Thursday, January 22, 2004

I wrote today.

I've been thinking recently about mathematical writing. It's a tricky business. Ideally, mathematical writing should simultaneously be concise, precise, elegant, lucid, and correct. Each goal is difficult in its own right. Further, there is tension in these goals; sometimes it is easier to be lucid by writing (at least initially) an imprecise or slightly incorrect statement which conveys the idea of a proof, then filling in corrections later. The conventions used by most authors are a compromise. It is hard to write proofs which are precise and correct without sounding stilted, because ordinary speech patterns are often too vague. There is a reason why we write let k be an integer instead of k is an integer; the former sentence defines the type of k, while the latter sentence observes a fact which might be a consequence of something else (e.g. Let k = m + n where m and n are integers. Then k is an integer).

There are other conventions, too, which students of mathematics absorb over time. The phrase for suppose not usually starts a proof by contradiction; the phrase without loss of generality means that the proof treats a special case to which all other cases can be reduced trivially. Even the vague-sounding phrase almost everywhere has a precise meaning: a statement holds a.e. (almost everywhere) if it is true except on a set of zero measure.

I recently read a draft of a paper which severely abused the conventions of mathematical style. It was not a mathematical paper, and I think the author only adopted the style because he thought it sounded impressive. I read the paper slowly, marking as I went, and when I finished I walked to the bathroom and washed my hands and face with hot water until I felt better. I was in the bathroom for several minutes.

My favorite mathematical authors usually write two or three descriptions of their ideas. First, they describe the idea at a high level, often in intuitive language: this quantity describes how close a matrix is to being singular; this theorem describes why we cannot comb a sphere covered with hair without creating a part somewhere. Then there may be a special illustrative case, something to show the idea without the baggage of technical details -- though the technical details may be mentioned so that the reader is alerted to their existence. Finally, there is a theorem and a proof, which should ideally be written using a concise and suggestive notation. If done well, the reader is left pondering the ideas presented, and does not have to battle constant confusion because the author has decided to make n approach zero as an integer epsilon goes to infinity.

I cope gracefully enough with authors who present their intuition clumsily. Similarly, I'm entirely sympathetic to authors who occasionally resort to awkward devices or notation in their proofs. But those who write a vague and imprecise proof sketch and claim they are done irritate me immensely, as do those who write enormous quantities of unmotivated (and often irrelevant) algebra without even attempting to help the reader develop a mental road map first.

Concise, precise, elegant, lucid, and correct -- that's my ideal. I read few papers, mathematical or otherwise, that do well in all categories. I'm still critical of my own ability to be simultaneously concise and lucid, but I leave room for self-forgiveness. And I make progress, however slowly.

If only I could apply the same criteria to political speeches!

  • Currently drinking: Hot water with lime