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Assume the following is given in a published paper:

  • There is an important result.

  • The important result might contain a gap, as possibly some people agree.

  • The authors never respond to questions on that and so maybe they do not want to make it public that there is a gap, which can not be filled.

  • It is not likely to give a direct counterexample to the important result, but only to point out possible gaps.

What to do then? The problem is obviously that some people might quote and use the result in new papers. How about the idea that such gaps could be marked in the published version of the paper as a motivation for future research? Of course a recent editor of the journal should choose a new referee to decide whether to mark such a paper. Also I think this should not have bad consequences for the authors as there might be reasons one is not aware of.

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  • Related, if not duplicate Can I challenge a paper already published in a peer reviewed conference/journal? Commented Feb 12, 2017 at 10:12
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    Do you have a source for the North Korean example? Or did you just make it? Commented Feb 12, 2017 at 14:38
  • I have heard of it, but since I have no official secure source I removed that part. Commented Feb 12, 2017 at 17:11
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    @MadJack: One possible motivation would be the different culture of mathematics. My understanding is that in most other academic disciplines, it is somewhere between feasible and common to write a paper(/comment/letter/note/response...) pointing out a perceived error in someone else's published paper. In mathematics, this almost never happens. Commented Feb 12, 2017 at 19:42
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    I'd suggest talking with some experts in the relevant research area, to see whether they agree that there is a gap and whether they have convinced themselves that, perhaps despite a gap, the result is valid. Commented Feb 12, 2017 at 23:52

2 Answers 2

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If the paper has not been published yet and the gap is brought to the attention of the editor and/or referee, they should inform the authors of the problem and ask for it to be corrected. If the authors don't correct the gap then the paper should be rejected.

If the paper has already been published, anyone else who finds the gap, brought it to the attention of the authors and didn't receive a satisfactory reply has two main formal options that I can think of:

  1. Write to the journal which published the paper. The journal can decide what to do, with an obvious step being to retract the paper. For an important result that was published in a serious journal, I expect the journal to take the matter quite seriously and solicit opinions from experts (which is essentially the same as the "gap referee" idea in your question) about whether such a drastic measure should be taken. In the event of a decision to retract, the journal will issue a statement that it no longer endorses the correctness of the paper, and probably add an explanation of the reasoning behind the decision.

  2. Alternatively, the discoverer of the gap can write and publish a paper pointing out the gap.

In both of these approaches, the existence of the gap has entered the official scientific record and most other researchers in the field will become aware of it, so the risk of people erroneously relying on an unsubstantiated result is reduced, though not completely eliminated. The risk can be further reduced by publicizing the gap through other informal channels such as blogs, seminar and conference talks, emails to groups of colleagues, posts on MathOverflow, etc.

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  • Do you know an example where 1. has happened in a bigger journal? I think it might be better to somehow mark the paper than to remove it since also wrong papers can have some good impact. Commented Feb 12, 2017 at 14:11
  • @Mare I don't know of an example, sorry (the only one that comes to mind is Wiles's proof of Fermat's Last Theorem where famously a gap was found and then corrected some time later), and am myself curious to know if/where this has happened. In any case a retraction doesn't mean the paper is removed - it will still exist in printed copies of the journal (and possibly online but with a prominent retraction notice being displayed like you suggest). Commented Feb 12, 2017 at 17:07
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    I guess the thing with Wiles was that Wiles responded to that gap, while the situation might be more complicated when authors never reply to possible problems with gaps. Commented Feb 12, 2017 at 17:12
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    It is much more natural when its directly related to the journal and linked close to the article in the journal. I dont really feel comfortable doing 1. or 2. when finding a possible gap, when I just can say that I dont see the reason of a lemma or something without spotting a real mistake. Concrete example is mathoverflow.net/questions/261906/… . Commented Feb 12, 2017 at 17:26
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    There is a list of examples at math.stackexchange.com/questions/139503/… Commented Feb 12, 2017 at 17:29
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In some prominant cases, gaps are not very important, as long as the overall proof idea and the original theorem are both correct.

You can try to fill the gaps. If the proofs must be significantly changed, then you might have a major contribution that may be publishable on the same level journals. If the proof idea is still correct, then you still have some contribution to be published on some specialized journals.

Clarification: Here is a very famous example that fillable gaps are usually not important. This 366-page paper claims to provide a detailed and complete proof for Poincaré Conjecture, and suggests that the original Perelman's proof has missing details or incomplete (pp. 9). However, most professional mathematicians believe that those gaps filled in the 366-page paper are not very important.

If a gap cannot be filled, then, at least some lemmata or some claims in the proof are wrong, and it is a mistake, not merely a gap. In math we often use gap to describe a proof that has skipped steps, very hard to follow, or sloppy.

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    Actually, gaps in proofs leave the results in doubt. If you don't have a complete proof, you don't have a theorem. Gaps are very important, sorry. You surprise me with this flippant remark. Commented Apr 4, 2025 at 19:02
  • @Buffy Please refer to the edits that I made. Commented Apr 4, 2025 at 23:50
  • This answer could be improved by clarifying how one determines/shows that "the overall proof idea and the original theorem are both correct." when the proof has gaps. I’m also not sure what to make of the current closing paragraph – if it’s known that a gap can be filled, it’s not a gap. Commented Apr 5, 2025 at 4:24
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    @MisterMiyagi If a gap cannot be filled, then, the proof is wrong, and it is a mistake. In math we usually use "gap" to discribe something that is likely to be correct, but incomplete or sloppy. Commented Apr 5, 2025 at 7:16
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    Hmmm, "usually not important". Sorry. Is your intuition good enough to know when a gap is fillable? Mine sure isn't. A famous example isn't "usually". Some gaps are left in presentations to students of proved theorems as those gaps are either tedious or beyond the student's current state of skill, but those aren't gaps in the proof. One "waves hands" over some details, but a proof can't do that and still be a proof. Commented Apr 5, 2025 at 9:54

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