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I am a young researcher and sometimes I face an uncomfortable situation : I find an error in a research paper! Of course, most of the time, it is all just my misunderstanding but it happens that after checking tens of times, there is indeed an error and therefore I am not sure what I should do. "Am I the first one that notices this mistake or someone has already reported it? and then where?" "Hass this error been corrected in a later version? How can I know? ", "Should I send an email to the authors ? But still how could I be diplomatic enough not to bother them (and maybe the mistake is still mine, I am not an expert of the field)? " The situation gets worse when

-The paper is already old. Personally I would find annoying if someone asked me questions on a project finished years ago. And for very old papers the authors can just be retired already.

-It comes from a different field like physics or chemistry. Where they use a different language. How could I explain myself correctly and without creating a "diplomatic crisis"?

So I am here to ask advice to the experimented researchers how to deal with this awkward situation.

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    $\begingroup$ It depends what you call error. They forgot to discard the empty set? There's a gap in a computation? The main theorem is false? etc. It's hard to say something general. $\endgroup$ – YCor Apr 27 '18 at 10:11
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    $\begingroup$ I think that most mathematicians find that most of their work has received less reading, comments, questions, etc. from the mathematical community than they would like. So although there is something annoying about receiving a comment on some old work, there is also something fun about it, that someone has read it and is really trying to understand it (and probably is using it for something). So I wouldn't worry too much about this. $\endgroup$ – Will Sawin Apr 27 '18 at 11:10
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    $\begingroup$ Also, if you phrase your question in terms of a request for clarification, and you don't type something like "Your paper is wrong, you idiot", then at worst you will be just one more person making demands on the professor's time that they would prefer not to fulfil. Successful professors have a lot of these and they are unlikely to remember your question, unless it turns out to be a serious mistake. Thus, if you have checked it multiple times and are confident you have a good question, I say ask away. $\endgroup$ – Will Sawin Apr 27 '18 at 11:12
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    $\begingroup$ a good range of answers can be found at academia.stackexchange.com/questions/18911/… $\endgroup$ – Carlo Beenakker Apr 27 '18 at 11:20
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    $\begingroup$ Is it appropriate to mention names here? One of my most positive experiences as a young mathematician was asking a big-shot in my field about what seemed to be a mistake in his paper. (It was very minor.) I approached him very timidly, but he spoke to me with great friendliness, readily acknowledged the mistake, and kept in touch with me to make sure I understood the resolution. I've always appreciated his graciousness. $\endgroup$ – LSpice Apr 27 '18 at 13:55
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Of course, reporting an error benefits the author, so I advise to write to him/her. A polite form is: Dear X, I am reading your interesting paper Y, and I have difficulty in understanding why A implies B on p. C. I will appreciate it very much if you explain this to me (help me to understand).

I received such letters several times. In most cases I was able to explain. In one case I published a correction. And of course I am thankful to all who wrote these letters.

When I was a student I wrote to one famous mathematician: Dear X. Let me bring to your attention that conjecture W which you state in your paper Y seems to be proved by Picard in 1880 in his paper Z. Do you think his proof is correct?

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    $\begingroup$ How did the famous mathematician respond? $\endgroup$ – Todd Trimble Apr 27 '18 at 13:15
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    $\begingroup$ He did not respond. 3 months later I saw his obituary in the BAMS:-( $\endgroup$ – Alexandre Eremenko Apr 27 '18 at 13:18
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    $\begingroup$ Gee, and your wording did not seem harsh at all. Gerhard "Better Not Send A Telegram" Paseman, 2018.04.27. $\endgroup$ – Gerhard Paseman Apr 27 '18 at 13:55
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    $\begingroup$ Then there's the matter of how to be diplomatic when the error reflects a rush to claim priority over the person whose expertise renders the mistake embarrassingly obvious. $\endgroup$ – Adam Epstein Apr 28 '18 at 9:45
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I have discussed about this at the following link:

https://sites.google.com/site/wrongmathematicalpapers/

In particular, I propose the idea that we can establish a journal whose sole purpose is about reporting significant errors in papers already published in recognised journals (so the referee process in those journals is supposed to be rigorous, so the supposed journal - which does not exist yet - will not consider predator journals with bad referee procedures) and/or giving corrections which save the claimed results in those papers.

This journal is also a good bookkeeper about wrong published results in mathematics, so a novice will not mistakenly use them without finding an alternative proofs of the claimed results. Such a journal will potentially solve much of the problems you mentioned. I also proposed that the journal can harmoniously cooperate with other normal journals.

This journal will be of great benefits to mathematics, I can imagine, and I hope that mathematicians will seriously consider about it. I myself am contacting some publishers about the ideas to see if they will be interested in.

Best regards,

Tuyen

Addition: This is meant to be a comment to Alexander Eremenko's answer, but I do not have 50 points so cannot do so. Yes, I also encourage first to contact the author. However, if it turns out that either the published paper was wrong (with no way to save the results from the ideas in the paper) or the author was not responsive, then you can consider the journal I proposed (of course, I also encourage to submit to the journal I proposed only after you contacted the journal which published the original paper if they are willing to publish an erratum and they refused to do so). Even if other journals decided to publish an erratum, then still the journal I proposed is still good, it can post a short announcement on the erratum, so it is as intended a bookkeeper of wrong published papers. More detail can be read in the link.

Addition 2: This is meant to be a comment to Adam Epstein's comment, again I cannot give a direct comment because of low points. Maybe not many people notice the mistakes in the published papers, in particular if they are not in the same research field. Anyway, when a mistake is found, I think it is easy to sort out how to present it, for example in the journal I proposed. You are worry about what the person who made the mistakes feel, but you can also turn the cards and think what the same person feels if his/her (wrong) publications get him/her some advantages over peers. There have been people in science whose careers have been very negatively affected by having wrong publications. As far as I know, mathematics seems to be more tolerant about this issue.

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    $\begingroup$ Note that there are sites in other sciences for reporting errors (e.g. Pubpeer). This sounds more flexible than the journal format. What's the point of a journal? get credit at reporting errors? $\endgroup$ – YCor Apr 27 '18 at 17:47
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    $\begingroup$ The point of having a journal is to make sure that the error is genuine. So there will be referees to check that the paper which says that another paper is wrong is correct. Otherwise, anyone can write something, saying that a result is wrong, and put it online. How can you judge whether such a claim is correct? Many times, many people thought that some results are wrong, but actually the people who thought the results are wrong are wrong. $\endgroup$ – Tuyen Apr 27 '18 at 18:26
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    $\begingroup$ Also, having a journal is to prevent another extreme. Someone puts a statement that such and such results are wrong, and you may say: Well, this is just an online claim, I do not believe in it. There are also other benefits of having such a journal, please see my answer above and more detail in the link. $\endgroup$ – Tuyen Apr 27 '18 at 18:53
  • $\begingroup$ The status of a journal is better than an online webpage. That is why people usually ask you whether your preprint on arXiv is published or accepted. Even though arXiv and journals serve basically the same purpose of providing you with new knowledge. $\endgroup$ – Tuyen Apr 27 '18 at 19:01
  • $\begingroup$ Not less important, the journal protects authors who point out mistakes in other papers, especially of big names. Not to imply anything, but just in case such authors need protection. You can see this point by yourself. $\endgroup$ – Tuyen Apr 27 '18 at 19:16

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