Reason: Learning Mathematics as a Compressed Language System
Exploring the ideas of language, symbols, and the learning of mathematical thought
Author’s Preface
This essay offers a descriptive account of how mathematics is taught, learned, and understood—not as a self-contained symbolic system, but as a specialized form of writing embedded within language. The central claim is that a substantial part of learning formal mathematics involves becoming fluent in a symbolic system that functions like a specialized language and spelling code. This system compresses verbal reasoning into compact forms optimized for precision and manipulation, yet it remains unintelligible apart from the linguistic scaffolding that defines and explains its elements.
Contrary to popular claims that mathematics is a “universal language,” this essay argues that mathematical symbols are not self-explanatory and cannot communicate meaning in isolation. Their intelligibility and utility depend on language—spoken or written—for instruction, interpretation, and use. Each linguistic group has to scaffold the symbols with their own language. The position developed here aims to clarify how mathematics operates as a mode of symbolic expression grounded in linguistic mediation. The discussion draws on research in mathematics education, cognitive psychology, and the philosophy of language and mathematics.
Introduction
Mathematics is often described as a language of symbols, an abstract system that transcends the constraints of ordinary speech. Yet in every setting where mathematics is taught or practiced, language plays a central role. Explanations, instructions, definitions, and proofs are expressed through natural language, and even symbolic expressions are introduced, clarified, and manipulated using verbal formulations. Mathematical symbols, for all their formal power, do not stand alone—they function as abbreviations of verbal reasoning.
This essay explores the idea that mathematical notation operates as a compressed spelling system: a way of writing that condenses linguistic statements into symbolic form. While the full experience of learning mathematics includes concepts, reasoning, syntax and abstraction, a necessary component of that experience is learning to decode and produce symbolic expressions that act as condensed versions of verbal formulations. The essay will examine how symbols depend on language for meaning, how mathematical communication is structured by linguistic framing, and how this hybrid system shapes understanding.
Discussion
Mathematical Symbols as Compressed Verbal Forms
Most mathematical symbols can be understood as condensed equivalents of verbal expressions. The equality sign (=) compresses the phrase “is equal to”; the summation symbol (∑) abbreviates “the sum of…”; quantifiers such as ∀ and ∃ correspond to “for all” and “there exists,” respectively. These are not freestanding signs with intrinsic meaning. Their interpretation depends on prior verbal instruction and contextual explanation (Lakoff & Núñez, 2000; Duval, 2006).
This view contrasts with the notion that mathematical symbols form a language unto themselves. They are better described as visual representations of already linguistically formed ideas. As Devlin (2000) notes, symbols in mathematics are useful precisely because they are not tied to sound, allowing for manipulation and abstraction—but their meaning arises only when grounded in language.
Language as Interpretive Medium
Research in mathematics education confirms that language is central to mathematical learning. Anna Sfard (2008) argues that becoming competent in mathematics involves entering a specialized discourse, one that combines verbal, symbolic, and visual forms. These forms are not equal in communicative power: symbols require verbal scaffolding to be understood and used effectively.
Even manipulatives and visual tools require explanation. A learner watching someone combine algebra tiles or arrange number blocks gains little unless the instructor names the operations, explains the relationships, and defines the goal. As Goldin-Meadow (2003) demonstrates, even gestures gain meaning in mathematics only when synchronized with speech. The idea of discovering structure purely through visual or physical intuition is misleading; linguistic mediation is always present, whether external or internal.
The Role of Internal Language in Symbol Use
Cognitive research reinforces the centrality of language in symbolic reasoning. Baddeley’s (2000) model of working memory includes the phonological loop, which temporarily stores and rehearses verbal information. Mathematical problem-solving often depends on this loop: learners internally articulate steps, interpret notation to themselves, and encode relationships in verbal form before acting symbolically.
Thus, even when manipulating symbols, individuals rely on internalized language to guide their actions. The process of forming a hypothesis, checking a calculation, or interpreting a graph is rarely, if ever, purely visual or logical. It is linguistically mediated cognition, where symbols serve as tools for expressing—and compressing—already-formed verbal ideas.
Dialogue and the Limits of Symbols
Two individuals cannot sustain a mathematical dialogue using symbols alone. Even highly formal proofs are embedded within prose. Collaboration, instruction, and critique depend on linguistic explanations that frame, justify, and negotiate meaning. Without verbal mediation, symbols become ambiguous, especially when there is no shared agreement on assumptions, definitions, or context.
This point is supported by the work of Radford (2003), who shows that mathematical meaning arises from the interplay of symbols, gestures, and speech. No single form suffices on its own. Mathematical communication requires multimodal integration, and language is the primary organizing medium.
Parentheses, brackets, and operator precedence—mathematical equivalents of punctuation—help structure symbolic expressions but do not convey meaning independently. Just as a sentence becomes ambiguous or unintelligible when stripped of grammar, a symbolic expression loses its coherence when detached from linguistic context. Halliday’s (1978) concept of "language as social semiotic" applies directly here: symbols gain meaning not as isolated tokens but as elements in a structured linguistic expression.
Notation as a Specialized Writing System
Mathematics should not be described as a separate language, but as a specialized writing system within language. It is orthographic in the broad sense—though the technical term orthography may be unfamiliar to many readers. In this context, an orthographic system refers to a rule-based method for writing—one that dictates how symbols are used, arranged, and interpreted.
Mathematical notation, like other writing systems, is governed by rules of form and structure: placement, syntax, operator order, and grouping conventions. It does not represent sound (as in alphabetic scripts), but it does encode meaning through spatial and visual form. Its function is not to replace language but to compress it, allowing for efficient expression and manipulation once linguistic understanding is in place (Olson, 1994).
The relationship is not reciprocal. While any symbolic expression must be interpreted linguistically to gain meaning, not every linguistic statement can be compactly expressed symbolically. In this sense, mathematical symbols are a subdomain of linguistic expression, not an alternative to it.
Mathematical Vocabulary: Specialized, but Linguistic
The language of mathematics includes technical terms, but the bulk of its vocabulary consists of ordinary words—such as “function,” “group,” “normal,” “limit,” or “order.” These are familiar terms given formal and often specialized definitions in mathematical contexts. Their meanings are not derived from the symbols themselves but from verbal exposition, often grounded in metaphor and analogy (Lakoff & Núñez, 2000).
Even more technical terms like “homeomorphism” or “eigenvalue” are learned through linguistic explanation, not by inspection of notation. The lexicon of mathematics remains a subset of the broader language, enriched by specialized use but not structurally distinct from it.
No Instruction Without Language
Symbol-only instruction cannot function at all. Mathematical meaning is never transmitted through symbols alone. Every attempt to teach or communicate mathematical ideas must involve language, either in explicit form or through implicit translation by the learner. The idea of “purely symbolic reasoning” does not correspond to how people actually understand or use mathematics.
Mathematical understanding is not reducible to spelling, nor is symbol fluency the whole of mathematical competence. But without linguistic mediation, the symbolic layer is inert. The acquisition of symbolic fluency is, in part, the acquisition of a compressed symbolic spelling system, one whose interpretation always depends on language.
Summary
Mathematics, in its formal notation, does not operate as a language apart from natural speech. It is a highly compressed writing system, structured for clarity, generality, and manipulation, but dependent on verbal explanation for its acquisition, interpretation, and communication. Mathematical symbols are not meaningful in isolation; they function as abbreviations for verbal reasoning, requiring linguistic context to become intelligible. Learning mathematics therefore involves, among other things, learning a new system of spelling—a rule-governed way of writing that encodes relationships and operations in a compact visual form. This symbolic system is powerful, but it is not self-sufficient. Its meaning always arises from language.
Readings
Baddeley, A. D. (2000). The episodic buffer: A new component of working memory? Trends in Cognitive Sciences, 4(11), 417–423.
This article introduces the episodic buffer in Baddeley’s model of working memory, demonstrating how verbal and visual information are integrated. It underscores the role of the phonological loop in maintaining and manipulating mathematical symbols via subvocal rehearsal, supporting the essay’s view that symbolic reasoning is always linguistically mediated.
URL: https://doi.org/10.1016/S1364-6613(00)01538-2
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131.
Duval analyzes how understanding mathematics requires coordinating multiple semiotic registers—symbols, language, diagrams, gestures—and shows that symbol manipulation alone does not yield comprehension. This aligns with the essay’s claim that mathematical notation depends on verbal interpretation for meaning.
URL: https://doi.org/10.1007/s10649-006-0400-z
Devlin, K. (2000). The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip. New York: Basic Books.
Devlin argues that mathematical cognition evolved alongside language and social communication, suggesting that our capacity for abstract numeric thought is rooted in linguistic skills. His perspective reinforces the essay’s argument that symbols function as compressed spellings of verbal structures.
URL: https://www.amazon.ca/Math-Gene-Mathematical-Thinking-Evolved/dp/0465016197
Goldin-Meadow, S. (2003). Hearing Gesture: How Our Hands Help Us Think. Cambridge, MA: Harvard University Press.
Goldin-Meadow shows through classroom studies that gesture and speech operate together to support mathematical reasoning. Gestures alone are insufficient; they gain meaning only when integrated with language, underscoring the essay’s point that nonverbal instruction must be linguistically framed.
URL: https://www.hup.harvard.edu/books/9780674018372
Halliday, M. A. K. (1978). Language as Social Semiotic: The Social Interpretation of Language and Meaning. London: Edward Arnold.
Halliday’s framework treats language as a social semiotic system in which meaning arises through shared conventions. Applied to mathematics, this view supports the essay’s assertion that symbols require linguistic and social context to function, not standalone interpretation.
URL: https://www.cambridge.org/core/journals/language-in-society/article/abs/m-a-k-halliday-language-as-social-semiotic-the-social-interpretation-of-language-and-meaning-london-edward-arnold-1978-pp-256/12CEC8267EFAC14C9CAFAC76112A318F
Lakoff, G., & Núñez, R. E. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.
Lakoff and Núñez argue that mathematical concepts derive from conceptual metaphors grounded in physical experience and structured by language. Their work underpins the essay’s thesis that mathematical notation is a linguistically embedded system of metaphorical thought.
URL: https://pages.ucsd.edu/~rnunez/COGS252_Readings/Preface_Intro.PDF
Olson, D. R. (1994). The World on Paper: The Conceptual and Cognitive Implications of Writing and Reading. Cambridge: Cambridge University Press.
Olson examines how different writing systems shape cognition, treating mathematical notation as a specialized script. He shows that understanding such a script requires prior mastery of the language-based concepts it encodes, echoing the essay’s view of notation as a compressed symbolic spelling system.
URL: https://www.cambridge.org/core/journals/language-in-society/article/abs/david-r-olson-the-world-on-paper-the-conceptual-and-cognitive-implications-of-reading-and-writing-cambridge-new-york-cambridge-university-press-1994-pp-xix-318-hb-2495/FA2962F436FD6FE9FB6BA7DD8057888E
Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students' types of generalization. Mathematical Thinking and Learning, 5(1), 37–70.
Radford provides evidence that mathematical generalization emerges through the interaction of gesture, language, and symbols. His semiotic-cultural approach supports the essay’s claim that symbols are communicative artifacts whose meaning is negotiated through linguistic and gestural mediation.
URL: https://www.researchgate.net/publication/228553085_Gestures_Speech_and_the_Sprouting_of_Signs_A_Semiotic-Cultural_Approach_to_Students%27_Types_of_Generalization
Sfard, A. (2008). Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing. Cambridge: Cambridge University Press.
Sfard contends that mathematical learning is the acquisition of a specialized discourse, combining symbols, words, and argument structures. Her work affirms the essay’s main thesis: mathematics is not an autonomous language but a discursive mode built on natural language.
URL: https://www.researchgate.net/publication/281665057_Thinking_as_communicating_Human_development_the_growth_of_discourses_and_mathematizing