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Question 1. A paper was rejected because of 'not suitable level of rigour' without a single example of a mathematical error/imprecision. What can the author do in this situation?

It sounds a matter of fact: writing that a work is not mathematically rigorous requires at least one example of a mathematical error/imprecision - for instance, an inaccurate definition. What can the author do, if this rule has been violated?

Needless to say, the author should first recheck the whole work once again. Assume this has been done and no errors/imprecisions have been found. Assume also that the editors refuse to clarify anything.

An instinctive reaction could be then to submit to another journal. But the most probable outcome is an automatic rejection - most likely, the same people are going to handle the work and think: 'oh, that is the same erroneous work, not even corrected'.

The author seems to be in a stalemate.

One can report the case to SciRev: this will not help her or him but might help others.

Question 2. Can you share any of your own experience?

Positive or negative cases are welcome, especially transparent and refutable ones, that is, supplied with a complete correspondence with a journal. Here is just one. (Ethics of publishing received peer reviews.)

On different reliability standards (Section 2).

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    $\begingroup$ Something can be written so vaguely that pointing out mistakes is not possible. $\endgroup$
    – user2520938
    Commented Aug 1, 2021 at 9:07
  • $\begingroup$ I think Question 1 is not very interesting and prompted down-votes. I like however question 2: A few answers can be helpful for example for writing better referee reports. $\endgroup$
    – Roland Bacher
    Commented Aug 1, 2021 at 9:41
  • $\begingroup$ @user2520938 It sounds a matter of fact: writing that 'something is written vaguely' requires at least one example of a mathematical error/imprecision. For instance, an inaccurate definition or undefined object, but not necessarily a mathematical mistake (as clearly indicated in the question). If a work is indeed very vague, then pointing out such imprecision would hardly take more than a minute. $\endgroup$
    – Mikhail Skopenkov
    Commented Aug 2, 2021 at 4:56
  • $\begingroup$ Question was closed 'as offtopic'. However, there are plenty of similar questions on mathoverflow: e.g., mathoverflow.net/questions/82465/resubmitting-a-paper, mathoverflow.net/questions/146543/… etc. Can one researcher ask other ones for advice on mathoverflow or not? Can one reopen the question? $\endgroup$
    – Mikhail Skopenkov
    Commented Aug 2, 2021 at 5:35
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    $\begingroup$ @MikhailSkopenkov: Without you showing us the article under discussion, it is impossible for us to give you any meaningful advice. This discussion is doomed to be too abstract and generic to be useful. $\endgroup$
    – Alex M.
    Commented Aug 2, 2021 at 6:48

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Rejection is mostly a negative experience for the author. I have however experienced two rejections for which I am very grateful:

My first experience is a paper which got rejected mainly (I think) for too much handwaving on page so and so. Trying to reduce the handwaving led to the discovery of a gap which my coauthor and I were unable to fill. (I ignore if the referee had a good nose or was actually aware of the gap.) We put the whole thing on hold for two or three years before finding a way around the problem and submitting a correct paper (which got accepted).

I guess we would just have let the whole thing rot if the paper with the wrong proof got accepted in the first place. I think the referee did really a great job: he just pointed out the stink without putting our noses into it.

This suggests perhaps some advice for writing useful referee reports : Try not to humiliate authors but point out the faulty part in a nice way. (I do not always follow my advice: some papers are just too irritating!)

My second experience involved a paper about a somewhat strange bijection of power-series which preserved rationality and algebraicity (over fields of positive characteristic) in one direction. An experimental section with a large bunch of examples suggested that this should also be true in the opposite direction. The paper got rejected by "Experimental maths". The rejection made me so angry that I found very quickly a proof for the missing direction. The resulting paper had no longer any experimental flavour and got accepted (by a different journal). I am sure that rejection spurred me. Without it, I would certainly not have found the proof for the missing direction.

There is of course a big difference in these two cases: Rejection was completely justified in the first case.

I think the initial paper could have very well been accepted in the second case. I was however very lucky that it got rejected: The resulting anger gave me a lot of energy and resulted in a much better paper.

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  • $\begingroup$ Thank for your answer. It is unclear how your examples are relevant here: as you write, the first rejection was completely justified and the second (experimental) paper did not pretend to be rigorous. This question is not just on rejections but on unjustified assertions on correctness of a work. BTW, in the case in question (users.mccme.ru/mskopenkov/other/lmp.html), a previous version of the work has been rejected by another journal with a clear and convincing report (scirev.org/reviews/…). $\endgroup$
    – Mikhail Skopenkov
    Commented Aug 2, 2021 at 5:05
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    $\begingroup$ I just wanted to give some positive feedback on rejection. You will probably not get any helpful answers in your case. However, sometimes papers get rejected not because they are wrong but because they do not appeal to the referee (style: a bunch of technical computations leading to a result which does not seem interesting, $\endgroup$
    – Roland Bacher
    Commented Aug 2, 2021 at 9:10
  • $\begingroup$ I think there are three roads to publication of a paper: (1) It contains good results, (2) The author is famous (generally well-correlated with (1), (3) The paper is very well written and is interesting to read (the most frequent case, not everyone publishes a correct proof of the Riemann hypothesis every day). $\endgroup$
    – Roland Bacher
    Commented Aug 2, 2021 at 9:23
  • $\begingroup$ Thank you for your clarification. Notice that this question concerns works rejected 'as being wrong' rather than 'not appealing to the referee'. And writing that a work is wrong just because it does not appeal to one is no-no. Also, you are right that the question will not get any answers - just because it is closed. First let us reopen and then see what happens. $\endgroup$
    – Mikhail Skopenkov
    Commented Aug 3, 2021 at 7:57

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