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The title says it all. Suppose you are refereeing a paper where the author A makes strong statements about other papers by a different author B, like: the proof of Theorem 1 in paper [B] is wrong and we provide an alternate proof.

One possibility would be to diligently read the other paper, compare the results, and decide. This doubles (or triples) the refereeing effort, and puts the referee in a more difficult position, opposing both author B and the authority of the other journal and its referees, in case he decides to support A's claim. I am not sure this is the proper course of action, especially when the disputed result is not of such importance to deserve an exceptional attention.

Another possibility would be to ask the managing editor to involve (also) the other author B in the refereeing process, asking his opinion. The potential dangers in following this way are clear I think.

What do you do in such situations?

Waldemar
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    Why not simply ask the author to write in his paper specifically what is wrong with the proof in [B]? This would certainly make it better and easier for both the referee and the readers. – Andrei Smolensky Jul 17 '14 at 10:32
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    Good question. Does A carefully point out the error in B's proof? Is A's proof of B's theorem sufficiently different and interesting to warrant publication even if B's proof turns out to be correct or easy to repair? – alvarezpaiva Jul 17 '14 at 10:32
  • My question is just hypothetical! :) anyway, let's say that the claim of author A is very vague and does not point at the mistake. And there are enough results in the paper to justify its publication. – Piero D'Ancona Jul 17 '14 at 10:35
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    Point out the potentially troubling and unsupported statements and recommend their removal or weakening. If the author provides data supporting the alternate proof, good. No support behind B's proof being wrong? Don't say it. It would be like asserting Andrew Wiles' proof missed covering the exponent being the sixth Fermat prime. (We have only found five so far.) – The Masked Avenger Jul 17 '14 at 10:47
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    I think, as the comments indicate, this is a difficult question to answer in the abstract. What to do depends too much on the specific circumstances. It's worth noting that Voevodsky discusses a real example of this in his talk on proof checking: http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf – Deane Yang Jul 17 '14 at 15:33
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    This is a delicate situation since: (a) people are easily offended, and especially so when something non-flattering about them appears in print; (b) a general policy where mistakes in published papers can never be pointed out is obviously silly. Regarding your second paragraph, I personally have always subscribed to the theory that responsibility for the correctness of a paper rests solely with the author, not the referee (or editor or anyone else). – Christian Remling Jul 17 '14 at 17:27
  • Firstly, there is nothing "strong" in saying that some other's proof is wrong, of course if it is indeed so. Secondly, to check whether it is so or not, at least for yourself, can be done only via one way -- to read that proof under question – Victor Jul 18 '14 at 10:15
  • https://www.pubpeer.com/journals is a recently launched great website to debunk false research results. I have even seen a few math papers there, but otherwise Nature and Science dominate the scene. – John Jiang Sep 24 '15 at 18:37
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    I'd start by asking A (or if you're worried about protecting anonymity asking the editor to ask A) whether he's communicated the alleged error to B and what B's response was. – Steven Landsburg Oct 03 '18 at 02:26

1 Answers1

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I am surprised that this question never received an answer, though it did receive useful comments. My view is that a claim like "Theorem 1 in paper B is wrong" is a claim, just like any other in paper A, and it's part of the job of the referee to check claims. So, I think that, as the referee, you have to decide whether or not this claim should remain in the paper, and that decision should be based on whether or not the claim is correct. If, indeed, paper B has an error and no erratum resolving it, then it's a real service of paper A to point it out and supply a correct proof. On the other hand, if there's no mistake in paper B, then by allowing this claim in paper A, you're allowing an error to get into the published literature.

Regarding "doubling the referee effort," consider that not all claims receive an equal amount of justification in paper A, and not all require an equal amount of effort to check. For example, paper A might have a lemma whose proof is left to the reader. As the referee, you'd want to check that you think the lemma is correct, and if you can't (or can't do so within the time you've allotted for refereeing), then you would ask in the referee report for more details. Do the same with this claim. If, when you read that proof in paper B, the error is obvious, then you've verified the claim. If you can't find the error in paper B, then it would be entirely appropriate to ask author A to supply more detail backing up their claim. This course of action was suggested by the first comment, but the third comment makes it clear the OP wanted more. Odds are that the error is subtle if the authors and referees of B missed it, so having A put more details about the error would help future readers. When in doubt, it's always wise to ask yourself what would help the reader? If there is a subtle error out there, having A devote more discussion to it will definitely help future readers.

A second guiding question is: what would help the author? It sometimes happens, when you referee a paper, that you find a mistake but the author is convinced there is not a mistake. A good referee will help the author realize their error. Likewise, perhaps the author is convinced that there's an error in paper B but you don't think there is. Again, helping the author figure this out will make them a better mathematician and will improve paper A that you've been asked to referee. Relatedly, it could be that there is a mistake in B but it's a very minor mistake, like B forgot to include a case. It would be appropriate for you, as the referee, to make suggestions (or even to insist) regarding A's wording (e.g., ask A to change it from "there is a mistake" to "Incidentally the case ... is missing from B, but is easily resolved"). When you do this, you can help prevent A from saying something stupid (e.g., acting like a minor mistake is a huge deal), you can help A navigate the politics of the situation (an aspect pointed out in Christian's comment), and you can improve A's paper and the experience of its readers, in seeing how best to handle a situation like this.

I don't think it's necessary to get B involved, though you could suggest to A that they alert B to the error (or that A should ask B for clarification if the proof is unclear). If you are concerned that author B is especially prickly and might make a problem with A or with the editor, you can alert the editor to the claim A is making and ask the editor's advice. Don't assume the editor will read A's paper or your referee report in full.

Lastly, it might be appropriate to think about power dynamics. If author A is junior and author B is senior, it would be good to help A find the right phrasing to navigate pointing out this error in a way that won't cause a backlash and hurt A's career. Or, if A is senior and B is junior, then maybe the language of how the error is pointed out could be softened. I think it's safe to assume that no professional mathematician wants to make a mistake, or to have one pointed out publicly, so in general it's probably wise to err on the side of softer language in cases like this. Let's all try to make math a little bit more welcoming, encouraging, and friendly.

David White
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    I think you're putting too much responsibility on the referee here. This situation is definitely one that the managing editor should deal with, even before the paper gets sent to referees. – Sam Hopkins Feb 25 '24 at 23:28
  • @SamHopkins I would agree with you if Theorem 1 in B was a famous result and the error was unfixable. But, the OP says A has a proof of the result. And, if that's not the main thrust of A's paper, I can't imagine that the editor would even necessarily know this claim was in paper A, until the referee found it. For example, one of my first papers pointed out someone's error and fixed it. This was buried on page 24 of 45. It wasn't a big deal but I did feel there was value in pointing out the wrong statement. http://tinyurl.com/2fsszvpf – David White Feb 25 '24 at 23:33
  • I note that your paper has "Note that in [31], Lurie’s axiom is claimed to hold for positive flat symmetric spectra. This is an error, as acknowledged in [33]. Indeed, the example given in Proposition 4.2 of [49] demonstrates this failure conclusively, for both the positive and the positive flat model structures." So both the original author acknowledged the error, and an explicit counterexample existed in the literature. The question is what to do if the paper being refereed is the first to bring it up. – David Roberts Feb 26 '24 at 03:23
  • @DavidRoberts In my case, the referee didn't even mention this line. But my point to Sam is that the handling editor would have no easy to way to even know that was in the paper until the referee gave it a careful read. – David White Feb 26 '24 at 03:30