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Language of mathematics


The language of mathematics is the system used by mathematicians to communicate mathematical ideas among themselves. This language consists of a substrate of some natural language (for example English) using technical terms and grammatical conventions that are peculiar to mathematical discourse (see Mathematical jargon), supplemented by a highly specialized symbolic notation for mathematical formulas.

Like natural languages in general, discourse using the language of mathematics can employ a scala of registers. Research articles in academic journals are sources for detailed theoretical discussions about ideas concerning mathematics and its implications for society.

Here are some definitions of language:

These definitions describe language in terms of the following components:

Each of these components is also found in the language of mathematics.

Mathematical notation has assimilated symbols from many different alphabets and typefaces. It also includes symbols that are specific to mathematics, such as

Mathematical notation is central to the power of modern mathematics. Though the algebra of Al-Khwārizmī did not use such symbols, it solved equations using many more rules than are used today with symbolic notation, and had great difficulty working with multiple variables (which using symbolic notation can simply be called , etc.). Sometimes formulas cannot be understood without a written or spoken explanation, but often they are sufficient by themselves, and sometimes they are difficult to read aloud or information is lost in the translation to words, as when several parenthetical factors are involved or when a complex structure like a matrix is manipulated.



My own attitude, which I share with many of my colleagues, is simply that mathematics is a language. Like English, or Latin, or Chinese, there are certain concepts for which mathematics is particularly well suited: it would be as foolish to attempt to write a love poem in the language of mathematics as to prove the Fundamental Theorem of Algebra using the English language.
Mathematics would appear to be both more and less than a language for while being limited in its linguistic capabilities it also seems to involve a form of thinking that has something in common with art and music. - Ford & Peat (1988)
  • a systematic means of communicating by the use of sounds or conventional symbols
  • a system of words used in a particular discipline
  • a system of abstract codes which represent antecedent events and concepts
  • the code we all use to express ourselves and communicate to others - Speech & Language Therapy Glossary of Terms
  • a set (finite or infinite) of sentences, each finite in length and constructed out of a finite set of elements - Noam Chomsky.
  • A vocabulary of symbols or words
  • A grammar consisting of rules of how these symbols may be used
  • A 'syntax' or propositional structure, which places the symbols in linear structures.
  • A 'Discourse' or 'narrative,' consisting of strings of syntactic propositions
  • A community of people who use and understand these symbols
  • A range of meanings that can be communicated with these symbols
  • Mathematics describes the real world: many areas of mathematics originated with attempts to describe and solve real world phenomena - from measuring farms (geometry) to falling apples (calculus) to gambling (probability). Mathematics is widely used in modern physics and engineering, and has been hugely successful in helping us to understand more about the universe around us from its largest scales (physical cosmology) to its smallest (quantum mechanics). Indeed, the very success of mathematics in this respect has been a source of puzzlement for some philosophers (see The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner).
  • Mathematics describes abstract structures: on the other hand, there are areas of pure mathematics which deal with abstract structures, which have no known physical counterparts at all. However, it is difficult to give any categorical examples here, as even the most abstract structures can be co-opted as models in some branch of physics (see Calabi-Yau spaces and string theory).
  • Mathematics describes mathematics: mathematics can be used reflexively to describe itself—this is an area of mathematics called metamathematics.
  • Knight, Isabel F. (1968). The Geometric Spirit: The Abbe de Condillac and the French Enlightenment. New Haven: Yale University Press. 
  • R. L. E. Schwarzenberger (2000), The Language of Geometry, published in A Mathematical Spectrum Miscellany, Applied Probability Trust.
  • Alan Ford & F. David Peat (1988), The Role of Language in Science, Foundations of Physics Vol 18.
  • Kay O'Halloran, Mathematical Discourse: Language, Symbolism and Visual Images, Continuum, 2004.
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