1. Existence as a material totality

Material existence exhibits forms of organization in which components acquire properties and possibilities for transformation through their relations. A molecule, an organism, or a social formation is a determinate reality: it has a composition, an internal organization, conditions of reproduction, and a history. Understanding such realities requires investigating both their parts and the mediations that articulate them. The analysis of components and the reconstruction of the totality are complementary moments of knowledge.

We shall call an objective organic structure an organization of material relations through which components reciprocally condition one another and participate in the reproduction or transformation of a delimited totality. Structure designates the articulation of relations and constraints; organic indicates constitutive dependence among differentiated moments; objective expresses the fact that this organization and its effects exist independently of whether a subject knows them adequately.

In social systems, this objectivity includes relations produced by human activity. Money, property, and the state depend on historical practices and institutions, yet their effects exceed each individual’s understanding and will. Independence from an individual consciousness is compatible with a relation’s social genesis.

Organic organization also encompasses conflicts, asymmetries, antagonisms, and failures of reproduction. An organization may sustain itself through relations that undermine its own conditions of existence. Its investigation must therefore address relative stability, the accumulation of tensions, and the possibility of qualitative transformation. The concept is used here in a relational and historical sense, without attributing a conscious purpose or a biological organization to every system.

Selecting a totality of reference is a methodological step that requires justification. Investigating a cell, a watershed, or a circuit of capital reproduction involves delimiting components, exchanges, scales, and relevant variables. This delimitation is revisable: its consequences must be confronted with the object’s behavior. An open system can be a legitimate unit of analysis, provided that the decisive exchanges with its environment are represented.

2. Marxist foundations: relations, genesis, and reproduction

In the 1857 introduction to the Grundrisse, Marx examines production as the activity of socially determined individuals. General abstractions identify shared features, but knowledge of a historical organization requires reconstructing its specific determinations. The concrete totality reproduced in thought results from articulating those determinations; its conceptual representation must be distinguished from the real process to which it refers (Marx, 1973a).

In the chapter on capital, Marx characterizes the developed bourgeois system as an organization whose relations mutually presuppose one another. Its constitution as a totality involves producing or subordinating the elements it needs to reproduce itself (Marx, 1973b). Two problems are thereby articulated: how an organization arises and how it operates once constituted. Its present mechanisms of reproduction can transform the historical conditions that initially made the system possible.

This relationship between genesis and functioning makes it possible to understand that a structure has a history even when it reproduces itself regularly. Present regularity may depend on institutions, infrastructures, habits, and distributions of power produced through earlier processes. Historical dependence must be investigated through identifiable mechanisms: material accumulation, learning, irreversible investment, institutional change, or modifications of the conditions of reproduction.

In his notes on the dialectics of nature, Engels places reciprocal action at the center of the examination of matter in motion and relates causal analysis to the delimitation of processes within a broader interconnection (Engels, 1925/1974). The later systematizations of Rosental and Iudin (1965) and Frolov (1984) condense this orientation into the principle of universal interconnection. Their formulations belong to specific developments within the Marxist tradition and should be read alongside the primary works.

The philosophical application that follows develops this orientation through contemporary mathematical tools. The use of manifolds, operators, or probabilistic models requires definitions and proofs appropriate to those tools. Their value for dialectical materialism depends on whether they make real determinations knowable and explain concrete transformations.

3. Universal interconnection and reciprocal action

Universal interconnection expresses an ontological orientation: material realities exist through conditions, relations, and processes that connect them with other realities. Its methodological consequence is to investigate the mediations relevant to each problem. A relation may be direct or indirect, persistent or transient, constitutive or circumstantial; its importance may also change with the scale or historical regime.

Several senses of relation should be distinguished. A causal interaction involves mechanisms of production or modification; a structural dependence determines which configurations are possible; a statistical correlation describes an association between variables; a formal correspondence relates mathematical representations. Confusing these senses produces apparent explanations: an association may arise from a common cause, and an algebraic resemblance may occur in materially different processes.

Reciprocal action requires identifying how effects are linked and with what delays. It can be represented, for example, by a coupled system:

Here, x and y represent states or sets of variables; u, external conditions; and θ, model parameters. Reciprocity is made concrete in the dependencies of F and G. If both functions are differentiable, cross-derivatives help investigate local influence, although a causal dependence also requires justification through the mechanism and research design.

An interaction may be reciprocal and profoundly asymmetric. Nor must it be instantaneous: delays, memory, and accumulation modify its effects. In political economy, for example, productive organization conditions state power, while legal and political decisions alter the conditions of accumulation. An explanation must specify actors, resources, institutions, and sequences, rather than stopping at an assertion of mutual influence.

The objectivity of laws is compatible with contingency and probabilistic models. A regularity can operate through distributions, tendencies, or constraints without uniquely determining every event. Investigating necessity and contingency requires specifying the conditions under which a regularity holds and the processes that may modify those conditions.

4. Essence as the unity of mathematical layers

Essence is understood here as the articulated unity of the structural determinations that constitute a system, make its identity intelligible, and condition its transformation. Its mathematical formalization unfolds through layers: each specifies which relations are treated as constitutive, which variations are admissible, and under which correspondences the structure is preserved. This development follows the framework of Geometry, History, and Lawfulness (Gómez Julián, 2026), specifying the conditions of preservation.

The unity of essence admits multiple determinations. A property that may vary when studying one layer can be decisive in another. The length of a circle is irrelevant to establishing its topological equivalence to another circle, but it is constitutive of its intrinsic metric geometry. The level of abstraction depends on the question and must preserve the determinations relevant to answering it.

The layers are enriched by adding structure. Choosing a topology on a set and choosing a metric on a manifold are acts of specification. By contrast, a bijection, a homeomorphism, or an isometry compares objects that are already structured. This distinction prevents the construction of an object from being confused with the criterion of equivalence between objects.

For a fixed smooth Riemannian manifold, the relevant automorphism groups satisfy:

A Riemannian isometry is a diffeomorphism that preserves the metric tensor; a diffeomorphism preserves the underlying topological structure. The inclusions express increasing restrictions on transformations. Whether they are strict depends on the space considered. Riemannian definitions and their relationship with curvature are developed in Lee (1997). Moreover, preserving certain invariants is a necessary condition in many problems, but it may be insufficient to establish a complete equivalence.

The operational layer requires specifying what is preserved. If a group G acts on M through ρ and another group G′ acts on M′ through ρ′, an equivalence of actions requires a correspondence φ between the groups and a correspondence f between the spaces satisfying:

This condition must be accompanied by appropriate regularity and invertibility properties. The abstract group, its action, and the preserved structure are distinct pieces of information. Symmetries may be discrete or continuous; a structure may have only the identity symmetry. Lie groups are particularly useful for regular continuous symmetries, but they do not exhaust the possible forms of operational organization.

In a dynamical formulation, if Φ_t and Ψ_t represent evolutions, a conjugacy preserving the time parameter satisfies:

The equivalence changes if changes in the time scale are allowed, if only orbits are compared, or if the evolution is stochastic. The philosophical question of the persistence of essence must be translated into an explicit preservation criterion appropriate to the process under investigation.

5. An example: two circles and two dynamics

Consider a circle of radius r and another of radius R, each equipped with its intrinsic arc-length metric. Their angular parametrizations allow us to write:

If the radii are positive, the two circles have the same cardinality and are homeomorphic and diffeomorphic. Their total lengths, however, are 2πr and 2πR. When r ≠ R, there is no global isometry between the circles with the specified metrics: an isometry would preserve total length. This difference directly illustrates how one topological determination admits different metric determinations.

The abstract isometry group of each circle remains isomorphic to O(2). Knowing only the abstract group is therefore also insufficient to recover the radius. The metric and the relevant action are required. The operational layer provides information about admissible transformations without replacing the full set of structural data.

Now fix a single circle and compare two angular evolutions:

The first describes uniform rotation without equilibrium points; the second has equilibria at 0 and π, with different stability properties. Although the space and its metric are the same, the dynamical vector fields differ. The existence of equilibria prevents these evolutions from being conjugate through a dynamics-preserving bijection.

The example establishes a central consequence: the geometry of state space does not, by itself, determine the law of evolution. Specifying a system may require adding potentials, vector fields, forces, constraints, boundary conditions, or probabilistic rules. Investigating how a material organization produces its lawfulness requires representing precisely these data and their relations.

6. Coupling, history, and the composition of the totality

For several components to form a totality, they must interact through relations compatible with their material nature. These relations can be represented through networks, differential equations, jump processes, flow balances, or combinations of such tools. The use of a system of equations must be justified by the question and the object’s properties.

A formulation incorporating memory can be written schematically as:

The kernel K(t,s) determines how earlier states exert an influence; u(t) represents external inputs. In an equation with memory, specifying only the instantaneous state may be insufficient. Assumptions concerning the functions and kernel must support the existence, uniqueness, and stability of solutions within the domain employed.

History can also be formalized as a set of events I with a causal precedence relation ≼. A partial order admits incomparable events and requires reflexivity, transitivity, and antisymmetry. Its past function can be defined by:

The order is recovered through i ≼ j if and only if i ∈ H(j). Conversely, a past function recovers a partial order when, in addition to reflexivity and transitivity, mutual precedence is required to imply identity. Without this last condition, one generally obtains a preorder.

A partial order can be represented by a category with exactly one morphism from i to j whenever i ≼ j. A general transition category admits more information: it may contain several transformations between the same states. Reducing it to accessibility forgets differences between transitions; identifying it with a partial order requires additional conditions, such as being thin and skeletal (Riehl, 2016).

Path homotopy preserves a different kind of information. Two paths with the same endpoints may be continuously deformed into one another while passing through different states, consuming different resources, or activating irreversible mechanisms. Their homotopy class does not automatically reconstruct the causal order or the complete material history. If a process returns to the same observable state, the return events must be distinguished to preserve the historical sequence.

Path dependence can be represented by enlarging the state to incorporate relevant memory. This enlargement is a modeling decision: it requires establishing which records of the past affect the future and testing their predictive contribution. Information accumulation, causal precedence, and irreversibility are related notions whose connections require specific conditions.

7. Contradiction, emergence, and qualitative transformation

Dialectical categories guide the investigation of processes in which an organization reproduces itself through relations that can also transform it. Their fruitfulness requires connecting philosophical interpretation with mechanisms and observable criteria. A social contradiction, a dynamical instability, and a logical contradiction belong to different registers; establishing their relationships requires a specific argument.

The unity and struggle of opposites makes it possible to investigate constitutive dependencies containing opposing tendencies. An analysis of a concrete process must identify which poles presuppose one another, which conditions sustain their unity, and which mechanisms intensify or modify their relationship. The existence of feedback alone does not resolve these questions.

The transformation of quantity into quality finds one possible representation in changes to a system’s stability or organization. Consider the normal form of a supercritical pitchfork bifurcation:

The equilibria satisfy x(μ − x²) = 0. For μ < 0, the only real equilibrium is x = 0, which is asymptotically stable. For μ > 0, the stable equilibria x = ±√μ appear, while the origin becomes unstable. A continuous variation in the parameter qualitatively changes the organization of equilibria.

This result belongs to the model and its assumptions, including the symmetry expressed by the normal form. Identifying a bifurcation in a real phenomenon requires estimating parameters, observing changes, and considering alternatives. Moreover, qualitative transformation does not always require a discontinuous jump in the observed variable: in this example, the new branches emerge continuously from the origin.

The negation of the negation can contribute to examining processes in which a transformation supersedes an earlier organization while preserving certain relations in a new configuration. Its application requires reconstructing what is abolished, what is preserved, and what emerges. The sequence must be established through the material process, without imposing a repetition of stages or a necessary outcome in advance.

Emergence designates properties of organization whose investigation requires considering relations between components and scales. One must specify whether the term refers to a collective property, a reduction that is impossible under certain constraints, or a new causal capacity. Nonlinearity may promote collective behavior, but linear couplings also generate collective modes and global constraints. The relationship between totality and parts requires defining the observable and the criterion of composition.

Complexity and chaos should likewise be distinguished. Deterministic chaos refers to properties of certain dynamics, including sensitivity to initial conditions within the regime considered. A complex system need not be chaotic. For smooth autonomous flows in planar regions with bounded orbits and under the relevant assumptions, the Poincaré–Bendixson theorem restricts limiting behaviors; this restriction cannot simply be transferred to forced systems, discrete maps, or delay systems.

8. Critical phenomena, scales, and renormalization

The physics of critical phenomena makes it possible to investigate how microscopic interactions produce collective regularities. In the liquid–gas transition, phase coexistence occurs along a line ending at a critical point. As that endpoint is approached, the density difference between the phases disappears. Ordinary coexistence and criticality are related but conceptually distinct phenomena (Tong, n.d.-a).

In a model of a continuous transition, a common description of the correlation length is:

The quantity ξ measures a spatial correlation scale; ξ₀ is an amplitude; and ν is a critical exponent. The relation is asymptotic and describes an approach to the critical point within the specified regime. In a finite sample, system size limits observation of the ideal divergence.

The thermodynamic limit considers increasing the number of components and the volume while maintaining an appropriate density. It makes it possible to formulate singularities that do not arise in the same way in regular finite systems. A threshold is a reference value of a variable or parameter; the two concepts perform different explanatory functions.

Renormalization constructs effective descriptions by changing scale and examining how the relevant parameters transform. Near suitable fixed points, different microscopic systems can share exponents and scaling relations. This universality must be delimited through dimension, symmetries, the range of interactions, and other relevant properties (Wilson, 1983; Lesne & Laguës, 2012).

A renormalization group acts on effective descriptions and should not automatically be confused with an invertible physical symmetry of each microscopic state. Averaging may discard information. Comparison across scales therefore encompasses both preserved regularities and degrees of freedom that have been eliminated.

The philosophical contribution lies in showing how a material unity admits levels of description with effective lawfulness. Sharing a universality class identifies collective regularities within a regime; it leaves open chemical, biological, and historical differences that are decisive for other questions. Structural comparison gains explanatory force when it specifies both what is preserved and the domain of validity.

9. Fractals, measures, and multifractal spectra

Fractal geometry provides tools for describing organizations whose spatial occupancy exhibits scaling behavior. Exact self-similarity, statistical self-similarity, and approximate regularities over finite intervals must be distinguished. In natural materials, observational resolution and characteristic scales restrict the interval over which a scaling law is defensible (Lesne, 1998).

For box counting, let N(ε) be the number of boxes of side length ε needed to cover a bounded set. When the relevant limit exists:

This defines the box-counting dimension. Hausdorff dimension is constructed through coverings and Hausdorff measures; its equality with D_B requires conditions. An exponent estimated over a finite range likewise does not establish the existence of the ideal limit.

The ternary Cantor set provides a calculable example. At stage n, there are 2ⁿ intervals of length 3⁻ⁿ. Its self-similarity dimension satisfies:

In this case, that dimension coincides with the Hausdorff and box-counting dimensions; its topological dimension is zero. The example distinguishes the number of components in an approximation, topological organization, and metric scaling behavior. A fractal dimension does not, by itself, characterize the entire structure.

Introducing a distribution on a space requires a measure. A measure space is a triple (X, Σ, μ), where Σ is a sigma-algebra and μ is a measure. For disjoint measurable sets, countable additivity states:

Disjointness is a relation between sets. A metric space, by contrast, is defined through a distance d. Metric and measure can be articulated—for example, when studying balls and their measures—but they represent different data (Viaclovsky, 2003).

For a multifractal measure, a local exponent can describe, when the limit exists, the behavior of mass around a point:

The spectrum f(h) records the dimension of sets with a given local exponent. Its definition is different from an operator’s eigenvalue spectrum. Multifractal formalization can represent scaling heterogeneity, provided that the measure and estimation procedure are appropriate.

10. Dissipation, coherence, and Vitiello’s case

Open systems exchange energy or matter with their environment. Under appropriate conditions, these exchanges sustain persistent organizations far from equilibrium. The theory of dissipative structures investigates such mechanisms and their stability conditions; the production of local organization is compatible with entropy production in the relevant whole (Prigogine, 1977).

Vitiello (2014) investigates a correspondence between an electrodynamic representation under specific conditions, damped and amplified oscillators, squeezed coherent states, and self-similar fractal structures. His construction uses a constant magnetic field and a harmonic scalar potential. The result provides a delimited case of formal articulation between physical descriptions.

Part of the mechanism can be examined through the pair of equations:

With appropriate parameters, one term represents damping and the other amplification. Combined coordinates relate this representation to the coupled equations used in the construction. The correspondence shows the importance of including the degrees of freedom and exchanges with the environment required by the model.

For the damped oscillator considered in isolation, with m, k, γ > 0, the mechanical energy satisfies:

The second equality follows by multiplying the oscillator’s equation by its velocity. It identifies a loss of mechanical energy from the subsystem; its physical interpretation requires a balance with the environment. An enlarged representation may conserve a global quantity even though one of its parts exhibits dissipation.

The proposed philosophical interpretation treats these results as examples of how lawfulness becomes intelligible through the reconstruction of relations and conditions. Extending a correspondence to other objects would require specifying new maps, verifying what they preserve, and testing their consequences. The case supports a relational mode of investigation; the universal scope of an ontology requires a further argument.

11. Self-organized criticality and 1/f-type noise

Bak, Tang, and Wiesenfeld (1987) proposed self-organized criticality as a mechanism for investigating the emergence of scaling behavior in extended dynamical systems. Investigating a particular model requires specifying the driving, dissipation, local rules, and relationships between driving and relaxation times. Avalanches provide observables whose size and duration can be compared across scales.

Self-organized criticality is a mechanistic hypothesis that must be tested for each system. Self-organization encompasses broader processes, while chaos, fractality, and criticality refer to distinct properties. An observed power law admits alternative mechanisms; identification requires dynamical evidence and model comparisons.

In a time series, a 1/f-type noise relation is formulated through the power spectral density:

The case α ≈ 1 is commonly called pink noise within the interval studied. The spectrum describes the distribution of power across frequencies. An ideal model proportional to 1/f at all frequencies presents integrability problems, so cutoffs or limits of the observed regime must be specified.

An event-size distribution is a different object. For an ideal continuous power-law tail density:

Normalization requires τ > 1; its mean is finite when τ > 2, and its variance is finite when τ > 3. These conditions follow by integrating the moments over a tail without an upper cutoff. With a finite cutoff, those moments may exist even when they diverge in the idealization.

The spectral exponent α and the size exponent τ are not interchangeable. Nor can the absence of a mean be inferred merely from the presence of a power law. The empirical design must establish which variable is measured, over which domain, and with what uncertainty.

12. From concepts to statistical investigation

Materialist research must connect the proposed determinations with operations of observation and testing. Each layer requires specifying the object, the available data, the equivalence criterion, and the admissible error. This requirement makes the relationship between a philosophical claim and its scientific representation assessable.

At the set-theoretic layer, counting depends on rules for identifying units and distinguishing repeated observations. At the topological layer, an observed network depends on how connections are detected and which relations remain unrecorded. At the metric layer, a distance must justify its units, weightings, and sensitivity. The historical layer requires records that allow events to be ordered or transitions to be estimated. These choices condition one another and must be documented.

12.1. Information geometry and the statistical metric

A regular family of distributions p(x; θ) can be equipped with Fisher information:

Under conditions of regularity and local identifiability, this matrix defines a Riemannian metric on the parameter domain considered. If it is singular, restrictions or a different formulation are required. Its interpretation is statistical: it measures the local distinguishability of distributions within the model.

For the normal family with parameters (μ, σ), where σ > 0, direct calculation of the moments of the score vector gives:

The same absolute change in the mean is statistically more distinguishable when dispersion is smaller. The formula expresses a determination of the probabilistic model. Interpreting it as the geometry of a physical or social process requires justifying the link between parameters, observables, and mechanisms. Locating this structure at the metric layer does not remove that interpretive requirement.

An intrinsic Fréchet mean minimizes expected or empirical squared distance. Its existence and uniqueness depend on the geometry and the distribution. On a finite-dimensional Hadamard manifold, a distribution with finite second moment admits a unique mean; other geometries may admit multiple minima. The choice of an intrinsic estimator should be compared with appropriate competitors, using criteria external to the procedure whenever possible (Gómez Julián, 2026).

12.2. Scaling laws: estimation and comparison

Evaluating a power-law tail calls for estimating its lower threshold, fitting the model with an appropriate method, quantifying uncertainty, and comparing alternatives such as a lognormal or a power law with a cutoff. A straight-looking logarithmic plot is insufficient. If observations are dependent, a procedure based on independence must be adapted and justified (Clauset et al., 2009).

12.3. A protocol for testing across layers

Synthetic experiments can verify a procedure when the generating process is known. If the evaluation favors the chosen geometry by construction, success establishes adequacy only within that scenario. A claim about real objects requires external data, appropriate controls, and sensitivity analysis. Discretization error, algorithmic tolerances, and finite-size effects are also part of the evaluation.

13. Reproducible calculation: spatial symmetry and irreversibility

Consider three states {0, 1, 2} with the discrete topology and distance one between distinct states. This geometry is symmetric under every permutation. Define a discrete-time Markov chain through a transition matrix whose rows indicate the starting state:

All entries are positive, and each row sums to one. The chain is irreducible and aperiodic. The columns also sum to one, so the uniform distribution π = (1/3, 1/3, 1/3) is stationary. Cyclic rotation of the labels preserves the dynamics; a reflection reversing the cycle exchanges the probabilities 0.7 and 0.2 and is no longer a dynamical symmetry of P.

The stationary current between two states is defined as:

For the oriented edges 0 → 1, 1 → 2, and 2 → 0, the result is J = (0.7 − 0.2)/3 = 1/6. The distribution remains stationary while a current circulates. Stationary reversibility would require detailed balance, namely π_iP_ij = π_jP_ji for each pair; that condition fails.

The divergence between forward and backward pair fluxes provides a per-step indicator of irreversibility:

This quantity is a nonnegative Kullback–Leibler divergence, expressed in nats per step. Its interpretation as physical entropy production requires additional assumptions connecting transitions with thermodynamic exchanges. The purely probabilistic calculation already establishes asymmetry in trajectory probabilities in the stationary regime.

Compare this with Q = (P + Pᵀ)/2. Its diagonal entries are 0.1 and its off-diagonal entries are 0.45. It preserves the same space, distance, and stationary distribution, but satisfies detailed balance. All its currents and its irreversibility indicator are zero.

13.1. Complete verification code

The following calculation uses only the Python 3 standard library. It checks normalization, stationarity, and the analytical results. The matrices are models constructed for this example; they are not presented as estimates from empirical data.

from math import log, isclose
from itertools import permutations

P = [[0.1, 0.7, 0.2],
     [0.2, 0.1, 0.7],
     [0.7, 0.2, 0.1]]
Q = [[(P[i][j] + P[j][i]) / 2 for j in range(3)]
     for i in range(3)]
pi = [1 / 3] * 3

for name, A in [("P", P), ("Q", Q)]:
    assert all(isclose(sum(row), 1.0) for row in A)
    assert all(value > 0 for row in A for value in row)
    stationary = [sum(pi[i] * A[i][j] for i in range(3))
                  for j in range(3)]
    assert all(isclose(a, b) for a, b in zip(stationary, pi))
    J = [[pi[i] * A[i][j] - pi[j] * A[j][i]
          for j in range(3)] for i in range(3)]
    irr = sum(pi[i] * A[i][j] *
              log((pi[i] * A[i][j]) / (pi[j] * A[j][i]))
              for i in range(3) for j in range(3))
    symmetries = [s for s in permutations(range(3))
                  if all(isclose(A[s[i]][s[j]], A[i][j])
                         for i in range(3) for j in range(3))]
    expected_J = 1 / 6 if name == "P" else 0.0
    expected_I = log(3.5) / 2 if name == "P" else 0.0
    assert isclose(J[0][1], expected_J, abs_tol=1e-12)
    assert isclose(irr, expected_I, abs_tol=1e-12)
    assert len(symmetries) == (3 if name == "P" else 6)
    print(f"{name}: J01={J[0][1]:.6f}; "
          f"I={irr:.6f}; symmetries={len(symmetries)}")

P: J01=0.166667; I=0.626381; symmetries=3
Q: J01=0.000000; I=0.000000; symmetries=6

13.2. What the example identifies

The stationary distribution alone does not distinguish these two dynamics. Transition frequencies and trajectory probabilities do. With a finite trajectory, transitions would have to be estimated and error quantified while accounting for temporal dependence. Unobserved events, state aggregation, and sampling intervals can conceal currents.

The conclusion is structural in scope: a symmetric metric is compatible with irreversible dynamics. Irreversibility does not, by itself, require replacing distance with an asymmetric function. It may reside in the evolution operator and trajectory probabilities. A geometry of directed costs or a nonreversible Finsler formulation may be useful for other problems, but requires additional justification.

The three-state model is a finite space and does not require a positive-dimensional Riemannian manifold. This also shows that the layers form an adaptable analytical framework: each object requires the structures that actually apply to it.

14. Nature and society: material unity and specificity

Investigating universal relations must preserve differences of organization. A biological system requires consideration of metabolism, regulation, development, and evolution; a social relation requires incorporating practices, institutions, conflict, and historical conditions. Physical exchanges constitute material conditions, while each level’s explanatory capacities depend on its specific mechanisms.

In the cardiovascular system, tissue heterogeneity and flow distribution require investigating functionally differentiated organizations. Sharma (2009) examines dynamical behaviors and tools from complexity research within this domain. A scaling exponent or a temporal irregularity must be interpreted alongside physiology and measurement conditions. Variability may perform different functions depending on the process and its regime.

In political economy, the unity of production, circulation, and social reproduction includes property and class relations. Access to means of production, wage labor, credit, and state power condition one another through determinate institutions. A network representation or a system of equations must incorporate these differences if it is to explain accumulation, crises, or social transformation.

The historical analysis of an institution distinguishes the conditions of its emergence, its mechanisms of reproduction, and the possibilities for superseding it. Reconstructing the genesis of slavery, for example, requires investigating coercion, property, and concrete productive relations. Explaining it does not imply morally justifying it or postulating a universal succession of stages through which every society must pass.

A relation can facilitate certain productive capacities while imposing suffering, domination, or the destruction of other capacities. Historical judgments about progress therefore require explicit criteria, identifiable actors, and comparative evidence. Understanding a social totality includes the conflicts through which its members reproduce or transform their conditions of existence.

15. Synthesis: essence, lawfulness, and history

The materialist conception of existence as an objective organic structure holds that determinate realities are constituted through relations, maintain a relative unity, and transform under material conditions. Knowing them requires reconstructing components and connections, distinguishing scales, and examining mechanisms of reproduction and change.

Essence can be formalized as a unity of set-theoretic, topological, differentiable, metric, operational, and historical-dynamical determinations. Each layer establishes a domain of relations and a preservation criterion. Investigating an object requires determining which layers are relevant, how they are articulated, and which variations modify its identity with respect to the question posed. The persistence of certain relations may coexist with decisive transformations in others.

The components of a totality may thus have different natures and participate in a common organization through compatible relations. Their mathematical integration requires specifying couplings, domains, constraints, and conditions of evolution. The system’s unity is expressed through this organization and its capacities, without requiring a universal isomorphism among all its components. Shared structures acquire explanatory scope when one establishes what they preserve and why they are materially relevant.

Critical phenomena, fractal structures, and the correspondences investigated by Vitiello provide precise problems for investigating relationships between scales and formalizations. History adds constraints that are not automatically captured by a topology, a distance, or an abstract group. The Markov example makes this difference visible: the same space and distribution admit different currents and different degrees of irreversibility.

Symmetries reveal structural regularities, but their connection to conserved quantities requires dynamical conditions. Within a variational framework, Noether’s theorem connects continuous symmetries of the action with conservation laws under its corresponding assumptions (Tong, n.d.-b). Determining a complete lawfulness requires incorporating the data that specify the process, in addition to its geometric organization.

The cognitive strength of dialectical materialism resides here in a practice of inquiry: reconstructing material unity through its differentiated determinations, following its historical formation, and subjecting its representations to testing. The universality of connections becomes concrete by explaining which operate, how they operate, and under which conditions they transform the totality.

Glossary of precise terms

Automorphism. An invertible correspondence from a structure to itself that preserves the specified relations and operations.

Dynamical conjugacy. An invertible correspondence intertwining two evolutions with the chosen regularity criterion and time parametrization.

Essence. The articulated unity of the object’s constitutive determinations; formalizing it requires specifying the relevant layers and relations.

Invariant. A property or quantity preserved under an explicit class of transformations. A partial set of invariants may be insufficient to classify objects.

Isomorphism. An invertible equivalence preserving the structure of a specified category. The term requires identifying the objects and relations being compared.

Lawfulness. The organization of regularities, constraints, and transformation rules of the process considered. Its representation may be deterministic or probabilistic.

Riemannian metric. A smooth field of positive-definite inner products on a manifold’s tangent spaces. It may exist globally even when curvature prevents its reduction to a Euclidean metric through local coordinates on an open region.

Symmetry. A transformation preserving a specified structure or law. The symmetry of a space, an equation, and a solution are different pieces of information.

Totality. A delimited organization of components and relations with conditions of reproduction and transformation. Its methodological delimitation requires material justification.

References