Reason: Mathematics as an Evolving Symbolic Language, Not a Realm of Eternal Truths
Against Platonism and in Defense of Mathematics as Human Practice and Linguistic Construction
Author’s Preface
This essay develops a view of mathematics not as a discovery of eternal truths, but as a symbolic practice grounded in cognition, language, and experience. It challenges the enduring influence of metaphysical idealism and formalist abstraction by situating mathematics within the continuum of human reasoning—fallible, evolving, and linguistically mediated. While mathematical formalism offers precision and generativity, it does not exist outside of human interpretation, learning, and application. Every rule, transformation, or proof must be understood, validated, and communicated by minds embedded in time, culture, and language.
The reflections presented here do not deny the power of formal systems, but question the philosophical assumptions often made about their status. Rather than positing mathematics as a window into an abstract realm, this essay argues that mathematics, like all language, is a product of symbolic activity: invented, learned, practiced, and modified through human interaction. It is this view—not one of diminished rigor but of clarified grounding—that guides the following, which is a condensation of deeper and more extensive reflection:
Introduction
Mathematics is widely viewed as the crown jewel of human reasoning: a system of formal truths so precise and universal that it transcends language, culture, and even cognition. Yet this view, still popular in some philosophical and educational circles, fails to withstand scrutiny. The symbolic systems we call mathematics are neither eternal nor metaphysically pure; they are linguistically encoded, cognitively embedded, and socially transmitted. This essay argues that mathematics is not a domain of timeless truths but a specialized dialect of language. Its rules are learned, its meaning is interpreted, and its mastery depends on developmental, neurological, and social factors. The discussion draws on insights from philosophy of language, cognitive science, comparative psychology, and educational practice to dismantle the illusion of mathematics as metaphysical and to recast it as symbolic activity grounded in human cognition.
Discussion
1. Mathematics as Language Mathematics satisfies all the formal criteria for being a language. It is discrete, compositional, generative, modular, and systematic. It operates with a finite set of symbols governed by formation and transformation rules. These symbols are assigned meanings and are embedded within systems that facilitate inference, transformation, and abstraction. Like any language, it must be learned and interpreted. Its expressive power lies in its concision, unambiguity, and rule-based manipulability, not in any access to metaphysical truth.
2. Formal vs. Natural Language The distinction between mathematical and natural language is one of degree, not kind. Natural language is pragmatically rich and interpretively flexible. Mathematical language constrains ambiguity through syntax, semantic rigidity, and explicit transformation rules. Both are symbolic, learned, and context-sensitive. Mathematics minimizes interpretive overhead in exchange for greater abstraction and manipulation efficiency. Yet even the most advanced mathematical notation relies on natural language for justification, instruction, and contextual grounding.
3. Internalization and Learning Mathematical knowledge is not given—it is acquired. Children do not learn algebra by memorizing axioms; they learn through example, feedback, and procedural repetition. The “click” of understanding is cognitive, not formal. Most learners internalize rules tacitly before they can state them explicitly. Procedural fluency precedes metalinguistic awareness. Like language acquisition, mathematical learning involves trial, error, correction, and pattern recognition. There is no magical access to formal rules; they are absorbed, practiced, and only later abstracted.
4. Rules and Semantics Mathematical rules are not syntactic manipulations devoid of meaning. They presuppose semantic grounding. For example, the rule a + b = b + a assumes the commutativity of addition—an assumption dependent on the interpretation of "+" in a particular domain. Confusing rules with their symbolic instantiations (e.g., mistaking "a + b = b + a" for the rule itself) obscures the semantic structure of formal systems. Rules are generative operations in a metalanguage, not merely symbolic identities.
5. Expressive Equivalence and Cognitive Load Anything expressible in mathematics can, in principle, be expressed in natural language. However, doing so typically results in increased verbosity, ambiguity, and cognitive overhead. Mathematical notation is optimized for symbolic density and manipulation. Translating equations like E = mc^2 into English yields cumbersome, less tractable descriptions. The advantage of formal notation lies not in deeper meaning but in cognitive tractability and operational clarity.
6. Formal Systems and Inference Mathematical and logical systems encode inference rules within the language itself. These rules are explicitly defined and universally applicable within their formal domains. Unlike in natural language, where inference is contextual and pragmatic, formal systems embed inference into the structure of expression. Yet these rules are still learned, tested, and revised. They are not self-evident, and their validity is not metaphysical—it is practical, demonstrated through reliability and consistency.
7. Rejecting Syntax-Only Views The claim that mathematics is purely syntactic is incoherent. Syntax may define permissible symbol arrangements, but it does not confer meaning or validity. The interpretation of formal systems is always semantically charged. Even Gödel numbering—a technique often cited as syntactic formalism—requires interpretive assumptions to be meaningful. Without semantics, syntax is manipulation without insight.
8. Situated Reasoning and Context Dependence All reasoning, including formal reasoning, is situated. Rules are adopted, interpreted, and validated through practice. Different formalisms (e.g., propositional logic, modal logic, constructive logic) reflect contextual choices, not absolute standards. Mathematical truth emerges from a history of use, not a transcendent reality. That systems evolve, contradict, and compete is evidence of practice-driven development, not metaphysical disarray.
9. Comparative Cognition and Non-Linguistic Reasoning Animals such as corvids, elephants, and apes exhibit reasoning capabilities—problem solving, tool use, spatial inference—without formal language. This shows that intelligent behavior does not require symbolic formalisms. However, formal systems like mathematics do require linguistic scaffolding: inner speech, labeling, and verbal modeling are indispensable. Without language, there is no formal proof or symbolic manipulation.
10. Neural Substrates of Symbolic Thought Mathematical ability is grounded in neurobiology. Brain damage, developmental disorders, and aging can impair symbolic reasoning. Acalculia illustrates that symbolic arithmetic is not metaphysically pure—it is biologically contingent. The idea of mathematical truth existing independently of brains collapses in light of such evidence. Mathematics is not accessed—it is enacted.
11. Platonism and Its Residual Forms Though widely discredited, metaphysical Platonism persists in philosophy and even among practicing mathematicians. Claims that mathematical truths are “discovered” assume ontological status independent of mind. This is untenable. The objectivity of mathematics derives from shared conventions, not metaphysical grounding. The “unreasonable effectiveness” of mathematics in the sciences is a reflection of its design as a modeling tool, not evidence of eternal truths.
12. Evolution and Verification of Mathematical Systems Mathematics is dynamic. New fields emerge, new rules are proposed, and entire frameworks are discarded. Verification is not metaphysical but procedural—proof, counterexample, and practical application. Verification is a communal activity embedded in discourse, judgment, and interpretation. Mathematics progresses not through revelation but through refinement.
13. Consciousness, Understanding, and Meaning Formal systems are empty without consciousness. Symbols mean nothing unless interpreted. Understanding is not reducible to symbol manipulation; it involves attention, memory, prior knowledge, and metacognition. Meaning is relational, not atomic. Every symbol draws its meaning from a web of associations, usage patterns, and contextual cues. This is true in mathematics as much as in language.
14. Mathematics as Human Practice Mathematics is not separate from human life—it is embedded in it. It is learned, taught, explained, forgotten, recovered, and applied. It reflects the needs, limits, and possibilities of human cognition. Its canonical forms—textbooks, notation, theorems—are tools for organizing and transmitting knowledge, not access points to another realm.
Summary
Mathematics is a rule-governed, meaning-laden symbolic system, optimized for abstract reasoning but dependent on human cognition and cultural scaffolding. It is not an ontological domain of eternal truths but a functional subset of language. Its rules are not divinely imposed but discovered, tested, and refined through use. Its meanings are not intrinsic but assigned, interpreted, and understood in context. It differs from natural language only in its constraints, not in its nature. The mystique of mathematics as metaphysical is a residue of philosophical error. What remains is a powerful, flexible, and indispensable linguistic tool—one whose real significance lies not in its timelessness but in its cognitive and practical utility.
Readings List (APA Format):
Aczel, A. D. (2000). The mystery of the Aleph: Mathematics, the Kabbalah, and the search for infinity. Four Walls Eight Windows.
→ A narrative exploration of mathematical infinity, blending intellectual history and conceptual discussion without technical depth.
Devlin, K. (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. Basic Books.
→ A cognitively grounded, accessible account of mathematics as a human language capacity.
Ernest, P. (1998). Social constructivism as a philosophy of mathematics. State University of New York Press.
→ Explores mathematics as a socially embedded linguistic practice, not a system of eternal truths. Written for a broad academic audience.
Hersh, R. (1997). What is mathematics, really? Oxford University Press.
→ A readable argument against Platonism, presenting mathematics as a human cultural activity rooted in language and thought.
Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.
→ Advocates a cognitive-linguistic view of mathematics, emphasizing metaphor and embodied cognition. Explicitly anti-Platonist and non-technical in tone.
Livingston, E. (1986). The ethnomethodological foundations of mathematics. Routledge & Kegan Paul.
→ Investigates how mathematical reasoning is carried out in real-world practice. Qualitative and sociologically framed.
Mazur, B. (2003). Imagining numbers (particularly the square root of minus fifteen). Farrar, Straus and Giroux.
→ An informal but philosophically aware meditation on how we understand abstract mathematical objects through language and mental imagery.
Norton, J. D. (2007). Causation as folk science. Philosophers’ Imprint, 7(6), 1–22.
→ While focused on causation, this paper provides a model of concept critique that parallels many arguments about mathematical reasoning and everyday cognition.
Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. Basic Books.
→ Discusses learning formal systems (especially mathematical ones) through intuitive engagement and symbolic play, not rigid instruction.
Wigner, E. P. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics, 13(1), 1–14.
→ Classic essay exploring the philosophical puzzle of why abstract mathematics applies so well to the physical world. Clear, concise, and widely accessible.