Referee says the proof is wrong, but it is not, what to do?

Question

About 3 months ago, my co-author and I submitted a mathematical paper to a journal. Yesterday, we got the editor's response and the paper has been rejected. The reason is that the referee thinks that our results are wrong or trivial.

However, we carefully read the referee's report and we conclude that the "errors" pointed out by the referee are not errors at all and that our proofs are correct. In all modesty, we think that the referee understood very poorly even the statements of our theorems.

Personally, this is the first time that such a situation occurs to me. I already have four papers published in such journal and the reports I got from the referees have been always really accurate.

What do you think is the best thing to do? Forget about and try another journal? Write an e-mail to the editor explaining why we disagree with the decision? Other ideas?

EDIT: Thank you very much for all your answers. At the end, we submitted our paper to another journal with only some slight edits we have already planned to do independently by the referee's report.

Answer 1

I must disagree with some of the answers.

Some assume that it is a problem with the person asking the question and giving suggestive answers with a very limited number of fixes ("I would strongly advise performing a major revision on your presentation of the proofs"; "two potential problems. One is that you may be in error, [...]. The other is presentation[...]").

If we are attentive enough, we all know well what the source of our problems are: The magic word is feedback. If people complain consistently that our papers are hard to read, the suggestion to work it over has a point. If again and again referees find errors in the manuscript, we will take much more care to avoid errors (proof-read, double-check, triple-check).

But if we get an outlier answer and cannot fathom what exactly should be the problem, it is very, very likely that we are not the source of the problem. Working the paper over in this case is a waste of time.

The reason(s) that a referee rejects a paper are virtually unlimited. He is wrong, she does not care, he is busy, she is right, but explains it badly. Whatever.

But what you can do is to read very carefully the answer and ask yourself under what circumstances a person would give the answer. Is it aggressive, dismissive, bored, sad? It is rather short or more detailed? Does it look like the referee invested time or it is hastily written with boilerplate passages? From this you could often discern what the real reason for the rejection is.

Everyone will sooner or later have a paper rejected for inexplicable reasons, it is normal. Nothing to be afraid of.

That said, the best strategy already mentioned by other answers is to cut the losses and simply go to another journal. I would advise to challenge the decision if and only if the editor made it clear that the decision is not final and then if and only if it is objectively important to have it published in this specific journal. In that case do everything in your ability to check and recheck your paper and hone it to perfection because you will only get exactly one chance to convince the editor.

Answer 2

While you might be able to appeal to the editor, once you've been rejected, you're probably not going to get a change in decision. I would instead recommend focusing on a new submission to another journal.

Before you do so, however, I would strongly advise performing a major revision on your presentation of the proofs. Assuming that your proofs are in fact correct, you clearly did not convey them in such a way that their correctness and significance were obvious to the reviewer. The reviewer may have been a sloppy or shallow reader, but so in fact will be many of your other readers after publication.

In short, take the feedback as an indicator of how your paper needs to be improved before you submit it to another journal.

Answer 3

The reason is that the referee thinks that our results are wrong or trivial.

This first caught my attention. Trivial and wrong are two completely different things. How did you come to that conclusion?

the "errors" pointed out by the referee

This leads me to believe that the issue is mainly about errors. But:

In all modesty, we think that the referee understood very poorly even the statements of our theorems.

has, two possibilities:

• you are right. Math is quite hard and he did not invest enough time to understand it. Or maybe he does not even have the ability to. Which should not be "your" problem.
• He is very good in the field you are working and sees your results really as trivial. He therefore did not read it properly and marked wrong errors (it may looked to him like a "trivial paper with errors"...)

In both cases, I'd really recommend, to

Write an e-mail to the editor explaining why we disagree with the decision?

That's the most "academic" solution in my eyes as problems should be discussed until you agree on the same conclusion (even dough this is surely not possible everywhere, it is in math). Regarding the second case, it is may even possible, that you missed a point. You know, we all make errors all the time. But submitting a paper where someone (who actually should understand the topic) spotted errors and just ignoring (not able to convince him otherwise) his opinion is quite, let's say, self-confident (especially in math), and in general not advised to do;)

Don't get this wrong; most probably you are right and he is wrong. Ok. But you surely don't want it the other way around. Better make sure it is not.

Instead of writing directly to the editor, it is may a good advice to let your paper check by a colleague in your field, who is neither the referee nor the author to check for a third opinion.

Others suggested to rewrite your paper for more comprehensibility. I would not conclude that already; too many other factors may play a role. Depending on the editor's answer (say he did not get the points as he find it confusingly written) I might rewrite the parts.

Another journal is surely a good idea if that somehow won't work out, but I don't think you are in a hurry, right?

Answer 4

The editor and the referee are not the people you need to be talking to in this situation. Their jobs involve managing many submissions and they likely lack the time or inclination to reconsider every person who's been rejected. For every paper published, many other papers must not be, so even when not making mistakes you can fall victim of being less novel than the most novel papers submitted recently.

You have two potential problems. One is that you may be in error, in which case the peer reviewer was correct to reject you and you really need to fix your paper before resubmitting it anywhere. Given your proximity to the paper, it's easy to get too close to the problem and become blind to such errors, so getting new peers involved to openly review the paper or leaving it for a period of time and coming back to it with a fresh pair of eyes after a break might help. Not knowing you nor the peer reviewer, all an objective observer should see is two peers of similar standing in disagreement with no way to judge who is correct without factual details.

The other is presentation. If we assume the paper is correct and novel, it may not have been sufficiently presented as such. Even if the peer reviewer was a lazy reader, their misunderstanding of the paper may have been helped along by lack of clear communication. The only real solution to this is further iteration and, again, external input. Open review is the best way to constructively improve your paper at this point.

If you want to double down on the paper being perfect in every way and just misunderstood, submit to other journals. This is a gamble, as they may just as easily accept it having missed what the peer reviewer saw or reject it for a variety of reasons out of your control. As others have stated, those involved in rejecting your paper may not be impressed if that succeeds, may not care or never find out, or they might recognize that they got it wrong because everyone makes mistakes.

One submission of one paper to one journal being rejected is not a big deal, it's normal, so don't let that demoralize you.

Answer 5

To complement jakebeal's answer, I want to explain why I think that appealing to the editor is not the best choice.

As Mayou36 said, there are (basically) two possibilites: either the referee is wrong or you are. If the referee is wrong or even not qualified to judge your work, as you suggest, then this journal is probably not the place where I'd want to see my work published (of course, bad referee choices can happen, but sill). If you are wrong, you are already in an embarrassing situation. You don't want to make it worse by arguing with the editor.

No one here can say whether you are wrong or the referee. As others pointed out, try to make sure that it's not you before taking any other action.

In any case, you should at least make some minor changes. Academic communities are small, and it's not too unlikely that your next referee knows the one you're complaining about. You want him to take at least a second look before outright rejecting your paper as 'the same wrong stuff again'.

I once heard this story about someone submitting a paper with a serious flaw in one proof. One referee pointed it out, but the author just submitted the same version to another journal. It ended up in the office right next to the first referee's. As you can imagine, this did not improve the chances for acception.

Answer 6

I've had this happen several times. These papers were all in a technical subject where it was hard to find a referee. I would get a referee report making elementary errors while accusing me of making elementary errors. Often the referee had poor command of the English language. In one of the cases I even had an email back-and-forth with the referee through the journal editor. This was a total waste of time, as the referee was simply not able to understand basic concepts as he was accusing me of the same.

I have found that it's best just to resubmit the paper to another journal. On one of these papers I did substantially revise the exposition, but in the other cases I did not. I will have to disagree with Jake Beal over the value of revising.. I don't think the exposition is usually the issue in these cases. As the asker pointed out, such referees often don't even understand the statements of the theorems.

In all cases the next referee behaved normally. That is not to say the next referee always accepted the paper, but the referee always understood what the paper was about and when a referee rejected it he/she always had intelligible reasons.

Answer 7

You should not revise your paper as Jakebeal suggests just because of the negative outcome of the review process so far. Revisions should only be done if upon taking another look, you think you can improve the paper, or if the paper has pretty much been accepted on the condition that you make certain improvements. In the latter case you may face the difficult decision of making reversion you don't fully agree with, you then have to balance the option of publication with such revisions against the possibility of a rejection.

I have had a mathematical physics paper rejected on totally flawed grounds once. I did appeal the decision, but this went nowhere. I submitted the paper to another journal, where it was again rejected directly by the editor. I later found out that this editor also worked for the journal I originally submitted my paper to, he had probably seen the appeal there. I then submitted the paper to another journal (with a similar impact factor as the original journal) where it was accepted without even a referee report sent to me to address problems.

The paper was not revised, the referee report did not contain any useful information for me to act on. The misunderstanding that led the referee to think that my paper was flawed was caused by the referee not being familiar with the jargon used in my field. It was also quite clear to me that the referee had skipped all of the sections about the physics and had only read the section where the results are derived. There the referee was misled by the jargon. If the referee had just read the entire paper the problem would not have occurred. Adding more explanations would in principle be possible but this has to happen at the start of the paper that the referee didn't read.

So, while you should always consider if the paper can be improved, and not hesitate forging ahead and make changes even if it inconvenient (e.g. if the paper is already under review), making revisions by guessing what an incompetent reviewer got in his/her head to try to get to a different decision the next time your paper is reviewed in that incompetent way, is not a good way to go about revising your paper.

Answer 8

"Wrong" proofs may be wrong in the sense that

1. some small step is logically incorrect, or

2. what is written there is not a Bourbaki-style proof (Bourbaki-style = small-step, well-formulated), or

3. there is a typo, or

4. bad, incomprehensible English,

or a combination of 1.-4. holds. Items of which kinds (1.-4.) does the referee mention? Strictly speaking, any reason 1.-4. is a valid reason for rejection. E.g., I really, really hate to read a paper with a typo in a formula which makes me ponder a whole day long until I notice that a typo is the culprit. Regardless of the case, you should take the chance to improve the proof. However, a strong rejection might be objectively too harsh a measure; it really depends on the case whether a weak rejection would be more justified. If any of 1.-4. holds, don't complain as it would not contribute to your reputation. Only if neither 1.-4. is present and the referee's report is bogus, you can complain.

A related aside: recently, some people started asking for a machine-checkable proof in a theorem prover (e.g., Isabelle, just to name one). If you want to immunize your paper against the "wrong proof" claim completely:

• write a machine-checkable proof, and
• make it available to reviewers somehow [put it online, supply a file, etc.], and
• submit to another venue or resubmit.

Whether you submit elsewhere or resubmit would be the next issue; it really depends.

Notice that above, I abstained from speaking about "trivial proofs" or "trivial results": these would be too hard to formalize. Even trivial theorems might be published, but only if they are really genuine and are of great importance.

Answer 9

... try another journal? Write an e-mail to the editor explaining why we disagree with the decision?

I would pick up on this. I do not know how compact or concise or how deductive your proof is. I don't know how concise and deductive (rather than inductive) it can be. Let's suppose it is both concise and deductive, then spell out each step to the editor. Make sure each step is undeniably justified and spell that out.

If the result is "unimportant", make the case for how it is salient. What other mathematical work does this influence? How does your result help some other mathematical effort? If your result serves only your mathematical effort (like in previous papers of yours), then you have the problem of making sure those are salient as well.

I used to be a reviewer for the Audio Engineering Society. There was a guy from Hong Kong that submitted and published paper after paper after paper on the topic called "Wavetable synthesis," but they were only continuations of his own work and quite self-referential. Finally the editorial folks got tired of it, because it got to the question of "Who the hell cares?" or "Who gives a rat's ass?" You need to be able to answer that question.

Saying, "This work is important to increase my publication record so that I can get tenure" might not impress them. (I'm not saying you're saying that.)