Some Hints on Mathematical Style

David Goss

Many years ago, just after my degree, I had the good fortune to be given some hints on mathematical writing by J.-P. Serre. Through the years I have found myself trying to repeat this very sound advice to other mathematicians who are also starting out. Recently, I have been involved in the publishing of a proceedings volume, as well as being an editor of the Journal of Number Theory. Many of the papers coming my way are from young authors; so I have written down these hints in order to speed the process along.

This is a second (and, most probably, final) version of these “hints”. I have added comments from a number of mathematicians who read a first version. These hints are presented as a source of ideas on mathematical style. Feel free to use them in any way that you deem useful.

• Two basic rules are: 1). Have mercy on the reader, and, 2). Have mercy on the editor/publisher. We will illustrate these as we move along.

• General Flow of the Paper.

– Definition: All basic definitions should be given if they are not a standard part of the literature. It is perhaps best to err on the side of making life easier on the reader by including a bit too much as opposed to too little (Rule 1).

∗ Some redundancy should be built into the paper so that one or two misprints cannot destroy the understandability. The analogy is with “error-correcting codes” which allow transmission of information through noisy and defective channels.

– As a very general rule, the definitions should go before the results that they are used in (Rule 1).

– The “quantifiers” should always be clear (Rule 1). Some examples

to avoid:

∗ “We have f(x) = g(x) (x ∈ X).” What does the parenthesis mean? That f(x) = g(x) for all x ∈ X, or, for some x ∈ X?

∗ What does “f_{t,u}(x, y) = O (g_{t,u}(x, y))” mean? Very often it means that t, u, y are fixed and x is allowed to vary. Quantifier problems are especially troublesome with “big O” notation.

∗ The word “constant” is terribly ambiguous. It is important to make explicit exactly which variables the constant depends on.

– Theorem/Proposition/Lemma/Corollary: Give clear and unambiguous statements of results. These are what other people are reading your paper for; so you should ensure that these, at least, can be understood by the reader (Rule 1).

∗ The statement of the Theorem/Proposition/Lemma/Corollary should not include comments (except for an occasional brief remark in parenthesis) or examples.

– If you use or quote an important result of another person, you should give a reference. You should avoid giving the impression that such a result is obvious, a generally accepted fact, due to you, and so on.

∗ A reference to a book should always give the page!

∗ Try to avoid using “by the proof of” when the proof is in the paper and the statements can be rewritten to be directly quoted.

∗ A “well-known” result that is not in the literature should be

proved if needed (Rule 1).

– Proof: A proof should give enough information to make the theorem believable and leave the reader with the confidence (as well as the ability) to fill in details should it be necessary (Rule 1).

• Other comments:

– One should, of course, observe the usual conventions in terms of spelling, punctuation, and the other basic elements of style. Use complete sentences, with subject, verb, and complement (Rule 1).

∗ A verb should not be replaced by a symbol. It is bad to write: “... if x = 2, y = 3, z = 4” meaning “... if x = 2 and y = 3, then z is equal to 4”.

∗ It is also bad to write: “... we prove ζ_{Q}(2n) ∈ π^{2n}Q” instead of:

“... we prove that \frac{ζ_{Q}(2n)}{π^{2n}} belongs to Q” (or “is rational”).

– Use the present – not the past – form.

∗ As an example of bad writing, we have: “We have proved that f(x) was equal to g(x)...”. This is corrected to: “We have proved that f(x) is equal to g(x)...”.

– Long computations that can easily be carried out by the reader should be avoided. The ideas and results should be given with an invitation to the reader to do the calculation should it be desired (Rule 1).

∗ The exception to this rule is when the long computation is an essential part of the argument. In this case, it should be given in full (Rule 1).

– One should avoid giving the reader the impression that the subject matter can be mastered only with great pain. In fact, this is an ideal way to lose readers (or audiences!).

– One should avoid using abbreviations like “w.r.t.” (with respect to), “iff” (if and only if), and “w.l.o.g.” (without loss of generality). They simply do not look very nice (and “iff” is offensive! – Rules 1 and 2).

– You should not begin a sentence with a math symbol. This can confuse the printer as well as the reader (Rules 1 and 2).

∗ As a example of such bad writing, we have: “... we want to prove the continuity of f(x) = 2 cos^{2} x · sin x. cos x being continuous....”. This is corrected to: “...f(x) = 2 cos^{2} x · sin x. Since cos x is continuous...”.

– If your paper raises a natural question, and you don’t know the answer, by all means say so! This may turn out to be more interesting than the theorems that you prove.

∗ Conversely, refrain from making “conjectures” too hastily. Use instead the words “question” or “problem”. Remember that a good “question” should be answerable by “yes” or “no”. To ask “under what conditions does A hold” is not a question worth printing.

– It is often helpful to begin a new section of the paper with a summary of the general setting.

– After the paper is finished, it should be reread (and, perhaps, rewritten) from the reader’s point of view (Rule 1).

– A good way to begin is to use a standard classic of mathematical exposition (e.g., Bourbaki-Algebra, works by Serre or Milnor) as a basic model.

• Some further sources to look at:

– P. Halmos: How to write mathematics, Enseign. Math., 16, (1970), 123–152.

– W. Strunk, Jr., & E. B. White: The Elements of Style, Macmillan Paperbacks Edition, (1962).

– D. Knuth et al.: Mathematical Writing, MAA Notes #14, (1989).

– Some conventions on citations and pronouns may be found in: S. Zucker: Variation of a mixed Hodge structure II, Inventiones Math. 80, (1985), p. 545.

• Finally, I quote from a letter Serre wrote commenting on my original version: “It strikes me that mathematical writing is similar to using a language. To be understood you have to follow some grammatical rules. However, in our case, nobody has taken the trouble of writing down the grammar; we get it as a baby does from parents, by imitation of others. Some mathematicians have a good ear; some not (and some prefer the slangy expressions such as “iff”). That’s life.”