Mathematical musings: On writing and reading mathematics

• An author who writes perfectly precisely is writing for the benefit of readers who cannot think for themselves.

  Commentary. I recall some quote like this attributed to Keynes, but cannot track it down.

• In my experience most mathematicians are intellectually lazy and especially dislike reading experimental papers. (Francis Crick, in What Mad Pursuit).

  Commentary. Let us spin intellectually lazy into, say, specialize in a different kind of intellectual energy!

• Seek not the diamond in the rough.

  Commentary. In principle a badly-written paper may conceal a gem, but don't bet on it. Of course the first proof may be far from optimal, but if an author can't contrive to make a result look interesting and important, you can bet it isn't.

• When writing the first paper on a topic, take care to avoid cleverness.

  Commentary. Most often we work on problems within an established topic, where we can suppose that simple ideas have been worked through, so that new results require some cleverness. In the lucky event you discover a promising looking new topic, please first think carefully about what can be said as simple consequences of known results, to write down what I call baseline facts, before seeking clever ad hoc arguments. This is both a service to subsequent investigators, and insurance against looking foolish if someone discovers a simple derivation from known results.

• In writing a textbook or monograph on a topic on which other books already exist, there are only two acceptable overall architectures.

  And here they are. If you seek to bring some new perspective (angle, insight, application domain, etc) on the topic, then you must be able to state clearly in the introduction what your new perspective actually is, and follow through in the text. Alternatively you may take a "best standard treatment" approach, mostly choosing from and polishing existing accounts. Note this requires you to have actually read existing books .......

  Commentary. This seems self-evident -- why else write a book? -- but amazingly many books fit neither style.

• In non-technical writing, choose or interpolate between two acceptable writing styles.

  If you choose to emphasize telling true stories, emulate Steven Jay Gould. If you choose to emphasize facts and reasoned opinions, emulate The Economist. That's it. You are not Homer, de Toqueville, Hammett, Jon Stewart or Snoop Dogg, so don't imitate them. Trust me on this. I understand that writing a book ipso facto implies literary pretensions, but keep them in check.

• Consider tennis and mitosis: clear explanations are not enough.

  Longer piece, here

• Outside a mathematics department, "theory" does not mean "theorem".