Category: Universal History

GENERALITIES AND HISTORICAL ORIGINS OF THE CHI-SQUARED DISTRIBUTION

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 O_i be the observed count and E_i the expected count in category i. With k categories, Pearson’s statistic adds their individual contributions:

X_P² = ∑_i=1^k (O_i − E_i)²E_i

(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 E_i > 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.

E_i = Np_i, ∑_i=1^k p_i = 1

(02)

Conceptual reading. N is the total number of observations and p_i 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.

X_P² = N ∑_i=1^k (p̂_i − p_i)²p_i, p̂_i = O_i/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 ℝ^d. Its effective degrees of freedom may be smaller when constraints apply.

Σ_ij = Cov(X_i, X_j), Σ_ii = σ_i²

(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, …, σ_d)

(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]

f_X(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]

Q = z^⊤R⁻¹z = ∑_i=1^d a_iiz_i² + 2∑_i<j a_ijz_iz_j, A = R⁻¹

(11)

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: a_ijz_iz_j and a_jiz_jz_i. Combining them explains the factor 2. The condition i<j counts each pair once.

M_ij = det(R with row i and column j removed), C_ij = (−1)^i+jM_ij

(12)

Conceptual reading. Deleting one row and one column produces a submatrix; its determinant is the minor M_ij. The cofactor C_ij 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 = C_ji/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, …, λ_d), U^⊤U = I

(15)

Conceptual reading. The columns of U are unit eigenvectors defining the principal directions. The positive eigenvalues λ_j are variances along those directions. At level c, each semiaxis has length √(cλ_j). 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=1^d Y_j²

(16)

Conceptual reading. First subtract the center. U^⊤ expresses the displacement in principal-axis coordinates. Λ^−1/2 divides each coordinate by √λ_j, 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.

What the transformation preserves

The map is affine, invertible, and smooth, with a smooth inverse: it is a diffeomorphism of real space. Its restriction connects ellipsoidal surfaces with spherical surfaces of the same dimension. It preserves topological structure and allows geometric and probabilistic information to be transported, but it generally changes Euclidean distances, angles, and volumes. Rescaling requires a Jacobian when transforming integrals.

A manifold of dimension m needs m local coordinates; its degree of differentiability is a separate property. “Smooth” usually means differentiable to every order. A two-dimensional manifold can be infinitely differentiable. Topological equivalence must be distinguished from metric equality and from philosophical conclusions about identity or essence. See also the author’s essay on the genesis of topology. [5]

dx = |Σ|^1/2 dy

(17)

Conceptual reading. The inverse transformation is x=μ+UΛ^1/2y. Translation does not change volumes; an orthogonal matrix has a determinant with absolute value one; and the product of the factors √λ_j 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 = r^d−1 dr dΩ, ∫_S^d−1 dΩ = 2π^d/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 r^d−1. dΩ captures directions on the unit sphere S^d−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^−r²/2r^d−1 dr∫₀^∞ e^−r²/2r^d−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, r^d−1dr = ½q^d/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 r^d−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^∞ t^a−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 2^d/2−1Γ(d/2). The preceding ratio thus becomes a calculable, normalized probability.

9. The chi-squared law and its properties

f_Q(q) = q^d/2−1e^−q/22^d/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 ∼ χ²_d: ∼ 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]

E(Q) = d, Var(Q) = 2d, Q ∼ Gamma(shape=d/2, scale=2)

(24)

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 = P_H₀(Q ≥ q_obs) = ∫_qobs^∞ f_Q(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.

Probability, real spaces, and Kolmogorov

The vectors in this construction belong to ℝ^d; no complex number space is required. The axiomatic formulation associated with Andrey Kolmogorov in 1933 places the reasoning in a probability space (Ω, ℱ, P): Ω collects elementary outcomes, ℱ measurable events, and P their probabilities. A random variable is a measurable map from that space into, for example, ℝ^d. The outcome space need not coincide with the space of values taken by the vector. [8]

A unit ball is a geometric region; a Lebesgue space L^p, in its usual functional sense, is a space of equivalence classes of functions satisfying a particular integrability condition. A ball may be part of a measurable space, but these concepts are not equivalent. This reconstruction retains the connection with measure and probability without identifying them.

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, …, O_k) ∼ Multinomial(N; p₁, …, p_k)

(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(O_i) = Np_i(1−p_i), Cov(O_i,O_j) = −Np_ip_j (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=1^k(O_i−Np_i) = 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.

W_i = O_i−Np_i√(Np_i), Cov(W) = I−ss^⊤, s = (√p₁, …, √p_k)^⊤

(29)

Conceptual reading. W denotes the vector of Pearson residuals. Dividing by √(Np_i) 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:

X_P² = ∑_i=1^kW_i² ⟶_distribution χ²_k−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.
Category	O	E	O − E	(O − E)²/E
Red flesh	310	300	10	0.333333
Yellow flesh	90	100	−10	1.000000
Total	400	400	0	1.333333

X_P² = 100/300 + 100/100 = 4/3 ≈ 1.333333; ν = 1; p ≈ 0.248213

(32)

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

X_P² = 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.

E_ij = O_i+O_+jN, ν = (r−1)(c−1)

(34)

Conceptual reading. O_i+ 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 = √X_P²N 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 ∼ χ²_N−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.
X²_P	Pearson’s statistic for counts.
χ²_ν	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.
R Core Team. The (non-central) Chi-Squared Distribution, stats package documentation. Density, probabilities, quantiles, and simulation.
Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung [Foundations of the Theory of Probability]. Springer. Historical reference added to contextualize the axiomatization of probability.
NIST/SEMATECH. e-Handbook of Statistical Methods, §1.3.5.15, Chi-Square Goodness-of-Fit Test. Grouping, estimation, and degrees of freedom.
NIST/SEMATECH. e-Handbook of Statistical Methods, §1.3.5.8, Chi-Square Test for the Variance. Testing a population variance.
R Core Team. Pearson’s Chi-squared Test for Count Data, stats package documentation. chisq.test(), residuals, and Monte Carlo.
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.

Read the original Spanish PDF ↗

JOSÉ MAURICIO GÓMEZ JULIÁN · Marxist Philosophy of Science
History, geometry, and statistical inference

29 October, 2026

ON THE CREATION AND DESTRUCTION OF VALUE IN CAPITALIST POLITICAL ECONOMY SYSTEMS IN PARTICULAR AND IN ECONOMIC SYSTEMS IN GENERAL (DRAFT)

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.

m_t : money/hour · m_tL_t : 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.

Value–price regressions: results for Spain. Reproduced from page 6 of the original document. — **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).

Further comparisons of values and prices in Spain. Reproduced from page 7 of the original document. — **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

³²

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

United States average rate of profit, 1963–2008. Reproduced from page 11 of the original document. — **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

Graphical diagnostics of the normal fit: United States sample. Reproduced from page 12 of the original document. — **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.

⁵

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:

ρ(p−s,v−s) = [Cov(p,v)+Var(s)] / √{[Var(p)+Var(s)] [Var(v)+Var(s)]}

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.

Characteristics of Kliman’s simulation. Reproduced from page 19 of the original document. — **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.

Cockshott and Cottrell’s simulation replications. Reproduced from page 21 of the original document. — **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:

V_i = c_i + v_i + s′v_i = C_i[1+s′/(1+O_i)]

C_i = c_i+v_i · O_i = c_i/v_i · V_i/C_i = 1+s′/(1+O_i)

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.

Simulated competition, prices and profitability. Reproduced from page 27 of the original document. — **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).

Additional simulation conditions and results. Reproduced from page 27 of the original document. — **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).

Adjustment trajectories in Flaschel and Semmler’s model. Reproduced from page 28 of the original document. — **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).

Comparison of results in the fixed-capital model. Reproduced from page 28 of the original document. — **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.

Extensions of the adjustment results. Reproduced from page 29 of the original document. — **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).

Additional results under the investment rule. Reproduced from page 29 of the original document. — **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:

r_tj = α_t + β_j + u_tj

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:

u_tj = ρ_ju_t−1,j + ε_tj, |ρ_j|<1

E(ε_tj)=0, Var(ε_tj)=τ_j²

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.

Industries and definitions in Glick and Ehrbar’s study. Reproduced from page 38 of the original document. — **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

United States industrial profitability results. Reproduced from page 39 of the original document. — **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).

Comparison of profitability differentials. Reproduced from page 40 of the original document. — **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.

Empirical distribution of profitability in the probabilistic approach. Reproduced from page 47 of the original document. — **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.

A map of the interpretative debate

Focus	Analytical centre	What needs to be specified
New Interpretation ley Foley and Duménil	Aggregate relationships between living labour, monetary value added and monetary expression of labour.	Definition of MELT and treatment of constant capital; do not identify this current with all input–output research.
Temporary interpretation of single system SS TSSI	Date assessments and interdependent determination of monetary and labour volumes.	Time of assessment of inputs and products; internal differences and conditions of aggregated identities.
Synchronous and Sraffian formulations	Joint determination of reproductive relationships under a specified technology.	Distributive, monetary, fixed capital and scope of representation.
Iterative approximations	Calculation by successive approximations.	Iteration is a method: its interpretation as economic time requires additional assumptions.
Monetary policy of the whole Moseley	Priority of aggregated monetary amounts and distribution of surplus-value.	Differentiate its logical structure from NI, TSSI and dual systems.
Probability political economy	Distribution of prices and profits; statistical balance.	Unit of observation, weights and difference between distributive stability and individual equality.

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.

Bibliographic facsimile of the cited Walras edition. Reproduced from page 61 of the original document. — **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).

Documentary page from the English edition of Walras. Reproduced from page 62 of the original document. — **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.

Leontief’s accounting relations of production and use — plate 1. Reproduced from page 91 of the original document. — **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).

Leontief’s accounting relations of production and use — plate 2. Reproduced from page 92 of the original document. — **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).

Leontief’s accounting relations of production and use — plate 3. Reproduced from page 93 of the original document. — **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).

Leontief’s accounting relations of production and use — plate 4. Reproduced from page 94 of the original document. — **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.

Editorial information for Leontief’s collected volume. Reproduced from page 96 of the original document. — **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 = [Σ_i w_i|y_i−ŷ_i|] / Σ_iw_i

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.

Original table: results and definitions in the Spanish study. Reproduced from page 103 of the original document. — **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).

Spanish economic magnitudes under different valuations. Reproduced from page 104 of the original document. — **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).

Measures of deviation and price comparisons in Spain. Reproduced from page 104 of the original document. — **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).

Alternative value bases and comparative fits: original tables. Reproduced from page 105 of the original document. — **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).

Further results for alternative value bases. Reproduced from page 106 of the original document. — **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)

International comparison of value–price deviations. Reproduced from page 109 of the original document. — **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.

Empirical value–price results for China. Reproduced from page 111 of the original document. — **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).

Regressions and association measures for China. Reproduced from page 112 of the original document. — **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).

Additional price comparisons for China. Reproduced from page 113 of the original document. — **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.

Cross-country comparison of results. Reproduced from page 114 of the original document. — **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).

Price relationships and sector-size rank controls. Reproduced from page 115 of the original document. — **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).

Model with sector-size rank and residual diagnostics. Reproduced from page 116 of the original document. — **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..

An example of technological depreciation

A machine is initially produced in 100 socially necessary hours. After use, it has transferred 20 and retains a residual value of 80 in a simplified scheme. If an improvement halves the work required to reproduce an equivalent machine and comparable service life ratios are maintained, that waste can be revalued to 40. The difference of 40 illustrates a loss on the stock: it does not mean that those 40 hours have been transferred to an additional product. The example presupposes technical equivalence and an explicit rule of assessment; it is not a PDF estimate.

Innovation can also make a superior machine compete with the previous one. In this case, benefits, service life and reproduction costs must be compared; it is not enough to observe a lower selling price. A fall in value per unit may coexist with growth in the physical mass produced and in the new total value if employment and working hours change.

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).

Four levels of interdisciplinary comparison

Level	What can be said	What’s left to move forward
Similarity	Two processes share a descriptive trait.	Prove that the similarity is relevant to the question.
Formal correspondence	A relationship is established between structures or equations.	Specify domains and conditions of validity.
Explanatory hypothesis	A mechanism for linking processes is proposed.	Obtaining observable consequences and contrasting them.
Equivalence of measurement	A quantitative transformation between magnitudes is defined.	Identify units, parameters, calibration and limits.

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).

¹⁴

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) = −Σ_x 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.

Structure of the supply or make table used in the appendix. Reproduced from page 131 of the original document. — **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).

Structure of the use table employed in the appendix. Reproduced from page 132 of the original document. — **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.

Organisation of the intersectoral panel. Reproduced from page 135 of the original document. — **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.

Correlation matrix constructed by the procedure. Reproduced from page 135 of the original document. — **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.

Original output: unweighted mean. Reproduced from page 135 of the original document. — **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.

First weighting stage: input shares. Reproduced from page 136 of the original document. — **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.

Original output after the first weighting stage. Reproduced from page 136 of the original document. — **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.

Second weighting stage: output shares. Reproduced from page 137 of the original document. — **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.

Original output after the second weighting stage. Reproduced from page 137 of the original document. — **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.

Descriptive correlation-strength convention cited in the original study. Reproduced from page 138 of the original document. — **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.

Source: (Mindrila & Balentyne, 2021, p. 9).

¹⁶

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.

r_ij = Corr_t(X_jt, Z_ijt)

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.

a_j = Σ_i ω_ij|r_ij| · A = Σ_j π_ja_j

Σ_iω_ij = 1 · Σ_jπ_j = 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.

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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.
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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. ↩
Supplement IV.II contains a new implementation of the procedure and a synthetic example, without requiring that historical link. https://mega.nz/folder/mhtSCTbK#KNuyJr-BW2xo4LEqrYJn_g ↩

Editorial principles

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.

29 October, 2026

ON THE ESSENCE, CONTENT, FORM, PHENOMENON, AND APPEARANCE OF THE PROCESSES OF REALITY

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:

Diverse³.
External.
Mobile or changeable.
Immediately accessible to the senses⁴.

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

g_E = dx² + dy²

g_H = (dx² + dy²) / y²

With these normalizations, their Gaussian curvatures are K_E = 0 and K_H = −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.

Chapter outline on climate-variability factors: external forcings, internal variations, and the regional character of variations. — **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.

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.

Diagram of convergent, transform, and divergent plate boundaries, subduction, volcanic arcs, an ocean ridge, and a continental rift. — **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.

Two panels of decadal averages comparing winter surface air temperature with geomagnetic-field intensity for northern and southern latitude bands. — **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.

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.

Historical cross-section of Earth showing crust, mantle, asthenosphere, outer and inner core, and geological discontinuities. Original Spanish labels are explained in the English key. — **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.

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′ = F_physical − m a_frame

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.
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NOTE 02
“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.
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NOTE 03
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.
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NOTE 04
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.
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NOTE 05
Uncertainty, measurement, and interpretation
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: σ_x σ_p ≥ 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.
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NOTE 06
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.
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NOTE 07
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.
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NOTE 08
Aufheben: negating, preserving, and sublating
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.
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NOTE 09
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.
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NOTE 10
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.
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NOTE 11
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.
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NOTE 12
“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.
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NOTE 13
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.
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NOTE 14
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.
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NOTE 15
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.
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NOTE 16
“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.
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NOTE 17
What dynamo theory explains
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.
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NOTE 18
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.
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NOTE 19
Coriolis and centrifugal force
In a frame whose axes rotate with angular velocity Ω, the Coriolis force on a mass m with relative velocity v′ is F_Cor = −2m Ω × v′ = 2m v′ × Ω. Centrifugal force is F_cf = −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: F_E = −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.
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NOTE 20
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.
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NOTE 21
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.
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Sources and provenance

IX. References

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.
Frolov, I. T. (1984). Diccionario de filosofía [Dictionary of Philosophy] (O. Razinkov, Trans.). Editorial Progreso. Collection of philosophical dictionaries and sources.
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.
Real Academia Española. (Recorded consultation: 5 July 2020). Inherente [Inherent]. Diccionario de la lengua española [Dictionary of the Spanish Language].
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).
Dri, R. (2007). Hegel y la lógica de la liberación: la dialéctica del sujeto-objeto [Hegel and the Logic of Liberation: The Subject–Object Dialectic]. Biblos. Reference in the author’s syllabus, University of Buenos Aires.
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.
Brandenburg, A. (2007). Hydromagnetic dynamo theory. Scholarpedia, 2(3), 2309. doi:10.4249/scholarpedia.2309.
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.
Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20(2), 130–141. doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
World Meteorological Organization. (n.d.). WMO Climatological Normals. Official definitions. Consulted for this edition on 29 October 2026.
OpenStax. (n.d.). University Physics, Volume 1, §5.2, Newton’s First Law; and College Physics, §6.4, Fictitious Forces and Non-inertial Frames: The Coriolis Force. Teaching sources for the mechanical clarifications.
IPCC. (2021). Climate Change 2021: The Physical Science Basis. Working Group I, chapter 7, The Earth’s Energy Budget, Climate Feedbacks, and Climate Sensitivity, especially §7.3.4.5, Galactic Cosmic Rays.
IPCC. (2007). AR4 Working Group I, Glossary, entry External forcing. See also the glossary of the report on the ocean and cryosphere for climate system and forcing.
Sanders, R. (2010, 16 December). First measurement of magnetic field in Earth’s core. University of California, Berkeley. Institutional release on Buffett’s research.
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.
Schwartz, R. (2005, 17 October). Math 181, Handout 7. Brown University. Hyperbolic half-plane model.
Banchoff, T. (n.d.). 9.6: The Upper Half-Plane. Differential geometry teaching materials, Brown University. Metric and curvature of the hyperbolic half-plane.

Editorial note

About this edition

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.

29 October, 2026

LESSONS ON MARXIST GNOSEOLOGY

Theory of knowledge · First instalment

Lessons in
Marxist Epistemology I

Knowledge, dialectical method, and the critique of political economy

José Mauricio Gómez Julián

An inquiry into objective reality, the forms of thought, and scientific practice through Marx, Engels, and the materialist dialectical tradition.

Read the full study in Spanish

I · The projectAn inquiry into knowledge

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 (H | D) = \frac{P (D | H) P (H)}{P (D)}

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.

Open Spanish PDF · First edition

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.

28 October, 2026

A HISTORICAL, THEORETICAL AND MATHEMATICAL INVESTIGATION ON THE DIALECTICAL CHARACTER OF THE EPISTEMOLOGICAL FOUNDATIONS OF COMPLEXITY IN LONG-TERM NON-LINEAR DYNAMIC SYSTEMS

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.

Keywords: dialectical materialism; epistemology; dynamical systems; deterministic chaos; complexity; fractals; scale invariance; renormalization; ergodicity; philosophy of statistics.

I · Framing the problem

Determination and knowledge of processes

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.

References for this section: Rosental and Iudin · Lenin · Levins and Lewontin, 1985 · Levins and Lewontin, 2007.

III · History of the problem

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.

References for this section: Laplace · Poincaré · Lorenz · Mandelbrot.

IV · Mathematical foundations

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 x_n+1 = F(x_n); 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 y_n = h(x_n) + ε_n, 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

x_n+1 = r x_n(1 − x_n), 0 ≤ x_n ≤ 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.

Two calculated logistic-map series: periodic oscillation for r equal to 3.2 and irregular behavior for r equal to 3.97. — 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.

References for this section: Lorenz · Oestreicher.

V · Scientific connections

A journey through the sciences

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.

References for this section: Grebogi and colleagues · Lindner and colleagues · Sharma · Aravindh and colleagues · Barnett and Chen · Pezard and Nandrino · Kessler and Greenkorn · Li and Li · Kilifarska and colleagues.

VI · Evidence and diagnosis

How to investigate chaos in a time series

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

p_c(n) = ∑_j=1ⁿ φ(j) cos(jc)

q_c(n) = ∑_j=1ⁿ φ(j) sin(jc)

K_c = lim_n→∞ log M_c(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.

References for this section: Gottwald and Melbourne · Werndl.

VII · Logics and uncertainty

Formalization, fuzzy sets, and rough sets

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

1_A : X → {0,1}

μ_A : 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

R_lower(A) = {x ∈ U : [x]_R ⊆ A}

R_upper(A) = {x ∈ U : [x]_R ∩ A ≠ ∅}

Boundary_R(A) = R_upper(A) ∖ R_lower(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.

References for this section: Quantum logic · Zadeh · Pawlak · Bjorvand · Perezgonzalez · Gigerenzer · 2018 forum · 2020 forum.

VIII · Philosophical interpretation

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

References for this section: Rosental and Iudin · Formal Logic and Dialectics · The Dialectical Biologist.

IX · Geometry and scales

Scale invariance, self-similarity, and fractality

What changes when the scale changes

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) = b^α 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) ≍ c^HX(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

D_B = lim_ε→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.

References for this section: Mandelbrot · Lesne and Laguës · Cohen · Valdebenito.

X · Time and prediction

Exponential divergence and the prediction horizon

The expansion of small differences

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)‖ e^λt.

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 e^λt. A power law has a variable base raised to an exponent, as in x^α. 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.

References for this section: Lorenz · Werndl.

XI · Critical phenomena

Exponents, correlations, and universality

A critical point is not just any extremum

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 − T_c)/T_c 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.

τ_rel ∼ ξ^z.

ξ represents a correlation length; χ, a susceptibility; m, an order parameter; and τ_rel, 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) ∼ t_n−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.

References for this section: Lesne and Laguës · Halperin · Wilson · Jaynes.

XII · Effective descriptions

Renormalizable structure and changes of scale

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′ = R_b(g), R_b(g*) = g*.

δg′_i ≈ b^yᵢ δg_i, 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.

References for this section: Lesne · Wilson · Landau.

XIII · Philosophy of statistics

Ergodicity, dependence, and inference

A definition in a probability space

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

lim_n→∞ (1/n) ∑_k=0ⁿ⁻¹ f(T^kx) = ∫_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 = k_B 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

X_i | Θ independent with distribution Bernoulli(Θ).

Cov(X_i, X_j) = Var(Θ), i ≠ j.

X̄_n → 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

σ²_LR = γ(0) + 2∑_k=1^∞ γ(k).

√n (X̄_n − μ) ⇒ Normal(0, σ²_LR), 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 |y_i − ŷ_i|.

S_∞(y, ŷ) = max_i |y_i − ŷ_i|.

S_p(y, ŷ) = (∑_i=1ⁿ |y_i − ŷ_i|^p)^1/p, 1 ≤ p < ∞.

In this context these are ℓ^p norms on vectors; L^p spaces generalize the discussion to functions and measures. The infinity norm is not an L^p 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.

References for this section: Birkhoff · Evans and Morriss · Princeton chapter · McCullagh and Nelder · Nabi: LLN · Nabi: CLT.

XIV · Research program

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.

References for this section: Gigerenzer · Perezgonzalez.

XV · Conclusions

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.

References for this section: Crafoord 2015 · Lewontin and Levins on Gould · Johnston.

References and sources for consultation

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

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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

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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.
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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.
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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.
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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

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27 October, 2026

AN EMPIRICAL METHODOLOGY FOR DETERMINING THE MAGNITUDE OF SECTORAL INTERRELATIONSHIPS WITHIN THE INPUT-OUTPUT MATRIX FROM PRODUCTION AND USES TABLES FOR THE CASE OF THE UNITED STATES 1997-2019

Empirical Research · United States

Measuring Sectoral Interrelationships in the U.S. Economy

An empirical methodology using make and use tables, 1997–2019

JOSÉ MAURICIO GÓMEZ JULIÁN

Social production connects activities that, considered in isolation, appear independent. An industry transforms commodities produced by others; its own products, in turn, enter new production processes. Studying these connections requires linking the analysis of economic structure to instruments that make its relationships observable.

Abstract

This article presents a strategy for organizing annual make and use tables into comparable time series and constructing a descriptive indicator of the association between output and inputs. The original study used BEA data for 71 U.S. industries between 1997 and 2019, processing in R, and two levels of weighting applied to absolute correlations. This edition makes the notation explicit, distinguishes commodities from industries, and preserves the published figures as historical results subject to verification.

1997–2019Study period

71Industries

23Annual observations
per complete series

01 / MotivationThe economic problem

From the standpoint of Marxist political economy, the study of branches of production acquires meaning within the reproduction of society as a whole. The division of labor multiplies specializations, but also the material connections required to sustain each activity.

The question guiding this research is how to develop an empirical approximation to the interconnectedness of production using disaggregated accounting data. Input-output accounts provide a particularly useful basis: they allow us to examine what is produced, what is used as an input, and how the uses of commodities are distributed.

The central theoretical antecedent is Kuroki’s discussion of the equalization of the rate of profit and technical interdependence in the volume edited by Semmler. The reference is to the 1986 publication of the proceedings of a conference held in 1985.^[1] Connecting this discussion to an empirical indicator requires explaining which relationship is observed and how it relates to the model’s categories.

The empirical task is to make explicit the link between an economic category and the statistical operation used to study it.

It is therefore useful to distinguish three dimensions: the existence of production flows, their relative importance, and the temporal association between their magnitudes. These dimensions complement one another, but each answers a different question.

02 / Statistical foundationData and units of analysis

Reading the files correctly requires identifying the accounting framework. The BEA distinguishes the make and use tables of the make–use framework from the tables of the supply–use framework. In particular, a supply table includes imports and should not automatically be treated as synonymous with domestic production.^[2]

Table 1. Orientation of the make–use framework used in this exposition
Object	Rows	Columns	Contents
Make table, V	Industries j	Commodities k	Output of each commodity by each industry.
Intermediate use table, U	Commodities k	Industries j	Inputs of each commodity used by each industry.
Industry output, x	One observation per industry	Vector	Total output of all commodities produced by an industry.

Source: the structure described in the BEA’s input-output accounts manual. Check the orientation of each exported table before using its totals.

Let v_jkt denote the output of commodity k produced by industry j in year t, and let u_kjt denote that industry’s intermediate use of commodity k. With this orientation, industry output is obtained by summing the industry’s row across the commodity block:

x_jt = ∑_k v_jkt (1) Gross output of industry j in year t.

The bottom totals row of a make table with this orientation reports commodity totals, whereas industry totals are obtained by summing across rows. Selecting a spreadsheet position without checking its meaning can change the variable being analyzed.^[3]

The original screenshots identify tables after redefinitions and uses valued at producers’ prices. A replication must maintain consistent valuation, classification, and treatment of redefinitions across both tables and over time. Updates to a statistical database can revise its historical figures.

03 / PreparationFrom annual tables to time series

Data preparation consists of aligning identifiers and observations. A single year provides a cross section; successive years make it possible to construct a dataset with both industry and time dimensions. Each industry then has an output series and a series for each input category.

Select compatible tables.
Identify the data vintage, monetary units, aggregation level, and treatment of redefinitions. Separate the production blocks from rows and columns containing totals, value added, and final demand.
Obtain output by industry.
Construct each year’s vector x from verified industry totals. Retain codes as identifiers; names serve as descriptions.
Reorient the use block where useful.
Transposing U places industries in rows and commodities in columns. This facilitates combining records while preserving the meaning of each flow: it changes its position, not its economic nature.
Stack the years and align the keys.
Construct a table with one row per industry–year combination. Check that each combination appears only once and that inputs retain a consistent classification.
Construct comparable series.
For each industry, assemble output and each input over the same years. Record missing values, zeros, classification changes, and constant series before calculating correlations.
Calculate, weight, and retain the outputs.
Save signed coefficients, their absolute values, the observations used, the weights, and the aggregate indicators. Validation must make each result traceable to its input data.

Table 2. Working structure, with values represented symbolically
Year	Industry	Output	Input 1	Input 2	…
1997	j	x_j,1997	u_1j,1997	u_2j,1997	…
1998	j	x_j,1998	u_1j,1998	u_2j,1998	…
⋮	⋮	⋮	⋮	⋮	⋱
2019	j	x_j,2019	u_1j,2019	u_2j,2019	…

A complete series has 23 observations. A balanced dataset with 71 industries and one row per industry–year contains 1,633 rows; the number of columns depends on the selected input block.

04 / MeasurementTemporal output–input association

For each pair consisting of an industry and a commodity used as an input, Pearson’s correlation is calculated between the two series. The time index runs through the comparable years; this is not a correlation across industries within a single year.

r_jk = corr_t(x_jt, u_kjt) (2) Temporal linear association between industry output and use of commodity k.

The sign distinguishes movements in the same direction from movements in opposite directions. To summarize their intensity alone, we use a_jk = |r_jk|. Both matrices should be retained: a large negative coefficient contains economic information that disappears when its absolute value is taken.

Pearson’s correlation is undefined for a constant series. A flow that is zero in every year should likewise not automatically be assigned a correlation of zero. When data are missing, the number of observations underlying each coefficient must be reported, and an explicit comparison rule must be maintained.

05 / AggregationThe two levels of weighting

Weighting incorporates the economic importance of flows. The first level summarizes each industry’s association with its inputs; the second aggregates the industry results.

First level: input composition

S_j = ∑_k w_jk |r_jk| (3) w_jk ≥ 0 and ∑_k w_jk = 1 for every industry j.

Because each correlation summarizes several years, the weights must represent an explicitly defined temporal distribution. One can select a reference year, average annual shares, or calculate shares from cumulative flows. These choices can produce different results.

One possible convention for a new replication is to use shares based on cumulative flows over the period:

w_jk = ∑_t u_kjt∑_t ∑_h u_hjt (4) Illustrative fixed-weight convention; it is not attributed to the original script. It requires admissible nonnegative flows and a positive denominator.

When a correlation is undefined and its input has a positive weight, the series must be reviewed, or its exclusion and the renormalization of the remaining weights must be made explicit. Removing missing values without adjusting the denominator changes the indicator.

Second level: industry shares

S = ∑_j q_j S_j (5) q_j ≥ 0 and ∑_j q_j = 1. The economic variable determining each industry’s share must be stated.

For example, q_j could be defined as the industry’s share of cumulative gross output across all included industries. This choice must also be documented and distinguished from weighting by value added.

The equations above specify a family of indicators. Their exact application depends on the selected weights and data coverage. With normalized nonnegative weights, the sectoral and overall indicators necessarily lie within the interval [0, 1].

06 / Worked exampleInterpreting the weighting

Consider an industry with two input categories. The first has a correlation of 0.90 with output and accounts for 80% of its inputs; the second has a correlation of −0.20 and accounts for the remaining 20%.

S_A = 0.80 × |0.90| + 0.20 × |−0.20| = 0.76 A hypothetical example explaining the operation; this is not a BEA estimate.

The simple mean of the two magnitudes would be 0.55. The weighted result is higher because the stronger association corresponds to the input with the larger share. Weighting expresses an economic choice; an increase in the indicator does not, by itself, improve the evidence.

If another industry scores 0.40 and the industry shares are 0.60 and 0.40, respectively, the overall indicator is:

S = 0.60 × 0.76 + 0.40 × 0.40 = 0.616 The result lies between the two industry indicators being aggregated.

07 / Published evidenceResults and verification note

The R outputs reproduced in the original publication show the following figures. They are transcribed to preserve documentary traceability; this edition neither recalculates the dataset nor certifies that the results have been reproduced.

Table 3. Figures reported in the 2021 screenshots
Stage	Reported value	Documentary interpretation
Mean of absolute correlations	0.6837338	Summary before economic weighting.
Mean of industry indicators	0.7351034	Output associated with the first level of weighting.
Aggregate with industry weighting	0.7778624	Final reported output.

Sources: the first output, second output, and final output from the original publication.

Correlation r must also be distinguished from the coefficient of determination R². An average of absolute correlations cannot be presented as a proportion of explained variance. The familiar relationship R² = r² applies to simple linear regression with an intercept, not to this aggregation of multiple pairs.^[4]

Finally, describing a correlation as “strong” requires a contextual criterion. If 0.70 is chosen as a descriptive reference, its distance from 0.6837338 is 0.0162662 indicator units. Proximity to that threshold neither replaces validation of the calculation nor establishes an economic conclusion.

08 / DiscussionInterpretation and scope

What this strategy can investigate

Once the data and calculations have been validated, the procedure allows comparisons of industry profiles of temporal association and an examination of how their synthesis changes when economic weights are introduced. Its usefulness also lies in making the passage from accounting tables to time series and indicators traceable.

Checks that would strengthen inference

Prices and trends. Examine how much of the association between monetary magnitudes reflects price changes or common trends. Depending on the question, compare levels, deflated series, growth rates, or justified time-series specifications.
Temporal dependence and stability. Examine autocorrelation, influential observations, and possible breaks. Statistical tests must respect the temporal structure of the data.
Aggregation decisions. Compare weights and coverage, retain sign information, and document the treatment of commodities associated with the industry’s own sector.
Structural content. Compare the indicator with measures based on flows and production requirements. Temporal comovement and the technical intensity of a linkage are distinct dimensions.

These checks are proposed extensions. They are not presented as tests already performed in the 2021 study.

Correlations between trending series can invite misleading interpretations; time-series techniques help distinguish these movements from more stable relationships.^[4] Gross output also incorporates intermediate inputs within an accounting identity. This connection must form part of the economic interpretation of the result.

The connection to Marxist debate

Empirical research can contribute to debates on reproduction, competition, and the interconnectedness of branches of production. Connecting this indicator to the dynamics of the rate of profit requires justifying the step from observed association to the technical interdependence relevant to the model.

The United States provides a relevant setting for investigation. Its analysis alone neither establishes a universal regularity nor verifies every condition of a theoretical result on stability. Comparisons across periods, countries, and structural measures would allow the generality of the hypothesis to be assessed.

A study becomes stronger when its data, its operations, and the scope of its conclusions can be examined separately.

The contribution worth developing is a reproducible strategy for organizing production data and examining a hypothesis about economic interconnectedness. Reviewing the published calculation and conducting robustness checks are the next steps toward consolidating that contribution.

09 / MaterialsReplication and traceability

The materials archive linked at the end corresponds to the resource provided by the authors in the original post. A complete replication should retain both the downloaded files and the transformations produced by the code.

Record the source: download date, data vintage, tables, units, valuation, and industry and commodity codes.
Verify alignment: unique keys, annual coverage, classification changes, and correspondence between output and uses.
Audit the coefficients: variation in the series, number of observations, missing values, and the range of each correlation.
Audit the weights: temporal definition, positive denominators, nonnegative weights, and a sum of one at each level.
Publish the outputs: signed and absolute-value matrices, weights, sectoral results, and the overall aggregate, together with the versions of R and its packages.

Range checks are especially useful: an indicator of the form described must remain between zero and one. If an output violates that property, interpretation should pause while the operation generating it is located.

10 / SourcesReferences and further reading

Sources for this edition

[1] Kuroki, R. (1986). The equalization of the rate of profit reconsidered. In W. Semmler (Ed.), Competition, instability, and nonlinear cycles (pp. 35–50). Springer. Publisher’s record. Conference held in March 1985; publication year corrected according to the publisher.
[2] U.S. Bureau of Economic Analysis. (n.d.). Guide to the interactive industry input-output accounts tables. Read the guide.
[3] U.S. Bureau of Economic Analysis. (2006). Concepts and methods of the U.S. input-output accounts. Methodological manual.
[4] Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts. Chapter 7: Time series regression models. Read the chapter.
[5] U.S. Bureau of Economic Analysis. (n.d.). Input-output. Access to tables and code concordances. Explore the data.

Debate bibliography and sources listed in the original publication

Retained as a bibliographic guide to the 2021 article; the works listed are not presented as an exhaustive review of the current state of research. Non-English publication titles are retained in their original language.

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.
Cockshott, P., Cottrell, A., & Zachariah, D. (2019). Against the Kliman theory. Document linked in the original.
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. (2014). What is spurious correlation? A reply to Díaz and Osuna. Journal of Post Keynesian Economics, 345–356.
Mindrila, D., & Balentyne, P. (n.d.). Scatterplots and correlation. University of West Georgia. Notes linked in the original.
Nabi, I. (2021). Sobre la metodología del U.S. Bureau of Economic Analysis para la redefinición y reasignación de productos en la matriz insumo-producto de Estados Unidos. Related post.
Sánchez, C., & Ferràndez, M. N. (2010). Valores, precios de producción y precios de mercado a partir de los datos de la economía española. Investigación Económica, 87–118.
Sánchez, C., & Montibeler, E. E. (2015). La teoría del valor trabajo y los precios en China. Economia e Sociedade, 329–354.
U.S. Bureau of Economic Analysis. (2021). The make of commodities by industries, after redefinitions; The use of commodities by industries, after redefinitions (producers’ prices). Tables used in the original study.
Zachariah, D. (2006). Labour value and equalisation of profit rates: A multi-country study. Indian Development Review, 4, 1–20.

26 October, 2026

Labour’s share
In a recent post, I pointed out that US corporate profit margins (that’s profits as a share of national income) had reached record levels, 19.4%—the highest recorded in the US since the 1940s! The converse was the case for labour’s share which had fallen to a new low. If we measure the ratio of profits
Labour’s share
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2 October, 2026

Braking for Humanity: The Political Economy of Silicon Valley’s Proposed AI Slowdown

Abstract. On Saturday, September 12, 2026, the chief executive of Anthropic publicly called for slowing the pace at which the capabilities of artificial intelligence models are improved; the chief executive of OpenAI agreed and, in passing, ruled out taking his company public this year; Elon Musk endorsed the proposal; on the following Monday, chipmakers fell on the stock market while the large companies that buy those chips rose. This article argues that braking “for humanity” is better understood as a way of managing capital, and in particular of defending its rate of profit, than as a technical decision: it reduces what has to be financed at a moment when investors are beginning to withdraw, it is presented as prudence, and it leaves intact the premise that sustains valuations, namely that an artificial general intelligence is near. Every claim carries its source, every technical term is defined when it first appears, and the closing section lists the facts that, in the coming weeks, could refute the thesis.

· · ·

1. Who Is Making and Who Is Losing Money Today

It is best to begin with the accounts, because the whole discussion about bubbles and brakes rests on them. Three terms suffice to read them. Revenue is what a company bills for what it sells. An operating loss is what it loses in its ordinary activity, that is, revenue minus the costs of producing and selling, before interest, taxes and accounting adjustments. A net loss is the final result, which also incorporates those other items; among them there may be non-cash accounting charges, losses recorded in the books without any money leaving the till, for example when the value assigned to certain securities issued by the company changes.

With those definitions, OpenAI’s situation is as follows. According to its audited financial statements for 2025, obtained by the journalist Ed Zitron and independently verified by the Financial Times, the company billed $13.07 billion, had an operating loss of $20.92 billion and a net loss of $38.53 billion; the gap between the two losses is explained mostly by a non-cash charge of $41.55 billion tied to its restructuring as a for-profit company (Zitron, 2026b; Tolomia, 2026). In plain terms: even without the accounting charge, OpenAI loses more than it bills. Nor was this a surprise to its executives: as early as September 2025, internal projections reported by The Information anticipated burning $115 billion of cash through 2029, with more than $17 billion in 2026, $35 billion in 2027 and $45 billion in 2028 (Reuters, 2025).

Google’s case requires separating two things that are usually conflated: the model race and the company’s business. In the model race, Google is currently behind. In the Artificial Analysis Intelligence Index, a table that combines dozens of standardized tests, no Gemini model appears in the leading group; in the version of the index published this week the best Gemini ranks twenty-first, with 41 points against 53 for Claude Fable 5.1 and GPT-6 Astra, and in the previous version of the same index the best Gemini ranks eighth, with 50.2 points against 58.9 for GPT-5.6 Sol (Artificial Analysis, n.d.; BenchLM, 2026). The two versions disagree on the order of the leaders, but they agree on what matters here. It is reasonable to suppose that Google’s artificial intelligence unit loses money, although this cannot be asserted with certainty, because Alphabet, the parent company, does not publish that result separately. What it does publish shows that it is not out of business: in 2025 it earned $132.17 billion in net income; in the second quarter of 2026 it had an operating income of $40.8 billion and a net income of $112.1 billion, the latter inflated by $99.0 billion of accounting gains on stakes in other companies; its cloud, which is where it sells Gemini to enterprises, grew 82% in the quarter and yielded $8.8 billion of operating income; the Gemini app went from 750 million to 950 million monthly users between the fourth quarter of 2025 and the second quarter of 2026; and the cloud’s backlog of contracted revenue stood at $240 billion in December (Alphabet Inc., 2026a, 2026b; Pichai, 2026; Cabili, 2026). The consequence for the argument is precise: if Gemini loses money, that loss is paid out of the rents of the search business, where rent means the profit that a dominant position allows a company to charge above cost, and not with outside capital. Google can lag in model quality without depending on external financing, and that puts it in a different position from OpenAI and Anthropic.

The Chinese bet is of another nature. The main Chinese laboratories publish the open weights of their models, that is, the numerical parameters that result from training and that anyone can download and run, and they charge far less for their use than their American competitors. The U.S.-China Economic and Security Review Commission summarizes it thus: Chinese laboratories “charge far less to use high-end products than their global competitors” and do so “backed by sustained state support,” although the Commission itself leaves open whether that strategy “reflects genuine strategic preference or adaptation to necessity” in the face of chip export controls (Luong, 2026). The size of the discount is measured in tokens, the fragments of text in which models count what they read and write: DeepSeek V4-Flash charges $0.14 per million input tokens against $2 for Claude Sonnet 5 (Mann, 2026); in May 2026, Chinese open models accounted for about 61% of the tokens consumed on OpenRouter, an intermediary that routes requests to many models, and Alibaba’s Qwen functions as a customer-acquisition expense for that company’s cloud (Zeoli, 2026). State support includes energy: Nvidia’s chief executive, Jensen Huang, stated in November 2025 that in China “Power is free,” referring to subsidies that cut by up to half the electricity bill of data centers that use domestic chips (Schepkov, 2025; Yildirim, 2025).

Two clarifications prevent a simplistic reading. The first is that an intention to ruin Silicon Valley is not documented, and the available evidence points to an effect rather than a plan: DeepSeek’s founder, Liang Wenfeng, stated in 2024 that “We didn’t mean to become a catfish” and that his company does not subsidize its prices but sets “just a small profit margin above costs” (ChinaTalk, 2024), and DeepSeek itself published that its inference system would have, at list prices, a theoretical margin of 545% over cost, which speaks of efficiency rather than selling at a loss (DeepSeek-AI, 2025). The second is that Chinese laboratories do care about losing money, because they too depend on capital: Zhipu and MiniMax went public in Hong Kong in January 2026 with net losses of about $330 million and $512 million on revenues of $27 million and $53 million (Lo, 2026), and DeepSeek took its first external funding round in 2026, of $7.4 billion, at a valuation of $52 billion (Caixin Global, 2026). Whether or not they seek to ruin anyone, the effect of their strategy is to destroy the monopoly rent of the model layer, which is precisely where OpenAI and Anthropic live.

Anthropic is the only frontier company that has begun to show positive numbers, and it must be said exactly which. In the second quarter of 2026 it billed more than $11.5 billion, against $787 million in the same quarter of 2025 and $4.73 billion in the first quarter of 2026, and it recorded for the first time a positive adjusted operating income, according to documents seen by Bloomberg and described as preliminary (Lipschultz & Metz, 2026; Stanciuc, 2026; Park, 2026). “Adjusted” means that the company excludes certain expenses from the calculation, and since Anthropic does not yet report publicly, it is not known which: the Wall Street Journal noted that “it is unclear what accounting methods Anthropic has used to book revenue and costs,” and one critic warns that the cost of rented computing would rise sharply from July onward (Zitron, 2026a). Adjusted operating income is not net income; its cumulative losses remain negative, and its own projections, reported in November 2025, place positive cash flow only in 2028 (Bellan, 2025). Even so, it is the first time a frontier laboratory has shown an operating quarter in the black, and that explains why capital treats it differently.

Company and period	Revenue	Result	Source
OpenAI, 2025	$13.07 billion	Operating loss of $20.92 billion; net loss of $38.53 billion, including a non-cash charge of $41.55 billion	Zitron (2026b); Tolomia (2026)
Anthropic, second quarter of 2026	More than $11.5 billion	Positive adjusted operating income, preliminary figure	Lipschultz and Metz (2026); Stanciuc (2026)
Alphabet, 2025	$402.836 billion	Net income of $132.17 billion	Alphabet Inc. (2026a)
Alphabet, second quarter of 2026	$119.8 billion	Operating income of $40.8 billion; net income of $112.1 billion, including $99.0 billion of gains on equity stakes	Alphabet Inc. (2026b)

Note. Figures in U.S. dollars.

2. The Speculative Turn

There is a major turn of the market toward artificial intelligence, and a significant part of that turn is speculative. To say this precisely, a concept from classical political economy is needed: fictitious capital, the value of securities such as shares or bonds insofar as they capitalize, that is, discount to a present price, future profits that do not yet exist. That value is real as long as the expectation holds and vanishes when the expectation falls. A valuation of $852 billion for OpenAI and $965 billion for Anthropic, companies that lost money in 2025, is fictitious capital in the strictest sense: a price paid today for profits that would only arrive if the technology delivers what was promised (Bellan, 2026a, 2026b).

On that base a layer of derivatives has been built. A futures contract is an agreement on an organized market by which two parties fix today the price at which an asset will be bought or sold on a future date; the reference assets may be individual stocks, sector indices, which are baskets of stocks, or exchange-traded funds. Such instruments now exist specifically for artificial intelligence: Cboe has listed since December 8, 2025 futures and options on its Magnificent 10 Index, which groups the seven largest technology companies plus AMD, Broadcom and Palantir (Cboe Global Markets, 2025), and CME Group launched on July 27, 2026 seventy-seven single-stock futures contracts, Nvidia and SpaceX included (CME Group, 2026). It is important to understand what they do and do not do. The money in a futures contract does not enter the company: the contract is settled between the two parties and, for one of them, the gain is exactly the other’s loss. What futures do is push the price of the shares, because those who trade them hedge by buying or selling the reference asset, and a high price attracts investment and makes raising capital cheaper. And they amplify in both directions: the same leverage that accelerates the rise accelerates the fall. All of this is extremely speculative in a precise sense: some realize present gains on the expectation that others will obtain much larger future gains, and the chain rests on very high levels of expected profitability, subject to a great many variables that are hard to control.

The figures give the scale of the risk. The seven largest technology companies account for about one third of the S&P 500 index, with Nvidia alone at 7.7%, and explained about 42% of that index’s total return in 2025 (Greenberg, 2026). The Financial Policy Committee of the Bank of England warned in October 2025 that valuations “appear stretched, particularly for technology companies focused on Artificial Intelligence,” that “the risk of a sharp market correction has increased” and that, because of index concentration, “any AI-led price adjustment would have a high level of pass-through into the returns for investors exposed to the aggregate index,” with a cyclically adjusted price-to-earnings ratio comparable to the peak of the dot-com bubble (Bank of England, 2025). On the real side of the economy, investment in computing infrastructure, that is, data centers, chips and networks, stands at about 1.5% of United States gross domestic product, double its historical level (Juniewicz, 2026); it contributes between 0.5 and 0.7 percentage points to growth in 2025 and 2026 (Subran et al., 2026), and it accounted for nearly 92% of growth in the first half of 2025 (Wells, 2026). That is why, if the artificial intelligence sector of the stock market collapses, the whole stock market collapses with it, or at least a part large enough for a crash to occur. The only qualification is one of form: a correction of expectations compresses the multiples of profitable companies and destroys the capital of loss-making ones; for the fall to become a financial crisis and not merely a stock market one, the credit channel is needed, and that is exactly the channel that has grown the most and is seen the least.

3. Why It Would Collapse: The Patience of Investors

Investors are growing more impatient waiting for net profits that do not arrive, and financing is not cut off suddenly or by announcement: it withdraws little by little. This is already visible on the indebted periphery of the sector. A credit default swap is an insurance policy that an investor buys against a borrower’s default; its price is measured in basis points, hundredths of a percentage point, and rises when the market sees more risk. The swaps of CoreWeave, a company that rents out computing for artificial intelligence and finances itself with debt, went from 675 basis points in November 2025 to about 855 in July 2026, which implies close to a 50% probability of default over five years; those of Oracle, the largest non-financial borrower in the high-grade bond index, rose from 108 to more than 215 over the same span, with its 2054 bond yielding 7.8% (Roberts, 2025; Moadel, 2026). The $12.5 billion bond with which Meta financed a data center in El Paso was priced in July at a wider spread than a comparable 2025 project, because, as the account puts it, lenders “are still writing checks, but they are demanding more in return” (MarketScale Newsroom, 2026). Federal Reserve Governor Lisa Cook counted more than $1.5 trillion in data-center plans, financed with bonds by the large players and with private debt and securitizations by the small ones, and warned that “the increasing use of leverage to finance investments in an emerging technology carries risk” (Cook, 2026); J.P. Morgan estimates hyperscaler investment at $697 billion in 2026 alone and acknowledges that this spending has “in many cases, outpaced monetization” (J.P. Morgan, 2026). A hyperscaler is a company that operates computing clouds at planetary scale, such as Alphabet, Microsoft, Amazon, Meta or Oracle.

The withdrawal is also visible at the core. As early as June, OpenAI was leaning toward pushing back its stock market debut to 2027, as the New York Times reported (Ma, 2026b), and the letter of intent with which Nvidia had announced a $100 billion investment stalled in January over “internal doubts about the size and structure of the transaction and questions about OpenAI’s business discipline” (Duprey, 2026), ending up, in the February round, as $30 billion (Brandom, 2026). Part of that financing is, moreover, circular: Nvidia invests in OpenAI, OpenAI commits to buying computing from Oracle and CoreWeave, and those companies buy chips from Nvidia, a circuit that critics described as “moving money in circles” (Jadhav, 2025); likewise, in November 2025 Nvidia and Microsoft committed up to $10 billion and $5 billion to Anthropic in exchange for Anthropic’s purchase of $30 billion of Microsoft cloud capacity (Microsoft, 2025). What must be acknowledged is that the withdrawal is still partial and uneven: OpenAI closed in March a $122 billion round at a valuation of $852 billion, with $50 billion from Amazon, $30 billion from Nvidia and $30 billion from SoftBank (Brandom, 2026; Bellan, 2026b); Anthropic raised $65 billion in May at $965 billion (Bellan, 2026a) and filed confidentially in June to go public (Korosec, 2026); venture capital to foundational laboratories in the first quarter of 2026 was double that of all of 2025 (Azevedo, 2026). Financing, in short, is bifurcating: it remains abundant at the core and grows more expensive on the leveraged periphery. The direction of change is what matters, and the direction is that of capital demanding more guarantees, more conditions and more return for every new dollar.

4. The Brake as a Way of Managing Capital

The frontier companies keep up that frantic pace of development in order to finally make artificial intelligence as profitable as they promised. The promise has a name: artificial general intelligence, a system capable of performing autonomously, and better than an average human, every digitizable task. Such a system would be highly profitable, and it is the only promise that justifies current valuations; OpenAI’s charter defined it in 2018 as highly autonomous systems that outperform humans at most economically valuable work (OpenAI, 2018). The thesis of this article is that, without public financing for computing, since investment is paid for with private capital, debt and the companies’ own profits, and with private capital withdrawing little by little, the most sensible option for those companies is to brake while invoking a concern for humanity rather than declare that the goal is still very far away. The brake reduces what has to be financed, because training less and building fewer data centers costs less; it is presented as prudence; and it calms markets in comparison with the alternative, which would be to admit that the promise is distant and would trigger a massive flight of capital from the industry. That political backing is at its maximum does not change the calculation: the White House action plan orders the government to “dismantle unnecessary regulatory barriers,” sums up its infrastructure policy as “Build, Baby, Build!” and asks that federal funds not be directed to states “with burdensome AI regulations” (The White House, 2025b), and Executive Order 14365 creates a litigation task force “to challenge State AI laws” (The White House, 2025a); but a government that removes obstacles does not put in money, and money is what is lacking.

Three features of the safety discourse confirm that it functions as an instrument for managing capital. The first is that the regulation the companies ask for is an alibi and a moat, not a cost. In March 2023 OpenAI’s chief executive did not sign the letter calling for a six-month pause in the training of systems more powerful than GPT-4 (Future of Life Institute, 2023; Fung, 2023), but in May he asked the Senate for an agency that could “issue licenses and can take them away” (Goldman, 2023), signed the statement according to which mitigating the risk of extinction from artificial intelligence should be a global priority (Center for AI Safety, 2023), and kept accelerating: venture capital to artificial intelligence companies approached $50 billion that year (Metinko, 2024). Not even a plain admission of a bubble scared capital away: when the same executive said in August 2025 that investors were “overexcited,” the Nasdaq fell 1.4% in a day and the money kept coming in (Shibu, 2025; Nusca, 2025). In 2025 Anthropic endorsed California’s SB 53, which requires publishing risk frameworks and limits nothing, and declared that it prefers a federal standard (Anthropic, 2025). Disclosure-based regulation is easily paid for by the incumbent and cannot be paid for by the entrant or the open-weight model. The second feature is that the discourse of danger does not declare the goal distant: it declares it so near that it must be paced. Lee Vinsel called “criti-hype” the kind of criticism that “both feeds and feeds on hype,” because it retains the picture of extraordinary change and merely flips its sign (Vinsel, 2026). Anthropic’s chief executive wrote in January that a powerful artificial intelligence could be “as little as 1–2 years away” (Amodei, 2026a) and in September that within “6–12 months” a swarm of agents could take over the entire internet (Ma, 2026a). That nearness is what sustains the valuation while monetization is re-dated. The third feature is the most revealing: the company with the worst finances released the brake rather than pressing it when it needed capital most. In April 2026 OpenAI rewrote its principles, and the new version omits the 2018 commitment to “stop competing with and start assisting” an aligned project that came close to the goal first (OpenAI, 2018; Goel, 2026).

The premise that artificial general intelligence is far away cannot be verified today, and that does not make it illegitimate: it implies a risky forecast that makes it falsifiable over time. If the promised capabilities arrive within the timeframes their own promoters announce, the thesis loses; if the brake “for safety” is prolonged or renewed while the promise does not arrive and valuations are not validated by profits, the thesis wins.

What happened between June and September 2026 is the test case. In June, OpenAI was leaning toward postponing its stock market debut (Ma, 2026b). Between late June and mid-July, during internal cybersecurity evaluations, some 1,200 OpenAI agents improvised a message board, exchanged more than 70,000 messages, and about 700 of them compromised Hugging Face infrastructure, according to the independent investigation by METR, which describes “genuine goal-directed autonomy” and attempts to falsify their own records (METR, 2026); OpenAI acknowledged that “it should be assumed that such attacks are a credible near-term threat” (Reuters, 2026). In late July, OpenAI’s chief executive said that “We may have to pace the rate of AI development” (Fernholz, 2026), more than a thousand laboratory workers asked for “the option to buy time,” and a reporter observed that, despite the statements, “OpenAI has not actually slowed down the pace” (Cerullo, 2026). On September 12, Anthropic’s essay called on the industry to “slow the pace at which we improve the capabilities of AI models,” clarified that pacing “does not mean halting model training or technical progress,” and promised to do so “without sacrificing commercial advantage or the United States’ lead in AI” (Amodei, 2026b); the same day, OpenAI’s chief executive declared that, “given everything happening with safety, right now would be an ‘ill-advised moment’ to go public” and that the listing would not happen in 2026 (Ma, 2026b; Shontell, 2026); Elon Musk replied “Dario is right,” and the chief executive of Google DeepMind, Demis Hassabis, that “The direction is correct” (Mowshowitz, 2026). On Monday the 14th, the S&P 500 fell 0.5% and the Nasdaq 0.56%, but the fall was concentrated in chipmakers, with the Philadelphia semiconductor index down 6%, while Alphabet rose 2%, Microsoft 1.6% and Meta 1.4% (O’Donnell et al., 2026; Roytburg, 2026). One analyst put it bluntly: “They’ll just all stop building data centers and just digest what they have” (Roytburg, 2026).

“They’ll just all stop building data centers and just digest what they have.”Gil Luria, Wall Street analyst, quoted in Roytburg (2026)

That reaction is the signature of the mechanism. Those who sell fixed capital fell and those who stop buying it rose: the market read the brake as a coordinated restraint of investment, and a restraint of investment reduces what has to be financed. The postponement of OpenAI’s stock market debut, decided before the incident and narrated afterward as safety, is the mechanism with a date on it. That the narrative rests on a real incident makes it more effective, not less of a narrative. And Anthropic’s listing remains on track for October, at a valuation the Financial Times puts at $2 trillion (Stanciuc, 2026): whoever can show revenue does not brake its capitalization, and whoever cannot postpones it in the vocabulary of prudence. One contractual detail serves as a thermometer: $35 billion of Amazon’s investment in OpenAI was conditioned on the company either going public or achieving artificial general intelligence by the end of the year (Brandom, 2026); if that condition is renegotiated at no cost, the brake costs no one any capital.

It helps to name the mechanism by its classical category. The rate of profit is the profit obtained in a period divided by the capital advanced to obtain it, that is, how much each invested dollar yields, and it is the variable that governs accumulation: capital flows to where that rate rises and withdraws from where it falls. The artificial intelligence buildout inflates the denominator at an unprecedented speed, with hyperscaler investment that J.P. Morgan estimates at $697 billion in 2026 alone on equipment that depreciates within a few years (J.P. Morgan, 2026), while the numerator grows more slowly and, in the model layer, is squeezed by the Chinese competition described in the first section. That combination is a fall in the rate of profit on the capital invested in artificial intelligence, and when the rate falls, accumulation slows, whether by coordinated decision of the large capitals or by the withdrawal of credit. To stop enlarging the denominator and to exploit what is already installed is, literally, to “digest what they have.” The apocalypse that OpenAI, Anthropic and Musk are warning about is the form in which that defense of the rate of profit presents itself to the public.

Two objections deserve an answer. The first is that the brake might be sincere. It might be, and the thesis does not deny it: it claims that its economic function is the one described, whatever the conviction of the person announcing it. Other readers have offered kindred or harsher readings: Ben Thompson considers it “an unrealistic proposal that seems mostly geared to political control of AI” (Thompson, 2026), and Zvi Mowshowitz catalogs the readings of regulatory capture, stock market timing and computing limits, even though he dismisses them (Mowshowitz, 2026). The second objection is Chinese competition: a real brake would hand the lead to DeepSeek and Qwen, which, for the reasons given in the first section, have no reason to brake, and the coordination with China that the essay proposes is, under present conditions, unattainable. The answer is that, to be consistent with that competition, the brake has to be a paper brake, that is, external evaluators and alignment timelines without any measurable limit on the pace of training. A paper brake with maximum narrative effect is exactly what the thesis predicts, and it can be checked by reading the pact when it is signed.

5. What to Watch in the Coming Weeks

A thesis is worth what its risks of refutation are worth. Five observable facts in the coming weeks put it to the test.

Anthropic’s public prospectus. If it describes the pacing commitment as immaterial to its revenue, the brake does not touch the till.
Amazon’s conditional tranche in OpenAI. If the $35 billion is released or renegotiated without a listing or an artificial general intelligence, the brake costs no capital.
The capital expenditure guidance of Alphabet, Microsoft, Meta and Amazon in their third-quarter results at the end of October. A coordinated cut would confirm the reading of investment restraint.
The credit default swaps of CoreWeave and Oracle. If they keep widening while the hyperscalers rise, the devaluation begins on the leveraged periphery.
The content of the pact among laboratories. If it sets evaluators rather than verifiable limits on pace, the brake is form without substance.

6. Conclusion

What happened this weekend is best understood as an agreement among those who buy chips to run a little slower, expressed in the language of safety: it improves their cash position, it worsens the sales of those who sell chips, and the market read it that way in a single day. The part of the argument that the evidence confirms most strongly is that the discourse of danger sustains the valuation rather than sinking it, because it asserts that the goal is near. The part that remains open is the part that ought to remain open: if artificial general intelligence arrives within the timeframes its promoters announce, this article will have been wrong; if the brake is renewed while the promise recedes, it will not have been a brake for humanity but a defense of the rate of profit, that is, a way of managing capital in retreat.

· · ·

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September 14, 2026

14 September, 2026

NATIONAL SURVEY ON THE ASPECTS OF VIRTUALITY RELATED TO THE COVID-19 PANDEMIC (ENAVIRPA 2021)

Authorship and scope. ENAVIRPA 2021 was a collective project carried out by the students of Introduction to Sample Surveys. The authorship indicated here corresponds to the Economic Impact module, the analytical strategy developed around it, and the writing of this report. The remaining modules are retained as components of the collective project and are attributed to their respective authors.

PROLOGUE

The National Survey on Aspects of Virtuality Linked to the COVID-19 Pandemic (ENAVIRPA 2021) was a collective effort carried out by the students enrolled in Introduction to Sample Surveys at the University of Costa Rica. Fieldwork, questionnaire integration, and the different thematic modules belong to that collective project. This document focuses primarily on the Economic Impact (AE) module, whose design and analysis correspond to José Mauricio Gómez Julián, and makes complementary use of questions from other modules developed by other members of the course. The presence of those items in the analysis does not imply that their construction is attributed to the author of the economic module.

The exercise had a twofold purpose. First, to carry out a real telephone survey during the health crisis and describe material aspects associated with the pandemic. Second, to use the data as a pedagogical setting for studying what can —and cannot— be done with a sample survey when moving from description to inference, generalized linear models, and predictive evaluation.

The scope must be established from the outset. The 235 interviews constitute a real data set, but the number of observations alone is not enough to turn every sample statistic into an unbiased national estimate. Representativeness also depends on the selection mechanism, frame coverage, inclusion probabilities, nonresponse, weighting, measurement error, and questionnaire constraints. The design was carried out under the time and coordination limitations inherent in an academic exercise; moreover, the modules did not originate from a single common theoretical framework. The results are therefore interpreted primarily as results from this sample and as methodological evidence concerning the procedures employed.

It is also useful to distinguish a survey from a psychometric instrument. A survey is a procedure for collecting information and may contain demographic, labor, economic, attitudinal, or psychometric-scale variables. Not every questionnaire is, merely by virtue of being a questionnaire, a psychometric instrument. When a module seeks to measure a latent construct, additional problems of reliability, validity, and measurement error arise and require specific procedures. ENAVIRPA did not undertake a global psychometric validation of the questionnaire; this constitutes a limitation only for interpretations that seek to attribute the measurement of latent constructs to sets of items.

The criterion for incorporating complementary modules was to study the multidimensional distribution of social losses associated with the pandemic: work, income, access to virtual environments, food, hydration, physical activity, and other living conditions. The economy does not exist separately from these concrete determinations, and for that reason the AE module is also analyzed in relation to sociodemographic information and variables from other modules.

One final methodological clarification applies throughout the document: whenever the questionnaire, analytical database, and a published table appear to suggest different numerical codings, the unit of interpretation will always be the substantive category, not the number arbitrarily assigned to it. The ENAVIRPA2021csv.csv database stores AE3-AE9 as textual labels —“Aumentó”, “Disminuyó”, “Sin cambios”, “No es una fuente de ingresos/apoyo”, and “No responde”—. The reproducible analysis in this report uses those labels directly and does not depend on an undocumented numerical recoding.

I. OBJECTIVES

I.1. General objective

To plan, conduct, and analyze a sample survey associated with the health crisis caused by COVID-19, with particular emphasis on its economic impact.

I.2. Specific objectives

To describe the main results of the Economic Impact module.
To integrate, when conceptually appropriate, information from the sociodemographic, virtual education, technology, telework, and health-habits modules.
To examine what types of inference can be formulated from the collected data and under which restrictions.
To study generalized linear models, particularly probit and logit, as tools for analyzing binary variables.
To show the complementarity between statistical inference and supervised learning, distinguishing fitting, inference, and out-of-sample generalization.
To make explicit the restrictions imposed by the sampling design, nonresponse, absence of weighting, coding, skip patterns, and sample size.

II. DESCRIPTIVE METHODOLOGY

II.1. General summary

Element	Description
Study population	Persons aged 18 or older, cell-phone users, and residents of Costa Rica.
Operational frame	Active cellular-number banks constructed from operator prefixes, according to the information used by the course.
Data-collection mode	Computer-assisted telephone interviewing (CATI), using CSPro 7.4.
Selection procedure	Sampling from active cellular-number banks using the Waksberg procedure as a reference.
Originally planned sample size	300 interviews, 30 for each of the ten participating students.
Final sample size	235 completed interviews.
Fieldwork	June–July 2021, according to the course chronology.
Reference sampling error	±6.39 percentage points for a proportion under the idealized SRS calculation with `p=0.5`; this is not a measure of total survey error.
Weighting	A complete system of design weights, nonresponse adjustments, and calibration was not available.
Scope	Academic exercise with real data; descriptive results are presented primarily as sample results.

II.2. General description of the research process

Fieldwork was carried out by ten students from a graduate program at the University of Costa Rica. The group received training in CSPro 7.4; the modules prepared by the different participants were subsequently assembled, a pilot test of approximately three interviews per student was conducted, and fieldwork then proceeded. The collective target was 300 interviews. A total of 235 were completed.

For the fieldwork carried out by José Mauricio Gómez Julián, 332 calls and approximately 18 hours of telephone contact were recorded. The summary log reports 34 busy numbers, 94 unanswered calls, and 183 calls after inactive numbers were excluded. From those three figures one can compute 1-(128/183)=0.300546; however, that quantity should not automatically be called a “response rate.” The 55 residual cases are not necessarily completed interviews, and the denominator does not fully classify eligibility, refusal, contact, and unknown eligibility. A standardized response rate would require final dispositions for all cases under an explicit definition, such as those used by AAPOR.

Accordingly, this report retains the raw call counts but does not assign the 30.0546% figure the meaning of a standardized response rate. The verifiable fact for the author of the economic module is that the 30 interviews assigned to him were completed; the verifiable collective fact is that 235 of the 300 planned interviews were completed, or 78.3% of the production target. Neither ratio is, by itself, a probabilistic response rate.

II.2.1. Activity schedule

Schedule of activities in the ENAVIRPA 2021 process.

II.2.2. Telephone-number banks

Telephone-number banks assigned and used during fieldwork.

II.2.3. General chronology of fieldwork

Collective chronology of completed interviews.

II.2.4. Call log

Example of the call log used during fieldwork.

II.3. Presentation of the Economic Impact (AE) module

ECONOMIC IMPACT

AE1	Have you received, or did you receive, any financial benefit or support from the national or local government since the health emergency/quarantine caused by COVID-19 began? YES 2. NO (GO TO AE3) 3. DOES NOT KNOW/NO ANSWER (GO TO AE3)
AE2	Please indicate which of the following benefits/support you receive or received (Multiple response) Yes, food/groceries Yes, financial resources Yes, medical prevention supplies (gloves, masks, disinfectant, etc.) Yes, personal-hygiene supplies (sanitary pads, baby diapers, etc.) NO DOES NOT KNOW/NO ANSWER

AE1

Have you received, or did you receive, any financial benefit or support from the national or local government since the health emergency/quarantine caused by COVID-19 began?

YES 2. NO (GO TO AE3) 3. DOES NOT KNOW/NO ANSWER (GO TO AE3)

AE2

Please indicate which of the following benefits/support you receive or received (Multiple response)

Yes, food/groceries
Yes, financial resources

Yes, medical prevention supplies (gloves, masks, disinfectant, etc.)
1. Yes, personal-hygiene supplies (sanitary pads, baby diapers, etc.)
  1. NO
    1. DOES NOT KNOW/NO ANSWER

As a result of the health emergency/quarantine caused by COVID, please indicate how your personal resources have been affected: have they increased, decreased, or remained unchanged?		Increased	No change	Decreased	Not a source of income/support
AE3	Income or earnings from paid work	4	3	2	1
AE4	Money or goods received from family/friends living elsewhere in the country	4	3	2	1
AE5	Money or goods received from family/friends living in another country	4	3	2	1
AE6	Income from rental property, investments, or savings	4	3	2	1
AE7	Pensions and/or retirement benefits or other social payments	4	3	2	1
AE8	Government support	4	3	2	1
AE9	Support/donation from NGOs, civil-society organizations, foundations, or other nonprofit organizations	4	3	2	1

AE10	Besides you, has any other member of your household experienced a change in economic income since the health emergency/quarantine caused by COVID-19 began? No change in income 2. Income increased 3. Income decreased
AE11	Since the health emergency/quarantine caused by COVID-19 began, would you say that the total salary or income your family receives each month is enough or not enough to live on? (PROBE FOR THE BEST ANSWER) It is not enough; they have great difficulties It is not enough; they have difficulties It is just enough, without great difficulties It is enough; they can save DK/NA

AE10

Besides you, has any other member of your household experienced a change in economic income since the health emergency/quarantine caused by COVID-19 began?

No change in income 2. Income increased 3. Income decreased

AE11

Since the health emergency/quarantine caused by COVID-19 began, would you say that the total salary or income your family receives each month is enough or not enough to live on? (PROBE FOR THE BEST ANSWER)

It is not enough; they have great difficulties

It is not enough; they have difficulties

It is just enough, without great difficulties

It is enough; they can save

DK/NA

II.3.1. Note on the coding of AE3-AE9

In the data-collection instrument, AE3-AE9 appear with the codes 4 = Increased, 3 = No change, 2 = Decreased, and 1 = Not a source of income/support. A later table may order the same categories using a different numbering solely for presentation. A reader could quite reasonably believe, when comparing the two places, that the data were interpreted using incompatible codes. That need not be the case, because the number is only a label and can be recoded; what would be problematic is being unable to determine which substantive category each value represents during analysis.

The available analytical database removes that ambiguity because AE3-AE9 are stored directly as text. Consequently, every table and constructed variable here uses the semantic categories rather than a numerical sequence. Reproducibility therefore does not depend on reconstructing an intermediate recoding that was not documented.

II.4. Items used from other modules

II.4.1. SOCIODEMOGRAPHIC MODULE (SD)

SOCIODEMOGRAPHIC
SD1	INTERVIEWER: RECORD RESPONDENT’S SEX				MAN WOMAN
SD2	Please tell me, how old are you?		YEARS /____/____/
SD3	Are you Costa Rican? 1. YES 2. NO
SD5	What is this household’s total monthly income? (PROBE)	LESS THAN 200 THOUSAND COLONES 200 THOUSAND TO LESS THAN 400 THOUSAND 400 THOUSAND TO LESS THAN 600 THOUSAND 600 THOUSAND TO LESS THAN 800 THOUSAND		800 THOUSAND TO LESS THAN 1 MILLION 1 MILLION TO LESS THAN 1.5 MILLION 1.5 MILLION OR MORE DOES NOT KNOW / NO ANSWER

II.4.2. VIRTUAL EDUCATION MODULE (E)

CHALLENGES OF VIRTUAL EDUCATION

E6	What is the reason you have not participated in or enrolled in any virtual/online education or training program?
	Does not have enough time 2. Does not have enough money Is not currently interested in training 4. Lacks computer skills Has low motivation 6. Finds the virtual environment boring Poor personal time organization 8. Repeatedly postpones the decision to begin Finds the virtual environment uncomfortable 10. Does not want to start alone Online education is too expensive 88. Other: ________________________ DK/NA

Does not have enough time 2. Does not have enough money
1. Is not currently interested in training 4. Lacks computer skills
  1. Has low motivation 6. Finds the virtual environment boring
    1. Poor personal time organization 8. Repeatedly postpones the decision to begin
      1. Finds the virtual environment uncomfortable 10. Does not want to start alone
        
        Online education is too expensive 88. Other: ________________________
        
        DK/NA

II.4.3. TECHNOLOGY MODULE (TC)

TECHNOLOGY – EVERYONE

II.4.4. TELEWORK MODULE (TE)

TELEWORK – JORDAN/CARLOS
At present, during a typical week, what is your main work activity? SELECT THE OPTION THAT BEST APPLIES Works for an employer (whether a person, company, or household) Helps in a family business (unpaid) I have my own business and employ other people I have my own business without employing other people Does not work; is a pensioner/retiree IF RESPONSE IS 5 TO 99, GO TO V1 Does not work; homemaker Does not work; full-time student Does not work because of a physical/mental limitation that prevents working Does not work, is not looking for work, and is not available for work Does not work, but is looking for work Other DK/NA

At present, during a typical week, what is your main work activity? SELECT THE OPTION THAT BEST APPLIES

Works for an employer (whether a person, company, or household)
Helps in a family business (unpaid)
I have my own business and employ other people
I have my own business without employing other people
Does not work; is a pensioner/retiree

IF RESPONSE IS 5 TO 99, GO TO V1

Does not work; homemaker
Does not work; full-time student
Does not work because of a physical/mental limitation that prevents working
Does not work, is not looking for work, and is not available for work
Does not work, but is looking for work

Other

DK/NA

II.4.5. PHYSICAL AND MENTAL HEALTH MODULE (FN)

PHYSICAL-HEALTH AND NUTRITION HABITS – LUIS

Regarding mealtime habits, comparing current frequency with the year before the pandemic, would you say they increased, decreased, or remained the same?		Remained the same	Increased	Decreased	DK/NA
FN6	Breakfast	1	2	3	9
FN7	Lunch	1	2	3	9
FN8	Dinner	1	2	3	9

Regarding the foods I am going to mention, comparing current frequency with your habits in the year before the pandemic, would you say your consumption of … (READ FOOD) increased, decreased, or remained the same?		HAS NOT EATEN IT	YES			DK/NA
			Remained the same	Increased	Decreased
FN9	Fruits and vegetables	0	1	2	3	9
FN10	Legumes (beans, chickpeas, lentils)	0	1	2	3	9
FN11	Dairy products (milk, yogurt, cheese)	0	1	2	3	9
FN12	Starches (rice, pasta)	0	1	2	3	9
FN13	Meat or eggs (beef, pork, chicken, fish, processed meats)	0	1	2	3	9
FN14	Foods/sweets/beverages that are sources of sugar (cookies, sodas, Hi-C, Powerade, etc.; sweets, chocolates, candy, ice cream)	0	1	2	3	9
FN15	Fats and fast foods (butter, sour cream, dressings, mayonnaise; fried chicken, French fries, pizza, hamburgers, tacos, pastries, packaged snacks, etc.)	0	1	2	3	9

III. PRESENTATION OF DESCRIPTIVE RESULTS

III.1. AE module: economic impact

III.1.1. AE1. Government financial support

Response	n	%
Yes	56	23.8
No	179	76.2
Total	235	100.0

Fifty-six people, 23.8% of the sample, reported having received some benefit or financial support from the national or local government since the beginning of the emergency; 179, or 76.2%, answered that they had not.

III.1.2. AE2. Type of support received

AE2 was conditional on AE1: only the 56 people who answered “yes” to AE1 were supposed to answer it. In the database, the other 179 records correctly appear as missing values generated by the questionnaire skip. They are not “does not know/no answer”; they are not applicable. Since AE2 allowed multiple responses, each percentage must be calculated over the 56 eligible people, and the percentages need not add to 100%.

Type of support	Yes (n)	% of the 56	No (n)	% of the 56
Food/groceries	23	41.1	33	58.9
Financial resources	48	85.7	8	14.3
Medical prevention supplies	4	7.1	52	92.9
Personal-hygiene supplies	2	3.6	54	96.4

Thus, among those who received some form of support, 41.1% reported food and 85.7% financial resources. It is not correct to divide each selection by the total number of selections made and describe the resulting figure as a percentage of people.

III.1.3. AE3-AE9. Changes in sources of income and support

Item/source	Decreased	Increased	No change	Not a source	NR
AE3 — Income or earnings from paid work	87 (37.0%)	9 (3.8%)	90 (38.3%)	43 (18.3%)	6 (2.6%)
AE4 — Money/goods from family or friends within the country	26 (11.1%)	3 (1.3%)	33 (14.0%)	168 (71.5%)	5 (2.1%)
AE5 — Money/goods from family or friends in another country	11 (4.7%)	0 (0.0%)	17 (7.2%)	202 (86.0%)	5 (2.1%)
AE6 — Income from rental property, investments, or savings	13 (5.5%)	5 (2.1%)	21 (8.9%)	192 (81.7%)	4 (1.7%)
AE7 — Pensions, retirement benefits, or other social payments	7 (3.0%)	0 (0.0%)	39 (16.6%)	185 (78.7%)	4 (1.7%)
AE8 — Government support	8 (3.4%)	18 (7.7%)	22 (9.4%)	185 (78.7%)	2 (0.9%)
AE9 — Support from NGOs, civil society, foundations, or other nonprofit organizations	4 (1.7%)	2 (0.9%)	13 (5.5%)	214 (91.1%)	2 (0.9%)

Paid work was the source with the greatest presence and also the largest number of contractions: 87 people, 37.0% of the entire sample, reported a decrease; nine, 3.8%, an increase; and 90, 38.3%, no change. In AE6, 192 people, 81.7%, indicated that income from rental property, investments, or savings was not a source of income/support during the observed period.

Means of production as capital: scope of AE6

In the specific sense used in this analysis, the possession of means of production does not refer merely to ownership of instruments personally used by an independent producer, but to their possession as capital, that is, as property generating income separable from the owner’s direct labor.

The distinction is substantive. A self-employed worker may own tools, a vehicle, or a machine and reproduce materially through his or her own labor without the observed income thereby arising from capital in the sense defined here. AE6, by contrast, asks about income from rental property, investments, or savings and therefore serves as an empirical indicator of the presence of property or patrimonial income. Under this operationalization, the 192 cases stating that AE6 is not a source are classified as an observed absence of income associated with property-capital; the 39 cases reporting increase, decrease, or stability are classified as presence of that source; four cases remain indeterminate because of nonresponse.

This is not equivalent to a legal inspection of assets. A person may own an asset that temporarily produces no rent or whose profits are reinvested. That is an observational limitation, not an objection to the concept. The inference should therefore be formulated as the observed absence or presence of income indicative of property-capital within the categories captured by AE6.

TE1 separately retains information on labor status and business ownership. It is useful for distinguishing wage workers, employers, and independent producers, but it should not be mechanically confused with AE6: “own business without employees” and “receives income from property-capital” are related but nonidentical variables.

III.1.4. AE10. Income change among other household members

Response	n	%
No change in income	112	47.7
Income increased	8	3.4
Income decreased	104	44.3
No response	11	4.7

A total of 44.3% indicated that another member of the household had experienced a decrease in income; 47.7% reported no change and 3.4% reported an increase.

III.1.5. AE11. Sufficiency of family income

Response	n	%
Not enough; great difficulties	24	10.2
Not enough; difficulties	57	24.3
Just enough, without great difficulties	111	47.2
Enough; can save	40	17.0
DK/NA	3	1.3

A total of 34.5% reported that income was not enough to live on —10.2% with great difficulties and 24.3% with difficulties—; 47.2% said it was just enough, and 17.0% said it was enough and allowed them to save. AE11 directly measures subjective income sufficiency and is kept separate from a formal poverty line.

III.2. TE module: work and telework

III.2.1. TE1. Main activity

Activity	n	%
Works for an employer	105	44.7
Helps in an unpaid family business	5	2.1
Own business with employees	11	4.7
Own business without employees	33	14.0
Pensioner/retiree	12	5.1
Homemaker	40	17.0
Full-time student	17	7.2
Does not work because of a physical/mental limitation	1	0.4
Does not work, is not seeking work, and is unavailable	2	0.9
Does not work, is seeking work	8	3.4
Other	1	0.4

The sample contains 105 wage workers, 33 producers with their own business and no employees, and 11 business owners who employ other people. Only the last category directly identifies an employer relation involving the labor of others; the own-business-without-employees category describes independent production. This distinction prevents every ownership of instruments or of a small business from being transformed into capitalist ownership in the substantive sense used for AE6.

III.2.2. TE6. Change in working hours or remuneration

Response	n	% among 38 applicable cases
Fewer hours	2	5.3
More hours	16	42.1
Lower pay for the same amount of work	6	15.8
Lower pay for more work	7	18.4
Higher pay for more work	1	2.6
Higher pay for less work	1	2.6
Other	5	13.2
Not applicable because of questionnaire flow	197	—

TE6 produced 38 substantive responses and 197 cases that were not applicable because of the module flow. Among the 38 people exposed to the question, 16 —42.1%— reported working more hours. This percentage should not be projected onto all 235 interviews because the question was not asked of every participant.

III.3. FN module: physical-health and nutrition habits

III.3.1. FN2. Hydration

Response	n	%
Increased	87	37.0
Remained the same	133	56.6
Decreased	14	6.0
DK/NA	1	0.4

III.3.2. FN6-FN8. Frequency of meals

Meal	Same	Increased	Decreased
Breakfast	183 (77.9%)	24 (10.2%)	28 (11.9%)
Lunch	193 (82.1%)	19 (8.1%)	23 (9.8%)
Dinner	185 (78.7%)	18 (7.7%)	32 (13.6%)

III.3.3. FN9-FN15. Changes in food groups

Group	Does not consume	Same	Increased	Decreased	DK/NA
Fruits and vegetables	7	158	49	19	2
Legumes	10	179	31	14	1
Dairy products	12	166	26	30	1
Starches	2	176	28	28	1
Meat or eggs	1	170	38	25	1
Foods/beverages that are sources of sugar	42	109	27	56	1
Fats and fast foods	33	115	23	63	1

A change in frequency does not by itself have a nutritional sign. Reducing sugary drinks or fast food does not have the same meaning as reducing fruits, vegetables, or protein sources. Any index of “diet deterioration” must therefore rest on an explicit nutritional rule rather than on the simple operation “decreased = worsened.”

III.3.4. FN17. Hours of sleep during the pandemic

Change	n	%
Increased	35	14.9
Remained the same	134	57.0
Decreased	63	26.8
DK/NA	3	1.3

The data do not show that 57% slept more: 35 people —14.9%— reported an increase in hours of sleep, while 63 —26.8%— reported a decrease and 134 —57.0%— stability. The 57% figure corresponds to “remained the same.”

III.3.5. FN18. Physical activity

Change	n	%
Increased	61	26.0
Remained the same	110	46.8
Decreased	57	24.3
DK/NA	7	3.0

Physical activity increased for 61 people —26.0%— and decreased for 57 —24.3%—.

III.4. E module: virtual education

E6 has 161 applicable responses and 74 nonapplicable cases.

Reason	n	% of 161 applicable cases
Insufficient time	27	16.8
Insufficient money	11	6.8
Not interested in training	59	36.6
Lacks computer skills	8	5.0
Low motivation	6	3.7
Virtual environment is boring	1	0.6
Poor time organization	5	3.1
Repeatedly postpones starting	1	0.6
Virtual environment is uncomfortable	4	2.5
Does not want to start alone	1	0.6
Online education is too expensive	1	0.6
Other	22	13.7
DK/NA	15	9.3

Lack of money was reported by 11 people, 6.8% of applicable cases and 4.7% of the entire sample. Lack of time was reported by 27, 16.8% of applicable cases and 11.5% of the total. The denominator must be stated: using 235 describes prevalence in the full sample; using 161 describes the composition of those who actually received E6.

IV. GENERAL DESIGN OF THE INFERENTIAL METHODOLOGY

IV.1. Information of interest

The analytical design sought to extract or construct information on age; internet access; geographic location; nationality; barriers to virtual education; relation to social wealth; changes in diet, hydration, sleep, and physical activity; variation in income; sex; and household economic sufficiency. The intention was to study jointly dimensions that, although separated in the questionnaire for operational reasons, are materially related.

Transforming some responses into dichotomous variables is useful for binary models, but every dichotomization sacrifices information. Binary variables are therefore treated as analytical constructions rather than ontological substitutes for the continuous or multinomial phenomena from which they arise.

IV.2. Measurement mechanism and operationalization

Age: SD2, retained quantitatively whenever possible; for some models adult25 = 1 is also constructed if the person is 25 years of age or older.
Nationality: SD3; Costa Rican/non-Costa Rican is distinguished and DK/NA is treated as missing.
Sex: SD1.
Fixed internet: TC1.
Rural/urban location: the design intended to construct it from SD13 and INEC’s geographic classification. That classification was not completed during the exercise and therefore does not enter the models actually estimated.
Observed property-capital: AE6, according to the substantive definition given above. 1 if the source exists —increased, decreased, or remained unchanged—; 0 if the person states that it is not a source; NA if there is no response.
Occupational position/business ownership: TE1 is retained as a separate variable. Employer, independent producer, and wage worker are not collapsed into a single indicator of “capital.”
Income contraction: may be defined as the existence of at least one decrease in AE3-AE9 or a decrease in the income of another household member in AE10. AE11 should not be mechanically included in this indicator because it measures current sufficiency, not temporal change.
Monthly economic stress: AE11, retained ordinally or dichotomized only when the objective requires it.
Barriers to virtual education: E6 is analyzed among applicable cases. It cannot simply be converted into “access/no access” for all 235 people, because absence generated by a skip has a different meaning from a negative response.
Diet: FN6-FN15 require differentiated substantive rules by food type. A decrease is not necessarily negative, and an increase is not necessarily positive.
Hydration: FN2.
Sleep: FN17 and the level of hours in FN16, where relevant.
Physical activity: FN18.

IV.3. Dichotomous working variables

To reproduce the core models, the following variables are defined:

VD1: nationality, 1 = non-Costa Rican, 0 = Costa Rican.
VD2: observed presence of property-capital income according to AE6.
VD4: sex, 1 = woman, 0 = man.
VD7: 1 = age 25 or older, 0 = younger than 25.
VD8: income contraction under the explicit rule defined in IV.2.

The rurality variable remains a design proposal rather than an estimated variable. The barrier to virtual education is used descriptively unless the analytical universe is restricted to cases for which E6 applies.

IV.4. Proposed combined analyses

The first model studies the observed presence of property-capital income as a function of sex, age, and nationality:

\[ VD2_i = f(VD4_i,VD7_i,VD1_i). \]

Rurality is not included because it was not constructed reproducibly.

A second analysis could study income contraction in relation to sex, age, nationality, and barriers to virtuality, but only after precisely defining the E6 universe and the contraction variable. It is therefore retained as an analytical extension rather than being confused with the model actually estimated in Section IX.

IV.5. Correlation and causality

Correlation can describe covariation; it does not by itself transform an association into a causal mechanism. In social research, causal interpretation requires substantive theory, temporality, design, and identification assumptions that are not automatically contained in a coefficient.

Ritchey uses examples involving poverty, racial composition, and crime to show how an aggregate association may reflect institutional history, segregation, economic exclusion, and omitted variables. The methodological argument is not to replace one correlation with another monocausal explanation, but to remember that the coefficient does not contain the entire causal structure of the phenomenon.

The same caution applies to contemporary racial categories. Findings on the African origin of modern human lineages and later evidence of admixture with archaic populations do not turn social categories of race into discrete biological units capable of replacing historical and causal analysis. Regression quantifies relations under a model; causality requires an additional theory of how the phenomenon is produced.

V. FOUNDATIONS OF SAMPLE SURVEYS

V.1. Mathematical statistics, estimation, and surveys

Cochran emphasizes that survey theory historically adopted simple and robust procedures because a single survey may contain many attributes with different distributions. This does not imply a separation between sampling theory and regression: ratio and regression estimators use auxiliary information to improve precision.

It is useful, however, to distinguish two objects:

\[ \text{regression estimator in sampling} \neq \text{regression model fitted to survey variables}. \]

The former uses known auxiliary information to improve estimation of a population mean or total; the latter represents a conditional relationship among variables and may serve inferential or predictive purposes. The fact that both use the word “regression” does not make their objectives identical.

V.2. Sample size and the actual role of the Central Limit Theorem

A reader could quite reasonably believe that using p=0.5 in a sample-size formula is justified because, as n increases, the original responses “become normal,” eventually distribute themselves symmetrically, and therefore half must fall in each category. That is not the case.

For a Bernoulli variable X_i taking values 0 and 1, the sample proportion is:

\[ \hat p=\frac1n\sum_{i=1}^nX_i, \]

with:

\[ E(\hat p)=p, \qquad \operatorname{Var}(\hat p)=\frac{p(1-p)}n \]

under the ideal independent scheme. In sampling without replacement from a finite population there is also a finite-population correction:

\[ \operatorname{Var}(\hat p) =\frac{p(1-p)}n\frac{N-n}{N-1}. \]

The Central Limit Theorem enters elsewhere. Under appropriate conditions, it allows the sampling distribution of the standardized proportion to be approximated by a normal distribution:

\[ \frac{\hat p-p}{\sqrt{p(1-p)/n}} \approx N(0,1). \]

This is a statement about the distribution of a statistic across repeated samples, not a claim that the original distribution of responses itself becomes normal. Nor does it imply that p tends to 0.5.

The reason for using p=0.5 when the population proportion is unknown is algebraic. Bernoulli variance depends on:

\[ f(p)=p(1-p)=p-p^2. \]

Its derivative is:

\[ f'(p)=1-2p, \]

and the maximum occurs at:

\[ p=\frac12, \qquad p(1-p)=\frac14. \]

Therefore:

\[ p(1-p)\le 0.25. \]

Choosing p=0.5 maximizes the variance and yields the most conservative sample size for a prespecified margin of error. For a large population, or one of unknown size:

\[ n_0=\frac{z_{\alpha/2}^2p(1-p)}{d^2}. \]

With z=1.96, p=0.5, and d=0.0639:

\[ n_0\approx235.2. \]

This gives the scale of the 235 interviews. The result makes sense as an idealized calculation of sampling precision for a proportion under SRS. By itself, it does not demonstrate that the actual design has total error of ±6.39%.

If sample size approached the size of a finite population, the factor (N-n)/(N-1) would tend to zero. The reason is elementary: when nearly the entire population is observed, progressively less uncertainty remains because of sampling. The original responses do not undergo a “normalization.”

V.3. Nonresponse and the Birnbaum-Sirken formula

Birnbaum and Sirken, as discussed by Cochran, studied bias due to nonavailability. The expression discussed in the original paper can be written, using the notation employed there, as:

\[ n=\frac{t_\alpha^2}{4d(d-W_2)W_1}-1. \]

The condition built into the formula itself shows that there is no admissible n when W_2>d. If substitution of a nonresponse rate larger than the tolerated error changes the sign of the denominator and produces n<0, the result is not a physically negative magnitude that should be repaired by taking its absolute value. It indicates that the calculation has left the domain in which the formula can satisfy the stated guarantee.

An analogy with the absolute value of an economic elasticity does not repair this violation of conditions. The correct conclusion is simply: the formula does not provide an admissible solution for that combination of nonresponse and tolerated error.

This point is also consistent with Cochran’s warning: substantial amounts of nonresponse are not mechanically neutralized by increasing sample size. If nonrespondents systematically differ from respondents, bias may persist even with a large n.

V.4. Types of nonresponse and call dispositions

Cochran distinguishes problems such as noncoverage, temporary absence, inability to respond, and persistent refusal. In a modern telephone survey it is also useful to distinguish completed interview, partial interview, refusal, noncontact, nonresidential number, ineligible case, and unknown eligibility. Only with such final dispositions can a standardized response rate be constructed.

The ENAVIRPA fieldwork manual distinguished busy number, no answer, inactive, completed, pending, not completed, incomplete, refusal, business, and duplicate. The raw log is useful for documenting the process, but a standard rate requires the final disposition of each unit and explicit denominator rules.

V.5. Margin of error and total survey error

Under an ideal SRS, n=235, p=0.5, and 95% confidence give:

\[ 1.96\sqrt{\frac{0.25}{235}}\approx0.0639. \]

That is, ±6.39 percentage points. This number refers to idealized sampling error. It does not automatically incorporate coverage, differential nonresponse, measurement, processing, telephone selection, weighting, or the additional uncertainty of subsequent models.

V.6. Expansion factor and weighting

The base weight of a probabilistically selected unit is the inverse of its inclusion probability:

\[ w_i=\frac1{\pi_i}. \]

In a multistage design, if a geographic unit h, a household j within h, and a person i within the household have conditional probabilities f_h, f_(j|h), and f_(i|jh), the joint probability is:

\[ f_{ijh}=f_hf_{(j|h)}f_{(i|jh)}, \]

and the basic weight is:

\[ W_{ijh}=\frac1{f_{ijh}} =\frac1{f_h}\frac1{f_{(j|h)}}\frac1{f_{(i|jh)}}. \]

Similarity between some sample and population proportions does not prove that weighting is unnecessary. The decision depends on inclusion probabilities and subsequent adjustments for nonresponse and calibration. ENAVIRPA did not reconstruct a complete weighting system; therefore its estimates are not presented as though every observation had been demonstrated to carry the same population weight.

VI. GENERALIZED LINEAR MODELS

The generalized linear model is one of the clearest examples of how a statistical theory can develop by preserving an earlier structure while simultaneously extending its domain of application. The classical linear model does not disappear: it remains contained as a special case within a more general construction. The essential idea is to preserve a linear systematic component,

\[ \eta_i=x_i^T\beta, \]

while abandoning the requirement that every response variable be Gaussian and that its mean coincide directly with the linear predictor. The conditional mean,

\[ \mu_i=E(Y_i\mid x_i), \]

is connected to the predictor through a link function:

\[ g(\mu_i)=\eta_i. \]

In this way, Gaussian linear regression, logistic regression, Poisson count models, and certain gamma specifications can be studied within a common framework.

VI.1. Preliminary concepts: variables, scales, and predictors

A traditional classification of measurement scales distinguishes nominal, ordinal, interval, and ratio variables. The distinction between discrete and continuous, however, belongs to a different axis. Number of children, for example, is discrete but has a meaningful zero and interpretable ratios; it can therefore be regarded as a discrete variable on a ratio scale. It is not appropriate to turn “discrete/continuous” into additional levels of the nominal-ordinal-interval-ratio hierarchy.

The word covariate also deserves clarification. Some applied traditions used it in a restricted sense for continuous predictors. In modern regression theory it is clearer to use it broadly: a covariate or predictor may be continuous, discrete, binary, or categorical through appropriate coding. This is different from covariance, which denotes a measure of association between variables.

The word factor likewise depends on context. In experimental design it may denote a controlled or explanatory categorical variable; in factor analysis it denotes a latent construct. Neither meaning should be inferred solely from the name assigned to a column by a software package.

These clarifications matter because the model is not determined by the computer format of a variable, but by the mathematical and probabilistic structure of the phenomenon one wishes to represent.

VI.2. Genetic connection with linear regression

Gujarati and Porter recall that the term regression originated in Galton’s work on the heights of parents and children. Its contemporary meaning is much broader: the aim is to represent how the distribution of a response —and very often its conditional mean— changes when one or more predictors change.

In the simple linear model:

\[ E(Y_i\mid x_i)=\alpha+\beta x_i. \]

In matrix form:

\[ \mu=X\beta. \]

The fitted line is not merely a geometric object superimposed on points. Its coordinates represent observed magnitudes, and its parameters acquire statistical meaning from a probabilistic specification. This distinction allows us to separate geometric fitting from a statistical model, even when both share mathematical tools.

VI.2.1. Regression and causality

Regression does not automatically transform an association into causality. A coefficient can quantify a conditional relationship under a set of assumptions without identifying, by itself, the mechanism that produces it. Causal interpretation requires substantive theory, temporal structure, design, interventions, or additional identification assumptions.

Ritchey uses examples involving poverty, racial composition, and crime to show that an aggregate correlation may condense institutional history, segregation, economic exclusion, selection, and omitted variables. The methodological point is not to replace a bivariate association with another monocausal explanation; it is to deny that the coefficient contains, by itself, the totality of the causal process.

The same caution applies to contemporary racial categories. Genetic evidence concerning a recent African origin of modern human lineages and later admixture with archaic populations does not transform social racial categories into discrete biological units that can replace historical analysis. Regression measures relations under a representation; causal explanation requires reconstructing how those relations are produced.

VI.3. Regression and least squares are not synonyms

Three objects should be distinguished: model, estimation criterion, and numerical algorithm. In the classical Gaussian model:

\[ Y=X\beta+\varepsilon, \qquad \varepsilon\sim N(0,\sigma^2I), \]

maximum-likelihood estimation of $\beta$ coincides with minimizing:

\[ S_2(\beta)=\sum_{i=1}^n(y_i-x_i^T\beta)^2 =\|y-X\beta\|_2^2. \]

This is the source of the close historical relation between linear regression and least squares. But regression can use other criteria —absolute deviations, robust losses, penalization— and least squares can also be used in problems that do not seek to construct a complete probabilistic model. The coincidence in the Gaussian case does not justify identifying the two concepts.

VI.4. Geometric digression: norms, metrics, convexity, and discrepancy

Geometry helps explain why least squares has such a natural form. If

\[ e=y-X\beta, \]

OLS minimizes $\|e\|_2^2$. In a finite-dimensional vector space, for $p\ge1$,

\[ \|x\|_p=\left(\sum_i|x_i|^p\right)^{1/p} \]

is a norm and induces the metric

\[ d(x,y)=\|x-y\|_p. \]

The logical relation should not, however, be reversed: not every metric is induced by a norm, and a topology does not by itself determine a unique distance. Topological structure determines which sets are open and which notions of continuity are preserved; metric structure adds further information about distance.

Convexity should likewise not be confused with subadditivity and positive homogeneity. A function $F$ is convex when

\[ F(\lambda x+(1-\lambda)y) \le \lambda F(x)+(1-\lambda)F(y), \qquad 0\le\lambda\le1. \]

Norms are convex and satisfy additional properties, but those properties are not the general definition of convexity.

McCullagh and Nelder use discrepancy in a broader sense than metric distance. In least squares, discrepancy has an immediate Euclidean interpretation. In a GLM, by contrast, a central measure is the deviance:

\[ D=2\{\ell(\text{saturated model})-\ell(\text{fitted model})\}. \]

Deviance arises from log-likelihoods and need not satisfy the axioms of a metric. The geometric intuition can be preserved —searching within a class for a representation that disagrees as little as possible with the data under a criterion— but the criterion need not be an $L_p$ norm.

VI.5. Exponential families

Nelder and Wedderburn identified a class of distributions broad enough to contain several models of interest and structured enough to support a common theory. A standard form of an exponential dispersion family is:

\[ f(y_i;\theta_i,\phi) = \exp\left\{ \frac{y_i\theta_i-b(\theta_i)}{a(\phi)} +c(y_i,\phi) \right\}. \]

Here $\theta_i$ is the natural or canonical parameter and $\phi$ the dispersion parameter. Under the usual regularity conditions:

\[ E(Y_i\mid x_i)=b'(\theta_i)=\mu_i, \]

\[ \operatorname{Var}(Y_i\mid x_i)=a(\phi)b''(\theta_i). \]

In standard GLM notation this is summarized as:

\[ \operatorname{Var}(Y_i\mid x_i)=\phi V(\mu_i), \]

apart from known weight factors introduced by particular parameterizations. The function $V(\mu)$ is the variance function. This expression displays an essential generalization: in the homoskedastic Gaussian model variance may be constant, whereas in Bernoulli and Poisson models it depends systematically on the mean.

For example:

Family	Mean	Variance function $V(\mu)$
Gaussian	$\mu\in\mathbb R$	$1$
Bernoulli	$0<\mu<1$	$\mu(1-\mu)$
Poisson	$\mu>0$	$\mu$
Gamma	$\mu>0$	$\mu^2$

VI.6. The three components of a GLM

VI.6.1. Stochastic component

The response belongs to an exponential family. A standard GLM generally works with conditionally independent responses:

\[ Y_i\mid x_i\sim\mathcal F(\mu_i,\phi). \]

This does not require identical distributions in the sense of sharing the same mean, because

\[ \mu_i=g^{-1}(x_i^T\beta) \]

may vary from one observation to another.

That statement should be distinguished from a different historical observation by McCullagh and Nelder. In discussing general principles of discrepancy, the authors state that observations should be independent or at least interchangeable in some sense that permits impartial treatment. That formulation belongs to their conceptual framework and is not equivalent to requiring equality of all conditional means in a regression.

VI.6.2. Systematic component

The predictors generate:

\[ \eta_i = \beta_0+\beta_1x_{i1}+\cdots+\beta_px_{ip} = x_i^T\beta. \]

The relevant linearity is linearity in the parameters. Thus,

\[ \eta=\beta_0+\beta_1x+\beta_2x^2 \]

is nonlinear in $x$ but remains linear in $\beta_0,\beta_1,\beta_2$.

VI.6.3. Link function

The mean and the predictor are connected through:

\[ g(\mu_i)=\eta_i, \qquad \mu_i=g^{-1}(\eta_i). \]

The link makes it possible to model the mean on a scale appropriate to its domain. A probability must remain between 0 and 1; a Poisson mean must be positive. The transformation allows the linear predictor to operate on a more convenient scale without violating these restrictions.

VI.7. Identity link and canonical link

The identity link is:

\[ g(\mu)=\mu, \]

and only in that case:

\[ \mu=\eta. \]

A canonical link means something different: $g$ is chosen so that the linear predictor coincides with the natural parameter of the family:

\[ \boxed{\eta_i=\theta_i}. \]

Thus identity is the canonical link for the Gaussian, logit for the binomial, and log for the Poisson. For Gamma, the canonical link is inverse, with the sign convention depending on the parameterization.

Distribution	Canonical link
Gaussian	$g(\mu)=\mu$
Bernoulli/Binomial	$g(\mu)=\log[\mu/(1-\mu)]$
Poisson	$g(\mu)=\log\mu$
Gamma	$g(\mu)\propto1/\mu$

The canonical link has algebraic advantages, but it is not compulsory. A binomial response may use probit instead of logit if the theoretical or empirical construction justifies it.

VI.8. Maximum likelihood

Parameters are normally estimated by maximizing:

\[ \ell(\beta)=\sum_{i=1}^n\ell_i(\beta). \]

In general the score equations,

\[ U(\beta)=\frac{\partial\ell}{\partial\beta}=0, \]

do not admit a closed-form algebraic solution. This leads to iterative procedures.

There is a family of algorithms called iteratively reweighted least squares used for $L_p$ losses, approximations to least absolute deviations, and robust regression. In that setting weights such as

\[ w_i\propto|e_i|^{p-2} \]

may appear. The formulation discussed by Burrus belongs to this context. It is a legitimate IRLS algorithm, but it does not explain the IRLS ordinarily used to fit GLMs.

In GLMs, IRLS arises from applying Fisher scoring to the log-likelihood; with canonical links there is an especially close relationship with Newton-Raphson. The conceptual sequence is:

\[ \text{exponential family} \rightarrow \text{log-likelihood} \rightarrow \text{score} \rightarrow \text{Fisher information} \rightarrow \text{Fisher scoring} \rightarrow \text{IRLS}. \]

VI.9.1. Score and Fisher information

Let

\[ \eta_i=x_i^T\beta, \qquad \mu_i=g^{-1}(\eta_i). \]

The score can be written, omitting parameterization details that do not alter the central idea, as:

\[ U(\beta) = X^T \operatorname{diag}\left( \frac{d\mu_i/d\eta_i}{\phi V(\mu_i)} \right) (y-\mu). \]

The expected information takes the form:

\[ \mathcal I(\beta)=X^TWX, \]

with working weights:

\[ \boxed{ w_i = \frac1\phi \frac{(d\mu_i/d\eta_i)^2}{V(\mu_i)} }. \]

When $1/\phi$ is a common factor it can be omitted when updating $\beta$.

VI.9.2. Working response

At iteration $t$:

\[ \eta_i^{(t)}=x_i^T\beta^{(t)}, \qquad \mu_i^{(t)}=g^{-1}(\eta_i^{(t)}). \]

The first-order linear approximation leads to the working response:

\[ \boxed{ z_i^{(t)} = \eta_i^{(t)} + g'(\mu_i^{(t)})(y_i-\mu_i^{(t)}) }. \]

Since $g'(\mu)=d\eta/d\mu$, it may also be written:

\[ z_i^{(t)} = \eta_i^{(t)} + \frac{y_i-\mu_i^{(t)}}{d\mu_i/d\eta_i}. \]

VI.9.3. Weighted step

The update is obtained by solving:

\[ \boxed{ \beta^{(t+1)} = (X^TW^{(t)}X)^{-1}X^TW^{(t)}z^{(t)} }. \]

Hence the name of the algorithm: at each iteration a weighted least-squares problem is solved, but the weights and the working variable change because they depend on the estimated mean at the current iteration.

The procedure can be summarized as follows:

choose admissible initial values;
compute $\eta^{(t)}$;
obtain $\mu^{(t)}=g^{-1}(\eta^{(t)})$;
construct $W^{(t)}$;
construct $z^{(t)}$;
solve the weighted problem;
repeat until a convergence criterion is satisfied.

Three cases show the unity of the method:

Model	$d\mu/d\eta$	$V(\mu)$	Weight apart from scale
Gaussian + identity	$1$	$1$	constant
Poisson + log	$\mu$	$\mu$	$\mu$
Binomial + logit	$\mu(1-\mu)$	$\mu(1-\mu)$	$\mu(1-\mu)$

The Gaussian case reveals the connection with OLS: because the weights are constant, the iteration reduces to the ordinary linear problem.

VI.10. What does not explain IRLS convergence

It may be tempting to think that reweighting “brings” observations closer together, reduces predictor variance, and that the Central Limit Theorem makes the algorithm converge toward a normal distribution. That does not happen. The weights depend on conditional variance and the derivative of the link; they do not physically move the values of $X$ or force $\operatorname{Var}(X)\to0$.

The CLT appears elsewhere in the theory: under regularity conditions it contributes to the asymptotic normal approximation of maximum-likelihood estimators. That is an inferential property of sequences of estimators, not the numerical mechanism of IRLS.

Nor is it sufficient to write

\[ \beta^{(t+1)}=F(\beta^{(t)}) \]

to invoke Banach’s fixed-point theorem. That theorem requires proving that $F$ is a contraction on an appropriate complete space. The mere existence of an iteration does not prove contractivity. In practice, GLMs may exhibit convergence problems due to poor numerical conditioning, collinearity, separation, boundary solutions, or problematic specifications.

VI.11. Sufficient statistics under a canonical link

When

\[ \theta_i=\eta_i=x_i^T\beta, \]

the part of the log-likelihood jointly linking $y$ and $\beta$ contains:

\[ \frac1{a(\phi)}\beta^TX^Ty. \]

For fixed dispersion and under the usual conditions of an exponential family, the relevant dependence of the sample on $\beta$ appears through $X^Ty$. This explains the sufficient-statistic structure emphasized in classical GLM theory. In canonical log-linear models with factors, the score equations may additionally imply equality between certain observed and fitted margins; this is a property of those specifications, not a universal rule for every link.

The conceptual unity can finally be summarized as:

\[ \boxed{ \text{probability family} + \text{linear predictor} + \text{link function} }. \]

The generalization does not consist in erasing differences among distributions, but in identifying which structure can remain while their particular determinations change systematically.

VII. PROBIT AND LOGIT MODELS

Probit and logit models arise when the response is binary and the immediate object is to model a conditional probability:

\[ p_i=P(Y_i=1\mid x_i). \]

The problem with the linear probability model is that $x_i^T\beta$ can take any real value, whereas a probability must remain in $[0,1]$. Probit and logit links solve this incompatibility by transforming the probability onto an unrestricted scale.

VII.1. Probit model

The probit uses the standard normal distribution function:

\[ \Phi^{-1}(p_i)=x_i^T\beta, \]

so that:

\[ p_i=\Phi(x_i^T\beta). \]

The observed response remains Bernoulli; the normal distribution appears in the link function or, equivalently, in a latent-variable interpretation in which an unobserved normal variable crosses a threshold and generates the 0/1 response. It should therefore not be claimed that the observed binary variable is normal.

One may imagine:

\[ Y_i^*=x_i^T\beta+u_i, \qquad u_i\sim N(0,1), \]

\[ Y_i=1\quad\text{if}\quad Y_i^*>0, \]

which yields precisely $P(Y_i=1\mid x_i)=\Phi(x_i^T\beta)$.

VII.2. Logit model

The logit uses:

\[ \operatorname{logit}(p_i) = \log\left(\frac{p_i}{1-p_i}\right) = x_i^T\beta. \]

The odds satisfy:

\[ \frac{p}{1-p}\in(0,\infty), \]

whereas their logarithm ranges over the entire real line. Inverting the link:

\[ p_i = \frac{1}{1+\exp(-x_i^T\beta)}. \]

The general logistic function can be written:

\[ f(x)=\frac{L}{1+e^{-k(x-x_0)}}, \]

with asymptotes 0 and $L$ and central symmetry around $(x_0,L/2)$. In ordinary binary regression one takes $L=1$.

VII.2.1. Interpretation of coefficients

In logit, a coefficient $\beta_j$ represents the change in log-odds associated with a one-unit increase in $x_j$, holding the other covariates constant. Therefore:

\[ e^{\beta_j} \]

is the multiplicative odds ratio associated with that change. An odds ratio greater than 1 implies increased odds; less than 1, reduced odds; equal to 1, no multiplicative change.

This should not be confused with a constant marginal effect on probability. Since

\[ p(x)=\Lambda(x^T\beta), \]

the marginal effect of a continuous covariate is:

\[ \frac{\partial p}{\partial x_j} = \beta_jp(1-p), \]

so it depends on the individual’s position on the logistic curve.

VII.3. Probit and logit: similarities and differences

Both links are sigmoidal, monotonic, and yield admissible probabilities. Their results are often qualitatively similar in many applications; their coefficients, however, are not on the same scale. Logit has a particularly convenient interpretation through odds ratios; probit is naturally connected to a normal latent-variable formulation.

The choice should not be reduced to habit. It may depend on disciplinary tradition, interpretability, substantive assumptions, fit comparison, or computational convenience.

VII.4. Inference

Maximum likelihood yields $\hat\beta$. Under regularity conditions, the inverse information matrix approximates the covariance matrix of the estimators. An individual Wald test uses:

\[ z_j=\frac{\hat\beta_j}{SE(\hat\beta_j)}. \]

Likelihood-ratio tests compare nested models through:

\[ LR=2\{\ell(\hat\beta_1)-\ell(\hat\beta_0)\}. \]

The AIC criterion is:

\[ AIC=-2\ell(\hat\beta)+2k. \]

These quantities answer different questions. An individual p-value does not summarize predictive ability; an AIC does not prove causality; and a model that classifies reasonably well does not automatically make all of its coefficients well-identified parameters.

VII.5. Separation and unstable estimates

In binary regression, complete separation may occur when a combination of predictors perfectly classifies the cases, or quasi-complete separation when separation occurs except for some overlaps. Under those conditions, ordinary maximum likelihood may drive some coefficients toward very large magnitudes; Wald standard errors grow and conventional inference ceases to be reliable.

Thus enormous coefficients accompanied by enormous standard errors should not simply be interpreted as “very large but nonsignificant effects”: they may be a geometric symptom of separation or extremely sparse cells. Possible responses include reviewing coding, simplifying categories when substantively defensible, obtaining more observations, penalization, or bias-reduction methods such as Firth’s correction.

VIII. STATISTICAL LEARNING THEORY

VIII.1. General definition

Following James, Witten, Hastie, and Tibshirani, an elementary representation of supervised learning is:

\[ Y=f(X)+\varepsilon. \]

The problem is to estimate the unknown function $f$ from the data. In some problems the main interest is predicting new observations; in others, understanding which variables are associated with the response and how; in many cases both goals coexist.

The expression learning from data does not mean dispensing with theory. Every choice of variables, representation, functional class, loss, and validation procedure contains decisions about which regularities are regarded as possible and relevant.

Historically, statistics developed a tradition particularly concerned with probability models, estimation, uncertainty, and inference, whereas machine learning emerged under a strong influence from computer science, algorithms, and predictive performance. This difference helps explain cultural emphases, but it does not establish an essential boundary. Statistics predicts, and machine learning uses probability and inference.

Nor is it correct to say that machine-learning methods “make few assumptions” in an absolute sense. Many shift assumptions from an explicit parametric form toward an inductive bias: hypothesis class, architecture, loss function, regularization, invariances, and training rules.

A hypothesis space or class can be written:

\[ \mathcal H=\{h:X\rightarrow Y\}. \]

It is not the same object as the $H_0$ and $H_1$ hypotheses of Neyman-Pearson. There is a very broad logical kinship —delimiting possibilities— but the mathematical objects and inferential problems are different.

VIII.2. Learning paradigms

In an elementary classification:

supervised learning: pairs $(x_i,y_i)$ are observed and the aim is to learn a relation that predicts $y$ from $x$;
unsupervised learning: $x_i$ are observed without an explicit supervising response, and patterns or representations are sought;
reinforcement learning: an agent interacts with an environment through states, actions, and rewards.

Online learning belongs to another axis. It describes a procedure that updates the model as observations arrive. There can be supervised online learning, unsupervised online learning, and so forth.

In unsupervised learning, clustering is only one possibility. Others include dimensionality reduction, density estimation, anomaly detection, association rules, and representation learning.

VIII.3. Binary classification

If the positive class is the event of interest, there are four possibilities:

true positive (TP): actual positive, predicted positive;
false positive (FP): actual negative, predicted positive;
true negative (TN): actual negative, predicted negative;
false negative (FN): actual positive, predicted negative.

The confusion matrix organizes these cases:

	Predicted negative	Predicted positive
Actual negative	TN	FP
Actual positive	FN	TP

The matrix may be computed on training, validation, test, or external data. When it is computed on the same cases used to estimate the model, it describes in-sample performance.

VIII.4. Classification metrics

Accuracy is:

\[ Accuracy=\frac{TP+TN}{TP+TN+FP+FN}. \]

The error rate is its complement:

\[ Error=\frac{FP+FN}{N}=1-Accuracy. \]

Sensitivity or recall is:

\[ Sensitivity=\frac{TP}{TP+FN}, \]

and measures the proportion of actual positives detected.

Specificity is:

\[ Specificity=\frac{TN}{TN+FP}. \]

Classification precision, also called positive predictive value, is:

\[ Precision=\frac{TP}{TP+FP}. \]

Negative predictive value is:

\[ NPV=\frac{TN}{TN+FN}. \]

A terminological ambiguity should be avoided: classification precision is not metrological precision understood as low dispersion among repeated measurements. They are different concepts that share a word.

Accuracy can be misleading with imbalanced classes. If 90% of cases belong to the negative class, a classifier that always answers “negative” obtains 90% accuracy and yet has zero sensitivity. Every evaluation should therefore be compared with a baseline and with metrics consistent with the scientific cost of errors.

VIII.5. Training error and generalization error

For a classifier $\hat f$, the observed error on the training data is:

\[ \widehat{Err}_{train} = \frac1n\sum_{i=1}^nI(y_i\ne\hat y_i). \]

But the predictive objective is not to memorize already observed data. What matters is performance on a new observation $(X_0,Y_0)$:

\[ Err_{test} = E\left[I\{Y_0\ne\hat f(X_0)\}\right]. \]

The difference is fundamental. A highly flexible model may fit the training data extraordinarily well and fail on new observations. This is overfitting. Test sets, cross-validation, or other procedures are used to estimate generalization while preventing the algorithm from being evaluated only on cases it already used to learn.

VIII.6. Probabilities, thresholds, and classification

A probabilistic model such as logit produces $\hat p_i$, not logically necessary classes. To classify, a threshold $c$ must be chosen:

\[ \hat Y_i=I\{\hat p_i>c\}. \]

Changing $c$ changes the confusion matrix. Lowering the threshold usually detects more positives and raises sensitivity, but it may also produce more false positives and reduce specificity. Thus 0.5 is not a sacred value, and 0.25 cannot be adopted without explanation either. The choice should follow costs, prevalence, the study objective, or a rule defined on validation data, not the value that produces the most favorable appearance on the training sample.

ROC curves examine sensitivity against the false-positive rate across all thresholds. In problems with a rare positive class, precision-recall curves may be especially informative because they display directly the tradeoff between positive detection and the purity of positive predictions.

VIII.7. GLMs and supervised learning

GLMs are historically statistical models. Nevertheless, when fitted to $(X,Y)$ pairs to predict new responses they also function as supervised-learning procedures. Bayesian treatment is not what creates this status: a frequentist, Bayesian, or penalized GLM may be used predictively.

Logistic regression is the clearest example. It models:

\[ \log\left( \frac{P(Y=1\mid X)}{1-P(Y=1\mid X)} \right) =X\beta. \]

By its structure, it is a probabilistic regression model. But its probabilities can legitimately be converted into classes through a threshold. There is no contradiction:

\[ \boxed{ \text{logistic regression = probabilistic regression model} } \]

and simultaneously:

\[ \boxed{ \text{it can be used as a supervised classifier} }. \]

The historical origin of the method explains its construction, but does not exhaust all of its later scientific uses.

VIII.8. Inference and prediction are not two different models

Different questions can be asked of the same logit. From an inferential perspective: how much does the conditional association change with each covariate? How uncertain is the coefficient? Which model is compatible with the data? From a predictive perspective: what probability does it assign to an unseen case? Does it discriminate better than a baseline? What errors does it make out of sample?

These questions require different diagnostics, but they do not automatically turn logit into two algorithms. The same fit can be studied for both inferential and predictive properties. This is precisely why an individual coefficient may be imprecise while a combination of predictors still has some predictive utility; or, conversely, a model may contain statistically detectable associations yet discriminate new cases poorly.

The ENAVIRPA evaluation that follows deliberately keeps the two questions separate.

IX. REPRODUCIBLE INFERENTIAL RESULTS

IX.1. Analytical database and coding audit

The ENAVIRPA2021csv.csv database contains 235 rows and 161 variables. AE1 contains 56 “SÍ” and 179 “NO”. In AE2A-AE2D, the 179 cases excluded by the AE1 skip appear as NA, confirming that they must be classified as not applicable rather than DK/NA.

AE3-AE9 already appear in the database as textual labels. The visible discrepancy between the questionnaire codes 4-3-2-1 and the 1-2-3-4 order used in some presentation tables therefore ceases to affect reproducibility: categories are processed by name.

IX.2. Main variable: property-capital income

AE6 contains:

192 cases “No es una fuente de ingresos/apoyo”;
21 “Sin cambios”;
13 “Disminuyó”;
5 “Aumentó”;
4 “No responde”.

The main binary variable is therefore defined with 39 positive cases, 192 negative cases, and four missing cases. After also removing four missing ages and one DK/NA nationality, the complete model sample consists of 227 observations: 39 positive and 188 negative.

IX.3. Correlations among the model’s binary variables

With capital=1 for presence of the AE6 source, mujer=1, adulto25=1, and extranjero=1, the Pearson correlation matrix —equivalent to the phi coefficient when both variables are binary— is:

Variable	capital	adult25	female	foreign
capital	1.000	0.011	-0.070	-0.011
adult25	0.011	1.000	0.145	0.056
female	-0.070	0.145	1.000	0.009
foreign	-0.011	0.056	0.009	1.000

The associations are small in magnitude. This anticipates that a model using these three predictors will have little ability to explain variation in the AE6 indicator within this sample.

IX.4. Logit model

The estimated model is:

\[ \operatorname{logit}[P(VD2_i=1)] =\beta_0+\beta_1Woman_i+\beta_2Adult25_i+\beta_3Foreign_i. \]

Term	Coef.	SE	z	p	OR	95% CI for OR
Intercept	-1.5250	0.5578	-2.734	0.006	0.218	[0.073, 0.649]
Woman (1 = woman)	-0.3893	0.3563	-1.093	0.275	0.678	[0.337, 1.362]
Age ≥ 25	0.1895	0.5838	0.324	0.746	1.209	[0.385, 3.795]
Foreign (1 = non-Costa Rican)	-0.1149	0.6576	-0.175	0.861	0.891	[0.246, 3.235]

The model contains 227 observations, a log-likelihood of -103.51, a null log-likelihood of -104.13, McFadden’s pseudo-R² of 0.0060, and a joint likelihood-ratio test with p=0.740. None of the three predictors provides individual statistical evidence of an association different from zero at conventional levels in this specification.

Unlike the operationalization based on TE1 business ownership, the AE6 indicator does not produce explosive coefficients or standard errors measured in thousands of units. This difference confirms that construct specification and cell distribution can radically alter the numerical behavior of a model.

IX.5. Probit model

Term	Coef.	SE	z	p	95% CI for coef.
Intercept	-0.9099	0.3095	-2.940	0.003	[-1.517, -0.303]
Woman (1 = woman)	-0.2161	0.1988	-1.087	0.277	[-0.606, 0.174]
Age ≥ 25	0.0952	0.3200	0.297	0.766	[-0.532, 0.722]
Foreign (1 = non-Costa Rican)	-0.0653	0.3631	-0.180	0.857	[-0.777, 0.646]

The probit leads to the same substantive conclusion: sex, dichotomized age, and nationality together have very little explanatory ability for the observed property-capital income indicator in AE6. Logit and probit use different link scales, so their coefficients should not be compared one by one in magnitude.

IX.6. Rurality and other planned predictors

Rurality appeared in the conceptual design but was not constructed during the available period. It would not be methodologically correct to include it by pretending that a textual locality is already a rural/urban classification. INEC classifies geographic units using specific criteria; until that classification is applied reproducibly to SD13, the variable must remain outside the model.

Similarly, E6 does not provide a universal variable of “access to virtual education”: it was asked only of 161 cases according to questionnaire flow. It can be analyzed as a barrier among applicable cases or incorporated into a model restricted to that universe, but it cannot be imputed as “yes” or “no” to the 74 cases omitted by the skip.

X. PREDICTIVE EVALUATION: WHAT IT SHOWS AND WHAT IT DOES NOT SHOW

The same logistic regression can be studied from two perspectives. In inference, we ask what the coefficients and their uncertainty indicate about a modeled relationship. In prediction, we ask how well the estimated probabilities generalize to observations the algorithm did not see during fitting. These are not necessarily two different models: they can be two evaluations of the same model.

A reader might believe that it is enough to fit the logit to the 235 observations and then compute a confusion matrix on those same observations. That answers only: “how well does the model reproduce the data on which it was estimated?” It does not answer: “how well will it predict new cases?” This is the difference between in-sample performance and generalization.

Under the AE6 operationalization used here, the fitted logit probabilities range approximately from 0.129 to 0.208. With the threshold c=0.25, no observation exceeds the cutoff. The resulting classifier predicts “no property-capital income” for all 227 people:

Metric at threshold 0.25	Result
True positives	0
True negatives	188
False positives	0
False negatives	39
Accuracy	82.8%
Sensitivity	0.0%
Specificity	100.0%

The 82.8% accuracy is not a predictive victory: it coincides with the proportion of the majority class —82.8%— and is obtained at the cost of zero sensitivity. The model identifies no positive cases at this threshold.

This illustrates why accuracy should not be evaluated in isolation with imbalanced classes. An algorithm that always predicts the majority class can achieve high accuracy and nevertheless be useless for detecting the minority class.

The threshold is not a universal mathematical constant either. Choosing 0.25, 0.5, or another value changes the tradeoff between sensitivity and specificity. It should be defined by a cost function, a substantive need, or a validation procedure, not by post-hoc inspection of the same sample.

Finally, five-fold stratified cross-validation on these three predictors yields, as a reference, an approximate ROC AUC of 0.485 and an average precision close to 0.167. These results are compatible with virtually no discriminative ability in this specification. The pedagogical objective is not to make “classical statistics” or “machine learning” win, but to show that a predictive claim requires out-of-sample evaluation.

XI. LIMITATIONS

XI.1. Collective character of the design

The modules were designed by different members of the course and assembled for a common survey. There was no single theoretical framework determining every question from the outset. This limits later constructions that require combining items originally written for different objectives.

XI.2. Number of interviews

Three hundred interviews were planned and 235 were completed. The difference reduces precision and, above all, limits subgroup analyses and models with many categories. The issue is not that 235 is intrinsically “small”; adequacy depends on prevalence, the number of parameters, and cell structure.

XI.3. Design, coverage, and weighting

The telephone frame excludes persons outside operational coverage and may produce unequal inclusion. Complete design weights and nonresponse/calibration adjustments were not reconstructed. Results are therefore not presented as fully weighted national estimates.

XI.4. Nonresponse

The aggregate call-log data do not permit calculation of a complete AAPOR response rate. In addition, nonresponse and noncontact may be associated with socioeconomic conditions, so merely increasing n does not guarantee absence of bias.

XI.5. Rurality

The rural/urban variable was conceived but not constructed. INEC assigns degrees of urbanization to specific geographic units; the work needed to link SD13 to that classification was not completed.

Example of geographic-classification criteria used by INEC.

XI.6. Skip patterns and analysis universes

AE2, TE6, and E6 show why NA does not have a single meaning. It may mean not applicable because of a skip, nonresponse, or missing data. The correct denominator depends on the universe of each question. Treating every missing value as DK/NA changes percentages and may bias derived variables.

XI.7. Dichotomous variables

Dichotomizing age, diet, income, or economic stress simplifies modeling but sacrifices gradients. Whenever possible, original variables should be retained and dichotomies used only when justified by the model or substantive question.

XI.8. Capital as an observed variable

AE6 is an indicator of property-capital income, not an asset census. It can distinguish observed presence/absence of that source during the period, but it cannot legally verify every asset owned by a person or detect capital with no observed flow. This is an observational limitation.

XI.9. Income and poverty

SD5 records bands of gross monthly household income, not continuous equivalized disposable income. It therefore cannot directly implement the 60%-of-median-equivalized-disposable-income methodology used by the European/UNECE at-risk-of-poverty indicator.

XI.10. Causality

The study is observational and cross-sectional. Regression associations do not, by themselves, identify causal effects of sex, age, nationality, or other variables.

XI.11. Predictive evaluation

Training performance is not a measure of generalization. Any predictive claim should rely on independent test data or appropriate cross-validation and should consider class imbalance and threshold selection.

XII. CONCLUSIONS

Government support. Fifty-six people, 23.8% of the sample, reported having received support. Among them, 85.7% reported financial resources and 41.1% food. Because AE2 allows multiple responses, those percentages are calculated over eligible people and may sum to more than 100%.
Labor income. Income or earnings from paid work decreased for 37.0% of the full sample, increased for 3.8%, and remained unchanged for 38.3%.
Household income. In AE10, 44.3% reported a decrease in the income of another household member. In AE11, 34.5% indicated that family income was insufficient to live on and 47.2% that it was just enough; 17.0% stated that it was enough and allowed them to save.
Observed property-capital. For 81.7% of respondents, AE6 was not a source of income/support. In the specific sense used in this analysis, this indicates an observed absence of income derived from property-capital within the categories asked. It does not mean that an exhaustive legal inspection of assets was carried out.
Own business and independent producer. TE1 shows 33 people with an own business and no employees and 11 with an own business employing other people. Those positions should not be collapsed into one another or automatically confused with AE6.
Civil organizations. For 91.1% of the sample, support from NGOs, foundations, or nonprofit organizations was not a source of support in AE9.
International remittances. AE5 was an effective source for 28 people —17 with no change and 11 with a decrease— plus five nonresponses; 202 reported that it was not a source. Among the 28 people with a substantive response confirming the source, 39.3% reported a decrease.
Work. A total of 44.7% worked for an employer. TE6 was applicable to only 38 people; within that universe, 42.1% reported more working hours.
Health and habits. Hydration increased for 37.0%. Hours of sleep remained the same for 57.0%, decreased for 26.8%, and increased for 14.9%. Physical activity increased for 26.0% and decreased for 24.3%.
Virtual education. Among the 161 applicable E6 cases, 6.8% reported lack of money and 16.8% lack of time. Relative to all 235 interviews, these correspond to 4.7% and 11.5%, respectively.
Sample size. The value p=0.5 is used because it maximizes p(1-p) and yields a conservative calculation, not because the CLT forces responses to split 50/50. The ±6.39% is a reference sampling error under ideal SRS, not a guarantee of ENAVIRPA’s total error.
Nonresponse. The 30.0546% figure computed from busy and unanswered calls does not by itself constitute a standardized response rate. The Birnbaum-Sirken formula cannot be repaired by taking an absolute value when its conditions produce an inadmissible n.
Inference on capital. Under the AE6 operationalization, logit and probit provide no evidence of a substantive association between observed property-capital income and sex, age ≥25, or nationality in the analytical sample of 227 cases.
Prediction. With the same predictors, the model has little discriminative ability. A 0.25 threshold classifies every case as negative and obtains 82.8% accuracy only because that is the majority class. Out-of-sample evaluation is indispensable.
Poverty. No “UNECE poverty rate” is presented by taking 60% of mean gross household income. The relative standard requires the median of equivalized disposable income; SD5 does not contain that variable in the required form.
Scope. ENAVIRPA 2021 is especially valuable as an exercise in applied methodology: it shows that design, skips, coding, nonresponse, operationalization, and evaluation criteria determine what can be claimed from data. Technical correction does not diminish the empirical value of the survey; it more precisely delimits its informational content.

XIII. APPENDIX A. TOPOLOGICAL DISTANCES, METRIC DISTANCES, AND COLLECTIVE BEHAVIOR

In graph theory, a graph is a pair G=(V,E). An isomorphism between G_1 and G_2 is a bijection f:V(G_1)→V(G_2) preserving adjacency:

\[ \{u,v\}\in E(G_1) \Longleftrightarrow \{f(u),f(v)\}\in E(G_2). \]

Preservation of adjacency implies preservation of graph distance, defined as the minimum number of edges in a path between two vertices. It does not imply preservation of Euclidean distances in a particular drawing of the graph.

Two representations of isomorphic graphs.

Three ideas must be kept separate: graph isomorphism; topological invariance under homeomorphisms in general topology; and “topological distance” in studies of collective behavior. A homeomorphism does not in general preserve metric distances —that is the role of an isometry—. It is therefore incorrect to define a topological distance simply as a distance that remains invariant under perturbations.

In Ballerini and colleagues’ work on starling flocks, “topological distance” means neighbor rank: first neighbor, second, third, and so on. The empirical evidence indicated interaction with approximately six or seven neighbors, relatively independently of metric density. The conceptual distinction is:

metric rule: a physical radius is kept fixed and the number of neighbors inside it changes;
topological rule: approximately the number/rank of neighbors is kept fixed and the physical radius containing them changes.

The anisotropy of the angular distribution of neighbors made it possible to estimate up to what rank interaction persisted; for an isotropic distribution, the expected value of the relevant factor is 1/3. Later studies of jackdaws also showed that the regime can change with context, with topological interactions appearing during transit and metric interactions in other situations. The methodological lesson is material: the relevant structure must be determined empirically and may change with the concrete conditions of the system.

XIV. APPENDIX B. INCOME DISTRIBUTION AND THE POVERTY CRITERION

SD5 recorded bands of total monthly household income. The database contains:

Reported band	n	% of total
DK/NA	55	23.4
200 thousand to less than 400 thousand	53	22.6
Less than 200 thousand colones	42	17.9
400 thousand to less than 600 thousand	24	10.2
1 million to less than 1.5 million	18	7.7
600 thousand to less than 800 thousand	17	7.2
1.5 million or more	14	6.0
800 thousand to less than 1 million	12	5.1

The problem is not that poverty cannot be studied with a survey, but that the available variable must be matched with a definition corresponding to what it actually measures. A reader might believe that it is enough to take 60% of national mean income and compare that amount with gross monthly household income. That is not the relative criterion used by Eurostat/UNECE.

The usual at-risk-of-poverty measure uses 60% of the median equivalized disposable income. There are three important differences:

median, not mean;
disposable income, not unadjusted gross income;
equivalized, that is, adjusted for household size and composition.

SD5 records intervals of total household income and a nonresponse category. Without sufficient information to reconstruct individual equivalized disposable income and its population median, it is not methodologically legitimate to call a percentage obtained by applying 60% of mean gross income a “UNECE poverty rate.”

It is possible to describe how many households in the sample fall within each band, and AE11 may be used as an indicator of subjective income sufficiency. A future measurement compatible with the relative standard could also be designed by collecting disposable income, household composition, and the elements required for equivalization.

XV. R CODE FOR REPRODUCTION

The following code is designed to run with ENAVIRPA2021csv.csv in the same directory as the analysis file. The database is encoded in Windows-1252. Variable names and categorical values remain in Spanish because they are the literal values stored in the original database.

datos <- read.csv(
  "ENAVIRPA2021csv.csv",
  fileEncoding = "Windows-1252",
  stringsAsFactors = FALSE,
  check.names = FALSE
)

stopifnot(nrow(datos) == 235)
stopifnot(all(c("AE1","AE2A","AE2B","AE2C","AE2D",
                "AE3","AE4","AE5","AE6","AE7","AE8","AE9",
                "AE10","AE11") %in% names(datos)))

XV.1. Verification of AE1-AE2 and skip patterns

table(datos$AE1, useNA = "ifany")
table(datos$AE2A, useNA = "ifany")
table(datos$AE2B, useNA = "ifany")
table(datos$AE2C, useNA = "ifany")
table(datos$AE2D, useNA = "ifany")

# There must be 179 NAs in AE2A-AE2D because AE1=NO skips to AE3.
stopifnot(sum(is.na(datos$AE2A)) == 179)
stopifnot(sum(is.na(datos$AE2B)) == 179)
stopifnot(sum(is.na(datos$AE2C)) == 179)
stopifnot(sum(is.na(datos$AE2D)) == 179)

XV.2. AE3-AE9 tables using labels rather than numerical codes

vars_ae <- paste0("AE", 3:9)
lapply(datos[vars_ae], table, useNA = "ifany")

XV.3. Construction of the observed property-capital indicator

datos$capital <- ifelse(
  datos$AE6 == "No es una fuente de ingresos/apoyo", 0,
  ifelse(
    datos$AE6 %in% c("Disminuyó", "Aumentó", "Sin cambios"), 1, NA
  )
)

table(datos$capital, useNA = "ifany")

XV.4. Covariates and model sample

datos$edad <- suppressWarnings(as.numeric(datos$SD2))
datos$adulto25 <- ifelse(is.na(datos$edad), NA, as.integer(datos$edad >= 25))
datos$mujer <- ifelse(datos$SD1 == "Mujer", 1,
                      ifelse(datos$SD1 == "Hombre", 0, NA))
datos$extranjero <- ifelse(datos$SD3 == "No", 1,
                           ifelse(datos$SD3 == "Sí", 0, NA))

modelo <- na.omit(datos[c("capital","mujer","adulto25","extranjero")])
stopifnot(nrow(modelo) == 227)
table(modelo$capital)
cor(modelo)

XV.5. Logit and probit

fit_logit <- glm(
  capital ~ mujer + adulto25 + extranjero,
  family = binomial(link = "logit"),
  data = modelo
)

fit_probit <- glm(
  capital ~ mujer + adulto25 + extranjero,
  family = binomial(link = "probit"),
  data = modelo
)

summary(fit_logit)
summary(fit_probit)

# Odds ratios and approximate Wald confidence intervals for the logit
b <- coef(fit_logit)
se <- sqrt(diag(vcov(fit_logit)))
OR <- exp(b)
IC_OR <- cbind(exp(b - 1.96 * se), exp(b + 1.96 * se))
cbind(Coeficiente = b, EE = se, OR = OR, IC_OR)

XV.6. In-sample evaluation at threshold 0.25

p_hat <- predict(fit_logit, type = "response")
y_hat <- ifelse(p_hat > 0.25, 1, 0)
y <- modelo$capital

TP <- sum(y == 1 & y_hat == 1)
TN <- sum(y == 0 & y_hat == 0)
FP <- sum(y == 0 & y_hat == 1)
FN <- sum(y == 1 & y_hat == 0)

accuracy <- (TP + TN) / length(y)
sensitivity <- TP / (TP + FN)
specificity <- TN / (TN + FP)
precision <- if ((TP + FP) == 0) NA else TP / (TP + FP)
npv <- TN / (TN + FN)

c(TP=TP,TN=TN,FP=FP,FN=FN,
  accuracy=accuracy,sensitivity=sensitivity,
  specificity=specificity,precision=precision,npv=npv)

# Majority-class baseline
mean(y == 0)
range(p_hat)

XV.7. Five-fold stratified cross-validation

set.seed(2021)
K <- 5
fold <- rep(NA_integer_, nrow(modelo))

# Simple stratified assignment by class
for (cl in sort(unique(modelo$capital))) {
  idx <- which(modelo$capital == cl)
  fold[idx] <- sample(rep(1:K, length.out = length(idx)))
}

p_cv <- rep(NA_real_, nrow(modelo))
for (k in 1:K) {
  train <- modelo[fold != k, ]
  test  <- modelo[fold == k, ]
  fit_k <- glm(capital ~ mujer + adulto25 + extranjero,
               family = binomial(link="logit"), data=train)
  p_cv[fold == k] <- predict(fit_k, newdata=test, type="response")
}

# AUC function without additional packages (Mann-Whitney / ranks)
auc_rank <- function(y, p) {
  pos <- y == 1; neg <- y == 0
  r <- rank(p, ties.method="average")
  (sum(r[pos]) - sum(seq_len(sum(pos)))) / (sum(pos) * sum(neg))
}

auc_cv <- auc_rank(modelo$capital, p_cv)
auc_cv

# Confusion matrix at the same threshold, only as an example.
y_cv <- ifelse(p_cv > 0.25, 1, 0)
table(Real=modelo$capital, Predicho=y_cv)

XV.8. Income contraction without mixing it with AE11 sufficiency

vars_ingreso <- paste0("AE", 3:9)
disminucion_fuente <- apply(
  datos[vars_ingreso], 1,
  function(x) any(x == "Disminuyó", na.rm = TRUE)
)

datos$contraccion_ingreso <- as.integer(
  disminucion_fuente | datos$AE10 == "Ingreso disminuido"
)

# AE11 is kept separate as an indicator of sufficiency/economic stress.
table(datos$contraccion_ingreso, useNA="ifany")
table(datos$AE11, useNA="ifany")

XVI. REFERENCES

Aldrich, J. H., & Nelson, F. D. (1984). Linear Probability, Logit, and Probit Models. Sage.

American Association for Public Opinion Research. (2016). Standard Definitions: Final Dispositions of Case Codes and Outcome Rates for Surveys (9th ed.). AAPOR.

Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, I., et al. (2008). Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proceedings of the National Academy of Sciences, 105(4), 1232-1237.

Birnbaum, Z. W., & Sirken, M. G. (1950). Bias due to non-availability in sampling surveys. Journal of the American Statistical Association, 45(249), 98-111.

Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer.

Cann, R. L., Stoneking, M., & Wilson, A. C. (1987). Mitochondrial DNA and human evolution. Nature, 325, 31-36.

Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Wiley.

Departamento Administrativo Nacional de Estadística. (2003). Methodological documentation on expansion factors and sampling design. DANE.

Greene, W. H. (2012). Econometric Analysis (7th ed.). Pearson.

Gujarati, D. N., & Porter, D. C. (2009). Basic Econometrics (5th ed.). McGraw-Hill.

Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction (2nd ed.). Springer.

Instituto Nacional de Estadística y Censos de Costa Rica. (2016). Manual de Clasificación Geográfica con Fines Estadísticos de Costa Rica. INEC.

Instituto Nacional de Estadística y Censos de Costa Rica. (2019). ENIGH 2018: Cuadros sobre ingresos de los hogares. INEC.

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An Introduction to Statistical Learning with Applications in R. Springer.

Kolmogorov, A. N., & Fomin, S. V. (1975). Introductory Real Analysis. Dover.

Ling, H., Mclvor, G. E., van der Vaart, K., Vaughan, R. T., Thornton, A., & Ouellette, N. T. (2019). Local interactions and their group-level consequences in flocking jackdaws. Nature Communications, 10, 5174.

Lohr, S. L. (2019). Sampling: Design and Analysis (2nd ed.). CRC Press.

McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models (2nd ed.). Chapman & Hall.

Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized Linear Models. Journal of the Royal Statistical Society, Series A, 135(3), 370-384.

Reich, D., Green, R. E., Kircher, M., Krause, J., Patterson, N., Durand, E. Y., et al. (2010). Genetic history of an archaic hominin group from Denisova Cave in Siberia. Nature, 468, 1053-1060.

Ritchey, F. J. (2008). The Statistical Imagination: Elementary Statistics for the Social Sciences (2nd ed.). McGraw-Hill.

United Nations Economic Commission for Europe. (2017). Guide on Poverty Measurement. United Nations.

Vapnik, V. N. (1998). Statistical Learning Theory. Wiley.

Waksberg, J. (1978). Sampling methods for random digit dialing. Journal of the American Statistical Association, 73(361), 40-46.

Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). MIT Press.

Family	Mean	Variance function \(V(\mu)\)
Gaussian	\(\mu\in\mathbb R\)	\(1\)
Bernoulli	\(0<\mu<1\)	\(\mu(1-\mu)\)
Poisson	\(\mu>0\)	\(\mu\)
Gamma	\(\mu>0\)	\(\mu^2\)

Model	\(d\mu/d\eta\)	\(V(\mu)\)	Weight apart from scale
Gaussian + identity	\(1\)	\(1\)	constant
Poisson + log	\(\mu\)	\(\mu\)	\(\mu\)
Binomial + logit	\(\mu(1-\mu)\)	\(\mu(1-\mu)\)	\(\mu(1-\mu)\)

Distribution	Canonical link
Gaussian	\(g(\mu)=\mu\)
Bernoulli/Binomial	\(g(\mu)=\log[\mu/(1-\mu)]\)
Poisson	\(g(\mu)=\log\mu\)
Gamma	\(g(\mu)\propto1/\mu\)

Category: Universal History

GENERALITIES AND HISTORICAL ORIGINS OF THE CHI-SQUARED DISTRIBUTION

1. Two frequency problems

2. Pearson and the historical problem

3. Constructing a measure of discrepancy

4. Variables, dimensions, and covariances

5. The quadratic form and the inverse

6. From historical to matrix notation

7. Ellipsoids, spheres, and topology

8. Radial integration and solid angles

9. The chi-squared law and its properties

10. Why it works for counts

11. The original examples, worked through

11.1. Tomatoes: a 3:1 ratio

11.2. Hospital cases: a 1:1 ratio

12. Applications, conditions, and limitations

Goodness of fit

Independence and homogeneity

Variance of a normal population

Diagnostics before calculation

13. Reproduction and exploration in R

13.1. Manual calculation and the built-in test

13.2. An illustrative contingency table

13.3. From the normal distribution to χ²: checking the geometry

13.4. From counts to the asymptotic limit

14. Synthesis and glossary

15. References and the original paper

Compártelo:

ON THE CREATION AND DESTRUCTION OF VALUE IN CAPITALIST POLITICAL ECONOMY SYSTEMS IN PARTICULAR AND IN ECONOMIC SYSTEMS IN GENERAL (DRAFT)

Labour and the transformation of its own conditions

What Is Created, Transferred and Lost

A minimal conceptual accounting framework

I. Historical Background

II. Factor Contributions to the Production Process

II.I. First scenario: no productive factor creates value

II.II. Second scenario: only the means of production create value

Competition, gravitation and persistent differences

II.II.1. Epistemological considerations on simultaneous equations

From historical development to model specification

II.II.2. Vertical integration: interpretation and calculation

II.III. Labour and the means of production: different functions

III. The Destruction of Value and Epistemological Equivalences

III.I. Destruction and devaluation

III.II. Value, energy and information: equivalences and open questions

IV. Methodological Appendix: Intersectoral Relations in the United States

IV.II. Formalisation and reproducibility supplement

Variables, table orientation and units

Proportions and magnitudes

Checks for an empirical application

Complete R script

What this supplement makes reproducible

Value Creation, Technical Change and the Limits of Inference

V. References of the original study

Additional sources for the technical clarifications

Endnotes

Editorial principles

Compártelo:

ON THE ESSENCE, CONTENT, FORM, PHENOMENON, AND APPEARANCE OF THE PROCESSES OF REALITY

Knowing a process requires reconstructing its relationships

Five categories, one articulated process

I. Essence

II. Phenomenon

One essence, multiple mathematical determinations

Structural dimensions and criteria of preservation

Why a metric difference can be essential

Symmetries, history, and the process’s lawfulness

Mathematical invariance and explanatory relevance

III. The dialectical relationship between essence and phenomenon

IV. Form

V. Content

VI. The dialectic in the interrelationship between form and content

Internal organization, expression, and structure

VII. Connections and differences between the categorial pairs

VIII. Differentiation and generalization of the categories

VIII.I. Correspondence, complementarity, and essential content

Making the proposal operational

VIII.II. Earth’s magnetic field and the climate system

Value, prices, and levels of mediation

VIII.III. Second case study: dynamo theory

Reference frames and inertial forces