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There is an interesting discussion taking place at MO over the suitability of this question: Complex structure on $S^6$ gets published in Journ. Math. Phys. The question is about the correctness of a paper that has recently appeared in a peer-reviewed journal, purporting to settle a long-standing open problem regarding existence of a complex structure on the 6-dimensional sphere $S^6$.

The basic question here is whether the community thinks this type of question could be suitable for MO, but a subsidiary consideration is how to edit this particular question to remove at least some of the features that some commenters found objectionable. Namely, they found the question brusque and somewhat confrontational or less than respectful to the paper's author. Let me then take a crack at rewording the question:

A paper purporting to solve a major open problem has recently appeared:

  • Gabor Etesi, Complex structure on the six dimensional sphere from a spontaneous symmetry breaking, Journ. Math. Phys. 56, 043508-1-043508-21 (2015). Link: http://arxiv.org/abs/math/0505634

An earlier preprint by this author on this topic, http://arxiv.org/abs/math/0505634 (now 10 years old), had some problems [insert specific criticisms here]. My question is whether these difficulties have been satisfactorily addressed in the current paper, and if so, whether this paper has indeed answered this outstanding question (whether a complex structure on $S^6$ exists).

This rewording could perhaps be improved, but the idea would be to try and pinpoint the specific mathematical considerations within the body of the question, and do so in a more or less polite and decorous way.

Onto the more general question: personally, I'm somewhat torn. We've generally declared to would-be solvers of the Riemann Hypothesis (etc.) that MathOverflow is not to be used as a vetting service, and that certainly seems like wise policy to me. Some users (I'll quote Joseph O'Rourke in his comment) take it a little further and say "It is inappropriate to discuss the flaws in a paper in a public forum." Other commenters seem perplexed by this stance.

I'd like to hear more discussion on this. I can definitely see that we do not want to be in the business of raking people over the coals, whether they be cranks or serious researchers. But if we manage to take the personal element out of it and get straight down to the business of pinpointing really specific mathematical difficulties encountered in peer-reviewed published papers, it's hard for me to think of really convincing counterarguments. Anyway, what do you think about this?

Last I checked there are three votes to reopen, so maybe it's good to discuss this openly here.

Oh, by the way: what about Community Wiki? There's another closely related question Is there a complex structure on the 6-sphere? which mentions the same paper that is CW. (I think one could argue that a suitably reworded question with specific mathematical content would not be a duplicate.)

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    $\begingroup$ "It is inappropriate to discuss the flaws in a paper in a public forum". Actually, I am one who is perplexed by this stance: after all, a published paper is a public document (well, let's assume it is open-access), so why avoiding a public discussion on it? Criticizing the paper does not necessarily mean criticizing the author! – $\endgroup$ – Francesco Polizzi Jun 25 '15 at 14:06
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    $\begingroup$ It was certainly my impression that we had a settled policy of not accepting questions of this general form. I do not have time at the moment to gather references for that, nor do I claim that that policy is necessarily correct, but I think that it has some consensus behind it. $\endgroup$ – Neil Strickland Jun 25 '15 at 14:37
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    $\begingroup$ Todd, you said that published papers are fair game here: meta.mathoverflow.net/questions/1265 $\endgroup$ – user25199 Jun 25 '15 at 15:41
  • $\begingroup$ @Carl Thanks for that. As might be obvious, in the present post I was leaning more towards the position that I more explicitly took there, so at least there is some consistency of where I seem to stand personally. $\endgroup$ – Todd Trimble Jun 25 '15 at 16:29
  • $\begingroup$ @NeilStrickland I think there has been some back-and-forth on this. The consensus might be more about evaluating preprints than about papers in (reputable) peer-reviewed journals (which then become part of established literature). Carl's comment brought me back to an earlier 'tea' discussion where some of this was discussed: tea.mathoverflow.net/discussion/973/…. $\endgroup$ – Todd Trimble Jun 25 '15 at 16:35
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    $\begingroup$ I think it will also be appropriate, if the question stands, to remove the non-math discussion from the question's comment section (perhaps keep those comments on record here). $\endgroup$ – Yoav Kallus Jun 25 '15 at 22:26
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    $\begingroup$ @YoavKallus Thanks for the suggestion. I think I might be more inclined to archive such comments at tea.mathoverflow.net (which still exists but doesn't attract too much attention). $\endgroup$ – Todd Trimble Jun 25 '15 at 23:27
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    $\begingroup$ @ToddTrimble I think the format above is better than the question as it currently stands. One should link to the journal version of the article primarily, with the current version of the arXiv paper as a backup, if they are the same, or similar enough. Then one should link to a specific old version of the arXiv paper, not just the generic identifier. I hesitate to edit without some sort of consensus, at least with a moderator such as yourself :-) $\endgroup$ – David Roberts Jun 26 '15 at 1:09
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    $\begingroup$ Ah well, I did it anyway. Forgiveness rather than permission, etc. $\endgroup$ – David Roberts Jun 26 '15 at 1:13
  • $\begingroup$ @DavidRoberts Seems like a good idea. Thanks for taking the initiative. :-) $\endgroup$ – Todd Trimble Jun 26 '15 at 1:15
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    $\begingroup$ I still think a more focussed approach like you outline above would be more in line with MO practive than currently but I hope @MishaVerbitsky comes here to discuss (I'm certainly not going to unilaterally make changes that major). $\endgroup$ – David Roberts Jun 26 '15 at 1:47
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    $\begingroup$ If the question is not appropriate to MO, is there an alternate location that might be more appropriate to discuss it? The arguments Misha Verbitsky gives that it is an important question to discuss seem convincing... $\endgroup$ – Will Sawin Jun 26 '15 at 4:04
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    $\begingroup$ IMO the question should be still more focused. "My question is in which way these difficulties have been addressed in the current paper." Is a question I would consider acceptable. In any case, even if the general second question is preserved I consider it as better to ask rather "in which way" than "whether." For on thing yes/no questions are rarely good MO questions. $\endgroup$ – user9072 Jun 26 '15 at 7:52
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    $\begingroup$ @WillSawin there is this Selected Paper Network one of whose goals seems to be to evaluate papers. (I did not follow this, as I do not think it is a good idea, but then there are many things on the internet I do not consider a good idea.) On your second remark: perhaps M.V. could just do what he'd have done in the "dark ages" before MO. $\endgroup$ – user9072 Jun 26 '15 at 8:05
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    $\begingroup$ My cited comment was inartfully phrased. I support quid's proposed standard ("If there is a specific question about a specific step in a proof..., then it is on topic."), and TMA's rephrasing of my comment. $\endgroup$ – Joseph O'Rourke Jun 26 '15 at 23:02
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A scientific paper is like a politician --- much of its reputation is based on hearsay, and superficial impression, not on cold analytic examination. Very few have willingness and time to investigate thoroughly. Thus, a baseless accusation can easily ruin a reputation. So, it is important to make accusations as precise as possible.

If the OP read the paper 10 years ago and found the mistakes then, but is unwilling (for whatever reason) to re-read the paper in sufficient detail to check if these mistakes are still there, then the very least he can do is to describe the mistakes that he found back 10 years go, and ask if they were fixed and how. Then the author (or any other interested party) at least has a chance of deflecting the (implicit) accusation of incorrectness.

As the question stands, it should be closed.

Addition: Carl makes a correct suggestion in the comments. It is a duty of a questioner to inform the author about the question to give an opportunity to respond. It is preferable to do so before posting a question, and to incorporate the response into the question.

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    $\begingroup$ Perhaps the author should be contacted in cases like these, so they have a chance to reply? $\endgroup$ – user25199 Jun 25 '15 at 15:50
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    $\begingroup$ @Carl I agree. Preferably before the question is posted. $\endgroup$ – Boris Bukh Jun 25 '15 at 15:53
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    $\begingroup$ And like a politician, bad publicity can be good publicity. If Etesi's result is affirmed in answers- well then, what could be a better advertisement for his paper? $\endgroup$ – Daniel Moskovich Jun 28 '15 at 7:11
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    $\begingroup$ @DanielMoskovich The maxim "Every publicity is a good publicity" is about a benefit to a politician, not to politics. We should strive for better. $\endgroup$ – Boris Bukh Jun 29 '15 at 8:56
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This meta question is phrased as a general question about the appropriateness of discussing published papers on MO. That's a fair question, but I think that it's actually quite a different question from the question of whether Misha Verbitsky's question was appropriate.

The paper by Gabor Etesi has a history. Namely, Verbitsky read an earlier version of the paper, found that it was flawed, and now (seeing that the paper has been published) is wondering if anyone has studied the paper carefully and believes that it is correct.

In my mind, this is a very different scenario from a question that simply asks, without any further context, "Is the following published paper correct?" That type of question is IMO not appropriate as it stands for the simple reason that it doesn't provide any explanation of why the question is being asked. Is there a specific point that the asker is having trouble with (in which case details should be given, after which the question probably becomes suitable for MO)? Is the asker just generally suspicious of the author or journal in question and wants someone else to study the paper first before spending any effort on it (in which case the question might be inappropriate for MO)?

To put it another way, I don't think that it makes sense to adjudicate such cases based primarily on whether the paper is published or not. The appropriateness of the question should instead depend on factors such as whether the asker has specific, concrete objections.

Returning to Verbitsky's question—at the time that I'm typing this, it has been edited so that there is no indication that Verbitsky read an earlier draft and found serious problems. Though the edited version might have the merit of being less confrontational, I believe that it's actually less appropriate than the original, because the reasons for Verbitsky's skepticism are even less clear.

In the end, I think that Todd Trimble's rewrite is a good one, so I'm not suggesting an entirely different rewrite, but I do think it's important to frame the general question properly. To repeat, I don't think it makes sense to formulate a general policy that uses the publication status of a paper as the primarily criterion for judging the appropriateness of questions about it.

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    $\begingroup$ I agree that the question as it currently stands is still not a very good question for MO (and could be closed as a duplicate), although it's more polite. I hope Misha Verbitsky will consider bringing back some of the context that led to the asking of the question in the first place. $\endgroup$ – Todd Trimble Jun 25 '15 at 23:56
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    $\begingroup$ I strongly agree. One could add a little paragraph at the end, along the lines of "I read the paper when it first appeared 10 years ago. Back then it had this flaw: <insert description>. Does anyone know if the published version of the proof is correct and the result is reliable?" I would not see this as confrontational if the mathematical problems in the original version are made clear. Without details it looks more ad hominem (which I believe was not the intention). $\endgroup$ – Joonas Ilmavirta Jun 26 '15 at 6:59
  • $\begingroup$ I could not agree more with TC. $\endgroup$ – pro Jun 27 '15 at 6:03
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The distinction between "published" and "preprint" is fuzzy and not all that relevant. I propose to adopt the same policy for both that is:

[W]e should avoid discussing preprints papers in general terms. If there is a specific question about a specific step in a proof and suitable context is given so that reading the paper is not a prerequisite, then it is on topic.

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    $\begingroup$ A first approximation to the meaning of "published" is published in a 'reputable' peer-reviewed journal. Modulo the word 'reputable', I think that's a fairly crisp meaning. The distinction is I think somewhat relevant because one of the big points of such publication is having some public measure of trust and reliability. $\endgroup$ – Todd Trimble Jun 25 '15 at 19:26
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    $\begingroup$ "one of the big points of such publication is having some public measure of trust and reliability" I do not get the point at all. If anything, I would take this as argument for there being higher hurdles for questioning the validity, not the converse as you seem to propose. If somebody, especially an expert, has some issue with a published paper they might contact an editor. $\endgroup$ – user9072 Jun 25 '15 at 19:40
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    $\begingroup$ I just meant that flaws in a published paper have more serious repercussions if people are more prone to putting trust in them over non-vetted preprints, and thus the urgency of settling any doubts about them might be felt to be more acute. "If somebody, especially an expert, has some issue with a published paper they might contact an editor." -- that's a point worth keeping in mind! $\endgroup$ – Todd Trimble Jun 25 '15 at 19:49
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    $\begingroup$ @Todd The reputability of JMP for "mathematicians" is cast into doubt by the question in its current form, and in general there is a continuous spectrum of journal reputability, so I think the claim of fuzziness remains. As I commented just now at the main question, the important issue is the mathematics, not the reputability of the journal. $\endgroup$ – user25199 Jun 26 '15 at 7:55
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I think Joseph O'Rourke meant to say something else. I will guess that something like the following is intended.

MathOverflow is not a forum for judging, critiquing, or refereeing preprints or published articles. MathOverflow is for answering specific questions in mathematics, including but not limited to judging, critiquing, or refereeing a part of a proof or statement, possibly a whole proof or statement if concise enough for this forum. To keep the quality high and the helpful and welcoming atmosphere of the forum, subjective opinions and similar statements which detract from such atmosphere and quality are discouraged.

With the appropriate specific details included, Todd's suggested edit seems fine to me. It is about the math, not about fame, infamy, or if a paper is wrong. I hope Misha Verbitsky agrees to the change in tone, for the sake of the forum.

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    $\begingroup$ It seems that the community often accepts questions of the form "what is the status of Conjecture X" or even "what are the recent developments in Field Y." I think that "is this paper correct" is a much more specific type of mathematical question than these two. $\endgroup$ – Sam Hopkins Jun 25 '15 at 17:44
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    $\begingroup$ Indeed. However, the first two are slightly discouraged as sometimes being too broad. Also, the first two usually involve less work to answer because someone reading it either has an informed opinion to share or saw a web page on it. Answering the last question involves more work, and usually one has to wait a few years after publication to find someone who can answer it reliably. I believe a major criticism of the MO question on Mochizuki and ABC was " too soon to ask" . $\endgroup$ – The Masked Avenger Jun 25 '15 at 17:50

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