course, these observations limit the scope of Bortkiewicz’s model and its conditions; they do not prove that Marx’s procedure through simultaneous equations is adequate. It is therefore necessary to begin by supporting the epistemological validity of the historical approach to the process of transforming values into prices of production. In this regard, Richard Levins points out that:
In the bookstores of my youth it was common to find titles that included the term “foundations of science”. These were not books about science at all, but about logic, inheritors of the Russell-Whitehead program to derive mathematics from logic and science from mathematics. There was still hope that objectivity could be achieved through analytical methods: clear definitions, unequivocal categories, precise measurements, and the discovery of algorithms that could replace the whim of human judgment. But the programme as a whole has been a failure, as it should have been. The foundations of science lie in history and sociology, not formal analysis. (Levins, 1993, p. 547).
Now, to understand how this applies to Political Economy, we must first understand what value is in terms of social metabolism. In such terms, when it is claimed that only labour creates value, it is claimed that labour is the only common quality of commodities which explains their quantitative equivalence in exchange. (Emmanuel, 1972, p. 439). It is precisely the foundation of this thesis that must have a historical and sociological character, i.e. the study of the transition from non-commercial production societies, to those of simple merchant production to those of capitalist production as a historical and sociological process, not fundamentally mathematical-formal, although the mathematical instrumental is a useful tool, but, after all, a tool, not a foundation. Ironically, it is also Emmanuel who clearly shows how a historical materialist should not reason: That means that the dream of economists of all times of finding an invariable pattern of the value of commodities, with which the relation of all commodities would give us an absolute value for each of them, is unrealizable, since there is more than one factor of quantification and the case (theoretical or historical, no matter) is abandoned in which the tools of the independent producer are negligible or inalienable and in which the only factor of value is labour. (Emmanuel, 1972, p. 432). For him, and for many other Marxists (as will be seen below), the transition from simple merchant productive societies to capitalist productive societies is a mere ‘case’, meaning a mere exception, that is, a process that is not continuous and is naturally disconnected, as a reality ‘with gaps’ that are naturally inherent in it. However, on the same page Emmanuel points out relatively correctly that Bortkiewicz’s failure is not mathematical but logical. Thus, to substantiate his approach (which has certain minimal historical elements) he makes two quotes, one by Engels and the other by Marx.
The exchange of commodities for their values or approximately for their values therefore presupposes a much lower phase than the change based on prices of production, which requires a fairly high level of capitalist development… it is therefore absolutely correct to regard commodity values, not only theoretically but historically, as the prius of prices of production. This refers to the regimes in which the means of production belong to the worker, a situation which occurs in both the ancient and modern world with respect to the labourer cultivating his own land and to the craftsman. Marx in Volume III, Chapter X of Capital, quoted by (Emmanuel, 1972, p. 426).
In other words, Marx’s law of value governs in general, insofar as economic laws always govern, the whole period of simple commodity production, that is, until it is modified by the appearance of the capitalist mode of production. Until then, prices gravitate according to the values determined by Marx’s law and oscillate around them. Marx’s law of value has, therefore, a general economic validity, covering the whole period from the beginning of change… to the 15th century CE. 426).
Unfortunately, because he does not understand the importance of the historical method, Emmanuel wonders whether Marx and Engels were right or were fundamentally wrong, that is not the problem. What is important is to ask why men like Marx and Engels felt obliged to use a historical argument that lends itself so much to discussion. (Emmanuel, 1972, p. 426-427). Emmanuel’s answer is given by the same authors to whom he asks the question in question.
“Although jurisprudence was the subject of my specialised studies, I regarded it only as a discipline subordinate to philosophy and history (…) The general result I reached, and which, once obtained, served as a guide to my studies, may be briefly formulated as follows: in the social production of their lives, people enter into definite relations that are necessary and independent of their will: relations of production corresponding to a definite stage of development of their material productive forces. Taken together, these relations of production constitute the economic structure of society, the real basis upon which a legal and political superstructure arises and to which definite forms of social consciousness correspond. The mode of production of material life conditions the social, political and intellectual process of life in general. It is not people’s consciousness that determines their being; rather, their social being determines their consciousness. At a certain stage of their development, society’s material productive forces come into conflict with the existing relations of production—or, in their legal expression, with the property relations within which they have hitherto developed. From forms of development of the productive forces, these relations turn into their fetters. An era of social revolution then begins. As the economic basis changes, the entire immense superstructure is transformed more or less rapidly. When examining such transformations, it is always necessary to distinguish between the material transformation of the economic conditions of production, which can be established with the precision of natural science, and the legal, political, religious, artistic or philosophical—in short, ideological—forms through which people become conscious of this conflict and struggle to resolve it. Just as an individual cannot be judged by what they think of themselves, neither can such an era of transformation be judged by its consciousness. On the contrary, that consciousness must be explained from the contradictions of material life, from the conflict between social productive forces and relations of production. A social formation never disappears before all the productive forces for which it has room have developed; and new, higher relations of production never appear before the material conditions of their existence have matured within the old society itself. Humanity therefore always poses only problems it can solve, since closer examination shows that a problem itself arises only when the material conditions for resolving it already exist or are at least coming into being. In broad outline, the Asiatic, ancient, feudal and modern bourgeois modes of production may be designated as progressive epochs in the economic formation of society. Bourgeois relations of production are the last antagonistic form of the social process of production: antagonistic not in the sense of individual antagonism, but of an antagonism arising from individuals’ social conditions of life. Yet the productive forces developing within bourgeois society also provide the material conditions for resolving this antagonism. With this social formation, therefore, the prehistory of human society comes to a close. Frederick Engels, with whom I maintained a constant written exchange of ideas from the publication of his brilliant outline of a critique of economic categories (…) in the Deutsch-Französische Jahrbücher, had reached the same result by a different route—see his book The Condition of the Working Class in England. When he also settled in Brussels in the spring of 1845, we agreed to formulate our conception in opposition to the ideological conception of German philosophy: in fact, to settle accounts with our previous philosophical consciousness. This intention was carried out in the form of a critique of post-Hegelian philosophy” (Marx, A Contribution to the Critique of Political Economy, 1989, pp. 7–9).
For his part, Engels, quoted in the same source, points out that:
The development of the materialist conception, even if it were only in the light of a single historical example, was a scientific work that would have required long years of quiet study, for it is clear that nothing is achieved here with simple phrases, that only the existence of abundant historical materials, critically examined and totally dominated, makes possible the solution of this problem. No; this book is oriented from the outset to a systematic synthesis of the whole of economic science, to a coherent development of the laws of bourgeois production and bourgeois change. And since economists are nothing more than interpreters and apologists of these laws, developing them is, at the same time, criticizing all economic literature. Since Hegel’s death, hardly any attempt had been made to develop a science in its own internal connection. The official Hegelian school had only learned from the teacher’s dialectics the manipulation of the simplest procedures, which he applied to right and sinister, and also with a not infrequently laughable clumsiness. For her, Hegel’s entire legacy was reduced to a simple pattern by which all possible topics could be constructed, and to an index of words and turns that no longer had more mission than to be placed at the right time, that is, when positive ideas and knowledge were lacking. As a professor from Bonn said, these Hegelians knew nothing, but they could write about anything. And so it was, indeed. Yet, in spite of their presumptuousness, these gentlemen were so conscious of their deficiency that they avoided, as far as possible, the great problems; the old pedantic science maintained its positions by the superiority of its positive knowledge. Only when Feuerbach came and gave way to the speculative concept did Hegelianism slowly languish until it disappeared, and it was believed that the reign of the old metaphysics, with its unchanging categories, had re-established itself in science. 157-158). 10
The historical basis of the aforementioned situation is laid down by Engels as follows:
“The matter had its logical explanation. The regime of the Hegelian successors, which had become lost in mere phrases, was naturally followed by a period in which the positive content of science once again outweighed its formal aspect. At the same time Germany, in keeping with the tremendous bourgeois progress achieved since 1848, threw itself with extraordinary energy into the natural sciences. As these sciences became fashionable, and since speculative tendencies had never acquired much importance in them, the old metaphysical way of thinking regained ground, descending to the extreme vulgarity of Wolff. Hegel had been forgotten, and a new materialism of natural science developed which differs theoretically hardly at all from that of the eighteenth century, and which in most cases has no greater advantage than a more abundant body of natural-scientific material, principally chemical and physiological. The narrow philistine mentality of pre-Kantian times reappears, reproduced in its utmost vulgarity, in Büchner and Vogt; and even Moleschott himself, who swears by Feuerbach, repeatedly loses his way, in a highly amusing fashion, among the simplest categories. Naturally, the stiff old nag of bourgeois common sense stops perplexed before the ditch separating substance from appearance and cause from effect; and anyone wishing to hunt with greyhounds across the rugged terrain of abstract thought should not do so on the back of a nag” (Marx, A Contribution to the Critique of Political Economy, 1989, p. 159).
It was then, as Engels relates, that the methodological and epistemological problems appeared:
There was, therefore, another problem here which had nothing to do with political economy. On the one hand was the Hegelian dialectic, under the completely abstract “speculative” form, which Hegel left behind; on the other hand, the ordinary method, which was once again fashionable, the method, in its metaphysical essence, wolffian, which was precisely used by bourgeois economists to write their thick and incoherent books. This latter method had been so theoretically destroyed by Kant, and above all by Hegel, that only inertia and the absence of another simple method could explain that it still persisted practically. On the other hand, the Hegelian method was utterly useless in its existing form. It was essentially an idealistic method, and here we had to develop a more materialistic view of the world than all previous ones. That method sprang from pure thinking, and here we had to start from the dumbest facts. A method which, according to Hegel’s own confession, part of nothingness to arrive at nothingness, through nothingness (…) was nevertheless improper in this way. And yet, it was the only element of the existing logical material that could at least be used. It had not been criticized, it had not been overcome; none of the adversaries of the great dialectic had been able to open the gap in the airy building; it had fallen into oblivion, because the Hegelian school did not know what to do with it. The first thing was to thoroughly criticize the Hegelian method. What put Hegel’s discursive mode above all other philosophers was the formidable historical sense that served as its basis. As abstract and idealistic as his form was, the development of his ideas always went in parallel with the development of universal history, which in reality should only have been the touchstone of that one. And even if the real relationship was reversed and turned upside down, the real content nevertheless permeated all philosophy, all the more so because Hegel distinguished himself from his disciples in that he did not boast, as they did, of ignorance, but was one of the most learned minds of all time. He was the first to try to highlight in history a process of development, an internal connection; and as pilgrim as many things of his philosophy of history may seem to us today, the greatness of the fundamental conception is still something admirable, if we compare with him his predecessors or those after him who have been allowed to make general considerations about history. In phenomenology, in aesthetics, in the history of philosophy, everywhere we see this great conception of history reflected, and everywhere we find the matter treated historically, in a certain connection with history, even if this connection appears reversed in an abstract way. This conception of history, which made time, was the direct theoretical premise of the new materialistic conception, and this already provided a starting point for the logical method as well. If, even from the standpoint of “pure thinking”, this forgotten dialectic had led to such results, and if, moreover, it had ended up playing with all the previous logic and metaphysics, there must undoubtedly have been more to it than sophisticated and pedantic subtlety. However, the criticism of this method, which he had undertaken and which still reverses all official philosophy, was no small thing. (Marx, Contribution to the Criticism of Political Economy, 1989, p. 159-160).
Thus, Engels relates that the historical process by which Marx solved such methodological problems (the method to be used, which was the dialectic in its materialistic form) and epistemological (how to approach reality with that method) was as follows:
Marx was and is the only one who could devote himself to the task of extracting from Hegelian logic the core of Hegel’s true discoveries in this field, and to re-establishing the dialectical method stripped of its idealistic garb, in the simple way that it appears as the only exact form of thought development. In our view, the method on which Marx’s critique of political economy rests is an outcome that hardly outweighs the importance of the fundamental materialist conception. Even if the method was discovered, the criticism of political economy could be engaged in two ways: the historical or the logical. As in history, as in its literary reflection, things also develop, in great ways, from the simplest to the most complex relationships, the historical development of publications on political economy provided a natural conduit for criticism, and, in general terms, economic categories appeared here in the same order as in their logical development. This form seems to have the advantage of greater clarity, since it follows real development, but in practice it would only be, at best, more popular. History often unfolds in leaps and bounds, and if it were to be followed throughout its course, it would be necessary not only to collect many materials of little importance, but also to break the logical illusion many times over. Moreover, the history of political economy could not be written without that of bourgeois society, which would make the task endless for lack of preparatory work. So the only method indicated was the logical one. But this is, in fact, nothing more than the historical method, stripped of its historical form alone and of disturbing contingencies. Where history begins, there must also begin the discursive process, and its further development will be nothing more than the reflection, in an abstract and theoretically consistent form, of the historical trajectory; a corrected, but corrected reflection according to the laws that provide the actual historical trajectory itself; and thus, each factor can be studied at the point of development of its full maturity, in its classical form. With this method, we start from the first and simplest relationship that exists historically, in fact, for us; that is, here, from the first economic relationship that we encounter. Let’s go ahead and analyze it. Already in the mere fact that it’s a relationship, it implies that it has two sides that relate to each other. Each of these two sides is studied separately, from which its relationship to each other and its interaction emerge. Contradictions will emerge, which will have to be resolved. However, since we are not following an abstract discursive process, which operates exclusively in our heads, but a real succession of events, which have actually occurred in some time or are still occurring, these contradictions will have developed in practice and in it they will probably have also found their solution. We will examine the nature of this solution and see that it is achieved by creating a new relationship, whose two opposing sides we will have to develop now, and so on. (Marx, Contribution to the Criticism of Political Economy, 1989, p. 160-161).
Thanks to the master’s degree in methodology and epistemology given by Marx and Engels, combined with the one proposed by (Levins, 1993, p. 547), there would seem to be no room for doubt such as that of (Emmanuel, 1972, p. 426-427).
Arghiri Emmanuel points out that the quote he makes of reference can be considered (and further points out that other Marxists have expressed the same) as an authority argument in favour of his approach (Emmanuel differs from Bortkiewicz essentially because he manages to solve the problem, using simultaneous equations, making the wage rate the independent variable of the economic system, so he seeks to express that there is a dual essence within market prices, i.e. values and prices of production – which implies that capital also creates value, although labour appears as the determining productive factor – albeit determining only at the level of the system of equations, not the logic-of-the production process), but this confusion of quote (and those of Engels) lies in the fact that the economic formula, in which Engels says that prices and prices move around the value of goods in terms of time, is the expression of economic value, but not in terms of economic value, is not in the name of social expression, as M. One of Emmanuel’s problems is the same as that of the Baeza Valley, 1978, p. 170) blames Shaikh: The mistake of Shaikh, is not to consider that money itself can be exchanged above or below its value, when it is a commodity. Hence his claim to Marx’s procedure focuses not on showing the functions of money as a unit of value and a price pattern, but on the formal aspects of the problem. Valle, which Valle points out as a source of divergences between values-prices and surplus-value-profit, but also serves to understand the role played by the existence of MELT and that it follows that the working values of which Marx and Engels speak in pre-capitalist production societies (simple commodity units – hence Engels places this set of societies from the beginning of change until the 15th century A.D. -), from the formation of the general commodity equivalent, evidently have their correspondence in the market via price expression (the immediately perceptible characteristic), which does not imply (changes in the price of money – which can be exchanged above or below prices) – but it is obvious that values are proportional to the value of money (which is the value of money) in a rigorous, time-to-hours analysis (which is the MELT) that they are proportional to the value of money (which is the value of money) – but that they are, of course, proportional to the value of money (which is the value of money). 11
From historical development to model specification
The pending link requires three levels of distinction. Historical research examines the emergence and expansion of commercial relations, wage labour, concentration of means of production and competition. In logic we reconstruct how those relationships make intelligible the categories of value, money, capital and profit. In mathematics, variables, dates, constraints and solution procedures are chosen. None of these levels can automatically replace the others.
The historical precedence of certain forms of exchange does not prove that the first iteration of a calculation reproduces a precapitalist society. Nor does the convergence in reprices of production show that the actual transition has followed those steps. In order to maintain a temporary correspondence, production periods, investment and pricing rules, dated stock valuation, initial conditions and identified shocks are needed.
MELT allows for dimensional consistency between work and money under a given definition. The change in the monetary coefficient, the change in labour requirements and the redistribution of surpluses are different phenomena. Clarifying them allows comparing the proposals of Emmanuel, Shaikh, Moseley and the TSSI without confusing coincidence of numerical results with equivalence between theories. The iterative procedure to be examined below should be judged under this level separation.
Thus, after analysing Emmanuel’s approaches, it is appropriate to do the same with Shaikh’s solution to Bortkiewicz’s criticism, which is of an iterative nature. A masterful description of this process exposes it (Moseley, 2015, p. 244-245), who explains that:
A key assumption of this iterative procedure is the invariance (or ‘irritation rule’) assumption that the total price (for the economy as a whole) remains constant in the second step and in all subsequent steps, and equal to the total direct price (the total value expressed in monetary terms) in the first step. As already mentioned, this assumption is different from the Bortkiewicz–Sweezy invariance postulate (total benefit = total surplus-value). Another key assumption is that in the second step and in all subsequent steps, the total benefit (for the economy as a whole) is determined differently from step 1A. Total profit (or total surplus-value) is not determined by the variable capital product and the surplus-value rate (as in 1A), but is determined by subtracting the revised constant and variable total capital from the total price. Since the total price remains constant and the constant capital and the variable capital vary from one iteration to another, the total profit will also vary inversely to the constant and variable capital. This procedure is repeated step by step until changes in prices of production between two steps become insignificant and prices of production converge to long-term equilibrium prices. At each step, the physical quantities of the means of production, livelihoods and luxury goods remain the same, and are in fact the fundamental data in this iterative procedure, from which production values and prices are derived, similar to Bortkiewicz’s – Sweet Interpretation. The end results of this iterative process are prices of production which are the same as those derived from the Bortkiewicz–Sweezy method (except for a proportionality factor due to the different invariance postulate). Shaikh calls Bortkiewicz–Sweezy’s prices the “correct prices”, which his method is able to replicate. The rate of profit is also the same for both methods (= 25%). The advantage of this iterative method, according to Shaikh, is that these results are obtained, not by an alternative to Marx’s transformation procedure, but by successive applications of Marx’s procedure. The usual method of Bortkiewicz–Sweezy “’ skips the Marx transformation procedure and goes directly from values to long-term prices of production and the associated rate of profit, so that Marx’s transformation plays no role in determining prices of production and the rate of production profit. Shaikh argues on the contrary that, according to his method, Marx’s transformation plays a crucial connection role between values and prices of production.
However, Moseley’s conclusions regarding Shaikh’s approach is that In short, I conclude that Shaikh’s interpretation is not a very promising defense of Marx’s theory, nor a promising path for the further development of Marx’s theory. Shaikh’s interpretation is essentially the same as Bortkiewicz–Sweezy’s interpretation, with a “dual system” and simple reproduction as a logical framework and the same system of simultaneous equations to be solved for prices of production and the rate of profit. This is not a different interpretation of the logical structure of Marx’s theory, but just a different method of solving this system of simultaneous equations. (Moseley, 2015, p. 251). Moseley’s objection requires a distinction between equivalence of solutions and economic interpretation and will be discussed in more detail in Section III.II.I. As will be seen, Shaikh’s method must be evaluated by its economic problem and its calculation conditions, which, due to the unpalatability existing between empirical procedures and theoretical interpretations of the results [on which particular in-depth research needs to be carried out, since this issue occurs not only in Political Economy but also in Quantum Mechanics, as is verified in various passages of (Nabi, Lessons from Marxian Gnoseology I, 2021) which summarizes elements of the foundational debate concerning the interpretation of wave-particle duality; nevertheless, this topic is addressed with some depth in Section III.II.] does not delegitimize its use in scientific research of economic phenomena and, in fact, in modern terms speaking, it encounters the analysis of the insettimo-productive genome, which is also used by Marxists in the sense of the vertical and semantic, and which is no longer used by Marxists in the formal sense of the term Pasin Pasin Pasin integra integra sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección integra sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección sección Marx sección In this sense, the approach of the TSSI offers a temporary interpretation that within economic dynamics (or even as an external influence on the economic system) there can be variations in both the level of prices of production (values deviated from themselves – due to the process described by Marx in Volume III of Capital – expressed in monetary terms) and direct prices (values-labour expressed in monetary terms) that affect the
the expected gains. Work requirements may change with technology; monetary expressions may also change by monetary variations under a given MELT definition. Market prices can fluctuate with supply and demand, while prices of production designate regulatory relationships under specified conditions. Both concepts should not be confused when describing current fluctuations.
Contracts, inventories, production times, financial conditions and price adjustments may prevent an undertaking from immediately transferring a change in costs to the selling price. This affects their results and can result in losses on assets or commodities. An accounting adjustment is a real institutional fact, but recognizing it does not prove that every accounting rule directly measures a magnitude of value. The historical, replacement or other valuation must be declared and business loss, redistribution and change in social value must be distinguished.
(Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 312) points out that limanThe definition of ‘value’ used by Kliman is different from that used by us and most collaborators in the empirical literature; it is a secondary question in the current context, but nevertheless of some importance. From our point of view, Marxist “values” correspond to what some have called “vertically integrated labour coefficients”. In principle, these can be obtained by multiplying the reverse Leontief matrix in the vector of sectoral direct labour coefficients. In practice, they are obtained by an iterative procedure: the first approximation to Vi is the direct contribution of labour in sector i (usually as represented by the wage mass), while the n−th approximation is the sum of direct labour in sector i and the value of non-labour inputs consumed in that sector, as assessed in step n−1. As you will see, they are different definitions, not different definitions, which is different and different. 12
Vertical integration can be calculated by matrix inversion or by a convergent sequence. This computational equivalence does not imply a rupture between a simultaneous theory and its application. If the technology represented by A remains fixed and the convergence conditions hold, both procedures solve the same problem:
λ = ℓ(I−A)−1 = ℓ + ℓA + ℓA² + ℓA³ + ⋯, ρ(A)<1
λ(0) = ℓ, λ(k+1) = ℓ + λ(k)A
The index k counts iterations of the algorithm. The sum decomposes direct and indirect labour requirements by rounds of production. To turn it into a sequence of historical periods, dates and economic rules not contained in the identity must be introduced. The initial calculation using direct labour has a mathematical function; it does not prove that it represents a society without fixed capital. This clarification preserves the investigation of possible compatibilities while avoiding the use of an algorithm’s structure as proof of economic history.
The compatibility between a historical explanation and the empirical results discussed remains a working hypothesis that requires explicit correspondence. The temporary and single-system representation is a theoretical option that must be compared with its alternatives; it is not deduced from the iterative nature of the calculations. The correlations obtained by Cockshott and Cottrell, Sánchez, Valle and others are not annulled by that precision, but their interpretation is also not automatically resolved.
Interestingly, not all TSSI supporters have a historical approach to the matter, at least not Andrew Kliman. On the basis of (Borisov & Zhamin, 2009, p. 199), PMS is production based on private ownership of the means of production and on the personal labour of producers, who produce articles for sale on the market. The most typical representatives of simple commodity production today are small farmers and craftsmen who do not exploit other people’s labour. This type of production constitutes an economic regime which has existed in various economic and social formations. By its nature, simple commodity production has a dual character. As much as it is based on private ownership over the means of production, the small peasant or craftsman is an owner and this brings him closer to the capitalist. On the other hand, simple commodity production has its roots in personal labour, the producer is a worker, and this brings him closer to the proletariat. Simple commodity production and capitalist production are distinguished by the source from which private property emanates over the means of production. In the first case, such ownership is based on the personal work of the producer; in the second case, it is based on the exploitation of wage-labourers. In the simple commodity economy, commodities are only the products of human labour, while in the capitalist economy, man’s own labour-power becomes commodities. Simple commodity production develops under the action of the law of value, a law which, by fluctuating commodity prices and competition in the market, spontaneously regulates the distribution of the means of production and labour-power among the different branches of production.
In simple commodity production, the action of this law leads to the spontaneous development of the productive forces. The magnitude of the value of the commodity is determined by the socially necessary labour. Producers who use a higher technique and who organize production better produce their goods at less expense than are socially necessary. On the contrary, the sale of goods is made at prices corresponding to socially necessary labour. This makes the producer gain a surplus of money and enriches himself, which leads other producers to introduce more advanced production methods. Simple commodity production is unstable. The spontaneous action of the law of value, price fluctuations and competition cause the simple merchant economy to break down and producers to differentiate: some (the minority) get rich; others (the majority) go bankrupt. The process of differentiating producers under certain historical conditions – namely, when private ownership of the means of production exists and labour becomes a commodity – originates with the bourgeoisie and the proletariat, gives birth to capitalism, forms the initial basis for the birth of the capitalist mode of production. But this does not mean the absolute disappearance of simple commodity production. The PMS is not merely a theoretical construction, it is the origin of capitalism itself in historical terms, in the only terms in which the origin of any society can be analyzed.
However, an acquaintance told the author of this investigation that on May 27, 2018 he had the opportunity to converse with Andrew Kliman through the social network known as Facebook. There, the person in question put to Kliman that the physicalistic conception of exchange value, specifically that of the heterodox economist Adolfo Rodríguez Herrera, who maintains that exchange value is only one category of the capitalist mode of production, was wrong, since it is a category (exchange value, as well as use value, it may be said) inherent in all production intended for exchange, since even surplus-value already exists in precapitalist production societies and that, therefore, not only wage labour is abstract labour. His answer was, textually, “I think you’re right about the exchange rate. Nevertheless, there was never a society in which “simple commercial production” was the dominant mode of production. Moreover, the physicist may have simply meant that value is not a concept applicable to all societies. I also think you’re right about the abstract work. But that is a more complex discussion. Obviously, although it is not impossible that the physicist wanted to say what Kliman proposes, what is important is what he said, since acting as trtranslator (or believing to act as ) in scientific debates would not seem most appropriate for the sake of scientific rigor itself. However, the usefulness of products, the exchange and the generalization of the commercial form have different historical ranges; their relationship must be investigated, not taken for granted.
II.II.2. Vertical integration: interpretation and calculation
Vertical integration accounts for the labour required directly and indirectly along production chains. Zachariah, Fröhlich, Cockshott, Cottrell and Valle Baeza discuss its use in research on value. Pasinetti expresses the calculation as the product of the vector of direct labour coefficients and the Leontief inverse. Using a row vector ℓ, a matrix of technical coefficients A and the identity I:
λ = ℓ(I−A)−1
Each column of the inverse represents direct and indirect production requirements needed for one unit of final demand, under the model’s conventions. Multiplying those requirements by compatible labour coefficients and summing gives the corresponding total labour. If labour coefficients are measured in hours per physical unit of output, integrated coefficients retain the relevant labour unit; they are not necessarily monetary costs.
The word “vertical” refers to reconstructing the production chain that culminates in the final good; it does not derive from columns being drawn vertically. Nor are the results universally restricted to (0,1): their magnitude depends on units, technology and whether coefficients are direct or total. A change of units alters the number without changing the economic relation represented. This is a vector–matrix product whose components are calculated through inner products, not a single inner product whose result is a vector.
The accounting and analytical usefulness of the procedure must be assessed together with its assumptions: sectoral homogeneity, given coefficients, and the treatment of imports, secondary production and fixed capital. The monetary construction of the tables additionally requires examining how observed prices relate to the estimation of labour requirements.
The draft’s arithmetic example uses the vector a₀=(3,4,5) and the matrix with rows (1,2,3), (4,5,6) and (7,8,9). Their correct product is (54,66,78), not (54,66,88). That matrix is singular: it illustrates multiplication but cannot be identified as an inverse matrix. To retain the arithmetic lesson and also provide an economically admissible example, consider:
ℓ = (1, 2), A = [[0.20, 0.10], [0.10, 0.20]]
(I−A)−1 = (1/0.63) [[0.80, 0.10], [0.10, 0.80]]
λ = (1.5873016, 2.6984127)
The spectral radius of A is 0.30. The series converges and total labour requirements exceed direct requirements because indirect rounds are included. The two components are expressed in labour units compatible with the two products. This second example is new and pedagogical; it is not an estimate for the United States economy.
Without attributing general invalidity to the use of simultaneous equations in the formalisation of Marxian theory, vertical integration, not at the mathematical level but at the logical-conceptual level, describes a real process of political economy systems. Moreover, the construction of the input–output matrix is essentially an accounting construction, under an accounting logic, the utility of which is not decided by classifying Leontief within a school. The proof of this is that the first chapter of his work InInput-Output Economics (which could be translated as Input–Output Economic Science or Input–Output Economic Analysis – the latter seeking more to preserve meaning than to achieve word-for-word accuracy in translation-), located from pages 3 to 18, only presents empirical data of an accounting nature and from them proposes its mathematical model, not vice versa.
Plate 19 · original document, p. 91
Leontief’s accounting relations of production and use — plate 1. Documentary facsimile. The main discussion and its clarifications explain its role in the argument.Scroll horizontally to read the table at a larger size.
Source: (Leontief, 1986, p. 6).
Plate 20 · original document, p. 92
Leontief’s accounting relations of production and use — plate 2. Documentary facsimile. The main discussion and its clarifications explain its role in the argument.Scroll horizontally to read the table at a larger size.
Source: (Leontief, 1986, p. 8).
Plate 21 · original document, p. 93
Leontief’s accounting relations of production and use — plate 3. Documentary facsimile. The main discussion and its clarifications explain its role in the argument.Scroll horizontally to read the table at a larger size.
Source: (Leontief, 1986, p. 7).
Plate 22 · original document, p. 94
Leontief’s accounting relations of production and use — plate 4. Documentary facsimile. The main discussion and its clarifications explain its role in the argument.Scroll horizontally to read the table at a larger size.
Source: (Leontief, 1986, p. 9).
It is even in the chapter where he makes theoretical-mathematical approaches and developments. In his own words, The input–output analysis is a method for systematically quantifying the interrelationships between the various sectors of a complex economic system. In practical terms, the economic system to which it applies can be as large as a nation or even the entire world economy, or as small as the economy of a metropolitan area or even a single company. (Leontief, 1986, p. 19). In fact, the first chapter of his work was drafted in 1981, while the second was drafted in 1985 (the following were drafted after the two dates mentioned) as shown below; furthermore, if the reader observes the previously submitted accounting matrices, they contain data from dates prior to 1981 (birth date of the first chapter of his work), specifically 1947. Linear models represent some interdependent relationships and can serve as approximations; they do not necessarily reconstruct all the dynamics of a complex system. The relevance of non-linearities and phenomena studied by Lorenz requires the formulation of another question and another specification.
Plate 23 · original document, p. 96
Editorial information for Leontief’s collected volume. Documentary facsimile. The main discussion and its clarifications explain its role in the argument.
Source: (Leontief, 1986).
The input-to-product tables collect statistical and accounting information; requirements models are derived from them by conventions and assumptions. The information observed should be distinguished from the estimated technical relations and from the economic theory with which they are interpreted. Its calculation can be either direct or iterative. Neither of these two possibilities alone proves the validity of a historical, temporal or simultaneous interpretation of value. The comparison between NI, TSSI, Moseley and other approaches should be made at the conceptual, dimensional, temporal and empirical levels.
II.III. Labour and the means of production: different functions
The third scenario examines the hypothesis that both labour and the means of production create new value. Discussing it requires distinguishing the production of use-values from the creation of value. Machines can greatly increase material productivity while continuing, in Marxian theory, to transfer previously produced value. The objection attributed to Steve Keen—why a machine’s loss of value should equal the value it contributes—raises precisely this distinction between physical capacity and the social determination of value.
The Marxian answer is not that only unpaid labour creates value. Living labour adds v+s; the wage represents the price of a capacity whose exercise can produce more than the equivalent needed to reproduce it. The portion replacing v is also created within the process. Constant capital transfers the value of the means consumed under the relevant social conditions of valuation; “constant” does not mean that their prices or value can never change.
The bodily reproduction of workers—food, rest, care and the maintenance of health—is an indispensable material condition. Education, conversation, networks and learning resources can develop capacities without every improvement being paid for directly by the employer who subsequently uses it. This does not make the reproduction of labour-power costless or external to society: households, institutions and other workers bear its costs. Its determination includes historical and social components as well as physiological needs.
Equipment repairs, maintenance and improvements also mobilise labour and resources. Distinguishing these outlays from labour’s capacity to create value avoids grounding the labour theory merely in an opposition between a regenerating organism and a machine requiring repair. The exclusivity of labour as the source of new value must be argued within a theory of the social relations of production, and its empirical implications subjected to testing. Biology supplies material conditions; it does not, by itself, provide a sufficient demonstration of that exclusivity.
What is the function that shapes the transfer of value, for example, of the means of production, i.e. constant capital? The answer to such a question requires a distinction between economic wear and tear, productive use, useful life and the accounting method of depreciation used for the capital in question and precisely for this reason depreciation does not necessarily have to be linear (i.e. constant). It should be recalled that Marx put forward a general theory concerning capitalist political economy, so in his mind it would not seem intuitive to expect that they would provide rigorously detailed answers to questions such as whether constant capital transferred its value in an arbitrary way or otherwise (the simplifier hypothesis used provisionally by Marx in his work is implicitly that value transfer is constant), since the object of study is essentially total social capital (what a well-trained neoclassical would call MacMacroeconomics).
Of course, there is research to back this up. Thus, relevanteA relevant analysis which is necessary to confirm the explanatory capacity of the VAT is to compare its values obtained against other possible alternative base values (steel value, maize value, among others).
As is known, some authors such as Hogdson (1982) or Roemer (1981) have advocated the possibility of other core values as a way to criticize TVT. In Table 5, following Cockshott and Cottrell (1997), we present the main results of the calculations for the 65 alternative base values, as many as the 65 sectors of the TIO-2000, and not only for some sectors as is commonly done in the literature: steel, energy (see Tsoulfidis and Maniatis 2002 and Cockshott and Cottrell 1997). (Sánchez & Ferràndez, Valores, prices of production and market prices from data on the Spanish economy, 2010, p. 98-99).
Following is the table referred to in the previous reference, on which some questions must be summarised. First, DAM is the word used to express the abbreviation of Absolute Average Deviation and is nothing more than the distance between each value and the average (it is a way of characterizing the variation in a dataset); as one StackExchange Cross Validated user (whose name appears on the site referred to as andyl) points out, Today, statistical values are predominantly calculated using computer programs (Excel, …), no longer with hand-held calculators. Therefore, I would say that calculating the ‘average deviation’ is no more cumbersome than calculating the ‘standard deviation’. Although the standard deviation may have “… mathematical properties that make it more useful in statistics”, it is, in fact, a distortion of the concept of variance of an average, since it gives additional weight to data points away from the average. It may take some time, but I, for my part, expect statisticians to reuse the “average deviation” more often when discussing the distribution between data points; it represents more accurately how we actually think about the distribution. (StackExchange, 2014). In the same line, it is pointed out in (Investopedia, 2020) that Generally, according to mathematicians, when a data set has a normal distribution, that is, there are not many atypical values, the standard deviation is the preferred indicator of variability. But when there are large atypical values, the standard deviation will record higher levels of dispersion or center deviation than the average absolute deviation. The operational differences between the DAM and the standard deviation are rooted in its calculation process. The standard deviation is calculated by three steps: 1) Find the simple arithmetic mean, or simple average, of the data points (summing them up and dividing the total by the number of observations), 2) Subtract the average for each observation and square the difference of each result, 3) Determine the average of those differences to the square and then the square root of the average. The DAM is calculated by three other steps: 1) The average of all observations is calculated; 2) The difference between the average and each observation is calculated; 3) The average of the absolute values of those differences is calculated. As can be seen, the differences are rooted in steps 2 and 3, since in step 2 of the DAM it does not rise to the square, while in step 3, also belonging to the DAM, the average is calculated as the absolute value of differences (in estimating the standard deviation not only a square root intervenes, but previously – as already noted – for the solution to exist and belong to the domain of the real numbers, such differences rise to the square – the standard deviation is, in fact, a specific application of the function distance in Euclidean spaces, in which the differences are raised to the square to guarantee the positiveness of the result, since there are no negative distances-).
Secondly, the weighted average absolute deviation DAMP is calculated by applying justified weightings to absolute differences between two comparable magnitudes:
DAMP = [Σi wi|yi−ŷi|] / Σiwi
The Spanish abbreviations DAM and DAMP in the source tables correspond to MAD and WMAD in English.
Here and ŷ are the magnitudes that are compared: for example, observed and estimated prices or explicitly standardised price vectors. ŷ does not necessarily designate an arithmetic mean. If the weights add up to one, the denominator is simplified. The formula can express a weighted absolute error; its conversion into a percentage requires declaring a normalisation. The resource of fast.ai cited in the document is preserved as a historical reference to the operational expression, not as a basis for any economic inference.
As they point out (Sánchez & Ferràndez, Valores, prices of production and market prices based on data on the Spanish economy, 2010, p. 91), In the debate on the measures used, Steedman and Tomkins (1998) question the use of distance measurements (DAM, DAMP), proposing measures supposedly independent of the units used to measure prices: an angular measure (θ), the coefficient of variation (CV) and a related distance measure (d). For its part, Petrovic (1987) by means of the RC%E1 states that this measure is numerically independent; in fact, Steedman and Tomkins (1998) find that its tangent index of θ, is related to the RC%E, and Diaz-Calleja and Osuna (2009), from their critical perspective with this type of empirical studies,2 show how the measures used: DAM, DAMP, Pearson’s uncentred R, among others, are related as distance measures (…) RC%E refers to the square root of the percentage of error (…) The authors are critical of the empirical work that relates values and prices: by the indetermination of the correlation between main values and unit prices, by incorrect specification in regression models and by aggregation problems. We will not address these criticisms in this paper. The current study relates sectoral and non-unitary values and prices, the possibility of a spurious correlation is assessed and discarded in the section Work values versus alternative base values. Thirdly, these other measures will be defined synthetically. 13
The angular measure of Steedman and Tomkins must be distinguished from an average of directions. For a positive vector of dimensionless reasons x and the unit vector 1, the angle θ measures deviation from proportionality. With mean μ and population standard deviation σ of its components:
cos θ = μ / √(μ²+σ²), tan θ = σ/μ
The identity μ=√(μ2+σ2) cos θ expresses the mean as a function of that angle; it does not define an independent angle mean. In order to apply these relations, the reasons for forming x, normalisation and convention of variance must be explained. The expressions of the cited articles shall not be transcribed by confusing diagonal arrays of values, their inverters and vectors.
The coefficient of variation is CV=σ/μ, or s/x̄ in a sample convention. Expressed as a percentage is multiplied by a hundred. This corrects the inversion of the coefficient shown in the draft. Their interpretation requires a scale of reason and caution when the average approaches zero.
CV = σ/μ · CV(%) = 100σ/μ
If the two compared vectors are normalized to length one, their Euclidean distance is related to the angle by:
d = 2 sin(θ/2) = √[2(1−cos θ)]
The radical encompasses the whole expression 2(1−cos θ). For 0<θ<π/2, d<θ<tan θ is met; in the case of proportionality the three measures are zero. The independence of units or numerals depends on how the reasons are constructed, not to call any comparison anguangular. The investigations of Sánchez and Ferràndez bring together several relevant measures, without exhausting all possible dispersion measures.
The non-centred determination coefficient, frequently used in unintercepted regressions, should be distinguished from the usual centred coefficient. The first compares the sum of residual squares with the sum of squares of the zero-dependent variable; the second uses deviations from its mean. Their magnitudes are not automatically comparable or generally equivalent to the square of a Pearson correlation. Excluding interception requires a theoretical and diagnostic justification; it is not justified only by wanting to measure a direct relationship.
Plate 24 · original document, p. 103
Original table: results and definitions in the Spanish study. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Ferràndez, Valores, precios de producción y precios de mercado a partir de los datos de la economía española, 2010, p. 89).
Plate 25 · original document, p. 104
Spanish economic magnitudes under different valuations. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Ferràndez, Valores, precios de producción y precios de mercado a partir de los datos de la economía española, 2010, p. 95).
Plate 26 · original document, p. 104
Measures of deviation and price comparisons in Spain. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Ferràndez, Valores, precios de producción y precios de mercado a partir de los datos de la economía española, 2010, p. 97).
Plate 27 · original document, p. 105
Alternative value bases and comparative fits: original tables. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Ferràndez, Valores, precios de producción y precios de mercado a partir de los datos de la economía española, 2010, p. 100).
Plate 28 · original document, p. 106
Further results for alternative value bases. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Ferràndez, Valores, precios de producción y precios de mercado a partir de los datos de la economía española, 2010, p. 101).
As can be seen in the study of the correlations within productive processes (which is precisely what an input–output analysis of an economy entails, since they evaluate only, and empirically, production at a specific time), it is intuitive, logical and theoretically valid (also, apparently, empirically valid) to study only a moment of time (although, of course, also a study with panel data seeking to generalize the results found by S&F and the other authors who have investigated this phenomenon).
S&F anticipated in its research that &The first objective is to provide empirical evidence about the explanatory capacity of values to determine market price movements by comparing these results with those offered by other alternative base values. (Sánchez & Ferràndez, Values, prices of production and market prices from data on the Spanish economy, 2010, p. 87). Thus, in relation to the grounds for dismissing a spurious correlation in particular, and as an essential conclusion to the econometric results in particular, S&F notes that:
Firstly, it is noted that no base value (1 to 65) has a distance measurement and determination coefficient greater than that computed by the work requirements [ao]. For the above, the greatest explanatory power of alternative base values seems to be statistically dismissed. For the United Kingdom, Cockshott and Cottrell (1997) even perform multiple auxiliary regressions, explaining market prices together with several alternative base values (several explanatory variables). The elasticities of the work are not only more significant (statistical t) and closer to the unit, but the signs and impact of the other base values are sometimes negative and the elasticities are not kept close to the unit (the log-log regressions have been performed 65 + 1 to verify this, in Table 5 only R2 is presented, which are again strictly comparable). In the Spanish case, at the same time, this increased explanatory capacity of values occurs regardless of the index used. As noted above, the TTT has been criticized as being arbitrary or ambiguous, as it depends on the statistical index used; however, the results obtained relativise at least this type of criticism. However, these results also raise doubts as to the existence of a spurious correlation with the size of the sector (see Table 1). Kliman 2002 and replicas in Cockshott and Cottrell 1997, 2005), as we observed very low adjustments for some alternative base values. If there is a predominant false or spurious correlation by the size of the sector, even if there is no real association between values and prices, alternative values should obtain a significantly high R2. However, this is not the case, since it is found that operating with the same methodology there are goodness … of adjustment of 2% for agriculture and 86% for construction, while vertically integrated labour requirements obtain a goodness of 97% against market prices. It must be stressed that the prices obtained have been multiplied by sectoral gross production, so that they are sectoral prices. The relationship between values and prices within VAT is therefore strengthened and we make a brief international comparison. (Sánchez & Ferràndez, Valores, prices of production and market prices based on data from the Spanish economy, 2010, p. 99-102)
Plate 29 · original document, p. 109
International comparison of value–price deviations. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Ferràndez, Valores, precios de producción y precios de mercado a partir de los datos de la economía española, 2010, p. 103).
Thus, they put forward in their final conclusions that:
From a traditional reading, the results of our research confirm the greater capacity of labour values to explain the whole price structure. Using MET, the values approach market prices by 11% and prices of production by 18.5%, while the latter approach market prices by 15.8%. At the global level of the Spanish economy, it seems then that values explain market price movements well. This better explanatory capacity of TFT is maintained: (a) regardless of the deviation or regression measures used, since the same general conclusions are reached using different indices in this work, which empirically refutes criticisms by some authors such as Steedman and Tomkins that raise problems with the measurement indices and (b) with the use of alternative base values, as the comparative study finds that none of them explain market prices better than the results produced by TFT (measured with R2, DAMP, d, among others). In this respect, criticisms of a possible spurious correlation between values and prices are also relativised. Moreover, the strong association between values and prices is not only observed under the input–output scheme, but the same conclusions are also drawn from a stochastic perspective. Under this scheme it has been found that in Spain the deviation between values and market prices does not exceed 17%. The research has also yielded results on the fundamental variables of the Spanish economy according to the ILO-2000. The rate of profit, with different prices, is between 16 and 13 percent, the rate of surplus-value between 67 and 77 percent, and the capital composition is greater than 4 and greater than 6 if we estimate it as a vertically integrated composition. Although very close to the measures g and p, the differences in prices should be highlighted. Thus, a variable as important as the rate of profit is higher in value than in price: 16% in market prices, 13.3% in prices of production and 17.2% in values. The Spanish economy has been booming at the beginning of the century, with real gross domestic product (GDP) growth above 4%, so it seems that the value measurement is more sensitive to this boom situation than other prices indicate. On the other hand, and from a stochastic perspective, the distribution of profits shows a clear synchronicity with the growth of the Spanish economy in certain sectors such as construction and services; furthermore, the estimation of profitability, including fixed capital, is somewhat anticipated by the theory: that the average level of this profitability is lower (16%) than if only the working capital (25%) is weighted, however, the first estimate shows less bimodality in its distribution (using the FDP). The above results seem to suggest that the labour value theory remains a valuable scheme both theoretically and empirically, capable of explaining the phenomena of capitalist economies. However, there are lines of research that need to be further developed in connection with this work: on the one hand, it is necessary to examine the evolution of values over time, not only with the aim of observing price deviations but also with the intention of verifying whether the theoretical prediction of a trend decline in labour costs is fulfilled; on the other hand, it is also necessary to clarify and clarify technical aspects relating to the measures used (deviation and correlation) which we have not addressed in this work. Last but not least, the task remains to continue to suggest interpretative lines that open up the debate and help to give the TNP greater theoretical consistency. (Sánchez & Ferràndez, Valores, prices of production and market prices based on data from the Spanish economy, 2010, p. 110-111).
These results have also been replicated for the case of China in (Sánchez & Montibeler, The theory of labour value and prices in China, 2015), as shown below.
Plate 30 · original document, p. 111
Empirical value–price results for China. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Montibeler, La teoría del valor trabajo y los precios en China, 2015, p. 337).
Plate 31 · original document, p. 112
Regressions and association measures for China. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Montibeler, La teoría del valor trabajo y los precios en China, 2015, p. 338).
Plate 32 · original document, p. 113
Additional price comparisons for China. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Montibeler, La teoría del valor trabajo y los precios en China, 2015, p. 338).
In addition, such empirical results are compared between China, the United States, Greece and Spain.
Plate 33 · original document, p. 114
Cross-country comparison of results. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Montibeler, La teoría del valor trabajo y los precios en China, 2015, p. 339).
Plate 34 · original document, p. 115
Price relationships and sector-size rank controls. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Montibeler, La teoría del valor trabajo y los precios en China, 2015, p. 342).
The variable Rank, shown in the graph above, answers that Analogically, it is possible in cross-sectional analysis (III, between prices p and d, vgr.), to approximate to construct a variable that identifies the order of the size of the sectors. This rank (R), ranks each sector from smaller to larger according to its level of production and is incorporated by forming the cross-cutting model (…) (Sánchez & Montibeler, The theory of labour value and prices in China, 2015, p. 341).
Furthermore, with regard to the results of the econometric model, the authors point out that Models 1 and 2 show how the inclusion of the rank variable does not make direct prices irrelevant to explain market prices. The models present a waste with normal and homocedastic distribution. (Sánchez & Montibeler, The theory of labour value and prices in China, 2015, p. 342). On the dispersion presented by some points, they comment that The dispersion between these variables shows in turn some atypical points that are modeled in the regressions of Table 7. (Sánchez & Montibeler, The theory of labour value and prices in China, 2015, p. 341).
Plate 35 · original document, p. 116
Model with sector-size rank and residual diagnostics. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (Sánchez & Montibeler, La teoría del valor trabajo y los precios en China, 2015, p. 348).
Finally, the authors conclude that: The results of the great price closeness in the case of China add up to those of other recent work. The weighted average absolute deviation between direct and market prices is 15.13%, while between direct and prices of production is only 9.07%. These results are not altered by changing the deviation measure or distance, meaning and order of proximity are not significantly affected. It seems that, for one of the most important economies on the planet, the force of attraction that values have towards different prices is quite strong, specifically the variations in values determine the variations in current prices by 97%. The regression analysis between different prices also shows this conclusion, in line with what has been found in various countries such as the USA, Greece, Korea, Spain, etc. (Sánchez & Montibeler, The theory of labour value and prices in China, 2015, p. 349).
III. The Destruction of Value and Epistemological Equivalences
III.I. Destruction and devaluation
Mechanics and thermodynamics make it possible to distinguish conservation, transformation and energy exchange. A mechanical system subject to conservative forces and without explicit temporal dependence can retain its mechanical energy. An isolated thermodynamic system retains its total energy; when the system exchanges energy with the environment, its energy can change without violating the conservation principle. It is not correct to identify unconditionally a conservative system and an isolated system, nor to say that outside isolation energy conservation ceases to govern.
ΔU = Q − W · sign convention: W is work done by the system
Generalized coordinates and conjugate moments belong to the description of mechanical states; their use requires declaring the model hypotheses. The fact that a physical quantity is preserved does not establish that there is an economic law of conservation of value. Socially necessary labour, commercial validation and reproduction conditions are historical and social determinations, not units of mechanical energy.
Therefore, keeping technology fixed is not enough to demonstrate universal conservation of value, and technological change should not be defined as an energy input into an isolated economic system. The economic hypothesis that is of interest is more precise: when the social labour necessary to produce a commodity or equivalent equipment decreases, existing assets can be devalued even if they remain materially intact. The examination of this technological depreciation requires a distinction between stocks, flows, losses and transfers.
The question of the innovation agent refers to social work, its institutional conditions and the competitive process. The same productive development that increases material wealth can reduce unit value and depreciate previous assets. The relationship is economic and historical; it is not deduced from a literal equivalence between human labour and physical energy.
Social work is involved in innovation that can devalue previous assets; this does not make it the only possible cause of any loss of value. This occurs through the processes of technological innovation, which are characterized by being incessant within the capitalist mode of production (or capitalist political economy system, are equivalent terms). This is because technological innovation is past work that multiplies the productivity of the labour-power, because the role of technological innovation is to distribute that physical and mental wear and tear of the worker (both living labour and objective work in means of production, without confusing living labour with the value of the labour-power) into a greater quantity of goods in relation to the quantity of goods in which it was distributed before the innovation in question. This destruction of value is what Marx called moral depreciation. As noted in (Flores Morador, 2013, p. 2), MarMarx uses the term “moral” in the modern sense of “cultural”. The term, very common since the 14th century and in later times, means “pertaining to character or temperament”, from the Latin moralis “appropriate behavior of a person in society”, literally “pertaining to manners..
Marx reasoned moral depreciation in the following terms:
But in addition to material wear, every machine is subject to what we can call moral depreciation. Machinery loses exchange value insofar as machines of the same construction can be reproduced at a cheaper price or better constructed to compete with them. … In both cases, the value of a machine, however new and strong it may still be, is determined not by the actual working time actually materialized in it, but by the working time needed to reproduce it or to reproduce another better machine. I mean, the machine gets more or less depreciated.
The shorter the period during which its total value is reproduced, the lower the risk of moral depreciation and tear, and the longer the working day, the shorter that period. When machinery is installed in any branch of production, a series of new methods are put into practice, one after the other, to achieve its reproduction in more cheap conditions. 211.) (…) and a whole series of reforms, which affect not only loose parts or appliances, but their block construction. This is why it is during the first period of your life that this special reason for prolonging the working day acts more acutely. (Marx, Capital, 2010, p. 332-333).
III.II. Value, energy and information: equivalences and open questions
Flores Morador (2013) proposes to study the value of work as information. The question deserves to be examined both by its criticism of a physicalistic reading of Marx and by the scope of the informational alternative. A correspondence between hours and bits needs definition and contrast; it is not presented here as a result already obtained. Flores Morador’s argument attributes to Marx a problematic combination of three elements, which are preserved and discussed below.
1) The exchange value is defined as the chronometrically weighted average amount of simple labour (per each capital producing the eighth commodity in the total social capital) socially necessary to produce a given commodity. 2) Fixed capital is referred to several times as physical capital (remember that for Marx’s time there was no fixed capital that was non-physical (what in modern accounting is called intangible fixed asset), while now objective research, development and innovation processes (which involve more abstract processes – a consequence of the development of intellectual work as a result of technological change – which although non-physical do not cease to be objective) are crystallized in the so-called intellectual property rights (which are the main component of the so-called fixed intangible assets) 3) Every idealistic (in its philosophical sense) conception of the production process is rejected.
The above is based on the content of the investigation. There, Flores Morador argues that “(…) For Marx, technologies are tools or machines and both are physical things. He was interested in studying the intrinsic value of labor in the capitalist production process. He recognizes that the life of a machine depends first of all on two physical factors: 1) erosion by use and 2) corrosion by abandonment. … For Marx, the productivity of a technology is inversely proportional to the value it transfers to the product. The longer the life of the machine, the greater the mass of the products over which the value transmitted by the machine is distributed, and the smaller the portion of that value added to each commodity 2. We discover here some inconsequences; Marx acknowledges the transfer of physical information and matter from the technological device to the product, which is clearly wrong (…) When Marx speaks of “value transfer”, he speaks of physical erosion by use and corrosion by misuse (…) However, Marx also acknowledges a third “moral” factor that depreciates the productivity of a machine (…) this “moral depreciation” of a technology refers to the cognitive condition of human technological capabilities at some point in history and not to any physical property. According to Marx, a machine carries the sign of a cognitive know-how. This knowledge cannot be measured in reference to any physical property and has nothing to do with the tears and wear of a device. If any measurement can be made, it must be the measurement of cognitive ability expressed through practical applications. From the standpoint of the 21st century, it is easy to conclude that the problem with Marx’s view is that it is too narrow and that the only essential depreciation of value that is interesting for a theory of labour value is that of moral depreciation. (Flores Morador, 2013, p. 1-3).
Firstly, it seems that what the aforementioned author does not seem to see is that the essence of what Marx proposes is the social nature of work which creates value, together with the material conditions of its exercise (which undergoes modifications as the technological level of a society varies).
Secondly, the cited author raises as very narrow Marx’s conception of the value due to su physicalism (which he attributes to Marx), because the author is not taking into account that one of the purposes pursued by Marx’s research at the time was to strip Political Economy of all idealistic or metaphysical clothing (as the case may be), which in the context of the nineteenth century (especially considering the influence on abstract thought that the various theoretical conceptions of Classical Mechanics had on thinkers in that time) makes it entirely natural for Marx to make repeated references to capital as a physical, autonomous existence of the world of ideas, since his desire was to express, as was already seen in his work Contribution to Political Economy, which was precisely conditioned by the process of the world of ideas at that time.
Thirdly, Flores Morador states that ((…) this moral depreciation of a technology refers to the cognitive condition of human technological capabilities at some point in history and not to any physical property., which is a blatant confusion between the implications that a certain technological level has in the ideology of human society with what technology itself is. On this point, it must undoubtedly be said that the name Marx assigned to this type of depreciation (moral depreciation and tear) gives room to this type of reading, so here it is considered appropriate that, instead of calling it moral depreciation and tear, this depreciation via technological innovation should be called as technological depreciation or competitive depreciation, because within capitalism the engine of the incessant technological change is competition.
Fourth, Flores Morador proposes to introduce value as information. The passage quoted in the draft is an incoherent translation informaciónInformation is not information, it is not matter or information, so it is kept here as a documentary indication of the problem and not as a usable proposition. The substantive argument must be reconstructed by distinguishing between information, matter and energy, without retrospectively attributing further technical notions to Marx. That reconstruction does not yet demonstrate a quantitative equivalence with labour.
The contemporary measurement of information requires a distinction between its probabilistic definition, its coding and its material supports. A bit is not an economic magnitude by definition. Historical reference to quantum physics or teleportation does not demonstrate a general conversion between information and value. The valid point of the argument is methodological: no later technical concepts should be attributed to Marx in retrospect without reconstructing his mediations.
Fifthly, as mentioned above, not to consider that the meaning of the word informainformation is significantly different is to ignore precisely the impact that the incessant technological innovation that has arisen since Marx to date has had on the ideology of society.
Sixth, a unit of measurement must be distinguished from the magnitude it measures. Time, energy and information are not different forms of the same unit. They can be related by specific models for example, a power relates energy and time, but that relationship needs parameters and conditions. To express a labour value in bits it would be necessary to define what information is measured, how it is socially produced and what empirical correspondence allows it to be related to hours. The possibility cannot be taken for granted by philosophical similarity.
Seventh, the interpretation discussed seeks to justify the revision of the Marxist philosophical system (unfinished in many respects) under the logic of philosophical idealism and metaphysics. The argument can be examined in what it poses after its criticism of Marxian notions of value. It begins by pointing out that Rafael Capurro introduced a very interesting connection between the technological significance of information and the phenomenological field of philosophy. (Flores Morador, 2013, p. 4), but who is Rafael Capurro? He is an Uruguayan philosopher whose works focus on the science and ethics of information, but also on psycho-analysis, whose philosophical foundation is essentially idealistic in nature, since Sigmund Freud In investigating the causes of the pathological processes of the psyche, he decidedly rejected all attempts – on a vulgar materialistic basis in its essence – to explain the changes in the content of psychic acts due to physiological causes. But at the same time, he completely departed from the materialistic conception of the world, renounced the objective methods of researching the psyche and created a theory that is, as a whole, arbitrary, artificial and subjectivist. Its essence lies in separating the psyche from the material conditions and causes that give rise to it. The psyche is conceived as something independent, which exists in parallel to material processes (Psychophysical Parallelism) and which is directed by special, unknowable and perpetual psychic forces, located beyond the limits of consciousness (Unconscious). Over the human soul, as a fatality, there are immutable psychic conflicts derived from unconscious tendencies to pleasure (primarily sexual), to aggression towards the principle of reality to which consciousness adapts. Freud subjects to psychoanalysis all the psychic states, all the actions of man, as well as all historical events and social phenomena, i.e., he interprets them as a manifestation of unconscious passions, primarily sexual. 197).
Of course, someone might argue that just because Freud and psychoanalysis are essentially an idealistic (in the philosophical sense) approach to reality, does not necessarily mean that Flores Morador is. This in itself is true, although in the case of the author studied it is not. On the one hand, not only does Flores Morador conceive of value and technology basically as cultural states (which follows from previously made references to the author), but he also posits that According to Capurro, information is fragmented intentionality. Capurro understands the modern era of computing as a postmodern phenomenon, which is already found in the philosophy of Husserl and Heidegger. Another important difference is that the communication of information leaves behind the opposition between object and subject and replaces it within intersubjectivity and context; in the new reality the information content is not attached to a topic. 4). There are several points to point out here.
First, it is clear that defining information as fragmented intentionality is categorically an idealistic approach to reality. Second, his main philosophers to back up his hypotheses are Husserl and Heidegger, when Husserl was the one who coined the philosophy of intentionality (defined as the power of minds to deal with something, i.e., to represent or defend things, properties and states of things) and Heidegger is one of the founders and principal representatives of German existentialism (in fact, Heidegger was Husserl’s assistant). The above can be verified in (Rosental & Iudin, 1971, p. 211). Thirdly, to conceive of the opposition between object and subject and that of intersubjectivities in a context, as well as the fact that the information content is not related to a particular subject, allows to discuss the philosophical roots of Flores Morador’s vision, without that affiliation replacing the evaluation of his arguments, since the first element is idealistic and the second is metaphysical.
To avoid any doubt, the author in question points out that Let us examine here more closely what kind of ‘substance’ work is. It cannot be considered a natural substance, as if it were a natural energy, because the physical energy involved in the physical act of work is completely consumed in the act of work. If some labour energy can be found in the product of labour, it cannot be of a physical nature. We can then conclude that the physical and moral spheres of reality are independent of each other. (Flores Morador, 2013, p. 4).
Flores Morador’s interest in this essay lies in opening up the discussion between depreciation, knowledge and technology. It does not here state the absolute historical priority of its proposal without a specific bibliographic review. A general equivalence between value, energy and information is also not accepted as proven. His study poses a fertile hypothesis that requires conceptual reconstruction, formalisation and contrast.
In information theory, Shannon’s amount of information must be differentiated from semantic content, technical utility and economic value. For a discrete variable, entropy is defined from probabilities:
H(X) = −Σx p(x) log2p(x) [bits]
A more bit file doesn’t necessarily have more utility or incorporate more socially necessary work. Copying instructions, producing knowledge, maintaining infrastructure and developing technology are different processes. The existence of physical media of information does not in itself convert hours, jules and bits into interchangeable units. A materialist inquiry must study the concrete supports, work and social relations of that information production.
The distinction between materiality, objectivity, physicality and ideality is also not resolved by calling all intellectual work “non-physical”. Cognitive activity has bodily, technical and social conditions; legal and institutional relations have their own levels of description. The evaluation of Husserl, Heidegger, Capurro or psychoanalysis must address specific arguments, avoiding using an interlocutor’s philosophical affiliation as sufficient proof of falsehood.
The scientist and the worker may share a salary relationship and subordinate positions in relation to the means of production, although their working conditions, remuneration, autonomy and ownership require concrete analysis. Class membership is not determined solely by schooling; nor is an occupational label sufficient to solve all cases.
Finally, a price change may express redistribution, monetary changes, changes in output, expectations or losses. There is no universal rule that allows every price drop to be read as the destruction of value or excludes any loss in the absence of innovation. Productive consumption can transfer value; loss of utility, unproductive deterioration and material destruction require a different treatment. The limits of the system and the operation allowing transfer, conservation, realisation or loss should be declared on an aggregate scale.
· · ·
IV. Methodological Appendix: Intersectoral Relations in the United States
The content of the research, with methodological precision carried out by (NABI & B.A., AN EMPIRIC METHODOLOGY FOR THE DETERMINATION OF THE MAGNITY OF SEKTORY INTERLATIONS IN THE INSUM-PRODUCT MATRICE OF PRODUCT FROM PRODUCT FROM THE COURTS OF PRODUCT and US FOR THE CASES OF UNITED STATES 1997-2019, 2021), is then integrated, designed to verify the degree of reciprocal relationship between industrial sectors or productive branches. 15
The input–output matrix and supply and use tables represent production relations, intermediate consumption and product destinations under national accounting conventions. They are central instruments for studying economic structure over a period and, through harmonisation, its temporal transformation. The matrix does not in itself incorporate all dimensions of economic life or class relations.
Marxian research uses these instruments together with other traditions of structural analysis and national accounting. They are not the exclusive property of the New Interpretation, nor does their use automatically classify a researcher in that stream. The distinction between statistical description and statistical-mathematical analysis in footnote 15 is retained to specify the type of work carried out.
There are theoretical studies, such as Kuroki’s, in which it is claimed that in the long term there is a process of equalisation of industrial profit rates if and only if the sectors of the political economy system are highly interrelated. In this respect, Kuroki points out that Then we could conclude that only the system in which both sectors use relatively much of each other’s production as their own input and, therefore, the degree of technical interdependence is large, has a stable uniform rate of profit (i.e. for example, the economy where much wheat is needed to make iron and at the same time much iron is used to produce wheat). (Kuroki, 1985, p. 48-49). In this research, an association is investigated in a concrete sample that could be relevant to that condition, at least for the case of planetary hegemonic political economy of recent decades, since it is assumed to be representative. For this reason, the case of the United States will be studied, just as at the time Marx studied England for the same methodological reasons.
Row and column orientations shall be declared for each table. In a table of product uses by industry, the rows identify the products and the industrial columns used, to which final demand components and other aggregates are added according to the format. The supply tables record the production of products by industry. The terminology production, offer, make and supply requires consideration of the version and coverage used.
The supply and use tables, the subsequent redefinitions tables and the derived symmetric matrices are not identical objects. Transposition refocuses rows and columns; turning products into industries requires additional methodology. The selected table and its aggregates determine what can be added, compared or correlated.
See also (Kliman, The law of value and laws of statistics: sectoral values and prices in the US economy, 1977-97, 2002), (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005), (Kliman, Reply to Cockshott and Cottrell, 2005), (Zachariah, 2006), (Sánchez & Ferràndez, Values, prices of production and market prices based on Spanish economic data, 2010), (Cockshott, Cottrell, & Valle Baeza, The Empirics of the Theory of Value: Reply to Nitzan and Bichler, 2014), (Sánchez & Montibeler, Theory of labour and prices in China, 2015) and (Zachariah, Cottrell, & Zachariah, Against the Labour theory, 2019)), the study is based solely on the data from the Spanish economy, (Cockshott, Cottrell, & Valle Baeza, The Empirics of the Theory of Value: Reply to Nitzan and Bichler, 2014), (Sánchez & Montibeler, Theory of labour value and prices in China, 2015) and (Cockshott, Cottrell, Zachariah, 2019) in the study of universal value and market prices based on the data from the Spanish economy, whereas a specifically, while a specific, multi-annual, multi-annual, multi-annual, multi-annual, multi-annual, multi-annual, multi-annual, multi-annual study of theoretical research is being carried out in the context of a particular, while a specific, multi-annual, multi-annual, multi-annual, multi-annual, multi-annual, multi-annual study on-annual, multi-annual study on-annual study on-annual, on-annual study on-annual basis.
As is verified in (NABI, US METHODOLOGY BUREAU OF ECONOMIC ANALYSIS FOR the redefinition and reassignment of products in the US INSUMO-PRODUCT MATRICS, 2021), in the U.S. SCN, it is appropriate to distinguish the compilation of supply and use tables, the redefinitions and reassignments and the subsequent derivation of requirements tables by explicit assumptions. Thus, as is shown in the source cited, just as the MIP allows a deeper study of a system of capitalist political economy, the redefinition and reassignment allow a deeper study of the MIP as a result of achieving within this statistical structure a greater degree of homogeneity between its components or, from a more general conception of systems, a lower degree of heterogeneity.
Economically speaking, it is valid to conceive of the interrelation between two variables as the degree of association between them (regardless of the orientation of such an association) and, in that sense, it is then valid to think that a coefficient of correlation between products and their inputs (since such inputs are provided by the other industries (and according to them are placed within the MIP)), is a descriptive indicator of temporary association which must be contrasted with measures of technical interrelation (which is a qualitative notion as such – that of industrial interrelation-), considering also that the form taken by the MIP is linear (not so its form reveals its essence, but it is the designed statistical instrument that exists and can be used in empirical research, one of linear conception) and that all the above-mentioned research points in the same direction.
The empirical methodology presented here seeks to construct time series with the help of the statistical programme R that allow measuring the interrelation between industries by correlation between products and inputs (the people employed to produce them) by industry.
The system of national accounts (hereinafter, SCN) of the United States (hereinafter, U.S.) is presented through production tables and use tables (separately), under the label of Supply Table for the case of the table containing commodity production broken down by industrial sector and Use Table for the case of the table containing data on inputs consumed by each industrial sector in the production of each commodity for each industrial sector i−e, where i denotes rows and columns.
The specific objective of the historical methodology is to organize production and inputs for 71 US industries during 1997-2019. This level of aggregation corresponds to the available selection used by the study, not to a universal statement on the maximum detail of all macroeconomic statistics.
Annual tables make it possible to build a panel when codes, coverage and valuation are maintained or harmonised. The R procedure automates reorganisation and calculation; it does not guarantee conceptual comparability by itself. Classification checks, statistical reviews, imports, prices and product-industry correspondence are needed. Those decisions should precede the interpretation of the correlations.
Following are two images. One image shows the structure of the production frame in a sufficiently small proportion, while the other does the same with the use frame.
Plate 36 · original document, p. 131
Structure of the supply or make table used in the appendix. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (U.S. Bureau of Economic Analysis, 2021).
Plate 37 · original document, p. 132
Structure of the use table employed in the appendix. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
Source: (U.S. Bureau of Economic Analysis, 2021).
Thus, starting from the production tables (production tables) from 1997 to 2019, as well as the use tables (intermediate consumption or productive consumption tables), the following steps should be carried out.
1. The last row is drawn from the production table, which contains the total production of each of the industries that make up the economic system (for the United States in the period considered, there are 23 production tables, one per year, from 1997 to 2019) broken down by the industrial sectors that produce it. Thus, each of the 23 production tables containing the annual data on 71 industries will have a vector row consisting of 71 elements. 2. The use table is taken and transposed, i.e. the location of the contents of its rows is exchanged for that of its columns and vice versa. The above is done in order to place the intermediate consumption or productive consumption originally in the columns of the table of uses in the rows of the table of production transposed to the original table of production. This will be useful to facilitate the automation of the construction of the frame containing the desired time series.
3. By combining the row vector extracted from the production table and the transposed use table, 71 new tables are generated. Each of these tables will be composed in its first column for the time period in question (from 1997 to 2019), in its second column each of the industrial sectors providing inputs to others and themselves for production will be located, in the third column the number of the industrial sector will be indicated according to its position in the production-use tables (since it is twenty-three years of study for industries in general and for each in particular, the number indicating the position of each industrial sector will be repeated twenty-three times), in the fourth column the total production of each sector will be located and finally in n−4 (in this case the remaining 71 columns) each of the industrial sectors generating global social production will be located. Thus, 71 tables with the configuration described above (one per industrial sector) will be formed and from that table the product-added correlations will be obtained, which will be executed in the next step. 4. The table constructed in step 3, which contains panel data for the 71 industrial sectors that make up the US economy for the period 1997-2019, can be divided into 71 parts, where each part contains panel data for each of the 71 industrial sectors individually. Thus, on each of these 71 data panels generated, correlation is calculated (the column vector of the total output of each industry correlated over time with each of the n-th column vectors representing the contribution at the level of intermediate inputs that each of the industrial sectors supplied to the product in question), by storing it in the statistical program R within a vector-line data structure and, finally, by lining up the vectors to form the Pearson correlation matrix.
5. A weighted average of the Pearson correlation coefficients for each of the 71 industrial sectors is then calculated (expressing the weighted average correlation coefficient for each industrial sector), where the weighting factor is the relative share of each input in total intermediate consumption. 6. Finally, each of these weighted average intra-sectoral coefficients (pertaining to each sector) is re-weighted to obtain the weighted average correlation coefficient of all industrial sectors, i.e. the correlation coefficient that summarizes the interrelationship between all industrial sectors of the US political economy system. Here, the weighting factor is the relative participation of each industrial sector in the economic system.
It is clear that, as in this research, what is of interest is to know the magnitude of the sectoral interrelation and not the meaning of that interrelation (expressed in the sign of the correlation coefficient obtained), when estimating the averages the calculations are made with the absolute values of these coefficients. The following are images which contain, in an orderly and minimal and sufficient manner, the statistical tables resulting from the procedures described in steps 3 to 6.
Plate 38 · original document, p. 135
Organisation of the intersectoral panel. Historical output of the procedure; this is not the synthetic example in the reproducibility supplement.Scroll horizontally to read the table at a larger size.
Source: Author’s calculations following the methodology described in step 3.
Plate 39 · original document, p. 135
Correlation matrix constructed by the procedure. Historical output of the procedure; this is not the synthetic example in the reproducibility supplement.Scroll horizontally to read the table at a larger size.
Source: Author’s calculations following the methodology described in step 4.
Plate 40 · original document, p. 135
Original output: unweighted mean. Historical output of the procedure; this is not the synthetic example in the reproducibility supplement.
Source: Author’s calculations using the preceding figure and the methodology described in step 4.
Plate 41 · original document, p. 136
First weighting stage: input shares. Historical output of the procedure; this is not the synthetic example in the reproducibility supplement.Scroll horizontally to read the table at a larger size.
Source: Author’s calculations following the methodology described in step 5.
Plate 42 · original document, p. 136
Original output after the first weighting stage. Historical output of the procedure; this is not the synthetic example in the reproducibility supplement.
Source: Author’s calculations using the preceding figure and the methodology described in step 5.
Plate 43 · original document, p. 137
Second weighting stage: output shares. Historical output of the procedure; this is not the synthetic example in the reproducibility supplement.Scroll horizontally to read the table at a larger size.
Source: Author’s calculations following the methodology described in step 6.
Plate 44 · original document, p. 137
Original output after the second weighting stage. Historical output of the procedure; this is not the synthetic example in the reproducibility supplement.
Source: Author’s calculations using the preceding figure and the methodology described in step 6.
The original calculation reports an average of correlation magnitudes close to 0.683 before weighing, about 0.735 after the first weighing and about 0.778 after the second. The pictures above preserve the exits from the studio. The data set has not been recalculated here; the figures are presented as historical results of the procedure described.
Those numbers are not causality percentages or determination coefficients. A weighted average of absolute correlation values does not necessarily constitute the correlation of two added variables. The threshold of 0.70 is also not universal: the importance of a partnership depends on the context, design, uncertainty and question. If you wanted to measure the distance from 0.683 to 0.700, this would be 0.017; 0.027 corresponds to reaching 0.710. Differences are coefficient points, not relative percentage increases.
The following table of Mindrila and Balentyne is preserved as a descriptive convention used in the original study. It does not turn the proximity to a threshold into a proof of technical interdependence or evidence of a profit equalisation mechanism.
Plate 45 · original document, p. 138
Descriptive correlation-strength convention cited in the original study. The original study material is retained; the surrounding discussion explains its interpretation and limitations.Scroll horizontally to read the table at a larger size.
The defendable conclusion is that the procedure finds an appreciable temporal association between production and inputs in the sample and under the chosen weights. The stability of a model such as Kuroki’s is not automatically verified by that association, nor can it be ascribed to a universal assumption of any developed capitalist economy. In order to investigate the connection, measures of production structure compatible with the model and an examination of its other hypotheses are needed.
The historical link to the code and archives cited is preserved in footnote 16. In order for this edition to be understood without relying on that link, a complete formalisation of the procedure, a new script in R and reproducible synthetic data are included below. The synthetic example does not replace the original data or reproduce historical figures by affirmation.
IV.II. Formalisation and reproducibility supplement
This section explains the operations of the Annex and provides a new implementation. The historical material retains its results; the following code enables the procedure to be understood, revised and applied to harmonised data. It is not claimed to have recovered or executed the 2021 link script.
Variables, table orientation and units
Seajt the output of the purchasing industry in year t, and Zijt the value of the input of the industry or product i used by j. Period, geographical coverage, valuation and classification should be the same. A table of products by industry does not become an industry by industry just by transposing it; if the universes differ, a methodological correspondence is needed.
The original panel covers 19972019, 23 years. If the two relevant dimensions have 71 comparable categories, 71×71 series pairs can be formed, each with up to 23 observations. Production of j is repeated by comparing it with each supplier i. Null inputs, constant series and absences require different rules.
rij = Corrt(Xjt, Zijt)
For an explicit weighting rule, inputs shares within intermediate consumption of j and productive shares πj are defined. The script offers weights calculated with period totals or the last year. They are transparent choices of this implementation, not an affirmed reconstruction of all the undocumented decisions of the historical archive.
aj = Σi ωij|rij| · A = Σj πjaj
Σiωij = 1 · Σjπj = 1
With non-negative weights, the index A falls between zero and one when all the necessary components are defined. It summarizes the magnitude of association. The signed version retains R instead of R. Neither is automatically a correlation between aggregate output and aggregate inputs; nor is its square an explanatory causal percentage.
Proportions and magnitudes
A common monetary factor can change the levels of all series and generate co-movements. Therefore, a temporary correlation in nominal values should be compared, where data permit, with series in volume or consistently deflated, growth rates and specifications that consider common trends. Transformation should not be chosen merely because the desired correlation increases or decreases.
Input ratios, technical coefficients and chain rates answer other questions. One sector may show a high temporal correlation with a small participation input; another may technically depend on an input whose series is almost constant. Temporary association and productive need are not equivalent.
Checks for an empirical application
This is a problem
Checking
Consequences of interpretation
Classifications
Harmonize codes per year and verify sectoral mergers or separations.
Avoid attributing a change in classification to production.
Prices and trends
Comparison of nominal levels, volumes and justified transformations.
Evaluates sensitivity to inflation and common trends.
Temporary dependency
Examining residues, persistence and structural changes.
It conditions uncertainty and inference.
Weighing
Compare last year, average or cumulative, declaring the rule.
It distinguishes general association and weight of large sectors.
Sign and zeros
Report signed and absolute results; do not confuse zero with missing data.
Avoid hiding negative relationships or making observations.
Constant series
Declare indefinite correlations and their weight
It prevents selective re-normalisation without reporting.
Technical interdependence
Contrast with directly and totally compatible requirements.
It does not replace a condition of stability with a correlation.
Generalisation
Comparing periods and economies with equivalent information.
It delimits the geographical and temporal range.
Complete R script
The code uses base R. The input table contains year, purchasing industry, production, supplier and input. The embedded example is synthetic and deliberately small. The exit serves to verify the functioning of the procedure; it should not be cited as evidence from the United States. This edition does not communicate a new execution of the BEA panel or additional empirical validation.
# Reproducible complement to the HTML edition.
# Author of this edition: JOSÉ MAURICIO GÓMEZ JULIÁN.
# New implementation of the procedure described; not the historical script.
# No additional packages are required. The synthetic data are not BEA data.
interrelation <- function(panel_data, years = sort(unique(panel_data$year)),
weight_mode = c("pooled", "last"), min_obs = 3L) {
weight_mode <- match.arg(weight_mode)
required <- c("year", "industry", "output", "input_industry", "input_value")
if (!is.data.frame(panel_data) || !all(required %in% names(panel_data)))
stop("Required columns: ", paste(required, collapse = ", "))
d <- panel_data[, required]
d$industry <- as.character(d$industry)
d$input_industry <- as.character(d$input_industry)
if (anyNA(d) || any(!is.finite(d$output)) || any(!is.finite(d$input_value)))
stop("Missing or non-finite data are not silently imputed.")
if (any(d$output <= 0) || any(d$input_value < 0))
stop("Output must be positive and inputs non-negative. Handle accounting adjustments explicitly.")
d <- d[d$year %in% years, , drop = FALSE]
if (length(years) < min_obs) stop("The time span is insufficient for correlation analysis.")
dest <- sort(unique(d$industry)); suppliers <- sort(unique(d$input_industry))
keys <- paste(d$year, d$industry, d$input_industry, sep = "|")
if (anyDuplicated(keys)) stop("Duplicate year-industry-supplier rows were found.")
expected <- length(years) * length(dest) * length(suppliers)
if (nrow(d) != expected) stop("Incomplete panel: distinguish zeros from missing observations.")
out <- unique(d[, c("year", "industry", "output")])
if (anyDuplicated(paste(out$year, out$industry, sep = "|")))
stop("Output for an industry-year cannot differ across suppliers.")
R <- matrix(NA_real_, length(dest), length(suppliers),
dimnames = list(dest, suppliers))
W <- matrix(0, length(dest), length(suppliers), dimnames = dimnames(R))
for (j in dest) {
for (i in suppliers) {
z <- d[d$industry == j & d$input_industry == i, , drop = FALSE]
z <- z[match(years, z$year), , drop = FALSE]
if (nrow(z) >= min_obs && sd(z$output) > 0 && sd(z$input_value) > 0)
R[j, i] <- cor(z$output, z$input_value, method = "pearson")
use <- if (weight_mode == "last") z$year == max(years) else rep(TRUE, nrow(z))
W[j, i] <- sum(z$input_value[use])
}
if (sum(W[j, ]) <= 0) stop("An industry has no positive intermediate consumption.")
W[j, ] <- W[j, ] / sum(W[j, ])
}
# An indefinite correlation with positive weight produces NA, not renormalisation.
safe_mean <- function(x, w) {
active <- w > 0
if (any(!is.finite(x[active]))) return(NA_real_)
sum(x[active] * w[active]) / sum(w[active])
}
sector_abs <- vapply(dest, function(j) safe_mean(abs(R[j, ]), W[j, ]), numeric(1))
sector_signed <- vapply(dest, function(j) safe_mean(R[j, ], W[j, ]), numeric(1))
ow <- if (weight_mode == "last") out[out$year == max(years), ] else out
totals <- tapply(ow$output, ow$industry, sum)[dest]
shares <- as.numeric(totals / sum(totals)); names(shares) <- dest
list(correlations = R, input_weights = W, output_weights = shares,
by_industry = data.frame(industry = dest,
association_abs = unname(sector_abs), association_signed = unname(sector_signed),
output_share = shares, row.names = NULL),
unweighted_abs = if (anyNA(R)) NA_real_ else mean(abs(R)),
first_weight_abs = if (anyNA(sector_abs)) NA_real_ else mean(sector_abs),
final_abs = safe_mean(sector_abs, shares),
final_signed = safe_mean(sector_signed, shares),
settings = list(years = years, weight_mode = weight_mode,
min_obs = min_obs, missing_policy = "strict"))
}
# Entry scheme after harmonisation of supply tables and uses.
# This is the year of industry output input industry input value.
# One row a year, buyer and supplier industry, zeros are observations.
# Do not mix domestic supply with import uses without declaring the adjustment.
synthetic <- expand.grid(year = 2001:2008, industry = c("A", "B"),
input_industry = c("A", "B"), stringsAsFactors = FALSE)
t <- synthetic$year - 2000
synthetic$output <- ifelse(synthetic$industry == "A", 100 + 9*t + t^2, 150 + 6*t + 2*t^2)
synthetic$input_value <- ifelse(synthetic$input_industry == "A",
0.15*synthetic$output + t,
0.22*synthetic$output + 2*sin(t))
demo <- interrelation(synthetic)
print(demo$by_industry)
print(c(absolute_index = demo$final_abs, signed_index = demo$final_signed))
stopifnot(abs(sum(demo$output_weights) - 1) < 1e-12,
all(abs(rowSums(demo$input_weights) - 1) < 1e-12),
is.finite(demo$final_abs), demo$final_abs >= 0, demo$final_abs <= 1)
# Sensitivity to temporary transformations: units are preserved
# the weightings separately; dividends are not calculated.
# The following function diagnoses association in logarithmic growth rates.
growth_correlations <- function(panel_data) {
dest <- sort(unique(panel_data$industry)); suppliers <- sort(unique(panel_data$input_industry))
ans <- matrix(NA_real_, length(dest), length(suppliers), dimnames = list(dest, suppliers))
for (j in dest) for (i in suppliers) {
z <- panel_data[panel_data$industry == j & panel_data$input_industry == i, ]
z <- z[order(z$year), ]
if (any(z$output <= 0) || any(z$input_value <= 0)) next
if (any(diff(z$year) != 1)) next
a <- diff(log(z$output)); b <- diff(log(z$input_value))
if (length(a) >= 3 && sd(a) > 0 && sd(b) > 0) ans[j, i] <- cor(a, b)
}
ans
}
print(growth_correlations(synthetic))
# Use with real harmonised data:
# The following information is provided by the Commission to the European Parliament and the Council:
# This is the total amount of the loan, which is calculated on the basis of the amount of the loan.
# The following information is provided by the Commission to the European Parliament and to the Council:
# The historical results of the PDF are not replaced by those in the example.
What this supplement makes reproducible
With compatible input data, the functions calculate correlation matrices, holdings, sectoral and aggregate indices, sign results and a growth rate diagnosis. The original files referred to in footnote 16 are necessary for an exact replication of the 2021 figures if they contain decisions or treatments not specified in the PDF. The explanation, the formulas, the tables that do contain the PDF and the instructional code are fully available in this HTML.
Findings and scope
Value Creation, Technical Change and the Limits of Inference
The tour allows you to keep the initial question and to specify its conditions. Marxian theory attributes to living labour the creation of new value and distinguishes that function from the transfer of value from the means consumed. Innovation changes productivity and social conditions of reproduction: it can expand material output, reduce value per unit and devaluate existing assets. No logical contradiction arises from the mere fact that these movements coexist.
The correlations between estimated values and prices, documented in the studies discussed for various economies, are relevant evidence for the comparison of explanations. Its interpretation requires specifying how values are constructed, what prices are used, what aggregation and numerals are adopted and what statistical transformations are carried out. The relative explanatory power over alternative bases is an interesting proof; it must be accompanied by diagnoses and an explicit discussion of shared restrictions.
The gravity of profitability does not amount to immediate equality, nor does the persistence of differences exclude any competitive pressure. Time series, cross-cutting and dynamic models answer different questions. The simulation results are conditional on assumptions; the applied statistics do not convert an aggregate correlation into a general demonstration of causation. A rigorous historical approach requires both precautions.
Matrix representation and iterative solution can be complementary tools. Its relevance is not determined by the school genealogy of an author or by the simultaneous or sequential appearance of the calculation. It is decided by the economic content of the relations, their dates, units, mechanisms and ability to confront experience.
The extension of analysis to knowledge and information remains an open programme. A materialistic formulation must investigate the social production of capacities, codes, teams and institutions and build contrasting correspondences. The analogy with energy or with bits can guide questions; to answer them you need theoretical mediations and explicit measurements.
Formal accuracy strengthens criticism when categories retain their historical content and conclusions respect the scope of evidence.
The argument in perspective
V. References of the original study
The complete bibliography of the source document, with verified bibliographic corrections.
Publication titles are retained in their original languages to identify the cited editions accurately. The documentary figures retain their original labels; their captions and the accompanying discussion are in English.
Andrews, D. W. (1991). An Empirical Process Central Limit Theorem for Dependent Non-identically Distributed Random Variables . Journal of Multivariate Analysis, 187-203.
Berk, K. (1973). A CENTRAL LIMIT THEOREM FOR m-DEPENDENT RANDOM VARIABLES WITH UNBOUNDED m. The Annals of Probability, 1(2), 352-354.
Borisov, E. F., & Zhamin, V. A. (2009). Diccionario de Economía Política. (L. H. Juárez, Ed.) Nueva Guatemala de la Asunción, Guatemala, Guatemala: Tratados y Manuales Grijalbo.
Cockshott, P., & Cottrell, A. (2005). Robust correlations between prices and labor values. Cambridge Journal of Economics, 309-316.
Cockshott, P., Cottrell, A., & Valle Baeza, A. (2014). The Empirics of the Labour Theory of Value: Reply to Nitzan and Bichler. Investigación Económica, 115-134.
Dedecker, J., & Prieur, C. (2007). An empirical central limit theorem for dependent sequences. Stochastic Processes and their Applications, 117, 121-142.
Díaz, E., & Osuna, R. (2007). Indeterminacy in price–value correlation measures. Empirical Economics, 389-399.
Emmanuel, A. (1972). El Intercambio Desigual. Ensayo sobre los antagonismos en las relaciones económicas internacionales. México, D.F.: Sigloveintiuno editores, s.a.
Farjoun, E., & Machover, M. (1983). Laws of Chaos. A Probabilistic Approach to Political Economy. Londres: Verso Editions and NLB.
Flaschel, P., & Semmler, W. (1985). The Dynamic Equalization of Profit Rates for Input-Output Models with Fixed Capital. En Varios, & W. Semmler (Ed.), Competition, Instability, and Nonlinear Cycles (pp. 1-34). New York: Springer-Verlag.
Fröhlich, N. (2013). Labour values, prices of production and the missing equalisation tendency of profit rates: evidence from the German economy. Cambridge Journal of Economics, 37(5), 1107-1126. [Accepted in 2012; article published in 2013.]
Glick, M., & Ehrbar, H. (1988). Profit Rate Equalization in the U.S. and Europe: An Econometric Investigation. European Journal of Political Economy, 179-201.
Gloria-Palermo, S. (2010). Introducing Formalism in Economics: The Growth Model of John von Neumann. Panoeconomicus, 153-172.
Godwin, H., & Zaremba, S. (1961). A Central Limit Theorem for Partly Dependent Variables. The Annals of Mathematical Statistics, 32(3), 677-686.
Guerrero, D. (October-diciembre de 1997). UN MARX IMPOSIBLE: EL MARXISMO SIN TEORÍA LABORAL DEL VALOR. 57(222), 105-143.
Kliman, A. (2002). The law of value and laws of statistics: sectoral values and prices in the US economy, 1977-97. Cambridge Journal of Economics, 299-311.
Kliman, A. (2005). Reply to Cockshott and Cottrell. Cambridge Journal of Economics, 317-323.
Kliman, A. (2008). What is spurious correlation? A reply to Díaz and Osuna. Journal of Post Keynesian Economics, 31(2), 345-356. [The online edition is dated 2014.]
KO, M.-H., RYU, D.-H., KIM, T.-S., & CHOI, Y.-K. (2007). A CENTRAL LIMIT THEOREM FOR GENERAL WEIGHTED SUMS OF LNQD RANDOM VARIABLES AND ITS APPLICATION. ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, 37(1), 259-268.
Kuhn, T. (2011). La Estructura de las Revoluciones Científicas. México, D.F.: Fondo de Cultura Económica.
Kuroki, R. (1985). The Equalizartion of the Rate of Profit Reconsidered. En W. Semmler, Competition, Instability, and Nonlinear Cycles (pp. 35-50). New York: Springer-Velag.
Landau, L. D., & Lifshitz, E. M. (1994). Curso de Física Teórica. Mecánica (Segunda edición corregida ed.). (E. L. Vázquez, Trad.) Barcelona: Reverté, S.A.
Leontief, W. (1986). Input-Output Economics. Oxford, United States: Oxford University Press.
Levins, R. (December de 1993). A Response to Orzack and Sober: Formal Analysis and the Fluidity of Science. The Quarterly Review of Biology, 68(4), 547-55.
LI, X.-p. (2015). A Central Limit Theorem for m-dependent Random Variables under Sublinear Expectations. Acta Mathematicae Applicatae Sinica, 31(2), 435-444. doi:10.1007/s10255-015-0477-1
Parzen, E. (1957). A Central Limit Theorem for Multilinear Stochastic Processes. The Annals of Mathematical Statistics, 28(1), 252-256.
Pasinetti, L. (1984). Lecciones Sobre Teoría de la Producción. (L. Tormo, Trad.) México, D.F.: Fondo de Cultura Económica.
Real Academia Española. (18 de 03 de 2021). Diccionario de la lengua española. Retrieved from Edición del Tricentenario | Actualización 2020: https://dle.rae.es/transitar?m=form
Real Academia Española. (23 March 2021). Diccionario de la lengua española. Retrieved from Edición Tricentenario | Actualización 2020: https://dle.rae.es/ecualizar?m=form
Rosental, M. M., & Iudin, P. F. (1971). DICCIONARIO FILOSÓFICO. San Salvador: Tecolut.
Rosental, M., & Iudin, P. (1971). Diccionario Filosófico. San Salvador: Tecolut.
Sánchez, C., & Ferràndez, M. N. (October-diciembre de 2010). Valores, precios de producción y precios de mercado a partir de los datos de la economía española. Investigación Económica, 87-118. Retrieved from https://www.jstor.org/stable/42779601?seq=1
Sánchez, C., & Montibeler, E. E. (2015). La teoría del valor trabajo y los precios en China. Economia e Sociedade, 329-354.
Steedman, I., & Tomkins, J. (1998). On measuring the deviation of prices from values. Cambridge Journal of Economics, 379-385.
U.S. Bureau of Economic Analysis. (1 April 2021). The Domestic Supply of Commodities by Industries (Millions of dollars). Retrieved from Input-Output Accounts Data | Data Files. Supply Tables – Domestic supply of commodities by industry ● 1997-2019: 15 Industries iTable, 71 Industries iTable: https://apps.bea.gov/iTable/iTable.cfm?reqid=52&step=102&isuri=1&table_list=3&aggregation=sum
U.S. Bureau of Economic Analysis. (1 April 2021). The Domestic Supply of Commodities by Industries (Millions of dollars). Retrieved from Input-Output Accounts Data | Supplemental Estimate Tables. After Redefinition Tables. Make Tables/After Redefinitions – Production of commodities by industry after redefinition of secondary production ● 1997-2019: 71 Industries iTable: https://apps.bea.gov/iTable/iTable.cfm?reqid=58&step=102&isuri=1&table_list=5&aggregation=sum
U.S. Bureau of Economic Analysis. (1 April 2021). The Use of Commodities by Industries. Retrieved from Input-Output Accounts Data | Supplemental Estimate Tables. After Redefinition Tables. Use Tables/After Redefinitions/Producer Value – Use of commodities by industry after reallocation of inputs ● 1997-2019: 71 Industries iTable: https://apps.bea.gov/iTable/iTable.cfm?reqid=58&step=102&isuri=1&table_list=6&aggregation=sum
Valle Baeza, A. (1978). Valor y Precios de Producción. Investigación Económica, 169-203.
Walras, L. (1954). Elements of Pure Economics or The Theory of Social Wealth. (W. Jaffé, Trad.) Homewood, Ilinois, Estados Unidos: Richard D. Irwin, Inc.
Wooldridge, J. M. (2010). Introducción a la Econometría. Un Enfoque Moderno (Cuarta ed.). México, D.F.: Cengage Learning.
Zachariah, D. (June de 2006). Labour value and equalisation of profit rates: a multi-country study. Indian Development Review, 4, 1-20.
Additional sources for the technical clarifications
Marx, Karl. El capital, libro I, capítulos 6–8. Texto primario para fuerza de trabajo, valorización y funciones del capital. Consult source.
Stark, Philip B. The Normal Curve, the Central Limit Theorem, and Markov’s and Chebychev’s Inequalities for Random Variables. University of California, Berkeley. Consult source.
Verzani, John. Neumann series and iterative methods. College of Staten Island, City University of New York. Consult source.
NIST. Dataplot Reference Manual: Coefficient of Variation. Definición, condiciones de uso y relación con la desviación estándar relativa. Consult source.
OpenStax. University Physics, vol. 2, sección 3.3: First Law of Thermodynamics. Consult source.
Shannon, Claude E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27, 379–423, 623–656. Texto original. Consult source.
U.S. Bureau of Economic Analysis. What are direct and indirect requirement coefficients? Definiciones metodológicas oficiales. Consult source.
U.S. Bureau of Economic Analysis. What are requirement tables, and how are they formed? Consult source.
Fröhlich, Nils (2013). Labour values, prices of production and the missing equalisation tendency of profit rates: evidence from the German economy. Cambridge Journal of Economics, 37(5), 1107–1126. DOI: 10.1093/cje/bes066. Aceptado en 2012, publicado en 2013. Consult source.
Kliman, Andrew (2008). What is spurious correlation? A reply to Díaz and Osuna. Journal of Post Keynesian Economics, 31(2), 345–356. DOI: 10.2753/PKE0160-3477310208. Publicado en línea en 2014; se distingue esa fecha de la del volumen. Consult source.
Walras, Léon (1954). Elements of Pure Economics, or the Theory of Social Wealth. Traducción de William Jaffé. El índice incluye producción, formación de capital y circulación; se corrige la afirmación de su ausencia. Consult source.
Endnotes
While Kliman does not provide details of how he constructed the simulation, the research (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005) details such a procedure, since these authors asked Kliman for the simulation details and explain that the latter was kindly provided with them. ↩
The relationship between proportional thinking, units and measurement is developed in the conceptual threshold, the scaling analysis and the methodological complement of this edition. ↩
The fact that a trend is classical (which is what I consider to express with complete fidelity a truly Marxian theory – that is, a theory whose foundations stem uniquely from the work of Marx and Engels – does not mean that it does not evolve, quite the contrary, since all the fundamental principles of Marxism involve the dynamic and complex process of incessant transformation of reality. This implies that the basic and superstructure categories, alienation, exploitation, historical materialism, means of production, ideology, class consciousness, mode of production, political economy are indispensable in all analyses carried out by the researcher, which were defined by Marx and Engels, as well as by other Marxists, among whom the philosophers of the Soviet philosophical school (mainly in their post-Stalinist stage -who despite never leaving the Soviet Union at a political level, did not, however, abandon the philosophical academy-), Trotsky, Lenin, Rosdolsky, Gramsci, Ernest Lardoel, Alan Freeman, Raphael Menzivarin, Rafael Menzivar and Dagobert Gérrez; of course, that they are Marxists does not necessarily mean that the epistemological uniformity that is valid in the epistemological realm of Marxism is not necessarily compatible with all human authors (even those who are not fully aware of the humanities of Marxism) is not necessarily valid in their own right, and that they do not necessarily mean that the epistemological uniformity of the epistemological epistemology (which is not valid in the realm of Marxism) is not necessarily valid (even in the case of human beings of Marxists) is not fully compatible with human beings of Marxist (whould not beings that are not necessarily not necessarily, and that they do not necessarily mean that they are not necessarily not necessarily not necessarily not necessarily have to be, even human beings of the human beings of human beings of Marxist). To these great revolutionaries, theorists and, with your permission, dear comrades, I dedicate this investigation as a humble tribute to their revolutionary greatness. ↩
Negrites and curves were added to this investigation in order to highlight the main points of the exposure. ↩
This is not to be confused with the independence or not of values and prices, but rather with the vector of physical escalation relative to the value vector and the price vector (whether production or market prices). It is valid to assume such independence from the physical scale vector since there are no theoretical or logical reasons to think that the price will be related to the numerical magnitude which merely refers to the number of times it occurs in physical reality (the number of bank loans made, the number of cars produced, etc.) or, what is the same, its size in the statistical sense. Precision: Zero covariance does not imply independence in general. The absence of association with the scale factor is a condition that must be justified for the data and units chosen. ↩
There is a valid argument for taking industry size into account when measuring price-value correlations, but it is quite different from the ideas considered above. This point can be addressed by the concept of ‘specific price’ proposed by Farjoun and Machover (1983), i.e. the market price obtained per unit of labour embedded. (Cockshott & Cottrell, Robust correlations between prices and labour values, 2005, p. 311). ↩
This is the name by which Andrew Kliman refers to the set of investigations (converging conclusions at the same point) born within the NI on the explanatory capacity of values with respect to prices of production in cross-sectional data (the input–output matrices are data structures of this kind). ↩
On the perpetual inventory method they point out that This method adds gross investment and reduces depreciation and disposed capital. The depreciation shall be estimated using the straight line method. Scrap is estimated by a scrap function. (Glick & Ehrbar, 1988, p. 199). In turn, it is known as scrapping ( OCscrapping in English) as defined by the OECD as scrapping occurs when capital assets are withdrawn from share capital at the end of their lifetime. Waste is synonymous with ‘waste’ and ‘retirement’ (…) (OECD, 2005). In a nutshell, scrapping a fixed capital is what is accounted for as estimating the rescue value or residual value of a fixed asset. Accuracy of this edition: the withdrawal or scrapping of an asset and the estimation of its residual value are separate operations; they should not be identified only. ↩
On this they add that The national accounts of the USA The United States They are compiled from surveys periodically reviewed by the census results. Industries are classified according to the establishment. (Glick & Ehrbar, 1988, p. 199). ↩
Engels here is making a hyperbolic assumption to prove a point. The minimal study of Marx’s work verifies that he was not a single historical example. ↩
There is a debate within the Marxist schools about MELT whether in the sphere of production or in the sphere of circulation. However, that debate will not be discussed here and will be the subject of further investigation. ↩
The curves and negritas with curve are added by the author of this research to make clearer the discontinuity between the theoretical and empirical process used by the NI. ↩
What S&F refers to is that: It will be clear that neither θ nor the coefficient of variation depends in any way on the options of physical or numerical units of measurement. (Steedman & Tomkins, 1998, p. 381). ↩
Physical wear and tear of work equipment: Loss of its use value as a result of productive operation or inactivity. The material wear of any machine is double. One comes from the use, as in the case of coins that wear out as they circulate hand in hand; another comes from their inaction, as the inactive sword, which oxidizes in the coffin (C. Marx). Material or physical wear is reflected in the physical change of certain parts of the machine (destruction, rubbing, corrosion). As physical wear and tear occurs, the value of the means of labour is partly transferred to the newly produced articles. The value of the means of labour transferred to the product in proportion to their wear and tear constitutes depreciation, which is intended for the total or partial restoration of the means of labour. In addition to physical wear and tear, there is also moral depreciation and tear. (Borisov & Zhamin, 2009, p. 67). Depreciation: Money compensation for the value of the basic means of labour (machines, installations, buildings), value which gradually passes to the new product obtained in the production process or to the work done (services). In the production process, the fundamental means of labour are not only materially worn out but also in their value which they gradually transfer to the product. There is also a depletion of value as a consequence of the moral depletion (see) of the basic means of production, which is due, on the one hand, to the cheaper production of analogous means of production as a consequence of the increase in labour productivity and, on the other hand, to technical ageing as a consequence of the progress of science and technology. Unlike in the capitalist economy, where the depreciation of fixed capital (see) is the private affair of each entrepreneur or monopoly, in the socialist economy it is society which, according to a plan, sets depreciation rates which form an integral part of the cost price and are taken into account in the planned price formation. The annual depreciation ratio depends on two factors: the average annual value of the basic funds depreciated and the size of the depreciation ratio. The depreciation rate is set at such a percentage as to the value (price) of the basic funds and indicates the number of years in which the value of those funds has to be repaid. The depreciation rates are different, depending on the type of basic funds. In addition, in order to plan the overall depreciation deductions by undertakings and branches, the National Economic Council and other departments also determine the average depreciation rates. Accurate, scientifically established depreciation rates are of great importance for socialist economic practice. If they are low, the renewal of basic funds becomes slower, and thus indirectly slows down technical progress; whereas if they are high, the cost of production increases unnecessarily. (Borisov & Zhamin, 2009, p. 6-7). Precision of this edition: the transfer of production must not be confused with any deterioration due to inactivity, damage or loss of utility; each case requires a balance of losses and transfers. ↩
Here we distinguish statistical-mathematical from statistical in the sense that the first concept passes through areas that concern the application of Mathematical Statistics to concrete problems of reality (i.e., the scientific approach to data), while the second may be in the sense previously defined, may be in its purely descriptive sense or may even be what is popularly understood by it. ↩
The base document was published in 2021 as a draft under the pseudonym Isadore Nabi. This edition is signed under the name JOSÉ MAURICIO GÓMEZ JULIÁN. Historical names of earlier publications are preserved in their references to maintain their bibliographic location. All paragraphs, graphic materials, references and notes of the basic document have been integrated, regularizing headings, page-to-page continuity and references.
The corrections include the scope of the limit theorems; the distinctions between dependence, correlation and collinearity; the scaling of correlations; the temporal notation of the profitability model; the statistical nature of its innovations; the difference between iteration and economic time; the units of vertical integration; the matrix example; DAMP, coefficient of variation and angular distance; the relationship between conservation and energy exchange; the scope of cross-sectoral evidence and the distinction between value and price. The pending argumental link on transformation develops within the text.
Extensive citations are kept as debating materials and should not be confused with adherence to all their conclusions. Historical charts and figures are embedded reproductions; no data has been invented to replace them. The new formalisations, examples and script are identified as developments in this edition. The entire article can be read without downloading the PDF or uploading images, typographies or libraries from other servers.
Quotations are translated from the Spanish wording reproduced in the source. Where an English-language publication is cited, this English rendering is not presented as a verified transcription of its original wording.
Marxist Philosophy of SciencePhilosophy of Science · Dialectical Materialism
On the Essence, Content, Form, Phenomenon, and Appearance of the Processes of Reality
Categories, mediations, and knowledge of natural and social processes
JOSÉ MAURICIO GÓMEZ JULIÁN
Original essay · 10 April 2021 Complete revised and expanded edition · 29 October 2026 18,571 words · Approximate reading time: 98 min
Introduction
Knowing a process requires reconstructing its relationships
Reality is not exhausted by its immediate presentation. A temperature variation, a magnetic signal, a price, or motion observed from a platform can be recorded precisely and still require an explanation of the relationships that produce it. The question of essence begins when we investigate what makes a manifestation possible and why it takes a particular form.
This essay examines five categories—essence, content, form, phenomenon, and appearance—within dialectical materialism. Its purpose is not to offer five isolated definitions, but to reconstruct their relationships, differences, and usefulness for studying natural and social processes. A process’s material unity does not eliminate its determinations; analytically distinguishing those determinations does not turn them into separate entities.
The central problem is methodological: how can essence–phenomenon and content–form be articulated when the object of research contains multiple components, scales, relationships, and instances of mediation? Philosophical tradition provides general categories; scientific practice requires specifying which relationships explain the process, how they are organized, and through what evidence they can be known.
The proposed contribution is to develop the notion of essential content and clarify the distinction between phenomenon and appearance. Essential content identifies a fundamental articulation within the content of a delimited system. The phenomenon–appearance distinction makes it possible to investigate objective manifestation alongside the conditions of a partial presentation or insufficient understanding. Both proposals are offered as research instruments open to discussion and testing.
The discussion begins with Rosental, Iudin, and Frolov, examines the relationship between material history and logical reconstruction, and develops cases involving climate, geomagnetism, the geodynamo, reference frames, topology, and political economy. Interdisciplinary comparisons are fruitful when they clarify questions and mediations. They do not remove the need to argue within each science or turn an analogy into a physical, mathematical, or economic equivalence.
The elaboration on essence and mathematical structures develops this articulation across several levels: topology is one dimension of the essential, whose relevance is assessed alongside differentiable structure, metric, symmetries, and historical-dynamical determinations. The discussion draws on the author’s manuscript on geometry, history, and lawfulness while expressly distinguishing mathematical results, philosophical interpretation, and research proposals.
Complete revised and expanded edition of the essay published on 10 April 2021. It includes the full development of its arguments, four source illustrations, all 21 revised notes, and the original bibliography with supplementary references. Reformulated passages are integrated into the discussion; the historical Spanish-language document remains available at the end.
Reading guide
Five categories, one articulated process
The table provides an initial orientation. Its definitions are developed and examined in the following sections; they should not be understood as five material compartments or as a universal spatial sequence.
Working distinctions used in this edition
Category
Guiding question
Necessary qualification
Essence
Which fundamental relationships explain the delimited process?
It is not a separate thing or, by definition, a spatial center.
Content
Which elements, processes, and relationships constitute the object under consideration?
It is not reducible to an unorganized sum.
Form
How is that content organized and expressed under particular conditions?
It includes modes of organization, not merely outline or visible aspect.
Phenomenon
How does the process manifest itself, and what can be investigated through that manifestation?
It may require instrumental and theoretical mediation.
Appearance
Which partial presentation or insufficient interpretation requires explanation?
It is not reducible to subjective fantasy and is not always entirely false.
Essential content does not add a sixth thing to the object. It designates a reasoned selection of fundamental determinations within a broader content. Its usefulness depends on the question and on the criteria used to establish that selection, not on a merely verbal decision.
The discussion can be read in three movements: categorial foundations; a proposal for differentiation; and applications and methodological consequences. The contents allow readers to return to a definition, compare cases, or consult a note without leaving the same document.
· · ·
I. Essence
Following the philosophical reasoning of Rosental and Iudin (1971, p. 148), we should begin by stating that both essence and phenomenon are philosophical categories reflecting aspects necessarily inherent in every object of reality. We can define essence as the set of the object’s properties and relationships that are:
The deepest: fundamental relationships that explain the object’s internal structure, identity, and transformations. Depth here is explanatory, not necessarily spatial.
Relatively stable: relationships that maintain a degree of persistence within a timescale and a domain of conditions. Such stability does not mean immobility or perpetual equilibrium. It may consist in the reproduction of an organization, dynamic stability, or the persistence of relationships during a transformation. Conditions that allow an internal equilibrium to be maintained or restored constitute a particular case of this stability, not its universal definition1.
This set, which we shall call the essential set, generally determines the object with respect to the following aspects:
Origin: the conditions of emergence, generative causes, and historical relationships that make the object possible. In ordinary usage, “origin” may mean beginning, motive, birth, source, or root; here its causal and historical meaning must be specified, without thereby introducing a moral cause or a predetermined purpose.
Character: the set of qualities and circumstances through which a thing differs from others in its mode of being or existing2.
Direction of development: the tendencies, possibilities, and constraints of its subsequent evolution. A tendency is not a single destiny and does not exclude external conditions, contingencies, or bifurcations.
Within this interpretive framework, essence can be understood as a mediation between starting points and outcomes: it lies in the object’s process of development. The expression “intermediate place” does not designate the geometrical center of a trajectory, but an articulation of relationships. Nor should the order of exposition of a philosophy be confused with a chronological sequence of material events.
In a materialist reading of the relationship between Hegel and Marx, the decisive difference concerns the foundation attributed to the process: material and social determinations are the starting point of conceptual reconstruction, while the concept is an outcome of knowledge. The opposition cannot simply be reduced to reversing two boxes—“being” and “concept”—since Hegelian logic develops more complex mediations. What is retained from the initial formulation is the question of dominance within a relationship: which conditions explain the genesis, reproduction, and transformation of the totality?
Aufheben, or a sublation that negates and preserves, makes it possible to examine this transformed continuity. In this essay, the dominance of origin is understood as a hypothesis of explanatory priority to be justified in each investigation, not as a law according to which every beginning necessarily dominates every outcome. Marx (2007, pp. 14–18) develops a concrete treatment of this question when analyzing the relationships between production, consumption, distribution, and exchange:
Nothing, then, is simpler for a Hegelian than to identify production and consumption. This occurred not only among socialist essayists, but also among prosaic economists such as Say, who think, for example, that if we consider a people, its production would be its consumption. Or likewise humanity in the abstract. Storch demonstrated Say’s error by pointing out that a people, for instance, does not merely consume its production, but also creates means of production, fixed capital, and so on. (…) Moreover, to consider society as a single subject is to consider it falsely, speculatively. In a subject, production and consumption appear as moments of one act. What matters most here is to emphasize that, whether production and consumption are considered as activities of one subject or of many individuals, in each case they appear as moments of a process in which production is the actual starting point and therefore also the predominant moment.
Consumption as need is itself an internal moment of productive activity. But the latter is the starting point of realization and therefore its predominant factor, the act in which the whole process repeats itself. The individual produces an object and, by consuming it, returns to himself, but as a productive individual who reproduces himself. In this way, consumption appears as a moment of production. In society, by contrast, the relationship between producer and product, once the product is finished, is external, and the object’s return to the subject depends on the subject’s relationships with other individuals. The subject does not take possession of it immediately. Moreover, immediate appropriation of the product is not the subject’s purpose when producing within society.
Between the producer and the products stands distribution, which determines, through social laws, the share of the world of products that falls to the producer, thus intervening between production and consumption (…) Now, does distribution exist as an autonomous sphere alongside production and outside it? (…) Although it appears as a presupposition for the new period of production, it is itself in turn a product of production: not merely of historical production in general, but of historically determinate production.
Essence is thus a “place” in a relational sense: an articulation of relationships among components and with their conditions of existence. A delimited system may have internal and external relationships; if the universe is adopted as a totality by definition, no exterior to that totality is postulated. This conceptual distinction does not itself provide a physical theory of the universe or allow its systems to be counted through an undefined expression such as n − 1.
We call essential or nuclear those relationships whose explanatory function is fundamental to the problem investigated. Their importance cannot be deduced solely from the size of components or from a “relative weight” quantifiable in every case. It must be argued through their causal involvement, the constraints they impose on the system’s behavior, and their ability to explain the object’s production and reproduction. Our present inability to measure that importance does not eliminate the possibility of an objective hierarchy; neither does it authorize us to assert a specific hierarchy without evidence.
Components involved in essential relationships may condition other internal or external relationships. Such dominance is relative to the process, scale, and conditions: a relationship secondary to one question may be decisive for another. A possible connection between Hegelian mediation and Soviet definitions of essence as a set of internal connections is proposed here as a philosophical interpretation; demonstrating a historical genealogy would require further textual study.
Thus, Rosental and Iudin (1971, p. 147) define essence as follows:
Meaning of the given thing, what the thing is in itself, as distinct from all other things and from the variable states it undergoes under the influence of particular circumstances. The concept of “essence” is highly important for every philosophical system and for distinguishing philosophical systems according to their answers to the questions of how essence relates to being and how the essence of things relates to consciousness and thought. For objective idealism, being, reality, and existence depend on the essence of things, understood as something independent, immutable, and absolute. In this case, the essences of things form a peculiar ideal reality that generates and governs all things (Plato, Hegel). For subjective idealist tendencies, essence is a creation of the subject, who projects it outward and represents it in the guise of things.
The only correct standpoint consists in recognizing the reality of things’ objective essence and its reflection in consciousness. Essence does not exist outside things, but within and through them, as their principal general property, as their law. Human knowledge gradually apprehends the essence of the objective world, penetrating it ever more deeply. This knowledge is used to act back upon the objective world with a view to its practical transformation (…)
II. Phenomenon
An analytical introduction to the category of phenomenon must begin by stating that essence and phenomenon constitute a unity: just as there can be no “pure” essences that do not appear, neither are there phenomena devoid of essence. This means that even what appears contains at least some elements of the essential set. A perceptual interpretation can therefore be insufficient without entirely lacking a relationship to the object. Phenomenon should be understood as the set of the object’s properties and relationships that are:
This set represents the way essence manifests or reveals itself.
The definition of essence is further developed in Gómez Julián (2021a, pp. 89–90):
The formulation developed in Lessons in Marxian Gnoseology I (Gómez Julián, 2021a, pp. 89–90) holds that essence should not be imagined as an isolated thing, but as a place where relationships are articulated. Several components and kinds of connection may be involved: each component’s internal relationships, relationships between components, and their relationships with the totality. The unity of essence refers to the integration of the process investigated; it does not require that this process have a single simple cause.
The limiting example of a self-contained totality helps us think about internal relationships without postulating an exterior. The original reference to the Big Bang should be read as a conceptual illustration of that problem, not as a cosmological claim that the universe was a particle situated in a container, that it had an ordinary “external surface,” or that we know all of its initial conditions.
The relationship to mathematics requires distinguishing several structural determinations: topological—such as continuity and connectedness—differentiable, metric—such as distances, angles, and curvature, depending on the framework—and algebraic-operational. A topological space is a set equipped with a topology; it is not, by mathematical definition, the “place of the whole essence.” Organizing a process may additionally require a specified history and dynamics. The unity of essence does not require reducing those relationships to one type of structure.
The comparison highlights organizations not exhausted by visible shape, without identifying topology with essence or geometry with appearance. Both disciplines investigate local and global structures; metric differences may be intrinsic and explanatorily fundamental. The following subsection develops how these dimensions are articulated and how mathematical preservation differs from causal or explanatory relevance.
There is also a problem of establishing hierarchies: distinguishing components may be easier than identifying the relationships that articulate them, and characterizing relationships between components may be easier than reconstructing every internal process within each component. These differences depend on the object, scales, information, and available techniques. In quantum theory, they cannot simply be attributed to imprecise instruments: the formalism’s own constraints must be respected and distinguished from its philosophical interpretations5.
The methodological thesis retained here is that investigating relationships more deeply may require more precise measurements, better models, and new experimental devices. It does not follow that every property can be known simultaneously with arbitrary precision, or that every difficulty in knowledge is merely technological.
Conceptual elaboration · Essence and mathematical structures
One essence, multiple mathematical determinations
The relationship between essence and mathematics is not exhausted by topology. An organization can be investigated through set-theoretic, topological, differentiable, metric, algebraic-operational, and historical-causal determinations. Within this essay’s philosophical framework, these dimensions formalize different aspects of essential relationships; their articulation must be justified with respect to the concrete process. No independent essence is postulated for each discipline, and not every material system is assumed to require representation as a Riemannian manifold.
The manuscript Geometry, History, and Lawfulness: Dialectical Materialist Foundations of Spaces with Metric Structure develops this perspective through a hierarchy of preservations and the question of the historical constitution of symmetries (Gómez Julián, 2026, §§1.7–1.8, 4.2.1, and 8). Its conceptual orientation is incorporated here: investigate which relationships are preserved, which change, and how different determinations of one object are articulated. That orientation must be distinguished from the validity of each theorem, conjecture, or extension proposed in the manuscript.
Structural dimensions and criteria of preservation
Articulable dimensions of mathematical description; neither material compartments nor a universal chain
Dimension
What it characterizes
Relevant preservation or equivalence
Set-theoretic
Cardinality and correspondences between elements, when sets without additional structure are considered.
Bijections preserve cardinality; they do not by themselves guarantee continuity, smoothness, or distances.
Topological
Continuity, neighborhoods, connectedness, compactness, and other properties defined by the topology.
Homeomorphisms preserve topological structure. This preservation does not determine all differentiable or metric information.
Differentiable
Compatible smooth atlases, tangent spaces, and the framework for formulating differential variations.
Diffeomorphisms preserve differentiable structure. The structure makes it possible to formulate dynamics but does not itself select an evolution law.
Metric-geometric
Lengths, distances, and, in Riemannian geometry, angles and intrinsic curvature.
Isometries preserve the metric. Intrinsic properties derived from it are not merely features of an external silhouette.
Algebraic-operational
Composition of transformations and symmetries of specified structures.
Groups, their actions, and their invariants are investigated. Being a Lie group requires the appropriate conditions, not merely the word “symmetry.”
Historical-causal and dynamical
Interaction order, inherited constraints, path dependence, and transition rules.
Which maps preserve those relationships and are compatible with evolution must be specified. Irreversible transitions do not automatically constitute a group of invertible transformations.
The table distinguishes aspects that may depend on one another. On a Riemannian manifold, for example, the metric tensor determines its Levi-Civita connection—the torsion-free connection compatible with the metric—and, through it, curvature. These objects should not be counted as invariably independent data. Nor is differentiable structure obtained simply by renaming topology (Meinrenken, 2002, §§1 and 12–18).
On a fixed smooth manifold M equipped with a Riemannian metric g, transformations preserving additional structure satisfy the following inclusions:
Structure preserved on one object
Isom(M, g) ⊆ Diff(M) ⊆ Homeo(M) ⊆ Bij(M)
The inclusions express additional preservations; they need not be strict for every object. Adding a structure does not mean producing it through a bijection, and a Lie group is not a universal final rung following every metric.
Why a metric difference can be essential
Consider one set, the upper half-plane U = {(x, y) ∈ ℝ² : y > 0}, with its usual topology and differentiable structure. We can equip it with the restricted Euclidean metric and the Poincaré hyperbolic metric:
One topological and differentiable underlying space
gE = dx² + dy²
gH = (dx² + dy²) / y²
With these normalizations, their Gaussian curvatures are KE = 0 and KH = −1.
Both metrics induce the same topology on U, but they are not isometric: they differ in an intrinsic property, curvature. The identity of the topological and differentiable underlying space does not erase differences in distance and geodesic geometry. The example uses the hyperbolic model described by Schwartz (2005) and its curvature formulation presented by Banchoff (§9.6); compatibility of Riemannian distance with manifold topology is established in Meinrenken (2002, Theorem 14.7).
A difference not distinguished by topological classification can therefore be fundamental to a geometric explanation or to a model whose dynamics depend on the metric. It is not merely an altered drawing or a passive coordinate change: distinct metric tensors have been specified on the same space. “Phenomenal from the topological standpoint” can only mean that a distinction does not enter that equivalence; it does not demonstrate that the distinction is inessential to the object or process investigated.
Symmetries, history, and the process’s lawfulness
Symmetries provide a way to investigate what remains under determinate transformations. What is preserved must nevertheless be specified: a topology, a metric, a causal order, an action, or an evolution law. Admissibly transforming a state, preserving geometry, and being a symmetry of dynamics are not interchangeable descriptions. Historical information can impose constraints not contained in topology considered alone.
The author’s manuscript proposes reconstructing operational lawfulness by articulating topology, differentiable structure, history, and metric. This elaboration takes up that question as a research program, without asserting as a universal mathematical consequence that any history and any metric produce a unique physical symmetry group. Noninvertible transitions may call for semigroups or other composition structures; these must be justified for the model rather than treating every historical transformation as a Lie-group element.
Mathematical invariance and explanatory relevance
Two questions must be distinguished: what does a class of transformations preserve? and which relationships are fundamental to explaining this process? The first determines equivalences and invariants relative to specified structures. The second requires justification of mechanisms, dependencies, constraints, and evidence. Even the group consisting only of the identity leaves every property invariant, without that observation itself selecting which properties explain the process.
The unity of essence can thus be understood as an articulation of determinations whose mathematical reconstruction requires several levels when the problem demands them. Selecting essential content may jointly involve topological, differentiable, metric, algebraic, historical, and dynamical relationships. The purpose is not to multiply names for one datum, but to avoid mistaking preservation of one structure for complete knowledge of the object.
III. The dialectical relationship between essence and phenomenon
Essence thus appears as determining, and phenomenon as determined; phenomenon is given immediately, whereas essence is concealed. In terms of features, phenomenon is richer than essence, but essence is deeper than phenomenon. The essence of an object is always one, although it manifests in many phenomena; phenomenon is more changeable than essence. One phenomenon may manifest several essences, even opposing ones; phenomenon may express essence in a distorted or inadequate manner, which is the appearance of objects within the universal system called reality. Explanatory unity concerns the delimited object: that a phenomenon results from several processes does not oblige us to attribute mutually contradictory essences to one thing in the same sense. Partial appearance must be explained through the relationships that produce it.
When manifestation is interpreted partially, insufficiently, or in a distorted way, the investigator may reach mistaken conclusions about the process. The problem is confusing what is immediately observed with a sufficient explanation of reality, not denying reality to the observable. In this edition, “real” designates what actually exists, and “reality” the concrete totality of processes, relationships, and conditions of existence; neither term is restricted to what is directly perceptible. A measurement is real, but its causal significance requires investigation.
Contradiction, however, exists not only between essence and phenomenon, but also within essence itself; these are the object’s fundamental contradictions, which determine its general development6.
(Rosental & Iudin, 1971, p. 148). Essence is mutable.
The contradiction between essence and phenomenon conditions the complex and contradictory character of the process of knowledge: ‘(…) if the form of manifestation and the essence of things directly coincided, all science would be superfluous’ (K. Marx and F. Engels, vol. XXV, part II, p. 384). To penetrate without limit from phenomenon toward essence, discover the essence of things behind external phenomena, and find the grounds for why essence manifests in one way rather than another: such is the aim of knowledge. Through immediate contemplation, human beings become acquainted with things on the surface, with phenomena. Knowledge of essence is achieved through abstract thought.
In science, the transition from knowledge of phenomenon to knowledge of essence takes the specific form of a transition from experiment (observation) to explanation through description.
(Rosental & Iudin, 1971, p. 148).
IV. Form
Following Rosental and Iudin (1971, pp. 190–191), both form and content are philosophical categories that disclose the internal sources of the unity, integrity, and development of material objects. These categories thus reflect aspects necessarily inherent in every object of reality. “Inherent” means something united to a thing by its nature in such a way that it cannot be separated from it (Real Academia Española, 2020).
Form therefore also partially expresses content7, but an analysis based solely on form—even assuming it has been adequately identified—will generally lead, at best, to a significantly incomplete and therefore significantly defective, harmfully one-sided account.
V. Content
Content should be understood as
(…) the set of elements and processes constituting the basis of objects and conditioning their existence, development, and the replacement of their forms. The category of form expresses the internal connection and mode of organization and interaction of the phenomenon’s elements and processes, both with one another and with external conditions. The development of form and content is the development of two aspects of the same phenomenon: the differentiation of a unity, giving rise to contradictions and conflicts that lead to the abandonment of the form and the modification of the content.
(Rosental & Iudin, 1971, p. 191).
In this sense,
The unity of form and content is relative and transitory, and is disrupted by changes, conflicts, and the struggle between them. The source of the [contradictions] between form and content lies in the difference between their functions in development.
(Rosental & Iudin, 1971, p. 191).
VI. The dialectic in the interrelationship between form and content
Following the Soviet authors, content is the basis of development, while form is the thing’s mode of existence. Content has its own movement—it contains the thing’s essence—while form depends on content. Content contains possibilities of development beyond those form may permit; content is the guiding element of development, while form has relative independence and may facilitate or obstruct development. A change of form results from a change in content itself, which conditions content’s guiding role in development. Form never remains invariant. But a change or elimination of form does not always occur abruptly; more often it results from a gradual intensification of contradictions between form and content. Changes of form are also influenced by external conditions, factors, and connections that do not directly belong to content.
Form has a relative independence that grows stronger with the history of a given form. The stability of form is a factor securing the progressive development of content. Yet that same stability, which stimulates development in early stages, becomes a source of conservatism over time. Contradictions between form and content are not contradictions between a passive and an active side. The real process unfolds through their reciprocal action as opposites actively influencing the development of the thing or object. The mismatch between form and content caused by form lagging behind content is highly important for development, but characterizes only one of its contradictions.
The resolution of contradictions between form and content depends on their character, degree of development, and the conditions under which they arise. Resolution may occur by changing form in accordance with changes in content, changing content in accordance with a new form, rejecting form, subordinating an old form to new content, and so on. In passing from one qualitative state to another, the old form is either eliminated or transformed, with the qualification that it cannot be eliminated before the premises and elements of a more developed form have been prepared within it.
This is the dialectical process of Aufheben8, in which the old form is rarely discarded completely and absolutely. The new form does not always become dominant at once, but begins to prevail gradually. Old forms support development less effectively than new ones, and for this reason the new form comes to occupy an increasingly important place. This character of the old form’s Aufheben also makes regressive development—the restoration of old forms—possible. The dialectic of content and form is strikingly manifested in society’s constant renewal and development.
As Frolov (1984, pp. 85–86) explains, form and content are categories that
(…) reflect the interconnection of two aspects of natural and social reality: the elements and processes composing the object or phenomenon, ordered in a particular way—that is, content—and the mode of existence and expression of this content and its various modifications—that is, form. The concept of form is also used in the sense of the internal organization of content; the problem of form is thereby further developed in the category of structure. In pre-Marxist philosophy, especially idealist philosophy, form was reduced to structure, while content was identified with a certain unordered collection of elements and properties—with “matter.” This helped establish and preserve, over a prolonged historical period, idealist representations of the primacy of form over content. In materialist dialectics, structure and internal ordering are considered necessary components of content.
As structure and organization are modified, the object’s content and its physical and chemical properties change substantially. In registering the multiple modifications of content and its modes of existence and manifestation, form also possesses a structure. The interaction of content and form necessarily includes, within development, both the action of the various components of content upon form and the action of form upon content, taking account of their objective relationship of subordination. In this interaction, content represents the object’s guiding, determining side, while form changes depending on changes in content and the concrete conditions of its existence. In turn, form, possessing relative autonomy, actively acts back upon content: a form corresponding to content accelerates its development, whereas a form that no longer corresponds to changed content impedes its development.
The interrelationship of content and form is a typical example of the interconnection of dialectical opposites, characterized both by their unity and by the contradictions and conflicts between them. The unity of content and form is relative and temporary; mutual conflicts and contradictions inevitably arise in the course of development. As a result, a mismatch between content and form emerges, ultimately resolved by the “shedding” of the old form and the emergence of a new form appropriate to the changed content. The emergence, development, and overcoming of contradictions between content and form, and their struggle—mutual transitions between content and form, the “filling” of an old form with new content, form acting back upon content, and so on—constitute an important component of the dialectical theory of development. The relationship of content and form is particularly complex and contradictory, in the dialectical sense, in the development of society.
Contradictions between content and form in the spheres of the mode of production, base and superstructure, and economy and politics constitute a highly important driving force of social development.
At the level of thought, Frolov adds:
In thought, content and form are fundamental concepts of the science of logic. In materialist dialectics as logic—dialectical logic—the content of thought means natural and social reality insofar as it appears before us as the object of theoretical scientific thought: as a set of concepts, representations, theories, various abstractions, and idealizations established in the historical development of knowledge. Forms of thought are different ways in which the social subject engages with the object on the ideal plane, directed toward reproducing the regularities and properties of objective reality in the content of thought: categories, ascent from the abstract to the concrete, various forms of inference, and so on. The set of forms of thought organizes cognitive content in a particular way and guides the subject’s thinking in the processes of acquiring new knowledge.
Internal organization, expression, and structure
Form here has a broader meaning than silhouette: it includes organization and modes of expression. Depending on the question, an internal structure may be analyzed as the form of organization of a content and, simultaneously, as part of the content of a broader totality. This analytical relativity does not mean arbitrariness: the object delimited and the relationships investigated must be specified.
Nor should form–content interaction be reduced to an active part acting upon a purely passive one. The explanatory priority assigned to content is compatible with feedback effects, organizational constraints, and mutually conditioning changes. An institution, a structure of circulation, or a technical device may facilitate or block material possibilities without thereby becoming a self-sufficient cause.
VII. Connections and differences between the categorial pairs
The Soviet sources examined reveal analogies and connections between essence–phenomenon and content–form. This essay’s hypothesis is that these connections make their methodological articulation possible; it does not assume that the entire Marxist tradition has asserted a strict identity between the two pairs. Nor are these categories inhabitants of two unrelated worlds.
To examine the problem, we turn to Rosental’s The Problems of Dialectics in Marx’s Capital (1961). The following quotations retain passages on relationships between concepts, degrees of knowledge, value, price, and the development of forms. What these sources state must be distinguished from the inferences proposed here on their basis.
Therefore, concepts reflecting concrete reality must be connected with one another and transform into one another; otherwise they would not faithfully reflect reality in perpetual development. All this explains why Lenin considered the essential content of logic to lie in the relationships between concepts, their transformations, and their connection.
(Rosental, The Problems of Dialectics in Marx’s Capital, 1961, p. 414).
As Capital demonstrates, the categories of phenomenon and essence express degrees and the deepening of knowledge, which moves from external, superficial aspects toward internal, essential ones. This movement from phenomenon toward deeper essence gives the process of knowledge its dialectical character.
(Rosental, The Problems of Dialectics in Marx’s Capital, 1961, pp. 279–280). Here essence and phenomenon are understood as theoretical descriptions of the development of knowledge.
We have just examined the logical relationship between the categories of chance and necessity. But these categories, which register particular aspects of the whole, are closely connected through relationships and transformations with categories such as essence and phenomenon, the singular—or particular—and the general, content and form, cause and effect, interaction, and so on. For example, the relationship between value and price is intimately reflected in the mutual connection not only of necessity and chance, but also of essence and phenomenon, the general and the singular, and content and form. The problem of essence and phenomenon is closely connected with the abstract and the concrete. Linking essence with phenomenon, law with its concrete forms of expression, involves an advance from the abstract to the concrete.
The intermediate links on which Marx insists so strongly are discovered precisely in the course of this advance, which will be discussed later.
(Rosental, The Problems of Dialectics in Marx’s Capital, 1961, p. 446). This passage makes explicit transformations between pairs of categories, including the two pairs investigated in this essay.
For example, if the category of content is directly connected with the category of form in dialectical logic, this is not arbitrary: in real phenomena and processes themselves, the aspects reflected in the concepts of content and form do not exist without one another and interpenetrate while also opposing one another.
(Rosental, The Problems of Dialectics in Marx’s Capital, 1961, p. 440). This explicitly establishes the direct connection between content and form and its gnoseological foundation. In this sense, it seems to imply that the categories “content” and “form” are used to study the historical trajectory of reality’s phenomena and processes; however, this still does not settle the question of the analytical sphere to which essence–phenomenon belongs.
Value—content—and exchange-value—form—constitute a unity. Commodity exchange encounters the limits of an outdated form of value and requires the emergence of a new, higher, and more complete form, generating a contradiction between content and form.
(Rosental, The Problems of Dialectics in Marx’s Capital, 1961, p. 442). The same implication noted in relation to page 440 is visible here.
If Kant is to be believed, phenomenon and essence, the singular and the universal, constitute two unconnected worlds, entirely separated from one another. For idealist “empiricists”—Berkeley, Hume, and others—sensations constitute the only reality, and nothing else exists outside them. For all idealists, in a word, the problem of phenomenon and essence, the singular and the universal, the sensible and the abstract, was insoluble.
(Rosental, The Problems of Dialectics in Marx’s Capital, 1961, p. 253). By referring to Kant and to the categories “the singular” and “the universal,” and describing a gnoseological problem the idealists cited by Rosental could not resolve, the passage makes it plausible to regard essence–phenomenon as categories used to study the historical trajectory of phenomena and processes through the historical evolution of thought.
Knowing the price of a commodity requires no analysis, but knowing value as the basis of price requires scientific analysis. Concerning forms of expression and their basis, Marx writes: “The former reproduce themselves directly and spontaneously as mental forms developing on their own; the latter must be discovered by science.” Phenomenon and essence therefore indicate degrees of deepening knowledge. Marx writes that discovering, for example, the concealed basis or secret of wages required a long historical period, but that once this basis is known, nothing is easier than understanding the necessity of that form of expression.
(Rosental, The Problems of Dialectics in Marx’s Capital, 1961, pp. 264–265). This reflection appears to support the implications inferred from page 253. The same author, a compiler of the 1965 Philosophical Dictionary as well as earlier and later editions9, states the following about the philosophical category of phenomenon:
(From the Greek φαινόμενον: representation, phenomenon.) A concept designating what is given in experience and known through the senses. In Kant’s philosophy, phenomenon is distinguished in principle from noumenon, which lies beyond the limits of experience and is inaccessible to human contemplation. Through the concept of phenomenon, Kant sought radically to separate essence and appearance, considering the former unknowable (agnosticism). From the standpoint of dialectical materialism, there is no sharp boundary between phenomenon and essence: essence comes to be known through phenomenon. (Phenomenalism, phenomenology.)
(Rosental & Iudin, Philosophical Dictionary, 1971, p. 171). This reference also appears to confirm the interpretation of essence–phenomenon reached through Rosental (The Problems of Dialectics in Marx’s Capital, 1961, p. 253).
The preceding reading distinguishes two analytical emphases without turning them into ontological compartments:
1. Content–form makes it possible to investigate the constitution, organization, and historical trajectory of real processes. It also allows the content and forms of thought to be studied, as Frolov’s exposition shows.
2. Essence–phenomenon makes it possible to investigate the objective relationship between determinations and manifestations and reconstruct the advance of knowledge toward deeper explanatory relationships. Its gnoseological function does not eliminate the objectivity of what is known.
The object’s real history and the history of knowledge of it are connected, but do not automatically coincide. Logical reconstruction can grasp relationships already operating before they were known and can revise inherited categories. Correspondence between the two pairs is therefore partial, argued, and problem-dependent, rather than an equivalence between interchangeable words.
VIII. Differentiation and generalization of the categories
VIII.I. Correspondence, complementarity, and essential content
In this essay, “Bohr–Marx correspondence principle” designates a materialist appropriation of the correspondence principle and its interpretation in Soviet philosophy. It is not the conventional name of a physical principle jointly formulated by Bohr and Marx. “Bohr–Marx complementarity” likewise names the author’s proposed philosophical generalization, whose scope must be distinguished from Bohr’s quantum complementarity.
The Soviet source formulates the correspondence principle as follows:
It is one of the fundamental methodological principles of scientific development. Its philosophical significance consists in expressing the movement of knowledge from relative truths toward an ever fuller absolute truth. It was first formulated during the period of the complete collapse of the concepts of classical physics (Bohr, 1913). According to the correspondence principle, whenever scientific conceptions undergo a rupture, the fundamental laws of the new theory created through that rupture possess characteristics such that, in a limiting case, assigning an appropriate value to a characteristic parameter in the new theory transforms them into the laws of the old theory. For example, the laws of quantum mechanics become those of classical mechanics when the magnitude of the quantum of action can be neglected. The correspondence principle can be traced in the history of mathematics, physics, and other sciences.
It reflects the law-governed connection between old and new theories, arising from the internal unity of qualitatively different levels of matter. Such unity conditions not only the integrity of science and its history, but also the immense heuristic role of the correspondence principle in penetrating a qualitatively new sphere of phenomena. A scientific interpretation of this principle discloses the dialectic of cognition and demonstrates the inconsistency of relativism.
(Rosental & Iudin, Philosophical Dictionary, 1971, p. 375).
The relationship between continuity and transformation in theories offers a methodological orientation: a generalization must clarify what it preserves, what it modifies, and under which conditions it recovers an earlier formulation. Adding names is insufficient. Nor does correspondence between scientific theories guarantee that any philosophical classification is correct; it poses a question to be resolved through argument and concrete applications.
Connections between essence–phenomenon and content–form thus raise the problem of their differences: what distinguishes each pair, what they share, and which mediations can be constructed without dissolving them into verbal identity.
The distinction between an objective process and its subjective reconstruction is retained here, while the reversal of labels in the initial formulation is corrected: both pairs may refer to objective relationships, and both participate in knowledge. Thought is not an immobile or necessarily “ossified” copy of the real; it is a historical, social, practical activity that abstracts, organizes, tests, and transforms its representations. The object’s material priority does not make that activity superfluous.
The effort to systematize dialectical logic—developed in different stages of Soviet philosophy and in the work of Elí de Gortari, among others—requires specifying how natural and social phenomena are described. A rigorous history of these programs cannot be reduced to 1917–1989 or treat them as a school without internal differences. That context nevertheless helps situate this essay’s ambition: to articulate philosophical categories and scientific practices without replacing each discipline’s specific work.
In meteorological systems with strange attractors, for example, simply calling a set of processes “content” and their organization “form” may be too general for some research questions. Variables, scales, connections, boundary conditions, and mechanisms must be identified. It does not follow that chaotic systems cannot be formalized. Lorenz’s work (1963) precisely demonstrates the investigation of nonperiodic behavior through deterministic nonlinear differential equations.
Formalizing a system, solving its equations, explaining its mechanisms, and accurately predicting its long-term trajectory must be distinguished. Sensitivity to initial conditions limits certain forecasts even when a model is mathematically defined. The relevant criticism of an excessively dichotomous description concerns its possible explanatory insufficiency for a concrete problem, not a supposed universal inability of formal logic, mathematics, or computers to study motion.
Expanding analytical layers may help construct material subsequently integrated through synthesis, in the methodological sense discussed by Marx and Rosental and developed in Lessons in Marxian Gnoseology I. The aim is to adapt questions and distinctions to the object by considering different angles of a natural or social investigation.
Not every generalization automatically offers advantages. Its fruitfulness may not be immediate, but it must be examinable: which distinctions does it introduce, which relationships does it help reconstruct, and which problems does it solve better than the previous classification? Expanding vocabulary is only one possible means for this work.
The proposal introduces an explicitly constructed category: essential content. It is “artificial” only in the sense of being a deliberate conceptual construction; its legitimacy depends on distinguishing relevant real relationships, not on having invented a word. It is intended to contribute to the systematization of dialectical materialist philosophy and communication between disciplines, without imposing a single classification that ignores their objects and methods.
Within content–form, essential content designates the articulated set of determinations of content fundamental to explaining the delimited process. It does not indiscriminately encompass everything contained in the system or identify the essential with what is most numerous, heaviest, or spatially innermost.
Essential content can therefore be mathematically articulated across several dimensions. The relevance of a topological relationship does not exclude a metric relationship, causal constraint, or evolution law from being equally fundamental to the process. All these levels need not be imposed on every object: which are involved and how they condition one another must be justified. The distinction developed in One essence, multiple mathematical determinations prevents essential content from being identified with an isolated topology.
Within essence–phenomenon, that same set may be investigated as the process’s essence: the questions concern how its determinations manifest themselves, through which mediations, and by what evidence they can be known. This duality is functional: one articulation of relationships is examined through different questions. A shift of perspective does not make every use of “content” synonymous with “essence.”
At the manifestation end of the relationships, appearance allows partial, superficial, or distorted presentations to be discussed, but should not be treated as automatically equivalent to form or phenomenon. The author’s studies of the negative binomial distribution and of eigenspaces, eigenvectors, and eigenvalues provide the application precedents mentioned in the document: Gómez Julián (2020; 2021b), originally published under the pseudonym Isadore Nabi. The proposal is made explicit here so readers can understand it without consulting those works.
Making the proposal operational
Using “essential content” requires specifying the process to be explained; its temporal and spatial scales; the content initially considered; the relationships proposed as fundamental; and the evidence that would allow this selection to be revised. The term must not function as a certificate of depth awarded by an investigator to their own explanation.
From general classification to a research question
Operation
Question
Expected outcome
Delimit
Which process, under which conditions?
An explicit object of research.
Describe content
Which components, processes, and relationships are involved?
An organized inventory open to revision.
Identify mediations
How are mechanisms and observables related?
A testable causal or explanatory reconstruction.
Propose essential content
Which relationships are fundamental to this explanation?
A reasoned hierarchy, not a label.
Test
Which results support or challenge the selection?
Limited confirmation, modification, or abandonment of the proposal.
Additional usefulness would be assessed by comparing explanations: does it distinguish previously conflated mechanisms, make omitted conditions explicit, generate new hypotheses, or avoid invalid inferences? If it merely renames an existing distinction, its contribution would be terminological. If it improves reconstruction of the process and guides testing, its scope would be methodological.
The essay calls “Bohr–Marx complementarity” the articulation of perspectives that makes it possible to study one totality through the material history of its relationships and through the logical reconstruction of those relationships. A single category may acquire differentiated functions according to the question without artificially duplicating objects.
This proposal retains the interest in connecting historical and logical trajectories, as well as essence–phenomenon and content–form. Its relationship to Bohr is analogical and methodological: it does not demonstrate formal equivalence with quantum-observable complementarity or attribute to Marx a theory he did not formulate. Soviet contributions provide philosophical precedents for the interpretation of correspondence; the extension proposed here must stand on its own coherence and usefulness.
One final theoretical difficulty nevertheless remains. The classical definition of appearance describes it as the
External manifestation, directly given to the senses, of the essence of things or, more precisely, of some part of essence. In this sense, appearance is equivalent to phenomenon. A subjective moment is present: phenomenon expresses the subject’s essence inadequately, in a distorted way—a partially submerged object appears broken, the Sun appears to revolve around Earth, and so on. Nevertheless, reducing appearance solely to this subjective moment would be mistaken, since it is connected in one way or another with objective essence, of which it is a manifestation. The subjective element giving rise to an erroneous representation of a phenomenon’s essence is itself often conditioned by objective factors. Knowledge must trace appearance back to its essence and explain how the latter manifests in the former. (Essence and phenomenon.)
(Rosental & Iudin, Philosophical Dictionary, 1971, pp. 20–21). This seems to contradict the preceding discussion flagrantly—or at least apparently.
In complex systems and latent-variable models—for example, hierarchical probabilistic models—access to relationships not directly observed requires mediations: indicators, measurement assumptions, estimation procedures, and tests. Quantum theories and hidden-variable proposals associated with de Broglie and Bohm likewise raise questions about relationships between formalism, observables, and interpretation.
A latent variable in a statistical model is not thereby a physical entity or a demonstrated essence. Likewise, a theory’s probabilistic character does not automatically decide its ontological interpretation. Classical statistical mechanics provides an important precedent for investigating mediation between microscopic and macroscopic descriptions, without thereby reducing quantum probability to classical ignorance.
The methodological conclusion is that mediation may increase and the categories used should make that increase explicit. Moving from data to explanatory structures depends not only on improving instruments, but also on revising identification assumptions, inferences, and conditions of validity.
In simple contexts, a relatively undifferentiated distinction between immediate presentation and explanation of the process may suffice. In more complex contexts, phenomenon and appearance should be distinguished: phenomenon is the objective, investigable manifestation; appearance is its partial presentation or an insufficient interpretation whose genesis requires explanation. This differentiation may also be useful in classical problems and does not depend exclusively on the involvement of probabilities.
The reference to Born and Schrödinger’s wave function identifies a domain in which the relationship between observable outcomes and theoretical description is particularly important. It does not authorize identifying quantum probability theory with a simple extension of classical statistics. The proposed philosophical generalization retains the requirement of correspondence: more detailed distinctions should allow less detailed classifications to be recovered when the question and conditions permit.
Categories are understood here as conceptual instruments for reconstructing articulations of relationships. The more complex the relevant relationships, the greater the possible need to differentiate them. This need is demonstrated through explanatory function, not by invoking complexity as a slogan.
Essence, content, and form allow different abstractions of the object and its modes of manifestation. They do not necessarily constitute successive spatial strata or a universal ladder always to be climbed in the same order. Research can advance, revisit its distinctions, and modify its initial delimitation.
In the social sciences, reference to Durkheim’s social facts reminds us that phenomena also have objective social conditions and relationships. “Social fact,” however, has a specific theoretical meaning in Durkheim; it is not simply synonymous with every social phenomenon. The essay’s materialist thesis is that manifestations and knowledge of them have material and historical conditions, even when far-reaching abstractions are required.
Just as the first generalization was justified through the two studies mentioned—one abstract and another concrete–abstract, in the sense developed by Enrique Dussel as the concrete-in-thought and discussed in Gómez Julián’s Lessons in Marxian Gnoseology I (2021a)—the second generalization, distinguishing “phenomenon” and “appearance,” will now be examined.
VIII.II. Earth’s magnetic field and the climate system
Consider the preceding discussion in relation to Kilifarska, Bakhmutov, and Melnyk’s study (2020) of connections between Earth’s magnetic field and climate, itself evidently a complex analysis.
To undertake this task, the concept of climate must first be introduced10. Kilifarska, Bakhmutov, and Melnyk (2020, p. 43) characterize it through prevailing meteorological conditions over prolonged periods—temperature, wind direction and speed, precipitation, and cloudiness—and their relationship with the hydrosphere, cryosphere, land surface, its absorption and emission properties, and the biosphere.
Broadly, climate is described through statistical properties of atmospheric conditions and the climate system, including means, variability, distributions, and extremes, for a time interval and a location or region. The customary reference to thirty years concerns the conventional period of World Meteorological Organization climatological normals; it does not mean that climate exists only after that interval or that thirty observations always constitute a large or sufficient sample.
The quantity and resolution of data, their autocorrelation and quality, and the research question condition the information available. Thirty years of daily observations are not equivalent to thirty independent observations. Distinguishing record duration, number of data points, and effective sample size is indispensable to prevent a climatological convention from becoming an incorrect statistical rule (WMO, climatological normals).
The components of the climate system interact continuously. Solar radiation is its principal external energy source; the energy balance also depends on how energy is absorbed, reflected, emitted, and redistributed. Atmospheric composition, aerosols, clouds, oceans, ice, and the surface participate in this balance.
Kilifarska, Bakhmutov, and Melnyk (2020, p. 44) refer to Herschel (1801) when reviewing historical precedents for hypotheses about solar variations and climate. That historical reference does not itself establish a causal attribution of contemporary climate change. Its relevance here is to show that climate explanations distinguish mechanisms, timescales, and mediated relationships.
The reference work organizes its discussion into external and internal variations. The former include changes in solar luminosity and magnetic activity, orbital parameters, cosmic rays, and lithospheric activity; the latter include oceans, albedo, sea ice, atmospheric composition, climate modes, and living systems. The following figure preserves the chapter outline presenting that organization.
In Kilifarska, Bakhmutov, and Melnyk’s account (2020, p. 44), possible external influences include particles of solar, galactic, or extragalactic origin; volcanism, tectonics, continental movement, and changes in relief; and geomagnetic and orbital changes. Ocean circulation, albedo, atmospheric composition, and the activity of living beings, including humans, are examined within the system’s processes. The full set of mechanisms discussed is thus retained without treating the book’s classification as the only possible taxonomy.
This expository organization must be distinguished from IPCC terminology. In IPCC glossaries, solar variations, volcanic eruptions, and anthropogenic changes in greenhouse gases or land use may act as external forcings of the climate system. “External” depends on the delimitation of the system and does not necessarily mean “extraterrestrial.” Nor is every influence a radiative forcing: forcing factors, response processes, feedbacks, and internal variability must be distinguished (IPCC, glossary).
Atmospheric composition illustrates the point: an imposed change in carbon dioxide can be a forcing, while a water-vapor response to a temperature change can function as a feedback. Classification requires specifying the process and its conditions, not merely naming the material component.
Figure 1. Organization of chapter 3 in Kilifarska, Bakhmutov, and Melnyk (2020, p. 43). The source image is retained; its classification is discussed in the text.Enlarge figure and read its interpretation
Reading the outline: 3.1, external forcings and mechanisms of influence (p. 44): solar luminosity (44), solar magnetic field (48), orbital parameters and Milankovitch cycles (50), galactic cosmic rays (51), lithospheric activity (53). 3.2, internal variations around an average climate state (58): oceans (59), planetary albedo (60), sea ice (60), atmospheric composition (61), climate modes (61), living systems (62). 3.3, regional character of variations and the challenge to hypotheses (63). References (64).
Consider the role of lithospheric activity11, particularly the movement of tectonic plates. The example is useful because it draws on familiar spatial representations without assuming that every explanatory relationship is geometrical. The following figures show tectonic processes and a comparison of temperature and geomagnetic-field variations, respectively.
Methodologically, these materials require different reconstructions: a tectonic diagram represents mechanisms and spatial relationships, whereas a time series represents measured or reconstructed variations. An image, a classification, and a correlation are not the same kind of evidence.
Figure 2. Tectonic processes reproduced by Kilifarska, Bakhmutov, and Melnyk (2020, p. 56, fig. 3.7). The image caption refers to Kious and Tilling, This Dynamic Earth, USGS, and an illustration by José F. Vigil.Enlarge figure and read its interpretation
The diagram connects convergent, transform, and divergent boundaries, seafloor spreading, subduction, volcanic arcs, mountain belts, ocean ridges, continental rifts, and mantle regions. Arrows represent relative movements. Its function in the essay is to illustrate spatial relationships and mechanisms; the image does not demonstrate a particular climatic influence.
Figure 3. Comparison of temperature and geomagnetic field in Kilifarska, Bakhmutov, and Melnyk (2020, p. 58, fig. 3.8), reproduced on p. 25 of the essay. The curves retain the source’s colors and legends.Enlarge figure and read its interpretation
The image presents decadal averages in latitude bands from 40° to 70° in both hemispheres over an approximately century-long interval. Magnetic curves use the intensity axis; thermal curves use the temperature axis. Legends distinguish magnetic reconstructions and temperature series. Visual coincidence of changes does not itself establish causality; inference requires examining data, scales, mechanisms, and alternative explanations.
The second of these two figures compares variations in winter surface air temperature and geomagnetic intensity over a century-long interval. Temperature curves appear in red and gray tones and magnetic curves in black. The source graph presents decadal averages for latitude bands in both hemispheres, making attention to its scales, data, and averaging procedures necessary.
These materials constitute only part of the book’s analysis. Their complexity does not exhaust possible explanations and must be assessed by comparing hypotheses, measurements, and results from other investigations. The object and the means of investigation simultaneously condition what can be known, although physical complexity must not be confused with limitations of the measurement procedure.
The coexistence or covariation of two curves does not demonstrate that one variable causes the other. An explanation requires an identifiable mechanism, compatible magnitudes, control of alternative factors, and assessment of uncertainties. The geomagnetic discussion is retained here as a case of mediation and comparison between hypotheses, not as an established explanation of contemporary warming. Specifically, the IPCC (2021, chapter 7, §7.3.4.5) assesses the contribution of galactic cosmic rays to effective radiative forcing between 1750 and 2019 as negligible. That assessment concerns that mechanism and period; it does not eliminate every conceivable geophysical question.
The philosophical interest of the case allows three questions to be posed about our categories’ ability to guide analysis.
The first question is whether simply naming “content” and “form” suffices to represent and explain the interaction of processes shown in both figures. The second is whether those two general designations suffice to reconstruct even one figure in detail. The third concerns the integrated analysis of the book’s chapter summarized in the reproduced outline: how would its variables, mechanisms, and scales be integrated?
The proposed answer is that these two designations alone do not provide all the necessary specification. This does not demonstrate that dialectical materialism possesses only two categories or that its classical categories are useless. It indicates the need for mediations and operational distinctions when seeking to guide a concrete investigation, as in the Marxian study of political economy.
The coexistence of many variables raises problems of classification, estimation, prediction, causal identification, and probabilistic modeling. The involvement of several sciences adds compatibility problems between concepts, instruments, and scales. Criticizing a theory’s philosophical presuppositions is not equivalent to automatically rejecting its empirical content because of its categories’ intellectual origin. Critical appropriation requires showing what is preserved, what is transformed, and which results support the transformation.
Now consider Earth’s layers alone, as shown in the following illustration.
Figure 4. Earth’s layers. Provenance recorded in the document: Un Blog de Dibujos / Noches en vela. The historical diagram and its original Spanish labels are preserved in full; an English reading key is provided below.Enlarge figure and read the English key
English key to the Spanish labels: corteza = crust; astenosfera = asthenosphere; manto externo / interno = outer / inner mantle as named in this historical diagram; núcleo externo / interno = outer / inner core; corteza continental / oceánica = continental / oceanic crust. The diagram also names the Conrad, Mohorovičić, Gutenberg, and Lehmann discontinuities, and one spelled “Dahn.” The drawing’s terminology is preserved as documentation of the original figure; it does not replace a current geological classification. The text and notes distinguish compositional and mechanical criteria and avoid automatically identifying physical layers with philosophical categories.
The illustration of Earth’s layers distinguishes three problems. First, explaining a process may require relationships among several components: in the geodynamo, the liquid outer core, solid inner core, and conditions of exchange with the mantle participate in different ways12. The “unity” of essence refers to an explanatory articulation, not to a requirement that every process have exactly two components or that the geographical core automatically be a philosophical essence.
Second, component selection depends on the problem. In a model restricted to certain core dynamics, some of the exterior may be treated through boundary conditions or approximations. It cannot be concluded that the crust or mantle is universally irrelevant to internal geodynamics. The mantle regulates thermal exchanges important to core evolution; different questions require different delimitations.
Third, a component’s observability does not determine its explanatory importance. In political economy, price is a manifestation to be explained and also a relevant datum for testing theoretical reconstructions; it is not discarded because it is immediate. The comparison with Earth poses a methodological question: which manifestations and conditions must be included to explain the delimited process? The answer requires arguments from each discipline.
The original text asks whether, in some analyses, selected content may coincide with essence, whereas in others a broader content must be distinguished from its essential articulation. Essential content makes this difference explicit. It does not authorize identifying one terrestrial layer with “form,” another with “content,” and another with “essence” without explaining their functions.
The reference to the commodity also requires precision: use-value, value, and exchange-value are not interchangeable names. Use-value concerns the commodity’s material usefulness; value refers, in Marx’s theory, to socially necessary abstract labor, while exchange-value expresses a relationship through which value manifests. Utility is not a substance that produces value when added to exchange-value. The question of hierarchy and mediations between these determinations is retained, while avoiding the assignment of universal dominance through analogy with geological layers.
The usefulness of distinguishing essence, content, form, and appearance must be assessed through the relationships it helps reconstruct and the questions it helps resolve. The image suggests a problem; it does not itself prove the philosophical classification.
The preceding generalizations are proposed as means for formulating more precise models and experimental practices: they could subdivide manifestations, distinguish relationships, and make mediations explicit according to each investigation’s theoretical and empirical needs. This is the usefulness sought, not an outcome demonstrated merely by introducing categories.
The development connects with Lessons in Marxian Gnoseology I and the aspiration to articulate its foundations with particular studies. Assessment would require applying the proposal, comparing explanations, and determining whether the distinction actually improves the design, testing, and interpretation of models. Subdivisions should respond to the object and question rather than multiplying without criteria.
Topology and geometry offer another instructive comparison13. Their vocabularies are not limited to open and closed sets: they include neighborhoods, boundary, connectedness, disjoint sets, and—in structures with a suitable metric or uniformity—boundedness. A set can be simultaneously open and closed, or neither; classification depends on the space and topology under consideration.
The richness of these categories distinguishes properties that a binary description does not develop by itself. However, topology should not be identified with “interior” or metric structure with “exterior”: these are types of mathematical structure. Mathematics also models transformations, flows, temporal evolution, and relationships between spaces; its definitions do not condemn it to describing only static reality.
The lesson drawn here for Marxist research is the value of producing precise, articulated distinctions according to the problem. The scope of formalization can be discussed philosophically without denying its ability to study dynamic processes. An additional category is justified if it clarifies previously conflated relationships; the number of terms alone does not ensure deeper knowledge.
This richness is not merely a multiplication of names: it distinguishes dimensions of essential relationships. Topological equivalence does not itself decide differentiable, metric, or dynamical equivalence. As the example of two metrics on one half-plane shows, what one classification does not distinguish may be fundamental to another explanation. “Metric” therefore does not mean “merely apparent,” and “topological” does not designate the whole of essence.
The question can be transferred to political economy: if value is investigated as a fundamental determination of commodities, how are the price of production and market price—called “commercial” price in the initial formulation—reconstructed together? What additional relationships would need to be considered for interest or rent?
It is not assumed that there are only two types of price or that all these determinations can be ordered along a single scale. The proposal is to use the developed distinctions to clarify groupings, levels, and mediations, alongside the historical and gnoseological conditions discussed in Lessons in Marxian Gnoseology I. Their explanatory ability must be demonstrated through a concrete economic analysis.
Value, prices, and levels of mediation
The question of the price of production, market price—called commercial price in the preceding passage—interest, and rent shows what the proposed distinction should contribute. These determinations are not simply placed in concentric circles around value. They require reconstructing competition, profit-rate equalization, conditions of realization, credit, property, and surplus distribution, depending on the object investigated.
A broader content of economic relationships can be considered, and within it we can ask which determinations are essential to explaining a specific outcome. The apparent autonomy of an income stream may then be analyzed as an objective social form of manifestation, without denying its real effects. The task is to reconstruct historically determinate mediations, not to declare illusory every phenomenon that fails to coincide immediately with the theory of value.
This elaboration preserves the essay’s interdisciplinary ambition: to learn from other sciences’ problems of organization, scale, and observability. An analogy’s fruitfulness is judged by the questions it helps clarify; the validity of each explanation is judged within the corresponding field.
VIII.III. Second case study: dynamo theory
Reference frames and inertial forces
What are fictitious forces? As will become clear, this concept is highly important for understanding natural phenomena from a dialectical materialist perspective. Consider the example discussed in Physics Stack Exchange (2012), for which several preliminary concepts must first be introduced briefly.
Following the gnoseological descriptions discussed in Gómez Julián’s Lessons in Marxian Gnoseology I (2021a, pp. 184–185), including Marx’s construction of simple and expanded reproduction schemes, the relevant preliminary distinction must be established: because the categories below concern non-inertial reference frames, inertial reference frames must first be introduced.
An inertial reference frame is one in which a body subject to zero net force remains at rest or moves at constant velocity in a straight line. This is the content of Newton’s first law. The second law relates net force, mass, and acceleration within the domain of Newtonian mechanics.
A frame translating at constant velocity relative to an inertial frame is also inertial. A frame accelerating or rotating relative to it is non-inertial; inertial terms are incorporated to write equations in that frame in a Newtonian-like form. Constant acceleration does not make a frame inertial either (OpenStax, University Physics, §5.2).
The document relates this distinction to schemes of simple and expanded reproduction of capital. The comparison may be retained as an invitation to examine conditions of reproduction and transformation, but not as a scientific equivalence: a change of physical reference frame is not the same process as an expansion in the scale of accumulation, and an economy with simple reproduction is not an economy without dynamics. Each case requires its own variables, laws, and conditions.
The passage cited from Einstein (2005, p. 66), paraphrased here from the document’s transcription, states the special principle of relativity:
If a coordinate system K is chosen in which physical laws have their simplest form, those laws are also valid in another system K′ translating uniformly relative to K. The qualification “special” restricts this equivalence to uniform translational motion; it does not simply extend it to non-uniform motion of K′ relative to K.
A paraphrase of the cited passage, not a verbatim reproduction of the 2005 translation. The principle is also satisfied by Galilean and Newtonian mechanics within the domain of the comparison.
With these concepts clarified, we can turn to the discussion of fictitious forces in Physics Stack Exchange (2012).
Describing a system from an accelerated—and therefore non-inertial—frame introduces terms called inertial or fictitious forces. “Fictitious” here means that the term does not represent an additional interaction between bodies of the same kind as a contact or gravitational force in the inertial description. Its frame dependence does not make the relative motion it helps describe imaginary.
Consider a free ball with negligible net force in an idealized region. It is insufficient to say that space is empty: a vacuum may contain fields, and a ball may interact with them. The relevant assumption is the approximate absence of net force on the ball in the chosen inertial frame.
If the observer accelerates with acceleration a relative to that frame and the observer’s axes do not rotate, the measured relative acceleration of the ball is −a. In the observer’s frame, the form of the equation of motion can be preserved by adding F_inertial = −m a. The ball has acquired no new unknown interaction: the frame from which its motion is described has changed.
The correct inference therefore distinguishes two statements: there is relative acceleration with respect to the accelerated frame, but no additional physical force on the ball need be postulated in the inertial description. An objective result requires specifying the frame and the transformation between descriptions.
An accelerating frame without rotation
m a′ = Fphysical − m aframe
The prime indicates acceleration measured in the non-inertial frame; mass is treated as constant in the Newtonian regime.
Lumen Physics brings together everyday examples of accelerated frames: taking off in an aircraft, turning a corner in a car, rotating on a merry-go-round, and describing the circulation of a tropical cyclone. These examples show that the same motion can be described from different frames and that equations must be adapted to the chosen frame.
Language about what “seems to act” does not imply that consciousness creates the phenomenon. Relative motion can be measured objectively; the task is to distinguish physical interactions from terms associated with the acceleration or rotation of coordinates14. Inertial effects also appear with ideal, correctly designed instruments free from calibration error.
Theoretical mediation enters into interpretation: which frame is used, which forces are included, and which transformation connects one description with another? Instrumental corrections may be needed in a particular experimental situation, but inertial forces do not exist because of a defect in the apparatus. Nor does understanding the case require appeal to quantum-gravity problems15.
The gnoseological lesson is to review the descriptive model and measurement procedure together. Sometimes transforming coordinates or including inertial terms suffices; in other cases the instrument, its calibration, or its data processing must also be adapted. Theory and practice are reciprocal: observational conditions guide model formulation, while theoretical demands can motivate new experiments.
Incorporating inertial forces has several methodological implications. It allows equations to be written for accelerating and rotating observers, descriptions to be compared, and coordinates suited to the problem to be selected. Such a frame change is an operation within a physical theory; it is not itself a new theory of matter.
From the essay’s dialectical perspective, the example invites us to think of motion through relationships and mediations, without identifying all movement with spatial displacement. Elí de Gortari’s Dialectics of Physics is a reference for the philosophical program mentioned. Assessing that program does not replace deriving and testing each physical equation.
In a purely translational description, F_inertial = −m a_frame summarizes the corresponding term. Rotating frames additionally involve centrifugal and Coriolis terms and, if angular velocity varies, an Euler term. The complexity of the description is made explicit and can be tested against measurements.
The essay proposes an analogy with capitalist competition’s role in the relationship between values, prices of production, and market prices. Within Marx’s theory, the formation and redistribution of profit and conditions of product realization require mediations between value production and its market expressions. Circulation does not itself create the aggregate value attributed by the theory to social labor, although it participates in value’s realization and distribution and in capital’s reproduction.
The analogy warns that a manifestation cannot be adequately interpreted without reconstructing the system of relationships producing it. It does not establish that competition is literally an inertial force or that transforming values into prices is a coordinate transformation. “Labor,” “value,” “force,” and “energy” belong here to different theories: political economy’s abstract labor cannot simply be measured in joules, and social circulation cannot be eliminated by changing observers.
Retaining the comparison therefore requires specifying its scope: it helps formulate questions about mediations but does not demonstrate a mathematical identity between physics and economics. Relationships between values and prices require economic arguments of their own; accelerated frames require their physical derivations. A more complete description may be better suited to the object without “more holistic” being, by itself, a criterion of verification.
The relationship F_inertial = −m a_frame clearly illustrates this limit. The term depends on the frame’s acceleration and allows relative motion to be described; it does not prove an economic thesis. Its usefulness for analysis and synthesis16 consists in showing that an observable result must be interpreted through explicit conditions rather than as an arbitrary appearance.
The geodynamo: generation, organization, and manifestation
Dynamo theory17 investigates how the motion of an electrically conducting fluid can maintain a magnetic field. On Earth, thermal and compositional convection18, rotation and its Coriolis effects19, electromagnetic induction, and dissipative losses are involved. Merely adding “convection + Coriolis” as two isolated ingredients is insufficient: their couplings and the conditions enabling dynamo action must be considered.
The liquid outer core plays a central role in field generation. High temperatures prevent explaining Earth’s field as that of an ordinary giant permanent magnet. The solid inner core may influence the field through its growth, thermal and compositional exchange, and coupling with the surrounding fluid; its stabilizing role should not be presented as a universal consequence already demonstrated by one study20.
The problem is particularly well suited to distinguishing internal field, surface manifestations, and inferential mediations: a signal measured at the surface does not reproduce the entire structure of the field in the core.
The geodynamo as an application of the categorial distinction
Analytical level
Concrete question
Manifestation
Which field variations are measured at the surface or reconstructed from records?
Content
Which fluids, conducting properties, motions, exchanges, and boundaries are involved?
Form of organization
How are flow and field structured under rotation and geometrical constraints?
Proposed essential content
Which couplings explain the maintenance and evolution of the field in the model considered?
Testing
Which constraints are imposed by geomagnetic, paleomagnetic, seismic, and geodetic data?
The classification does not replace magnetohydrodynamic equations. It makes visible which question corresponds to each description and how the parts of the explanation connect.
The example also clarifies the articulation of levels: describing continuity and connections in the fluid domain does not itself determine conductivity, boundary geometry, rotational conditions, thermal and compositional evolution, or the law connecting flow with field. Depending on the question and model, some of these relationships may jointly belong to its essential content. Philosophical classification does not replace physical data or make topology a sufficient explanation of the geodynamo.
The relationship between internal generation and external manifestation shows both the process’s unity and the need to distinguish its mediations. Knowing a surface field requires measurements; explaining its origin additionally requires models of motion, material properties, energy exchanges, and observational constraints. None of these tasks can be replaced solely by the name of a philosophical category.
The philosophical difficulty of the example lies in structures and processes interacting dynamically and nonlinearly. Relationships may produce collective patterns, reorganizations, and outcomes not understood by enumerating isolated components. Speaking of new information requires specifying the quantity or organizational change being studied; not every interaction creates information in the same technical sense.
Self-organization is a problem to investigate, not a property to be assigned by definition to any system called complex. When a pattern is said to self-organize, the energy or matter exchanges, constraints, and mechanisms enabling its formation must be explained. The dialectical perspective can guide these questions without replacing investigation of their answers.
Essence thus interacts with form through mediation, via content, which interacts with both as a first mediation: immediately in the sense of being less mediated, rather than necessarily unmediated in an absolute sense. Complexity also arises because all these elements are, colloquially, “together and mixed up,” though not so thoroughly as to make a logical distinction for analysis and synthesis impossible, since their unity is relative rather than absolute21.
Phenomenon is an objective manifestation of the process and may provide information about its fundamental relationships. Content encompasses components and processes with different functions: some directly participate in the essential explanation of the delimited problem; others provide conditions, mediations, or effects. “More internal” and “more external” may serve as metaphors for dependence or mediation, but their spatial relevance depends on each object.
Geological classification provides an example of subdivision. Gass, Smith, and Wilson (2008, p. 52, according to the reference used in the document) distinguish mantle regions in the cited discussion. General geological terminology should distinguish upper mantle, transition zone, and lower mantle when the problem requires it; the lower mantle extends to the boundary with the outer core. These are material divisions based on compositional, mineralogical, or mechanical criteria, not universal philosophical assignments.
The possibility of establishing more detailed divisions supports the methodological demand to specify classification criteria. It does not oblige us to assign a fixed philosophical category to each layer or demonstrate that the essential explanation always lies in the component with the smallest radius.
In the terminology proposed by this study, “appearance” names a partial presentation or superficial understanding that may lead to an inadequate interpretation of the object. The superficial is not necessarily false: it may be true and still insufficient to explain the process. Phenomenon retains its objectivity and provides both a starting point and a test for knowledge.
Deeper investigation means reconstructing relationships of form, content, and essence, identifying mediations, and returning to manifestations with a more determinate explanation. There is no obligation always to follow a rigid sequence of “appearance → phenomenon → form → content → essence.” Actual research can proceed recursively and correct its initial hierarchy. Intellectual apprehension does not close knowledge either: explanation must remain testable and revisable.
· · ·
Methodological elaboration
From categories to testable research
The proposal’s value depends on its ability to guide effective research operations. A philosophical taxonomy can organize questions, detect reductions, and reveal previously implicit mediations. It does not replace the identification of mechanisms, derivation of consequences, or confrontation with evidence. Articulating the sciences requires a working relationship between philosophy and disciplines, not the subordination of results to predetermined labels.
1. Delimiting the system and the question
It must be specified whether the investigation concerns a historical genesis, present dynamics, a response to perturbations, or a statistical regularity. Boundary conditions, scales, and variables are selected in relation to that question. What is external to a modeled system may be internal to a broader totality without making the choice of boundaries irrelevant.
2. The relationship between observation and explanation
Records, indicators, derived results, and postulated entities are distinguished. A model may reproduce observations through different mechanisms; we must therefore ask which aspects are identified and which remain compatible with alternative explanations. Moving toward the essential is a task of explanatory discrimination, not a movement away from data.
3. Criteria of explanatory importance
Selecting essential relationships may draw on established mechanisms, structural dependencies, invariances within a domain, and predictable changes when certain conditions are modified. No isolated criterion holds universally. High correlation is insufficient to demonstrate causality; a necessary condition is not always a sufficient explanation. Relevance must be argued in relation to the process.
4. Articulation between levels
A macroscopic regularity cannot always be deduced by simply adding microscopic components. Aggregation, interaction, constraints, the emergence of patterns, and feedback are investigated. Likewise, the priority of certain social relationships does not eliminate the relative autonomy or efficacy of institutions, legal forms, or representations. Determining mediations helps avoid both reductionism and an unspecified invocation of totality.
5. Revising the categories
A classification must change when it conflates levels, prevents a mechanism from being recognized, or fails to distinguish incompatible results. In the methodological use defended here, correspondence requires explaining how what remains valid in an earlier formulation is preserved within a more determinate reconstruction. The proposed complementarity requires perspectives to provide articulable information rather than merely coexist without criteria for relating them.
Identifying essential determinations also requires specifying which mathematical structure is preserved and for which question it is explanatory. Shared topology does not ensure shared metric or dynamics; shared symmetries do not replace knowledge of mechanisms and conditions. This distinction prevents a partial equivalence from becoming an exhaustive explanation.
Conclusions
Essence, content, form, phenomenon, and appearance are distinctions for investigating an articulated, changing reality. Differentiating them does not break the process’s unity; it prevents that unity from becoming undifferentiated identity. Essential content seeks to identify fundamental relationships within broader contents and examine how they are organized, manifest, and become known.
Essence thus appears as a relational unity susceptible to multiple mathematical determinations. Topology may formalize one of them; differentiability, metric, transformation structures, and the process’s history may contribute others. Their articulation, rather than identifying one layer with the whole essence, guides the selection of essential content and explanation of its manifestations.
Distinguishing phenomenon from appearance allows both objective manifestations and conditions of their partial presentation or insufficient interpretation to be investigated. The objectivity of a social appearance or relative motion does not oblige us to accept every immediate interpretation as true. Explanation reconstructs why the process presents itself in that way and which relationships make it intelligible.
Climatic, geodynamic, mathematical, and economic cases present different problems of scale, mediation, and hierarchy. Comparing them enriches philosophical questions when their scientific differences are preserved. An explanation’s depth does not depend on locating it beneath a surface, multiplying categories, or proclaiming a totality: it depends on the relationships reconstructed, conditions of validity, and evidence against which it can be tested.
The program remains open. Its robustness requires clear definitions, historical reconstruction of categories, precise disciplinary applications, and a willingness to correct hypotheses. Scientific and social practice can transform research questions and categories; this is one consequence of taking the historical and processual character of knowledge seriously.
Notes · 01–21
Notes and supplementary discussions
The document’s numbering is retained. These notes are part of the argument; their clarifications are integrated with the revised formulations in the main text.
NOTE 01
Equilibrium means a state in which certain contributions balance, not their physical destruction. In translational mechanical equilibrium, net force is zero; when rotation is investigated, torque balance must also be considered. Dynamic equilibrium and steady states require criteria suited to the system. Balancing tendencies does not mean annihilating components or demonstrate that every object tends toward equilibrium. The original formulation of “relative mutual destruction” is retained in its intended sense of compensation, with this physical qualification.
“Mode of being” here designates the determinate manner in which an object exists. Its aspect before an observer is a manifestation of that mode, but does not exhaust its existence or relationships. Distinguishing character from presentation prevents defining being exclusively through appearance.
The document associates essence with greater homogeneity and stability. Such homogeneity should be understood as a possible organizational coherence, not as identity among components: a stable structure may depend on heterogeneous elements, asymmetries, and differentiated relationships. Its minimum stability must be investigated, not assumed by definition.
Access to essential relationships is usually mediated by abstractions, models, inferences, and instruments. The oppositions “internal/external,” “stable/changeable,” and “immediate/mediated” express analytical emphases, not mutually exclusive properties universally attributable to two classes of things. An observable phenomenon may also require complex instrumental mediations.
The original note cites a popular definition of the uncertainty relation attributed to Wikipedia (2021): the impossibility of simultaneously determining certain pairs of quantities, such as position and linear momentum, with arbitrary precision. The reference is retained in the historical bibliographical record; research sources on different uncertainty relations are added to support the discussion.
Rosental and Iudin (1971, p. 398), in the transcription used by the document, state:
“A thesis in quantum mechanics formulated by Heisenberg in 1927 concerning the impossibility of simultaneously determining a particle’s position and momentum exactly. Uncertainty relations take the form of quantitative relationships between the so-called indeterminacies of conjugate variables: position and momentum, and also time and energy. The smaller the indeterminacy of a particle’s position, the greater that of its momentum, and vice versa. An analogous relationship exists between determinations of a particle’s time and energy. The uncertainty relation is an objective characteristic of microcosmic phenomena connected with their corpuscular and wave nature; the ‘indeterminacies’ are inherent in the micro-object’s real state and do not in any way denote a limit to knowledge. Heisenberg and Bohr inferred the uncertainty relation from the influence of the instrument determining a particle’s position on its momentum—for example, the effect on an electron’s momentum of the aperture in a diaphragm through which it passes—and from the influence of the instrument determining its momentum on its position in space. Similarly, one describes how instruments measuring time affect a particle’s energy and how instruments measuring energy affect the possibility of determining time exactly. Positivist philosophical conclusions have been inferred from the uncertainty relation, such as denying the causal conditioning of an elementary particle’s states and denying the objectivity of the microcosm, its independence from cognitive activity—so-called ‘instrumental’ idealism [Instrument]. Criticism of such idealist distortions of quantum mechanics from materialist positions has helped clarify its true meaning.”
This quotation documents the source’s philosophical interpretation, which must be distinguished from mathematical constraints. For position and momentum, the preparation uncertainty relation bounds the product of their standard deviations in the same state: σx σp ≥ ℏ/2. It does not merely describe poor instrument quality. Measurement error and disturbance of another observable are related but different problems (Busch, Lahti, & Werner, 2014).
The image of a photon colliding with an electron introduces disturbance associated with measurement, but is not a universal derivation of all uncertainty relations. Nor can it be concluded that measuring a gravitational perturbation eliminates those constraints and reduces every error to zero. A proposed gravitational measurement would have to specify its interaction, observables, and own limits; it does not itself invalidate quantum formalism.
Operationally defining a procedure does not require adopting an idealist ontology. Likewise, rejecting idealism does not permit disregarding verified physical constraints. The world’s objectivity is compatible with precise limits on what a theory simultaneously attributes to its states and measurements. The energy–time relationship requires specific treatment: in the usual formalism, time does not always play the same operator role as position, so the position–momentum relation should not be transferred mechanically.
The original concern with causality, objectivity, and the human capacity to know remains open as a philosophical discussion. It must be distinguished from specific empirical or mathematical statements, with each subjected to the appropriate argument.
The document replaces “global” with “general” in its quotation from Rosental and Iudin to emphasize that essential relationships are not considered a single absolute determinant. This intervention is made explicit when retaining that version: the replacement should not be attributed to the source’s literal wording. External conditions, mediations, and contingencies can also participate in development.
A form may partially express a content. The degree of partiality and recoverable information depend on the problem. It is not assumed that every form always expresses its content “minimally” or that the same proportion of information applies to all phenomena.
The document cites Hegel (1968, pp. 97–98) to highlight the German verb’s double meaning: it signifies both preserving or maintaining and, at the same time, causing to cease or bringing to an end. This is precisely the linguistic tension of Aufheben; there is no need to claim that German lacked a word including preservation. The spelling is standardized from the variant “Auffheben” that also appeared.
On this reading, eliminating and preserving are not two wholly separate acts: a determination is negated while moments of it are preserved in a transformed configuration. “Eliminate–preserve–sublate” helps explain this, but should not become a universal three-step recipe. Preservation and negation are determined by the process and are not necessarily symmetrical.
According to the passage cited by the author, Dri (2007, p. 48) explains that in the Phenomenology of Spirit the subject traverses configurations in which the preceding one is eliminated, preserved, and sublated. The observation on page 49 distinguishes this transformation from a mere mixture of heterogeneous ingredients. Translating Aufheben as the Spanish superación—or English “sublation”—therefore requires a conceptual account making both preservation and negation visible.
“Synthesis,” however, should not in every usage be reduced to mixing two flavors. In this essay, analysis and synthesis designate operations of distinguishing relationships and reconstructing them within a totality. A totality’s unity does not imply homogeneity in all its components: internal heterogeneity may constitute its organization.
The original comparisons with microstates and macrostates, relationships between values and prices, and wave–particle duality are retained as problems of articulation between descriptions. They are not identities between theories or proofs by analogy of a thesis about Hegel. A totality may be an individual, a collective, or a delimited system; in each case, the relationships allowing it to be treated as such must be identified.
To situate Mark Moiseevich Rosental and his compilations, the document refers to Philosophy in the Soviet Union and the biographical entry on Rosental at the Proyecto Filosofía en español. Both links and the reference to different dictionary editions are retained.
Kilifarska, Bakhmutov, and Melnyk use concepts and discussions connected to climate literature, including the IPCC, according to the reference to their preface (p. viii) recorded in the document. This does not mean that every hypothesis in the book represents an IPCC conclusion. The Panel was established in 1988 by the World Meteorological Organization and the United Nations Environment Programme; its assessments cover climate change’s scientific basis, impacts, adaptation, and mitigation. Shared vocabulary is distinguished here from validation of a particular hypothesis. The Spanish typo alienados (“alienated”) is corrected to alineados (“aligned”), the intended meaning in the original note.
The lithosphere is the mechanically rigid envelope comprising the crust and uppermost mantle. Its thickness varies and is not generally limited to 10–50 km; it should not be confused with crustal thickness. The compositional distinction crust–mantle and the mechanical distinction lithosphere–asthenosphere follow different criteria. This difference is relevant to the argument about levels of classification.
“Inner core” and “outer core” compare positions within Earth’s core: both lie inside the planet. The document’s observation prevents confusing “outer” with being outside Earth. Their explanatory importance, however, does not follow exclusively from spatial position.
The argument does not claim that Marxism possesses only essence–phenomenon and content–form or investigates reality with two categories. It holds that certain questions may require additional distinctions compatible with reconstructing the whole. Objects appear through phenomena and are investigated as processes with histories and conditions of manifestation. Knowledge changes because it is partial and revisable, and because the processes investigated may themselves change. This double historicity justifies examining and, when necessary, reformulating abstract categories.
The elaboration on mathematical structures of the essential sharpens this criterion: philosophical categories are not identified one-to-one with mathematical branches. Preserving continuity does not exhaust preserving metric relationships, dynamics, or historical constraints. Different equivalences must be defined and their explanatory relevance justified.
In Newtonian mechanics, net force determines the change in linear momentum; with constant mass, F = m a. Interactions may produce changes in motion and deformations. An inertial force is a term dependent on the accelerated frame, not an additional interaction always requiring an equivalent real force acting on the same body. A description’s use of that term does not imply distortion of a properly conducted measurement. “Fictitious force,” “inertial force,” and “pseudoforce” are the designations compared here.
The document raises a question about compatibility between physical theories that should be delimited: quantum mechanics and special relativity are articulated in quantum field theories; constructing a complete, empirically established quantum theory of gravitation is a different problem. The validity of theories in different domains does not require one automatically to refute another. Quantum-gravity research has a history far longer than one decade and is unnecessary for deriving classical inertial forces. The note is retained with this distinction between domains, approximations, and open questions.
“Analytical–synthetic” refers to distinguishing relationships and reconstructing their articulation, following the methodological usage developed in Lessons in Marxian Gnoseology I. Synthesis, mixture, and Aufheben should not be equated without explanation. Abstraction retains its connection with empirical testing and practice.
The original note refers to a Wikipedia entry (2021) drawing on Axel Brandenburg’s article (2007), Hydromagnetic Dynamo Theory. The specialist reference is incorporated directly. The theory investigates how a conducting fluid in motion can regenerate a magnetic field and compensate for dissipative losses over prolonged periods.
Earth, the Sun, and other astrophysical bodies present different conditions. The geodynamo involves metallic fluid in the outer core; the solar case involves plasma and processes distributed across regions of the star, which should not be reduced without qualification to a single location. The original mention of Mercury and the Jovian planets is retained as part of the problem’s planetary scope, not as a claim of an identical mechanism in all of them.
Magnetohydrodynamic equations connect fluid motion with the electromagnetic field. Rotation or electrical conductivity alone is insufficient: flow, medium properties, scales, and conditions for dynamo excitation and maintenance must be investigated. The Spanish typo convence (“convinces”) is corrected to convecta (“convects”) in the description of the fluid.
Convection is transport associated with fluid motion and is one mechanism of heat transfer alongside conduction and radiation. Evaporation is not synonymous with convection. In the core, compositional differences can drive motion in addition to thermal buoyancy; thermal and compositional convection are therefore distinguished.
In a frame whose axes rotate with angular velocity Ω, the Coriolis force on a mass m with relative velocity v′ is FCor = −2m Ω × v′ = 2m v′ × Ω. Centrifugal force is Fcf = −m Ω × (Ω × r). The document’s expression, m Ω × (r × Ω), is equivalent to the latter. The conventions and relative character of velocity are made explicit here.
Centrifugal force depends on position relative to the axis and angular velocity. Coriolis force depends on relative velocity and frame rotation, not on “two angular velocities,” as a repeated phrase in the original note suggested. If Ω varies with time, the Euler term also appears: FE = −m (dΩ/dt) × r. These terms correspond to defined conditions of the frame and do not require acceleration of varying magnitude: uniform rotation already produces a non-inertial frame.
A merry-go-round helps distinguish them. A person maintained at a fixed radius requires an interaction producing centripetal acceleration in the inertial description; the centrifugal term appears in the rotating description. If the person walks radially relative to the platform, relative velocity gives rise to the Coriolis term. Contact forces and motion constraints must be included to determine the actual trajectory; it cannot be inferred solely from the observer’s sensation.
Tangential and angular velocity are different quantities. If the person is initially at rest relative to the platform, their inertial tangential velocity is Ω × r. As radius changes, evolution depends on interactions and applied torques, not on a universal rule conserving that tangential velocity. A pedagogical explanation is available in OpenStax, §6.4.
Finally, an expression’s lack of explicit position dependence does not demonstrate that it is unrelated to every topological consideration or prove that it describes an effect without cause. Relationships between metric structure, geometry, topology, and causality cannot be deduced from the presence or absence of a single variable in an equation. The document’s StackExchange (2014) reference was insufficiently identified; it is retained in the bibliographical provenance record, while an identifiable teaching source supports the clarifications provided here.
Internal field, surface signals, and energy sources
Buffett (2010) obtains an indirect estimate of the internal magnetic field from the dissipation of tidally driven motion and its effects on Earth’s nutation. The distinction between internal field and surface signals is relevant to mediation: observables constrain a reconstruction without constituting a complete copy of the mechanism.
Aubert, Labrosse, and Poitou (2009) investigate geodynamo evolution models combining core thermodynamics and numerical simulations. In the discussion cited by the document, increased power associated with inner-core nucleation may accompany a shift in the field-generating region, so an internal increase need not be reflected with equal intensity at the surface. Mediations can attenuate a signal without making the process unreal.
The original note also refers to Science 2.0 under a date recorded as 2020. The corresponding information also appears in a University of California, Berkeley release by Robert Sanders dated 16 December 2010. It reports Buffett’s estimate that around 60% of the power considered comes from the exclusion of light elements during inner-core growth. This estimate belongs to that discussion and date, rather than representing a universal, invariant proportion of all terrestrial energy.
Inner-core growth may contribute to compositional buoyancy in the outer core. Its influence should not be reduced to a substance that “stabilizes” the field by itself; nor does estimating an energy contribution demonstrate a philosophical law whereby one of two components always dominates every object’s evolution. The case investigates differentiated functions and dependencies; the hierarchy must be established through models and evidence.
The unity of components is historical and conditioned. Analytically distinguishing them does not require their material separation at the object’s genesis or guarantee a unique decomposition. Research must justify its division and then reconstruct the interactions that division makes it possible to investigate.
Information on historical editions is retained as recorded in the document. Supplementary sources support clarifications in this edition; unidentified details are flagged rather than completed by conjecture.
Bibliography of the original main text
Einstein, A. (2005). The Foundation of the General Theory of Relativity. In 100 Years of Gravity and Accelerated Frames: The Deepest Insights of Einstein and Yang-Mills (pp. 65–119). World Scientific. Translation cited by the document; original work published in 1916.
Gass, I., Smith, P., & Wilson, R. (2008). Introducción a las ciencias de la Tierra [Introduction to the Earth Sciences]. Reverté. Edition and pages follow the reference used in the document.
Kilifarska, N. A., Bakhmutov, V. G., & Melnyk, G. V. (2020). The Hidden Link between Earth’s Magnetic Field and Climate. Elsevier. Source of the reproduced climatic and geomagnetic materials.
Lumen Physics. (Consulted by the original document on 10 April 2021). Fictitious Forces and Non-inertial Frames: The Coriolis Force. Teaching chapter cited.
Marx, K. (2007). Elementos fundamentales para la crítica de la economía política (Grundrisse), 1857–1858 [Foundations of the Critique of Political Economy] (Vol. I). Siglo XXI.
Gómez Julián, J. M. (2020). Algunas reflexiones sobre la distribución binomial negativa II: un análisis teórico y aplicado [Some Reflections on the Negative Binomial Distribution II: A Theoretical and Applied Analysis]. Document published under the pseudonym Isadore Nabi. Reference document. Historical URL recorded in the PDF: original archive.
Gómez Julián, J. M. (2021a). Lecciones de gnoseología marxiana I [Lessons in Marxian Gnoseology I]. Document published under the pseudonym Isadore Nabi. Link recorded in the historical document.
Gómez Julián, J. M. (2021b, 2 April). Una interpretación multidisciplinaria de los espacios característicos, vectores característicos y valores característicos [A Multidisciplinary Interpretation of Eigenspaces, Eigenvectors, and Eigenvalues]. Published under the pseudonym Isadore Nabi. Reference document. Historical URL: original archive.
Rosental, M. (1961). Los problemas de la dialéctica en «El Capital» de Marx [The Problems of Dialectics in Marx’s Capital]. Ediciones Pueblos Unidos.
Rosental, M., & Iudin, P. (1971). Diccionario filosófico [Philosophical Dictionary]. Tecolut. Page references to the cited edition are retained. Digital reference collection.
Physics Stack Exchange. (2012, 24 April). How fictitious are fictitious forces?. Discussion cited by the document. A source of discussion and explanation, distinguished from the added research and teaching sources.
Physics Stack Exchange. (2012, 8 February). Why can’t we think of free fall as upside down rocket?. Cited discussion.
Un Blog de Dibujos. (Recorded consultation: 9 April 2021). Capas de la Tierra [Earth’s Layers]. Noches en vela. Provenance indicated in the document. The consultation date is not identified with the page’s publication date.
Sources cited in notes and indirect references
Hegel, G. W. F. (1968). Ciencia de la lógica [Science of Logic]. Solar / Hachette. Reference cited in the note on Aufheben (pp. 97–98).
De Gortari, E. (1964). Dialéctica de la física [Dialectics of Physics]. Universidad Nacional Autónoma de México. UNAM historical catalogue. The 1964 edition is identified; the document did not specify a particular edition.
Buffett, B. A. (2010). Tidal dissipation and the strength of the Earth’s internal magnetic field. Nature, 468, 952–954. doi:10.1038/nature09643.
Aubert, J., Labrosse, S., & Poitou, C. (2009). Modelling the palaeo-evolution of the geodynamo. Geophysical Journal International, 179(3), 1414–1428. doi:10.1111/j.1365-246X.2009.04361.x.
News Staff. (n.d.). First Measurement of Magnetic Field Inside Earth’s Core. Science 2.0. Popular-science page mentioned in note 20. The document assigned this reference to 2020 without a complete bibliographical entry; that date is not retained as verified information.
Wikipedia. (2021; historical reference without an identified revision). Entries concerning the uncertainty principle and dynamo theory. The document did not record complete titles, revisions, and links; provenance is retained without inventing a particular revision. This edition’s clarifications refer to Busch, Lahti, and Werner, and to Brandenburg.
Physics Stack Exchange. (2014; incomplete historical reference). Discussion of centrifugal and Coriolis forces cited in note 19. The document provides no title, author, or link allowing unambiguous identification. OpenStax is added for the physical discussion; missing details are not invented.
Indirect references retained: Herschel (1801), mentioned through Kilifarska, Bakhmutov, and Melnyk (2020); Marx and Engels, vol. XXV, part II, p. 384, cited through Rosental and Iudin (1971, p. 148). Mentions of Lenin, Dussel, Born, Schrödinger, de Broglie, Bohm, Durkheim, Kant, Berkeley, and Hume identify interlocutors or precedents for the argument; no additional direct quotations or bibliographical entries for unspecified works are attributed to them here.
Supplementary references for this edition
Busch, P., Lahti, P., & Werner, R. F. (2014). Measurement uncertainty relations. Journal of Mathematical Physics, 55, 042111. doi:10.1063/1.4871444. Authors’ version.
Gómez Julián, J. M. (2026). Geometría, historia y legalidad: fundamentos dialéctico-materialistas de los espacios con estructura métrica [Geometry, History, and Lawfulness: Dialectical Materialist Foundations of Spaces with Metric Structure]. Working manuscript supplied by the author; date recorded in its metadata: 1 January 2026. Relevant sections: §§1.7–1.8, 4.2.1, and 8. Its philosophical contribution is distinguished here from the status of its results and conjectures.
Meinrenken, E. (2002). Riemannian Geometry. Course notes, University of Toronto, Spring 2002. Author’s text. See §§12–18 and Theorem 14.7.
The historical essay, published on 10 April 2021 under the pseudonym Isadore Nabi, is presented under the current authorship of JOSÉ MAURICIO GÓMEZ JULIÁN. Self-citations are standardized under that name, while the bibliography retains the provenance necessary to identify earlier publications.
This complete edition develops the eight argumentative sections, all subdivisions and cases, the four figures, the 21 notes, and the references of the 41-page document. Note numbering is retained, while the contents are adapted for navigation. Illustrations come from the PDF and retain their information and reference colors. Captions add context; figures can be enlarged within this page. The English key to Figure 4 explains its original Spanish labels.
Spelling, punctuation, word forms, and line breaks have been standardized. Unfinished transitions on pages 21, 24, and 34 are completed through explanatory development. The reference to Frolov (1984, p. 142) on page 5 was followed by an empty quotation: no quotation is fabricated to fill it; the author is retained and the question developed through passages actually present and identified references. Extended quotations are transmitted from the historical document with typographical normalization; indirect attributions are identified.
Substantive revisions clarify: the relative stability and mutability of essence; reality and perception; the interconnection of objective and cognitive levels; the authorial status of the Bohr–Marx designations; formalization of chaotic systems; climatological periods versus sample size; forcing classifications; the scope of the geomagnetic case; limits of geological and economic analogies; inertial frames and frame-dependent forces; quantum uncertainty; Aufheben; and criteria of material organization. Physically incorrect formulations are reformulated while retaining the problem they sought to examine, and bibliographical details left incomplete in the original are identified.
The orientation tables, operational explanations, methodological section, and conclusions develop this edition’s argument. They are not presented as quotations from historical authors. The downloadable file preserves the source document for comparison of formulations and pagination.
This English edition translates the complete revised Spanish HTML, including its cases, elaborations, notes, references, and figure commentary. Original-language bibliographical titles are retained with English reading translations where helpful. The archival PDF remains in Spanish and is reproduced byte for byte; it is supplementary to the full English reading text above.
This revision adds an elaboration on the articulation of set-theoretic, topological, differentiable, metric, algebraic-operational, and historical-dynamical determinations, drawing on the author’s manuscript Geometry, History, and Lawfulness. It adds non-isometric metrics on one topological and differentiable underlying space, distinguishes invariance from explanatory relevance, and connects that distinction with essential content, the geodynamo, methodology, and conclusions. The manuscript’s philosophical program is distinguished from established results; no general theorem determining every physical lawfulness is imported.
Understanding a changing reality also requires investigating how the categories through which we know it develop. How are experience, abstraction, and practice related? What distinguishes an immediate appearance from the relations that produce it? How can a theory reconstruct a historical process while keeping the order of its concepts grounded in that process? These questions guide the Lessons in Marxist Epistemology.
This first instalment develops a thematic study of Mark Rosental’s Los problemas de la dialéctica en «El Capital» de Marx—the problems of dialectics in Marx’s Capital. It brings that work into conversation with Marx and Engels, the philosophical dictionaries of Rosental and Iudin and of Frolov, and contributions by Hegel, Lenin, and Enrique Dussel. Selected passages are accompanied by my own comments, which extend, clarify, or discuss their arguments. [1]
The preface places this investigation within a five-part project: three studies of works by Rosental; a fourth on Eli de Gortari’s Introducción a la lógica dialéctica; and a fifth combining Richard Levins’s A Response to Orzack and Sober: Formal Analysis and the Fluidity of Science with Alkistis Elliott-Graves’s The Value of Imprecise Prediction. A subsequent synthesis is intended to preserve the traceability of the sources and make their relationship to my own elaborations identifiable.
Political economy is central because Capital allows us to study method at work: categories taking shape, relations being reconstructed, and observable forms being explained. From that starting point, the inquiry reaches into logic, the philosophy of mathematics, statistics, and physics. Their connection lies in the problem of how thought can grasp a reality that exists and changes independently of any particular account of it.
II · OrientationFidelity to method and the openness of knowledge
I use Marxist to identify an orientation that seeks to preserve and develop the logical, epistemological, and methodological foundations elaborated by Marx, Engels, and other classical authors in this tradition. Its content is assessed through the actual articulation of those foundations in research. A label or a declared political allegiance cannot settle that question by itself.
“This methodological fidelity does not mean restricting knowledge to the works of the classics or considering all their conclusions infallible.”
José Mauricio Gómez Julián · Revised manuscript, preface · Translated from Spanish
The continuity at issue allows particular results to be corrected, later knowledge to be incorporated, and research to extend to new objects. The expression classical Marxism identifies the methodological and conceptual core whose definition is itself part of the investigation. Establishing that core requires examining arguments and research practices; it cannot be inferred from a linguistic suffix.
For English readers, this also explains the choice of Marxist in the title. The intended scope includes Marx, Engels, other classical Marxists, and the further development of their research orientation. The methodological commitment is made explicit in the argument rather than delegated to the distinction between the words Marxian and Marxist.
This approach informs my critical use of Soviet sources. Their epistemological contributions do not entail agreement with every historical or political position they hold. In the preface, I explicitly distinguish my interpretation of the Soviet experience from interpretations advanced by authors whose work nevertheless provides important resources for this investigation.
· · ·
III · The object and its knowledgeEssence, appearance, and practice
The materialist starting point is the objective existence of the processes we investigate. Knowledge develops through our relations with them: perception, instruments, concepts, experiments, and social practices. Abstraction participates in this relationship by identifying determinations that immediate observation does not deliver in an already organized form.
The distinction between essence and appearance addresses this work of explanation. Observable forms manifest relations that need to be reconstructed. Investigating value, price, wages, or profit therefore requires explaining the mediations between their determinations and the ways in which they appear. Describing a visible outcome leaves open the question of the process that produces it.
In this vocabulary, a determination is a feature or relation through which the object acquires a specific content. A mediation is a connection or process needed to explain how determinations are related. These terms direct attention to the structure and development of the object under investigation.
Practice plays both a constitutive and a testing role: it helps form the problems, develop the instruments, and assess knowledge. Scientific activity has a material and social history. Its objectivity requires investigating how theoretical constructions relate to the actual processes they concern.
The study consequently connects sensory knowledge and abstract thought, theory and practice, relative and absolute truth. The possibility of deepening an explanation presupposes that knowledge can develop and be corrected while reality retains its objective existence. Necessity and chance are also examined within this framework, as questions about processes and their interrelations.
IV · Operations of methodReconstructing a concrete totality
Analysis distinguishes determinations within a reality initially apprehended in a confused way. Synthesis investigates their articulation and reconstructs a totality of relations. Their unity matters: dividing an object into parts and listing those parts does not yet explain how that object exists or changes.
The movement from the abstract to the concrete acquires its meaning here. Concrete reality is the starting point of knowledge; the concrete as reconstructed in thought is a result of its elaboration. Abstractions allow us to return to the object with a richer understanding of its determinations. Dussel develops this distinction in his reading of the method of the Grundrisse. [2][4]
Consider the categories of political economy. A society may first be apprehended as a complex whole. Analysis distinguishes relations such as commodity exchange, money, wage labour, and capital. Their reconstruction must explain how those relations are connected and under what historical conditions they exist. In this sense, concrete refers to a unity of multiple determinations, not simply to whatever can be immediately seen or touched.
Induction and deduction are examined as related operations within this investigation. The same applies to intuition and the logical assessment of arguments. Each procedure has functions and conditions of application that must be specified in relation to the problem. The discussion of mathematical intuition, constructive approaches, and the finite and infinite extends this methodological inquiry.
The distinction between inquiry and exposition is equally important. The order that makes a process intelligible need not reproduce every chronological stage of its development or every step taken to investigate it. Exposition must justify its mediations and retain its grounding in historical reality. [3]
V · Political economyCategories expressing historical relations
Economic categories acquire their content within determinate social relations. Commodity, money, capital, wage labour, and surplus value must be studied through their conditions of existence and transformation. The investigation asks how these forms are historically produced and how they are connected within social reproduction.
The study addresses the origins of capitalism, the formation of the industrial reserve army, and the transformation of values into prices of production. It also examines contradictions between concrete and abstract labour, use value and exchange value, and social production and private appropriation. The emergence of the general equivalent of commodities is considered within this development.
Several distinctions that organize the argument
Concrete labour concerns the specific useful activity that produces a particular use value; abstract labour concerns labour in the social determination relevant to commodity value. The distinction connects the useful character of production with the social relations through which products are equated.
Values and prices of production enter an inquiry into surplus-value production, its distribution among capitals, and the formation of an average rate of profit. The industrial reserve army is investigated as part of capitalist accumulation and its changing demand for labour. These questions require the historical and conceptual development provided in the full study.
Competition and technological change are particularly important in my comments on Rosental. Their examination connects productivity, surplus value, redistribution, and profitability. Production and circulation are treated as moments of a totality, preserving the distinction between the coexistence of different phases across a system and the succession of phases traversed by a particular commodity.
From this concern I propose studying input–output matrices dynamically, using comparable sequences of tables and models to investigate technical transformations and sectoral relations. The comparison with the tables in Volume III of Capital concerns the representation of productive relations and aggregate magnitudes. The choice of simulations or statistical procedures requires prior examination of the problem and the data.
The critique of political economy thus remains connected to history, quantitative investigation, and social action. In the preface, the pursuit of truth is explicitly articulated with concern for human suffering, justice, ethics, and morality.
· · ·
VI · Philosophy of statisticsProbability, evidence, and objectivity
The discussion of total probability, inverse probability, and objective Bayesianism belongs to a broader question: how can we investigate processes when available knowledge is partial and relationships are expressed probabilistically? The document connects this problem with Bernoulli, Laplace, Kolmogorov, and controversies between statistical schools.
Probability calculus and the interpretation of its components need to be distinguished. Axiomatization provides a framework of mathematical relations; research must establish what its objects represent, what information enters a model, and how its application is justified. Philosophical disagreements can affect the choice of procedures and the meaning of inferences even when identities of the calculus are shared. [5][8]
P(D) > 0 · Conditionals defined in the model
Bayes’s theorem in hypothesis-and-data notation, written for events. The posterior combines the model of the evidence with prior information. See DeGroot and Schervish, cited in the study.
Here, P(H) is the prior probability assigned to the hypothesis, P(D|H) represents the probability of the data under that hypothesis, and P(H|D) is the posterior probability conditional on the data and the model. The denominator P(D) normalizes the result. When the hypotheses form a discrete, mutually exclusive, exhaustive set, it can be obtained by summing the data probabilities under those hypotheses, weighted by their prior probabilities.
The likelihood treats the probability or density of the observed data as a function of the parameter or hypothesis, keeping the data fixed. It must be distinguished both from a probability distribution over the parameter and from the reliability of an empirical record. For continuous observations, Bayesian updating is expressed with the corresponding densities and appropriate normalization. [9]
Jon Williamson’s objective Bayesianism provides a framework for discussing constraints on rational degrees of belief and their relationship to evidence. My elaboration seeks to connect these questions with a materialist interpretation of objectivity and chance. The calculus, its inferential uses, and this philosophical interpretation are studied together while retaining their respective functions. [6]
The connection with quantum mechanics
Instruments, complementarity, measurement, and causality extend the investigation into physics. The study draws on Alexei Kojevnikov’s historical reconstruction of probability, Marxism, and ensemble interpretations of quantum mechanics. From that starting point, I examine objectivity, interpretation, and determinism within the document’s broader materialist argument. [7]
This discussion distinguishes mathematical descriptions, experimental arrangements, and philosophical interpretations. Its place in the study is to investigate how probabilistic representations relate to the objective processes and instruments involved in scientific knowledge.
VII · Structure and knowledgeInvariants, levels, and formal limits
The elaboration on essence incorporates a formalization through mathematical structures and classes of transformations. The question is which relations are preserved, and which can change, while retaining the identity under consideration. Specifying that choice helps define the level at which a system is investigated.
Structural level
Transformation
What is preserved
Set-theoretic
Bijection
Cardinality
Topological
Homeomorphism
Continuity structure; connectedness and compactness
Differentiable
Diffeomorphism
Smooth structure
Riemannian
Isometry
Metric tensor
Essence is investigated through constitutive relations and invariants. The organization of admissible transformations also allows an operational essence to be studied.
An invariant is a property preserved under the specified class of transformations. A topological comparison and a metric comparison therefore ask different questions of the same object. The choice of structure and transformations determines which kind of identity is being considered.
This articulation situates topology within a broader account. Set-theoretic, topological, differentiable, metric, and operational determinations permit distinct investigations. Their relevance must be established through the structure of the object and the transformations under which its identity is examined.
Prediction, decision, and proof
The revised manuscript also distinguishes limits of prediction, undecidability, and incompleteness. Lorenz’s 1963 work exhibits sensitivity to initial conditions in a deterministic model. The spectral-gap undecidability results rule out a universal decision algorithm for the families covered by their constructions. Gödel’s incompleteness results establish limits on proof in formal theories satisfying their conditions of application.
For the first incompleteness theorem, these conditions include consistency, effective axiomatizability, and sufficient arithmetic strength. The relevant conclusion concerns sentences that cannot be proved or refuted within the theory. The existence of a mathematical model, the possibility of predicting its behaviour, the availability of a universal decision procedure, and the reach of its proofs are consequently distinct questions.
I integrate these limitations with the history of our intellectual instruments and the objective existence of processes. Knowledge of essential relations and solutions to particular problems remain possible. Research must specify the reach of its resources and the mediations through which its results refer to reality.
VIII · Reading guideRoutes through the complete study
The full document supports a systematic reading or an approach organized around a particular problem. Its chapters are interconnected: economic applications return to the methodological categories, and discussions of probability reopen questions of objectivity, intuition, formalization, and practice. The Spanish headings below help locate these discussions in the document.
Preface · PrólogoPurpose and standpointThe project of the lessons, the selection of sources, the meaning of “Marxist,” and the connection between knowledge, ethics, and social commitment.
Chapters II and IIIFoundations and operations of knowledgeChapter II, Fundamentación objetiva (material) de la gnoseología marxista, develops essence and appearance, necessity and chance, instruments, analysis and synthesis, induction and deduction, intuition, truth, and practice. Chapter III, Las limitaciones de la lógica formal, continues the examination of formal logic.
Chapters IV–VIIHistory and the reconstruction of the concreteThe origins of classical political economy and capitalism, the industrial reserve army, and the relationship between abstraction and historical inquiry. Chapter VII is Lo abstracto y lo concreto como etapas gnoseológicas de la investigación científica.
Chapter VIIILaws, probability, and the interpretation of physicsLas leyes históricas que rigen el movimiento en general leads into general and specific laws, total and inverse probability, Bayes’s theorem, objective Bayesianism, and interpretations of quantum mechanics.
Chapters IX–XIIValue, prices, and contradictionsChapter IX examines the proceso de transformación de valores en precios de producción. The following chapters address socialism and communism as historical necessities, concrete and abstract labour, use value and exchange value, and the general equivalent of commodities.
Chapter XIII and notesSources and clarificationsThe references identify the works studied. Notes accompany the arguments, source discussions, and my own elaborations throughout the document.
Spanish terms used in the study · A short reading aid
Spanish
English and context
Gnoseología
Theory of knowledge; rendered here as epistemology.
Esencia / fenómeno
Essence / phenomenon or appearance: essential relations and their forms of manifestation.
Necesidad / casualidad
Necessity / chance. Casualidad means chance; causalidad means causality.
Lo abstracto / lo concreto
The abstract / the concrete, examined as moments in the reconstruction of a determinate totality.
Trabajo concreto / trabajo abstracto
Concrete labour / abstract labour.
Plusvalía / precios de producción
Surplus value / prices of production.
Ejército industrial de reserva
Industrial reserve army, investigated in relation to capitalist accumulation.
Probabilidad inversa
Inverse probability, within the discussion of inference and objective Bayesianism.
IX · SourcesBibliography of the study
The first nine references identify the sources cited in this presentation. The expandable section preserves the remaining bibliography of the manuscript. Titles and publication details retain the language of the source entries so readers can identify the editions used; English descriptions in the body are reading aids.
Rosental, M. (1961). Los problemas de la dialéctica en "EL CAPITAL" de Marx. Montevideo: Ediciones Pueblos Unidos.
Dussel, E. (1991). 2. El método dialéctico de lo abstracto a lo concreto (20, 41-33, 14; 21,3-31,38) :(Cuaderno M. desde la página 14 del manuscrito, terminado a mediados de septiembre de 1857). En E. Dussel, La producción teórica de Marx: un comentario a los grundrisse (págs. 48-63). México D.F.: Siglo XXI Editores. Obtenido de Consult source
Marx, K. (1989). Contribución a la Crítica de la Economía Política. Moscú: Editorial Progreso.
Marx, K. (2007). Elementos Fundamentales para la Crítica de la Economía Política (Grundrisse) 1857-1858 (Vol. I). (J. Aricó, M. Murmis, P. Scaron, Edits., & P. Scaron, Trad.) México, D.F.: Siglo XXI Editores.
Kolmogórov, A. (1956). Foundations of the Theory of Probability (Segunda Edición ed.). New York: Chelsea Publishing Company.
Williamson, J. (2010). In Defence of Objective Bayesianism. Oxford: Oxford University Press.
Kojevnikov, A. (19 de Junio de 2019). PROBABILITY, MARXISM, AND QUANTUM ENSEMBLES. Obtenido de The University of British Columbia: Consult source
Efron, B. (1978). Controversies in the Foundations of Statistics. The American Mathematical Monthly, 231-246.
DeGroot, M., & Schervish, M. (2012). Probability and Statistics. Boston: Pearson Education.
Remaining bibliography · 38 references
Bayes, T. (23 de Diciembre de 1763). An Essay towards solving a Problem in the Doctrine of Chances. Philosophical Transactions of the Royal Society of London, 370-418.
Bernoulli, J. (2006). The Art of Conjecturing (Together to a Friend on Sets in Court Tennis). Maryland: John Hopkins University Press.
Eremenko, A. (30 de Abril de 2020). Stack Exchange, History of Sciences and Mathematics. Obtenido de What was Kolmogorov’s point of view in the philosophy of mathematics?: Consult source
Feller, W. (1968). An Introduction to Probability Theory and Its Applications (Tercera ed., Vol. I). New York: John Wiley & Sons, Inc.
Filosofía en español. (9 de Febrero de 2018). Diccionario filosófico abreviado. Obtenido de URSS: Consult source
Fröhlich, N. (2012). Labour values, prices of production and the missing equalisation tendency of profit rates: evidence from the German economy. Cambridge Journal of Economics, 37(5), 1107-1126.
Frolov, I. T. (1984). Diccionario de filosofía. (O. Razinkov, Trad.) Moscú: Editorial Progreso. Obtenido de Consult source
Fundación del Español Urgente. (23 de Marzo de 2021). Formación de gentilicios extranjeros. Obtenido de Lista de topónimos y gentilicios: Consult source
Fundación del Español Urgente. (23 de Marzo de 2021). -ista (sufijo). Obtenido de Sufijos: Consult source
Gigerenzer, G. (2004). Mindless Statistics. The Journal of Socio-Economics, 587-606.
Greene, W. H. (2012). Econometric Analysis (International Edition). Essex: Pearson Education Limited.
Guerrero Jiménez, D. (2018). TRABAJO IMPRODUCTIVO, CRECIMIENTO Y TERCIARIZACIÓN (30 AÑOS DESPUÉS DE MARX Y KEYNES). International Journal of Political Economy, 1-16. Obtenido de Consult source
Haldane, J. B. (1945). Science and Everyday Life. Allahabad,: Kitab Mahal Publishers.
Hegel, F. (1968). Ciencia de la Lógica. Buenos Aires: Solar / Hachette.
Johnsen, J. (17 de Enero de 2019). What is the difference between positivism and empiricism? Obtenido de Quora: Consult source
Kohan, N., & Brito, P. (1 de Febrero de 2009). Marxismo para principiantes. Obtenido de nodo50: Consult source
Laplace, P.-S. (2015). Ensayo Filosófico Sobre Probabilidades. Ciudad de México: Biblioteca Digital del Instituto Latinoamericano de Comunicación Educativa. Obtenido de Consult source
Lenin, V. (1974). Cuadernos Filosóficos. Madrid: Editorial Ayuso.
Loughborough University. (21 de Febrero de 2008). Total Probability and Bayes’ Theorem. Obtenido de The theorem of total probability: Consult source
Maibaum, G. (1988). Teoría de Probabilidades y Estadística Matemática. (M. Á. Pérez, Trad.) La Habana, Cuba: Editorial Pueblo y Educación.
Marx, K. (1894). Capital. A Critique of Political Economy (Vol. III). New York: International Publishers.
Marx, K. (2010). El Capital (Vol. I). México, D.F.: Fondo de Cultura Económica.
Marx, K., & Engels, F. (1987). Karl Marx and Friedrich Engels Collected Works (Vol. XLII). Moscú: Progress Publishers.
Mittelhammer, R. (2013). Mathematical Statistics for Economics and Business (Segunda ed.). New York: Springer.
Nabi, I. (2020). Algunas Reflexiones Sobre la Distribución Binomial Negativa II (Un Análisis Teórico y Aplicado). Documento Inédito. Obtenido de Consult source
Nabi, I. (21 de Marzo de 2021). Sobre el papel y la viabilidad de la violencia en la lucha social de las mujeres en particular y en las luchas sociales en general. Obtenido de El Blog de Isadore Nabi: Consult source
Nabi, I., & B.A., A. (1 de Abril de 2021). UNA METODOLOGÍA EMPÍRICA PARA LA DETERMINACIÓN DE LA MAGNITUD DE LAS INTERRELACIONES SECTORIALES DENTRO DE LA MATRIZ INSUMO-PRODUCTO DESDE LOS CUADROS DE PRODUCCIÓN Y USOS PARA EL CASO DE ESTADOS UNIDOS 1997-2019. Obtenido de El Blog de Isadore Nabi: Consult source
North Carolina State University. (27 de Septiembre de 2020). People – Department of History . Obtenido de Dr Edith D Sylla: Consult source
Perezgonzalez, J. (3 de Marzo de 2015). Fisher, Neyman-Pearson or NHST? A tutorial for teaching data testing. (L. Roberts, Ed.) Frontiers in Psychology, 6(223), 1-11. doi:10.3389/fpsyg.2015.00223
Poisson, S.-D. (2013). Researches into the Probabilities of Judgments in Criminal and Civil Cases. (O. Sheynin, Ed.) Berlin: arXiv. Obtenido de Consult source
Rosental, M. M., & Iudin, P. F. (1971). DICCIONARIO FILOSÓFICO. San Salvador: Tecolut.
Rosental, M., & Iudin, P. (1959). Diccionario filosófico abreviado. Montevideo: Ediciones Pueblos Unidos.
Russell, K. (29 de Enero de 2014). University of Manitoba. Obtenido de Hypothesis testing: Consult source
StackExchange Philosophy. (15 de Junio de 2015). How empiricism and positivism is distinguished? What's their differences? Obtenido de Philosophy: Consult source
TECH2 NEWS STAFF. (28 de Noviembre de 2019). SCIENTISTS MAY HAVE DISCOVERED A FIFTH FUNDAMENTAL 'FORCE OF NATURE,' THEY’RE CALLING IT X17. Obtenido de TECH2: Consult source
Wikipedia. (27 de Septiembre de 2020). Population Genetics. Obtenido de J. B. S. Haldane: Consult source
Wikipedia. (23 de Septiembre de 2020). Statistics. Obtenido de Inverse Probability: Consult source
Wikipedia. (13 de Marzo de 2021). Relación de incertidumbre. Obtenido de Mecánica cuántica: Consult source
Research document · Spanish
Continue with the complete lessons
The full study develops the sources, arguments, economic applications, and research notes introduced here.
Original public edition in Spanish · 2021 · 289 pages. This English presentation draws on the revised manuscript. The linked PDF retains the title and content of the original edition.
José Mauricio Gómez Julián Marxist Philosophy of Science · Theory of knowledge
Marxist Philosophy of Science Mathematics · Computing · Method
Sets and Combinatorics in R
A computational exploration of Cartesian products, power sets, permutations, and combinations
JOSÉ MAURICIO GÓMEZ JULIÁN
Originally published on April 4, 2021 · Revised and expanded edition
Abstract
This article brings together definitions, counting arguments, and reproducible experiments in R to study constructions on finite sets. It examines Cartesian products and power sets, ordered selections with and without repetition, combinations, and multiset permutations. The discussion distinguishes mathematical objects from their computational representations, explains the role of order and multiplicity, clarifies the relationship between power sets and σ-algebras, and separates the calculation of cardinalities from exhaustive enumeration.
Counting means establishing which differences make two objects distinct. Before running a function or applying a formula, we need to decide whether order matters, whether an element may appear more than once, and whether we are constructing subsets, sequences, or collections with multiplicities. R provides an accessible laboratory for making these decisions visible and comparing their consequences with results we can inspect.
The empirical approach proposed here consists of constructing finite examples, comparing representations, and checking identities in specific cases. This activity has exploratory and educational value and helps check implementations. An identity that holds for every size also requires a general argument: running a program for several values does not, by itself, turn an observed regularity into a theorem.
1. Mathematical objects and their representation in R
A set is a collection whose defining feature is membership: writing an element more than once or changing the order of presentation does not create a different set. Thus, {a, b, a} and {b, a} represent the same set. By contrast, the sequences (a, b) and (b, a) are different, and the sequence (a, a) has two positions even though it uses only one distinct symbol.
A multiset records how often each type of element appears, without distinguishing positions. For example, {a, a, b} and {a, b, a} describe the same multiset, which differs from {a, b, b}. Here, braces are used informally to display multiplicities; the context will make clear whether they denote an ordinary set or a multiset.
Which distinctions each object preserves
Object
Order
Multiplicity
Example
Set
Does not distinguish positions
Membership only
{a, b} = {b, a, a}
Sequence or tuple
Distinguishes positions
May include repetitions
(a, b) ≠ (b, a)
Multiset
Does not distinguish positions
Preserves quantities
{a, a, b} ≠ {a, b, b}
An R vector has positions and can contain repeated values; it is therefore not automatically a mathematical set. In this article, we represent sets using finite numeric vectors or character vectors with no duplicates or missing values. The function unique() removes repeated values; setequal() compares membership while ignoring order and duplicates, whereas identical() compares R objects strictly. R’s set operations are documented in [1].
R code
x <- c("a", "b", "a")
A <- unique(x)
A
# [1] "a" "b"
setequal(x, c("b", "a"))
# [1] TRUE
identical(c("a", "b"), c("b", "a"))
# [1] FALSE
1.1. Basic operations and verifiable laws
The union A ∪ B contains elements belonging to at least one of the sets; the intersection A ∩ B retains their common elements; and the difference A ∖ B retains elements belonging to A but not to B. Taking a complement requires a specified universe Ω: Ac = Ω ∖ A. Without an explicit universe, the phrase “everything that is not in A” does not determine a particular set.
R code
S <- c("a", "b")
T <- c("b", "c")
Omega <- c("a", "b", "c", "d")
union(S, T) # "a" "b" "c"
intersect(S, T) # "b"
setdiff(S, T) # "a"
setdiff(T, S) # "c"
setdiff(Omega, S) # "c" "d"
all(S %in% union(S, T)) # TRUE: S is a subset of S union T
For finite sets, the identity |S ∪ T| = |S| + |T| − |S ∩ T| corrects the double counting of shared elements. De Morgan’s laws describe how complements interact with unions and intersections: (S ∪ T)c = Sc ∩ Tc and (S ∩ T)c = Sc ∪ Tc. These laws follow by examining the membership of an arbitrary element; the script also checks them in this example.
2. Preparing the environment and reading conventions
R is the language and computing environment; RStudio is a development interface for working with R. The code can also be run in the R console or with Rscript. The basic examples use R and its standard packages; additional enumerations use gtools, and rje::powerSet() is retained as an alternative way to construct the power set [7].
R code
# Run once if not already installed:
# install.packages(c("gtools", "rje"))
A <- c("a", "b", "c") # Small, inspectable example
B <- c(1, 2, 3)
reference_set <- 1:10 # Reference numerical example
n <- length(reference_set)
k <- 3 # Selection length, where applicable
Outputs preceded by # are comments indicating expected results, not additional instructions. The following sections contain teaching examples; the complete script assembles the helper functions and checks in executable order. Package names are made explicit with package::function() to identify where each operation comes from.
The visual representations and tables are built from text and accessible HTML. This allows readers to select symbols, zoom in, and consult formulas without relying on screenshots.
3. Cartesian products: constructing pairs and tuples
Given two sets A and B, their Cartesian product A × B is the set of all ordered pairs (a, b) such that a belongs to A and b belongs to B. If both sets are finite, each element of A can be paired with every element of B:
A × B = {(a, b) : a ∈ A, b ∈ B} |A × B| = |A| · |B|
Vertical bars denote cardinality.
A Cartesian product preserves coordinate positions. In general, A × B and B × A are different sets, even though they have the same cardinality. If both factors are the same set, pairs such as (a, a) may occur: sharing the same source set does not require the coordinates to be distinct.
A = {a, b, c}; B = {1, 2, 3}; nine ordered pairs
A / B
1
2
3
a
(a, 1)
(a, 2)
(a, 3)
b
(b, 1)
(b, 2)
(b, 3)
c
(c, 1)
(c, 2)
(c, 3)
In R, expand.grid() produces one row for each combination of coordinates. In the function’s documentation, “combination” refers to a configuration of variable values, not an unordered selection in the combinatorial sense used in later sections. We specify stringsAsFactors = FALSE to keep text as text, and omit auxiliary attributes with KEEP.OUT.ATTRS = FALSE[2].
R code
product <- expand.grid(
A = A, B = B,
KEEP.OUT.ATTRS = FALSE,
stringsAsFactors = FALSE
)
nrow(product)
# [1] 9
product_10 <- expand.grid(
x = reference_set, y = reference_set,
KEEP.OUT.ATTRS = FALSE,
stringsAsFactors = FALSE
)
nrow(product_10)
# [1] 100
For k coordinates chosen from A, we write Ak. If |A| = n, then |Ak| = nk. For k = 0, we adopt A0 = {()}, the set containing the empty tuple. This convention keeps the cardinality equal to 1 even when A is empty.
4. Power sets: all subsets of a set
The power set of A, written 𝒫(A), contains all its subsets, including ∅ and A itself. When constructing a subset, each element of A has two possibilities: it is either included or excluded. For n distinct elements, the multiplication principle gives:
𝒫(A) = {S : S ⊆ A} |𝒫(A)| = 2n
For A = {a, b, c}, the eight subsets can be grouped by size. This arrangement by levels makes the distribution 1, 3, 3, 1 visible; a complete Hasse diagram would also include the covering relations between adjacent levels.
{a, b, c}
{a, b}{a, c}{b, c}
{a}{b}{c}
∅
The power set arranged by cardinality
We can construct it using standard R by enumerating combinations of every size. The code uses indices to prevent combn() from interpreting a numeric vector of length one as a population size.
R code
A <- c("a", "b", "c")
PA <- list(A[0]) # A list that CONTAINS the empty subset
for (k in seq_len(length(A))) {
indices <- utils::combn(seq_along(A), k, simplify = FALSE)
PA <- c(PA, lapply(indices, function(i) A[i]))
}
length(PA)
# [1] 8
# Alternative using the package from the original version:
# PA <- rje::powerSet(A)
# P10 <- rje::powerSet(1:10)
# length(P10) # 1024
4.1. Why the power set is a σ-algebra
A σ-algebra ℱ on a universe Ω is a family of subsets that contains Ω and is closed under complements relative to Ω and under countable unions. These conditions imply that it also contains ∅. The power set 𝒫(Ω) satisfies these conditions because it contains every possible subset of Ω [6].
However, “power set” and “σ-algebra” are not synonyms. On Ω = {a, b, c}, both {∅, Ω} and {∅, {a}, {b, c}, Ω} are σ-algebras; neither equals the eight-element family 𝒫(Ω). The first distinguishes only the impossible event from the certain event; the second distinguishes the block {a} from the block {b, c}.
On a finite universe, every family of subsets is finite. Consequently, checking closure under binary unions, together with the other axioms, establishes closure under countable unions: only finitely many distinct subsets can occur in such a union. The script provides a checker for this finite setting. This procedure cannot be extended to infinite universes without further justification.
4.2. The connection with combinations
The subsets of A are partitioned by size. There is exactly one subset of size zero, n subsets of size one, and, in general, a binomial number of subsets of size k. Therefore:
∑k=0n C(n, k) = 2n
This counts every subset once, including the empty set and the full set.
The correspondence between a subset and its vector of membership indicators also establishes a bijection between 𝒫(A) and {0, 1}n. This is a structural connection with binary sequences, not merely a numerical coincidence.
5. Ordered selections without repetition
We choose k distinct elements from n available elements and assign each one a position. There are n possibilities for the first position, n − 1 for the second, and so on. For 0 ≤ k ≤ n:
P(n, k) = n(n − 1) ··· (n − k + 1) = n!(n − k)!
These constructions are also called variations, arrangements, or partial permutations without repetition, depending on the terminological tradition. When k = n, we obtain full permutations: P(n, n) = n!. If k > n, there are no possible selections without repetition, and for k = 0 there is a unique empty selection.
With A = {a, b, c} and k = 2, we obtain six sequences: (a, b), (a, c), (b, a), (b, c), (c, a), and (c, b). The same pair of symbols can appear in two different orders.
R code
n <- 10
k <- 3
prod(n - seq_len(k) + 1)
# [1] 720
# Small enumeration; requires gtools:
# p2 <- gtools::permutations(3, 2, v = A)
# nrow(p2) # 6
The falling product avoids calculating two large factorials separately, although it can also overflow for large sizes. The general function in the script handles k = 0 first, avoiding misleading descending ranges.
5.1. Summing over different lengths
Counting all nonempty sequences of lengths 1 through n means summing P(n, k), rather than calculating a single permutation count. For n = 10, the total is 9,864,100; including the empty tuple raises it to 9,864,101. Lengths distinguish objects: a sequence with two positions is never the same as a sequence with three.
6. Ordered selections with unrestricted repetition
If a symbol may reappear in every position, there are always n options, and the available set does not shrink. Sequences of length k form the Cartesian product Ak:
R(n, k) = nk
For A = {a, b, c} and k = 2, the nine results are (a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), and (c, c). Compared with the previous section, the pairs containing identical symbols have been added.
R code
n <- 10
k <- 3
n^k
# [1] 1000
# With unrestricted repetition, explicitly enabled:
# r2 <- gtools::permutations(
# 3, 2, v = A, repeats.allowed = TRUE
# )
# nrow(r2) # 9
# No external packages needed for the two-position case:
pairs <- expand.grid(x = A, y = A,
KEEP.OUT.ATTRS = FALSE,
stringsAsFactors = FALSE)
nrow(pairs)
# [1] 9
For n = 10 and k = 10, the count is 10,000,000,000. Here, it is best to calculate the cardinality and stop: exhaustive generation is not a reasonable example for an ordinary computer. The script’s memory safeguard prevents this enumeration.
7. Combinations without repetition: choosing subsets
We now choose k distinct elements without distinguishing order. Every subset of size k gives rise to k! ordered sequences. Dividing the preceding count by that number gives:
C(n, k) = nk = n!k!(n − k)!
Factorial domain: 0 ≤ k ≤ n. By combinatorial convention, C(n, k) = 0 when k > n.
With A = {a, b, c} and k = 2, there are three subsets: {a, b}, {a, c}, and {b, c}. A program’s writing {a, b} in the order of the input vector does not give the subset positions.
R code
choose(10, 3)
# [1] 120
A <- c("a", "b", "c")
indices <- utils::combn(seq_along(A), 2, simplify = FALSE)
subsets <- lapply(indices, function(i) A[i])
length(subsets)
# [1] 3
# Another representation: each column is a combination.
utils::combn(A, 2)
# [,1] [,2] [,3]
# [1,] "a" "a" "b"
# [2,] "b" "c" "c"
choose() calculates cardinalities; combn() generates selections. By default, the latter arranges results in columns, whereas gtools::combinations() arranges them in rows. Confusing these orientations leads to counting the length of a selection instead of the number of selections [3].
For a ten-element set, summing combinations of sizes 1 through 10 gives 1,023 nonempty subsets. Adding the empty set yields the 1,024 elements of its power set. The identity C(n, k) = C(n, n − k) also follows from the bijection that assigns each subset its complement.
8. Combinations with repetition: choosing multiplicities
There are n available types; we choose a total of k units and allow types to repeat, without distinguishing positions. An outcome can be described by nonnegative integer multiplicities x₁, …, xₙ satisfying:
x₁ + x₂ + ··· + xₙ = k, xᵢ ≥ 0
For n ≥ 1, the stars and bars method represents the k units as stars and separates the n categories with n − 1 bars. Choosing the positions of the stars among k + n − 1 places is equivalent to choosing the multiplicities:
M(n, k) = C(n + k − 1, k) = (n + k − 1)!k!(n − 1)!
With A = {a, b, c} and k = 2, the six selections are {a, a}, {a, b}, {a, c}, {b, b}, {b, c}, and {c, c}. In particular, {a, b} and {b, a} represent the same outcome. For n = 10 and k = 3, the count is 220.
R code
n <- 10
k <- 3
choose(n + k - 1, k)
# [1] 220
# Small enumeration; requires gtools:
# m2 <- gtools::combinations(
# 3, 2, v = A, repeats.allowed = TRUE
# )
# nrow(m2) # 6
As an additional visual example, with the four symbols {1, 2, 3, 4} and size 2, the selections are 11, 12, 13, 14, 22, 23, 24, 33, 34, and 44. These are ten combinations with repetition; there would be sixteen ordered sequences because they would distinguish, for example, 12 from 21. A tree enumerating only those ten branches belongs in this section.
9. APPLE: permuting a fixed multiset
The word APPLE has five positions but four types of letters: A, P, L, and E. The letter P occurs twice. Swapping those two Ps does not change the visible word. Its distinct rearrangements therefore number neither 5! nor the unrestricted five-letter sequences drawn from four symbols.
When n objects have fixed multiplicities m₁, …, mᵣ summing to n, each visible sequence is counted m₁! ··· mᵣ! times if we artificially distinguish every object. Removing that distinction gives:
The distinction has two parts: combinations with repetition choose the multiplicities; multiset permutations take the multiplicities as fixed and count the possible orders. The script checks the APPLE example through a small enumeration: it permutes the five positions and removes duplicate words. This strategy is educational; it is inefficient for large multisets because it generates duplicates before discarding them.
The correct formula depends on which distinctions are preserved by the object we want to count.
10. Comparison tables and totals by length
The following formulas answer different questions. For n = 10 and k = 3, comparing them shows how much the count changes when we alter the importance of order or the possibility of repetition.
Four selection problems; n = 10 and k = 3
Construction
Order
Repetition
Cardinality
Result
Ordered, without repetition
Matters
No
n! / (n − k)!
720
Ordered, with unrestricted repetition
Matters
Yes
nk
1,000
Combinations without repetition
Does not matter
No
C(n, k)
120
Combinations with repetition
Does not matter
Yes
C(n + k − 1, k)
220
The two formulas without repetition require k ≤ n to use their factorial expressions. The formula for combinations with repetition applies directly for n ≥ 1. All models include one empty selection when k = 0.
Counting and enumerating are different operations
Object
Count
Enumerate in small examples
Product A × B
length(A) * length(B)
expand.grid()
Power set
2^length(A)
The script’s power_set(); rje::powerSet()
Ordered, without repetition
count_ordered(n, k)
gtools::permutations()
Ordered, with repetition
n^k
gtools::permutations(..., repeats.allowed = TRUE)
Combinations without repetition
choose(n, k)
utils::combn()
Combinations with repetition
choose(n + k - 1, k)
gtools::combinations(..., repeats.allowed = TRUE)
10.1. The eleven sizes for n = 10
The k = 0 row is essential for identities that include the empty object. The table presents counts without generating the corresponding collections.
n = 10; k from zero to ten
k
Ordered without repetition
Ordered with repetition
Combinations without repetition
Combinations with repetition
0
1
1
1
1
1
10
10
10
10
2
90
100
45
55
3
720
1,000
120
220
4
5,040
10,000
210
715
5
30,240
100,000
252
2,002
6
151,200
1,000,000
210
5,005
7
604,800
10,000,000
120
11,440
8
1,814,400
100,000,000
45
24,310
9
3,628,800
1,000,000,000
10
48,620
10
3,628,800
10,000,000,000
1
92,378
10.2. Nonempty totals: k from 1 through 10
Sums over lengths, with n = 10
Family
Sum
Total
Ordered, without repetition
∑ P(10, k)
9,864,100
Ordered, with unrestricted repetition
∑ 10k
11,111,111,110
Combinations without repetition
210 − 1
1,023
Combinations with repetition
C(20, 10) − 1
184,755
The last result follows from the identity ∑k=0m C(n + k − 1, k) = C(n + m, m), for n ≥ 1. One justification adds an auxiliary category that absorbs the units needed to reach m: choosing up to m units from n types is equivalent to distributing exactly m units among n + 1 types. We subtract one from the final count to exclude the empty selection.
With unrestricted repetition and at least one available symbol, allowing k to grow without a bound produces infinitely many sequences and infinitely many finite multisets. The range of sizes must therefore be stated whenever a “total” is presented.
Knowing a cardinality does not require materializing every object. A hypothetical matrix with 1010 rows, ten columns, and eight bytes per cell would require approximately 800 decimal GB for the data alone, before accounting for auxiliary structures and temporary copies. This estimate applies to that specific representation; actual consumption depends on the data type and algorithm.
The script therefore sets teaching limits of 100,000 results and 1,000,000 estimated cells before enumeration. These limits do not guarantee that every admitted computation will be fast or use little memory; they prevent accidental execution of the largest examples. For larger tasks, consider lazy generation, partial traversal, or specialized algorithms.
11.1. Overflow and approximate arithmetic
R’s double numbers do not represent every arbitrarily large integer exactly. Rounding can occur even when the result is not infinite. A quotient of factorials does not avoid this problem either: intermediate factorials can overflow even when the final result is representable.
choose() calculates binomial coefficients, and lchoose() obtains their logarithms without first forming the full quantity; lfactorial() serves an analogous purpose for factorials [5]. To estimate an order of magnitude:
R code
# Base-10 logarithm of C(2000, 1000):
log10_binomial <- lchoose(2000, 1000) / log(10)
log10_binomial
# Approximately 600.3114
# Logarithm of P(n, k), within the domain 0 <= k <= n:
n <- 1000
k <- 500
log_P <- lfactorial(n) - lfactorial(n - k)
Working with logarithms reduces numerical range problems; it does not turn a calculation into exact integer arithmetic. Nor should an enormous logarithm automatically be exponentiated, since overflow may reappear. If every digit of a large integer is needed, an appropriate arbitrary-precision tool must be selected and validated for that purpose.
11.2. What the checks establish
The script compares counts with small enumerations, checks summation identities, and examines boundary cases. It also avoids confusing an empty list with a list containing the empty selection. These tests detect implementation errors, inconsistent conventions, and regressions. Explanations based on bijections and counting principles justify the results beyond the finite grid that is checked.
12. Mathematical interpretation and applications
The preceding constructions are related by specific functions and correspondences. Forgetting the order of a sequence of length k without repetition produces a subset of size k, and every subset has k! preimages. If repetitions are allowed, forgetting order produces a multiset; in this case, the number of sequences associated with each multiset depends on its multiplicities.
This difference explains why dividing nk by k! does not, in general, give the number of combinations with repetition. For A = {a, b} and k = 2, the sequences aa and bb each have a single visible order, whereas ab and ba are grouped into the same multiset. The groups do not all have the same size.
12.1. Probability: enumeration does not specify a distribution
Suppose we make two independent draws with replacement from a mechanism that produces a and b with probability 1/2 each. The sequences aa, ab, ba, and bb each have probability 1/4. After forgetting order, the three multisets {a, a}, {a, b}, and {b, b} have probabilities 1/4, 1/2, and 1/4. Choosing uniformly among these three multisets therefore does not reproduce the same experiment.
The formula “favorable outcomes / possible outcomes” requires the counted outcomes to be equally probable. Defining the sample space, the selection mechanism, and the distribution is an additional step beyond enumerating objects.
12.2. Experimental design and data science
Cartesian product: constructing every configuration in a parameter grid or every combination of factor levels. If the factors have different sizes, multiply their cardinalities.
Power set: describing possible selections of variables. A set of n variables has 2n subsets, including the model with no selected predictors.
Ordered selections without repetition: assigning distinct positions to different people, with one candidate per position and no reuse.
Combinations without repetition: selecting committees or subsets of observations when selection order is not part of the outcome.
Combinations with repetition: distributing indistinguishable units among identified categories, with nonnegative quantities and no upper bounds.
The assumptions must be reviewed for each application. Incompatible factor combinations, limited inventories, unequal probabilities, or additional constraints change the problem. A table of formulas guides the choice; the definition of the object and mechanism determines which formula is appropriate.
12.3. Epistemological scope
A computational representation selects which information to retain. Converting a sequence into a set removes order and multiplicity; converting it into a multiset removes order but preserves frequencies. These transformations show that a mathematical abstraction is more than a change of name: it changes the distinctions organizing the problem and, with them, the counting results.
Computational experimentation becomes more informative when that selection is explicit. Defining, representing, deducing, executing, and comparing are complementary stages of reproducible inquiry into finite objects.
13. Complete reproducible script
The following program brings the article together. Copy its contents into a text file named sets_combinatorics.R, save it as UTF-8, and run it with source("sets_combinatorics.R", encoding = "UTF-8") or Rscript sets_combinatorics.R. External packages are used only if installed; the program reports when it skips those blocks.
The counting functions validate nonnegative integer sizes. The enumerators represent each selection as a vector inside a list, including empty cases. The size safeguard runs before calling the enumerators. The representation domain is restricted to finite numeric vectors and character vectors without missing values; sets of more complex objects require an appropriate equality relation.
Open the complete script · functions, examples, and checksComplete script · copy from the first line through sessionInfo()
# SETS AND COMBINATORICS IN R
# JOSÉ MAURICIO GÓMEZ JULIÁN
# Article script. Save as sets_combinatorics.R (UTF-8).
# The basic examples require no external packages.
# install.packages(c("gtools", "rje")) # Only if the optional blocks are wanted.
# 1. Validation and counting functions -------------------------------
nonnegative_integer <- function(x, name) {
if (!is.numeric(x) || is.complex(x) || length(x) != 1L || is.na(x) ||
!is.finite(x) || x < 0 || x != floor(x)) {
stop(name, " must be a finite nonnegative integer.")
}
invisible(TRUE)
}
validate_nk <- function(n, k) {
nonnegative_integer(n, "n")
nonnegative_integer(k, "k")
}
normalize_set <- function(x) {
if ((!is.numeric(x) && !is.character(x)) || is.complex(x) ||
!is.null(dim(x)) || anyNA(x) ||
(is.numeric(x) && any(!is.finite(x)))) {
stop("Use a finite numeric or character vector, with no NA values or dimensions.")
}
unique(x)
}
count_ordered <- function(n, k, repetition = FALSE) {
validate_nk(n, k)
if (!is.logical(repetition) || length(repetition) != 1L ||
is.na(repetition)) stop("repetition must be TRUE or FALSE.")
if (k == 0) return(1)
if (repetition) return(n^k)
if (k > n) return(0)
# Falling product: avoids forming two factorials separately.
prod(n - seq_len(k) + 1)
}
count_combinations <- function(n, k, repetition = FALSE) {
validate_nk(n, k)
if (!is.logical(repetition) || length(repetition) != 1L ||
is.na(repetition)) stop("repetition must be TRUE or FALSE.")
if (k == 0) return(1)
if (n == 0) return(0)
if (repetition) return(choose(n + k - 1, k))
if (k > n) return(0)
choose(n, k)
}
limit_enumeration <- function(rows, columns,
max_rows = 1e5,
max_cells = 1e6) {
if (!is.finite(rows) || rows > max_rows ||
rows * max(1, columns) > max_cells) {
stop("Enumeration stopped by the teaching limit; calculate only the count.")
}
invisible(TRUE)
}
# All enumerations return lists of vectors.
# For k=0, return list(x[0]): a list containing an empty vector.
enumerate_ordered <- function(x, k, repetition = FALSE) {
x <- normalize_set(x)
n <- length(x)
total <- count_ordered(n, k, repetition)
limit_enumeration(total, k)
if (k == 0) return(list(x[0]))
if (total == 0) return(list())
if (!requireNamespace("gtools", quietly = TRUE)) {
stop("This enumerator requires gtools: install.packages('gtools').")
}
m <- gtools::permutations(n, k, v = x,
repeats.allowed = repetition)
lapply(seq_len(nrow(m)), function(i) unname(m[i, ]))
}
enumerate_combinations <- function(x, k, repetition = FALSE) {
x <- normalize_set(x)
n <- length(x)
total <- count_combinations(n, k, repetition)
limit_enumeration(total, k)
if (k == 0) return(list(x[0]))
if (total == 0) return(list())
if (!repetition) {
# Using indices prevents combn(5, 1) from interpreting 5 as 1:5.
indices <- utils::combn(seq_len(n), k, simplify = FALSE)
return(lapply(indices, function(i) x[i]))
}
if (!requireNamespace("gtools", quietly = TRUE)) {
stop("This enumerator requires gtools: install.packages('gtools').")
}
m <- gtools::combinations(n, k, v = x, repeats.allowed = TRUE)
lapply(seq_len(nrow(m)), function(i) unname(m[i, ]))
}
power_set <- function(x) {
x <- normalize_set(x)
n <- length(x)
limit_enumeration(2^n, n)
result <- list(x[0])
for (k in seq_len(n)) {
result <- c(result, enumerate_combinations(x, k))
}
result
}
format_family <- function(F) {
vapply(F, function(x) {
if (length(x) == 0) "{}" else paste0("{", paste(x, collapse = ", "), "}")
}, character(1))
}
# 2. Sets and Cartesian products -------------------------------
A <- c("a", "b", "c")
B <- c(1, 2, 3)
reference_set <- 1:10
n <- length(reference_set)
stopifnot(
setequal(c("a", "b", "a"), c("b", "a")),
!identical(c("a", "b"), c("b", "a")),
identical(normalize_set(c(1, 1, 2)), c(1, 2))
)
product <- expand.grid(A = A, B = B,
KEEP.OUT.ATTRS = FALSE, stringsAsFactors = FALSE)
stopifnot(nrow(product) == length(A) * length(B))
product_10 <- expand.grid(x = reference_set, y = reference_set,
KEEP.OUT.ATTRS = FALSE, stringsAsFactors = FALSE)
stopifnot(nrow(product_10) == 100)
print(product)
# Set operations and laws within an explicit universe.
S <- c("a", "b")
T <- c("b", "c")
Omega <- c("a", "b", "c", "d")
stopifnot(
length(union(S, T)) == length(S) + length(T) - length(intersect(S, T)),
setequal(setdiff(Omega, union(S, T)),
intersect(setdiff(Omega, S), setdiff(Omega, T))),
setequal(setdiff(Omega, intersect(S, T)),
union(setdiff(Omega, S), setdiff(Omega, T)))
)
# 3. Power set and finite sigma-algebra ----------------------
PA <- power_set(A)
P10 <- power_set(reference_set)
stopifnot(length(PA) == 8, length(P10) == 1024,
sum(lengths(PA) == 0) == 1)
print(format_family(PA))
contains_set <- function(x, F) {
any(vapply(F, function(y) setequal(x, y), logical(1)))
}
is_finite_sigma_algebra <- function(F, omega) {
if (!is.list(F)) return(FALSE)
if (!all(vapply(F, function(x) all(x %in% omega), logical(1)))) {
return(FALSE)
}
if (!contains_set(omega[0], F) || !contains_set(omega, F)) return(FALSE)
for (x in F) {
if (!contains_set(setdiff(omega, x), F)) return(FALSE)
for (y in F) if (!contains_set(union(x, y), F)) return(FALSE)
}
TRUE
}
F_min <- list(A[0], A)
F_intermediate <- list(A[0], "a", c("b", "c"), A)
stopifnot(is_finite_sigma_algebra(F_min, A),
is_finite_sigma_algebra(F_intermediate, A),
is_finite_sigma_algebra(PA, A),
!is_finite_sigma_algebra(list(A[0], "a", A), A))
# On a finite universe, closure under binary unions suffices here:
# any countable union involves only finitely many distinct subsets.
# 4. Small sequence enumeration using base R -----------------------
pairs <- expand.grid(x = A, y = A,
KEEP.OUT.ATTRS = FALSE, stringsAsFactors = FALSE)
without_repetition <- pairs[pairs$x != pairs$y, , drop = FALSE]
stopifnot(nrow(pairs) == 9, nrow(without_repetition) == 6,
length(enumerate_combinations(A, 2)) == 3)
# 5. Count table for n=10, k=0,...,10 ------------------------
ks <- 0:n
counts <- data.frame(
k = ks,
ordered_without = vapply(ks, function(k) count_ordered(n, k), numeric(1)),
ordered_with = vapply(ks, function(k) count_ordered(n, k, TRUE), numeric(1)),
combinations_without = vapply(ks, function(k) count_combinations(n, k), numeric(1)),
combinations_with = vapply(ks, function(k) count_combinations(n, k, TRUE), numeric(1))
)
print(counts, row.names = FALSE)
stopifnot(sum(counts$combinations_without) == 2^n,
sum(counts$combinations_without[-1]) == 1023,
sum(counts$ordered_without[-1]) == 9864100,
sum(counts$ordered_with[-1]) == 11111111110,
sum(counts$combinations_with[-1]) == 184755)
# Lists for all lengths are illustrated with n=3.
# Millions of sequences for n=10 are not enumerated.
combinations_by_k <- lapply(0:length(A), function(k) {
enumerate_combinations(A, k)
})
stopifnot(sum(lengths(combinations_by_k)) == 8)
# 6. APPLE: permutations of a multiset -----------------------
letters_in_word <- strsplit("APPLE", "", fixed = TRUE)[[1]]
multiplicities <- as.numeric(table(letters_in_word))
apple_count <- factorial(length(letters_in_word)) /
prod(factorial(multiplicities))
stopifnot(apple_count == 60)
# 7. Boundary cases and memory safeguards ------------------------
stopifnot(count_ordered(0, 0) == 1,
count_ordered(0, 0, TRUE) == 1,
count_ordered(0, 2, TRUE) == 0,
count_ordered(2, 3) == 0,
count_combinations(0, 0, TRUE) == 1,
count_combinations(0, 2, TRUE) == 0,
length(power_set(numeric(0))) == 1,
identical(enumerate_combinations(5, 1), list(5)),
length(enumerate_combinations(A, 0)) == 1)
expected_failure <- tryCatch({
limit_enumeration(10^10, 10)
FALSE
}, error = function(e) TRUE)
stopifnot(expected_failure)
# Large counts: logarithmic scales do not provide exact arithmetic.
log10_binomial <- lchoose(2000, 1000) / log(10)
print(log10_binomial)
# 8. Optional block: gtools ---------------------------------------
if (requireNamespace("gtools", quietly = TRUE)) {
p2 <- enumerate_ordered(A, 2)
r2 <- enumerate_ordered(A, 2, TRUE)
c2 <- enumerate_combinations(A, 2)
m2 <- enumerate_combinations(A, 2, TRUE)
stopifnot(length(p2) == 6, length(r2) == 9,
length(c2) == 3, length(m2) == 6)
# Compare generated objects with formulas on a small grid.
for (nn in 1:4) {
xx <- letters[seq_len(nn)]
for (kk in 0:4) {
for (rr in c(FALSE, TRUE)) {
stopifnot(
length(enumerate_ordered(xx, kk, rr)) ==
count_ordered(nn, kk, rr),
length(enumerate_combinations(xx, kk, rr)) ==
count_combinations(nn, kk, rr)
)
}
}
}
ordered_by_k <- lapply(0:length(A), function(k) {
enumerate_ordered(A, k)
})
stopifnot(sum(lengths(ordered_by_k)) == 16)
# APPLE: distinguish positions first; then remove duplicate words.
positions <- gtools::permutations(length(letters_in_word), length(letters_in_word))
words <- apply(positions, 1, function(i) paste0(letters_in_word[i], collapse = ""))
apple_words <- unique(words)
stopifnot(length(apple_words) == apple_count)
print(head(apple_words))
} else {
message("gtools not installed: its examples were skipped; base counts were computed.")
}
# 9. Optional block: rje ------------------------------------------
if (requireNamespace("rje", quietly = TRUE)) {
PA_rje <- rje::powerSet(A)
stopifnot(length(PA_rje) == 8,
all(vapply(PA_rje, function(x) contains_set(x, PA), logical(1))))
} else {
message("rje not installed: the base-R power-set construction was used.")
}
cat("Checks completed without errors.\n")
sessionInfo()
The output of sessionInfo() records the R version, platform, and packages attached or loaded through a namespace. Keep it alongside the results. The absence of gtools or rje does not count as a successful check of its respective block: those examples are skipped with an explicit message.
14. References and documentation
The following references support the functions used and the conceptual clarification concerning σ-algebras. The counting arguments are developed in the article through multiplication principles, bijections, and explanations of how outcomes are identified.
R Core Team.Set Operations. Official documentation for the base package. Union, intersection, difference, and set comparison operations. Read the documentation.
R Core Team.Create a Data Frame from All Combinations of Factor Variables. Documentation for expand.grid(). Read the documentation.
R Core Team.Generate All Combinations of n Elements, Taken m at a Time. Documentation for utils::combn(). Read the documentation.
gtools package documentation.Enumerate the Combinations or Permutations of the Elements of a Vector. Reference for combinations(), permutations(), and their arguments. Read the CRAN documentation.
R Core Team.Special Functions of Mathematics. Documentation for choose(), lchoose(), factorial(), and lfactorial(). Read the documentation.
Kehle, Christoph (2025).Introduction to Functional Analysis. Lecture notes for 18.102, Massachusetts Institute of Technology, Section 3.1, Definition 3.1 and example, pp. 25–26. Definition of a σ-algebra and the power-set example with counting measure. Read the lecture notes.
rje package documentation.Miscellaneous Useful Functions for Statistics. Reference index, powerSet entry; also available from R with help("powerSet", package = "rje"). Read the package index.
Marxist Philosophy of Science · Philosophy of science
· · ·
A historical, theoretical, and mathematical investigation into the dialectical character of the epistemological foundations of complexity in long-term nonlinear dynamical systems
Determinism, uncertainty, scales, and emergence: toward a materialist interpretation of the relationships between laws, phenomena, and the ways we come to know them.
JOSÉ MAURICIO GÓMEZ JULIÁN
Originally published on April 4, 2021 · Revised and expanded web edition
Abstract
The possibility that deterministic laws can produce irregular behavior and severe limits on prediction requires us to distinguish between the determination of processes, their mathematical representation, and the conditions under which they can be known. This investigation proposes a dialectical materialist interpretation of this problem by connecting the categories of essence, phenomenon, appearance, form, and content with nonlinear dynamics, fractal geometry, critical phenomena, renormalization, and probability theory. Its historical account encompasses Laplace, Poincaré, Lorenz, and Mandelbrot; the tradition of Hegel, Marx, Engels, and Lenin; Soviet philosophical systematization; and scientific contributions associated with dialectical biology. Applications in astronomy, cardiovascular physiology, computing, econometrics, neuroscience, psychiatry, hydrodynamics, and geodynamics are examined. On this basis, the essay develops distinctions between nonlinearity, chaos, and complexity; between geometric irregularity and dynamical instability; between scaling and temporal evolution; and between ergodicity, independence, and exchangeability. The thesis advanced is that dialectics can guide an investigation of relationships, mediations, transformations, and levels of organization, provided that its interpretations are confronted with specific models and evidence. The philosophical correspondences proposed are distinguished from the mathematical definitions and empirical findings that serve as their points of reference.
That a phenomenon has causes, that a model determines a unique evolution, and that an observer can predict it are different claims. Conflating them leads both to a naive determinism—according to which knowing the laws would be enough to anticipate any event—and to an equally hasty interpretation of irregularity as the absence of lawfulness. The study of nonlinear dynamical systems makes it possible to examine this distinction with precise tools while raising philosophical questions about the relationship between reality and its representations.
The central question is how determination, uncertainty, and transformation are articulated when processes involve multiple scales, feedbacks, instabilities, and levels of organization. This question requires consideration of both the material conditions of phenomena and the historical, technical, and conceptual conditions under which they are known. A measurement has finite resolution; a model selects variables; a simulation uses numerical procedures; an explanation identifies mechanisms within a delimited scope. None of these mediations is external to scientific practice.
The thesis of this essay is that dialectical materialism can help organize this investigation through an analysis of the relationships between wholes and parts, continuity and discontinuity, necessity and contingency, and structure and transformation. Its fruitfulness depends on how these relationships are specified. A suggestive analogy may open a question; establishing an explanatory correspondence requires definitions, mechanisms, conditions of application, and empirical or theoretical scrutiny.
The proposed relationship is reciprocal. Scientific developments allow philosophical categories to be revised when, isolated from research, they risk becoming general formulas. Philosophical reflection, in turn, makes it possible to question the assumptions of models, the selection of scales, the meaning of their limits, and extrapolations between disciplines. This reciprocity is understood as a research program requiring specific developments; claiming its universal and optimal usefulness would require comparisons that an investigation of this scope cannot assume have already been made.
The essay develops a historical and conceptual reconstruction, accompanied by mathematical examples and methodological discussion. Figures taken from earlier investigations are presented as documentary evidence from their respective authors. The pedagogical calculations introduced here clarify concepts; they do not constitute new estimates for the physical, biological, or social systems cited.
II · Philosophical background
Essence, phenomenon, appearance, form, and content
A tradition of problems and reconstructions
Questions about what things are, how they manifest themselves, and how their components relate to one another run through ancient philosophy and reappear, with different meanings, in modern systems. Substance, essence, and phenomenon belong to this history, although their meanings are not interchangeable. Investigating their connections requires recognition of the diversity of traditions: there is no single definition that can be transferred without mediation from ancient Greece to Hegel, or from Hegel to a contemporary scientific theory.
Hegelian logic provides a decisive reference for understanding the elaboration of relationships between essence and appearance, form and content, and universality and particularity. Marx and Engels critically rework this inheritance from a materialist standpoint. Their attention to historical relationships, the organization of processes, and the forms in which social relationships manifest themselves offers a starting point for the philosophy of science. Identifying that starting point requires studying their concrete analyses; an intellectual affiliation alone does not demonstrate the validity of a scientific proposition.
Lenin continues this debate in Materialism and Empirio-Criticism, written in 1908 and published in 1909, by discussing the relationships between matter, experience, sensation, and knowledge. His positions can be examined alongside the problems raised by the physics of his time, distinguishing the historical controversy from later technical formulations. A philosophical tradition’s relationship with the sciences involves interpretation, reception, and criticism as well as continuity.
Soviet philosophy and its historical context
The systematization of categories in Rosental and Iudin’s Philosophical Dictionary is a reference point of the original work. The cited Spanish edition of 1971 must be identified as an edition within an earlier publishing tradition. The foreword reproduced in that edition refers to the Twentieth and Twenty-Second Congresses of the Communist Party of the Soviet Union and to the reassessment of the consequences of the cult of personality. These events and the associated intellectual processes should not be dated to 1971 simply because that is the date of the copy consulted.
An assessment of this endeavor can recognize its thematic breadth—philosophy, history, political economy, biology, physics, logic, aesthetics, ethics, and art—while also examining its institutional conditions. Accepting its contributions does not require adopting every formulation without revision, and criticism of Soviet policies does not justify dismissing its philosophical production wholesale. A materialist recovery of this history requires a differentiated assessment of authors, arguments, institutions, and periods.
The recognition expressed here toward this tradition consists in continuing its problems, scrutinizing its answers, and developing the categories systematized within it. Its function in this investigation is to provide an initial vocabulary for studying relationships that scientific models allow us to specify in new ways.
Dialectical biology and relationships between levels
The history of relationships between Marxism and the natural sciences extends beyond the Soviet Union. J. B. S. Haldane contributed to the development of evolutionary genetics and the modern synthesis; Aleksandr Oparin and Haldane formulated proposals concerning the origin of life; Stephen Jay Gould and Richard Lewontin critically examined adaptationism; and Richard Levins investigated problems of evolution, ecology, and biological organization. These names designate distinct contributions and positions whose relationship with a dialectical perspective must be reconstructed in each case.
The joint works of Levins and Lewontin, particularly The Dialectical Biologist (1985) and Biology Under the Influence (2007), provide references for studying the reciprocity between organisms and environments, the selection of explanatory levels, and the commitments involved in modeling. The relationship between parts and wholes appears there as a research problem: components transform the conditions in which they act and are, in turn, transformed by them.
Landau in physics and Semenov in chemistry remain reference points within the scientific horizon considered by the original text. However, a scientist’s membership in a Soviet context is not enough to attribute every result to a specific philosophical methodology. Supporting such an attribution requires sources concerning their practices, arguments, and research decisions. The value of their results can be examined independently of that historiographical question.
Working definitions
Philosophical categories and research questions
Category
Proposed use in this essay
Guiding question
Essence
Relationships and mechanisms that an investigation identifies as constitutive of a process under specified conditions.
Which relationships explain the organization and reproduction of the phenomenon?
Phenomenon
A concrete manifestation of a process, accessible through practices of observation and intervention.
What happens, and how does it present itself under particular conditions?
Appearance
A mode of presentation that may partially reveal a relationship while also concealing its mediations.
Which interpretations does the observation suggest, and which relationships remain outside that presentation?
Form
The configuration, arrangement, and mode of organization of content.
How are the elements and their relationships organized?
Content
The components, interactions, and processes that constitute the object of study.
Which processes and relationships are organized in this way?
These definitions neither exhaust the history of the categories nor establish a universal equivalence between philosophical and scientific vocabularies. They propose explicit uses for this essay. The same observation may be analyzed as a phenomenon in one question and as part of a system’s content in another; this variation requires justification of the scale and the object of reference.
The distinction between essence and phenomenon makes it possible to investigate mechanisms that an isolated observation does not reveal. It does not authorize dismissing any result inconvenient to a theory as mere appearance. A claim about an essential relationship must improve explanation, organize evidence, and allow procedures for revision. Form and content likewise condition one another: changing the organization of components may modify what a system does even when certain material elements remain the same.
Knowing a process requires reconstructing its mediations: what is observed, the relationships that produce it, and the conditions under which an explanation is valid.
An interpretive proposition advanced in this essay
An initial application can be formulated for a convection model. State variables represent selected quantities; equations express proposed relationships between them; a trajectory shows an evolution; and an attractor describes an asymptotic organization under certain conditions. Philosophical categories can guide questions about these relationships, but an equation does not automatically represent essence, nor does a figure constitute mere appearance by definition. That identification depends on what the model can explain and on its relationship with the material system.
From Laplace’s intelligence to the geometries of irregularity
Laplace: lawfulness and incomplete knowledge
The Essai philosophique sur les probabilités first appeared in 1814. The French edition of 1840 and the English translation of 1902 belong to its publishing history and must be distinguished from that first publication. In Laplace’s account, an intelligence that knew all forces and positions could bring the movements of the universe together within a single description. The scope of this ideal depends on exhaustive knowledge that does not correspond to the situation of a human observer.
The very resort to probability theory expresses the distance between that ideal and the conditions of effective knowledge. One source of Laplace’s philosophical interest lies in his articulation of a conception of natural lawfulness with procedures for reasoning when information is incomplete. The thesis that every event has causes should not be confused with a demonstration that any phenomenon can be predicted operationally.
Laplace’s discussion includes the principle of sufficient reason, the relationship between present and past events, and the interpretation of chance in terms of our ignorance of causes. These ideas retain historical value, although their extension to every domain of reality must be argued separately. Deterministic classical mechanics provides one framework; the different interpretations of quantum physics and probability open controversies that are not settled by the success of a classical model.
Poincaré: small differences and large consequences
Poincaré’s investigations of celestial mechanics and the three-body problem transformed the understanding of dynamical systems. Science et méthode (1908) contains a celebrated account of how small initial differences can produce large later differences. The problem is not exclusively one of unknown laws: even with a known evolution rule, the finite precision of initial data can severely limit prediction.
A historical reconstruction must recognize the antecedents and multiple contributions of the qualitative theory of differential equations. Poincaré represents a decisive moment, but the history of chaos cannot be reduced to a sequence of isolated founders. Its subsequent development combines analytical results, geometrical methods, computational techniques, and experimental problems.
Lorenz: determinism and nonperiodic evolution
In Deterministic Nonperiodic Flow (1963), Edward N. Lorenz studies a simplified convection model whose dynamics shows the compatibility of deterministic evolution and irregularity. Its historical significance includes discussion of the limits of long-term weather prediction. The article is a landmark in chaos theory; identifying it as the birth of the entire science of complexity would erase other scientific genealogies.
Reference model
ẋ = σ(y − x)
ẏ = x(ρ − z) − y
ż = xy − βz
The variables x, y, and z represent mode amplitudes in the reduced model; σ, ρ, and β are parameters. A common choice for illustrating its dynamics is σ = 10, ρ = 28, and β = 8/3. Terms such as xy and xz introduce nonlinearity. The properties of one particular regime should not be attributed to every parameter value: the same system can display qualitatively different behaviors.
The example’s philosophical appeal lies in a relatively compact rule generating a temporal organization that cannot be reduced to a simple periodic succession. The difficulty of anticipating a particular trajectory coexists with the possibility of studying equilibria, stability, attractor geometry, and statistical properties. The limit of one form of prediction opens other questions about the same process.
Mandelbrot: form, scale, and irregularity
Fractal geometry, developed by Benoît Mandelbrot and other authors, makes it possible to investigate structures for which the smooth figures of elementary geometry are insufficient as descriptive models. Coastlines, mountains, clouds, trees, rough surfaces, and branching networks motivate questions about how a measurement changes as resolution varies. The 1982 edition of The Fractal Geometry of Nature and the 1983 edition cited in the original investigation must be distinguished bibliographically.
This contribution does not eliminate Euclidean geometry or imply that every natural object is fractal. It proposes concepts suited to describing certain irregularities and scaling laws. In material phenomena, these properties are usually verified within finite intervals of resolution; extending them to arbitrarily small or large sizes requires an additional idealization.
Relating geometry to form and dynamics to content provides an initial orientation, but requires mediation: an attractor’s geometry contains dynamical information, and geometrical constraints can condition material processes. The dialectical contribution consists in investigating these reciprocal relationships, avoiding the conversion of an analytical distinction into an absolute separation.
Dynamical systems, nonlinearity, chaos, and complexity
State, evolution, and observation
A dynamical system can be described by a state space X and an evolution rule. In discrete time, one writes xn+1 = F(xn); in continuous time, a common representation is ẋ = f(x, t). A state brings together the variables that the model considers necessary for describing evolution. Deciding which variables to include is part of the scientific problem: an overly reduced representation may lose relevant mechanisms, while an excessively detailed one may prevent the identification of regularities.
For a differential equation to define a unique deterministic evolution, appropriate existence and uniqueness conditions are required. Writing an equation without a random term is not, by itself, enough to guarantee these properties. The determinism of a model should be distinguished from the metaphysical thesis that all reality is determined in a particular way.
Nor does the observed quantity necessarily coincide with the complete state. It may be represented as yn = h(xn) + εn, where h is the observation function and ε represents measurement error under an explicit model. A one-dimensional record may originate from a system with many dimensions. Its apparent irregularity does not automatically identify the mechanism that produced it.
What nonlinearity means
In a linear system, a superposition principle holds: a linear combination of the relevant inputs or states corresponds to a combination of their responses. Nonlinear interactions can break this property. Feedback that depends on the product of variables, saturation, or a threshold constitutes a common form of nonlinearity.
Nonlinearity may produce equilibria, cycles, quasiperiodic oscillations, multiple stable states, prolonged transients, or chaos. It does not guarantee chaotic behavior. The continuous logistic equation, for example, describes a nonlinear evolution that, in its usual form, converges to an equilibrium under appropriate conditions. The discrete logistic map displays a variety of regimes depending on its parameter.
Pedagogical example · Logistic map
xn+1 = r xn(1 − xn), 0 ≤ xn ≤ 1, 0 ≤ r ≤ 4.
This example allows stability and bifurcations to be studied. For values of r between 1 and 3, the nonzero fixed point is stable; varying the parameter produces changes in stability and period doublings. Other intervals contain chaotic regimes and periodic windows. Increasing a parameter therefore does not amount to monotonically increasing a supposedly universal quantity of complexity.
Figure 15 · Example calculated for this editionLogistic map with initial x = 0.2718281828. The first 250 iterations are discarded and the next 90 are shown. The data are synthetic and deterministic; the figure illustrates different regimes and does not by itself constitute a mathematical proof of chaos. It does not represent measurements of the empirical systems cited.
Deterministic chaos and its criteria
Chaos designates a family of dynamical properties whose formulation depends on the mathematical framework. Sensitive dependence on initial conditions is central to many characterizations, but must be accompanied by assumptions that distinguish chaos from other forms of instability. A separation between trajectories that grows without bound in an unstable linear system does not, on that ground alone, constitute the same phenomenon as irregular recurrent dynamics on a bounded set.
Lyapunov exponents, entropy, and mixing properties allow different aspects of dynamics to be studied. Their use requires specification of the system, the relevant measure, the limits considered, and the definition adopted. The existence of multiple characterizations is a reason to make the vocabulary explicit, especially when transferring it to empirical domains.
Attractors and phase diagrams
Phase space organizes a model’s possible states; a trajectory describes its evolution. A phase diagram represents that organization and can be used for both regular and chaotic dynamics. In general terms, an attractor is an invariant set approached by states from a region of initial conditions; technical definitions specify invariance, attraction, and additional properties.
A fixed point, a limit cycle, and certain more complicated sets can act as attractors. A strange attractor is associated with irregular, often fractal, geometry. Its representation is not an imprecise sketch replacing phase space: it is a structure studied within it. The fact that a figure looks visually dense does not mean its properties are mathematically undefined.
Complexity and emergence
Complexity may refer to diversity of components, the organization of interactions, multiple scales, difficulty of description, computational cost, or the emergence of collective patterns. No single measure resolves all these problems. Chaos theory studies an important part of this field but does not exhaust the sciences of complexity; a system with few variables can be chaotic, while a network with many components can display regular collective dynamics.
Emergence is understood here as the appearance of collective properties whose description requires consideration of organization and interactions. Determining whether that emergence rules out a derivation in principle, reflects computational limitations, or expresses the autonomy of an explanatory level requires additional arguments. The need to study a collective level does not automatically demonstrate that its properties are incompatible with the laws governing the components.
Necessary distinctions for an interdisciplinary reading
Concept
What it characterizes
What it does not establish by itself
Model determinism
Uniqueness of evolution from a state under the conditions of the formulation.
Unlimited practical prediction.
Nonlinearity
Failure of the relevant superposition principle.
The existence of chaos.
Chaos
Properties of dynamical irregularity and instability defined within a specific framework.
The absence of causes or statistical regularities.
Fractality
Geometrical and scaling properties.
A positive Lyapunov exponent.
Complexity
Problems of organization, interaction, scale, or description.
A single universal dynamics or measure.
Randomness
Probabilistic or algorithmic properties that must be defined.
That an observed signal originates from dynamics lacking determination.
Disciplinary breadth is valuable when it makes it possible to compare problems and recognize differences. The following cases preserve the scope of the original investigation while specifying what kind of evidence each contributes. Sharing an equation or a figure does not make the phenomena being modeled identical. Comparison requires specifying which structure is shared, how the variables are interpreted, and which procedures establish its relevance.
1. Astronomy: variable stars and strange nonchaotic attractors
Strange nonchaotic attractors show why geometric irregularity and chaos must be distinguished. In 1984, Grebogi, Ott, Pelikan, and Yorke published a foundational investigation of attractors that are strange without being chaotic in the dynamical sense used. In quasiperiodically forced systems, the geometry can be fractal while the relevant exponents fail to display the positive exponential instability characteristic of other attractors.
The expression “nonchaotic” does not mean that a system is deterministic only because sufficient information is available, nor that it necessarily occupies a universal midpoint between order and chaos. Its deterministic character depends on its evolution rule. The distinction between strange and chaotic depends on geometrical and dynamical properties that must be specified, including the difference between exponents corresponding to the forcing and those corresponding to response variables.
In Strange Nonchaotic Stars, Lindner and colleagues (2015) study luminosity records obtained by the Kepler telescope. For variable stars such as KIC 5520878, they present evidence consistent with strange nonchaotic dynamics. One clue is the ratio between primary and secondary frequencies, which is close to the golden ratio. The article’s scope is an inference supported by observations and diagnostics; a frequency ratio close to a particular number does not, by itself, demonstrate the complete dynamics.
Mathematical clarification · Golden ratio
φ = (1 + √5)/2 ≈ 1.6180339887…
φ − 1 = 1/φ ≈ 0.6180339887…
A frequency ratio may be expressed as φ or its inverse, depending on the order of the quotient. An irrational number cannot be expressed as a ratio of two integers, and its decimal expansion is infinite and not eventually periodic; this does not make it an infinitesimal. The golden ratio is a finite positive number. Its classical geometrical construction must be distinguished from other theorems about the division of line segments, such as the angle-bisector theorem. The presence of a ratio close to φ in a dataset requires consideration of precision, uncertainty, and other possible ratios.
Figure 1Attractor reconstruction from variations in stellar luminosity. Lindner and colleagues (2015). Resource retained from the original PDF, p. 9. Its interpretation requires the article’s diagnostics in addition to the frequency ratio. Reference.
The epistemological interest of this case lies in the fact that the same object can display frequency regularities and geometric irregularity. Research requires combining descriptive domains rather than inferring the entire dynamics from a single visual characteristic. Johnston’s 2015 news article, cited in a note in the original work, is retained among the supplementary popular-science sources; the main scientific reference is the article by Lindner and colleagues.
2. Cardiovascular physiology: rhythms, networks, and organization
Vijay Sharma’s review, published in 2009, examines applications of dynamics and fractal geometry to the cardiovascular system. It considers regimes of equilibrium, periodicity, quasiperiodicity, deterministic chaos, and random behavior, as well as vascular networks and conduction processes. This research horizon is retained with the clarification that a pedagogical classification does not replace diagnosis of a particular record.
Figure 2Schematic representation of different dynamical behaviors. Sharma (2009). Original PDF, p. 10. The drawings are pedagogical categories, not clinical criteria. Reference.
Temporal variability may fulfill different functions depending on the process, scale, and physiological state under study. Greater irregularity therefore does not automatically indicate disease, nor does greater regularity guarantee healthy functioning. Physiological interpretations must rely on specific variables and protocols; the figures present conceptual schemes rather than clinical thresholds.
The relationship between vascular structure, cellular activity, and systemic regulation makes it possible to examine levels of organization. A network can be studied through its components, pathways and modules, and overall organization. Causality may involve constraints imposed by the collective level on local activity and the effects of local activity on those constraints. This reciprocity constitutes a material question for a dialectical interpretation of biological organization.
Figure 3Components, pathways, modules, and large-scale organization. Sharma (2009). Original PDF, p. 11. The figure introduces levels of biological organization. Reference.
3. Computing: Duffing oscillators and logical operations
Aravindh, Venkatesan, and Lakshmanan (2018) study quasiperiodically forced nonlinear systems capable of performing logical operations. Their investigation uses a double-well Duffing oscillator and analyzes how forcing, bias, and encoded inputs modify its response. Computing appears here as a function implemented by a material dynamical organization.
The Duffing oscillator is a second-order nonlinear differential equation. A reference form is ẍ + δẋ + αx + βx³ = γ cos(ωt). This general expression does not replace the specific formulation used in the cited article, which involves quasiperiodic forcing, inputs, and noise. To preserve that detail, the equation included in the original paper is also reproduced.
Figure 4aOscillator formulation used in the computing investigation. Aravindh, Venkatesan, and Lakshmanan (2018). Original PDF, p. 12. The equation that shared numbering with another figure in the 2021 paper is retained. Reference.Figure 4bRegime projections for a quasiperiodically forced system. Aravindh, Venkatesan, and Lakshmanan (2018). Original PDF, p. 12. Parameter values belong to the source model. Reference.
The different regimes are identified through complementary observations and diagnostics. The authors study the transition between a torus, a strange nonchaotic attractor, and chaos, as well as the persistence of particular logical functions under the noise levels considered. This robustness has a specific parametric and experimental scope: it does not imply universal insensitivity to every disturbance.
Figure 5Lyapunov exponent and variation of the control parameter. Aravindh, Venkatesan, and Lakshmanan (2018). Original PDF, p. 13. Torus, strange nonchaotic regime, and chaos are interpreted according to the authors’ criteria. Reference.Figure 6Sensitivity and scaling diagnostics in the studied regimes. Aravindh, Venkatesan, and Lakshmanan (2018). Original PDF, p. 13. The graph retains its labels and parameters. Reference.
The philosophical relevance is that a functional property at the logical level depends on dynamical mechanisms and conditions of organization. There is no need to suppose that the logical operation exists as an entity separate from its physical implementation. Nor does it follow that mathematical logic is refuted by the device’s behavior: the specification of a gate and the dynamics implementing it belong to related levels of description.
4. Econometrics: attractors, monetary aggregates, and data quality
Barnett and Chen (1988) investigate the use of tools from chaos theory and fractal attractors for econometric inference, with an application to Divisia monetary aggregates. Their work makes it possible to formulate questions about dynamical structure and nonlinearity in economic time series. The journal’s correct title is Mathematical and Computer Modelling; the bibliographic link provided here points to the corresponding article.
Figure 7Projections for comparing dynamical structures in econometrics. Barnett and Chen (1988). Original PDF, p. 15. Graphical patterns must be assessed alongside the method and data quality. Reference.Figure 8Further phase projections of the economic cases studied. Barnett and Chen (1988). Original PDF, p. 15. The second set of panels is retained. Reference.
A reconstructed phase-space representation may reveal patterns, but it does not automatically demonstrate that an economy possesses a low-dimensional attractor. Sample size, observation frequency, regime changes, noise, aggregation, and variable selection condition the inferences. A favorable signal-to-noise ratio helps identify structure but does not by itself resolve the problem of causal identification.
In political economy, moreover, institutions, expectations, decisions, and conflicts may modify the relevant rules of a process. A model with constant parameters must justify the period and conditions of its application. The relationship between structure and history requires examining the stability of mechanisms and institutional forms; identifying a regularity does not eliminate the possibility that social action will transform the conditions producing it.
5. Neuroscience: recurrent networks and the emergence of representations
Recurrent neural networks make it possible to model the evolution of activity patterns. A state represents the selected activations, and a rule describes how they change through connections and inputs. In certain models, stable states or attractors can be interpreted as memories or representations. The relevance of that interpretation depends on the task, training, variables, and relationship with neurobiological evidence.
The example illustrates a possibility of functional emergence: a property attributed to the whole depends on distributed interactions. It does not imply that any biological network functions like a particular recurrent-network model, or that every mental process can be reduced to a fixed point. Moving between explanatory levels requires specifying what is retained and what is omitted in each representation.
6. Psychiatry and psychopathology: the scope of a dynamical paradigm
Pezard and Nandrino (2001) examine the relevance of a dynamical perspective for psychopathology. The original work draws on their discussion of networks, attractors, cognition, and the possibility of modeling psychological processes, including a dynamical interpretation of aspects of Freudian psychodynamics. This problem is retained as a proposal for modeling and conceptual discussion.
Using terms such as transition, attractor, or stability requires observable variables, a relevant rule or model, and criteria for testing the claims. A metaphor can guide exploration, but it does not become a clinical explanation simply by incorporating mathematical vocabulary. The temporal and relational character of a phenomenon is a starting point for investigating it; it is not enough to establish that the phenomenon is chaotic.
7. Hydrodynamics: turbulence, continuity, and instability
The transition between laminar and turbulent flow shows the importance of scales, instabilities, and collective transport. Kessler and Greenkorn’s explanation considers momentum exchange through organized movements of fluid regions, alongside molecular processes. A fluid particle in a continuum description should not be identified with an individual molecule.
Figure 13Pedagogical representation of flow regimes. Elert, The Physics Hypertextbook. Original PDF, p. 34. The depicted thresholds are not generalized to every geometry. Reference.
The Reynolds number, Re = ρUL/η = UL/ν, compares inertial and viscous effects through density ρ, characteristic velocity U, length L, dynamic viscosity η, and kinematic viscosity ν. Transition thresholds depend on geometry, perturbations, and flow conditions. The retained illustration has a pedagogical function: its values do not establish a universal criterion applicable to every configuration.
The continuum hypothesis in fluid mechanics assumes a separation of scales that makes it possible to define macroscopic fields. A representative volume contains enough constituents for an average property to be meaningful while remaining small relative to the relevant scales of the process. This volume has no universal microscopic size; it depends on the substance and the conditions of the problem. Its status as an abstraction does not mean that fluids are materially composed of continuous points.
Turbulence does not, by itself, imply the failure of the continuum hypothesis: many descriptions of turbulence employ continuum equations. Separation between molecular and macroscopic scales is studied, among other tools, through the Knudsen number, Kn = ℓ/L, where ℓ is the mean free path. Closure problems in a turbulent description must be distinguished from problems of rarefaction or the validity of the continuum approximation. Supersonic flow does not automatically invalidate that approximation either.
Statistical mechanics provides frameworks for relating microscopic and macroscopic descriptions. This does not establish a simple sequence in which every difficulty in a continuum equation is resolved by moving to a statistical theory. Each description introduces its own assumptions, scales, and problems. The discussion of absolute zero also requires precision: the classical idealization of an absence of thermal agitation should not be transferred into a claim that all quantum motion ceases at that temperature.
Rayleigh–Taylor instability occurs when a configuration of densities and acceleration is unstable; an idealized example is a denser fluid positioned above a less dense one under gravity. Everyday observation of water and oil can guide intuition, but mixing or stirring them does not by itself identify this instability. Relative position, acceleration, viscosity, surface tension, and, in magnetohydrodynamics, the magnetic field are relevant conditions.
Figure 14Hydrodynamic and magnetohydrodynamic simulations. Li and Li (2006). Original PDF, p. 35. Differences between panels have the scope of the source models and parameters. Reference.
The simulations by Li and Li retained in the paper illustrate the study of these problems using adaptive mesh refinement. Differences between panels must be interpreted in accordance with the parameters and models of the source work. A simulation explores the consequences of a formulation and compares them with other results; the appearance of a figure does not demonstrate a general law of complexity.
8. Geodynamics: magnetic fields and dynamo models
Dynamo theory investigates how the movement of conducting materials can sustain magnetic fields. The geodynamo combines fluid dynamics, electromagnetic induction, rotation, and energy transfer. The original investigation refers to three-dimensional simulations developed in the 1990s, within a broader history of models for the generation of Earth’s magnetic field.
The ability to reproduce particular features through simulations provides evidence concerning the plausibility of mechanisms under the modeled conditions. Resolution and parameter constraints must be examined before extrapolating to the planet. Likewise, an explanation of geomagnetic-field generation and a hypothesis about its relationship with climate are distinct problems: the reference to Kilifarska, Bakmutov, and Melnyk (2020) is retained without presenting every connection discussed in that volume as equally established.
The 0–1 test: what it distinguishes and what it does not
Gottwald and Melbourne’s 0–1 test was designed to distinguish, under appropriate conditions, between regular and chaotic dynamics using a time series. The regular class includes both periodicity and quasiperiodicity. The procedure does not require explicit reconstruction of the entire phase space, which facilitates its use in systems of different dimensions and formulations. The object being diagnosed remains a defined dynamics; a finite signal does not, on its own, contain a complete demonstration of its origin.
A schematic account begins with a scalar series φ(j) and a family of translation variables for particular values of an auxiliary parameter c. The growth of their mean square displacement is studied, or that of a corrected version eliminating an oscillatory term.
Outline of the 0–1 test
pc(n) = ∑j=1n φ(j) cos(jc)
qc(n) = ∑j=1n φ(j) sin(jc)
Kc = limn→∞ log Mc(n) / log n
Under appropriate conditions, regular dynamics produces bounded translations and an indicator close to zero, while chaotic dynamics produces diffusive growth and an indicator close to one. The implementation uses versions and estimators that must be described, including the selection of auxiliary parameters and the aggregation of their results. The indicator should not be interpreted as a universal scale of the intensity of chaos: an intermediate value may reflect finite data, transients, noise, or other difficulties.
Figure 9Comparative diagnostics of regular and chaotic dynamics. Gottwald and Melbourne (2016). Original PDF, p. 17. The material comes from the study of the test’s application to defined systems. Reference.Figure 10Evaluation of the diagnostic with noise-contaminated data. Gottwald and Melbourne (2016). Original PDF, p. 18. Noise, sampling, and series-length conditions are part of the interpretation. Reference.
Record length, temporal sampling, and noise affect the diagnosis. Excessively fine sampling can introduce practical difficulties; excessively sparse sampling can lose the structure being sought. Before applying a test, it is useful to examine whether the record contains regime changes, trends, artifacts, or missing values, and whether the treatment applied alters the properties one intends to detect. Robustness must be assessed against justified preprocessing decisions.
Noise, signal, and identification
“Noise” may designate instrumental error, unmodeled disturbances, or a random component deliberately introduced into a description. The word does not identify a single cause. Separating signal from noise presupposes a research question: a fluctuation treated as noise by one model may be the phenomenon of interest in another. What matters is to make explicit what is being modeled, why, and with what evidence.
Detecting irregularity is not enough to distinguish deterministic chaos from a stochastic process. Diagnostics of nonlinearity, reconstructions, exponent estimates, surrogate-data tests, and sensitivity analyses can be combined. No list of techniques operates as a universal recipe; each procedure has conditions. With limited records, different classes of models may be consistent with similar observations.
Two problems of unpredictability
Werndl’s discussion distinguishes difficulties that are often brought together under the same word. One problem is the loss of precision in trajectories from initial states known only approximately. Another is the loss of probabilistic relevance of sufficiently past information for certain events, associated with mixing properties in the framework she studies. Her 2009 investigation critically examines which specific implications can be attributed to chaos.
Both problems require defining what is to be predicted, at what resolution, and under which distribution or set of initial conditions. A trajectory may be difficult to anticipate while still allowing useful estimates of frequencies, ranges, or invariant measures. The distinction between forecasting a point state and describing a distribution is central to epistemology.
Figure 11Historical table of concepts associated with the study of chaos. Oestreicher (2007). Original PDF, p. 24. Its summaries are supplemented by the clarifications developed in the essay. Reference.
The retained table from Oestreicher brings together historical concepts such as causality, determinism, prediction, linearity, attractors, feedback, and fractal dimension. Its value is orienting and documentary. This edition’s working definitions are developed in the text to prevent a historical summary from being read as a single axiomatization of chaos.
Formal logic, classical logic, and the critique of reductionism
The original investigation criticizes the attempt to exhaust complexity through dichotomous schemes. This concern requires distinguishing the formal character of a logic from the principles adopted by a particular logic. A formal system specifies a language and rules; classical logic constitutes a family of systems with particular properties. Nonclassical logics can also be formalized.
Chaos theory develops through mathematical reasoning that can be formulated in classical frameworks. Its results therefore do not, by themselves, prove the failure of all formal logic. The more productive philosophical critique targets unjustified applications of assumptions concerning linearity, independence, separation of scales, or stability; those assumptions are not equivalent to the law of excluded middle.
The text Formal Logic and Dialectics, retained in the original bibliography, expresses a philosophical position on the limits of reductionism and on emergence. It remains a reference for that controversy, with its argument distinguished from a proved mathematical result. Recognizing emergent levels and reciprocal relationships requires examining concrete models; the vocabulary of dialectics can organize that examination without replacing it.
Quantum logic and physical propositions
Quantum logic studies structures of propositions associated with quantum systems and their relationships with probability. In one common formulation, the closed subspaces of a Hilbert space form a non-Boolean structure in which classical distributivity fails. The philosophical interest concerns how properties and measurement operations are represented.
The noncommutativity of certain operators should not be confused with a supposed general failure of associativity, or with the claim that all quantum logic is simply many-valued logic. In the usual lattices, meet and join retain lattice properties, including associativity and commutativity. The structure being studied must be specified before transferring its properties to an epistemological thesis.
Fuzzy sets: degrees of membership
A classical indicator function for a set A assigns the value 1 to its elements and 0 to the rest. In Zadeh’s formulation, a fuzzy set can be described by a membership function with values in the interval [0,1]. This tool represents graded categories, such as “high temperature,” once their context of use has been specified.
Classical and fuzzy membership
1A : X → {0,1}
μA : X → [0,1]
A degree of membership is not, in general, a probability. Saying that a value belongs to a fuzzy set to degree 0.7 is not the same as saying that there is a probability of 0.7 that it belongs to a classical set. The function can be constructed from expert knowledge, data, or learning procedures; fuzzy-set theory need not be defined as the imposition of assumptions external to the data.
Rough sets: indiscernibility and boundaries
In Pawlak’s classical formulation, a universe U and an equivalence relation R group together objects that are indiscernible on the basis of certain information. A set can be approximated from below and above through equivalence classes. The lower approximation brings together objects whose class is contained in the set; the upper approximation includes those whose class intersects it.
Pawlak approximations
Rlower(A) = {x ∈ U : [x]R ⊆ A}
Rupper(A) = {x ∈ U : [x]R ∩ A ≠ ∅}
BoundaryR(A) = Rupper(A) ∖ Rlower(A)
Uncertainty is related here to the resolution of information and the ability to discern objects. This differs from assigning degrees of membership to a category. Fuzzy-rough and rough-fuzzy extensions combine tools, but their terminology must be accompanied by definitions: the order of the adjectives is not enough to establish how a model is constructed.
Forms of uncertainty and possible representations
Problem
Representation
Condition to be made explicit
Variability of outcomes or uncertain information about events
Probabilistic model
Event space, measure, and interpretation.
Gradual membership in a category
Fuzzy set
The meaning and construction of membership.
Indiscernibility according to available attributes
Rough set
Indiscernibility relation and universe.
Conflict or plurality of descriptions
A logical or epistemological framework appropriate to the problem
What counts as a proposition, evidence, and consequence.
The contributions by Karmakar and Ganesan cited from ResearchGate are retained as part of the debate that motivated the original work. Their status is that of opinions and guidance in a forum, not a demonstration of the general superiority of an approach. Choosing between techniques requires examining tasks, assumptions, interpretability, available information, and performance beyond the data used to fit them.
The proposal of fractal logic
Bjorvand (1995) proposes an approach to intelligent systems connected with fractals, neuroscience, evolutionary computing, and set theories. The author himself distinguishes his proposal from a logic in the strict mathematical sense. It is retained for its exploratory interest and its place in the original discussion, without attributing to it an axiomatic apparatus or inference rules that the document does not establish.
Its inclusion distinguishes complete formalization, a heuristic program, and computational application. A proposal can be valuable as a design orientation before it constitutes a fully developed formal theory. Evaluating it requires asking which operations it defines, which problems it solves, and how its results compare with alternatives.
Statistical testing and scientific reasoning
The analogy between a hypothesis test and a reductio ad absurdum can be pedagogically useful if its limits are stated. An outcome improbable under a hypothesis is not a logical contradiction; a p-value does not represent the probability that the hypothesis is true. Fisher’s and Neyman–Pearson’s procedures also have different aims and structures, as Perezgonzalez (2015) explains.
Expert judgment, historical evidence, exploratory description, and study design are part of scientific reasoning. Gigerenzer’s 2004 critique of ritualized testing supports the need to think through the problem and report uncertainty. Recognizing that role does not turn an expert opinion into conclusive evidence or contextual reasoning into formal meta-analysis. Prior knowledge must be identifiable, discussable, and revisable.
Relationships, contradictions, and transformations
Wholes and parts
A whole is constituted through relationships between components and with an environment. Studying it requires delimiting the system and recognizing that this delimitation depends on a question. One description may treat the environment as a fixed condition; another may incorporate it as a dynamical variable. This decision has material consequences for what can be explained and predicted.
The expression “the whole is greater than the sum of its parts” may indicate the relevance of interactions, but it must be developed. In modeling terms, it is useful to identify which terms are lost through aggregation, which constraints emerge, and which collective observables are necessary. A sum may be appropriate for certain quantities while remaining insufficient to describe an organization; that difference does not refute the mathematical operation.
Dialectical contradiction and logical contradiction
Dialectical contradiction is used here to investigate tendencies or relationships that presuppose and stand in tension with one another within a process. Its content must be formulated materially: which forces, mechanisms, interests, or constraints intervene, through which relationships, and under which conditions they produce transformations. A logical contradiction, by contrast, consists in affirming and denying a proposition under the same conditions of meaning.
Divergence between two nearby trajectories does not automatically amount to an antagonistic contradiction. Nor does every negative feedback or competition between variables constitute antagonism in the sense of social theory. The proposed philosophical correspondence can be formulated as an interpretive hypothesis: certain dynamics make explicit tensions between mechanisms whose articulation produces qualitative changes. Each application must identify those mechanisms and the relevance of the term.
Necessity and contingency
Necessity may refer to consequences following from established relationships and conditions; contingency may refer to conditions that could have been different or whose occurrence is not fixed by the adopted level of description. In a deterministic model, evolution from a state can be unique while the choice of that state remains uncertain for an observer. This coexistence does not require attributing incompatible properties to the same object in the same sense.
Dependence on initial conditions, multiple scales, and the existence of disturbances make it possible to investigate how concrete events occur. The materialist position defended here recognizes that the limits of knowledge are part of a historical relationship with objective processes. A measurement limit, however, does not by itself imply a definitive thesis about the ontology of chance.
Quantity, quality, and bifurcations
A continuous variation in a parameter can produce a qualitative change in dynamics: loss of stability, the appearance of cycles, or a modification of basins of attraction. Bifurcations provide mathematical examples of how a quantitative transformation can relate to qualitative reorganizations. The content of that correspondence is determined through concrete conditions of stability and structure.
A dialectical interpretation finds a domain of dialogue here, although the existence of a bifurcation does not demonstrate a universal philosophical law. It matters to examine when the bifurcation occurs, whether the change is continuous or discontinuous in the relevant observable, whether hysteresis exists, and how fluctuations and perturbations intervene. Precision enables philosophy to guide testable questions and to be transformed by their results.
A reciprocal methodological requirement
A dialectical reading must accept that research will refine its categories and that some initial correspondences will prove inadequate. The criterion of usefulness is its capacity to reveal relationships, formulate hypotheses, explain transitions, and organize evidence. The breadth of a philosophical outlook increases its responsibility to distinguish domains, conditions, and levels of validation.
The unity of an interpretation is constructed by explaining the mediations between domains; the diversity of the sciences requires those mediations to be investigated.
A methodological proposition advanced in this essay
A scaling operation modifies the reference quantities used to describe an object. In certain phenomena, the transformation preserves a form or statistical relationship after the relevant values have been rescaled. Invariance does not mean that every quantity remains numerically identical; it may mean that quantities transform according to a homogeneous law.
Example of homogeneity
f(bx) = bα f(x), b > 0.
In this expression, α is a scaling exponent. If the relationship holds for any positive factor b within the relevant domain, one speaks of continuous scale invariance in this sense. If it holds only for factors belonging to a family such as b0n, with integer n, it is considered discrete invariance. This difference concerns the admissible scaling factors, not whether the system’s time is necessarily continuous or discrete.
A continuous-time model can exhibit discrete scale invariance, and a structure constructed by iterations can approximate statistical properties described by continuous laws. Spatial scale, temporal scale, instrumental resolution, and numerical integration step are concepts that must be distinguished. Representing a quantity in a computer does not by itself make the material phenomenon discrete.
Exact self-similarity, statistical self-similarity, and self-affinity
Exact self-similarity occurs when particular parts reproduce the structure of the whole through similarity transformations. Statistical self-similarity preserves distributions or statistical properties under scaling without requiring point-by-point coincidence. Self-affinity permits different factors for different directions or variables. The distinction matters for surfaces, time series, and trajectories whose axes represent different quantities.
As a conceptual example of statistical self-affinity for an appropriate process, one may write X(ct) ≍ cHX(t), where ≍ indicates equality in distribution in the specified sense and H is an exponent. The notation does not assert that two particular realizations coincide exactly. Nor does it allow a value of H to be assigned automatically to any irregular series.
Different texts use self-similarity and scale invariance as closely related or equivalent terms within a convention. The critical task is to identify that convention and the properties actually established. There is no need to invoke a supposed axiomatic opposition between continuity and discreteness to recognize differences between exact self-similarity, statistical scaling, and discrete scaling laws.
Mathematical fractals and natural structures
Cantor sets, the Koch curve, the Sierpiński triangle, and space-filling curves illustrate structures whose geometrical properties differ from those of elementary smooth figures. Some are constructed through exact iterations. Recursion provides a construction mechanism, but should not be used as a universal definition of fractality.
Figure 12aGeometrical constructions and iterative processes. Lesne and Laguës (2012). Original PDF, p. 31. The examples used to introduce scaling and dimension are retained. Reference.Figure 12bOriginal caption for the fractal constructions. Lesne and Laguës (2012). Original PDF, p. 31. The caption was a separate image in the file and is retained in full. Reference.
The retained figure from Lesne and Laguës includes constructions and dimensions that should be studied separately. The middle-thirds Cantor set has dimension log 2 / log 3; the Koch curve, log 4 / log 3; and the Sierpiński triangle, log 3 / log 2. In each case, dimension expresses a precise relationship between the number of parts and the reduction factor for an ideal construction, under appropriate similarity and separation conditions.
Similarity dimension · Appropriate constructions
D = log N / log(1/r), 0 < r < 1.
N: number of copies · r: reduction factor of each copy.
The formula is not a recipe applicable to any irregular shape. Hausdorff dimension is defined through measures constructed from covers; box-counting dimension studies the scaling of the number of boxes required to cover a set. They may coincide in important examples, but they are not identical by definition. Having an integer dimension does not, on its own, exclude all properties usually associated with fractals.
Box-counting dimension · When the limit exists
DB = limε→0 log N(ε) / log(1/ε).
Material structures have cutoff scales. A tree has branches of finite sizes; a vascular network has constituents and biological constraints; a coastline depends on the resolution at which it is measured. Scaling behavior observed over an interval does not prove that an ideal construction continues indefinitely. Measurement must report the range of scales, the fitting procedure, and its uncertainty.
An explicit example: the Koch curve
In the Koch construction, each segment is replaced by four segments, each one-third of its length. After n iterations, the number of segments is 4n, their individual length is 3−n times the initial length, and the total polygonal length is proportional to (4/3)n. The divergence of that length in the limit coexists with the curve’s location within a bounded region.
The example distinguishes length, boundedness, and dimension. It also shows that difficulty of measurement does not depend solely on an observer’s ignorance: certain properties follow from a completely known construction. For a natural object, by contrast, the empirical question is whether an interval exists within which a similar relationship adequately describes observations, and which processes generate that behavior.
Uncertainty, dimension, and knowledge
Fractal dimension is not defined as the number of properties providing incomplete information about a state. It is a geometrical property whose interpretation depends on the dimension used. Incomplete observation may make estimation difficult, but must be distinguished from what is being estimated. Uncertainty about a property and the property itself belong to different levels.
Likewise, a margin of imprecision does not automatically amount to a standard deviation. It may be expressed as absolute error, an interval of resolution, calibration uncertainty, probabilistic dispersion, or estimator bias. Using a standard deviation requires a relevant random variable or distribution and satisfaction of the conditions of its definition. Some distributions do not even have finite variance.
The most defensible philosophical application consists in recognizing that what is observed depends on scales and mediations, without concluding that all scales reveal an identical essence in every sense. Some mechanisms change in relevance as resolution varies; others require new collective variables. The unity of a material object does not guarantee uniformity in its description.
The continuum hypothesis and the limits of its relevance
Cantor’s continuum hypothesis concerns cardinalities: it states that no set size lies strictly between that of the natural numbers and that of the real numbers. The results of Gödel and Cohen establish its independence from ZFC under the corresponding consistency assumptions. This independence allows axiomatic extensions containing the hypothesis or its negation to be studied; it does not require assuming the hypothesis is true.
This problem is not equivalent to the physical continuum hypothesis. Nor does it establish a mathematical gap preventing the study of relationships between discrete and continuous structures. The theory of real numbers, limits, and differential equations can be developed within standard frameworks without resolving the continuum hypothesis. Axiomatic limitations and the limitations of a measuring instrument should not be identified with one another.
Gödel’s second incompleteness theorem concerns, under formal conditions, the impossibility for a sufficiently expressive consistent system to prove its own consistency internally in the relevant formulation. “Consistency,” “truth,” and “falsity of the system” are not interchangeable expressions. This result does not, by itself, demonstrate a general conclusion about what a finite being can know about an infinite reality.
The original investigation uses philosophical images concerning the finite and infinite, parts and wholes, and even, as an analogy, the theological relationship between creature and creator. Their place can be preserved as a reflective resource, distinguished from a mathematical premise. The same applies to the metaphorical reference to uniform convergence: in analysis, this has a precise definition that does not follow from several traditions expressing similar intuitions.
A small separation between trajectories may grow approximately exponentially in particular regimes and directions. This formulation assumes a scale on which the linearized approximation of perturbations is relevant. On a bounded attractor, separation does not grow indefinitely according to the same law: it eventually encounters the constraints of accessible space.
Local approximation of separation
‖δx(t)‖ ≈ ‖δx(0)‖ eλt.
The exponent λ describes a growth rate under specified conditions. If it is positive for the relevant direction or largest exponent, reducing initial error extends the time during which a desired precision is maintained. This does not necessarily eliminate the practical limit, because the temporal benefit depends on a logarithm.
Approximate prediction horizon
T ≈ (1/λ) log(Δ/δ0), λ > 0.
δ0: initial error · Δ: prediction tolerance.
As a purely pedagogical example, if λ = 0.5 per unit of time, δ0 = 10−6, and Δ = 10−1, the approximation gives T ≈ 23.03 units. Reducing the initial error by a factor of one thousand adds approximately 13.82 units. These values do not describe any of the empirical systems cited; they show the relationship between expansion rate, precision, and tolerance.
Exponentials and power laws
An exponential law has its variable in the exponent, as in eλt. A power law has a variable base raised to an exponent, as in xα. They are different functional forms and have different asymptotic behaviors. A semilogarithmic representation can help study an exponential, and a logarithmic representation on both axes can help study a power law, always with the corresponding statistical care.
The fact that two forms include the word “exponent” does not make them identical. This clarification is decisive for relating dynamical sensitivity, scaling laws, and critical phenomena. Each can offer a philosophical problem concerning transformations and limits, but interpretation must preserve the mathematical structure under examination.
What can be predicted over the long term
A point-prediction horizon does not amount to the impossibility of all long-term knowledge. Stable collective observables, invariant distributions, or estimable visitation frequencies may exist. The answer depends on the system’s properties and on the type of question: trajectory, distribution, transition, extreme, or response to a perturbation.
The physical time covered by a series must be distinguished from sample size. More observations can improve some estimates without equivalently extending the temporal horizon of a prediction. Nor does a densely recorded sample over a short interval always amount to prolonged observation of relevant states. Dependence between observations changes the information contributed by each additional datum.
A dialectical reading can investigate the coexistence of stable constraints and sensitive trajectories. The content of that coexistence does not require describing chaos as a mere appearance destined to disappear with greater knowledge. Sensitivity is a property of the model or regime under study; its epistemic consequences emerge when it is related to finite precision and prediction objectives.
In calculus, a critical point may refer to a location where a derivative vanishes or does not exist. In phase-transition theory, the term acquires another meaning: it relates to changes of state and singularities in the thermodynamic limit, together with specific collective properties. A maximum or minimum of an arbitrary function is not, on that ground alone, a critical phenomenon in this physical sense.
Temperature, an external field, or a concentration can act as control parameters, while an order parameter describes aspects of macroscopic organization. Near certain continuous transitions, a correlation length grows and scaling laws appear. These properties need not occur in every transition or in every complex system.
Lesne and Laguës provide a principal reference for this journey through transitions, turbulence, and fractal structures; Halperin’s 2019 note retains its role as a historical introduction to the theory of dynamic critical phenomena. Wilson’s development of the renormalization group is also incorporated to support the relationships between scale and universality.
Critical exponents
If τ = (T − Tc)/Tc is a reduced distance from a critical temperature, some quantities may display asymptotic power-law relationships. The following expressions outline particular critical regimes; their symbols and conditions must be specified for each model.
Examples of critical laws
ξ ∼ |τ|−ν · χ ∼ |τ|−γ
m ∼ (−τ)β for τ < 0, on the relevant ordered branch.
τrel ∼ ξz.
ξ represents a correlation length; χ, a susceptibility; m, an order parameter; and τrel, a relaxation time. The exponents ν, γ, β, and z describe different behaviors. The exponent β in a critical law must not be confused with a parameter called β in another model, such as the Lorenz system. The symbol ∼ expresses asymptotic behavior rather than an exact identity at any distance from the critical point.
These relationships allow investigation of the articulation between microscopic changes and collective responses. From a dialectical perspective, they provide a basis for asking about the mediations that make qualitative transformations possible. Identifying exponents directly with laws of antagonism between variables would omit the physical and mathematical conditions giving them meaning.
Universality and universality classes
Critical universality refers to shared scaling behaviors among systems differing in microscopic details under relevant common conditions. Dimensionality, symmetries, interaction range, and, for critical dynamics, conservation laws and slow modes can matter. Universality has a structure of classes and conditions; it does not assert that any system preserves all its properties at every scale.
Explanation through renormalization examines which perturbations or parameters remain relevant as scale changes. Some microscopic details affect amplitudes or corrections without affecting the dominant exponents of a class. It becomes possible to understand a correspondence between systems without assuming that their components or particular histories are identical.
The location, scale, and shape parameters of a statistical distribution do not define critical universality. Nor does increasing sample size by itself establish a universality class. Relationships between statistical limits, aggregation, and macroscopic laws deserve specific investigation, but must be distinguished from phase-transition theory.
The t distribution and inferential precision
The original reference to Student can be retained as an example of the importance of inferential conditions. For independent, identically distributed normal observations with unknown variance, the statistic standardized using the sample standard deviation follows a t distribution with n − 1 degrees of freedom. As those degrees of freedom increase, the family approaches the standard normal.
A result with specific assumptions
(X̄ − μ) / (S/√n) ∼ tn−1.
This relationship is not a general correction to short-term physics, nor does it identify prolonged observation with a large sample. If data are dependent or fail to satisfy the relevant model, the exact distribution above may no longer apply. The example teaches us to retain assumptions alongside a formula and to distinguish different meanings of “long term.”
Stochastic, critical, and complex phenomena
These three terms should not be arranged as though they formed a single hierarchy of increasing complexity. A stochastic process is characterized through a probabilistic structure; a critical phenomenon through particular transition and scaling properties; and complexity addresses problems of organization and description. They can intersect in different ways. Deterministic dynamics can display criticality, and a stochastic model can be relatively simple.
The discussion with Jaynes about probability and knowledge retains its philosophical relevance if the meaning of randomness is made explicit. The existence of effective probabilistic models does not by itself demonstrate ontological chance, and ignorance of causes does not demonstrate universal determinism. A controversy about the interpretation of probability cannot be settled merely by listing systems with fluctuations.
Why a collective description needs other variables
A many-particle system can obey microscopic laws while requiring collective concepts to explain its macroscopic behavior. Temperature, pressure, and order parameters are not merely alternative names for an individual trajectory. Their usefulness emerges when studying the organization and regularities of large ensembles.
Landau’s reflection on the appearance of statistical laws as the number of degrees of freedom increases retains its central role here. The autonomy of a collective explanation can mean that it uses its own variables, limits, and regularities. Determining its relationship of derivation from, or compatibility with, a microscopic theory requires consideration of the approximations and conditions involved in passing between levels.
Mean field and fluctuations
A mean-field approximation replaces certain effects of interactions with an average field or a simplified effective description. It does not generally consist in averaging the numbers of degrees of freedom of two models. It reduces the problem through an assumption about collective influence whose validity must be examined.
Near some critical transitions, fluctuations and long-range correlations can make an elementary mean-field approximation inadequate. It is not correct, however, to claim that mean field is invalid at every transition: dimensionality and other conditions may permit behavior of this type, including situations above an upper critical dimension. The importance of fluctuations is a concrete physical and mathematical problem.
Renormalization transformations
A renormalization scheme can integrate out or average over small-scale degrees of freedom and rescale the resulting description. The operation transforms parameters and couplings. Iteration studies a flow in a space of descriptions or theories, where certain fixed points organize large-scale behavior.
Outline of a scale transformation
g′ = Rb(g), Rb(g*) = g*.
δg′i ≈ byᵢ δgi, near a fixed point and in appropriate coordinates.
The parameters g describe couplings and b a scale-change factor. In a linearized description, perturbations with positive exponents can grow under the flow toward large scales and become relevant; others diminish. Marginal cases require further analysis. This vocabulary explains why certain details matter for a regime while others primarily affect corrections.
The historical designation “renormalization group” does not require every averaging operation to be invertible. Loss of detail can lead to a semigroup structure. This point is philosophically relevant: an effective description may retain enough information for some questions while losing it for others.
Three uses that must remain distinct
The original work draws on Lesne’s discussion of uses of the term renormalization. That plurality is preserved with more explicit delimitation. First, scale transformations and the elimination of degrees of freedom in critical phenomena and dynamics. Second, reduction to reference quantities or nondimensionalization, common in hydrodynamics and model comparison. Third, effective parameters or parameters corrected for interactions, associated with describing contributions that modify a reference quantity.
These uses can be related, but they are not interchangeable. Standardizing a column of data does not, by itself, amount to constructing a renormalization group. Eliminating a divergence in a perturbative expansion requires a specific theory and procedure. In classical and quantum electrodynamics, discussions of electron mass and charge must be situated within their respective formulations; the historical reference does not make the electron a generic example of ordinary classical mechanics.
A perturbative expansion is a controlled expansion in an appropriate parameter. It does not necessarily mean temporal evolution toward increasing complexity. Divergence of a series, a physical singularity, and trajectory sensitivity are distinct problems, although each may require revising approximations.
Dialectical interpretation and limits of extrapolation
The philosophical interest of changing scale lies in the fact that relevant relationships may be reorganized when a description changes. Explaining collective behavior may require retaining some variables and discarding others. This operation makes it possible to ask which aspects of a process persist, which are transformed, and under which conditions an effective theory is valid.
The analogy with the relationship between essence and appearance is useful if it avoids assuming that averaging over details always reveals a deeper truth. In some problems, reduction eliminates a decisive mechanism; in others, it identifies a robust regularity. The explanatory character of the operation is assessed through the questions it resolves and the evidence it organizes.
In the social sciences, transferring renormalization requires defining transformations, variables, and comparison criteria. Verbal resemblance between institutional scales and physical scales is insufficient. A materialist program can investigate this possibility without anticipating either the identity of mechanisms or the results of the analysis.
Consider a probability space (X, 𝔅, μ) and a transformation T preserving μ. In a common formulation, T is ergodic if every measurable invariant set has measure zero or one, with the relevant specification of invariance. This property rules out a nontrivial decomposition into invariant regions of positive probability with respect to the measure considered.
Ergodicity · A standard formulation
μ(T−1A) = μ(A).
T−1A = A, up to null sets ⇒ μ(A) ∈ {0,1}.
The measure is part of the specification. A dynamical system may possess several invariant measures, and properties must be attributed to the corresponding dynamics–measure pair. A finite record does not by itself demonstrate an asymptotic property; it may provide evidence consistent or inconsistent with concrete hypotheses.
The ergodic theorem and averages
For an integrable function, Birkhoff’s pointwise ergodic theorem establishes almost-sure convergence of time averages in a measure-preserving system. Under ergodicity, the limit agrees almost everywhere with the ensemble average. Without ergodicity, the general limit is related to conditional expectation with respect to the invariant information.
Birkhoff · Ergodic case
limn→∞ (1/n) ∑k=0n−1 f(Tkx) = ∫X f dμ, for μ-almost every x.
The equality does not assert that every microstate has equal probability, or that any finite trajectory exactly represents the average. Nor does it require expressing a whole as a linear sum of independent components. It is a result about averages along a dynamics and a measure, with assumptions that must be retained alongside its conclusion.
Equiprobability and statistical mechanics
The postulate of equal a priori probability is associated with the microcanonical description of an isolated system in equilibrium, given constraints such as energy, volume, and particle number. It must be distinguished from ergodicity. Under the usual thermodynamic convention, a closed system can exchange energy; an isolated system exchanges neither matter nor energy. The example therefore requires specification of the type of system.
Specifying energy and other macroscopic constraints does not amount to knowing the exact positions and momenta of every particle. The latter determines a microstate in a classical description. In a continuous space, speaking of equal probability for individual points likewise does not replace the definition of a measure on accessible regions.
The Princeton chapter cited in the original presents as a fundamental postulate a microscopic expression of entropy through accessible phase-space volume. That formulation must be distinguished both from immediately attributing equiprobability and from an ergodic hypothesis. Its relationship with equilibrium distributions requires development of the conditions of the argument.
Entropy and accessible states · Outline
S = kB log Ω.
Here Ω represents a count of states or an appropriately normalized, dimensionless measure of the accessible set, depending on the formulation. In the continuous case, that normalization must be specified; one does not simply take the logarithm of a dimensional quantity. A statistical formulation can be empirically fruitful without every use resting on the same demonstration of ergodicity.
Ergodicity breaking
Nonergodicity may arise when dynamics is organized into invariant components that do not explore one another in the manner required by the measure considered. There are also situations of extremely slow relaxation that prevent exploration of accessible regions within observational times. This practical difficulty must be distinguished from a structural absence of ergodicity in the mathematical limit.
An investigator’s lack of information does not by itself make a system nonergodic. It may prevent diagnosis of the property or selection of an appropriate measure. Likewise, chaos and nonergodicity are not synonymous: there are ergodic chaotic dynamics and nonchaotic dynamics with ergodic properties relative to particular measures. Mixing is another property requiring definition.
Introducing indicator or dummy variables into a regression can represent categories, institutional changes, or periods. It does not automatically restore ergodicity or turn a nonstationary series into a stationary one. The choice must follow from a hypothesis about the process, the design, and appropriate diagnostics. A statistical regime change and an ergodic decomposition are not the same concept.
Independence, exchangeability, and stationarity
Distinct properties of a sequence of variables
Property
Reference meaning
Clarification
Independence
The joint distribution factorizes in the sense of the probabilistic definition.
It does not amount merely to absence of correlation.
Exchangeability
The joint distribution is invariant under finite permutations of the indices.
It can coexist with dependence.
Strict stationarity
Finite-dimensional distributions are invariant under the relevant temporal shifts.
It does not mean arbitrarily interchanging times.
Ergodicity
Absence of a nontrivial invariant decomposition with respect to a measure; it permits particular relationships between averages.
It is not equivalent to equiprobability or independence.
McCullagh and Nelder’s observation about scales and the treatment of data remains relevant to examining what makes observations comparable. Equivalence of units, however, does not imply independence, and the fact that two observations belong to the same topological space does not determine their probabilistic dependence. The topological structure and the probability structure must be connected explicitly.
Exchangeability expresses a symmetry of a joint distribution, not permission to attribute an identical value or identical history to every observation. It may justify certain symmetric treatments under a model. In time series, order may contain indispensable information; permuting the data may destroy precisely the relationships being investigated.
An example: dependence through a common variable
Suppose that a random parameter Θ is first obtained and that, conditional on it, independent Bernoulli variables are then generated with success probability Θ. The sequence is exchangeable. Its variables are nevertheless marginally dependent when Θ varies: they share a source of variability.
Mixture example · Bernoulli
Xi | Θ independent with distribution Bernoulli(Θ).
Cov(Xi, Xj) = Var(Θ), i ≠ j.
X̄n → Θ almost surely, under this construction.
If Θ is not constant, the average of a prolonged realization converges to its particular parameter, not necessarily to E[Θ], the average mixing all realizations. The example clarifies why exchangeability and dependence can coexist, and why a time average is not always identified with an overall average. The shared variable organizes differences between realizations that indiscriminate aggregation can conceal.
Laws of large numbers under dependence
There is no single law of large numbers with independence as its universal assumption. There are families of results with different hypotheses. For an integrable stationary ergodic sequence, a result of this kind follows from the ergodic theorem. Other formulations use mixing conditions, martingale structures, or other restrictions. Dependence requires further investigation; it does not automatically eliminate the possibility of convergence.
The formulation of a law must indicate the limit to which the average converges. It may be a constant, a conditional expectation, or another random variable. Nor is it enough to say that systematic error is absent: probabilistic and integrability assumptions have a mathematical function that this general expression does not replace.
Central limit theorems under dependence
Ergodicity alone does not guarantee a central limit theorem. In a stationary setting with sufficient dependence and moment conditions, an asymptotic variance can incorporate covariances at different lags. The convergence and value of that variance require additional assumptions; they do not follow merely from formally writing a sum.
Outline of long-run variance · Under sufficient conditions
σ²LR = γ(0) + 2∑k=1∞ γ(k).
√n (X̄n − μ) ⇒ Normal(0, σ²LR), when the CLT applies.
γ(k) denotes an autocovariance and ⇒ convergence in distribution. The expression shows why treating dependent observations as independent can distort uncertainties. Long-range dependence, heavy tails, or regime changes may require different normalizations or limits. Evaluating a model must identify which result applies.
Nabi’s 2021 compendium on laws of large numbers and central limit theorems for dependent variables are retained as continuations of the original program. They must be consulted in the corresponding documents; the present essay develops the conceptual problem and shows reference conditions without attributing results to those texts that have not been reproduced.
Norms, losses, and reference quantities
The discrepancy expressions included at the end of the original investigation allow a further clarification. For finite vectors, the following are losses or distances based on different norms:
Discrepancies between observation and representation
S1(y, ŷ) = ∑i=1n |yi − ŷi|.
S∞(y, ŷ) = maxi |yi − ŷi|.
Sp(y, ŷ) = (∑i=1n |yi − ŷi|p)1/p, 1 ≤ p < ∞.
In this context these are ℓp norms on vectors; Lp spaces generalize the discussion to functions and measures. The infinity norm is not an Lp norm with a finite value of p. These expressions do not assume that ŷ is an arithmetic mean and do not establish exchangeability of a joint distribution.
If a single constant is sought to represent data, different losses may select different centers. The sum of squared errors is minimized at the mean; the sum of absolute errors at a median; and the maximum absolute deviation at the midpoint between the minimum and maximum. For the data 0, 0, and 9, those choices are 3, 0, and 4.5, respectively. The example makes visible that measuring discrepancy involves choosing which differences matter.
Dimensional consistency is necessary to interpret a loss. Comparing variables with different units or scales requires justification of weights or transformations. This decision is not merely aesthetic: it changes the relative importance of discrepancies. The philosophy of statistics finds here a concrete problem concerning how the aims of knowledge enter into the construction of an analytical instrument.
A materialist interpretation of the inferential problem
The philosophical contribution can be formulated more precisely as a critique of automatically generalizing observations beyond their conditions. Distributions, dependencies, and forms of aggregation represent material relationships under a research question. Their examination requires attention to time, shared mechanisms, and transformations of the population under study.
Ergodicity constitutes an important theoretical option within a broader class of probabilistic and dynamical structures. Investigating alternatives does not mean abandoning its definition or making it a generic name for a limitation of knowledge. The original program of expanding the foundations of inference gains precision when it identifies which assumption is relaxed and which result replaces it.
Measurement, models, and criteria for testing claims
Delimiting the object and the objective
An analysis of complexity begins by specifying which phenomenon is being studied and which question is to be answered. Describing a form, reconstructing dynamics, identifying a causal mechanism, forecasting an event, and comparing interventions are different tasks. Model selection must follow the problem and the available knowledge while also considering constraints on data and computation.
Delimiting a system establishes components, relationships, and an environment. It is useful to report which variables are observed, which remain latent, which constraints are maintained, and which mechanisms might change during the study. In social processes, subjects’ actions can alter institutions and relationships; in physical systems, an experimental modification can change the dynamical regime. In both cases, uncertainty about parameters must be distinguished from a modification of model structure.
Identifying relevant scales and times
Spatial resolution, sampling frequency, total observation time, and transient duration condition the results. A series may be extensive in its number of records yet short in time relative to the system’s relaxation. A regularity may exist within an interval of scales and disappear beyond it. These conditions must be part of the explanation rather than relegated to a technical detail without epistemological meaning.
One strategy is to compare results across justified resolutions and windows. The stability of an estimate under these decisions may support its robustness. Variation may reveal a regime change or a limitation of the estimator. Interpretation depends on mechanisms and testing designs, not on a preference for a more uniform result.
Separating uncertainties
Sources of uncertainty and the questions they raise
Source
Problem
Relevant examination
Measurement
Resolution, calibration, missing records, or artifacts.
Provenance, instruments, and sensitivity of the result.
Sampling
Selection of observations and dependence.
Design, coverage, autocorrelation, and precision.
Parameters
Unknown values within an assumed structure.
Identification, estimation, and appropriate intervals.
Model
Omitted variables or inadequate mechanisms or assumptions.
Comparison of structures and observable consequences.
Computation
Discretization, rounding, and numerical tolerances.
Numerical convergence and procedural stability.
Evolution of the process
A change in regime or material relationships.
Continued validity of the description and explanation of the change.
These sources can interact. Sensitivity to initial conditions can amplify measurement errors, while an incomplete equation can produce errors that do not disappear as instrumental precision increases. An interpretation of uncertainty must examine these relationships and avoid assigning every disagreement between model and observation to a single cause.
Making explanations comparable
Model evaluation must specify observables, fitting criteria, and prediction objectives. A good representation of a series does not guarantee a correct causal explanation. It is useful to study predictions other than those used to construct the model, analyze sensitivity to assumptions, and compare alternatives sufficiently flexible to address the same question.
In dynamical systems, comparison may include stability, bifurcations, the distribution of observables, responses to perturbations, and scaling properties. Using a single measure can conceal important disagreements. Multidimensional robustness requires selecting relevant dimensions and understanding what each measures; multiplying indicators without interpretation does not by itself produce a more solid assessment.
Giving categories operational content
If a configuration is proposed to express a relationship between form and content, one must identify which components change, how their interaction is organized, and which observables respond. If a contradiction is postulated, the mutually conditioning tendencies, the mechanism of their tension, and the expected transformations must be described. If a whole is invoked, the relationships justifying the system boundary must be stated.
These requirements do not reduce philosophical categories to a fixed list of variables. They allow examination of how the categories guide specific investigations. They also make it possible to revise a category when it fails to distinguish relevant situations or introduces an equivalence unsupported by the models. The relationship between practice and theory includes this capacity for correction.
History and the social conditions of knowledge
Scientific production depends on instruments, institutions, access to information, funding, training, and controversies. The availability of computers made it possible to investigate dynamics whose mathematical formulations already existed; Kepler’s records allowed stellar behaviors to be analyzed at a previously inaccessible resolution. These material conditions are part of the history of problems and of the evidence available.
Recognizing them does not imply judging a result’s validity by its author’s political position or explaining every controversy through external interests. A materialist analysis must examine arguments and procedures alongside the conditions of their production and circulation. The relationship between science’s internal and social dimensions requires concrete historical investigation.
An open research program subject to testing
A continuation of this work could study how explanation changes in a particular system when passing from microscopic variables to collective observables. Another line could compare which inferential results persist when independence or ergodicity is relaxed. A third could historically reconstruct the use of dialectical categories in identifiable modeling decisions.
In each case, it would be useful to formulate a bounded question, define a body of evidence, and make explicit which finding would support, weaken, or modify the hypothesis. This specificity allows philosophical breadth to become research that can be communicated and debated. It also facilitates comparison of the proposed perspective’s contribution with other methodological orientations.
Complexity as a problem of relationships and mediations
The history examined shows that determinism and prediction are not equivalent concepts. An evolution defined by laws can be sensitive to initial differences that no instrument measures with unlimited precision. This sensitivity does not eliminate every possibility of knowledge: it redirects attention toward observables, distributions, structures, and scales whose behavior can be investigated in other ways.
The interdisciplinary journey also shows that chaos, fractality, and complexity describe related problems without constituting a single property. Strange nonchaotic attractors distinguish geometry from exponential instability; critical phenomena articulate fluctuations and scaling laws; collective models require variables whose relevance cannot be recognized by observing isolated components. Each case retains conditions limiting its extrapolation.
The categories of essence, phenomenon, appearance, form, and content can guide an investigation of these mediations if they are defined and made open to discussion. Their usefulness lies in the questions they allow us to formulate and the relationships they help reconstruct. Identifying mechanisms, constraints, and transformations offers a more precise ground for dialogue between dialectics and the sciences than immediate equivalence between vocabularies.
The analysis of scale and renormalization shows how certain descriptions can preserve regularities while modifying the relevance of details. The analysis of ergodicity and dependence shows that inference requires distinguishing symmetries, averages, and probabilistic structures. These developments provide concrete tools for examining the unity and differentiation of levels within a reality investigated through finite, historical practices.
The materialist position defended here holds that reality constrains representations and that knowledge develops through revisable mediations. Dialectics offers an orientation for studying interactions, tensions, and transformations; the scope of each interpretation depends on its articulation with results and evidence. The sciences, in turn, allow that orientation to be refined and transformed.
Problems remain open concerning comparison between scales, the identification of mechanisms in historical systems, the interpretation of chance, and statistical results under forms of dependence. Their development requires specific models and studies. The investigation continues wherever a category becomes a precise question about material relationships, conditions of knowledge, and possibilities of transformation.
XVI · Reference apparatus
Glossary and clarifying notes
Attractor
An invariant set attracting states from a region under a specified definition and conditions.
Strange nonchaotic attractor
A geometrically irregular structure that does not exhibit chaos under the adopted dynamical criterion; it requires distinguishing exponents and forcing or response variables.
Bifurcation
A qualitative change in dynamics as parameters vary.
Universality class
A family of systems with shared scaling behaviors under relevant common conditions.
Coarse-graining or block description
The construction of variables or descriptions grouping together smaller-scale details.
Dialectical contradiction
A relationship between tendencies or determinations that condition one another and stand in tension within a process; it requires material specification.
Sensitive dependence
A property relating initial proximity to later separation according to a relevant mathematical definition.
Ergodicity
A property of dynamics relative to a measure that rules out a nontrivial invariant decomposition.
Phase space
A space whose points represent states of the modeled system.
Critical exponent
An exponent characterizing an asymptotic behavioral law near a particular critical transition.
Lyapunov exponent
An asymptotic rate of expansion or contraction of infinitesimal perturbations under appropriate conditions.
Fractality
A collection of geometrical and scaling properties requiring specification of the definition of dimension and observational regime.
Exchangeability
Invariance of a joint distribution under finite permutations of its indices.
Mixing
A property of loss of dependence between events separated by evolution, formulated relative to a measure.
Renormalization
A family of procedures whose meaning must be specified: scale transformations, effective parameters, or other related operations in a defined context.
Relaxation time
A temporal scale associated with the approach or response of an observable or distribution under particular conditions.
Supplementary historical and methodological notes
Recognition and contributions. Lewontin received the 2015 Crafoord Prize in Biosciences jointly with Tomoko Ohta. The recognition concerns contributions to understanding genetic polymorphism; its relationship with the author’s philosophical trajectory is an additional question. The prize should not be described simply as a category of the Nobel Prizes. The reference to Levins’s contributions to metapopulation theory and the study of organisms in changing environments is also retained.
Gould, Lewontin, and Levins. The discussions of spandrels, punctuated equilibria, and the critique of adaptationism belong to related but distinct contributions. Lewontin and Levins’s essay on Gould remains relevant to studying the relationship between intellectual commitment and scientific practice. The cited joint works must be distinguished from individually authored works and other collaborations.
Publication and consultation dates. A download date, website update, or translation date does not replace the year of the original article. The editions used in the 2021 paper are retained when identifiable, and verified corrections are indicated, including Sharma (2009), Werndl (2009), Lindner and colleagues (2015), and Laplace’s first publication (1814).
Documentary figures. Illustrations from the original paper retain their data, colors, and labels. Their colors belong to the source scientific material rather than the interface palette. Added captions delimit their scope; they do not update or recalculate the authors’ results. The panels and equation with repeated numbering are identified separately here.
Idealization and reality. A mathematical object can have an infinite scaling structure; a material observation has finite resolution and range. Comparing them requires a measurement model. This difference does not make a model arbitrary, but prevents unconditional identification of its ideal construction with the material object.
Logical terminology. The broad philosophical use of “formalization” to mean making something explicit is not equivalent to possessing a complete formal calculus. Likewise, criticizing a dichotomy applied to a phenomenon does not demonstrate that classical logic lacks the means to represent degrees, probabilities, relationships, or temporal evolution.
Expert judgment. Scientific experience can guide hypotheses and decisions. Its value depends on domain relevance, evidence quality, and the possibility of revision. Forum contributions are retained for their historical role in the argument and distinguished from articles and proved results.
Use of “meta-analysis.” A cross-cutting reflection on theories is not necessarily equivalent to the statistical procedure of combining study results. To avoid ambiguity, “epistemological analysis” or “research program” is used when that is the intended meaning.
Comparison with the original. The 2021 edition contains the formulations and extended quotations motivating this revision. A reference’s presence in the bibliography does not mean that all its claims are adopted; its function may be historical, conceptual, documentary, or critical.
The sources of the original investigation are retained, with additional references for the clarifications and expansions. Forum and popular-science entries are identified as such. Dates of consulted editions are distinguished from first publications where relevant. Incomplete references in the original paper are flagged without inventing publication details. Bibliographic titles are retained in the language of the cited work.
Bibliography of the original investigation, with clarifications
Aravindh, M., Venkatesan, A., and Lakshmanan, M. (2018). Strange nonchaotic attractors for computation. Physical Review E, 97, 052212. DOI.
Barnett, W. A., and Chen, P. (1988). Deterministic chaos and fractal attractors as tools for nonparametric dynamical econometric inference: With an application to the Divisia monetary aggregates. Mathematical and Computer Modelling, 10(4), 275–296. DOI.
Bjorvand, A. T. (1995). A New Approach to Intelligent Systems Theory. Norwegian Institute of Technology, University of Trondheim. Project report, April 28. Document.
Elert, G. (n.d.). Flow regimes. The Physics Hypertextbook. The original investigation records consultation in 2020. Resource.
Gottwald, G. A., and Melbourne, I. (2016). The 0–1 test for chaos: A review. In C. Skokos, G. A. Gottwald, and J. Laskar (eds.), Chaos Detection and Predictability, Lecture Notes in Physics, 915, 221–247. Springer. Authors’ version. Its pagination is independent of the book edition.
Halperin, B. I. (2019). Theory of dynamic critical phenomena. Physics Today, 72(2), 42–43. DOI.
Jaynes, E. T. (2003). Probability Theory: The Logic of Science. Cambridge University Press.
Kessler, D. P., and Greenkorn, R. A. (1999). Momentum, Heat, and Mass Transfer Fundamentals. Marcel Dekker.
Kilifarska, N. A., Bakmutov, V. G., and Melnyk, G. V. (2020). The Hidden Link Between Earth’s Magnetic Field and Climate. Elsevier.
Landau, L. D. (1994). Física teórica. Física estadística, vol. 5, second Spanish-language edition cited in the original investigation. Reverté. Identification of the volume, edition, and collaborators must correspond to the copy used.
Laplace, P.-S. (1814). Essai philosophique sur les probabilités. First publication. The original investigation also uses A Philosophical Essay on Probabilities (1902), an English edition. 1814 copy. Translation details in the original entry require comparison with the English copy’s title page.
Lesne, A. (1998). Renormalization Methods: Critical Phenomena, Chaos, Fractal Structures. Translated by Leila Schneps. Wiley. Contents published by the author.
Lesne, A., and Laguës, M. (2012). Scale Invariance: From Phase Transitions to Turbulence. Springer.
Li, S., and Li, H. (2006). Parallel AMR Code for Compressible MHD or HD Equations. Los Alamos National Laboratory. Technical resource cited in the original paper. Archived copy.
Lindner, J. F., Kohar, V., Kia, B., Hippke, M., Learned, J. G., and Ditto, W. L. (2015). Strange nonchaotic stars. Physical Review Letters, 114, 054101. DOI · Authors’ preprint.
Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130–141. DOI.
Mandelbrot, B. B. (1983). The Fractal Geometry of Nature. W. H. Freeman. Edition cited in the original investigation; the 1982 edition is distinguished when reconstructing the work’s history.
Marxists Internet Archive (n.d.). Formal Logic and Dialectics, in the resource The Meaning of Hegel’s Logic. The original investigation records 2018 as the resource or consultation date. Text. It is used as a philosophical intervention, not as proof of a mathematical impossibility.
McCullagh, P., and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman & Hall.
Nabi, I. (2021). Diferentes abordajes para el TCL con variables dependientes [Different approaches to the CLT with dependent variables]. La Biblioteca del Pueblo / El Blog de Isadore Nabi. Link recorded in the original investigation.
Nabi, I. (2021). Diferentes abordajes para la LGN con variables dependientes [Different approaches to the LLN with dependent variables]. La Biblioteca del Pueblo / El Blog de Isadore Nabi. Link recorded in the original investigation.
Oestreicher, C. (2007). A history of chaos theory. Dialogues in Clinical Neuroscience, 9(3), 279–289. Full text.
Pezard, L., and Nandrino, J.-L. (2001). Paradigme dynamique en psychopathologie: la «théorie du chaos», de la physique à la psychiatrie. L’Encéphale, 27(3), 260–268. Bibliographic record.
Poincaré, H. (1908). Science et méthode. Section on chance. The original investigation cites the English version, “Chance,” in Science and Method, pp. 64–90. Cited extract. The dates of the French work and its translation must be distinguished.
Princeton University Press (n.d.). The Fundamental Postulate. Chapter distributed as a publishing resource. Cited document. The 2021 entry does not identify the author or volume; it is retained without conjecturally completing those details.
ResearchGate (2018). When should one use Fuzzy set theory and Rough set theory? Is there any clear-cut line of difference between them? Forum; contributions include Sandipan Karmakar’s intervention. Discussion.
ResearchGate (2020). What is the difference between Fuzzy rough sets and Rough fuzzy sets? Forum; Ganesan G.’s contribution is cited in the original investigation. Discussion.
Rosental, M., and Iudin, P. (1971). Diccionario filosófico. Tecolut. San Salvador edition cited in the original investigation. The history of the work and its editions predates that year.
Russell, K. (2014). Hypothesis testing. Teaching material, University of Manitoba. Historical link recorded in the original. Its discussion is supplemented here with sources on the differences between statistical testing and logical deduction.
Sharma, V. (2009). Deterministic chaos and fractal complexity in the dynamics of cardiovascular behavior: Perspectives on a new frontier. The Open Cardiovascular Medicine Journal, 3, 110–123. DOI · Publisher’s text.
Stanford Encyclopedia of Philosophy (n.d.). Quantum Logic and Probability Theory. Resource cited in the original investigation with the date 2002. Entry. Its version may change; historically consulted content must be distinguished from later revisions.
Valdebenito, E. (2019). Fractales: la geometría del caos. Document deposited in viXra. Cited document. Retained as an expository source with a function distinct from the mathematical definitions and proofs used in this edition.
Werndl, C. (2009). What are the new implications of chaos for unpredictability? The British Journal for the Philosophy of Science, 60(1), 195–220. DOI. A later deposit date for a version does not replace the article’s year.
Supplementary sources and extensions
Birkhoff, G. D. (1931). Proof of the ergodic theorem. Proceedings of the National Academy of Sciences, 17(12), 656–660. DOI.
Cohen, P. J. (1963). The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences, 50(6), 1143–1148. DOI.
Royal Swedish Academy of Sciences (2015). The Crafoord Prize in Biosciences 2015. Institutional announcement.
Evans, D. J., and Morriss, G. P. Statistical Mechanics of Nonequilibrium Liquids. Section 3.4: Ergodicity, Mixing and Lyapunov Exponents. ANU Press electronic edition. Section.
Gigerenzer, G. (2004). Mindless statistics. The Journal of Socio-Economics, 33, 587–606. DOI.
Grebogi, C., Ott, E., Pelikan, S., and Yorke, J. A. (1984). Strange attractors that are not chaotic. Physica D, 13, 261–268. DOI.
Johnston, H. (2015). “Golden stars” pulsate in a strange, non-chaotic way. Physics World, February 17. Popular-science source cited in a note in the original investigation. Article.
Lenin, V. I. (1909). Materialismo y empiriocriticismo [Materialism and Empirio-Criticism]. Work written in 1908; its composition is distinguished from first publication.
Levins, R., and Lewontin, R. C. (1985). The Dialectical Biologist. Harvard University Press.
Levins, R., and Lewontin, R. C. (2007). Biology Under the Influence: Dialectical Essays on Ecology, Agriculture, and Health. Monthly Review Press.
Lewontin, R. C., and Levins, R. (2002). Stephen Jay Gould—What Does It Mean to Be a Radical? Monthly Review, 54(6), November. Essay mentioned in the notes of the original paper. Journal issue.
Pawlak, Z. (1982). Rough sets. International Journal of Computer & Information Sciences, 11, 341–356. DOI.
Perezgonzalez, J. D. (2015). Fisher, Neyman–Pearson or NHST? A tutorial for teaching data testing. Frontiers in Psychology, 6, 223. DOI.
Wilson, K. G. (1983). The renormalization group and critical phenomena. Reviews of Modern Physics, 55, 583–600. DOI. Account related to his 1982 Nobel Lecture.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI.
Euclid. Elementos [Elements]. Spanish-language edition of 1994 cited in the note on the golden ratio in the original paper. The definition of division in extreme and mean ratio is distinguished from other geometrical results mentioned there; volume and publication details are not completed without consulting the copy.
Documentary archive
Original 2021 investigation
The original Spanish paper retains all 54 pages, extended quotations, notes, equations, figures, and bibliography. It allows this edition to be compared with the initial formulations. The revised version develops clarifications and expansions in the body of the essay; the original file remains as a documentary record.
Consult the complete original Spanish paper
If the browser’s PDF viewer does not appear, use the download link below.
Marxist Philosophy of Science · Mathematical statistics
On Principal Component Analysis (PCA)
Geometric intuition, algebraic foundations, statistical interpretation, and the epistemological scope of dimensionality reduction
JOSÉ MAURICIO GÓMEZ JULIÁN
Representing a phenomenon through many variables allows a rich description, but can also make its principal relationships harder to identify. Principal component analysis—PCA—constructs new coordinates to study the joint variation of those variables. The decisive question is how to reduce a representation’s complexity without losing sight of what is preserved, what is discarded, and under which conditions its interpretation is legitimate.
Abstract
This essay develops PCA through four connected perspectives: a geometric transformation of a cloud of observations; a constrained optimization problem; a matrix decomposition; and a tool for scientific exploration and interpretation. It retains the gene-expression and financial-assessment examples, component-retention criteria, graphical diagnostics, Mahalanobis distance, the discussion of NIPALS, and the comparison with linear regression. Mathematical derivation, reproducibility, and the conditions supporting inference are expanded.
Editorial note. All thematic strands and 51 figures from the 2021 document are preserved. Formulations requiring greater precision are revised in the main discussion; the original images are collected without cropping in the visual archive and retain their attributions. The historical Spanish PDF remains available at the end.
I. Variables, attributes, and dimensionality
A dataset organizes observations of particular units of study: people, animals, firms, territories, products, or states of a physical system. Each unit is described by variables. When p numerical variables are recorded for n units, the usual representation is a matrix X of size n × p: rows correspond to observations and columns to variables. Each row can be interpreted as a point in the feature space ℝp.
The word attribute requires clarification. In data science it may designate a measured feature; in R it also refers to metadata associated with an object, such as its dimensions, names, or class. Adler’s observation about the importance of attributes in R objects is relevant to understanding their computational representation, but such metadata should not automatically be identified with the statistical variables in an investigation. A class attribute can change how R interprets an object without adding a dimension to the measured phenomenon. [1]
The dimensionality of a representation is the number of coordinates used. Intrinsic dimensionality, by contrast, refers to the degrees of freedom needed to describe the underlying structure under the model being considered. A cloud lying approximately on a plane may have many coordinates while still admitting an approximately two-dimensional representation. With noise, effective dimensionality also depends on the approximation tolerance. There is not always a single minimum number independent of the purpose of the analysis.
Variable selection and component extraction
Dimensionality reduction encompasses different strategies. Variable selection retains a subset of the original columns; feature extraction constructs new variables from them. PCA belongs to the second family: a component may combine information from all the initial variables. By definition, it does not identify which columns should be removed or solve a combinatorial search over subsets. Such problems can be connected to PCA, but they have different formulations.
This distinction also limits the meaning of “essential information.” In ordinary PCA, the operational criterion is the variance represented by a linear subspace. Whether that variation is essential to a biological, economic, or social explanation requires an additional argument about measurement, scale, context, and the research question.
II. Bellman, Pearson, and the problem of representation
The expression curse of dimensionality, associated with Richard Bellman, brings together difficulties that arise as the number of dimensions increases: growing computational costs for certain procedures, relative sparsity of observations, reduced effectiveness of some notions of neighborhood, and the need to impose structure on the problem. On a grid with ten positions per axis, systematically exploring p dimensions produces 10p combinations. This illustrates exponential growth; it is not a universal law requiring every statistical method to use exactly that amount of data. [3]
The difficulty depends on the problem, distribution, regularity, and relationships among variables. It is therefore useful to distinguish feature space from the sample space of probability theory: the former organizes observable coordinates, whereas the latter contains the possible outcomes of an experiment. They can be connected through random variables, but the concepts are not interchangeable.
In 1901, Karl Pearson studied lines and planes of closest fit to systems of points. His geometric approach approximates a cloud by a line or plane by minimizing the sum of squared orthogonal distances. Hotelling’s later formulation helped establish principal component analysis as a multivariate technique. This genealogy predates the contemporary vocabulary of machine learning. [11][22]
Historical context and scientific criteria
The original document politically characterizes Pearson as a Marxist. The question of the relationship between intellectual biography, the history of statistics, and philosophy of science is retained here, but that classification requires a dedicated historical investigation and is not used as a premise in deriving PCA. The method’s mathematical validity and the historical assessment of its author belong to different levels of argument that must be supported separately.
Pearson’s line provides a useful intuition: an elongated cloud can be described relatively accurately by a coordinate along its dominant direction. Identifying that direction, however, does not establish a cause of the phenomenon. The fit describes a structure of variation in the measurements; explaining it requires additional theory and evidence.
III. Definition and optimization of PCA
Let x be a random vector of p variables, μ its mean, and Σ its covariance matrix, assuming finite second moments. The first principal component is a linear combination of the centered variables with maximum variance, subject to a normalization constraint on its coefficients. In sample analysis, the population covariance is replaced by its estimate S.
PC1 = a1T(x − μ) = ∑j=1p aj1(xj − μj) maximize aTΣa, subject to aTa = 1.
The vector a₁ defines a direction; the component is the variable obtained by projecting onto it. The unit-length constraint prevents artificially increasing variance by multiplying all coefficients by a constant.
Without this constraint, multiplying a by an arbitrarily large number would increase the combination’s variance without bound. Normalization makes variation along different directions comparable. The second component maximizes the remaining variance, with a unit coefficient vector orthogonal to the first. The procedure continues successively.
ak ∈ arg max aTΣa subject to aTa = 1 and aTaℓ = 0 for ℓ < k.
Each new direction is chosen in the orthogonal complement of the preceding ones. This formulation characterizes the solution independently of the algorithm used to compute it.
The Lagrangian for the first direction is L(a, λ) = aTΣa − λ(aTa − 1). The stationarity condition leads to Σa = λa. The principal directions are therefore eigenvectors of the covariance matrix, and their variances are the corresponding eigenvalues. The first direction is associated with the largest eigenvalue; the remaining directions are ordered in decreasing order. If eigenvalues are repeated, a particular basis for the associated subspace may not be unique.
The mathematical interpretation has two levels. Eigenvectors specify how to combine the original coordinates; eigenvalues quantify variation along those combinations. Observation scores show where each unit of study lies in the new coordinate system. Keeping these three objects separate avoids confusing weights, observations, and amounts of variation.
The direction tells us how to look at the data; the score locates an observation in that view; the eigenvalue measures how much variation appears along it.
IV. Two genes and six mice
Following Josh Starmer’s teaching example used in the original document, consider six mice described by expression measurements for two genes. The mice are the units of study and the genes are the variables. Transcription is the synthesis of RNA from DNA; it must be distinguished from translation, which produces proteins. The quantities are retained as teaching values: no experimental unit or biological precision undocumented by the example is attributed to them. [13]
Original example data · transposed to show one observation per row
Mouse
Gene 1
Gene 2
1
10
6
2
11
4
3
8
5
4
3
3
5
2
2.8
6
1
1
In a two-dimensional plot, each mouse occupies the position defined by its two measurements. The first three have relatively high values for both genes, while the remaining mice have lower values. The cloud allows us to ask questions about covariation and grouping, but a figure alone does not establish population differences or causal mechanisms.
The first step in centered PCA is to subtract each variable’s mean. For these values, the mean of gene 1 is approximately 5.8333 and that of gene 2 is 3.6333. The centroid moves to the origin; differences between pairs of observations remain unchanged.
xijc = xij − x̄j Xc = X − 1x̄T.
The superscript c denotes centering; 1 is a column vector of n ones. Centering is not the same as dividing by standard deviations.
Starmer’s original sequence is preserved in full in the visual archive. The new figure shows the same six centered points, with the principal direction computed from the table and the orthogonal residuals involved in the fitting criterion.
Figure from the revised edition. Projections computed for the six mice in the table. Centering preserves distances; projection onto PC1 introduces the indicated residuals.
The example teaches the geometric mechanics. An actual gene-expression study requires attention to measurement, sequencing depth, appropriate transformations, batch effects, replication, and uncertainty. PCA on an unprepared count table may reflect technical magnitudes before biological differences of interest.
V. Transformations, structure, and preservation
A transformation is a mapping between spaces. It may be linear, affine, or of another kind. It need not map a domain into the same set: in general, we write f: D → C, and its image is the set of values actually reached in the codomain. For ordinary PCA, translations, orthogonal coordinate changes, and projections onto subspaces are especially important.
Transformations discussed in the original and their properties
Transformation
Characterization
Important distinction
Translation
x ↦ x + b
Preserves distances and differences between points; generally not linear because it does not fix the origin.
Rotation
x ↦ Qx, with QTQ = I and det Q = 1
Preserves inner products, norms, angles, and distances.
Reflection
An orthogonal transformation reversing orientation
Also preserves distances. An orthogonal matrix may combine rotations and reflections.
Uniform dilation or contraction
x ↦ cx
Multiplies all distances by |c|; an isometry only when |c| = 1.
Stretching along one or more axes
x ↦ Dx, with D diagonal
Changes relative geometry when scale factors differ.
Shear
A linear deformation that shifts coordinates according to other coordinates
May preserve area or volume in particular cases, but generally alters angles and lengths.
Orthogonal projection
x ↦ Px, with PT = P and P² = P
Generally not invertible; dimensionality reduction may contract distances and map distinct points onto the same point.
Isomorphism, homeomorphism, and isometry
An isomorphism preserves the specified structure: for example, addition and scalar multiplication in vector spaces. A homeomorphism is a continuous bijection with a continuous inverse. An isometry preserves distances; when also surjective, it establishes an equivalence between metric spaces. These notions are related, but none should be used indiscriminately in place of the others.
Preserving topology does not imply preserving a norm or metric: multiplying all coordinates by two provides an elementary counterexample. The transformation is a homeomorphism of ℝp and changes every nonzero distance. Similarly, a translation preserves distances without being a linear isomorphism under the original vector operations, unless it is the zero translation.
In an inner-product space, the Euclidean norm is defined by ‖x‖ = √⟨x,x⟩ and the induced distance by d(x,y) = ‖x − y‖. The norm measures vector length; the metric measures distance between points. Hilbert spaces generalize this framework to complete inner-product spaces. These extensions situate PCA within a broader mathematical landscape, but they do not justify transferring its properties to any normed space without proof.
The philosophical discussion of matter and form
The original reflection on Aristotle and Plato can be retained as a question about the connection between a system’s elements and the relations organizing them. This philosophical interpretation should nevertheless be distinguished from the mathematical definition of structure. Classifying objects “up to isomorphism” means grouping them according to a specified structural equivalence; it does not mean that they belong to any classification whatsoever or to none at all.
From a materialist perspective, mathematical abstraction can be examined as an operation that selects properties of concrete phenomena to study particular relations. The critical requirement is to state what is abstracted and what is preserved. In PCA, certain properties are preserved exactly under a complete orthonormal change of basis; others are only approximated when coordinates are discarded. Philosophical analysis becomes more precise when it starts from that distinction.
VI. Orthogonal projection and reconstruction
Let a be a unit vector defining a line through the origin, and let z be a centered observation written as a column vector. Its coordinate along that line is t = aTz. Its projection in the original space is ẑ = aaTz. The difference e = z − ẑ is perpendicular to the projection direction.
z = ẑ + e, ⟨ẑ,e⟩ = 0 ‖z‖² = ‖ẑ‖² + ‖e‖².
The Pythagorean theorem decomposes squared length into a represented part and a residual part.
For a fixed centered cloud, the sum of ‖z‖² over all observations is constant. Maximizing the sum of squared projected lengths therefore equals minimizing the sum of squared orthogonal residuals. This equivalence requires the same geometry, the same dataset, and comparison of subspaces of equal dimension.
∑i ‖zi‖² = ∑i ti² + ∑i ‖ei‖².
The two formulations do not distinguish an “analytical” PCA from a “numerical” one. They are mathematically equivalent objectives that can be solved using different algorithms.
Squaring is essential. Minimizing a sum of unsquared distances generally defines a different problem and may lead to another solution. Nor must components be computed through the Minkowski or Cauchy–Schwarz inequalities: these tools help prove properties of norms and inner products, but do not replace a specification of the optimization procedure.
A projection is a linear transformation that can be defined without choosing coordinates. Once a basis is chosen, it has a precise matrix representation; there is no opposition between a “purely mathematical” projection and a “merely computational” matrix. For a subspace whose orthonormal columns are collected in Vk, the projection matrix is Pk = VkVkT.
The original discussion of projective geometry is retained as context, with a qualification: the orthogonal projection of PCA belongs here to Euclidean geometry and linear algebra. It need not be conflated with the whole of projective geometry, whose historical development was collective and cannot simply be attributed to Lobachevsky.
In several dimensions, the criterion is written as minimization of the Frobenius norm of the reconstruction residuals. Approximation by the first k components is optimal among approximations of rank at most k under this quadratic loss. “Optimal” always means optimal for a declared criterion, not for every possible scientific purpose.
This identity uses eigenvalues of the sample covariance with divisor n − 1. To reconstruct the original coordinates, add the mean; if variables were standardized, also reverse the scaling.
VII. Slope, unit vectors, and components
The original geometric explanation uses a line with slope 0.25, represented by a displacement of four units along the first axis and one along the second. Its direction vector is (4,1)T. Its length is √17 ≈ 4.1231; dividing its coordinates by that length yields a unit direction.
The vectors have unit norm and zero inner product. The rounded values 0.97 and 0.242 in the original material illustrate this normalization.
The line can be written as y = 0.25x; its slope equals the derivative of this linear function. But the coefficients of a principal direction are not generally “derivatives of the phenomenon.” Here, slope is the ratio of the direction vector’s coordinates. The unit eigenvector contains a normalized representation of that direction and is determined by the covariance structure of the data.
Normalizing (4,1) does not standardize each observation or transform the data into a normal distribution. Nor does it make a cloud’s mean zero. It produces a unit-length vector. With that direction, an observation’s first centered coordinate would be approximately t1 = 0.9701x1c + 0.2425x2c; the second would be t2 = −0.2425x1c + 0.9701x2c.
Illustration and exact calculation
The direction (4,1) belongs to the original graphical explanation. Axes computed exactly from the six-mouse table may differ from that teaching slope. For traceability, the new figure and numerical table in this edition use the SVD of the transcribed values, while the visual archive preserves the original construction.
The Cauchy–Schwarz inequality guarantees that |⟨x,a⟩| ≤ ‖x‖‖a‖ and helps bound the length of a projection. The Minkowski inequality expresses the triangle inequality for Lp norms, with p ≥ 1. Pythagoras provides an exact identity when components are orthogonal in an inner-product space. These relationships serve different purposes; they are not three alternative algorithms for “finding” components.
VIII. Eigenvalues, eigenvectors, and SVD
For the centered matrix Xc, with observations in rows, sample covariance is S = XcTXc/(n − 1). This matrix is real, symmetric, and positive semidefinite. By the spectral theorem, it admits an orthonormal basis of eigenvectors collected as columns of V. Its eigenvalues are real and nonnegative.
S = VΛVT, VTV = I Λ = diag(λ1, …, λp), λ1 ≥ ⋯ ≥ λp ≥ 0 T = XcV, Cov(T) = Λ.
V describes the directions and T contains the scores. The diagonal matrix Λ contains the variances and expresses sample uncorrelatedness between different components.
The singular value decomposition of the data matrix provides the same centered PCA solution. In reduced form, if the rank is r, we write Xc = UΣsVrT, with U of size n × r, Vr of size p × r, and diagonal Σs of size r × r. The subscript distinguishes singular values from population covariance.
The singular values of Xc are square roots of the eigenvalues of XcᵀXc. To obtain component variances, the divisor n − 1 must be included.
What distinguishes the two decompositions
An SVD exists for any real rectangular or square matrix. Diagonalization by similarity requires a square matrix and is not always possible; for general matrices, eigenvalues may be complex and an eigenvector basis need not be orthogonal. These general qualifications, however, do not describe a difficulty of the covariance matrix in PCA: symmetry guarantees orthogonal diagonalization, and positive semidefiniteness guarantees nonnegative eigenvalues.
In a general diagonalization A = PDP−1, the outer matrices are inverses. In an SVD A = UΣVT, the matrices U and V have different roles and are not generally each other’s inverses. If rectangular, they do not even admit a two-sided inverse. In their complete square versions they represent orthogonal transformations; a reduced SVD contains orthonormal bases for subspaces.
The connection between eigenvectors and singular vectors is particularly direct in PCA, because the right singular vectors of Xc are eigenvectors of S. These are not two incompatible theories, but connected formulations for computing the same representation under the same centering and scaling conventions. [2][4]
Explained variance and discarded variance
ρj = λj/∑ℓ=1p λℓ Ck = ∑j=1k ρj, Rk = 1 − Ck.
ρj is the proportion explained by component j; Ck is the cumulative proportion in the first k; Rk is the fraction of variance excluded from that representation. Positive total variance is assumed.
The eigenvalue of a retained component represents variation preserved along that axis, not “lost information.” Reconstruction loss is related to the sum of discarded eigenvalues. This quadratic relationship does not turn variance into a universal measure of meaning, causality, or information in Shannon’s sense.
IX. Centering, standardization, and matrix choice
Centering subtracts the mean. Standardization by z-scores subtracts the mean and divides by each variable’s standard deviation. Vector normalization, in the sense used above, divides by a norm. These are different operations, although all may informally be described as changes of scale.
zij = (xij − x̄j)/sj, sj > 0 R = ZTZ/(n − 1).
If sj is computed using divisor n − 1, the sample covariance of Z is the original variables’ correlation matrix R. Constant columns require prior treatment.
The z transformation produces mean zero and unit variance under the convention used. It does not impose normality: a skewed distribution remains skewed after an affine transformation with positive scale. Descriptive PCA can be computed without assuming multivariate normality; certain tests, intervals, or control limits do require additional assumptions.
Two legitimate decisions with different objectives
Basis of analysis
Interpretation
Substantive decision
Covariances
Centered variables are analyzed in their measurement scales.
May be appropriate when units are comparable and variance differences are meaningful.
Correlations
The covariance of standardized variables is analyzed.
Gives each variable the same initial variance; often relevant with heterogeneous units, although it may also amplify unreliable variables.
A variable measured in thousands may dominate one measured in fractions if changing units increases its variance. But the magnitude of a variable’s mean does not, by itself, determine its variance. A scaling decision requires examination of dispersion, units, and purpose, rather than merely large or small absolute values.
Standardization changes the geometric problem by weighting the original coordinates differently. It should not be presented as a neutral operation that always improves analysis. It also matters whether standard deviation uses n or n − 1. In the same sample, consistently applying either divisor to all variables introduces only a common factor between the two standardizations; directions and explained-variance proportions coincide, but score magnitudes and variances change.
The connection to the same author’s essay on rescaling remains relevant. The web edition retains that link and emphasizes these distinctions so that both texts support each other without reproducing the confusion between standardization and transformation to a normal distribution. [9]
X. Orthogonality, uncorrelatedness, and rank
Two nonzero vectors with zero inner product are orthogonal and therefore linearly independent. The converse is false: two vectors may be linearly independent without being perpendicular. “Orthogonal” and “perpendicular” do not designate properties reserved for three and two dimensions respectively; orthogonality applies in any dimension within an inner-product framework.
For the coefficients of distinct components, ajTak = 0. For the principal variables, covariance is ajTΣak = 0. These identities are connected by the eigenvalue equation. They must not be confused with statistical independence, which requires factorization of the joint distribution.
A counterexample to independence
If U is uniformly distributed on [−1,1], the variables U and U² have zero covariance by symmetry, but U² is completely determined by U. The absence of linear correlation does not eliminate dependence. Under a joint normal distribution, however, uncorrelatedness between components does imply independence.
The number of components with positive variance equals the rank of the centered matrix. For n observations and p variables, that rank cannot exceed min(n − 1, p). A complete covariance decomposition can provide p eigenvectors, but some eigenvalues will be zero when there is linear dependence or insufficient sample rank.
Retaining a complete orthonormal basis of ℝp preserves Euclidean distances under the coordinate change. Retaining all positive-variance components also allows exact reconstruction of the observed centered cloud. With fewer components, the approximation generally leaves residuals; these vanish only when the data already lie in the retained subspace.
Axis order is fixed by eigenvalues, except for ties; sign orientation is arbitrary. Simultaneously replacing aj with −aj and reversing its scores describes the same solution. Isolated signs should therefore not be compared across programs without first aligning conventions.
XI. How many components should be retained?
Component retention is a methodological decision. It depends on the objective, measurement quality, and cost of discarding variation. A plane may be useful for visualization and insufficient for another task. The original document brings together Kaiser, Cattell, MAP, Bartlett, Horn, and Lawley; all are retained here, with their respective scopes clarified. [15]
Cumulative variance and reconstruction error
One may choose the smallest k reaching a previously justified variance proportion. Thresholds such as 80%, 90%, or 95% are conventional decisions, not guarantees of scientific sufficiency. In prediction, selection should be assessed against out-of-sample performance; in reconstruction, it can be justified through a tolerable error in the relevant units.
Kaiser (1960): eigenvalues greater than one
In correlation-based PCA, each standardized variable contributes variance one, and the mean of the p eigenvalues is one. The rule retains components with λ > 1, meaning those explaining more variation than an average standardized variable. This rationale does not automatically transfer to a covariance matrix with arbitrary units. The rule may retain too many or too few components and should not operate as an automatic decision.
Cattell (1966): the scree plot
The scree test looks for a change in slope between the first eigenvalues and a flatter tail. It offers a visual reading of diminishing returns. An “elbow” may be ambiguous; different readers may identify different cutoffs, and some spectra show no clear bend. The original suggestion to examine a baseline along the tail can be retained as a graphical aid, while making its interpretive element explicit.
Velicer (1976): minimum average partial (MAP)
The MAP criterion examines residual correlations among variables after successively removing components and normalizing residuals to obtain partial correlations. It seeks a minimum in the average of squared off-diagonal values. The intuition is to stop when removing further components no longer usefully reduces residual association and begins extracting specific structure or noise.
MAP should not be confused with choosing a percentage of variance or with a universal test of latent causality. The original source’s recommendation to have several high-weight variables can be useful for stable interpretation, but it is not a mathematical identity defining the criterion. Accessible implementations exist, including those documented in psych, so the practical claim that the method is difficult to access needs updating. [24]
The family of tests mentioned in the original evaluates hypotheses about the structure of remaining eigenvalues, such as their equality, under specific distributional assumptions and approximations. It is not the same as the familiar sphericity test applied to the complete correlation matrix. When used sequentially, the exact hypothesis, degrees of freedom, sample size, and stopping criterion must be stated.
Failure to reject a hypothesis does not prove it true: it may also reflect low power. Nor does every statistically significant result require adding a useful dimension. Interpretation must distinguish testing population structure from making a practical decision about representation.
Horn (1965): parallel analysis
Parallel analysis compares observed eigenvalues with those obtained under a random reference of the same size n × p. Simulation or permutation reproduces a setting of absent association according to an explicit mechanism. Components whose eigenvalues exceed the corresponding reference values are retained. It is necessary to state whether comparison uses means or a quantile, such as the 95th percentile, and how many replicates are generated.
The logic addresses a limitation of Kaiser: even when the population correlation matrix is the identity, sample eigenvalues fluctuate and will not all equal one exactly. One must also specify whether parallel analysis concerns components or common factors, because the matrices and comparisons are not identical. [25]
Lawley (1940): likelihood ratios and factor analysis
Likelihood-ratio procedures associated with maximum-likelihood factor analysis assess whether a given number of factors adequately reproduces a covariance or correlation structure under a probabilistic model. They are retained here as part of the original overview, but are not presented as an automatic property of descriptive PCA.
A common-factor model distinguishes shared variation from specific variation; PCA decomposes the total variation in the analyzed variables. Confusing components with factors may lead to criteria or interpretations being applied outside their framework. When software offers both procedures, it is important to check which has actually been run.
A documented decision
Report the primary criterion, compare it with another of a different kind, examine stability, and justify discrepancies. No rule replaces scientific judgment. If the objective changes from visualization to prediction or a latent-factor model, the appropriate choice of dimensions may also change.
XII. Interpreting the financial example
The document uses Minitab output based on eight variables: income, education, age, residence, employment, savings, debt, and credit cards. The eigenvalue table and complete coefficient matrix from that output are reproduced below. The labels require checking how each variable was measured or coded; “residence,” for example, does not by itself identify whether the measurement concerns duration, housing status, or something else. [6]
Eigenvalues, proportions, and cumulative variance · Minitab financial example
PC
Eigenvalue
Proportion
Cumulative
PC1
3.5476
0.443
0.443
PC2
2.1320
0.266
0.710
PC3
1.0447
0.131
0.841
PC4
0.5315
0.066
0.907
PC5
0.4112
0.051
0.958
PC6
0.1665
0.021
0.979
PC7
0.1254
0.016
0.995
PC8
0.0411
0.005
1.000
Complete coefficient matrix · rounding from the original output
Variable
PC1
PC2
PC3
PC4
PC5
PC6
PC7
PC8
Income
0.314
0.145
-0.676
-0.347
-0.241
0.494
0.018
-0.030
Education
0.237
0.444
-0.401
0.240
0.622
-0.357
0.103
0.057
Age
0.484
-0.135
-0.004
-0.212
-0.175
-0.487
-0.657
-0.052
Residence
0.466
-0.277
0.091
0.116
-0.035
-0.085
0.487
-0.662
Employment
0.459
-0.304
0.122
-0.017
-0.014
-0.023
0.368
0.739
Savings
0.404
0.219
0.366
0.436
0.143
0.568
-0.348
-0.017
Debt
-0.067
-0.585
-0.078
-0.281
0.681
0.245
-0.196
-0.075
Credit cards
-0.123
-0.452
-0.468
0.703
-0.195
-0.022
-0.158
0.058
The first three components have eigenvalues greater than one and account for approximately 84.1% of variation. The first two account for around 71.0%. Representing only PC1 and PC2 therefore leaves about 29% of total variation out of view: a two-dimensional plot is informative, but does not exhaust a three-component solution.
The relatively large positive coefficients for age, residence, employment, and savings in PC1 motivate the interpretation of long-term financial stability presented in the original. Among other contributions, PC2 contrasts education and savings with debt and credit cards. Under the chosen sign convention, the latter have negative coefficients; reversing the axis preserves the same structure. PC3 captures a different contrast, with substantial negative coefficients for income, education, and credit cards.
Names such as “stability,” “credit history,” or “academic and income qualifications” are interpretive summaries, not directly observed variables or automatically validated constructs. Supporting them requires examining all loadings, coding, and domain knowledge. A component does not become a valid indicator of risk, solvency, or behavior merely by receiving a plausible label.
How to calculate a score
Because the example uses a correlation matrix, its coefficients are applied to standardized variables. For a particular observation, the first score is obtained as follows:
The formula uses coefficients published to three decimal places; small differences from the program’s higher-precision internal calculation may occur.
As an additional teaching check, a hypothetical observation one standard deviation above the mean in age, residence, employment, and savings, and exactly at the mean of the other variables, would have an approximate score of 0.484 + 0.466 + 0.459 + 0.404 = 1.813. This is not an additional observation from the Minitab dataset: it illustrates substituting values into the linear combination.
Scores belong to observations; coefficients belong to directions. They are not entries of the same matrix under different names. If directions occupy the columns of V, the transformation for all observations is T = ZV. The principal space is the subspace spanned by retained directions; it should not be loosely identified with a particular matrix representing it.
XIII. Mahalanobis distance
An observation may be unremarkable in each variable separately yet unusual because of the combination of its values. If two variables usually increase together, a point high on one and low on the other may depart from the joint pattern even when neither marginal value is extreme. This is the intuition behind the original example.
DM²(x) = (x − μ)TΣ−1(x − μ) DM(x) = √DM²(x).
The formula assumes invertible covariance. For sample data, μ and Σ are replaced by estimates, and threshold interpretation depends on how those estimates were obtained.
Mahalanobis distance weights departure from the center while accounting for scale and covariation. In principal coordinates, if all eigenvalues are positive, squared distance is the sum of squared scores divided by their variances.
DM²(x) = ∑j=1p tj²/λj.
A modest deviation along a very low-variance direction may be more unusual than a larger deviation along a high-variance direction.
If covariance is singular, the ordinary inverse does not exist. A pseudoinverse, regularization, or a subspace model may be considered, but each option changes which departures the diagnostic detects. The pseudoinverse ignores null-space directions; consequently, a subspace-restricted diagnostic should be complemented by residual inspection when points outside that subspace are of interest.
Classical distance uses mean and covariance estimates that can themselves be affected by the outliers one intends to detect. It is not universally “more powerful” under every form of contamination or distribution. Its conceptual advantage over marginal inspection is incorporating the multivariate pattern, but performance depends on the estimator and model.
With known population parameters and multivariate normality, squared distance follows a chi-squared distribution with degrees of freedom equal to dimensionality. When parameters are estimated from the same sample, appropriate limits may require different corrections. A cutoff line should not be applied without stating its rationale.
XIV. The scree plot
A scree plot orders eigenvalues from largest to smallest and plots them against component number. It represents the spectrum of the covariance or correlation matrix being analyzed. The vertical axis should specify whether it shows eigenvalues, variance proportions, or a transformation of those quantities.
Figure from the revised edition. Eigenvalues from the original financial table; the dashed line marks Kaiser’s reference for correlation-based analysis.
The financial example shows a substantial decline across the first components and a later tail of smaller contributions. The original interpretation of retaining three components is consistent with Kaiser and a cumulative proportion near 84.1%. It does not show that the remaining five components have no possible relevance: they may contain contrasts important to a particular objective.
An approximately uniform spectrum suggests that no direction concentrates much variation; in that case, strong reduction loses a substantial fraction. A dominant eigenvalue suggests a direction of common variation, but does not by itself establish a single cause. Spectrum shape also depends on scaling and noise.
XV. The score plot
A score plot represents observations in the coordinates of two or more components. In the PC1–PC2 plane, each point is a unit of study, not an original variable. Axes should show the proportion of variation they represent, allowing readers to see how much of the cloud is visible.
The plot is examined for groupings, gradients, separations, extreme positions, and observations near the center. The original figure identifies a point in a lower corner as a candidate for investigation. Proximity to the edge of the plane does not automatically establish an error, a new group, or a statistically significant outlier.
Distribution around the origin follows from centering the scores. It is not a test of normality and does not imply a circular cloud. Each component has its own variance; unequal graphical scales can visually alter angles and distances. When the figure’s geometry matters, axis scales should be stated and an appropriate aspect ratio preferably maintained.
If two points are separated in the plane, they also differ in the full space. If they appear close, they may still differ in unrepresented components. Likewise, a projected grouping may be useful for exploration without establishing stable clusters, natural categories, or population differences.
Coloring observations by known experimental labels may help interpret the representation, but those labels do not enter the ordinary PCA calculation. If the explicit objective is to separate known groups, an appropriate supervised technique should be considered and validated out of sample.
XVI. The loading or “influence” plot
The document’s “influence” plot corresponds to a loading plot: it represents coordinates associated with variables on two components. Accompanying the traditional label with “loading plot” helps avoid confusion with observation-influence diagnostics in regression.
The term loading has no universal convention. Some programs display the unit eigenvector coefficients; others display coefficients multiplied by the square root of the eigenvalue, or variable–component correlations. The representation type must be stated before comparing magnitudes, interpreting lengths, or applying thresholds.
These identities use an unwhitened component and consistent covariance and scaling conventions.
A unit-eigenvector coefficient near zero indicates a small contribution to that particular linear combination; it does not declare the variable unimportant throughout the analysis. The variable may contribute to other components or to a question not captured by principal variance. Eigenvector coefficients lie between −1 and 1, but in many dimensions all may be far smaller than one without the component being irrelevant.
In the financial example, age, residence, and employment point toward PC1; debt and credit cards have substantial weight in the negative PC2 direction. Interpretation concerns the joint pattern of signs and magnitudes, including smaller loadings when they contribute a relevant contrast. Interpretive consistency should be examined under reasonable changes to observations, scaling, or variable selection.
XVII. The biplot
A biplot brings together markers for observations and variables in one figure. It is grounded in a factorization of the data matrix’s reduced-rank approximation, not an arbitrary overlay of two plots with different units. Starting from X̂c,k = UkΣs,kVkT, singular values can be distributed between the two sets of markers in different ways.
G = UkΣs,kα, H = VkΣs,k1−α X̂c,k = GHT.
The choice of α changes the scaling of observations and variables. Consult the software’s convention before interpreting distances or angles.
Observation positions help identify patterns; variable vectors or markers help relate them to the original measurements. Under appropriate scaling, inner products allow approximate reconstruction of centered values; certain variable orientations may approximate correlations. These readings depend on normalization and approximation quality in the displayed axes.
The financial figure retains the original readings of PC1 and PC2 and the suggestion to investigate the isolated point. The additional clarification is that PC3 also contributes appreciably and is absent from the plane. Apparent closeness between an arrow and a point should not be interpreted as an ordinary distance between objects of the same kind.
A biplot should report the input matrix, preprocessing, displayed components, represented variance, and marker scaling. Without this information, a visually convincing figure can support incompatible interpretations.
XVIII. The outlier plot
The original plot shows a Mahalanobis distance for each observation and a detection reference. No point exceeds the displayed limit. This specific result is retained without turning it into proof that the dataset contains no possible anomaly: the diagnostic concerns a particular metric and threshold.
An observation above a reference line is a signal to investigate. It may reflect a recording error, measurement failure, a unit from another population, an exceptional condition, or a real and scientifically important event. Automatic removal can erase precisely the variation requiring explanation.
It is useful to distinguish two positions: far from the center within the retained subspace, and far from the subspace itself. The first can be examined through a weighted score distance; the second through a reconstruction residual, frequently summarized by a residual sum of squares.
Dk²(x) = ∑j=1k tj²/λj, Q(x) = ‖xc − VkVkTxc‖².
A limit for Dk² does not replace a limit for Q. Their reference distributions and corrections depend on the model and application.
A defensible practice is to verify data provenance, correct demonstrated errors, document decisions, and compare results with and without disputed cases when appropriate. When many observations are examined, it also matters that a series of marginal thresholds can accumulate false alarms.
XIX. Numerical computation and NIPALS
The variational formulation defines the solution sought; the algorithm determines how it is computed. PCA can be obtained through eigendecomposition of covariance, SVD of the prepared matrix, iterative methods, or approximations designed for large matrices. Gradient descent is a possibility for certain formulations, not the mandatory mechanism “orchestrating” every PCA.
Computational approaches and implementation decisions
Procedure
Idea
Main consideration
Eigendecomposition
Diagonalize S or R.
Forming XTX can amplify conditioning problems; useful when the variable matrix has manageable size.
Direct SVD
Decompose the centered or standardized matrix.
Avoids explicitly forming covariance and directly connects directions, scores, and reconstruction.
Truncated iterative methods
Approximate only the leading components.
Require tolerances and convergence checks; useful when few axes are wanted.
Randomized SVD
Approximate the dominant subspace through projections and refinement.
Accuracy depends on the spectrum and parameters such as oversampling and iterations; fix the seed when repeatability is required.
NIPALS
Extract components successively through alternating updates and deflation.
Implementations may incorporate missing-value handling; convergence and assumptions must be stated.
Incremental PCA
Update an approximation in batches.
Reduces memory requirements, but algorithmic and batch details affect the approximation.
NIPALS, step by step
NIPALS stands for Non-linear Iterative Partial Least Squares. The algorithm’s historical name does not automatically make the resulting representation a nonlinear PCA. In its PCA application, it successively seeks rank-one approximations to the residual matrix. The original discussion remains useful for understanding loadings and scores. [4]
Prepare the data and define an initial residual matrix E = Xc, or E = Z if variables were standardized.
Choose a nonzero column of E to initialize a score vector t.
Update the direction using p = ETt/(tTt), and normalize p to unit length.
Update scores using t = Ep and repeat until a declared tolerance is met.
Store t and p, remove their contribution using E ← E − tpT, and continue with another component if appropriate.
This sequence describes the complete-data case. Some implementations adjust products and updates over observed values to handle missing entries. Replacing missing values with zeros is not sufficient, nor can every missingness pattern be assumed harmless. Identifiability, the amount of available information, and the missingness mechanism affect the result.
Sequential extraction allows computation to stop once enough components have been obtained. Deflation and rounding may also accumulate numerical errors; orthogonality, convergence, and reconstruction residuals should therefore be monitored. The claim that “most programs use NIPALS” should be treated as a historical observation from the cited source, not a general property of current software.
A reproducible report states the algorithm actually used. Saying only that “PCA was performed in Python” or “in R” does not identify the input matrix, centering, scaling, missing-value treatment, or retention criterion. Software facilitates computation but does not itself resolve design decisions.
XX. PCA and linear regression
The original comparison with linear regression clarifies two meanings of “best-fitting line.” In ordinary regression of y on x, coefficients minimize errors in the response variable, measured vertically in the usual plot. In two-dimensional PCA, the line minimizes the sum of squared orthogonal distances, treating the coordinates jointly under the chosen metric.
Regression of y on x: minβ₀,β₁ ∑i (yi − β₀ − β₁xi)² Rank-one PCA: min‖a‖=1 ∑i ‖zi − aaTzi‖².
The first formulation selects a response. The second approximates a centered cloud without designating a dependent variable.
The distinction is therefore not that regression “lacks any orthogonality.” In least squares, the residual vector is orthogonal to the space spanned by the design columns under the usual conditions. This algebraic property is not equivalent to measuring geometrically perpendicular distances from each point to a line in variable space.
PCA can be related to orthogonal regression or total least squares, but interpreting errors in all variables requires assumptions about their scales and measurement precision. Symmetry of the criterion does not make units irrelevant: changing one axis’s scale can change the principal line.
Principal component regression also exists, using scores as predictors. Selecting components solely by their variance can discard a low-variation direction informative about the response. The number of components should be chosen with the predictive task and independent validation in view.
Geometric association does not establish causality either. The distinction between description, prediction, and explanation must remain explicit even when both techniques share algebraic tools or representations through lines and planes.
XXI. Three genes and higher dimensions
The original example adds a third gene while retaining the same six mice. The first three-dimensional table contains the following data. The third coordinate shows how a structure barely visible in one plane may become clearer when another variable is examined.
Three-dimensional extension of the gene-expression example
Mouse
Gene 1
Gene 2
Gene 3
1
10
6
12
2
11
4
9
3
8
5
10
4
3
3
2.5
5
2
2.8
1.3
6
1
1
2
Each unit now occupies a point in ℝ³. PC1 identifies the direction of greatest variation; PC2 is chosen in its orthogonal complement, and PC3 completes an orthonormal basis. The plane formed by PC1 and PC2 provides the best two-dimensional approximation under the Euclidean quadratic criterion applied to the prepared data.
Mice 1, 2, and 3 have higher third-gene values than mice 4, 5, and 6. This pattern may contribute to separation in principal coordinates. But a component combines all three variables: it need not coincide with a particular gene’s axis. Examining its coefficients describes how much each measurement contributes to the resulting direction.
The final Starmer figures included in the document also show illustrative variants in which third-gene values change. These variants are preserved in the visual archive. Percentages labeled in them—for example, 91% and 4%, or 79%, 15%, and 6%—belong to the teaching scenes in which they appear; they should not be transferred to another table or preprocessing scheme as universal results.
To avoid that mixture, the numerical results added here are recalculated exclusively from the transcribed tables using covariance-based PCA: centering without standardization. The code below makes it possible to check how the results differ if correlations are used instead.
Recalculated results · centered data without standardization
Data
PC
Eigenvalue
Variance (%)
Cumulative (%)
Two genes
PC1
21.284012
96.3368
96.3368
Two genes
PC2
0.809321
3.6632
100.0000
Three genes
PC1
42.455970
95.7509
95.7509
Three genes
PC2
1.357664
3.0619
98.8129
Three genes
PC3
0.526366
1.1871
100.0000
Recalculated scores · two genes, covariance-based PCA
Mouse
PC1
PC2
1
4.719998
-0.826949
2
4.988861
1.392896
3
2.500152
-0.558086
4
-2.881249
-0.356784
5
-3.890244
-0.504866
6
-5.437518
0.853790
Sign convention: the first coefficient of each direction is nonnegative. For two genes, a₁ ≈ (0.941711, 0.336424)ᵀ and a₂ ≈ (0.336424, −0.941711)ᵀ. Reversing a direction and its scores preserves the result.
The same argument generalizes to p genes, even when the original space cannot be drawn. A table with thousands of variables and few observations admits a PCA calculation, but its sample rank is limited by n − 1. The existence of computable components does not guarantee that estimated directions are stable or faithfully reproduce those of a population.
Observed separations may be consistent with biological differences, but also with technical or preparation differences. Identifying gene 3 as “responsible” for a grouping requires distinguishing geometric contribution from causal mechanism. PCA helps discover questions and hypotheses; design and validation determine which explanations are supported.
XXII. Scientific scope and multidimensional interpretation
The interest of PCA extends beyond a data-reduction recipe. It allows us to study how multiple measurements can be organized through relations that do not coincide with isolated variables. This move from original coordinates to new combinations can be philosophically fruitful, provided that a statistical construction is not turned into a demonstrated ontological essence.
Relations, totality, and mediation
From a materialist reading, variables are measurements produced through concrete practices, and data express aspects of natural or social processes. Analysis of their covariation may reveal regularities that marginal inspection misses. Yet the totality of a phenomenon is not exhausted by a covariance matrix: nonlinear relationships, temporality, heterogeneity, historical conditions, and interaction mechanisms may also matter.
The idea of “essential relations” should be treated as an interpretive hypothesis requiring investigation. An eigenvector summarizes a mathematical structure of variation; attributing to it the status of a fundamental relation of the system requires connecting that structure with substantive theory and independent evidence. This separation strengthens philosophical discussion by making its mediations explicit.
Six dimensions of assessment
Assessing the method at complementary levels
Dimension
Guiding question
Mathematical
What objective is optimized, under which constraints and metric?
Geometric
Are coordinates being changed or directions discarded? Which distances are preserved?
Statistical
Do the sample, measurements, and stability support the observed regularities?
Computational
Which algorithm, tolerances, and data treatment produce the solution?
Substantive
Does the component interpretation make sense in biology, economics, or the domain studied?
Epistemological
Is the aim description, prediction, or explanation? Which inferential step needs additional evidence?
Applications and limitations
In biology, PCA can explore joint patterns of expression or morphology. In economics, it can summarize related series or indicators. In industrial processes, it can help monitor multivariate behavior. In images, it can approximate variation through a lower-dimensional basis. In every case, usefulness depends on the objective, data preparation, and component interpretation.
Major limitations include sensitivity to extreme observations, dependence on units and scaling, the inability of a linear model to fully describe curved structures, and the absence of an explicit class-separation objective. Ordinary PCA also does not automatically distinguish signal from noise or identify causal variables or latent constructs on its own.
Validation may include sensitivity to preprocessing, resampling to examine stability, comparisons between subsamples, residual inspection, and out-of-sample evaluation. When eigenvalues are close, individual directions may change substantially under small perturbations while the joint subspace remains more stable. In that case, evaluate the subspace rather than attaching excessive significance to the name of an isolated axis.
Finally, a low-dimensional representation should not be used to naturalize classifications of people or groups. The financial example illustrates a statistical operation; any decision-making use requires validation appropriate to the context, the meaning of the variables, and the consequences of the decisions.
Scope of the result
PCA identifies the directions of greatest linear variation in a chosen representation. Scientific knowledge additionally requires explaining why that representation is relevant, which relations it can capture, and what evidence supports interpretation of its patterns.
XXIII. Reproducible implementation in Python and R
Reproducibility requires preserving data, matrix orientation, transformations, algorithm, and conventions. The following examples recalculate the gene-table results. Python uses NumPy and direct SVD; R uses prcomp. Neither assumes group labels or uses screenshot percentages as inputs.
Both distinguish centering without scaling from centering with sample standardization. The covariance divisor is n − 1. Component signs may differ between implementations; directions should be aligned before comparing coordinates. With spectral ties, compare the subspace rather than necessarily comparing individual columns.
Python: computation and verification
"""PCA for the six-mouse example. Requires Python and NumPy."""
import numpy as np
X = np.array([
[10, 6], [11, 4], [8, 5],
[3, 3], [2, 2.8], [1, 1]
], dtype=float)
X3 = np.column_stack((X, [12, 9, 10, 2.5, 1.3, 2]))
def pca_svd(data, standardize=False):
"""Rows: observations. Sample scaling uses ddof=1."""
x = np.asarray(data, dtype=float)
if x.ndim != 2 or x.shape[0] < 2:
raise ValueError("At least two observations are required.")
if not np.isfinite(x).all():
raise ValueError("Resolve missing/nonfinite values before PCA.")
mean = x.mean(axis=0)
scale = x.std(axis=0, ddof=1) if standardize else np.ones(x.shape[1])
if np.any(scale == 0):
raise ValueError("Do not standardize constant columns.")
z = (x - mean) / scale
_, singular, vt = np.linalg.svd(z, full_matrices=False)
directions = vt.T.copy()
# Reproducible sign convention: nonnegative first coefficient.
signs = np.where(directions[0, :] < 0, -1.0, 1.0)
directions *= signs
scores = z @ directions
eigenvalues = singular**2 / (x.shape[0] - 1)
total = eigenvalues.sum()
if total <= 0:
raise ValueError("Total variation must be positive.")
proportions = eigenvalues / total
return dict(z=z, mean=mean, scale=scale,
directions=directions, scores=scores,
eigenvalues=eigenvalues, proportions=proportions)
if __name__ == "__main__":
np.set_printoptions(precision=6, suppress=True)
for label, data in [("Two genes", X), ("Three genes", X3)]:
for scaling in (False, True):
r = pca_svd(data, standardize=scaling)
v, t, lam = r["directions"], r["scores"], r["eigenvalues"]
print(label, "correlations" if scaling else "covariances")
print("Eigenvalues:", lam)
print("Explained variance (%):", 100 * r["proportions"])
print("Directions (columns):\n", v)
print("Scores:\n", t)
# Independent properties checking the published result.
assert np.allclose(v.T @ v, np.eye(v.shape[1]))
assert np.allclose(np.cov(t, rowvar=False), np.diag(lam))
assert np.allclose(t @ v.T, r["z"])
assert np.isclose(lam.sum(), np.var(r["z"], axis=0, ddof=1).sum())
for k in range(1, v.shape[1] + 1):
approximation = t[:, :k] @ v[:, :k].T
error = np.sum((r["z"] - approximation)**2)
assert np.isclose(error, (data.shape[0] - 1) * lam[k:].sum())
# Uncorrelatedness and independence are different:
# if U is uniform on [-1,1], Cov(U,U²)=0 by symmetry,
# although U² is determined by U; no simulated test is needed.
The checks verify orthonormal directions, diagonal score covariance, full-component reconstruction, and agreement between discarded variation and squared error. These identities detect transposition or scaling errors that a plausible figure may conceal.
R: the same sample convention
# PCA of the same data. Requires base R.
X <- matrix(c(10,6, 11,4, 8,5, 3,3, 2,2.8, 1,1),
ncol = 2, byrow = TRUE)
colnames(X) <- c("Gene1", "Gene2")
rownames(X) <- paste0("Mouse", 1:6)
X3 <- cbind(X, Gene3 = c(12, 9, 10, 2.5, 1.3, 2))
for (data in list(X, X3)) {
for (scaling in c(FALSE, TRUE)) {
fit <- prcomp(data, center = TRUE, scale. = scaling)
print(fit$sdev^2) # sample eigenvalues
print(100 * fit$sdev^2 / sum(fit$sdev^2))
print(fit$rotation) # directions
print(fit$x) # scores
reconstruction <- fit$x %*% t(fit$rotation)
stopifnot(max(abs(reconstruction -
scale(data, center=TRUE, scale=scaling))) < 1e-8)
}
}
# Signs may differ from Python without changing the solution.
# For parallel analysis, install psych if desired:
# set.seed(2026)
# psych::fa.parallel(X3, fa="pc", n.iter=1000, quant=.95)
# With six observations, this is an illustration, not an empirical recommendation.
The parallel-analysis example remains commented out because it requires the additional psych package. The function explicitly specifies components through fa="pc". A teaching sample of six units should not support recommendations about stable empirical groupings or population dimensionality.
Applying the model to new observations
A new observation is transformed using the means, scales, and directions fitted on training data. They are not recalculated using the test set. If PCA belongs to a predictive pipeline, its fit and all learned preprocessing must occur within each training partition of cross-validation. Otherwise information leaks from the evaluation set.
Tnew = ((Xnew − 1x̄trainingT)Dtraining−1)Vk.
D contains the standard deviations used when fitting the model. Without standardization, D is the identity.
In implementations such as sklearn.decomposition.PCA, centering and scaling are separate operations: PCA should not be assumed to standardize automatically. If StandardScaler is used, its default variance convention employs divisor n; state this when comparing exact magnitudes with R or the code above. Whitening is another transformation: it divides principal coordinates by a measure of their dispersion and changes output variances.
XXIV. Complete archive of the 51 original figures
This archive preserves every graphical element extracted from the PDF’s 49 pages, in order of appearance and without cropping their content. It includes Pearson’s historical figure, the transformation table, Starmer’s sequence, Dunn’s diagrams, and the results and plots from Minitab and Universidad Carlos III de Madrid. A visual filter integrates the color presentation into the article; each figure’s link opens the original PDF page in color.
Reading the historical material
The screenshots retain the labels and conventions of the original document, including the source languages. Where they call a sum of squares an “eigenvalue,” distinguish the eigenvalue of the cross-product matrix from a component’s variance. Where dividing by the hypotenuse is described as “standardizing,” that step concerns vector normalization. The clarifications in the main text govern interpretation of the notation. Percentages from different scenes should not be combined.
A. Pearson and geometric transformations · figures 1–5Original figure 01 · PDF, page 3. Orthogonal approximation line in Pearson’s historical illustration. Source: Pearson (1901); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 02 · PDF, page 6. Expression of gene 1 in the six mice and calculation of its mean. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 03 · PDF, page 6. Expression of gene 2 and calculation of its mean. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 04 · PDF, page 7. Cloud center obtained from the two means. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 05 · PDF, page 7. Original table of geometric transformations and their characteristics. Source: Weisstein, Transformation (2021); reproduced from Isadore Nabi’s document.View the original page in colorB. Centering, fitting, and projection · figures 6–15Original figure 06 · PDF, page 8. Translation of the cloud to the coordinate origin. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 07 · PDF, page 9. Preservation of relative positions after centering. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 08 · PDF, page 10. First candidate line through the origin. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 09 · PDF, page 11. Rotation of the candidate line against the centered cloud. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 10 · PDF, page 11. Illustration of a projection between planes. Source: Weisstein, Projection (2021); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 11 · PDF, page 12. Two fitting criteria expressed through projections. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 12 · PDF, page 13. Perpendicular residual and projected distance from the origin. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 13 · PDF, page 14. Right triangle connecting the observation, projection, and residual. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 14 · PDF, page 14. Pythagorean identity: constant total and complementary parts. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 15 · PDF, page 15. Decomposition of a vector into a score and a residual. Source: Dunn (2021); reproduced from Isadore Nabi’s document.View the original page in colorC. Sums of squares and principal direction · figures 16–25Original figure 16 · PDF, page 16. Choice of a candidate line for measuring fit. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 17 · PDF, page 16. Projection of an observation onto the candidate direction. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 18 · PDF, page 17. Distance from a projection to the origin. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 19 · PDF, page 18. Sum of squared projections for all six mice. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 20 · PDF, page 18. Comparison of directions and the maximum sum-of-squares criterion. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 21 · PDF, page 20. Direction vector with coordinates 4 and 1 and approximate length 4.12. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 22 · PDF, page 22. Normalization of the direction by dividing by its length. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 23 · PDF, page 22. Normalized coefficients for the illustrative gene combination. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 24 · PDF, page 23. Singular vector or eigenvector: principal-direction terminology. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 25 · PDF, page 24. Sum of squares and singular value: the original screenshot’s convention. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorD. Second component and coordinates · figures 26–35Original figure 26 · PDF, page 24. Second direction perpendicular to PC1 in two dimensions. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 27 · PDF, page 25. Illustrative linear combination defining PC2. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 28 · PDF, page 25. Normalization of the second direction’s vector. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 29 · PDF, page 26. PC2 coefficients compared with those of PC1. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 30 · PDF, page 27. Loadings, scores, and sum of squares in the second direction. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 31 · PDF, page 28. Coordinate change for the mice into the PC1–PC2 plane. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 32 · PDF, page 29. Position of each observation according to its principal scores. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 33 · PDF, page 29. Conversion of sums of squares to variances using n − 1. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 34 · PDF, page 30. Illustrative explained variance and scree plot. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 35 · PDF, page 31. Geometric diagram of components and approximation in several dimensions. Source: Dunn (2021); reproduced from Isadore Nabi’s document.View the original page in colorE. Financial example and diagnostics · figures 36–43Original figure 36 · PDF, page 32. Complete financial-analysis output in Minitab. Source: Minitab (2019); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 37 · PDF, page 35. Matrix expression of linear combinations of variables. Source: Universidad Carlos III de Madrid (2006); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 38 · PDF, page 36. A marginally ordinary point that is unusual in the joint pattern. Source: Minitab (2019); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 39 · PDF, page 37. Scree plot for the financial example. Source: Minitab (2019); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 40 · PDF, page 38. Score plot for the financial example. Source: Minitab (2019); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 41 · PDF, page 39. Loading plot, called an influence plot in the original. Source: Minitab (2019); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 42 · PDF, page 40. Biplot for the financial example. Source: Minitab (2019); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 43 · PDF, page 41. Distances and reference limit in the outlier plot. Source: Minitab (2019); reproduced from Isadore Nabi’s document.View the original page in colorF. Three genes and geometric extension · figures 44–51Original figure 44 · PDF, page 43. Table of three genes measured in six mice. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 45 · PDF, page 44. Addition of a third axis to feature space. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 46 · PDF, page 44. High third-gene values for the first three mice. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 47 · PDF, page 45. Low third-gene values for the last three mice. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 48 · PDF, page 46. Illustrative variant with greater separation associated with gene 3. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 49 · PDF, page 46. Weights of an illustrative combination in three-dimensional space. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 50 · PDF, page 47. Another principal direction and its weights in three dimensions. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in colorOriginal figure 51 · PDF, page 47. Final variant with three explained-variance percentages. Source: Starmer (2018); reproduced from Isadore Nabi’s document.View the original page in color
XXV. Glossary and correspondence with the document
Vocabulary that keeps analytical levels separate
Term
Definition in this edition
Observation
A unit of study represented by a row of X and, after transformation, by a row of T.
Original variable
A measurement corresponding to a column of X.
Principal direction
A unit eigenvector of the chosen covariance or correlation matrix.
Principal component
A variable obtained as a centered linear combination along a principal direction.
Score
An observation’s value on a component.
Loading
A coefficient or variable–component association, according to the declared convention.
Eigenvalue
A component’s variance when referring to the corresponding covariance or correlation matrix.
Singular value
A nonnegative SVD quantity; its square is an eigenvalue of XcTXc.
Principal subspace
The space spanned by retained directions.
Projection
An observation’s orthogonal approximation within the subspace.
Residual
The difference between the prepared observation and its reconstruction.
Effective dimensionality
The number of directions considered sufficient under a declared approximation criterion.
Thematic correspondence · no original sections omitted
Content of the 2021 PDF
Development in this edition
I. General conceptual and mathematical foundations
I–III, VIII–X: attributes, dimensionality, Bellman, Pearson, linear combinations, constraints, and spectrum.
III.I–III.III. Results, eigenvalues, criteria, and components
XI–XII: all original criteria, the complete financial table, loadings, and scores.
III.IV. Mahalanobis distance
XIII: definition, geometry, spectral equivalence, singularity, and threshold assumptions.
III.V–III.IX. Scree, scores, loadings, biplot, and outliers
XIV–XVIII: all five diagnostics, their interpretation, and its conditions.
III.X. Final considerations
XIX–XX: NIPALS, other algorithms, and comparison with linear regression.
IV. Geometric intuition in ℝ³
XXI: three genes, six mice, graphical variants, and higher-dimensional extension.
V. References
XXVI: all original bibliographic entries, with editorial corrections and technical additions.
Document figures
XXIV: an archive of all 51 figures, with source page and access to the historical PDF.
Additional developments
Derivations, reconstruction, stability, epistemology, code, and traceability in XXII–XXIII and notes throughout preceding sections.
The discussion maintains the essay’s multidisciplinary ambition: connecting intuition, formalization, computation, and interpretation. Precise distinctions allow these perspectives to complement each other without attributing to a formula results belonging to another level of analysis.
XXVI. References and complete document
The original document’s bibliography is preserved. Jolliffe’s name and the editorial presentation of entries have been corrected; encyclopedic sources and forums remain as historical or teaching aids. The additions incorporate technical documentation and further academic work. Inclusion of a reference in the original is not presented as a new verification of its entire contents or as automatic support for every assertion it contains.
Adler, J. (2012). R in a Nutshell (2nd ed.). O’Reilly Media.
Alger, N. (2013, March 4). Intuitively, what is the difference between Eigendecomposition and Singular Value Decomposition? Answer on Mathematics Stack Exchange. Question and answers.
Bellman, R. (1972). Dynamic Programming (printing consulted in the original). Princeton University Press. Originally published in 1957.
Dunn, K. G. (2021). Process Improvement Using Data. Chapter 6: latent-variable methods; section on PCA. Author’s textbook. PDF document.
Jolliffe, I. T. (2002). Principal Component Analysis (2nd ed.). Springer.
Massachusetts Institute of Technology. (2011; consulted in 2021 in the original). Linear transformations and their matrices. MIT OpenCourseWare, Linear Algebra. Course and materials.
Nabi, I. (2020). Sobre los estimadores de Bayes, el análisis de grupos y las mixturas gaussianas [On Bayes estimators, cluster analysis, and Gaussian mixtures]. Unpublished document cited in the original edition.
Nabi, I. (2021, April 3). ¿Por qué se realiza un ajuste por re-escalamiento, normalización o estandarización sobre los datos en el contexto del aprendizaje automático? [Why are data rescaled, normalized, or standardized in machine learning?]. Marxist Philosophy of Science. Related article.
Nabi, I. (2021, April 2). Una interpretación multidisciplinaria de los espacios característicos, vectores característicos y valores característicos [A multidisciplinary interpretation of eigenspaces, eigenvectors, and eigenvalues]. Complete document.
Pearson, K. (1901). On lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11), 559–572. https://doi.org/10.1080/14786440109462720.
Mathematics Stack Exchange. (2015, March 13). Understanding proof of isometry implies isomorphism. Discussion cited in the original. Its scope depends on the specified structures and surjectivity assumptions.
Starmer, J. (2018, April 2, according to the original reference). Principal Component Analysis (PCA). StatQuest. Cited video. Author’s material index. Source of the gene-expression teaching sequences.
Universidad Carlos III de Madrid. (2006, November 7, according to the original reference). Análisis de componentes principales [Principal component analysis]. Cited teaching material.
Universitat de Girona. (2002, January 24, according to the original reference). Número de factores a conservar [Number of factors to retain]. Material on factor analysis. Historical address cited. The reference is retained; the revised discussion distinguishes components from common factors.
Weisstein, E. W. (consulted in 2021). Projection. MathWorld, Wolfram Research. Reference article.
Weisstein, E. W. (consulted in 2021). Transformation. MathWorld, Wolfram Research. Reference article.
Wikipedia. (2020, version cited in the original). Curse of dimensionality. Encyclopedic entry.
Wikipedia. (2020, version cited in the original). Isomorfismo [Isomorphism]. Spanish entry.
Wikipedia. (2021, version cited in the original). Isomorphism. English entry.
Wikipedia. (2021, version cited in the original). Transcripción genética [Genetic transcription]. Encyclopedic entry. Transcription synthesizes RNA from DNA; translation produces a polypeptide chain from messenger RNA.
Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology. Additional historical reference.
Jolliffe, I. T., & Cadima, J. (2016). Principal component analysis: a review and recent developments. Philosophical Transactions of the Royal Society A, 374, 20150202. DOI. Open-access full text.
Revelle, W. VSS: Very Simple Structure, MAP, and other criteria. Documentation for the psych package. Author’s documentation. Includes Velicer (1976), Determining the number of components from the matrix of partial correlations, Psychometrika, 41, 321–327.
Revelle, W. fa.parallel: Scree plots of data or correlation matrix compared to random matrices. Documentation for the psych package. Author’s documentation. Includes Horn (1965), A rationale and test for the number of factors in factor analysis, Psychometrika, 30, 179–185.
The historical document provides access to the original composition, color screenshots, and discussion as published in 2021. This English web edition expands the exposition and clarifies its mathematical, statistical, and philosophical distinctions while maintaining traceability between versions.
Political economy · Economic statistics · Methodological note
Product Redefinition and Reallocation in the U.S. Input–Output Matrix
Technology Assumptions and the Hybrid Procedure Used by the U.S. Bureau of Economic Analysis
JOSÉ MAURICIO GÓMEZ JULIÁN Originally published: Revised version:
· · ·
Abstract
An industry may produce goods and services with different input structures. To address this heterogeneity, the Bureau of Economic Analysis (BEA) combines the redefinition of selected secondary products and the reallocation of their inputs with a subsequent matrix derivation based on the industry-technology assumption. This note explains the procedure, distinguishes its component operations, and shows why it matters for interpreting interindustry relationships.
I. The Problem of Secondary Production
An industry need not produce only one type of good or service. A hotel, for example, may provide accommodation while also offering restaurant services. Although both activities belong to the same establishment, their requirements for food, labor, energy, and other resources may differ considerably.
In standard make and use tables, products are initially recorded under the industry that produces them. A distinction is drawn between primary products, which are characteristic of an industry, and secondary products. The difficulty arises when the input structure of a secondary activity differs substantially from that of the primary activity.[3]
The methodological question is therefore how to construct relationships between output and inputs that support the study of interindustry linkages without automatically assigning the same technology to heterogeneous activities. The procedure examined here addresses this problem by adjusting the industrial attribution of selected activities.
II. Two Technology Assumptions
Transforming the make-use system into a symmetric representation requires assumptions about the inputs attributable to secondary products. Two formulations are central: the industry-technology assumption (ITA) and the commodity-technology assumption (CTA).[1]
Table 1. What determines the input structure?
Assumption
Proposition
Implication
ITA Industry technology
Products made by an industry share its input structure.
Inputs to secondary production are estimated from the producing industry.
CTA Commodity technology
Each product has an input structure independent of the industry producing it.
Technology is associated with the product, even when it is secondary in its industry of origin.
Expository summary based on Guo, Lawson, and Planting (2002). ITA and CTA stand for industry-technology assumption and commodity-technology assumption, respectively.
These assumptions express different ways of representing production. They should not be interpreted as universal empirical identities. The choice affects the transformation of the tables and, consequently, the interpretation of the resulting coefficients.
III. The BEA’s Hybrid Procedure
In the documentation examined here, the BEA uses a hybrid approach: it combines selective adjustments informed by commodity technology with a subsequent derivation based on industry technology. It is useful to distinguish two broad stages without confusing them with the operations carried out within the first stage.[1][2]
First stage: redefining output and reallocating its inputs
Secondary products are selected when their inputs differ substantially from those of the primary product of the originating industry. Their output is attributed to the industry in which those products are primary: this operation constitutes the redefinition of output.[5][6]
The adjustment must be accompanied by the reallocation of the associated inputs, including intermediate inputs and value-added components. Moving output alone would leave the records of products and costs out of alignment. These adjustments produce the supplementary tables known as after redefinition tables.[2]
This stage approximates the CTA principle, but it does not amount to mechanically applying a pure CTA model to the entire economy. The manual also specifies that reallocations should use an input structure appropriate to the redefined activity, rather than necessarily applying the receiving industry’s entire input pattern.
Second stage: deriving the requirements tables
Starting from the adjusted tables, the mathematical derivation uses the ITA to handle the remaining secondary production. The existence of secondary products therefore does not imply that all must first be redefined: some remain in their producing industry and are treated according to this second assumption.[1]
Understanding the procedure
Standard tablesPrimary and secondary output is recorded in its industry of origin.
After redefinition tablesSelected secondary output is transferred and its associated inputs are reallocated.
Derivation of requirementsCoefficients and cumulative relationships are calculated using the specified technology assumptions.
Reading outline. The second item combines the two operations within the first stage; the third corresponds to the second broad stage.
IV. An Example: Hotels and Restaurants
The BEA uses restaurant services produced by hotels to illustrate redefinition from accommodation to food services.[3] The following example develops that idea using hypothetical figures in monetary units; it does not reproduce observations from the U.S. economy.
Table 2. Before redefinition
Industry / product
Accommodation
Restaurant services
Total
Accommodation
100
20
120
Food services
0
80
80
Total by product
100
100
200
The accommodation industry produces 100 of its primary product and 20 of restaurant services as a secondary product.
Table 3. After redefinition
Industry / product
Accommodation
Restaurant services
Total
Accommodation
100
0
100
Food services
0
100
100
Total by product
100
100
200
Restaurant output of 20 is transferred. Product totals and aggregate output remain constant; attribution across industries changes.
Suppose that this output of 20 is associated with 8 of food inputs, 2 of other intermediate inputs, and 10 of value added. When the output is redefined, these components must also be moved from the accommodation column to the food services column in the use table. The reallocated amount is 20, equal to the value of the transferred output.
Central idea · This note’s synthesis
Redefinition changes the industrial attribution of output while preserving product totals.
The procedure changes the statistical representation: the restaurant may continue to operate within the hotel. The receiving industry in the adjusted tables is an analytical category to which the corresponding output and inputs are attributed.
V. From Adjusted Tables to Requirements
Once the supplementary tables have been obtained, two distinctions must be kept separate: the distinction between direct and indirect effects, and the distinction between the domestic and imported origin of production.
Table 4. Two dimensions of interpretation
Dimension
Distinction
Production linkages
Direct requirements describe inputs used immediately. Requirements incorporating indirect effects also capture successive links in the supply chain.
Origin of production
Domestic requirements consider production within the country. The tables the BEA calls total requirements also incorporate the imported component.
In BEA terminology, “domestic” is not synonymous with “direct”: domestic requirements tables may also capture both direct and indirect effects. See Medeiros and Howells III (2017).
To construct domestic requirements, the BEA starts with intermediate uses after redefinition and subtracts the imported component. It then calculates direct coefficients and market shares, which it combines to obtain the different domestic requirements tables.[4]
A direct requirements table should therefore not be defined as though it already incorporated all indirect effects. These emerge when successive relationships between suppliers and purchasers are developed through the corresponding matrix structure.
VI. Scope and Methodological Interpretation
The purpose of these adjustments is to obtain a more homogeneous relationship between products and inputs. This analytical gain has a counterpart: industries after redefinition depart from the boundaries used in standard tables and from their immediate correspondence with the industrial classification. The BEA manual explicitly identifies this tension between homogeneity and classificatory consistency.[2]
An implication for analysis follows: before comparing sectoral coefficients or indicators, it is useful to check whether they come from tables before or after redefinition. The category “industry” does not necessarily retain the same statistical content in both representations.
Descriptive and predictive uses must also be distinguished. The tables may represent average technological relationships for a reference year. Using them to simulate large changes requires evaluating the assumptions of constant returns to scale and no input substitution, which may become restrictive when the hypothetical economy departs from the observed one.[4]
Greater homogeneity in a table does not, by itself, demonstrate that its coefficients describe invariant technologies. Economic interpretation requires attention both to the construction procedure and to the conditions of the particular application.
VII. Conclusions
The treatment of secondary production is a substantive methodological decision in constructing input–output tables. In the hybrid approach examined here, the first stage redefines selected activities and reallocates their inputs; the second uses the industry-technology assumption to derive requirements from the adjusted tables.
Redefinition changes attribution across industries while preserving product totals. The calculation of direct coefficients must, in turn, be distinguished from the derivation of cumulative relationships incorporating indirect effects. Neither of these distinctions can be replaced by the distinction between domestic and imported production.
Understanding these operations makes it possible to read the input–output matrix as a constructed representation of production relationships: a tool whose scope depends on its sources, classifications, and technology assumptions. That understanding is indispensable for a rigorous interpretation of interindustry relationships.
· · ·
VIII. References
Guo, J., Lawson, A. M., & Planting, M. A. (2002). From Make-Use to Symmetric I-O Tables: An Assessment of Alternative Technology Assumptions (WP2002-03). Paper presented at the 14th International Conference on Input-Output Techniques, Montreal. Full text at the BEA (PDF).
Horowitz, K. J., & Planting, M. A. (2006; updated April 2009). Concepts and Methods of the U.S. Input-Output Accounts. Bureau of Economic Analysis. In particular, chapters 4, 9, and 12. Methodological manual (PDF).
U.S. Bureau of Economic Analysis. (2011, May). Measuring the Nation’s Economy: An Industry Perspective. A Primer on BEA’s Industry Accounts. Revised version. Full document (PDF).
Medeiros, G., & Howells III, T. F. (2017, March). Introducing Domestic Requirements Tables for 1997–2015. Survey of Current Business. Bureau of Economic Analysis. Article and mathematical derivation (PDF).
U.S. Bureau of Economic Analysis. (2005, November 16). What are redefinitions, and where are they described? Frequently asked questions. Read the institutional explanation.
U.S. Bureau of Economic Analysis. (2018, April 13). Redefinition. Glossary. Read the definition.
