Writing Mathematics

Abstract: this is just a few notes to myself, amounting to little more than a glorified annotated bibliography for writing mathematics.

Guide to Writing Math

Kevin Lee's A Guide to Writing Math argues that a page-full of calculations without explanation "contains no math".

Lee compares such writing to limiting Moby Dick to a single sentence ("The whale wins."). This is apt: mathematical writing should take the reader along the (relevant) thought process when solving a given problem, i.e. to convey "mathematical reasoning and ideas clearly to another person."

The paper states various rules-of-thumb which are useful to the neophyte (e.g., don't begin sentences with formulas, or explicitly state all formulas and variables).

Mathematical exposition (e.g., writing a textbook or explaining some concept) is slightly different and requires slightly different rules.

Mathematical Writing

Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts' Mathematical Writing provides (on pages 3–8 of the pdf) 27 rules for mathematical writing.

I would like to quote the relevant rules (the bold emphasis is mine, italics is Knuth et al.'s):

5. The statement of a theorem should usually be self-contained, not depending on the assumptions in the preceding text.

[...]

10. Don't use the style of homework papers, in which a sequence of formulas is merely listed. Tie the concepts together with a running commentary.

11. Try to state things twice, in complementary ways, especially when giving a definition. This reinforces the reader's understanding. [...] All variables must be defined, at least informally, when they are first introduced.

12. Motivate the reader for what follows. [...]

Perhaps the most important principle of good writing is to keep the reader uppermost in mind: What does the reader know so far? What does the reader expect next and why?

When describing the work of other people it is sometimes safe to provide motivation by simply stating that it is "interesting" or "remarkable"; but it is best to let the results speak for themselves or to give reasons why the things seem interesting or remarkable.

When describing your own work, be humble and don't use superlatives of praise, either explicitly or implicitly, even if you are enthusiastic.

13. Many readers will skim over formulas on their first reading of your exposition. Therefore, your sentences should flow smoothly when all but the simplest formulas are replaced by "blah" or some other grunting noise.

14. Don't use the same notation for two different things. Conversely, use consistent notation for the same thing when it appears in several places. [...]

15. Don't get carried away by subscripts, especially when dealing with a set that doesn't need to be indexed; set element notation can be used to avoid subscripted subscripts. [...]

16. Display important formulas on a line by themselves. If you need to refer to some of these formulas from remote parts of the text, give reference numbers to all of the most important ones, even if they aren't referenced.

17. Sentences should be readable from left to right without ambiguity. Bad examples: "Smith remarked in a paper about the scarcity of data." "In the theory of rings, groups and other algebraic structures are treated."

18. Small numbers should be spelled out when used as adjectives, but not when used as names (i.e., when talking about numbers as numbers).

19. Capitalize names like Theorem 1, Lemma 2, Algorithm 3, Method 4.

[...]

27. When in doubt, read The Art of Computer Programming for outstanding examples of good style.

[That was a joke. Humor is best used in technical writing when readers can understand the joke only when they also understand a technical point that is being made. Here is another example from Linderholm:

"... ØD = Ø and NØ = N, which we may express by saying that Ø is absorbing on the left and neutral on the right, like British toilet paper."

Try to restrict yourself to jokes that will not seem silly on second or third reading. And don't overuse exclamation points!]