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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.

Mare
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1 Answers1

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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.

Dan Romik
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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. – Mare Feb 12 '17 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). – Dan Romik Feb 12 '17 at 17:07
  • 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. – Mare Feb 12 '17 at 17:12
  • How about the following idea: For every article the journal could have a small forum where people can discuss the results. If enough people veto that there might be a gap, then the journal can mark that at some point. – Mare Feb 12 '17 at 17:15
  • @Mare discussion forums for math papers have been tried before with various arxiv-style clones (and could be a useful thing for many reasons and not just to address gaps), but none were successful so far AFAIK. Sounds like a cool startup idea, go for it. – Dan Romik Feb 12 '17 at 17:21
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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 http://mathoverflow.net/questions/261906/tachikawa-conjecture-for-commutative-algebras-proven . – Mare Feb 12 '17 at 17:26
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    There is a list of examples at http://math.stackexchange.com/questions/139503/in-the-history-of-mathematics-has-there-ever-been-a-mistake – Alexander Woo Feb 12 '17 at 17:29
  • @Mare - Why don't you feel comfortable doing 1 or 2? – aparente001 Feb 13 '17 at 07:06
  • See https://arxiv.org/abs/0709.1291, which prompted two retractions, acknowledged by the author. – David Roberts Mar 11 '17 at 10:05