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An author has proven an interesting mathematical result, but notices that this result contradicts existing literature.

Does the author have to find an explicit error or counter-example to the existing literature before considering submission? Can she submit the proven result to a journal, if she cannot find an error in her own proof?

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    Isn't that the whole point of like...academia and science in general? – Eppicurt Nov 28 '17 at 11:07
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    It would be a good idea to first ask peers/colleagues to look at the proof, before sending it to a journal. Especially if the existing literature is widely accepted, chances are high that there is an error that the author simply can't find on her own. It is, of course, still possible that the literature is wrong, but better save than sorry I'd say. – Dirk Liebhold Nov 28 '17 at 11:09
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    If I understand correctly, you have two contradictory proofs (one yours, one in the literature) and you cannot find an error in either one. I am not a mathematician, but it sounds as if at the moment you do not have a result. Either proof could be the one with the error (though perhaps not with equal probability...), and you need to determine which is which before you can draw your conclusions. – user2390246 Nov 28 '17 at 11:23
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    An additional way to check for errors (yours, or the existing paper) would be to send your work to the author of the existing paper. – GEdgar Nov 28 '17 at 11:40
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    @StephanKolassa: can you explain why this should be a duplicate? The questions seem totally different to me. – yupsi Nov 28 '17 at 16:39
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Certainly the fact that there is a contradiction with previous literature must be prominently advertised; to do otherwise would be scientific misconduct. I would not be confident in publishing such a paper unless I found the mistake in the other paper, or could give a counterexample showing they are wrong.

After having carefully worked out the other paper and my own paper, and asked any experts with whom I have an established connection, I might contact the authors of the contradicting paper (if they are still active). After their response, or lack thereof, I would consider publishing a preprint and after that submitting to a journal. The journal is the slowest and most uncertain way of finding out where the problem is.

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    In response to your parenthetical statement: Surely it suffices that the authors of the contradicting paper are still active. – Peter Nov 29 '17 at 13:28
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    @Peter You should never trust a vampire, not even when they publish mathematics. (edited) – Tommi Brander Nov 29 '17 at 13:38
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Forgetting the issue of publication, when two mathematicians find contradictory results, I think they have the collective intellectual duty to try to figure out what is going on. Generally this should mean that one of the purported proofs is wrong; however, it could also be that an even earlier result (used by one or the other contradictory proofs) was incorrect; conceivably, it could even mean that a contradiction has been found in whatever foundations of mathematics were being used, but we probably shouldn't take this possibility too seriously.

Generally speaking, I would say the burden of figuring out the root of the problem lies with the author of the most recent result (were it only because the others of the other result might be retired or dead). So you can't just go ahead and say "I proved not-X" when X appears in the literature, you need to analyse why and where the proof of X is wrong.

There are exceptions, however. One extreme example would be that if you can find a numerical counterexample to Fermat's Last Theorem (that anybody can check with a computer), you don't need to explain where Wiles's proof was wrong (or even understand it). More generally, if your proof of not-X is conceptually much simpler and/or much shorter than the proof of X found in the literature, I would say that this is a valid reason to shift the burden of finding an error to the authors of the latter.

One valid reason (at least, valid from the point of view of intellectual honesty: it might be another matter to actually convince anyone) not to analyse the proof of X for error is if you don't understand the techniques used therein. If they are too complicated, this might fall under the "your proof is much more simple" category mentioned above. But a genuinely problematic situation might arise if two mathematicians from completely different domains were to prove contradictory results, neither being able to understand the intricacies of the other's proof; third parties would then need to get involved to resolve the contradiction.

But in any case, any contradictory result you are aware of should be explicitly mentioned in a publication, and whatever reason you have not to analyse their proof in search of the error should be explained.

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    however, it could also be that an even earlier result (used by one or the other contradictory proofs) was incorrect Happened to me once :-) So, yeah, brace yourself for some digging! – fedja Nov 29 '17 at 1:43
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    I actually did find a numerical counterexample to Fermat's Last Theorem, but this comment box was to small to contain it. Man, I can't believe I joined Academia just to post this pathetic one liner :-) – user83711 Dec 1 '17 at 0:46
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Let me just note that Voevodsky (2002 Fields Medal) describes such a situation that he experienced himself (http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf):

In October, 1998, Carlos Simpson submitted to the arXiv preprint server a paper called “Homotopy types of strict 3-groupoids”. It claimed to provide an argument that implied that the main result of the “∞-groupoids” paper, which M. Kapranov and I had published in 1989, can not be true. However, Kapranov and I had considered a similar critique ourselves and had convinced each other that it did not apply. I was sure that we were right until the Fall of 2013 (!!). I can see two factors that contributed to this outrageous situation:

  • Simpson claimed to have constructed a counterexample, but he was not able to show where in our paper the mistake was. Because of this, it was not clear whether we made a mistake somewhere in our paper or he made a mistake somewhere in his counterexample.
  • Mathematical research currently relies on a complex system of mutual trust based on reputations. By the time Simpson’s paper appeared, both Kapranov and I had strong reputations. Simpson’s paper created doubts in our result, which led to it being unused by other researchers, but no one came forward and challenged us on it.

EDIT (01/01/2018): Let me add another (IMHO, relevant and interesting) example. Asher Peres wrote (https://arxiv.org/abs/quant-ph/0205076):

" Early in 1981, the editor of Foundations of Physics asked me to be a referee for a manuscript by Nick Herbert, with title “FLASH —A superluminal communicator based upon a new kind of measurement.” It was obvious to me that the paper could not be correct, because it violated the special theory of relativity. However I was sure this was also obvious to the author. Anyway, nothing in the argument had any relation to relativity, so that the error had to be elsewhere...

I recommended to the editor of Foundations of Physics that this paper be published [5]. I wrote that it was obviously wrong, but I expected that it would elicit considerable interest and that finding the error would lead to significant progress in our understanding of physics. Soon afterwards, Wootters and Zurek [1] and Dieks [2] published, almost simultaneously, their versions of the no-cloning theorem...

There was another referee, GianCarlo Ghirardi, who recommended to reject Herbert’s paper. His anonymous referee’s report contained an argument which was a special case of the theorem in references [1, 2]. Perhaps Ghirardi thought that his objections were so obvious that they did not deserve to be published in the form of an article (he did publish them the following year [7])."

  • Why the hell does this answer get 22 upvotes when it basically says, in much less clear words, what I was saying from the beginning? Hmm. – Daniel Goldman Dec 7 '17 at 10:32
  • @DanielGoldman: I don't quite understand your comment at two levels. First, as far as I can see, I gave this example before you mentioned it here "from the beginning". Second, the situation that you complain of may be, amusingly, a demonstration of the same reputation factor that is discussed here: people upvote "much less clear words" than yours just because they were written by Voevodsky :-) – akhmeteli Dec 7 '17 at 11:28
  • @akheteli, I meant I said, from the beginning of the answer, that a conflict with existing literature should not prevent someone from publishing. But now that I read the "answer" again, you never actually answered the question. But yes; reputation is interesting on this site, especially in that answers tend to get pounded once they go negative. – Daniel Goldman Dec 7 '17 at 11:37
  • @DanielGoldman : The example from my "answer" is called "very interesting" in your answer. So maybe the specific and relevant example was more interesting for other people than general reasoning. – akhmeteli Dec 7 '17 at 12:04
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    I cannot determine, from your answer alone, what your suggestion is. Are you suggesting that this was an extreme case? Are you suggesting that he publish the result? You don't even suggest that the OP makes sure that the result is valid by dong any due diligence. But again, that is the nature of SE. – Daniel Goldman Dec 7 '17 at 12:33
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As a reviewer, I would definitely recommend rejecting such a paper.

I can certainly imagine that there will be such cases where there is an interesting conversation in the community to be held. However, this conversation does not need to happen via journal articles. The appropriate course of action would be to first discuss such a situation with a few experts. If none of them can resolve the situation, post the article on the arXiv and draw attention to via conference presentations, etc.

If an apparent contradiction receives significant attention and yet is not resolved, then publishing an article describing the conundrum might make sense. This article would be very different from the one described by the OP though.

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    Thank you for your answer. I agree with discussing the result with other experts before submitting and if possible 'advertise' it to get opinions. Follow up question : would you reject it immediately or would you try to understand the error ? I ask this to understand whether submitting the paper is or not a good way to get new opinions on the proof. – kantadou Nov 28 '17 at 13:27
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    If I were the referee I might try to understand the error, but I’d definitely be annoyed at the author for trying to make me do their work for them. – Noah Snyder Nov 28 '17 at 15:09
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    If the contradiction is genuine, there is no reason to reject the paper outright. Starting with this as your first thought seems exceedingly premature! – Wolfgang Bangerth Nov 28 '17 at 16:16
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    @WolfgangBangerth Well, in that case the original paper the question was about completely disappears and is replaced by a "ZFC is inconsistent" paper (or whatever other foundations are used). – Arno Nov 28 '17 at 16:27
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    This approach to publication would seem to bias the literature toward initial results. There's no reason that the literature should advocate one side of a controversy over the other due to happenstance of publication order. – Nat Nov 29 '17 at 13:55
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To publish a contradictory result (in maths, as opposed to natural science) without further explanation would be to say "there is an error in one of these proofs but I don't know in which". Unless the issue at stake is of very great importance that is probably not a statement that is interesting enough to publish. I would say you should be able to point out an error, a counter-example or a hidden assumption in the original work.

Where there is no straight-forward mistake in either proof a hidden assumption should be considered possible and may be of real importance. An example is Von Neumann's purported proof that the results of quantum mechanics could not be produced by a hidden variable theory. This was contradicted by the development by Bohm of his pilot wave theory (a hidden variable theory that does just that). It was then realised that Von Neumann's proof, while not containing an explicit error, applied only to local hidden variable theories, and the pilot wave theory is non-local. This is an important distinction that (whether Von Neumann himself understood it or not) had not been generally appreciated.

This was a case of a physical theory providing a counter-example to a mathematical result about physical theories, thereby revealing a hidden assumption.

In pure mathematics it would be very unusual to publish a contradictory result with no attempt to resolve the paradox. Even in the rare instance you are suggesting the axiomatic basis of the field may need to be revised, you would be expected to have an opinion on the correct resolution. (The history of set theory provides examples of this type).

https://en.wikipedia.org/wiki/Hidden_variable_theory#Bohm.27s_hidden_variable_theory

https://en.wikipedia.org/wiki/John_von_Neumann#Mathematical_formulation_of_quantum_mechanics

https://en.wikipedia.org/wiki/Paradoxes_of_set_theory

  • That's a great example. The graph of Bell's inequality looks like the graph of a hidden variable problem that I am familiar with so I immediately knew it was wrong but not why. – Joshua Nov 29 '17 at 19:10
  • It is doubtless different in mathematics, but philosophers took several centuries before they nailed the problem with the ontological argument (for God's existence). People could show that something was wrong somewhere, but only with the development of predicate logic at the beginning of last century was it possible to say exactly what. This is not a case of only have two proofs of conflicting results, since people could show that the ontological argument itself was flawed. That was surely worth saying, even though the tools weren't available to nail the problem down. – cfr Dec 1 '17 at 2:31
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We had a similar situation once we were trying to solve an interesting problem in Thermodynamics of interfaces. While all earlier reports claimed a specific quantity to be always negative, we consistently received a positive value.

We were skeptical and began to look critically at our work and also the earlier works. Such a situation in principle says something new is found which contradicts other. Most of the times you have to place your scientific arguments not just by stating why you are correct but also by at least speculation why others could have been wrong. (Please, keep in mind that others were well-renowned scientists who did good science. But we all make mistakes and most often correct them).

Why is it good to speculate what could be wrong in literature?

  1. Provides a stronger proof and explicitly claims other people are wrong.

  2. Shows authors knowledge that he had understood others work before he claims something about it. Particularly helpful if your speculations are more logical and scientifically sound. Even a marginally acceptable argument, if valid, is sufficient enough to convince referees and readers.

  3. Attracts more readers, often speculating other works requires citing them. Having cross-reference to earlier works is hugely a good practice. Attracts also the scientists whose theory you refute.

By the way, in our work, we were happy to find a flaw in literature. We are now planning to prove our theory using multiple methods (theory, simulation, and experiment) before we begin writing about it. (That is why I do not provide details in this answer)

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    The higher problem with math, as opposed to non-deductive sciences, is that all the given information is explicit in the paper. If the authors of paper A find a contradictory paper B, then they must be able to show a specific logical error in one paper or the other. Not doing so is therefore a failure on their part. – Daniel R. Collins Nov 28 '17 at 16:11
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    The question is about mathematics, not about science. – Arno Nov 28 '17 at 16:30
  • @Arno I understand your point. But as stated in previous comment by Daniel. R. Collins, the problem requires strongly to find a flaw in previous works, as the results in a mathematical paper provide all information explicitly. Which means you don't have to speculate but can rather provide solid proof of flaw if you are able to find one. And I definitely recommend doing so for the same reasons stated above in my answer. – Ramanathan Varadharajan Nov 29 '17 at 13:40
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    As a practitioner of mathematics, I find this to be a good and useful answer, and I disagree with the first two comments (mathematics is a science; the "non-deductive" sciences are not void of "deduction"). Also, the comment of @fedja to the answer of Gro-Tsen shows that instead of the logical error being in one or the other paper, it might be in an earlier paper. – Lee Mosher Nov 30 '17 at 17:13
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Been there, done that.

During my MSc thesis I found that some (quite perplex and rare) thing from a book did not work as described there. (To be fair, the book definitions were not wrong, but ambiguous, however, the examples clearly showed the intended approach.) The way I did it (slightly, but decisively different), it worked. I was very cautious (being a student) in formulating the thesis. But it worked out and I got my best grade on it.

Putting the careful formulations aside, the thesis was like "they try to do XYZ in a abc way. It fails, here is a counterexample. If you change XYZ to XYZT, it works, have a look." In my eyes it was and still is a rock-solid research result. That's how science is made.

PS: Oh, and this happened in Germany, so a MSc is not a sign of a failed PhD, but rather the standard degree.

  • Is the book or the thesis available online? – Obie 2.0 Nov 28 '17 at 22:38
  • Yes. I was not going to rub the issue in the faces of the book authors', even if they are absent from here, hence the rather non-descript formulation in my answer. But at least my thesis is online. – Oleg Lobachev Nov 28 '17 at 22:45
  • Given that your thesis disproved their result some time ago, surely they (as members of the mathematical community as a whole) must be aware of this result? – Obie 2.0 Nov 28 '17 at 22:47
  • The initial result was sort of marginal, the book was ~20 years old at the moment of writing the thesis. I would not be so sure, they are aware of me disproving them. I have seen a somewhat similar correction years after my thesis, and I am quite sure it was an independent discovery. So yes, the community is aware of the result. No, the community might not be aware, I was first. I have a priority time stamp on the thesis, which is heart-warming, but not much more. The thesis is also in German, this does not really help. – Oleg Lobachev Nov 28 '17 at 22:52
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    Presumably because thepost doesn't seem to answer the questions, which are "Does the author have to find an explicit error or counter-example to the existing literature before considering submission? Can she submit the proven result to a journal, if she cannot find an error in her own proof?" – David Richerby Nov 29 '17 at 12:10
0

Sometimes, on further examination of the contradiction, one finds that both cases are true but that these cases are subtly different. The von Neumann proof above is an example. A trivial (formal because I don't remember an actual) example would if one result used an axiom system "A" with a seemingly simple extension "y" and the other used system "A" with the seeming equivalent extension "z" so that A+x gives results different from A+y whereas x and y may yield the same results if using B in stead of A.

This is probably covered above in the post about some not so obvious result used by both papers being questionable.

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TL/DR: Peer reviewed literature is not doctrine. That a result contradicts "accepted truth" does not mean that it should not be published. All that matters is that the result be properly checked for errors. If there are no apparent errors, then let the academic community decide why there are two seemingly correct yet conflicting results.

If you find a result that contradicts existing thought, publish it. Whether or not a result contradicts standing thought should not affect its acceptance or rejection. Only the quality of the argument should have an impact. I say "should not" because it is sadly not always the case.

If you are not certain as to whether or not your conclusion is correct, find someone with whom you could discuss the matter and see if they agree. If you can find someone prominent with whom you can co-publish, and who can add to the argument to make it more robust, then you might want to look into that.

Automatically Wrong

There seems to be a bit of a suggestion that if it contradicts existing literature, it must be wrong. Now, I suppose in mathematics, it is usually relatively "simple" to verify a proof, but even then there can be errors. Make sure you know what you're talking about. Get some feedback. But do not assume that you are wrong because the current literature is against you. If this contradiction were in science rather than mathematics, even more the reason to not assume that you must be wrong.

Much of published literature, at least in science, overrides past literature. In science, we only assume that a theory is true, until we find evidence which contradicts it (Further discussion on the nature of science). Mathematical theory can be such evidence.

Again, if your thesis is well thought out, properly explained, and an expert in the field cannot find an error with your work, there is no reason to assume that your answer is wrong. Just because an existing answer is accepted by academia should not dissuade you.

Conflict Resolution

There has been a suggestion that before publishing the paper, it should be determined which position is correct. However, that might not be possible, or neither party may have an answer. Publishing a result, so long as it is not apparently flawed, allows the community of experts to become aware of the potential alternative and work on explaining the conflict. As a professor recently said, the main way in which experts in a field communicate is through peer review. To refrain from publishing because the question of whose position is correct has not been resolved would in some ways be withholding potentially useful information from the academic community.

Doctrine

David Richerby made an interesting case. He suggested that because the other theorem has already been accepted, it should now take additional effort to overturn it. However, there is nothing that seems to be wrong with either theorem (at least according to the OP). David's suggestion means that if the OP's proof were conceived and received first, and it were the other result that was later identified, somehow the burden of proof would switch and now it would require exceptional levels of justification to get that result published. In other words, the order in which the result was produced is the only thing that is changing, and yet somehow that change takes the OP's result from requiring extraordinary proof to requiring ordinary proof and takes the currently established result and demands that now it would need extraordinary proof.

When a result carries more weight, simply because it is already established as being "true" by a group of people, that result becomes doctrine. That is not how academia works. That is not how research works. The validity of a result, and the level of justification needed, is determined only by the result. So long as there are no apparent flaws, and the result has been reasonably checked for flaws, as all results for publication need to be, then the answer is obvious: publish and let the academic community work on resolving the conflict.

Dan Fox linked to a very interesting example. "In 1991, Kapranov and Voevodsky published a proof of a now famously false result." In 1998, someone came up with a contradictory result, but could not find the error in the original proof. Ignoring what would have been the apparent consensus advice of SE Academia, the result was published anyway. It was not until 2013 that an error was found in the original proof, and it might have taken a lot longer had Simpson not pushed the issue by publishing his own result.

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