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I'm writing a report regarding a paper in mathematics, which I have spent three months reviewing. During this period, I have tried to understand the ideas presented by the authors, but in vain.

The format of the paper and the length of the proofs are making it hard, even to an experienced reader, to understand what the authors are trying to prove. In addition, there are some lexical mistakes overall throughout the paper.

Now, while the question addressed in the paper is novel. All the above mentioned problems have pushed me to opt for a rejection. But then, I'm hesitating and thinking about recommending a major revision instead, given that as I mentioned, the idea of the paper is novel.

So, is it normal to recommend a rejection solely based on the problems that I have mentioned or should one opt for a major revision instead?

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6 Answers 6

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My understanding (which I'm not sure is universal) is that a recommendation of major revision should indicate that you have high confidence that the authors can make the necessary changes (and in particular the necessary changes are possible!) for the paper to be accepted.

Given that you don't know if the paper contains a true theorem, I think this paper does not meet that threshold.

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    "Given that you don't know if the paper contains a true theorem" --incompetency of a reviewer cannot be a reason for rejection. I am not saying this is the case, but it is a possibility.
    – yarchik
    Sep 9 at 15:30
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    @yarchik: You take your argument to the end and you have the situation of the ABC conjecture, where many experts claim there is a significant gap in Mochizuki's claimed proof and Mochizuki claims all these experts are incompetent. Sep 9 at 16:24
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    It would seem a well written paper would at least let the reader know in an obvious way that it contained a theorem, and its proof. Sep 9 at 20:02
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    @yarchik: Communication is a two-way street. If the reviewers can't understand the paper, neither will its target audience. Sep 9 at 22:17
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In a case like this where you after a lot of effort cannot make out the contribution of the paper, a "reject with the possibility of a resubmit" is in order. My reason is that by returning "major revision", you are indicating that the contribution has sufficient value for publication (eventually and after lots of changes have been made). But what I understand from your write-up is that you cannot make out the contribution.

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    I found "reject with the possibility of a resubmit" amusing. I always felt it is obnoxious from the editor to give that message, something like "look, your paper is bad, but I am so noble-minded that I will allow you to resubmit it at a later date" ... so much against the idea of peer-review that I never resubmitted (so it may be a good strategy, according to your goals, I don't know :D ). Sometimes it is better to call crap for what it is, crap: if it is not a worthwhile contribution, better to give a rejection as a signal to restart from scratch, without giving "hope" to the authors.
    – EarlGrey
    Sep 9 at 7:09
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    @EarlGrey: When the editor judges that a submission is crap, then yes, they shouldn’t give the authors false hope. But as this answer illustrates, there are plenty of situations where a paper can’t be judged acceptable now, but isn’t necessarily crap either — it has some chance of becoming acceptable after revision — and “reject-and-resubmit” is for those cases.
    – PLL
    Sep 9 at 15:25
  • @PLL I still think that if you see the diamond of the interesting idea sparkling in the crap paper, then it is your duty to do the maieutic thing of suggesting major revision, and showing where the major corrections have to be done. But I am an optimist ;)
    – EarlGrey
    Sep 9 at 20:06
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    @EarlGrey: Absolutely — if I can see a diamond, I’ll say so, and recommend major revision. And if I can see it’s just crap, I recommend rejection. But if I can’t tell whether there’s a diamond in there or not (because of bad writing or similar), that’s when I think revise-and-resubmit can be appropriate.
    – PLL
    Sep 9 at 20:10
  • I assumed that the paper was written by an inexperienced author and that would induce me to give some lee-way as a referee. One of the most difficult things to learn as an author is that what is obvious to me is not necessarily obvious to the reader and that notation I learned is not necessarily agreed upon short-hand. A reject with possibility of resubmission is a clear message: "We cannot understand your paper to make a judgment on it". It is of course to be a very rare thing to do. In this case, the questioner apparently tried and failed to understand the paper. Sep 9 at 22:09
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Let me add another angle that may be considered harsh and cynical.

Papers should be readable when they are submitted; ensuring this takes some effort and time, and the authors should invest it. This author did a disservice to the editor and the referees by submitting a paper in such a bad shape. They are wasting everyone's time. A rejection "punishes" them and gives them a negative game-theoretical outcome for this. Condoning this behavior with a major revision encourages them to do the same again and again.

Not setting incentives to discourage this behavior is harmful to the community.

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    Even if the paper has been prepared in good faith (and the outcome is still the bad shape depicted here), a well justified rejection will be a good thing for the future (for the authors papers and for their reviewers), I agree.
    – EarlGrey
    Sep 9 at 14:14
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That happened to me before. My suggestion is to give a major revision stating paper is incomprehensible in this state and requires major rewriting before it can be reviewed properly. I stated no other explanation in my case and that was good enough for the editor and authors. The paper was in a much better state in the next revision. If a paper is salvagable and contains novelty and/or results/experiments that can be of use to someone, it will be a disservice to the authors and the community to reject it outright.

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    My experience is not in Mathematics and therefore not completely relevant, but I had an editor tell me that after asking for two major revisions, it was impossible for the editor to reject the paper. I am glad it worked out in your case. Sep 9 at 22:11
  • I got a paper reject after two revisions as they were insistent on bad/cheaty experimental practices even though method was acceptable. Sep 10 at 6:38
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In the given situation the most important thing for me would be to ask myself whether I believe that what the authors did there is not only original but ultimately correct, and that things can be repaired so that this becomes clear.

If a paper is incomprehensible and it's therefore not clear whether the claims are correct, this is a good enough reason for a rejection. If a paper is badly written and hard to understand, but ultimately the main results of the paper are in all likelihood correct and worthwhile (at the level of the journal in question), I'd ask for a revision.

An additional consideration is whether the writing of the authors disqualifies them sufficiently that you don't believe they are able to make the paper publishable without an unacceptable amount of editor and reviewer help even if results are potentially worthwhile. That may merit a rejection as well (I have seen obvious inability to handle mathematical formalism correctly and lack of understanding of what constitutes a valid proof in several papers even if the ideas behind them were good), but that'd be a subjective call depending on how sure you are.

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What is the journal you are reviewing for? Is it for the practitioner on the topic or for a broader audience? If you think the proof will benefit from discussion with the practitioners, suggest major revision and give already a forecast "if the form will not improve substantially, both in lexicon and in clarity of exposed proof, I suggest rejection in the next revision".

If you think the idea is really novel, then go for rejection and suggest to improve a lot the manuscript and then submit it to a higher impact (pick whatever metrics you like) journal.

The editor will not like this review suggestion, but who cares.

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