The Mathematical Vocabulary Problem

The language of mathematics can throw up barriers to broad dissemination of information about mathematics.

Mathematical statements are supposed to be precise, devoid of the ambiguities of ordinary speech. The language is unusually dense and relies heavily on a specialized vocabulary. The meaning and position of every word and symbol make a difference.

Mathematician William Thurston once expressed the difference between reading mathematics and reading other subject matter in this way: “Mathematicians attach meaning to the exact phrasing of a sentence, much more than is conventional. The meanings of words are more precisely delimited. When I read articles or listen to speeches in the style of the humanities . . . I find I have great trouble concentrating and comprehending: I think I try to read more into the phrases and sentences than is meant to be there, because of habits developed in reading mathematics.”

Such habits can add to the difficulties that mathematicians face in trying to communicate with the public, when they have to surrender the clarity and economy of their usual modes of expression to the messiness of ordinary language. Comfortable with their specialized vocabulary, mathematicians too often fall into the trap of assuming their listeners or readers have equal facility, or at least some familiarity, with the language.

To complicate the situation, at least in English, mathematicians have appropriated simple, everyday words for their own purposes, using them in unexpected ways or assigning them specific, technical meanings to express abstract concepts.

Consider, for example, the term “function,” a notion fundamental to mathematics. The American Heritage Dictionary of the English Language offers the following definitions:

1. The action for which a person or thing is particularly fitted or employed.
2.            a. Assigned duty of activity.
               b. A specific occupation or role: in my function as chief editor.
3. An official ceremony or a formal social occasion.

4. Something closely related to another thing and dependent on it for existence, value, or significance.Growth is a function of nutrition.

The mathematical meaning comes next:

5. Mathematics
a. A variable so related to another that for each value assumed by one there is a value determined for the other.

b. A rule of correspondence between two sets such that there is a unique element in the second set assigned to each element of the first set.

It is followed by three more definitions:

6. Biology The physiological activity of an organ or body part.
7. Chemistry The characteristic behavior of a chemical compound, resulting from the presence of a specific functional group.

8. Computer Science A procedure within an application.

That’s a hefty load for one word to carry. Readers or listeners encountering the word “function” may understandably have difficulties sorting through so many definitions to ascertain the word’s meaning in a particular context. Even when such a word is properly defined near the beginning and the context is clear, a reader unfamiliar with the notion may later revert to other, more familiar meanings of the word, potentially creating confusion in the reader’s mind.

When I was a writer for Science News magazine, I could only on rare occasions get away with using the word “function” in my mathematics news articles without offering some sort of definition of the concept, expressed in words. My editors were there to ensure that my articles were accessible to as broad a range of readers as possible, and this meant keeping in mind that a reader’s notion of what a word means could differ enormously from the author’s intended meaning.

In the same way, mathematicians should realize that words they use routinely can echo in unexpected ways in the minds of their listeners or readers, particularly in ways that reflect different experiences and contexts. Such words include acute, base, chaos, chord, composite, concurrent, coordinate, degree, dimension, domain, exponent, factor, graph, group, linear, matrix, mean, network, obtuse, order, power, prism, proof, radical, range, relation, root, series, set, vector, and volume. Each has a precise mathematical meaning; each also has multiple alternative meanings.

On the other hand, the word “fractal,” coined by mathematicianBenoit Mandelbrot, is a noteworthy example of a term that works in both a mathematical and a popular context. Mathematics could use more such words.

People are genuinely curious about mathematics, despite the overwhelming fear of the subject that many may feel. Mathematicians who pay particular attention to how they express themselves and connect with their audiences through a common, nontechnical language can make important contributions to the public understanding of mathematics.

This article is part of a contribution by I. Peterson to the Proceedings, International Congress of Mathematicians, Hyderabad, India, Aug. 25, 2010. For more, see “Communicating Mathematics.”


Gowers, T., editor. 2008. The Princeton Companion to Mathematics. Princeton University Press.
Peterson, I. 1991. Searching for new mathematics. SIAM Review 13(March):37-42.
  1. More examples of words that can mean different things in mathematics than in ordinary language: area, complex, derivative, differentiate, fraction, imaginary, integral, irrational, random, rational, real, variable.

  2. Even when such a word is properly defined near the beginning…

    Well, as Richard R. Skemp pointed out so well decades ago, that’s entirely the wrong approach to communicating a concept to students.

  3. “That’s a hefty load for one word to carry”, but I’m guessing you’d find that “set” carries a much heavier load.