On writing

There are three rules for writing the novel. Unfortunately, no one knows what they are. (W. Somerset Maugham)

Everyone has to develop their own writing style, based on their own strengths and weaknesses, on the subject matter, on the target audience, and sometimes on the target medium.  As such, it is virtually impossible to prescribe rigid rules for writing that encompass all conceivable situations and styles.

Nevertheless, I do have some general advice on these topics:

• Writing a paper
  • Use the introduction to “sell” the key points of your paper; the results should be described accurately. One should also invest some effort in both organising and motivating the paper, and in particular in selecting good notation and giving appropriate amounts of detail. But one should not over-optimise the paper.
  • It also assists readability if you factor the paper into smaller pieces, for instance by making plenty of lemmas.
  • To reduce the time needed to write and organise a paper, I recommend writing a rapid prototype first.
  • For first time authors especially, it is important to try to write professionally, and in one’s own voice. One should take advantage of the English language, and not just rely purely on mathematical symbols.
  • The ratio between results and effort in one’s paper should be at a local maximum.
• Submitting a paper
  • Proofread and double-check your article before submission; you should be submitting a final draft, not a first draft.
  • Submit to an appropriate journal.

I should point out, of course, that my own writing style is not perfect, and I myself don’t always adhere to the above rules, often to my own detriment. If some of these suggestions seem too unsuitable for your particular paper, use common sense.

Dual to the art of writing a paper well, is the art of reading a paper well.  Here is some commentary of mine on this topic:

• On “compilation errors” in mathematical reading, and how to resolve them.
• On the use of implicit mathematical notational conventions to provide contextual clues when reading.
• On key “jumps in difficulty” in a mathematical argument, and how finding and understanding them is often key to understanding the argument as a whole.
• On “local” and “global” errors in mathematical papers, and how to detect them.
• Some general principles (with a worked example) on how to justify a complicated looking step in an argument, for instance by strategically guessing and introducing temporary notation.

Some further advice on mathematical exposition:

• Michèle Audin’s “Conseils aux auteurs de textes mathématiques“.
• Clark Barwick’s “Notes on mathematical writing“.
• Henry Cohn’s “Advice for amateur mathematicians on writing and publishing papers“.
• Keith Conrad’s “Advice on Mathematical Writing“.
• Oded Goldreich’s “How to write a paper“.
• David Goss’ “Some hints on mathematical style“
• Timothy Gowers on “writing examples first!” (see also this followup post)
• Paul Halmos’ “How to write mathematics” (the book also contains similar pieces by Dieudonné, Schiffer, and Steenrod); the article can be found here.
• “Mathematical Writing” – notes from a lecture course by Don Knuth, Tracy Larrabee, and Paul Roberts.
• Dick Lipton on an analogy between paper writing and city planning.
• James Milne’s (sardonic) “Tips for Authors“.
• Igor Pak’s, “How to write math papers clearly“.
• Ashley Reiter’s “Writing a research paper in mathematics“
• Jean-Pierre Serre’s “How to write mathematics badly“