# 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.

References:

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.

# Paul Halmos on Writing Mathematics

As a mathematician, Paul R. Halmos (1916-2006) made fundamental contributions to probability theory, statistics, functional analysis, mathematical logic, and other areas of mathematics. He was also known and widely recognized as a masterly mathematical expositor. And he served as editor (1981-1985) of the American Mathematical Monthly.

Halmos described his approach to writing in an essay published in the book How to Write Mathematics (American Mathematical Society, 1973). One paragraph presents the essence of the process:

“The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea. To do so, and to do it clearly, you must have something to say, and you must have someone to say it to, you must organize what you want to say, and you must arrange it in the order that you want it said in, you must write it, rewrite it, and re-rewrite it several times, and you must be willing to think hard about and work hard on mechanical details such as diction, notation, and punctuation.”

Halmos adds, “That’s all there is to it.”

Halmos then expands on what he sees as the key elements of good mathematical writing.

1. Say something. To have something to say is by far the most important ingredient of good exposition.
2. Speak to someone. Ask yourself who it is that you want to reach.
3. Organize. Arrange the material so as to minimize the resistance and maximize the insight of the reader.
4. Use consistent notation. The letters (or symbols) that you use to denote the concepts that you’ll discuss are worthy of thought and careful design.
5. Write in spirals. Write the first section, write the second section, rewrite the first section, rewrite the second section, write the third section, rewrite the first section, rewrite the second section, rewrite the third section, write the fourth section, and so on.
6. Watch your language. Good English style implies correct grammar, correct choice of words, correct punctuation, and common sense.
7. Be honest. Smooth the reader’s way, anticipating difficulties and forestalling them. Aim for clarity, not pedantry; understanding, not fuss.
8. Remove the irrelevant. Irrelevant assumptions, incorrect emphasis, or even the absence of correct emphasis can wreak havoc.
9. Use words correctly. Think about and use with care the small words of common sense and intuitive logic, and the specifically mathematical words (technical terms) that can have a profound effect on mathematical meaning.
10. Resist symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it.

Halmos concludes: “The basic problems of all expository communication are the same. . . . Content, aim, and organization, plus the vitally important details of grammar, diction, and notation—they, not showmanship, are the essential ingredients of good lectures, as well as good books.”

The 44-minute  film I Want to Be a Mathematician: A Conversation with Paul Halmos is based on an interview with Paul Halmos, in which he discusses various aspects of writing, teaching, and research (Trailer).–Ivars Peterson