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In papers there are often sketchy arguments in proofs that I find hard to understand. Filling in the gaps is laborious and time-consuming. According to the post www.mathoverflow.net/questions/40729/does-a-referee-have-to-check-carefully-the-proof, referees seem to be faced with this problem as well.

Presently I'm preparing my first paper for submission to a journal. Of course I approve the usual conventions and write more or less sketchy proofs myself in order to keep the paper short (around 15 pages). On the other hand, I checked the proofs carefully. So I could support the publishing process by this idea:

Submit two versions of the paper:

  • a short one, designated for publishing
  • a long one, assigned for the referee with proofs given in full detail

Is this a good idea that simplifies the referee's life (and, maybe, helps getting the paper accepted) or is it, in contrast, maybe even a no-go ?

I appreciate your opinions very, very much. Thanks in advance.

N.b. I intentionally ask the question on MO (and not on academia.stackexchange.com) because I think checking proofs is particular to mathematics and doesn't occur this way in most other fields of science.

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    $\begingroup$ Please do not perpetuate this idea that «sketchy proofs» are acceptable. Also, as a writer you should be ideally concered with making the life of your reader easier, not the referee's. Your paper will have one, two, very, very few referees but —hopefully!— many readers: optimze for the general case! $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jan 16, 2014 at 16:48
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    $\begingroup$ By "sketchy proof" I mean phrases like "Equation (4.8) shows that ... is isometric. Since ... the containments (4.2) follow from (3.3) and (2.5)." There is nothing wrong with it. But: a) Showing that the function is actually isometric requires a tedious computation in its own where using (4.8) is one step (the key step). A referee who really validates the proof should do this computation (should the reader also do ?). If the referee had the long version, he could just read that computation line by line. b) Understanding why (4.2) follows from (3.3), (2.5) is sometimes not that obvious. $\endgroup$
    – user45614
    Commented Jan 16, 2014 at 17:57
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    $\begingroup$ Anything that is not obvious should be in the paper. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jan 16, 2014 at 18:32
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    $\begingroup$ and 'obvious' means: it was obvious to you before you figured out the results in your paper, and it will be obvious to you 10 years from now after your detailed unpublished notes became waste-paper. $\endgroup$
    – Johannes Ebert
    Commented Jan 16, 2014 at 20:29
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    $\begingroup$ I am reasonably confident in my memory that the American Mathematical Monthly mentions that authors may wish to send along supplementary calculations for their submissions. But when I looked at their page just now, I couldn't find any information on this. (It is possible that they give you more information when you actually sign up to submit a paper to them -- as I have. Maybe later I'll get the chance to search through that material.) $\endgroup$
    – Pete L. Clark
    Commented Jan 16, 2014 at 21:09

8 Answers 8

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You should submit only one version of the paper. Think of the referee as a typical reader, not a judge providing a certificate of correctness. If the referee needs additional information, then other readers will need it too.

Regarding putting a proof in an appendix: There are exceptional cases when this is necessary, but most often it is not, and it can even be annoying to the reader (or seen as a warning sign of a bogus paper!) when important arguments are removed from their context.

Reading a math paper will always be time consuming, but I don't think there is a general convention that proofs should be sketchy. As an inexperienced writer, it is sometimes hard to know when something should be left out, and when leaving it out will be regarded as a gap. Feedback from teachers and supervisors on this matter can be contradictory and confusing. If you feel you still haven't got the knack for it, bear in mind that it is much easier for a referee to ask you to remove details than to figure out when they are missing. I would not endorse any general advice to write sketchy proofs in order to keep the paper short.

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I don't think the editor will assign the referees a different version of the paper; after all, they are supposed to be judging whether the journal should publish the paper they are reviewing or not. I think you could post an extended version on the arXiv and make sure the referees are aware of its existence. Some of them may then give you useful feedback about whether they found the extended version helpful and may suggest moving some material into the main paper or vice versa.

Having said that, I think your chances of having the paper accepted are improved by submitting the best version you can up front. If you're not sure whether your proofs are too short or too long, have someone (your advisor?) read the paper and give you comments before submitting. This can only help your chances.

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I don't understand why you "approve the usual conventions and write more or less sketchy proofs myself in order to keep the paper short", unless the journal you want to submit to has a size limitation.

Indeed, it was a custom in the past to write a very short note with a sketch of the proof (or without a proof at all), and publish it as a "research announcement". There were special journals for such short announcements (Comptes rendus in France, similar thing in Russia, and many more). I think the reason for this practice was to secure priority. Publication (and writing) of the full text could take a lot of time. Then usually several such short notes were followed by a large, long paper.

On my opinion, this practice is obsolete. Today you can post an arbitrarily long paper on the arXiv to secure priority, and the only reason to publish "sketch proofs" in a journal could be the journal policy of length limitation.

On my opinion, the author should do everything to make his/her paper easier to read, is s/he really wants the paper to be read by the others.

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    $\begingroup$ c) You can delay the disappointment of finding a flaw in your proof to after its publication. :) $\endgroup$
    – darij grinberg
    Commented Jan 16, 2014 at 22:31
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    $\begingroup$ There is a difference between short and incomplete. If authors manage to write short, intelligible, complete proofs, well, great. As that is very, very hard, intelligible and complete (which itself is not the same as with all the dready calculations...) are the two to keep. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jan 16, 2014 at 22:48
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    $\begingroup$ If a journal has a length policy which would force one to replace proofs by sketches in a paper, then clearly that journal is a bad match for the paper! $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jan 16, 2014 at 22:51
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    $\begingroup$ Mariano, a lot of journals have length policies. For example, JAMS: math.ucla.edu/~tao/submissions_old.html In an ideal world, journals would have no length limitations, and all proofs would be written out in full detail, but we do not live in an ideal world. $\endgroup$
    – Timothy Chow
    Commented Jan 17, 2014 at 2:50
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    $\begingroup$ @Timothy Chow, I spent the first 20 years of my career in Soviet Union, where all journals had strong length restrictions. Splitting a long paper always worked for me. $\endgroup$
    – Alexandre Eremenko
    Commented Jan 23, 2014 at 20:35
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I put the detailed version on the arXiv, and publish a short version. (Reviewers always like shorter proofs than I do, so I can never publish the long version.) I fix typos in arXiv versions, no matter how old, even if the corrections are too minor to submit an erratum to the published version.

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    $\begingroup$ Yes, I am finding myself doing this also, adding to the arxiv version things like auxiliary data, (short) source code and extended proofs such as the situation where two cases of a proof are analogous and in the published version one says "the disconnected case is dealt with similarly". $\endgroup$
    – Gordon Royle
    Commented Jan 17, 2014 at 0:03
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In my community, sketchy proofs are only acceptable in a survey paper, and in such cases, references are given for more rigorous treatments. We also tend to put the ugly, technical proofs in an appendix. I don't know your community, but you might consider putting your "long" proofs in an appendix (at the very least for the referees, perhaps as an arXiv version), and let the referees suggest what to cut out.

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Here is an illuminating quote from Terence Tao's journal submission guidelines; I could not have said it better:

"Give appropriate amounts of detail. A paper should dwell at length on the most important, innovative, and crucial components of the paper, and be brief on the routine, expected, and standard components of the paper. In particular, a paper should identify which of its components are the most interesting. Note that this means interesting to experts in the field, and not just interesting to yourself; for instance, if you have just learnt how to prove a standard lemma which is well known to the experts and already in the literature, this does not mean that you should provide the standard proof of this standard lemma, unless this serves some greater purpose in the paper (e.g. by motivating a less standard lemma). Conversely, some computations which you are very familiar with, but are not widely known in the field, should be expounded on detail, even if these details are “obvious” to you due to your extensive work in this area."

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When I write a paper, I include "notes" with additional details etc. The package "version" then makes it very easy to produce pdf files with or without the notes. I find this very useful for myself. I submit the version without notes, and sometimes put the version with notes on my website or the arXiv (and occasionally alert the journal to its existence).

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The referee gets to opine on the validity of the paper, and the editor on the readability and worthiness of publishing in the journal. At least, that's how it roughly goes. You have to make the paper accessible by expository skill, finding the right mix of detail to leave in or out.

I recommend asking the editor if it is best to put stuff in for the referee and editor to cut, or to submit something more concise with a promise to expand on whatever needs clarification. Even better than asking the editor is to ask colleagues who have submitted to that journal recently, or similar, and get their war stories and guidance.

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    $\begingroup$ That depends, I guess. As a referee, I have always been asked about validity and worthiness (and I have always opined on readability, regardless of being asked about it!) by editors. I thought this was the norm, really. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jan 16, 2014 at 22:55

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