Mathematics as Understanding, Not Platonism

On the Linguistic, Interpretive, and Psychological Nature of Mathematical Practice

Author’s Preface

This essay rejects the view that mathematics reveals truths from a realm of Platonic abstraction and instead argues that mathematics is a human activity, grounded in language, understanding, and interpretation. The argument develops from observations about how mathematics is learned, how proofs function as persuasive texts, and how fluency in mathematics resembles fluency in reading or language. While the formalisms of mathematics may seem austere and universal, they depend entirely on the capacities of human minds to learn, comprehend, and recognize internal coherence. This shift from Platonism to a linguistic and cognitive view of mathematics has profound implications for how mathematical knowledge is understood, taught, and applied.

Introduction

Mathematics has long enjoyed a reputation as the most objective of disciplines—a domain of thought free from human error, perspective, or bias. This aura of objectivity derives in part from a metaphysical assumption: that mathematical entities and truths exist independently of human minds, much like Platonic forms. According to this view, mathematical work consists of discovering truths that already exist in an abstract, non-empirical realm.

But this assumption is not necessary. Nor is it coherent. Mathematical practice, from basic arithmetic to advanced proof theory, is dependent on human understanding. It is transmitted through language, learned through instruction, and understood through interpretation. There is no mathematics apart from minds that comprehend it.

This essay develops an alternative account of mathematics as a human activity—one that operates through language, requires comprehension, and depends on shared standards of intelligibility and persuasion. It considers the nature of mathematical understanding, the variability of interpretive skill among individuals, and the inherently rhetorical and judgment-based nature of mathematical proof. The aim is not to diminish mathematics, but to locate it where it truly belongs: in the human mind and in the collective practices of understanding.

Discussion

Mathematics Requires Understanding

Mathematics is sometimes imagined as a pristine, objective formalism that exists apart from human interpretation. But such a view cannot withstand scrutiny. Mathematical reasoning always depends on understanding. One cannot engage with mathematics without grasping the significance of its symbols, structures, and logical relations.

Terms like “understanding” and “meaning” are notoriously slippery. They do not admit of precise definitions. Yet they are central to all linguistic activity, and mathematics is no exception. We know when we understand a mathematical idea—not because we perceive some metaphysical truth, but because we can work with it coherently, apply it in context, and explain it to others.

This capacity to understand varies widely. Some people develop it with ease; others struggle. The distribution of mathematical aptitude is not even. Fluency in mathematics, like fluency in any language, is a learned and embodied skill—not merely a function of exposure but of practice, context, and cognitive facility.

Mathematical Reading as Cognitive Fluency

The contrast between unskilled and skilled reading is instructive. Novice readers process words sequentially, decoding one at a time. Skilled readers perceive entire phrases or clauses as meaningful wholes. They interpret not just words, but assertions.

A similar contrast applies to mathematical comprehension. Beginners often read equations or expressions symbol by symbol, interpreting each part in isolation. Skilled mathematicians, by contrast, perceive larger structures. They grasp the form of an argument or the pattern in a set of operations at once.

This fluency enables faster and deeper understanding. It is not mechanical decoding; it is recognition of meaningful patterns. As in reading, it requires immersion in the language, familiarity with its grammar, and repeated experience with its forms. And, as in reading, the ability to interpret and comprehend varies across individuals and depends on both training and innate cognitive traits.

Formal Proofs and the Role of Persuasion

One of the most distinctive practices in mathematics is the construction and evaluation of formal proofs. A proof, conventionally defined, is a sequence of statements that follow from one another according to rules of inference, starting from axioms or previously established results. This gives the appearance of mechanical certainty.

But proof is not purely mechanical. It is creative. Once constructed, a proof must be evaluated by a reader who decides whether it conforms to the rules and whether the steps follow validly. This decision is interpretive. It depends on whether the reader is persuaded that the rules have been respected and that the conclusion has indeed been reached from the premises.

Thus, proof is not an object with a fixed, Platonic identity. It is a text that must be interpreted and judged. The persuasion involved is cognitive, not emotional. The reader must be convinced by the structure, coherence, and clarity of the argument. But the act of being convinced remains a psychological phenomenon. There is no reified entity called “the proof” outside of human minds evaluating it.

Proofs, then, are not metaphysical truths. They are structured linguistic performances designed to satisfy a community of trained readers. They depend on the internal logic of a system, but also on human comprehension. No proof is valid apart from the capacity of someone to recognize it as valid.

The Variability of Proofs and Reader Comprehension

Proofs vary in form and expression. They may be spelled out in exhaustive detail or compressed into a few elegant lines. Steps may be made explicit or left for the reader to fill in. This variability is not accidental. It reflects the communicative function of proof.

A proof must be understood. Its structure must be grasped, its transitions followed, and its conclusion seen as entailed. This process is not uniform across readers. A highly condensed proof may be obvious to an expert but completely opaque to a novice. Conversely, a detailed proof may help a beginner but seem tedious to someone more experienced.

This means that the clarity of a proof is relative. It depends not on an absolute standard but on the interpretive capacity of the reader. The common classroom phrase “the proof is left as an exercise” reveals this clearly. The author believes that the student has the ability to supply the missing steps. Whether this belief is accurate is a different matter.

No proof stands independently of its readers. It gains its validity only when readers are persuaded that it conforms to the rules of the system and that the conclusion follows from the premises. Even when a community agrees on a proof, what that agreement amounts to is shared persuasion—an epistemic consensus, not a metaphysical discovery.

Mathematical Practice as a Linguistic and Cognitive Activity

Mathematics, when stripped of its mystique, is a language. It has syntax, grammar, punctuation, operators, and conventions. It has assumed meanings for symbols, contextual rules for interpretation, and domain-specific vocabularies. Its practitioners undergo extensive training to master its forms, understand its assumptions, and apply its transformations correctly.

Just as in natural language, meaning in mathematics is not fixed by the symbols themselves. It arises from usage, context, and convention. The same symbol may have different meanings in different subfields. The equal sign (“=”) may denote identity, assignment, or equivalence depending on context. Glyphs and operators acquire meaning only within a shared framework of rules and expectations.

And just as in language, misunderstandings occur. Misinterpretation, ambiguity, and confusion are not foreign to mathematics. They are integral to its practice, and part of what makes clarity so prized. Precision in mathematical writing is valued not because it reflects a Platonic ideal, but because it reduces the risk of failed communication between minds.

Mathematics is, in this light, a highly structured, specialized dialect of human language—tightly constrained, symbolically rich, and cognitively demanding. It is not external to human experience. It is part of it.

Summary

This essay has argued that mathematics should be understood not as a reflection of Platonic forms, but as a human linguistic and cognitive activity. Mathematics depends on understanding. It requires fluency in symbolic manipulation, comprehension of internal rules, and shared conventions for communication. The interpretation of mathematical texts, especially proofs, is an act of judgment and persuasion—not a glimpse into a metaphysical realm.

Mathematical skill resembles language fluency. It is developed through immersion, repetition, and reflection. The ability to follow or produce a proof varies across individuals and contexts. There is no proof apart from a human mind persuaded by its structure.

Platonic metaphors obscure more than they reveal. They suggest a detachment from human cognition that does not exist. Mathematics is learned, used, evaluated, and transmitted through language. Its rigor arises not from metaphysics but from sustained human efforts to reduce ambiguity and ensure clarity.

In viewing mathematics this way—as understanding rather than abstraction—we locate it properly within the human world. We see it not as a portal to another realm, but as one of the most sophisticated and powerful tools ever created by embodied minds to communicate, model, and reason.

Readings

Lakoff, G., & Núñez, R. E. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.
A key text arguing that mathematics arises from bodily experience and metaphorical reasoning rather than abstract discovery. The authors challenge the Platonic view and develop a cognitive-linguistic theory of mathematical thought.

Cartwright, N. (1999). The Dappled World: A Study of the Boundaries of Science. Cambridge: Cambridge University Press.
Though not solely about mathematics, Cartwright critiques the applicability of formal models—including mathematical ones—outside their proper domains. She emphasizes the importance of context and interpretation in scientific and mathematical reasoning.

Polanyi, M. (1966). The Tacit Dimension. Chicago: University of Chicago Press.
Polanyi explores the idea that understanding always involves tacit knowledge that cannot be fully formalized. This perspective aligns with the argument that mathematical understanding is not reducible to formalism.

Hersh, R. (1997). What Is Mathematics, Really? Oxford: Oxford University Press.
A thorough critique of Platonism in mathematics. Hersh defends the idea that mathematics is a human construct rooted in shared practice and understanding rather than abstract reality.

Kline, M. (1980). Mathematics: The Loss of Certainty. New York: Oxford University Press.
A historical account of changing ideas in mathematics, with particular attention to crises of foundationalism and the collapse of certainty. This work provides context for understanding why the Platonic view has declined.