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In the light of the recent "proof wars" in symplectic geometry (in which some groups contend that proofs given by some other groups are wrong, see here, here and here) I thought it would be good to have some basic guidelines (or best practices) for writing proofs in math papers as a community wiki on MO. This would be specially useful for the PhD students.

I don't just mean individual proofs but also the whole interconnected web of arguments in a paper. I think this is important as modern math papers tend to be long and highly complex and therefore difficult to referee.

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    $\begingroup$ I have voted to close. This is better suited to a blog format. $\endgroup$
    – Steven Gubkin
    Commented May 26, 2014 at 20:53
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    $\begingroup$ What are these proof wars about? $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented May 26, 2014 at 21:24
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    $\begingroup$ The question would have been closed as "subjective and argumentative" in the old days of MO. There are some like Leslie Lamport who believe in highly detailed and structured proofs, not far from formal proofs. Others believe in much less detail. Impossible to find a consensus on "best practice"; this is a highly individual decision based on esthetics, personal ethics, and experience. $\endgroup$
    – Todd Trimble
    Commented May 26, 2014 at 21:39
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    $\begingroup$ @ToddTrimble I think there are still things that almost everybody agrees to. For example it is not generally accepted if the proof of a theorem in section 2 uses a lemma in section 3 and the proof of that lemma uses another lemma in section 2 (although that won't result in an apocalypse). Also the purpose for such guidelines is not "esthetics, personal ethics", etc but to make it easier for the referee/reviewer to analyze the paper. I agree that this is too broad but there are many things that naturally come to mind for example a notation glossary would make reading a 50+ page paper easier. $\endgroup$
    – Reza Rezazadegan
    Commented May 27, 2014 at 7:55
  • $\begingroup$ There is no excuse for writing only a prose proof: research.microsoft.com/en-us/um/people/lamport/pubs/proof.pdf $\endgroup$
    – 0 _
    Commented Jul 27, 2015 at 1:50

2 Answers 2

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As basic hygiene, please state theorems before, not after their proofs. I find nothing more annoying than the style where some seemingly random agitation is followed by

... and therefore we have proved:

Theorem 3.47 Let a regular gizmo have transversality property (3.25). Then the closed sub-gizmo (3.20) has property (3.30).

The problem with such a structure is that it may in fact be impossible to rewrite in the normal theorem-proof order, because only after half the proof does "(3.20)" turn out to be well-defined, let alone closed; or properties "(3.25)" and "(3.30)" may be impossible to even state outside the context of the proof, e.g. because they involve auxiliary objects whose existence was not assured a priori.

So instead, 1º) set up the minimal environment where the result may be meaningfully stated; then 2º) state and 3º) prove it. Or at least, make sure that this could be done.

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Here is my own two cents worth:

1-State the theorems you cite from the literature clearly using Latex's theorem environment and try your best not to cite it inline in the middle of your proof. Also always give the full address to the theorem (e.g. [5,Thm 3.2.1]). This makes it easier to see if what you are looking for really follows from the citation.

2-Dont count on any "well-known argument", "folklore result", "h-principle", etc. Give a precise statement and also citation if the latter exists. Otherwise state it clearly as an "axiom" (as in the Givental-Kim paper on equivariant Gromov-Witten invariants which was written before the theory had a firm foundation.)

3-If your proof involves a subproof that can be important on its own or if it is something you have any kind of "misgivings" about, state it as a separate lemma so that the whole argument would be easier to analyze.

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    $\begingroup$ Wait... I can't merely cite Dunford and Schwartz (3 volumes) for my results? I have to give the theorem number? $\endgroup$
    – Gerald Edgar
    Commented May 26, 2014 at 21:40
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    $\begingroup$ I personally feel sympathetic to such guidelines: I too am frequently frustrated by appeals to "well-known", etc., particularly when it's not well-known to me! So when I write on the nLab, for instance, I like to include lots of detail and/or references to the literature. With sufficient layers of hyperlinking this becomes more and more feasible; to me an optimal scenario would be a structured environment where a succinct proof is given at the top layer, but with links to sublayers so that one could "zoom in" to finer levels of detail. This was harder in the old days with paper journals. $\endgroup$
    – Todd Trimble
    Commented May 26, 2014 at 23:16

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