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A recent thread brought up the issue of the appropriateness of discussing correctness of a published paper on MO.

Correctness of Thierry Aubin's argument about positive Ricci curvature

I did a little digging on Meta, and found a few relevant threads:

Using Math Overflow to check whether or not a proof is correct

On discussion of published papers at MO

It would appear the consensus here is that discussion of the correctness of published papers should be fairly specific. Rather than discussing the correctness of the entire paper, the author of the MathOverflow thread should read the paper and ask specific questions about specific steps in the paper.

So we are aiming for less of a general "editorial" stance from MO, and more of a specific discussion of the component issues in a paper.

Is this a summary of the consensus, or am I missing something?

Thanks,

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    $\begingroup$ I would like an answer on the public record about the two Annals papers that Kevin Buzzard likes to point out are directly contradictory (without saying who got it wrong): which one is correct, and why is the wrong one wrong? How do you feel this sits in relation to your question? $\endgroup$
    – David Roberts Mod
    Commented Jul 9, 2022 at 6:53
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    $\begingroup$ @DavidRoberts: Alternatively, both papers could be wrong while contradicting each other. $\endgroup$
    – Ryan Budney
    Commented Jul 9, 2022 at 6:54
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    $\begingroup$ One paper claims A, the other claims not A, both in a generic mathematical setting (no foundational questions, everything in ZFC, i think); I don't think that in the exact context the statement A is independent of ZFC. But in principle that could be true in some other circumstance $\endgroup$
    – David Roberts Mod
    Commented Jul 9, 2022 at 6:59
  • $\begingroup$ @DavidRoberts: Certainly if their claims are the opposite of each other, that is one thing. But it's the nature of the argument, getting to the claim, that might lead to the greater contradiction. $\endgroup$
    – Ryan Budney
    Commented Jul 9, 2022 at 7:12
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    $\begingroup$ When "plusoneing/minusoneing" questions, one can read "this question shows research effort / does not show any research effort". I think this is what applies here. A question such as "Is Wiles' proof of Fermat valid?" shows no research effort, and this also applies to "is this paper correct?" without further context. $\endgroup$
    – YCor
    Commented Jul 10, 2022 at 14:08
  • $\begingroup$ "I have a proof, and a counterexample, so one must be right", A. Fox, personal communication $\endgroup$
    – J.J. Green
    Commented Nov 15, 2022 at 9:27

2 Answers 2

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At Ryan Budney's request, here is my comment converted to an answer.

If the paper is more than a handful of years old, and you can either: point to a specific potential issue with the argument; or point to a public statement of doubt from someone else, then asking about the correctness of the paper is totally fine for MO.

These requirements should disqualify 90%+ of the bad faith "is this proof of the Riemann hypothesis correct" posts.

Moreover, if it really is the case that in a certain field many papers from 30 years ago have nontrivial gaps or even outright errors, then it is beneficial to have a place on the Internet pointing out these mistakes, and MO is a good fit for that because of its very public nature. This is especially beneficial for younger mathematicians (maybe mature mathematicians are aware, through the grapevine, of which papers/authors to be wary of). There have been discussions recently about how making MO more welcoming to younger folks is a long term goal of the site.

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    $\begingroup$ I've never understood the problem with these verification posts to be bad faith—I think plenty of Riemann-hypothesis provers really think that they've got a proof; rather (I've understood) it's that MO simply isn't meant for checking correctness, of cranks' arguments, or professional mathematicians' arguments, or of anyone's arguments. I agree that documenting "folklore" errors is a different, and very important, function. $\endgroup$
    – LSpice
    Commented Jul 10, 2022 at 13:13
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    $\begingroup$ @LSpice: I think there's some easy ways to come to the conclusion these types of questions are in bad faith. Checking correctness of an argument is arguably the easiest task a mathematician can perform. Sure, a paper can be poorly written or a reader can be unprepared for reading a paper. But assuming the reader is ready and the paper is fairly well written, you should be able to check on your own. If you can't, then you could ask specific questions and not just "is this paper correct?" $\endgroup$
    – Ryan Budney
    Commented Jul 10, 2022 at 18:21
  • $\begingroup$ @LSpice Certainly some instances of this involve bad faith, for example when the question-asker refuses to admit they wrote the paper they are inquiring about the correctneses of. But I agree that the fundamental issue is that MO is not set up for this (and bad faith may be employed in an attempt to get around this). $\endgroup$
    – Will Sawin
    Commented Jul 12, 2022 at 14:24
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    $\begingroup$ Perhaps "bad faith" was the wrong phrase to use. I guess what I really meant was more along the lines of what YCor said in a comment above: often questions about correctness are bad because it is clear that the asker spent no time thinking about the question. $\endgroup$
    – Sam Hopkins
    Commented Jul 12, 2022 at 18:11
  • $\begingroup$ @RyanBudney (1/2) I might be late to the discussion but I want to counter one point that you do that I do not consider true and that may be conditioning your position towards the kind of questions discussed here. Checking correctness of an argument is NOT arguably always the easiest task a mathematician can perform. We avoid making these kind of general statements so flippantly. Checking correctness of an argument can be, depending on the particular argument and the expertise/specialisation of the particular mathematician, a harder task than other research tasks. $\endgroup$
    – Hvjurthuk
    Commented Jul 22, 2022 at 15:02
  • $\begingroup$ (2/2) And it is also an important task to perform by the collective and by every individual mathematician. Producing papers and proofs of topics around your area of research can be easier than understanding a proof or checking an argument of another area far from yours, and both tasks are important! To set a(n in)famous example: would you say that Scholze and Stix study of Mochizuki's claimed proof was easier for them than producing papers on their particular areas of study (and note that however they were the closest to Mochizuki's work out of Mochizuki's influence)? I do not think so. $\endgroup$
    – Hvjurthuk
    Commented Jul 22, 2022 at 15:05
  • $\begingroup$ Final remarks (1/2): the kind of questions discussed here, when done in provable good faith and real interest, should be accepted in the site because these questions are an essential and core part of what mathematicians do (or should do): checking, improving and expanding arguments from other mathematicians to discover new truths. I would eliminate out of the site questions asking for correctness of arguments made by untrustworthy sources like some fringe mathematicians appearing only in Vixra or GM. $\endgroup$
    – Hvjurthuk
    Commented Jul 22, 2022 at 15:11
  • $\begingroup$ Final remarks (2/2): However, I would definitely push for accepting questions about correctness of papers or preprints written by mathematicians with at lest one peer-reviewed publication like (e.g., de Branges). Keep in mind that many proofs often use folklorical conventions that may be unknown or uneasy to understand for people (experts on other areas) out of the areas where these works come from and this is a problem for the necessary openness of mathematics, which is a fundamental concept if we want our science to keep together and new bridges being set and built between different areas. $\endgroup$
    – Hvjurthuk
    Commented Jul 22, 2022 at 15:16
  • $\begingroup$ Epilogue: These questions are therefore not just valid, but necessary in an ever expanding and highly connected branch of knowledge. We speak different dialects of the same language and even between different groups working in close areas (or even the same) there may be some differences in treatment or terminology that might set up some difficulties. MO is the perfect place to solve that problem and fill the unintentional gaps that might be out there. Always in good faith. We should not kill such a fundamental conversation because constant reviewing of arguments is what our science is about. $\endgroup$
    – Hvjurthuk
    Commented Jul 22, 2022 at 15:22
  • $\begingroup$ @Hvjurthuk: Maybe I should have qualified my statement. But checking correctness of a properly-written proof is much the same task as a compiler, for a computer programming language. Certainly often proofs are often not written well-enough so that one can perform the task this easily. If a referee is not feeling particularly generous, at that stage they can write back to the editor with complaints. $\endgroup$
    – Ryan Budney
    Commented Jul 22, 2022 at 17:33
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The linked question can be equivalently reformulated as "what specifically is the gap in Aubin's paper that Paul Ehrlich alludes to?", and this is certainly acceptable - I don't see how it is different from any other question asking to clarify a specific passage in a published paper that is opaque or too brief for the reader.

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    $\begingroup$ I agree. In this specific case, there is a published claim of incorrectness, which makes the situation a little unusual. In general, though, my impression of the consensus view is the same as Ryan Budney's. $\endgroup$
    – Timothy Chow
    Commented Jul 8, 2022 at 23:18
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    $\begingroup$ It appears as if Paul Ehrlich is retired. The last institutional affiliation I can find for him us U. Missouri-Columbia. I find these "What was X thinking?" questions a little frustrating, as they would seemingly be better directed at X, than at MO . Perhaps it would be better to phrase it as "Did Paul Ehrlich have a valid point, and if so, what is it, specifically?" $\endgroup$
    – Ryan Budney
    Commented Jul 9, 2022 at 0:22
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    $\begingroup$ I've updated the post using your suggestions. $\endgroup$
    – C.F.G
    Commented Jul 10, 2022 at 2:32

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