Rethinking Mathematical Knowledge: A Nativist Approach

Abstract

This paper develops a nativist approach to mathematical knowledge as an alternative account within the philosophy of mathematics. Rather than beginning with the ontological status of mathematical objects, it argues that any satisfactory account of mathematical knowledge should address three closely related questions: how to understand and justify truth-talk in mathematical practice, how mathematical knowledge is acquired, and why some mathematical knowledge has indispensable applications in the sciences. The paper first examines representative nominalist and Platonist responses to these questions. Nominalist accounts, while avoiding the specific epistemic difficulty associated with abstract objects, face challenges in explaining the non-arbitrariness of mathematical practice and the indispensability of mathematics in certain scientific theories. Platonist accounts, while offering a straightforward semantics of mathematical truth, still face the familiar epistemic problem of explaining how we acquire knowledge of abstract mathematical entities and require a fuller account of the applicability of mathematics to the empirical sciences. Against this background, the paper proposes two nativist theses: that some mathematical knowledge is innate, and that mathematics may function more fundamentally as a cognitive pattern than as a cognitive capacity. On this basis, mathematical knowledge is construed as cognitively dependent, while mathematics itself is understood as underlying our representations of the world. The paper concludes that the nativist account provides a framework for reconsidering the legitimacy of truth-talk in mathematical practice, the acquisition of mathematical knowledge, and the applicability and indispensability of some mathematics to sciences.

1. Introduction

Mathematics is among the most philosophically puzzling domains of inquiry. On the one hand, it is one of the most autonomous and rigorous disciplines: mathematical theories are developed through definitions, axioms, and logical inference, and many mathematical results appear to possess a degree of certainty rarely found elsewhere. When asked what mathematics studies, mathematicians may naturally mention numbers, functions, sets, algebraic structures, spaces, or other mathematical objects. On the other hand, when we ask whether such objects exist, and in what sense they exist, there is no comparable consensus. A physicist can usually affirm the existence of planets, rocks, or quarks by appealing to direct or indirect empirical access. Mathematical objects, if any, are by contrast standardly understood as abstract objects, and it is difficult to see how such objects could be known through ordinary empirical means. This gives rise to the familiar question: where does mathematical knowledge come from?

The debate about mathematical objects has often been framed as a dispute between Platonists and nominalists. Platonists hold that mathematical objects are mind-independent entities, whereas nominalists deny, or at least avoid, such ontological commitment. The epistemic challenge to Platonism, made especially vivid by Benacerraf [1], asks how a referential theory of mathematical truth can be reconciled with a plausible account of knowledge acquisition if mathematical objects are abstract and causally inert. Although several philosophers have recently questioned whether the Benacerraf challenge is decisive [2,3], it has nevertheless shaped subsequent debates about mathematical knowledge and a priori justification.

This paper suggests that simply affirming or denying the existence of mathematical objects may not be the most fruitful starting point. Abstract mathematical objects, if any, cannot be verified by ordinary scientific means, yet treating mathematics as mere invention and/or conventions seems to fit poorly with certain important features of mathematical practice. I therefore explore a third approach: a nativist account of mathematical knowledge. This proposal is not entirely novel. Early empirical studies, such as Starkey’s study of infants’ number perception [4] and Wynn’s study of addition and subtraction in infants [5], provided evidence that elementary numerical capacities emerge very early in human development. Dehaene subsequently developed a broader cognitive account of numerical cognition, arguing that humans possess cognitive mechanisms that support numerical representation and calculation [6]. These empirical approaches, however, have focused primarily on number cognition and elementary arithmetic rather than on mathematics as a whole. A more ambitious project is pursued by Lakoff and Núñez, who treat mathematical ideas and reasoning more generally as cognitive achievements grounded in embodied conceptual structures [7]. Their account extends the cognitive approach beyond elementary numerical cognition, although it has also attracted substantial criticism from mathematicians and philosophers of mathematics, particularly concerning its treatment of mathematical objectivity, abstraction, and the nature of mathematical knowledge. In parallel with these developments in cognitive science, philosophers of mathematics have increasingly explored empirically informed approaches to mathematical epistemology. For example, Pantsar develops an empirically feasible approach to the epistemology of arithmetic [8], while Maddy gives sustained philosophical attention to psychological approaches to the a priori sciences [9]. The aim of this paper is not to reduce mathematics to psychology, but to ask whether a nativist framework can illuminate certain philosophical questions about mathematical knowledge.

The paper proceeds as follows. Section 2 introduces three questions that serve as benchmarks for accounts of mathematical knowledge. Section 3 and Section 4 examine, in a deliberately selective way, how representative nominalist and Platonist strategies respond to these questions. Section 5 develops two nativist theses, and Section 6 explains how these theses may help answer the three questions. The conclusion summarizes the advantages and limitations of the proposal.

2. Three Questions About Mathematical Knowledge

Focusing exclusively on mathematical objects may not be the best way to understand the nature of mathematics, since the existence of such objects is precisely what is disputed. A constructive shift is to focus instead on mathematical knowledge. We usually use “mathematics” to refer to the discipline itself and “mathematical knowledge” to refer to what is developed, transmitted, and used within that discipline. In this paper, mathematical knowledge is understood in a broad sense as referring to the body of mathematical theories, rather than in a strictly epistemological sense, such as justified true belief.

This clarification matters because mathematical statements such as 2 + 2 = 4 or the Pythagorean theorem, a² + b² = c², are obviously treated as true within mathematical practice. Yet the philosophical status of this truth remains contested. We know that 2 + 2 = 4 by appeal to definitions and rules of addition, and we know the Pythagorean theorem by proof. In these cases, “true” may be understood in a normative or practice-internal sense: a mathematical statement is accepted when it follows from accepted definitions, axioms, and inferential rules. But whether mathematical statements are true in a correspondence sense, and whether they require mathematical objects as truth-makers, remains a central issue. A satisfactory account of mathematical knowledge should therefore address not only the ontological status of mathematical objects but also broader features of mathematical practice. I take the following three questions as benchmarks.

Q1. The semantics/normative question: How to account for the truth-talk in mathematical practice?

Q2. The epistemic question: How do we acquire mathematical knowledge?

Q3. The question of indispensable application: Why does some mathematical knowledge have indispensable applications in the sciences?

This framework is admittedly simplified. It is not intended to capture every issue debated in the philosophy of mathematics, nor is it meant to provide a universal criterion for assessing all positions in the debate, including Platonism, nominalism, structuralism, fictionalism, naturalism, and so on. The simplification is nevertheless useful for three reasons. First, the aim of this paper is constructive: to develop a nativist account rather than to provide a comprehensive taxonomy of positions in the philosophy of mathematics. Second, these three questions capture central pressures that any account of mathematical knowledge must face. Third, they allow us to compare nominalist, Platonist, and nativist responses in a manageable way.

Q1 can be read in at least two different ways: either as a semantic question or as a normative question. On the first reading, Q1 asks whether mathematical statements have truth values and, if so, what makes them true. It therefore concerns the semantics of mathematical statements and is closely connected to debates about mathematical objects. If mathematical concepts refer to mathematical objects, and if mathematical facts are constituted by the properties and relations of those objects, then mathematical objects and facts may serve as truth-makers for mathematical statements. For example, “3 is prime” is true if the number 3 exists and possesses the property of being prime. If there are no mathematical objects, then mathematical statements require some alternative semantic interpretation. On the second reading, Q1 concerns how to understand and justify the use of truth-talk in mathematics. It asks whether the practice of treating mathematical statements as true or false is legitimate and what, if anything, grounds this legitimacy. It should be noted that this reading does not, by itself, commit one to the view that mathematical statements have truth values or that mathematical discourse is merely fictional.

Q2 arises differently for different positions. Those who affirm the existence of abstract mathematical objects must explain how we can know anything about them. Those who deny such objects avoid this version of the epistemic challenge, but they still need to explain why we possess mathematical knowledge at face value and why mathematical practice appears to be non-arbitrary. In other words, if the acquisition of mathematical knowledge is treated merely in terms of intellectual invention and/or sociocultural convention, how can mathematical practices develop in such similar and convergent ways across different intellectual and sociocultural contexts? And how can we account for the broad agreement on the basic axioms and inferential rules governing mathematical practice across different intellectual and sociocultural backgrounds? These two questions can be seen as variants of Q1 if we take a deflationary understanding of truth-talk as the basis of non-arbitrary mathematical practice. However, it should also be noted that Q2 does not necessarily depend on any particular response to the different readings of Q1, since it concerns only the fact that we possess mathematical knowledge.

Q3 concerns the applicability and indispensability of mathematics in the sciences. Applicability and indispensability should be distinguished: applicability does not entail indispensability; rather, applicability is a necessary condition for indispensability. Therefore, one must first explain why mathematics is applicable to the sciences if one believes that mathematics is indispensable to them. A few remarks about the notion of “indispensability” are necessary. Colyvan argues that an entity is indispensable to a theory when a version of the theory without that entity yields fewer theoretical results, especially as measured by differences in observational consequences [10].

A stronger construal of indispensability appeals to the idea of uniqueness. According to this stronger construal, x is indispensable to a theory T if (a) x is indispensable to T in the Colyvanian sense, and (b) the application of x in T is uniquely determined. For example, quantum mechanical phenomena can be characterized in terms of group-theoretic structures as well as structures emerging from Hilbert spaces, illustrating the underdetermination of mathematical applications [11]. Accordingly, both group-theoretic structures and Hilbert-space structures are indispensable to quantum mechanics in the Colyvanian sense. However, neither application is uniquely fixed if a formulation of quantum mechanics employing group-theoretic structures yields the same theoretical results as one employing Hilbert-space structures.

A recent work on the role of complex numbers in quantum theory further illustrates why the issue of indispensability matters1. Earlier work suggested that real-valued formulations of quantum theory could preserve much of the empirical content of the standard theory [13]. More recent work by Renou and colleagues argue that certain experimentally testable predictions distinguish standard complex quantum theory from real-valued alternatives [14]. Although such results do not by themselves settle the philosophical issue, they make the question of indispensability more pressing. A philosopher who denies the existence of mathematical objects must explain why allegedly fictional mathematical structures can be so deeply embedded in scientific practice; a philosopher who affirms the existence of mathematical objects must explain why abstract objects are so successfully applicable to empirical phenomena.

3. The Nominalist Responses

Nominalists deny, or at least avoid commitment to, the existence of abstract mathematical objects. This has an immediate advantage: if there are no abstract mathematical objects, then we need not explain how human beings acquire causal or empirical access to them. The Benacerraf-style epistemic challenge is therefore less direct for nominalism than for Platonism. Still, nominalists must address two residual tasks. First, they need an account of truth-talk in mathematical practice that does not rely on abstract truth-makers. Second, they need an account of why mathematical practice appears non-arbitrary and why mathematics is so successful in scientific practice.

One nominalist strategy is to reinterpret mathematical statements through paraphrase. Field, for example, introduces the operator “according to” and suggests that “5 and 7 are twin primes” may be read as “according to arithmetic, 5 and 7 are twin primes” [15]. Hellman offers a modal-structural strategy according to which such statements can be understood in terms of what would hold if arithmetic were true, together with claims about the possibility of such structures [16]. These proposals are sophisticated and should not be dismissed as simple fictionalism. They show how one might preserve much of ordinary mathematical discourse without straightforwardly quantifying over mathematical objects. However, such strategies still leave an explanatory question unanswered. If mathematical truth is understood in terms of what follows from a given mathematical theory, we must ask why that theory, rather than alternative theories, possesses the authority it does within mathematical practice. Field treats mathematical proof as a matter of logical consequence from accepted mathematical axioms, but this shifts attention to the status of those axioms [15]. For example, if ZFC theory is treated as foundational for much of modern mathematics, a nominalist account needs to explain why ZFC theory is accepted as a legitimate basis for mathematical reasoning. This is not a decisive objection, but it indicates that nominalism requires more than a semantic paraphrase; it also requires an account of the normativity of mathematical practice.

A related issue concerns the common perception among mathematicians that some parts of mathematics are discovered rather than invented. Gowers observes that mathematical practice involves a spectrum: some developments feel more like discoveries, others more like inventions, and the difference often concerns the degree of control mathematicians have over what is produced [17]. This perception is ontologically neutral; it does not by itself support Platonism over nominalism. Still, it suggests that mathematical practice is not experienced as arbitrary construction. A nominalist can accommodate this point, but doing so requires explaining how non-arbitrariness arises without appeal to independently existing mathematical objects.

Nominalists also have a relatively natural explanation of ordinary applicability. Mathematical theories can be treated as representational tools or conceptual frameworks. Balaguer, for instance, suggests that mathematics provides theoretical apparatuses with which to make assertions about the physical world [18]. This model works well in many contexts, especially where different mathematical models can be used to represent the same phenomenon. However, the strongest cases of mathematical application in the sciences raise a further question concerning indispensability. In some areas of physics, mathematics may function merely as a convenient tool for constructing and formulating physical theories, whereas in others, it may constrain the construction and formulation of the theories themselves. If certain mathematical structures cannot be easily replaced without a loss of explanatory or predictive power—in other words, if they are indispensable—then a nominalist account must explain why mathematics plays such a stronger role in scientific theorizing. For example, the work on complex numbers in quantum mechanics mentioned at the end of the previous section presents precisely the kind of case that nominalists must explain.

One nominalization project attempts to answer this challenge by reformulating scientific theories without commitment to mathematical objects. Field’s nominalization of Newtonian mechanics is the most important example [15]. The project is philosophically valuable because it makes the indispensability issue precise. Nevertheless, its scope remains limited. Even if some scientific theories can be nominalized, it is not obvious that all mathematically rich sciences can be treated in the same way. Moreover, if a nominalized theory systematically preserves the relational structure of the original mathematical theory, one may wonder whether mathematics has been eliminated or merely re-described. Thus, nominalism remains a serious option, but it faces pressure to explain the non-arbitrariness of mathematical practice and the strongest cases of mathematical indispensability.

4. The Platonist Responses

Standard Platonism holds that mathematical objects, or at least mathematical structures, exist independently of the human mind. It is widely contended that the main advantage of Platonism is its semantic clarity [19]. If mathematical objects and facts exist, then mathematical statements can be true in a straightforward correspondence sense: a mathematical statement is true when it corresponds to an appropriate mathematical fact2. This gives Platonism a powerful explanation of why mathematics appears to be objective in both metaphysical and epistemic senses and why mathematical statements appear to be true independently of our beliefs3.

The familiar difficulty is epistemic. If mathematical objects are abstract, non-spatiotemporal, and causally inert, how do we acquire knowledge of them? This is not exactly the same as the Benacerraf challenge. The Benacerraf challenge concerns how mathematical truth can be reconciled with a theory of mathematical knowledge. Q2, as framed here, more emphasizes the cognitive side of the problem: what capacities enable human beings to acquire mathematical knowledge if human beings and mathematical knowledge belong to two metaphysically distinct worlds? Platonism gives a clear answer to the semantic question, but it still needs a convincing story about mathematical cognition.

Full-blooded Platonism attempts to solve the epistemic mystery by expanding the mathematical universes. In Balaguer’s version, every consistent mathematical theory describes some part of mathematical reality [22]. If mathematical universes are sufficiently abundant, then whatever mathematical theory we formulate is more likely to correspond to, or successfully describe, some objects/structures that actually exist. This approach is attractive because it avoids the need to identify one privileged mathematical universe. Still, it does not fully answer Q2. Firstly, even if every consistent theory has an intended model, as full-blooded Platonism maintains, this does not by itself explain how mathematicians identify axioms, recognize consistency, and come to know the objects/structures they investigate. Simply put, the plurality of mathematical universes does not explain how we gain knowledge of them. Secondly and more importantly, to determine whether a theory is consistent, we have to start from a set of axioms and rules of inference. What Q2 ultimately addresses is how we acquire or come to accept these axioms and inferential rules. The epistemic question is therefore not eliminated, but shifted toward the epistemic origins of the axioms and inferential norms that underlie mathematical practice.

Q3 presents both the applicability problem and the indispensability problem for Platonists. It is sometimes argued that one advantage of Platonism is that it can explain the applicability of mathematics to science by adopting a unified semantic account of mathematical and scientific theories. On this view, true mathematical theories, under appropriate interpretations, can be used to account for various physical phenomena. However, the truth of a mathematical theory does not, by itself, explain why that theory should be relevant to physical phenomena. An account of applicability must explain how abstract mathematical objects/structures and concrete physical phenomena/systems can be related in ways that support explanation, prediction, and theory construction. This problem concerns the connection between the mathematical and physical worlds, and in this respect it resembles the Benacerraf challenge.

Most contemporary Platonists do not regard the indispensability problem as a problem in its own right. Instead, they argue that the indispensability of certain mathematical objects to the sciences is a primitive fact that supports the existence of those objects. In its Quine–Putnam form, the indispensability argument holds that we should be ontologically committed to the entities indispensable to our best scientific theories [23,24]. Therefore, we should be committed to the existence of certain mathematical objects insofar as they are indispensable to those theories4. Baker develops a related explanatory version, according to which certain mathematical objects should be accepted as real because they play indispensable explanatory roles in science [25,26]. These arguments are important because they connect mathematical realism with actual scientific practice rather than with purely metaphysical speculation.

Nevertheless, the indispensability argument does not remove all difficulties. First, the Quine–Putnam version depends on controversial assumptions about confirmation holism and the unity of scientific theories. hilosophers such as Maddy and Dupré have emphasized that scientific practice is often more fragmented and selective than a simple holistic picture suggests [27,28]. Second, explanatory indispensability does not by itself explain applicability. If mathematics plays an explanatory role in science, we still need an account of how abstract mathematical objects become explanatorily connected to empirical phenomena, as discussed above.

To sum up, the point is not that Platonism fails. Rather, Platonism offers a strong semantic account and an influential realist interpretation of mathematical practice, but it leaves open important questions about cognition and applicability. These questions motivate the search for an alternative framework that can preserve the non-arbitrariness of mathematics while offering a more naturalistic account of mathematical knowledge acquisition and application.

5. The Nativist Theses of Mathematical Knowledge

The previous sections suggest that both nominalism and Platonism have significant resources, but also that neither framework, by itself, fully answers the three benchmark questions. A different strategy is to suspend, at least provisionally, the ontological dispute about mathematical objects and ask instead about the metaphysical nature and cognitive basis of mathematical knowledge. This section proposes two nativist theses. The first is:

• TH1. Some mathematical knowledge is innate.

By “innate,” I do not mean that human beings are born with explicit knowledge of arithmetic, geometry, or set theory, nor do I mean that mathematical knowledge must necessarily be acquired through reasoning. Rather, I mean that some of the cognitive structures underlying mathematical cognition may be part of our evolved cognitive architecture. Moreover, it is possible that some aspects of our cognitive architecture are themselves organized in mathematically significant ways, thereby opening up the possibility of acquiring related mathematical knowledge, probably in a contingent sense. Therefore, the claim that mathematical knowledge is innate differs from the claim that mathematical knowledge is a priori. A priori knowledge is standardly understood as knowledge justified independently of experience, whereas the present thesis concerns the cognitive origins of mathematical knowledge. It allows that some mathematical knowledge may be present from birth, that some mathematical knowledge may be acquired through reasoning, and that experience may activate, develop, and refine innate cognitive structures that contribute to the acquisition of some mathematical knowledge.

This thesis bears a family resemblance to both Leibniz’s and Kant’s accounts of the relation between cognition and mathematics. Like Leibniz, the nativist account rejects the assumption that innate knowledge must consist in explicitly possessed ideas or propositions. Leibniz argues that the mind contains innate principles not as fully articulated beliefs, but as dispositions or latent structures that become manifest through experience and reflection. In a similar spirit, the present account holds that some of the cognitive foundations of mathematics may be innate without implying that human beings are born with explicit knowledge of mathematical theories. However, the nativist account is closer to Kant’s view that mathematical cognition is rooted in the fundamental organization of cognition itself5. Kant argues that mathematical knowledge is possible because certain a priori forms of intuition and structures of understanding provide the conditions under which experience can be represented and organized. The present account shares the general intuition that mathematics is not merely a system of representations imposed upon an otherwise independent world, but is connected to the cognitive structures through which human beings represent the world. More specifically, TH2, proposed in the latter part of this section, suggests that mathematics may function as a cognitive pattern: certain mathematically significant structures may shape how humans organize their experience and structure their cognition.

TH1 is deliberately modest. It does not claim that all mathematical knowledge is innate. The qualifier “some” is intended at two levels. At the practical level, it accommodates the common view among mathematicians that some aspects of mathematics are discovered, whereas others are products of invention. For example, it is regarded as uncontroversial that Euler discovered Euler’s formula, i.e., $e^{i\phi }=cos\phi +isin\phi$, and it is also uncontroversial to say that Weierstrass invented the Weierstrass function, i.e., $\mathrm{f}(\mathrm{x})={\displaystyle {\sum }_{n=0}^{\infty }}{a}^{n}cos\left(b^n\pi x\right)$. At a more general level, many mathematical concepts, methods, and theories are clearly products of historical development, creative invention, and formal refinement, such as numerical analysis. Indeed, many modern mathematical constructions are not plausibly merely innate in any straightforward sense. The nativist claim therefore only suggests that some primitive forms or bases of mathematical cognition may be innate, and that these primitive forms or bases may help explain why more sophisticated mathematical knowledge is possible and non-arbitrary.

TH1 also renders the question of which parts of mathematics, if any, are innate an empirical question—one that cannot be settled by philosophical reflection alone. Nor does the fact that a mathematical concept or theory emerged early in human history indicate that it is innate. Arithmetic appeared very early in human history and is a natural candidate for innate numerical cognition, but other structures may be developmentally or cognitively basic despite being formally articulated much later. Bayesian inference, for example, was formulated in the eighteenth century, yet psychological theories of perception suggest that Bayesian-like processes may underlie aspects of perceptual adjustment [29]. This does not show that Bayesian inference itself is innate; it only illustrates that the cognitive basis of mathematics cannot be inferred directly from the historical sequence of mathematical discoveries. Therefore, this question should be addressed by empirical sciences such as cognitive science, developmental psychology, and related disciplines.

One may object that the fact that human beings possess cognitive structures that enable mathematical cognition is itself evidence for the existence of mathematical objects. Otherwise, one might ask, how could evolutionary processes give rise to cognitive structures that track non-existent entities? However, this objection presupposes a strong notion of existence, according to which mathematical knowledge and mathematical objects must be independent of the evolutionary processes that produced our mathematical cognition. An analogy can illustrate why this inference does not follow. We can design a machine to perform a particular kind of intellectual task and, at the same time, equip it with a certain capacity for self-reflection concerning how it performs that task. Suppose that, eventually, the machine is able to discover the computational procedures underlying its own operation. Does the fact that the machine can identify the procedures by which it operates entail that those procedures existed independently of the process that produced the machine? We may accept that the procedures exist independently of the machine in the sense that they were created by us. However, these procedures are human inventions designed to simulate a particular cognitive capacity. In principle, they need not be identical to, or even resemble, the mechanisms underlying the cognitive capacity they simulate. Before we created such procedures, they did not exist as independent objects. Their existence depends on the process through which they were constructed. By analogy, the fact that human beings possess cognitive structures for mathematics does not entail that mathematics itself is independent of the evolutionary processes that gave rise to those mechanisms.

With these clarifications, we can take a further step toward the main task of the nativist account: clarifying what kind of cognitive achievement mathematics represents. I propose a distinction between cognitive capacity and cognitive pattern. A cognitive capacity is a function whose result is accessible to conscious agents, such as seeing, speaking, calculating, or proving. A cognitive pattern is a deeper structure that organizes cognition without necessarily becoming an object of conscious awareness. The distinction can be illustrated independently of mathematics by the case of vision. We have conscious access to what we see, but not to the full set of neural and computational processes that structure visual perception. The capacity to see is therefore one thing; the underlying pattern by which visual information is organized is another. I use this analogy only to clarify the distinction between cognitive capacity and cognitive pattern, not to draw a parallel between visual perception and mathematical knowledge or between their respective objects. This distinction leads to the second thesis:

• TH2. Mathematics may fulfill hybrid functions, operating more fundamentally as a cognitive pattern than as a cognitive capacity.

This thesis is intentionally cautious. Mathematics is clearly a cognitive capacity in one sense: human beings can define, calculate, generalize, construct and prove mathematical theories. The suggestion is not that mathematics is never a capacity, but that its deeper role may be pattern-like. It may help structure how we perceive quantities, spatial relations, order, similarity, probability, and other features of experience. If this is correct, explicit mathematical knowledge is only one manifestation of a more general mathematical organization of cognition.

Two asymmetries motivate this view. The first is the asymmetry between the vastness of mathematics as a discipline and the relatively small portion of mathematics that is ordinarily used in everyday life. Mathematics contains an enormous range of theories, structures, and techniques, many of which have no direct role in ordinary practical reasoning. Most individuals make use of elementary numerical and spatial abilities, but they do not ordinarily need set theory, topology, abstract algebra, functional analysis, or most of higher mathematics in daily life. This does not show that such mathematics is useless; many advanced mathematical theories have important roles in science, technology, and other theoretical contexts. The point is rather that if the primary evolutionary function of mathematical cognition were simply to provide human beings with explicit mathematical knowledge for practical use, it would be surprising that mathematical inquiry so vastly exceeds ordinary practical needs. The gap between the richness of mathematics and the limited range of mathematics required in everyday action suggests that explicit mathematical knowledge may not be the main evolutionary function of mathematical cognition.

A more plausible interpretation is that mathematical cognition has a deeper and more general role. Its function may not primarily be to deliver consciously accessible mathematical propositions, but to organize experience in ways that later make explicit mathematics possible. On this interpretation, the mathematical structures that appear in conscious reasoning are partial articulations of more basic cognitive patterns. For example, our ability to distinguish one object from many, to track order and magnitude, to recognize spatial continuity and separation, or to represent probabilistic relations may depend on forms of organization that are mathematical in a broad sense, even when they are not consciously represented as mathematical knowledge. The development of explicit mathematics may then be understood as a reflective elaboration of structures already operative in cognition.

The second asymmetry concerns accessibility. Mathematical ability varies greatly among individuals, and advanced mathematics remains difficult even for many highly educated people. By contrast, basic linguistic competence is normally acquired widely, early, and without explicit instruction. If mathematics were primarily a cognitive capacity comparable to language, one might expect explicit mathematical competence to be more evenly distributed and more naturally acquired. Yet this is not the case. While elementary numerical abilities emerge early, the capacity to understand and produce advanced mathematics is uneven, fragile, and heavily dependent on education, training, notation, and cultural scaffolding. This asymmetry suggests that explicit mathematical competence may be a secondary achievement rather than the primary cognitive function of mathematics.

This point should not be overstated. The comparison with language does not show that mathematics is not a cognitive capacity at all, nor does it imply that mathematical cognition is merely artificial or culturally imposed. Rather, it suggests a distinction between two levels of mathematical cognition. At one level, there may be basic mathematical patterns that structure experience and thought. At another level, there is explicit mathematical knowledge, which partially depends on education, notation, proof practices, and theoretical reflection. The former may be widely shared and developmentally early, while the latter may be unevenly distributed and culturally elaborated. The more cautious conclusion is therefore that mathematics may have a hybrid role. Some aspects of mathematical cognition appear as conscious capacities, while others may function as underlying patterns that structure perception and thought. The nativist account developed below uses this hybrid view to reconsider mathematical truth, acquisition and application.

6. The Nativist Responses

This section explains how TH1 and TH2 may help answer the three benchmark questions. The proposal is exploratory rather than conclusive. Its aim is to show that a nativist framework can make sense of important features of mathematical knowledge without requiring an immediate commitment to either Platonism or nominalism.

6.1. Response to the Normative Question

If mathematical knowledge is, at least partly, rooted in our cognitive architecture, then the nativist account need neither affirm nor deny the existence of mathematical objects. Therefore, for the nativist account, Q1 should be read as a normative question concerning the legitimacy and interpretation of truth-talk in mathematics. The difficulty lies in whether the nativist account can offer an explanation of truth-talk in mathematical practice while remaining agnostic about mathematical objects. I argue that it can, and the key move is to distinguish mind-dependence from cognitive-dependence. By “mind-dependence,” I mean that mathematics depends on intentional choices, conventions, or inventions of individual human minds, and by “cognitive dependence,” I mean that mathematics depends on non-arbitrary, species-shared cognitive structures. Under this distinction, if mathematics is merely mind-dependent, it may appear to be an arbitrary human construction. If it is cognitively dependent, however, it depends on stable and shared features of our cognitive architecture. Such dependence would not make mathematics fictional in the ordinary sense; rather, it would make mathematical knowledge constrained by the ways in which human cognition is shaped and structured.

This distinction also helps locate nativism between Platonism and nominalism. Platonism treats mathematical truth as grounded in mind-independent objects or structures. Nominalism typically avoids such commitment and may interpret mathematics through paraphrase, fiction, or structural possibility. Nativism suggests that at least some mathematical concepts and norms may be grounded in shared cognitive structures. These concepts and norms are not chosen by individual mathematicians, and so they can explain why mathematics is not arbitrary, even if one remains agnostic about whether mathematical objects exist.

This issue of non-arbitrariness is closely related to what Clarke-Doane calls the “problem of mathematical agreement.” Clarke-Doane observes that mathematical agreement is more puzzling than it may initially appear. Not even agreement about whether a concrete string is a proof in a fixed logic can be explained in the obvious way (by recursively checking) since what follows from what in a logic—and even whether a concrete alleged proof in a logic is one—depends on the metatheory one uses to check. For example, Priest’s inconsistent arithmetic disagrees with the classical arithmetic about this. Moreover, an appeal to evolution alone does not explain the convergence insofar as the extent of agreement over metatheory required for agreement over proof exceeds what evolutionary selection would seem to have directly selected for [30].

The nativist account does not require the stronger claim that evolution directly selected particular mathematical axioms, foundations, or standards of proof. Indeed, such a claim would be implausible, especially for historically recent and highly abstract mathematics. TH1 instead proposes that evolution may have produced shared cognitive structures that constrain the formation of some basic mathematical concepts and inferential dispositions. TH2 strengthens this proposal by suggesting that some mathematically significant organization may operate at the level of cognitive pattern rather than explicit mathematical belief. If human beings share such underlying structures, then they need not independently acquire every basic mathematical norm through convention or explicit agreement. Rather, similar cognitive architectures may dispose them to organize quantity, order, spatial relations, identity, and other features of experience in sufficiently similar ways to provide common starting points for mathematical practice. Education, notation, formal proof, and other forms of cultural scaffolding can subsequently stabilize and extend this initial convergence. The nativist account therefore does not claim to provide a complete evolutionary explanation of mathematical agreement. It offers, more modestly, an explanation of why substantial convergence is possible without supposing either that every mathematical norm is explicitly encoded by evolution or that agreement must be imposed entirely by convention.

This combination of TH1 and the coherence account of mathematical truth can provide an explanation of truth-talk in mathematical practice. In mathematical practice, a proposition is treated as true when it is rigorously inferred from a set of accepted propositions in accordance with established definitions, axioms, and inferential rules. In this sense, the coherence account explains the normative role of truth-talk within mathematical practice: mathematical propositions are accepted as true insofar as they cohere with an established mathematical framework through rigorous inference. However, this account leaves open the question of why mathematicians converge on particular basic concepts, axioms, and inferential norms in the first place. This is where TH1 enters the picture. TH1 suggests that mathematical knowledge may be grounded in innate cognitive structures. These structures do not amount to innate knowledge of particular mathematical axioms or theories; rather, they provide cognitive foundations that make certain mathematical concepts and norms possible. Because these foundations are shared features of human cognition rather than arbitrary choices, the formation of certain mathematical concepts and norms may be constrained by the structure of our cognition. These concepts and norms can therefore serve as rules for mathematical practice.

The connection between Q1 and the problem of mathematical agreement can now be made explicit. Q1 asks what licenses the normative practice of treating some mathematical propositions as true and others as false. The problem of agreement asks why different mathematical agents converge, to the extent that they do, on the concepts, inferential practices, and standards that make such judgments possible. These are not identical questions, but the nativist response to Q1 bears directly on the latter because both concern the source of mathematical non-arbitrariness. If mathematical norms were merely products of unconstrained individual choice, widespread and stable agreement would indeed be difficult to explain. If, however, mathematical practice develops under shared cognitive constraints, then the same feature that gives mathematical truth-talk its non-arbitrary normative basis also provides a partial explanation of interpersonal convergence. The explanation is only partial because shared cognitive structure need not determine a unique mathematical foundation, nor does it eliminate genuine disagreement about axioms, logics, or advanced mathematical theories. It nevertheless explains why such disagreement occurs against a substantial background of shared concepts and inferential practices rather than against a background of unrestricted mathematical variation.

In this sense, the most basic mathematical concepts and norms do not require further mathematical justification, because they constitute part of the framework within which mathematical justification takes place. This is the main contribution that TH1 makes to the restricted coherence account: it helps address the central challenges facing the restricted coherence account, namely relativism and circularity, while also providing a partial response to the problem of mathematical agreement. The nativist response is not to deny these worries but to mitigate them. Mathematical coherence is not coherence with any arbitrary set of beliefs; rather, it is coherence within practices constrained by logic, proof, public standards, and shared cognitive structures. The nativist account therefore suggests a way of explaining both the non-arbitrariness of mathematical truth-talk and the substantial convergence of mathematical practice without requiring a commitment to independently existing mathematical objects.

6.2. Response to the Epistemic Question

According to TH1, the acquisition of mathematical knowledge is neither purely empirical nor purely a priori. More specifically, as mentioned above, TH1 allows that some mathematical knowledge may be present from birth, while TH2 further suggests that experience may activate and shape innate cognitive structures, and that reasoning may refine them into explicit mathematical concepts and theories. The nativist account does not treat human cognition and mathematics as belonging to categorically different domains, a view that would render their connection either impossible or mysterious. Rather, it suggests that the relationship between human cognition and mathematics can be investigated empirically. Therefore, for the nativist account, Q2 is not merely an epistemic question. Rhetorically speaking, the nativist account suggests that human beings are equipped with a “mathematical brain” through evolution, and this possibility turns the epistemic question into an empirical question concerning the development and functioning of the cognitive architecture that enables mathematical cognition and the acquisition of explicit mathematical knowledge.

TH2 may further explain why mathematical acquisition differs from the acquisition of many other capacities. If mathematics functions partly as a cognitive pattern, then it may underlie the very processes through which explicit mathematical knowledge is acquired. This does not make the acquisition of mathematics mysterious; rather, it suggests that mathematical learning involves making implicit cognitive structures explicit through reflection, education, and formalization. The analogy with vision is again useful: we do not learn the mechanisms of vision in order to see, but we may later study vision scientifically. Similarly, we may employ mathematical structures cognitively before we formulate them as explicit mathematical concepts and theories.

TH1 also leaves room for creativity. Even if some primitive mathematical cognition is innate, advanced mathematics involves invention, abstraction, and formal construction. Reasoning is indispensable for proof and justification, but it does not by itself generate every mathematical concept. Just as grammar constrains intelligible language without producing every possible novel, mathematical structures constrain legitimate mathematical reasoning without determining every mathematical idea. This helps explain why mathematics can be both constrained and creative.

Empirical work on early numerical cognition supports the general plausibility of this approach. Wynn, for example, argues that young infants can represent and reason about numbers of things [31]. Evidence of numerical competence in nonhuman primates also suggests continuity between human mathematical cognition and broader cognitive capacities [32]. Such findings do not establish the full nativist account, but they provide a promising empirical foundation for the claim that some basic components of mathematical cognition may have innate origins.

6.3. Response to the Question of Indispensable Application

The nativist account also offers a speculative yet elegant explanation of mathematical applicability. If mathematics is partly a cognitive pattern, then mathematics applies to science not because abstract objects or theories correspond to the physical world, nor merely because we invent useful fictions. Rather, mathematics applies because scientific theories are constructed through human representations of the world, and those representations may already be structured in mathematically significant ways. In short, within the nativist account, the applicability of mathematics is less mysterious because our representations of the world are themselves partly shaped by mathematical structures.

To better illustrate this point, consider elementary arithmetic. When we see two apples in one group and three apples in another, we naturally represent the situation as involving two and three objects. A Platonist may say that these groups instantiate the numbers 2 and 3. The nativist account need not deny this possibility, but it offers a different explanation: our shared cognitive architecture may dispose us to represent such collections numerically. Our agreement in representing the groups of apples as two and three apples does not arise because an external entity, such as the numbers 2 and 3, fixes our representations uniformly. Rather, it arises because we share similar cognitive structures that shape how we represent collections of objects. The same applies when the two groups are combined and represented as 2 + 3 = 5. The stability of such representations may reflect not only external regularities but also the mathematical organization of human cognition. Scientific theories are based on observations, measurements, models, and inferences, all of which depend on human modes of representation. If these modes of representation are, at least partly, structured by mathematics, then the mathematical character of science becomes less surprising.

Once the applicability of mathematics is explained in this nativist manner, the question of indispensability also becomes much less puzzling: if scientific theories are constructed through mathematically structured representations, it becomes less surprising that some mathematical structures may function as indispensable tools in science. Nevertheless, the stronger claim that certain mathematical theories are indispensable to particular sciences should be stated cautiously. The nativist account suggests that some mathematical theories may be difficult to eliminate from scientific theories because they are embedded not only in theoretical formulations but also in the ways empirical phenomena are represented, measured, and organized. This does not provide a criterion for determining which mathematical theories are indispensable to which sciences. Moreover, not every indispensable mathematical theory is innate, nor do all mathematical theories employed in science necessarily reflect underlying cognitive structures. Therefore, the nativist account provides only a naturalistic framework for understanding why mathematics can be more than a dispensable instrument in scientific practice, while leaving room for further empirical investigation.

7. Conclusions

This paper has used three questions about mathematical knowledge to compare nominalist, Platonist, and nativist approaches. Nominalist accounts have the advantage of avoiding commitment to abstract mathematical objects and thereby avoiding one version of the epistemic challenge. However, they still face the task of explaining the non-arbitrariness of mathematical practice, as well as the strongest cases of mathematical indispensability in science. Platonist accounts offer a clear semantics of mathematical truth and a strong explanation of mathematical objectivity, but they continue to face questions about how finite cognitive agents acquire knowledge of abstract entities and how abstract mathematical structures are connected to empirical science.

The nativist account proposed here does not attempt to settle the ontological debate about mathematical objects. Instead, it shifts attention to the cognitive basis of mathematical knowledge. Its first thesis is that some primitive forms of mathematical knowledge or mathematical cognition may be innate. Its second thesis is that mathematics may function not only as a conscious cognitive capacity but also, and perhaps more fundamentally, as a cognitive pattern that structures perception and thought. These theses allow mathematical knowledge to be understood as cognitive-dependent without reducing it to arbitrary fiction.

On this basis, the nativist account offers three contributions. First, it supports a restricted coherence account of mathematical truth, according to which mathematical truth is grounded in proof-governed coherence within practices constrained by shared cognitive structures. Second, it reframes the epistemic question as partly empirical: understanding mathematical knowledge requires philosophical analysis together with research on mathematical cognition, development, and perception. Third, it offers a naturalistic explanation of mathematical applicability and indispensability: mathematics may be central to science because representation itself is partly shaped and structured by mathematics.

The account remains programmatic. It requires further clarification of which aspects of mathematical cognition are innate, how innate structures are transformed into explicit mathematical knowledge, and how far the explanation extends beyond elementary mathematics. Nevertheless, it provides a promising alternative to the familiar opposition between Platonism and nominalism. By treating mathematics as rooted in shared cognitive structures while remaining agnostic about mathematical objects, the nativist account preserves important features of mathematical objectivity and offers a plausible framework for future philosophical and empirical investigation.