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I received a referee report for my single authored paper recently in Mathematics. Both the referees have asked for a revision. I am having a problem with a proof of a theorem in the paper.

Reviewer A asks to add more details to the proof of the theorem. He/She says it is not possible for him/her to understand one step in the proof.

Reviewer B says delete the proof of the theorem as it is not necessary. It is easy for him/her to follow it without any details.

The editor has asked me to revise the paper taking into account both the reviewers. I am confused now what to do to the proof of the theorem. Shall I keep it and add more details or shall I delete the theorem? If I delete it, then Reviewer A will get offended and if I add more details, then Reviewer B may reject the paper.

What should I do?

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    Does this answer your question? How to effectively address conflicting suggestions from reviewers
    – Sursula
    Aug 28, 2022 at 7:15
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    I like Wrzlprmft♦'s "have your cake and eat it too" answer. I will add, if you are forced to choose one way to go, I would generally choose to add more detail. In my experience, it's more likely that a suggestion to remove detail is a poor suggestion (coming from an old-school macho mindset that authors prove their worth by making readers struggle) than it is that a suggestion to add detail is a poor suggestion.
    – Greg Martin
    Aug 28, 2022 at 19:18

3 Answers 3

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In this particular case (and if the journal allows for it), moving the respective part to an appendix (and expanding it) seems to be a solution that should satisfy both reviewers. Those readers who are like Reviewer B and do not require the proof can ignore it and those who are like Reviewer A can read the proof in its full glory.

Moreover, Reviewer A’s request itself gives you an excellent argument against the complete removal requested by Reviewer B: You can argue that you want to keep that part because you consider it relevant for some readers, as evidenced by Reviewer A’s request.

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    In math, I would consider this solution weird (also I don't know why the journal wouldn't allow it). You don't normally read math papers linearly and whoever wants to skip a proof will. There are some cases where this is done, but normally it's due to special circumstances (e.g., proving a known/folklore result, proving a result tangential to the main result of the paper, appendix written by someone else, ...) and I don't see anything in the question indicating this.
    – Kimball
    Aug 29, 2022 at 0:03
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    @Kimball: In math, I would consider this solution weird – … but then, so is a reviewer asking to delete a proof (as also hinted at in other comments). I concur that in a lemma–theorem–proof paper, such appendices seem unnecessary, but my guess is that this is not such a paper. On the other hand, as somebody writing papers with proofs for other fields, I am quite used to putting them in the appendix. Either way, this is for the asker to decide.
    – Wrzlprmft ♦
    Aug 29, 2022 at 5:18
  • @Kimball in the US alone, each year 2'000 PhDs in "math" are awarded (all branches, including statistics, see here: ams.org/journals/notices/202201/rnoti-p96.pdf ) per year. In the US alone, there are 53'000 PhDs awarded in subject other than "math". Unless differently stated, we can statistically ignore the math issues and conventions.
    – EarlGrey
    Aug 30, 2022 at 8:00
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    @EarlGrey But how many of those 53,000 PhDs are in subjects where the papers contain "proofs" of anything? Surely one will find proofs in theoretical CS papers, but Kimball's comment applies equally well to theoretical CS.
    – kaya3
    Aug 30, 2022 at 8:40
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    @EarlGrey If a paper has a theorem and a proof of it, then to borrow a phrase, we can statistically ignore the possibility that it isn't mathematics.
    – kaya3
    Aug 30, 2022 at 9:24
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Of course without knowing the paper and detailed reviews it's impossible to tell, but I want to provide a slightly different view than Wrzlprmft.

In my opinion, this is most likely not even a real case of conflicting reviews. The key information I tend to take away from requests such as "delete X" or "greatly expand X" is that the reviewer did not like the current version of X. Notably, I read statements such as "proof XYZ is too high-level and needs to be expanded considerably" not as an endorsement that the proof is useful in general, but as a statement that the current version is not.

If viewed through that lense, both reviewers have actually identified the same problem, they just propose different approaches to fix it. In that sense I doubt that either reviewer will be greatly upset if you take the other reviewer's suggestion - and in the response letter you can always point to the other reviewer to explain why you have chosen a different approach to fix the problematic proof than the one that was suggested.

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    Initially I was thinking along the same lines but note that this concerns a proof of theorem (in a math paper I assume). Omitting the proof is generally not an acceptable solution, reviewer B seems to say that the proof is so easy that a reader can understand it even from a much reduced version that what OP has written. Reviewer A disagrees and for the proof of a math paper I don't think they would accept no proof as an alternative to a proof that they consider incomplete/ lacking detail.
    – quarague
    Aug 28, 2022 at 19:52
  • @quarague Maybe, but then why would a reviewer even suggest removing it? Proofs don't only appear in math papers - many applied-ish discipline papers also contain proofs, but often more as sort of auxiliary material (to name an example I know, many programming languages papers contain proofs that are, sometimes, not at all central to the idea that is being presented).
    – xLeitix
    Aug 29, 2022 at 6:46
  • @quarague That said, I do agree that "expand the proof" has a lower overall chance of offending any of the two reviewers.
    – xLeitix
    Aug 29, 2022 at 6:48
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When an editor asks you to revise a paper taking reviewers’ comments into account, they mean just that — they do not mean “follow all the reviewers’ suggestions precisely”. In my experience as an author, on any issues bigger than typos, it’s fairly usual that I (and/or co-authors) don’t quite agree with a referee’s suggested edits, but their comment helps me understand what was sub-optimal about my/our original version, and so find an alternative way to improve that shortcoming. And when I’m a referee, reading revisions, I never expect authors to have followed my suggestions to the letter — just to have considered my concerns and addressed them in some reasonable way.

So in cases like yours, the kind of things which might help allay both referees’ concerns are:

  • Start the proof by explicitly signposting/acknowledging what is easy about it. “This proof will be routine for experts, as a fairly standard application of pseudolinear estimation techniques.” This allays two possible sources of Referee B’s unhappiness: it lets them know that they can safely skip the proof, and it makes clear that you’re not overstating the novelty of this theorem.

  • Within the proof, be conscious which details are more difficult (bearing in mind Referee A’s comments), and explain those clearly; but also, to help highlighting them, don’t belabour the parts that really are routine. A step-by-step argument like

    A = B (by the second unitary condition) ≤ C (by convexity) = D (by Kane’s lemma, part (iii)) < E (by Chen’s inequality).

    can often be clearer as

    It follows directly by unitarity, convexity of f, and Kane’s lemma that A ≤ D; then D < F by Chen’s inequality (with ε = 1/7, and the stochasticity hypothesis ensured by Lemma 3.2).

I don’t mean to suggest that those are necessarily applicable in your case — but rather to illustrate how even when the referees’ specific suggestions are incompatible, the concerns behind them may both be reasonable, and it can be possible to find a revision addressing them both at once.

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    The editor also likely has pre-written responses, depending on the level of revisions needed, so the response was no different than if both referee's had given the same report.
    – Rob
    Aug 29, 2022 at 8:16

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