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Imagine that you are in the following situation: You write up a proof which eventually gets published. There you need a result which is not so well-known but it is contained in another paper P; therefore you just cite it. You read P and come to the conclusion: It's awful. You need plenty of time to insert the details or even correct it. It may also happen that the proof is somewhat too complicated because in your situation it is much easier. Maybe you have found a shorter proof, but based on the ideas in P. Now what do you do? Several options come into my mind:

  • Just cite the paper without any further explanation.
  • Cite the paper but give a short hint how to simplify the arguments.
  • Cite the paper but give a more elaborate explanation of the arguments.
  • Write up the details of the proof of the desired result in your situation and remark somewhere that it was inspired by the paper P.

For each option there are pros and cons. For example, you don't want to blow up your proof with material which does not seem be so important. Also, you don't want to bore your readers. This favors the first options. On the other hand, you might want to be sure that the readers understand the argument and don't have to read P. This favors the last options. What do you think, which option is your favorite and why? Also, are there other appropriate options?

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    $\begingroup$ If you care enough to ask on MO, I'd say put the improved proof in your paper. "For the convenience of the reader, we now recall the proof of [Awful 2009, lemma 3]..." $\endgroup$
    – Laurent Berger
    Commented Nov 27, 2011 at 12:01
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    $\begingroup$ @Ryan: Yes, and the same is true for all other questions which were asked so far under these tags ;). $\endgroup$
    – Martin Brandenburg
    Commented Nov 27, 2011 at 12:18
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    $\begingroup$ Definitely choose option 4. If the result is not so well known anyway, it deserves to be appear in another paper. Also, if you can improve the proof, this is something everyone should see. $\endgroup$
    – Spiro Karigiannis
    Commented Nov 27, 2011 at 12:20
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    $\begingroup$ Meta thread in case there is need for further discussion tea.mathoverflow.net/discussion/1225 $\endgroup$
    – user9072
    Commented Nov 27, 2011 at 18:14
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    $\begingroup$ We all stand on the shoulders of giants. The first proof of a result can be awful for a number of reasons eg: (1) the proper language/formalism for the proof (which may trivialize it) was only invented later (2) the proof follows the intuition of how it was thought up and so may, for instance, have case-by-case analysis or ugly computations in coordinates (3) the author may have been in a rush to beat the competition and so the result is unpolished (4) common folklore of the time may be lost and so details that would have been obvious then are not so now Just be fair to the original. $\endgroup$
    – Benjamin Steinberg
    Commented Nov 28, 2011 at 4:11

3 Answers 3

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Improving existing proofs is an important and undervalued part of mathematics. We don't just want to know whether something is true; we want to know why it's true. So I think that if you have a better proof of something, you should find a way to share it with the world.

Here are a couple of thoughts about the practicalities, to add to Andrew's suggestion about the nLab.

First, you could put the simplified proof into an appendix to your paper. I quite like appendices, as both a reader and a writer. Used well, they help to keep the main part of the paper flowing, while providing crucial details to those who want them.

Second, it's entirely possible that the author of [Awful 2009] will referee your paper. So whatever you write, you need to keep them sweet. I think this also favours the appendix option.

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    $\begingroup$ @Tom: This is a very nice idea, thanks! $\endgroup$
    – Martin Brandenburg
    Commented Nov 28, 2011 at 16:28
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    $\begingroup$ I agree with Tom. I have given new, simpler proofs (albeit probably not of highly important results) in appendices. While it certainly is important (and essential) to cite the original source, when most of us look up a proof, we go to the version that's the clearest and most elegant, which is almost by necessity never the original source. $\endgroup$
    – Spiro Karigiannis
    Commented Nov 28, 2011 at 18:47
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Since this has been reopened, I'm going to repeat what I said in the meta thread before this question gets closed again:

stick the simplified proof on the nLab and cite that

This is a completely serious suggestion. You have an improved version of the proof (whether it is globally improved or just locally improved[1]). It's not "original research" so an "Established Journal" might be reluctant to publish it[2]. But hordes of other mathematicians will be interested in reading it so it should be put somewhere that they can find it. This seems a perfect fit for the nLab.

If you have it written in reasonably standard LaTeX, I can even help you get it into the right formatting.

[1] A global improvement is one that anyone reading the original proof would be interested in, a local improvement is one that anyone reading your result that depends on it would be interested in.

[2] Insert standard rant about journals here.

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    $\begingroup$ And of course you should also cite the original paper if you do this. $\endgroup$
    – Dan Petersen
    Commented Nov 28, 2011 at 8:16
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    $\begingroup$ I don't see why that needs saying. Surely that's just normal citation convention. Simplified proofs are often published, especially when they shed new light on the original, and in those cases it's quite normal to say "... due originally to X, but see Y for a gentle introduction" (or whatever). $\endgroup$
    – Andrew Stacey
    Commented Nov 28, 2011 at 9:52
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    $\begingroup$ But does the nLab really welcome contributions that might not conform with the nLab point of view? In light of nlab.mathforge.org/nlab/show/nPOV, I would think not. $\endgroup$
    – Joel David Hamkins
    Commented Nov 28, 2011 at 14:46
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    $\begingroup$ Re my first sentence: Tom Leinster has started a discussion at the nForum from this. You're welcome to join in (non-registered posts are allowed). math.ntnu.no/~stacey/Mathforge/nForum/… $\endgroup$
    – Andrew Stacey
    Commented Nov 28, 2011 at 20:27
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    $\begingroup$ With Andrew, I'd prefer to have this discussion at the nForum, but in case there's no migration there, let me say that mathematicians like Joel and Andy are heartily encouraged to IGNORE the nPOV, if it suits them. And having a personal web at ncatlab, to write up notes on what one jolly well pleases, is also strongly encouraged. And now I'm going to go to the nForum... $\endgroup$
    – Todd Trimble
    Commented Nov 29, 2011 at 0:01
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The first question is whether your are bringing new ideas to the party. If so, then you add your new proof and trust that the referees do not get to picky about space constraints. So you are in Case 4.

If there are no new ideas, but the proof can be significantly simplified in your situation things are a little greyer. I would still say that you are in Case 4.

If the problem is that the original paper is a dog's breakfast, then I feel you just have to hold your nose and cite it; you're in Case 1. (If you want to correct other people's work, someone must have homework lying around waiting to be marked...) In this case I think there is an argument that by reproving the original result you are making a kind of weak claim to it. Other authors might cite you when they should really cite the original.

In the case where you are offering more than improved exposition, it is mainly the space constraints that apply. I do not see any ethical issue then.

[In reference to some of the comments. Of course improving exposition is very valuable. But I cannot think of any reasonable journal that would accept an article consisting of an improved exposition of an accepted result, unless it introduced new ideas. I am not defending this situation, but it is a fact.]

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    $\begingroup$ I would say that terrible written papers do not deserve to be cited. And regarding space constraints - let the editor decide. If they tell you to shorten your paper, the version on arXive is still there for grad students to read. In other words - most of the time I would go with option 4. $\endgroup$
    – Vít Tuček
    Commented Nov 27, 2011 at 17:05
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    $\begingroup$ @r0b0t, you are certainly entitled to your opinion, but if a result is proved somewhere (one knows that) and one uses it then one has to cite that paper. This is not a judgement call based on the perceived expostitory quality of that paper. (One might add remarks or clarfication if one thinks it is needed, but one has to cite!) Of course, there are situations, like related work, where one has some flexibility to cite or not and there one might or might not take such considerations into account, but for a proper result one uses there is no room for any considerations whether one cites or not. $\endgroup$
    – user9072
    Commented Nov 27, 2011 at 19:52
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    $\begingroup$ -1: Surely if the original paper is a dog's breakfast, and you can simplify it greatly, the mathematical community deserves to have the simplification available to it. And of course, you should give full credit to the original author, but if other people cite the simplified paper and not the original, it's really in part the original author's fault for not writing it up better in the first place. $\endgroup$
    – Peter Shor
    Commented Nov 28, 2011 at 14:16
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    $\begingroup$ @rObOt : I strongly disagree even with your weaker statement. If an author has proven a result, then they deserve credit for it whether or not they are good writers. $\endgroup$
    – Andy Putman
    Commented Nov 28, 2011 at 17:24
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    $\begingroup$ +1 Andy. Sometimes the breakthrough of realizing something can be done, and bludgeoning the problem into submission, is what's needed to inspire people to write "elegant" or "better" proofs. Over the course of time (50+years) people may start to make references to the original briefer and briefer, but I think it should always be done, just out of good manners $\endgroup$
    – Yemon Choi
    Commented Nov 28, 2011 at 21:55

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