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Our team finished a paper 3 months ago. However, we use a result from another paper in a very top journal (one of the best). There are quite a lot misprints and somewhat mistakes. We can fix most of them but one inequality which they didn't have explanation why this inequality is true. We have spent 3 months to ask people (some extremely top mathematicians, they published many papers in the top 5 journals) and no one understand...We also asked the authors, but it seems that they don't want to explain that to us...We are really lost and don't know what to do...Shall we submit our paper? We really can not make sure that the inequality is true...

PS. This result is crucial in our proof...

Now, we have clarified the 'gap', there are some problems in the proof, the assertions are not accurate/correct. However, luckily, the final result is correct. After discussing with one of the authors, we confirmed that it is a gap and it took us quite some time to fix it.

Well, some people told me that the 'big guys' do not care about the details. I am not sure whether these incorrect details would lead to wrong theorem. As a young scholar, I have to say that I am a little bit disappointed at math research (well, maybe my field).

user92646
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    You use results from another paper that you do not fully understand in your own paper, is that correct? – BPND May 29 '17 at 07:41
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    Is there any proof of the inequality? Also, have you looked at any followup papers either by the authors or other people? Sometimes, later work explains some ideas better than the original authors... – Nick S May 29 '17 at 09:56
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    I wouldn't use the inequality. I think it's okay to use a theorem you don't fully understand (I've had a colleague who used a theorem with a prize-winning proof that took a long time for experts to verify). Yet, there are so many red flags here that go beyond you merely using an established theorem that you don't understand to the depth that the original authors do, you risk publishing an error. I think your best non-mutually exclusive options are: wait and see if the proof is published, or try and prove/disprove the theorem (in the second case, publishing the result independently). – Dr. Thomas C. King May 29 '17 at 11:04
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    Explain the inequality to a few physicists. If they believe it while they wave their hands around, and murmur something about Maxwell equations, you can use it. – Davidmh May 29 '17 at 21:27
  • @BPND I will say that there is no proof for this inequality...The authors stated many statement in a paragraph and we verified most of them. There are some followup papers, but no one really use the main result of this paper (let along explanation). – user92646 May 30 '17 at 06:40
  • @ThomasKing THX. Your suggestion is good. We are still asking people and we are still discussing...All people have discussed are now confused. – user92646 May 30 '17 at 06:42
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    If I were in your situation, I would explain, early in the paper (probably in the introduction, not in the middle of some proof) that I need this inequality, and I would give the inequality a name or equation-number. I would add that the inequality is asserted in [reference] but without proof, and I have been unable to find a proof. Then, in any theorems that depend on the inequality, I would explicitly assume it, by name or equation-number in the statement of the theorem. It doesn't seem pleasant, but it's honest. – Andreas Blass Nov 04 '17 at 00:21
  • @AndreasBlass THX, man. We have filled the gap, which took us quite a long time. – user92646 Nov 13 '17 at 23:24

4 Answers4

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Ask on mathoverflow!

It's a site like this, but for math at research level. There is incredible expertise in all fields of math (I know of) over there. Before you ask, look at "How to ask" and also take a look around at the site to get a feel for how the folks roll.

Dirk
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Should you submit your paper under this situation - possibly.

People assumed that Fermat's last theorem was true for centuries before it was proved. You don't necessarily need every detail of every proof.

If it seems reasonable that it is true then it's okay so long as you are clear on this assumption, and caveats thereof, and your paper is essentially going to be retracted if the inequality isn't true, or have conditions to it.

From your description however There are quite a lot misprints and somewhat mistakes this tells me that the paper is of questionable if not poor quality, and your mistake was doing further work that depended on this. In light of this I would not wish to publish without at least some verification of the inequality. Peer review in no way means something is true.

camelccc
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    Agree. Peer review doesn't mean it is true. Especially for some `big' guy's papers....Most journals and reviewers would not question much about these papers. – user92646 May 30 '17 at 07:00
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Well isn't it obvious?

You just say "This is a proof dependent on the truth of result X".

That will be true no matter if X is true or not.

And then it is still a useful paper.

Then if X is shown to be true, then your paper becomes a true proof.

zooby
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    It is not obvious. As word spreads that the results in paper A are not reliable, responsible researchers stop citing it, and other papers building on it end up being ignored as well. The end result is that, even if statements in paper B are clearly labeled as being dependent on the unverified parts of paper A, the usefulness of paper B may become negligible. Even if later the results of A are verified, people in the field may still remain ignorant of the results of paper B. – Andrés E. Caicedo Oct 01 '19 at 20:55
  • Well if that's the case, the paper will be rejected and no harm done. Anyway, result X could still be proven by someone else and then your paper is then a complete proof. Unless the person is concerned with being "the champion who put the final piece of the puzzle in the proof". – zooby Oct 01 '19 at 21:26
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If the paper clearly states the inequality is true and it was peer-reviewed and published in a reputable journal, you can assume it is true. The rule of thumb here is that you need to be able to fully understand the results of the paper you use, but not necessarily be able to reproduce ever result you cite.
For example: when in any field you use statistics (e.g. an inequality), you need to understand the final equations you are using. However it would be silly to require almost every scientist to be able to independently prove each and every equation from scratch.
The same is true for various measuring machines. As a scientist you need to know how to use your equipment and what it can and can't do. However beyond that, it is perfectly acceptable to cite that it works and treat it as a black box.

dimpol
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    But apparently the OP is unable to "fully understand the results" which is exactly why he asks how he should handle it. I slightly disagree with your first sentence. You shouldnt "assume it is true" if it is not proven. – BPND May 29 '17 at 09:05
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    I disagree with this answer. This might be good advice in some fields, but for mathematics, this advice applied to this situation is in my opinion a big NONO. The OP wants to use an inequality which, from what I understand in his post, was not proven in that paper and was not referenced to a reliable source. And that raises a big red flag: in mathematics a result like this should be published only when it is obvious, which seems it is not. The fact that the authors refuse to explain it is another red flag... If you combine this with "here are quite a lot misprints and somewhat mistakes"... – Nick S May 29 '17 at 09:44
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    , which means that the authors made many mistakes on simple things (so why should one assume that they didn't make a mistake on this more complicated thing?). Given all those red flags, the paper should probably not have been published (at least not in this form) and unless they find a more reliable source I would be hesitant to rely on this. – Nick S May 29 '17 at 09:49
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    If the paper clearly states the inequality is true and it was peer-reviewed and published in a reputable journal, you can assume it is true. — This is absolutely wrong. At most you can assume that a few people think it's probably true, modulo a bunch of picky details that they didn't bother checking but smell okay. – JeffE May 29 '17 at 19:27
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    However it would be silly to require almost every scientist to be able to independently prove each and every equation from scratch. — True. But reading, understanding, and verifying a proof is a lot easier than proving something from scratch. – JeffE May 29 '17 at 19:28
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    " and it was peer-reviewed and published in a reputable journal, you can assume it is true. "There is a lot of BS published in peer-reviewed and even high impact journals I guess in every scientific field. If I learned one thing during my research it's definitely not to trust anything in a paper just because it's in a certain journal. –  May 29 '17 at 22:19
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    @JeffE did you secretly prove that P neq NP (I.e., proving is a lot harder than verifying)? Jokes aside, I think the answer means u can take a published result as a given hypothesis in the absence of an error. Then, using it u get a result/onclusion modulo the hyp. If this inequality was wrong, too bad for its authors. – PsySp May 29 '17 at 23:18
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    @NickS - Why don't you move your comment to an answer? // Question, also for Nick: What about the review process? Could OP go ahead and submit, and see how the reviewers react? Just wondering. Thanks. – aparente001 May 30 '17 at 03:11
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    @NickS My argument is that basically any discipline of science, including most parts of mathematics are simply too big to be able to understand every nuance of all the work you (indirectly) rely on. At that point, trusting the results of peer-reviewed paper is the lesser evil and the only practical option. Otherwise we would have to require any paper in applied functional analysis to be co-authored by an expert in measure theory or even set-theory, just so we have our bases covered. – dimpol May 30 '17 at 09:10
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    @dimpol I disagree. The right thing is to always find reliable references. Yes, you cannot check everything, but unfortunately the quality of peer-review nowadays is pretty bad. And based on the OP post, the paper raises too many red flags to simply rely on the fact that it is peer reviewed. – Nick S May 30 '17 at 11:15
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    @dimpol There is a wide stretch from "You don't have to understand everything" to "If it's published, consider it correct" – sgf May 30 '17 at 21:37
  • @sgf I'd argue that stretch isn't that wide. Suppose the work of an author turns out to be incorrect because the author relied on citing peer-reviewed work in a respectable journal which turned out to be false/incorrect. Now as a supervisor/colleague/funder you want to discuss whether the author should have scrutinized the paper more, instead of trusting the results. What guidelines would you use to determine academic 'negligence'? Because without any clear rules, I'd either call the stretch pretty narrow or its width not really relevant to the daily practice of research – dimpol May 31 '17 at 07:05
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    @dimpol, perhaps from a point of simple convenience for the researcher, it's sensible to assume that anything that is published is correct, because you can't prove that they were negligent, but both for the academic enterprise as a whole and for any researcher that wants to actually contribute anything valuable it's highly problematic to stop being critical of other researchers' results just because they have been published, to the point where you might just as well con people for a living. – sgf May 31 '17 at 12:43
  • @sgf Of course being critical of the work of others is a good thing and scientists checking each others work helps science progress. However there is a difference between allowing papers to be checked and requiring papers to be checked. In my view, any researcher should have the option to trust the results in a peer-reviewed paper in a reputable journal. The reason for that is that requiring every cited paper to be checked (and of course every paper that that papers cites, etc.) is impractical and guidelines for 'sometimes check' can't be made rigorous. – dimpol May 31 '17 at 13:54
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    @dimpol Sure, but there's a difference between not checking a paper, and checking it, being unable to replicate its results, and then just taking its result as correct anyways. – sgf May 31 '17 at 20:53