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.
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).