Is mathematics a language?

'The laws of Nature are written in the language of mathematics…the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word.' – Galileo Galilei


I can unequivocally say that yes, mathematics is indeed a language. In fact, it is only one language of many that are not the same as ‘plain English’, such as music. And as a language, it is far more unambiguous than English. But mathematics has its own typical, technical terminology, the same as most professions.

Mathematics language

It is, however, much the same as learning a foreign language; it is not spoken in the home and is almost exclusively learnt at school. The basic mathematical concepts should be developed in much the same way as the rules and regulations of a spoken language. Good fundamental knowledge and understanding is the only manner in which to proceed. The way in which the structures and terminology of language is taught, mirrors those of mathematics – basic numbers, shapes, quantities, and the like. The teaching methodology of CRA (Concrete to Representative to Abstract) focuses on brain activity, where the child first learns to play with objects in order to apply those skills to mathematical language. It can be compared to the routine in which we learn a second language. If we don’t use the acquired language regularly, we won’t be able to sustain our fluency; the same with mathematics. We often learn only the basic facts, unable to apply the solutions.


The parts that make up a language

  • Vocabulary; consists of symbols and words
  • Grammar; how the vocabulary may be used
  • Syntax; the linear placements of vocabulary
  • Discourse or narrative; the sentences that are structured using the vocabulary
  • Community; those who understand the language
  • Meanings; the communication achieved by means of the vocabulary


Mathematical vocabulary

Signs and symbols must be decoded, requiring a certain skillset. They often present in the form of pictures. The benefit is of course that speakers of any language can understand them. Without mathematical notations, or symbols, the language loses its power. We are dealing with a visual language.


=         equality                           2+3=5

≈         approximately equal    √2≈1.41

ℝ        real numbers                 π∈ℝ, 7∈ℝ, √(-1)∉ℝ

∝        proportional to             f(x) ∝ g(x)

(a,b)   open interval                 (a,b) = {x | a < x < b}

English language terms can mean something entirely different in mathematical language. Look at the following examples:

Mathematics language

Mathematics language

The surface of any sphere is equal to four times the greatest circle in it [S = 4∏r2]

– Archimedes, On the Sphere and the Cylinder, 220 BC


Mathematical grammar

Unlike natural language, mathematical language is understood by all mathematicians, regardless of their first language, hence the term; ‘a universal language’. The notations used for formulas function independently, despite the fact that some writing systems follow a right-to-left line, and not the Latin alphabet in which mathematics is written.

The following formula is understandable to native speakers of English, Russian, Arabic, and every possible language. Natural language phrases consist of parts of speech, as do mathematics, the formula below being an example.

sin x + a cos2x ≥ 0

This formula may be regarded as a sentence or sentential phrase in which the greater than or equal symbol acts as the verb.

In Mathematics as profession and vocation, in Mathematics: Frontiers and Perspectives,(V. Arnold et al, ed.), AMS, 200, p. 154, Yu. Manin said that ‘The basis of all human culture is language, and mathematics is a special kind of linguistic activity’.



Schwartz & Kenney (1995) explains the ‘verbs’ of mathematics:

Inferring: to apply the gained results to the initial problem, and subsequently interpreting those results.

  • Modelling and formulating: to produce suitable descriptions and relationships so as to mathematize the original challenge.
  • Communicating: to report to a particular audience on the outcome of the problem.
  • Transforming and manipulating: to change the mathematical form of the original problem in order to convey corresponding forms.


The uses of mathematics

It describes or explains the actual world and its phenomena;

  • Gambling by means of probability;
  • Farm measurements through geometry;
  • The reason why apples fall to the ground through calculus, and so on.

Its applications range from quantum mechanics, physics, physical cosmology, to engineering.

'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 theFundamental Theorem of Algebrausing the English language.'

– R.L.E. Schwarzenberger



Mathematical language has been evolving for the last 2,500 years. To mention one important contributor; René Descartes (1596–1650), the first mathematician to use letters at the end of the alphabet to indicate an unknown quantity. He introduced the power notations x2, x3.

Robert Recorde (c.1510–1558) invented the sign of equality, published in his book The Whetstone of Witte (London, 1557):

'I will sette as I doe often in worke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicause noe.2. thynges, can be moare equalle.'

Everything I have read in my research for this article agrees: Mathematics is a Language.


Further reading

This book comes highly recommended:

Devlin, K. The Language of Mathematics: Making the Invisible Visible, Henry Holt and Company, 2000.


Thank you

There are two people I have to thank for their assistance with this article.

  • Mr George Pauer; a man curious about almost anything. His mathematical interest goes much deeper than this mere article. I thank my wonderful friend for reviewing my article.
  • Mr Floris Schoeman; a mathematics coach, amanuensis and mathematics genius. As a great supporter of The CV Branch, thank you for giving me the idea to write this article, and thank you for your explanation of how mathematical language learning and ‘normal’ language learning connect.

Mr Schoeman offers maths coaching with a difference. You may find him on:


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