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).