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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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:

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

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

1. 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.
2. 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.
3. 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).
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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

1. Aravindh, M., Venkatesan, A., and Lakshmanan, M. (2018). Strange nonchaotic attractors for computation. Physical Review E, 97, 052212. DOI.
2. 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.
3. Bjorvand, A. T. (1995). A New Approach to Intelligent Systems Theory. Norwegian Institute of Technology, University of Trondheim. Project report, April 28. Document.
4. Elert, G. (n.d.). Flow regimes. The Physics Hypertextbook. The original investigation records consultation in 2020. Resource.
5. 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.
6. Halperin, B. I. (2019). Theory of dynamic critical phenomena. Physics Today, 72(2), 42–43. DOI.
7. Jaynes, E. T. (2003). Probability Theory: The Logic of Science. Cambridge University Press.
8. Kessler, D. P., and Greenkorn, R. A. (1999). Momentum, Heat, and Mass Transfer Fundamentals. Marcel Dekker.
9. Kilifarska, N. A., Bakmutov, V. G., and Melnyk, G. V. (2020). The Hidden Link Between Earth’s Magnetic Field and Climate. Elsevier.
10. 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.
11. 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.
12. Lesne, A. (1998). Renormalization Methods: Critical Phenomena, Chaos, Fractal Structures. Translated by Leila Schneps. Wiley. Contents published by the author.
13. Lesne, A., and Laguës, M. (2012). Scale Invariance: From Phase Transitions to Turbulence. Springer.
14. 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.
15. 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.
16. Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130–141. DOI.
17. 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.
18. 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.
19. McCullagh, P., and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman & Hall.
20. 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.
21. 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.
22. Oestreicher, C. (2007). A history of chaos theory. Dialogues in Clinical Neuroscience, 9(3), 279–289. Full text.
23. 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.
24. 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.
25. 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.
26. 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.
27. 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.
28. 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.
29. 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.
30. 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.
31. 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.
32. 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.
33. 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

34. Birkhoff, G. D. (1931). Proof of the ergodic theorem. Proceedings of the National Academy of Sciences, 17(12), 656–660. DOI.
35. Cohen, P. J. (1963). The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences, 50(6), 1143–1148. DOI.
36. Royal Swedish Academy of Sciences (2015). The Crafoord Prize in Biosciences 2015. Institutional announcement.
37. 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.
38. Gigerenzer, G. (2004). Mindless statistics. The Journal of Socio-Economics, 33, 587–606. DOI.
39. Grebogi, C., Ott, E., Pelikan, S., and Yorke, J. A. (1984). Strange attractors that are not chaotic. Physica D, 13, 261–268. DOI.
40. 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.
41. Lenin, V. I. (1909). Materialismo y empiriocriticismo [Materialism and Empirio-Criticism]. Work written in 1908; its composition is distinguished from first publication.
42. Levins, R., and Lewontin, R. C. (1985). The Dialectical Biologist. Harvard University Press.
43. Levins, R., and Lewontin, R. C. (2007). Biology Under the Influence: Dialectical Essays on Ecology, Agriculture, and Health. Monthly Review Press.
44. Lewontin, R. C., and Levins, R. (2002). Stephen Jay Gould—What Does It Mean to Be a Radical? Monthly Review, 54(6), November. Essay mentioned in the notes of the original paper. Journal issue.
45. Pawlak, Z. (1982). Rough sets. International Journal of Computer & Information Sciences, 11, 341–356. DOI.
46. Perezgonzalez, J. D. (2015). Fisher, Neyman–Pearson or NHST? A tutorial for teaching data testing. Frontiers in Psychology, 6, 223. DOI.
47. Wilson, K. G. (1983). The renormalization group and critical phenomena. Reviews of Modern Physics, 55, 583–600. DOI. Account related to his 1982 Nobel Lecture.
48. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI.
49. 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.