Reason: Mathematics as a Difficult Dialect of Language
Counting, Measuring, and the Human Invention of Rules
Author’s Preface
This essay continues my exploration of reasoning, language, and their boundaries. Mathematics has long been presented as if it were something other than language, occupying an elevated or even metaphysical status. Philosophers have often treated it as belonging to a Platonic “third realm,” beyond the material world and beyond the mind, yet somehow accessible to both. This position, influential for centuries, persists in modern thought. But upon closer inspection, mathematics is best understood as a highly specialized human invention: a dialect of language that operates under strict conventions.
In earlier work I emphasized that counting, measurement, and the use of probability are not self-sufficient operations. They depend on prior acts of description and categorization. Here, I extend that observation by focusing on mathematics itself: its character as language, its reliance on human invention, its rules of transformation, and its limitations in describing the world.
Introduction
Mathematics is widely regarded as a realm apart—timeless, universal, and absolute. By contrast, natural languages are acknowledged as contingent, cultural, and evolving. Yet this contrast is misleading. Both mathematics and natural language share the same foundation: they are systems of symbols that must be learned, transmitted, and agreed upon within human communities. Both serve to describe aspects of the world.
The purpose of this essay is to argue that mathematics is best understood not as a Platonic object but as a difficult dialect of human language. It differs from everyday speech in its orthography, in the rigor of its rules, and in its explicit use of self-referential transformations. But in its essence, it remains language—a system of meaning and communication, inseparable from human judgment and invention.
Discussion
1. Counting and Measurement as Human Decisions
When mathematics is applied to the world, it begins with counting and measurement. Yet counting is never given by nature. A flock of sheep may be counted as “thirty animals,” or “ten rams and twenty ewes,” or “sixty legs.” Measurement is equally dependent on choice: length may be expressed in feet, meters, or cubits; time in hours, seconds, or heartbeats.
In each case, decisions must be made about what to count, how to measure, when and where to record, and for what purpose. These judgments precede the numbers. Without them, there is only the world in its fullness, not already partitioned into discrete units or magnitudes.
2. Mathematics as Language and Orthography
Mathematics is language with a specialized orthography. Its symbols condense long statements into compact forms. “Two plus two equals four” becomes “2 + 2 = 4.” This concision reduces ambiguity but requires a shared system of rules. Without the conventions, the symbols are meaningless scratches. With them, they become intelligible.
Like grammar in natural languages, mathematical rules establish how expressions may be constructed and transformed. The difference is that mathematics makes such transformations explicit and self-referential. From “x + 3 = 7” one may, by rule, obtain “x = 4.” In ordinary language, paraphrase exists, but with less rigidity and less guarantee of meaning-preservation.
3. The Illusion of a Platonic Realm
The consistency and apparent universality of mathematics have encouraged the illusion that it must exist outside language. Philosophers from Plato onward have posited a “third realm” of mathematical objects, timeless and absolute. But this realm is incoherent. No clear account has ever been given of how such a domain could interact with the human mind or the material world.
The simpler explanation is that mathematics is a human invention—an abstract extension of language that gains its power from precision and systematization. Its universality is an artifact of shared rules, not proof of an otherworldly existence.
4. Learning and Difficulty
Language must be learned; mathematics must be learned. Both are human inventions transmitted through culture. The difference is that ordinary language is absorbed naturally through immersion, while mathematics requires formal instruction. Its difficulty stems from the density of its conventions, the rigor of its transformations, and the elaborate structures it builds. Each new level of mathematics demands mastery of prior rules before advancing.
This difficulty should not be mistaken for evidence of transcendence. It is the difficulty of mastering a dialect with especially strict rules—a dialect designed for concision, clarity, and abstraction, but still a dialect of language.
5. Meaning and Understanding
Language is inseparable from meaning. Without meaning, it is only noise or marks. Mathematics shares this dependence. Its symbols acquire significance only within a system of conventions that link them to concepts and operations. Without learning the rules, “2 + 3 = 5” is no more than a set of arbitrary symbols. With the rules, it is intelligible and communicative.
Thus, mathematics is a system of meaning, not a collection of mystical entities. It is law-governed, conventional, and functional—grounded in human invention and sustained by shared understanding.
Summary
Mathematics is not a Platonic realm but a human invention: a dialect of language with a specialized orthography and self-referential rules. Its apparent universality comes from shared conventions, not metaphysical status. Counting and measurement always depend on prior judgments about what to include, how to record, and for what purpose. Like any language, mathematics must be learned, and it is especially difficult because of its density of rules and its distance from everyday experience. Ultimately, it is a system of meaning and communication, inseparable from human invention.
Readings
Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.
An accessible treatment of mathematics as a human cognitive construction, grounding abstract mathematical concepts in bodily experience and metaphor.
Polanyi, M. (1966). The tacit dimension. Garden City, NY: Doubleday.
Polanyi’s discussion of tacit knowledge is relevant here because mathematical conventions depend on shared, often unspoken practices of use and interpretation. Mathematics works not because of metaphysical existence but because of tacitly shared rules.
Cartwright, N. (1983). How the laws of physics lie. Oxford: Clarendon Press.
Although focused on physics, Cartwright’s analysis underscores that mathematical laws and models are selective descriptions, not mirrors of reality. They describe aspects of the world under limited circumstances.
Kline, M. (1980). Mathematics: The loss of certainty. New York: Oxford University Press.
A historical survey of mathematics showing how its supposed certainty has been undermined over time, reinforcing the view that mathematics is contingent and human-made.
Wilder, R. L. (1967). The origins of modern mathematics. New York: Scientific American.
A historical account emphasizing the cultural and linguistic invention of mathematical concepts, demonstrating that mathematics develops within human traditions rather than existing outside them.