Robert B. Ash is a retired mathematics professor from the University of Illinois.
Writing Math
Ash has posted a concise guide to writing good mathematics: “Remarks on Expository Writing in Mathematics” (pdf).
He begins his three-page essay by writing,
Successful graduate students in mathematics are able to reach an advanced level in one or more areas. Textbooks are an important part of this process. A skilled lecturer is able to illuminate and clarify many ideas, but if the pace of a course is fast enough to allow decent coverage, gaps will inevitably result. Students will depend on the text to fill these gaps, but the experience of most students is that the usual text is difficult for the novice to read. At one extreme, the text is a thousand page, twenty pound encyclopedia which cannot be read linearly in a finite amount of time. At the other extreme, the presentation in the book is essentially a seminar lecture with huge gaps.
He offers six main recommendations
- "Adopt a linear style"
- "Include solutions to exercises"
- "Discuss the intuitive content of results"
- "Replace abstract arguments by algorithmic procedures if possible"
- "Use the concrete example with all the features of the general case," and
- "Avoid serious gaps in the reasoning."
He concludes,
I hope to see a change in the reward structure and system of values at research-oriented universities so that teaching and expository writing become legitimate as a specialty. This will help to improve the current situation in which many advanced areas of mathematics are inaccessible to most students because no satisfactory exposition exists. I hope to see more mathematicians write lecture notes for their courses and post the results on the web for all to use.
A review
A review of a 1998 book of his, A Primer of Abstract Mathematics, says,
The strength of this book is the writing itself. Ideas are developed in a conversational (but not chatty) style. The proofs are terse and compact, but remain complete and readable, striking an impressive balance between informality and rigor. Ash is careful to point out important issues which may lurk behind the smooth exposition; for example, flagging the issue of well-defined operations when dealing with residue classes, or pointing out that an example, however "typical", is not a proof. (This doesn't prevent him from using one, however, in the case of the statement that there exist integers a and b such that ax + by = gcd(x, y), arguing that the lack of rigor is a small price to pay for the increased clarity of the exposition). Anyone who deals with the struggle to find the balance between including too much and too little when composing a mathematical argument will find many good models here.
On-line books
Ash's home page also contains links to three on-line books on algebra, one on complex variables, one on statistics, and one on "the computer algebra system Pari/GP, designed for computations in number theory."
