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I have found a counterexample to the main theorem of a 20-year-old paper. The paper does not seem to be well known but it was written by reputable mathematicians and published in a highly regarded journal. I believe that the main theorem is "mostly true" and the authors simply forgot to add hypotheses to their theorem which would exclude my counterexample. I am interested in applying their results in another paper, but I can't really trust the results.

The paper in question is very long and involves a great deal of technical machinery with which I am totally unfamiliar, so checking their proof in detail is out of the question. Also, an e-mail to the authors went unanswered.

Would a question along the lines of "To what extent are the results in Paper X correct" be appropriate for MO? I would genuinely like to know the answer to this question; however, I would be surprised if anyone other than the authors could answer it in a reasonable amount of time. Moreover (and more importantly), I question the appropriateness of publicly calling attention to the mistakes of others, especially when I am not prepared to fix them myself.

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    $\begingroup$ I think it is fine to ask about a specific point in a paper, preferably in a neutral way, aiming to understand what the authors had in mind. For example, you could also outline the result you want to use, and ask how (or if) your potential counterexample affects that statement. Given your thoughtful meta question, I'm sure you can write something tactful doing that. $\endgroup$ – Lucia Feb 25 '16 at 17:30
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    $\begingroup$ In my opinion it's, of course, appropriate to point out mistakes in papers. For, having strong proofs is what distinguishes mathematics from many other sciences. If you call attention to a mistake you also help other researchers not to build on a wrong result. Maybe you can send your counterexample as an erratum to the journal. If they take it under review, it's not unlikely that the referee will have a look at the original paper and will find out what the correct theorem should be there. $\endgroup$ – Todd Leason Feb 26 '16 at 1:59
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    $\begingroup$ You could look at the following question for comparison: mathoverflow.net/questions/218270 $\endgroup$ – Neil Strickland Feb 26 '16 at 5:49
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    $\begingroup$ Thanks to all. I believe I will go ahead then, although I will first confirm the mistake and double check that I indeed contacted the authors about this particular issue. $\endgroup$ – Frank Thorne Feb 28 '16 at 20:11
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One approach is to find a different stakeholder: see who cites the paper and uses the result. Then ask them if they are interested in your counterexample. If they are responsive, they may give you some help, either in showing you are right (your counterexample works and a rough indication where the original proof breaks) or in showing you are wrong (the authors meant something else by what they said, and your counterexample does not apply).

Discussing the quality of a paper, or even a collection of statements, is not suitable for MathOverflow, primarily because of length, and secondarily because of the potential of a flame war. Taking a particular statement, and showing a counterexample to the statement, if done briefly enough, is suitable. I would couch it (as Lucia suggests) as "Here is statement T, a theorem of Paper A. My example E seems to show T is false, for basic reason R. What am I missing? (Due to the technical nature of the proof of theorem T, I can't see where E breaks the proof.)" This puts you in a position of wanting to understand and asking for high level clarification, as opposed to critiquing (and inadvertantly suggesting the unreliability of the paper). If it doesn't smell like you are trying to prove RH, it should be well received on this forum. It might even help you find a stakeholder.

A little weaker, but reasonable, approach would be to ask where the "hidden hypotheses" are in paper A. It may be that the writing style obscures them from you, and someone familiar with the paper can show you where they are. However, this can easily sound critical. If your goal is to apply theorem T, it might be best (after having done your citation search) to ask for references to other uses of T.

Gerhard "Don't Mention Collatz Conjecture Either" Paseman, 2016.02.25.

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I propose to ask it as follows: In my view such a question written clearly is suitable for MO.


A question about X's paper on Y

The main theorem of X's paper on Y is

"quote (precisely) and if necessary explain the terms"

I have found the following apparent counterexample which goes as follows:

"explain clearly"

Is this indeed a counterexample or did I miss something?

Motivation : I believe that the main theorem is "mostly true" and the authors forgot to add hypotheses to their theorem which would exclude my counterexample, but as I am interested in applying their results in another paper I would like to clarify this matter.

Remark: The paper in question is very long and involves a great deal of technical machinery with which I am unfamiliar.

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Before you ask here:

Try to find errata for the paper. 20 years old makes that harder, since most things were not on the net then. Maybe you can contact an author of the paper?

Also look for papers that refer to this one. This is easier than it was 20 years ago.

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    $\begingroup$ Note: "an e-mail to the authors went unanswered." $\endgroup$ – Allen Knutson Feb 27 '16 at 23:15

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