[Math] Obsessive editing/revising of math papers

journalsmathematical-writing

I've been wanting to ask this question, because I have no insights into the way other mathematicians prepare papers (for eventual publication).

How much are editing, revising, updating, adding to, etc., part of the "normal" process of process of drafting math papers? Specifically, papers of moderate (10-15 pages) length. I've noticed I tend to do this for several months, and the thought has occurred to me that perhaps I'm being too "fussy" and that I'm wasting time.

Best Answer

My quick thoughts on the topic:

Most of the papers are not written carefully. Here are my major complains that apply to many (if not to most) of the papers that I have seen:

  • Proofs are too sketchy and difficult to follow. Very often they are not correct. Even, if the mistakes are minor and easy to correct, very often the proofs are not correct the way they are written.
  • Results often have complicated and abstract statements and there are no examples that would illustrate the main result.
  • Authors often write something like that: `It follows from [5]' and they refer to a 500 pages long book without specifying any particular results.
  • Introductions do not provide right motivation by placing the results within the existing literature.

I think it is extremely rude and unprofessional to write papers in the manner described above. This are some of the reasons why:

  • Reading a paper that is not well written takes much more time both for the referee and the readers. The author perhaps saved some of his or her precious time by writing a paper not very carefully, but he or she put a burden of filling details on the shoulders of those who want to read it.
  • Writing: `It follows from [5]' means that the author did not bother to check what particular result needs to be used and the reader, not familiar with [5], needs to spend hours finding the right result. Very often no result actually applies, because the result from [5] that the author had in mind, needs to be modified before it can be applied to the particular situation.

By writing a paper the author should (in my opinion):

  • Keep in mind that most of the readers are graduate students who have a very limited knowledge and maturity. Papers should help them learn the subject and so the papers should have all necessary details and relevant comments placing the result in a broader framework. For example the introductions should be like a short and well written survey on the subject. At last, but not least, the papers should have no mistakes.

In my own practice I do my best to follow the rules above. Whether I am successful or not, others will judge.

  • When I write a paper, I always try to write it in a way accessible to a (talented and motivated) graduate student.
  • When I have a result with a complete proof and if I could write a (10 pages long) paper within a week, it usually takes me at least a month (of hard work from early morning to late night) to write it, because of the standards that I try to follow.
  • At the stage of writing a paper, I usually write about 30 pages of scrap paper for every single page in the paper. This is, because I check carefully every proof several times, each time from scratch.
  • If something can be easily explained by adding a couple of lines, I do it. For example it is well known that every separable metric space can be isometrically embedded into $\ell^\infty$. One could quote this result from the literature, but since the proof is 1-2 lines long, why not to add such a proof? Of course, one needs to keep a certain balance and not add too many details. A good advice (in my opinion) is: if a paper could be written in 10 pages, it should be written on say, 13 pages.
  • When I quote a result from the literature I usually state the result as a lemma (instead of saying: by Theorem 3.12 in [5]). If the statement is not exactly as in [5] I explain why the modified statement is true.
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