Cognitive Platonism II: From Natural Regularities to Abstract Forms

1. Introduction

Mathematical structures can present themselves as discoveries rather than inventions. Once a relation has been established, its consequences seem independent of our wishes. No mathematician can decide that π should have another value, that 7 should stop being prime, or that a theorem should have a different consequence while preserving the same assumptions. This experience of necessity has historically provided one of the strongest motivations for mathematical Platonism and contemporary mathematical realism. If mathematical truths are discovered and cannot be altered by us, one can suppose that they exist independently of us. Contemporary arguments for mathematical realism have developed this intuition in more precise forms, including the indispensability arguments associated with Quine and Putnam and subsequently developed by Colyvan (Putnam 1971; Colyvan 2001). At the same time, the epistemological difficulty of explaining how embodied cognitive agents can know causally inert abstract objects has remained a central problem for Mathematical Platonism (Benacerraf, 1973).

Cognitive Platonism accepts the non-arbitrariness of mathematical truth but does not regard independent Platonic existence as a necessary conclusion (Dodig-Crnkovic 2025). The recurrence of stable mathematical relations within certain structures does not by itself establish that the corresponding abstract entities exist independently in a separate Platonic domain. What is given to us through interaction are regularities and relations, which cognitive agents subsequently abstract and articulate mathematically. An alternative account is possible. Natural processes exhibit stable relational regularities. Cognitive agents, themselves parts of nature, encounter those regularities through interaction. Through successive processes of abstraction, they construct increasingly general representations of them. Once such a relational structure has been articulated, its internal consequences constrain further reasoning and can themselves be discovered. This position is related to naturalistic and embodied approaches to mathematics. Lakoff and Núñez (2000), for example, argue that mathematical concepts develop through embodied cognitive mechanisms. Maddy (1990, 1997) has sought forms of mathematical realism compatible with philosophical naturalism. Structuralist approaches likewise shift attention from mathematical objects considered as independently individuated entities toward relational structures (Shapiro 1997).

The need for further elaboration has recently become apparent in Michael Levin's discussion of Platonic Space, where he explicitly compares his position with Cognitive Platonism, identifying substantial agreement but also several apparent differences (Levin 2026). The present paper uses that comparison as a point of departure for specifying more precisely what Cognitive Platonism claims about constraints, representation, mathematical discovery, and, most importantly, the ontological status of abstract forms. The central distinction proposed here is between the independent existence of natural regularities and the cognitive articulation of those regularities as abstract forms.

2. Nature Before Laws

Cognitive Platonism begins with a simple observation: nature affects cognitive agents before cognitive agents formulate laws about nature. Physical systems interact. Differences propagate. Organisms sense, respond, remember, compare, and learn. Only much later in biological and cultural evolution do cognitive systems formulate explicit representations, concepts, geometries, mathematical constants, and scientific laws. From this perspective, the familiar statement that “nature obeys laws” may reverse the epistemic and historical order through which laws become available to us. Nature exhibits regular behaviors and structures. Cognitive agents identify stable relations within natural processes and formulate them as laws. This does not make laws arbitrary or merely subjective. A natural law is constrained by the organization of the world. Nature exhibits the same behaviors and structures under the same conditions. Predictions work when models adequately capture relevant relations. Objectivity develops through increasingly reliable coordination among models, observations, interventions, and multiple observer-agents. This understanding has affinities with philosophical accounts that resist treating laws as governing prescriptions written into nature. Cartwright (1983), for example, has argued against an overly literal interpretation of universal laws as direct descriptions of concrete physical situations, while van Fraassen (1980) has defended an empiricist conception according to which successful scientific representation does not require commitment to every element of the ontology suggested by a theory.

The claim made by Cognitive Platonism is therefore neither naïve realism nor subjective constructivism. The regularities are not invented by observers, but their articulation as laws is an achievement of observer-agents interacting with an independently structured world (Dodig-Crnkovic, forthcoming). This position also distinguishes the central importance of mathematical description in many theories, especially in physics and computing, from the stronger ontological conclusion that mathematical structures exist independently of the physical processes they describe. Wigner's famous discussion of the “unreasonable effectiveness of mathematics in the natural sciences” drew attention to the remarkable success of mathematical structures in fundamental physics (Wigner 1960). However, more recently, Garte, Marshall, and Kauffman (2025) contrasted this effectiveness in physics with what they call the “reasonable ineffectiveness of mathematics in the biological sciences.” Their argument is not that biology lies outside mathematics or violates physical law, but that biological evolution, agency, and the continual generation of novel possibilities resist the kind of precise, exceptionless mathematical characterization familiar from some areas of fundamental physics. This contrast is important because the organization of natural systems changes profoundly across levels. Lower-level physical regularities do not cease to operate as organization complexifies, but they become embedded within additional layers of constraint, boundary conditions, feedback, memory, historical dependence, and cross-level interaction. Higher-level organization can thereby constrain and redirect lower-level dynamics without violating the physical laws governing those dynamics. Anderson’s (1972) classic argument that “more is different” challenged the assumption that knowledge of fundamental laws is sufficient for deriving the organizing principles characteristic of higher levels of nature. In biology, Noble's theory of biological relativity similarly rejects a privileged causal level and emphasizes reciprocal causation across organizational scales, including the effects of higher-level boundary conditions on lower-level processes (Noble 2012).

This multilevel organization provides one reason why simple mathematical regularities formulated at lower levels need not be sufficient for characterizing the behavior of increasingly complex biological systems. The issue is not whether living systems obey physical laws. They do. The issue is that those laws do not by themselves specify the historically developed organization within which physical processes occur. As organization accumulates through evolution and development, present dynamics depend increasingly on prior structure, memory, constraints, feedback relations, and interactions across multiple spatial and temporal scales. In this sense, the contrast between Wigner's “unreasonable effectiveness” and Garte, Marshall, and Kauffman's “reasonable ineffectiveness” should not be read as a contrast between mathematical and non-mathematical nature. It points instead to differences among organizational regimes of nature. Mathematics can capture exceptionally stable and general regularities in some physical domains, while biological organization introduces historical and multilevel dependencies that make similarly simple universal descriptions increasingly difficult. This is consistent with the broader account developed in Evolving Intelligence, in which higher levels of organization do not replace lower-level physical dynamics but constrain, integrate, and redirect them through progressively more complex organizational regimes (Dodig-Crnkovic, forthcoming). Living and cognitive systems remain physically embodied throughout, but their organization becomes explanatorily indispensable.

The resulting picture is therefore not one in which mathematics stopps applying as complexity increases. Rather, the forms of mathematical description required become increasingly dependent on the organization of the system being modeled. The explanatory relation between mathematics and nature is itself something to be investigated rather than assumed to consist in the straightforward instantiation of a pre-given mathematical order. Wigner's question thus remains fundamental: why does mathematics work so remarkably well where it does? It can be complemented by the question raised by Garte, Marshall, and Kauffman: why does its predictive and explanatory reach differ so markedly across domains and levels of organization? Cognitive Platonism does not regard either observation as sufficient evidence for an independently existing Platonic realm. Both can instead be approached from the same starting point: nature exhibits relational organization at multiple levels, and cognitive agents progressively abstract mathematical structures capable of representing those regularities with varying degrees of scope, precision, and explanatory power.

3. From Embodied Interaction to Abstract Representation

Representation itself is not an all-or-nothing capacity. At basal levels of cognition, interaction with the world is direct and embodied; the organization of the agent itself carries the history of interaction and guides ongoing action. Rodney Brooks’ behavior-based approach to intelligence provides a useful formulation of this condition. In Intelligence without Representation, Brooks (1991) challenged the assumption that intelligent behavior requires detailed internal representations of an external world, arguing instead for direct coupling between embodied agent and environment. His frequently cited principle that the world can function as its own best model captures the direct coupling between embodied agent and environment on the basal level (Brooks, 1990). Cognitive Platonism adopts this insight as a starting point. At the basal level, representation is embodied: the organization of an agent reflects its history of interaction and directly participates in ongoing action.

Increasing biological organization introduces memory, internal state differentiation, anticipation, distributed regulation, and target states. Multicellular developmental systems provide particularly clear examples. Levin's work on bioelectric regulation shows that tissues can maintain distributed anatomical target states that guide morphogenesis, repair, and regeneration. Such target states can be experimentally modified, resulting in predictable changes in subsequent anatomical development (Levin 2012, 2019, 2026). Representation therefore develops progressively. Nervous systems extend integration, prediction, memory, and simulation. At still higher levels of neural and social organization, representations themselves become objects of cognition, making possible counterfactual reasoning, symbolic manipulation, mathematics, scientific models, and cumulative cultural knowledge.

Abstract forms emerge through a succession of levels of abstraction from direct embodied interaction to symbolic and abstract representation. This development does not imply a discontinuity between a non-representational physical world and a pre-existing realm of representations. Rather, representation develops continuously through successive levels of organization and abstraction.

4. Mathematical Constants π and e: From Practical Regularities to Abstraction

The histories of π and e illustrate the distinction between encountering a regularity and articulating an abstract mathematical object. The relation represented by π was encountered through material practices involving circular objects, measurement, construction, and geometry. Ancient Babylonian and Egyptian mathematics already contained numerical approximations associated with circular measurement, while Greek mathematics progressively transformed such practical relations into geometrical objects suitable for proof. The long history of π illustrates the transition from practical approximation toward increasingly abstract mathematical investigation (Berggren, Borwein, and Borwein 2004; Boyer and Merzbach 2011). The constant π was not first encountered as an independently existing abstract object and only later applied to physical circles. Historically, access proceeded through circles, measurements, ratios, geometrical constructions, and progressively abstract mathematical relations. Once developed mathematically, however, π ceased to be merely a practical ratio associated with measured circles. It appeared in trigonometry, infinite series, analysis, probability, complex analysis, Fourier theory, and many other mathematical contexts. The abstraction became more general than the concrete interactions through which it was originally approached.

The history of e is even more revealing. Relations eventually unified through e emerged through several initially distinct mathematical practices. Logarithms arose in connection with calculation; compound-interest problems led Jacob Bernoulli to limiting expressions later recognized as involving e. Subsequent developments in calculus, exponential functions, infinite series, and complex analysis progressively revealed a common relational structure. Euler's work played a decisive role in consolidating these relations into the mathematical object now denoted by e (Maor 1994; Boyer and Merzbach 2011). No single historical episode corresponds to the appearance of e as an already fully articulated mathematical entity. Different regularities and operations were progressively brought into relation until their deeper unity became mathematically visible. Once these abstractions had been established, however, their consequences became mathematically necessary. One could discover properties of π and e that had never been observed in the practical situations from which the abstractions historically developed. This is genuine discovery.

The historical argument of course does not prove that mathematical entities are cognitively constructed rather than independently existing. A mathematical Platonist can consistently interpret the same history as the gradual discovery of pre-existing mathematical objects. The more modest conclusion is that the historical process is fully compatible with an abstraction-based account and therefore cannot, by itself, establish Platonic ontology.

5. The Pythagorean Theorem and the Plurality of Geometries

The Pythagorean theorem provides another instructive example. Relations among the sides of right triangles were known in different mathematical cultures long before the systematic axiomatic treatment associated with Greek geometry. Babylonian mathematics contains numerical examples of what are now called Pythagorean triples, while related geometrical knowledge developed independently in Indian and Chinese mathematics (Boyer and Merzbach 2011). Within Euclidean geometry, the relation expressed as a² + b² = c² is necessary. Once the relevant geometrical structure and assumptions are specified, the result can be proved independently of further measurement.

For a long period, Euclidean geometry was readily understood not merely as one mathematical geometry but as the necessary structure of physical space itself. The development of non-Euclidean geometries fundamentally altered that picture. Hyperbolic and elliptic geometries showed that internally consistent geometrical systems could be constructed in which Euclidean relations, including the familiar form of the Pythagorean theorem, no longer hold generally (Coxeter 1998).

6. Mathematical Necessity Without Ontological Platonism

A distinction is therefore needed between two propositions, where the first does not logically entail the second:

• − Given a mathematical relational structure, certain consequences follow necessarily
• − That relational structure exists independently of all cognitive agents

This is precisely where several positions in the philosophy of mathematics diverge. Mathematical Platonism or realism accepts the independent existence of mathematical entities or structures. Nominalist approaches attempt to preserve scientific and mathematical practice without such ontological commitments; Field's Science Without Numbers (1980) provides one of the most influential modern examples. Structuralism proposes that mathematics is fundamentally concerned with positions and relations within structures rather than independently characterized objects (Shapiro 1997). Naturalistic and embodied approaches attempt, in different ways, to account for mathematics within the natural and cognitive world rather than beginning from access to a separate abstract realm (Maddy, 1997; Lakoff and Núñez 2000).

Cognitive Platonism belongs to this naturalistic landscape but emphasizes the generative development of abstraction through observer-agent interaction. The important point is that cognitive construction does not mean arbitrary invention. Once a relational structure has been abstracted and stabilized, its consequences constrain the cognizing agent. A mathematician does not freely choose the value of π or e, nor can a proof be made to yield an arbitrary conclusion while its assumptions and inferential relations remain unchanged. Thus mathematical discovery can occur after cognitive abstraction has generated a sufficiently stable relational domain. We may therefore distinguish between:

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Construction of an abstract relational space

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Discovery of consequences within that space.

The second does not require the first to have taken place outside nature or independently of cognitive agents.

7. Cognitive Platonism and Levin's Platonic Space

Michael Levin's recent account of Platonic Space is close to Cognitive Platonism in several important respects. In his explicit comparison with Cognitive Platonism, Levin states substantial agreement concerning the causal efficacy of abstract forms, the absence of a pre-written biological script, feedback between physical embodiment and higher-level organization, substrate-invariant causal architectures, anticipation, and morphological computation (Levin, 2026). Several apparent differences are smaller than they first appear.

Constraints and Enablement

Levin contrasts an emphasis on constraints with what he calls enablement. In Cognitive Platonism and the broader info-computational framework, however, constraints are not understood merely as restrictions. Constraints at the same time enable organization precisely by reducing degrees of freedom and channeling dynamics toward particular interactions and possibilities. A constraint can therefore simultaneously exclude possibilities and generate new ones. Chemical bonding, molecular geometry, cellular boundaries, anatomical structures, and developmental organization are examples in which restriction of lower-level possibilities enables higher-level organization. On this interpretation, constraint and enablement are complementary rather than opposed. Some examples:

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Physics. Restricting Degrees of Freedom to Create Order. In Acoustic and Optical Cavities, forcing sound or light waves to bounce between two precise mirrors (a geometric constraint) cancels out chaotic, random wavelengths. This enables the synchronized, amplified phase needed to generate a laser beam.

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Chemistry. Structural Boundaries That Drive Reactivity. Enzyme Active Sites. An enzyme uses its rigid, physical pocket to trap specific substrates. By locking them into a forced orientation and reducing their spatial freedom, it drastically lowers the activation energy, enabling reactions.

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Biology: Limits That Spark Evolutionary Innovation. The Genetic Code. The rigid, universal syntax of DNA and RNA restricts how amino acids can be assembled. This structural limitation is what enables predictable protein folding, stable heredity, and the infinite diversity of life.

Representation

Levin also questions the idea that biological form should generally be understood as emerging without representation or control. His experimental work points to bioelectrically encoded anatomical target states that can be read, modified, and used to redirect morphogenesis. Cognitive Platonism does not deny such representations. Representation is understood developmentally and hierarchically. At basal levels, cognition can operate through direct embodied interaction in the sense emphasized by Brooks (1991). At subsequent levels, biological systems develop memory, internal target states, distributed control, prediction, counterfactual modeling, and eventually symbolic representation. Levin's anatomical target states therefore fit naturally within this succession of representational levels.

The Ontological Status of Abstract Patterns

The more substantive difference concerns the ontological status of abstract patterns. Levin argues that mathematical truths are discovered rather than constructed. He uses examples such as primality and e to suggest that mathematical structure is not reducible to physical history or to the cognitive activities through which humans become aware of it. In some formulations, mathematical patterns provide not merely descriptions of physical behavior but deeper explanations of why physical systems produce particular outcomes (Levin, 2026). Cognitive Platonism agrees with the non-arbitrariness of such mathematical relations but does not infer from this that they must exist as autonomous entities prior to cognitive abstraction. The difference between the two positions can therefore be expressed as a question: Does cognition discover an independently existing abstract form instantiated in nature, or does it abstract and articulate relational regularities already embodied in natural processes? The second alternative does not deny reality. What exists independently is the relationally structured natural world.

8. Physical Laws and Mathematical Laws

The distinction becomes especially significant when considering laws of nature. The conventional expression that “nature obeys physical laws” can be interpreted in more than one way.

On one interpretation, laws or mathematical structures have explanatory priority. Physical systems behave as they do because they instantiate or conform to those laws. A sufficiently strong form of this position approaches mathematical Platonism: abstract structures are ontologically primary, while physical processes instantiate them.

On another interpretation, physical processes come first. Nature exhibits structured dynamics. Observer-agents detect stable relations in those dynamics and abstract them into laws. Mathematical formalization then makes it possible to derive new consequences, construct models, predict outcomes, and connect apparently unrelated phenomena.

The two interpretations can frequently support the same scientific practice. Their difference concerns ontological direction.

Platonic interpretation: abstract structure → physical instantiation → observed regularity

Cognitive Platonist: natural interaction → regularity → abstraction → mathematical form

Levin's recent examples suggest a distinction between mathematical and physical necessity. He notes, for example, that physical constants can at least coherently be imagined having different values or dynamics, whereas mathematical constants such as e cannot simply change while the underlying mathematical relations remain the same. He also argues that some physical phenomena may have their strongest explanations in mathematical patterns that cannot themselves be derived from physics (Levin, 2026). This suggests an important independence of mathematical reasoning from physical explanation. It does not yet determine whether mathematical structures exist as entities independently of cognitive abstraction, nor exactly how mathematical structures relate ontologically to physical laws. That question remains open.

9. Why Mathematics Appears Platonic

The persistence of Platonism may itself be understandable within Cognitive Platonism. At the beginning of abstraction, the cognitive contribution is obvious. Humans measure, compare, classify, idealize, define, construct representations, and develop symbolic systems. As an abstract system becomes stabilized, however, its historical and cognitive origins fade from view. Its internal relations increasingly constrain the cognitive agent rather than appearing to be produced by that agent. New results become surprising. Proof forces conclusions that nobody chose. Mathematicians working independently converge upon the same relations. Structures developed for one purpose unexpectedly become applicable in entirely different domains. Mathematics thus comes to present itself phenomenologically as an independently existing landscape awaiting exploration. This experience should not be dismissed. Indeed, it may help explain the extraordinary historical persistence of mathematical Platonism.

Cognitive Platonism takes that experience seriously while offering another interpretation of it. An abstract structure that emerged through cognitive interaction can, once stabilized, support objective and non-arbitrary exploration. Its consequences are not under the arbitrary control of its constructors. Cognitive construction can therefore generate a domain in which subsequent activity takes the form of genuine discovery. In this way, Cognitive Platonism attempts to explain why mathematics can feel Platonic without assuming that the phenomenology of discovery settles the ontology of mathematical objects.

10. Explanatory Autonomy and Ontological Independence

The distinction can also be formulated in terms of explanation. Mathematical abstraction produces structures that can become relatively autonomous from the concrete circumstances in which they first developed. The theory of numbers, Euclidean and non-Euclidean geometries, calculus, probability theory, and complex analysis can be explored without continually returning to the empirical situations from which some of their concepts historically emerged. This explanatory and inferential autonomy is one of mathematics' greatest strengths. But explanatory autonomy does not necessarily entail ontological independence. The same distinction applies to scientific models more generally. Models can reveal relations that are difficult or impossible to perceive directly. They can generate novel predictions and disclose previously unknown connections. Their explanatory productivity does not require treating the model itself as an independently existing entity that governs the physical system. Cognitive Platonism proposes that abstract forms can be understood similarly. They are real in the sense that they have stable structure, constrain reasoning, enable prediction, coordinate action, and can have causal consequences when instantiated through cognitive and physical systems. Their reality need not consist in existence within a separate ontological realm.

11. Conclusions

Cognitive Platonism begins with nature rather than with an independently existing realm of Forms. Natural processes possess real relational organization independently of human cognition. Through interaction with those processes, cognitive agents generate information, stabilize representations, detect regularities, and progressively construct abstract relational structures. Mathematics represents one of the most developed forms of this process. Once established, mathematical structures possess internal necessities that support genuine discovery. The fact that mathematical truths are discovered rather than arbitrarily chosen therefore need not entail that mathematical entities exist independently in a Platonic realm.

The histories of π, e, geometry, and mathematical abstraction are compatible with both interpretations. A Platonist can understand them as progressive discovery of structures that existed independently all along. Cognitive Platonism understands them as progressive abstraction of relational regularities encountered through interaction, followed by discovery within the resulting abstract structures. Historical evidence by itself does not decide between these ontologies. The distinction is subtle but consequential.

What Cognitive Platonism minimally assumes to exist independently is the structured natural world with which cognitive agents interact. Forms are ways in which increasingly capable cognitive systems articulate, stabilize, generalize, and explore its relational organization. Whether an additional independently existing Platonic domain should be postulated remains an open metaphysical question. Cognitive Platonism does not claim to disprove it. It argues that such a domain is not required to account for mathematical necessity, mathematical discovery, or the extraordinary explanatory effectiveness of abstract forms.