Why Mathematics is a Language

May 31, 2022

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Mathematics is considered the heart of the beta-field, whereas languages are by definition alpha. One could say, therefore, that mathematics and language are direct opposites. Yet, mathematics is sometimes called the language of science and considering the definition of what makes a language, mathematics satisfies all the criteria!

There are several ways to define “language”. One could argue that a language is a system of words or codes used within a discipline, or a system of communication using symbols or sounds. Linguist Noam Chomsky has given language the definition of “a set of sentences constructed using a finite set of elements”, some other linguists believe a language is any system that is able to represent events and abstract concepts. However, a thing every definition of language has in common is that a language has to contain the following components:

  • A vocabulary of words and/or symbols;
  • Every word and/or symbol must have meaning;
  • Within the language, there is a set of rules, the grammar, that outline how the vocabulary is used;
  • There is a syntax, which organises the symbols into linear structures or propositions;
  • The strings of syntactic propositions forms a narrative or discourse;
  • There must be (or have been) (a group of) people who can understand and use the symbols.

When you think about language in this abstract definition, mathematics satisfies all the criteria. The symbols have meanings, there is a syntax, a grammar, and they are all the same throughout the world. The people who understand and use the symbols are the mathematicians, scientists, and all the others that use mathematics to communicate certain concepts or findings. The narratives of mathematics can be mathematics itself (meta-mathematics), real-world phenomena (the gas-extraction problem), and abstract concepts (understanding of cardinality/classification).

Mathematical Vocabulary

The vocabulary of a language consists of words and/or symbols unique to that language. A mathematical equation can be considered a sentence with nouns and verbs, just as in a “regular”, spoken language. Let us consider the following equation: 2+4=6. Think about it: when you say this out loud, you use your own language for the numbers and operators, “two added to four equals six”. In this English sentence, the nouns are “two, four, and six” and the verbs are “added”and “equals”. In this example, we are left with “to”, which is a preposition, but we will ignore that for now. If you compare this to math equations, we can conclude that “nouns” in mathematics include:

  • Arabic numerals (0, 1, 2, 20394.785);
  • Fractions (\frac{1}{4}, \frac{4}{7}, \frac{13}{3});
  • Variables (a, x, \theta);
  • Diagrams or visual elements (matrix, rectangle, circle);
  • Infinity (\infty);
  • Pi (\pi);
  • Imaginary numbers (i, -i);

and verbs include:

  • (In)equalities (=, <, >);
  • Actions (+, -, \cdotp, :);
  • Other operations (\sin, \cos).

Combining these nouns and verbs into an equation forms a sentence with infinitives, adjectives, etc. Also, as in other languages, a symbol’s role is determined by its context.

Mathematical Grammar and Syntax

The rules of mathematics are international; no matter in which country you follow a maths lesson, the structure of the equations is the same. There are several, grammatical rules that can be observed in mathematics:

  • Formulas are read from left to right;
  • The Latin or Greek alphabet is used for parameters and variables;
    • Integers are usually indicated by i, j, k, l, m, n;
    • For real numbers, one uses a, b, c, \alpha, \beta, \gamma;
    • w and z are commonly used for complex numbers;
    • x, y, z indicate unknowns (notice the double role z plays!);
    • Functions are usually named f, g, h;
    • For specific concepts, the Greek alphabet is typically used (\Gamma for the Gamma distribution);
  • For the order in which the symbols interact, parentheses and brackets are used;
  • The phrasing of integrals, function, and derivatives is always the same.

Because of this international, mathematical grammar and syntax, the language of mathematics can be used as a universal language all over the world. Every formula or equation has the same meaning, regardless of the language that accompanies it. Therefore, through mathematics, people can learn and communicate, even if they encounter other communication barriers.

Is Mathematics a Language, though?

Since there are many different definitions of a language, not everyone agrees that mathematics can be considered one. Sometimes, one of the criteria of a language is that it must be a spoken form of communication but maths is obviously a written form of communication. Of course, one could say simple mathematical equations aloud, e.g. 2 + 4 = 6, but other equations might be more difficult; for instance, how would you pronounce a matrix? Furthermore, when speaking mathematics, one would speak in his own language, for example English or Dutch, not a universal one. However, this is also not a completely solid argument, as this reasoning would not classify sign language as a real language, though many linguists accept signing as a true language. Though, I think one can view signing as the speaking language for mute/deaf people.

Y \cdotp e^S, I Think So!

All in all, I think one can make a strong case for mathematics as a language. With all its rules, conventions, and universal grammar, it certainly resembles the most robust definition of a language. I think the criterion that a language has to be spoken, can be dismissed, as, according to the abstract definition of a language as presented above, a language has to be understood and used by (a group of) people (in the past), and for this, a written language can also suffice. How interesting to see how two extreme opposites as language and mathematics are actually quite the same!

Sources

Ford, Alan, and F. David Peat. “The Role of Language in Science.” Foundations of Physics 18.12 (1988): 1233–42. 

Helmenstine, Anne Marie, Ph.D. “Why Mathematics Is a Language.” ThoughtCo, Aug. 27, 2020, thoughtco.com/why-mathematics-is-a-language-4158142.

Klima, Edward S., and Ursula Bellugi. “The Signs of Language. “Cambridge, MA: Harvard University Press, 1979. 

Riccomini, Paul J., et al. “The Language of Mathematics: The Importance of Teaching and Learning Mathematical Vocabulary.” Reading & Writing Quarterly 31.3 (2015): 235-52. Print.

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