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I am refereeing my first paper and I'm quite excited! But inexperienced and I would like to ask an advice to the Maths Community of MO. Let me tell you that I have already read Refereeing a Paper, but it seems that my question is quite different. Roughly:

What is the point after which you get nervous while refereeing a paper?

Specifically:

  1. I've found many English mistakes. (Well, I'm not a native English and so I can understand. So I am not nervous yet)

  2. I've found some maths inaccuracies like "let A be any set"... and then I have discovered that the proof of the first result works only for finite sets. (OK, those are only inaccuracies - I am not nervous yet)

  3. There are many references like "we use the notation of [X]", "this result is proved in [X]", where [X] is a BOOK, without specifying a precise section, or the number of the result... should I get this book and read all to find out the correct references? - just thinking of it, makes me a bit nervous..

  4. (most importantly). There is a mathematical more serious mistake. Something that might be fixed, but not obviously (my definition of obvious is three evenings, in this case). I am not saying that the paper is completely wrong but that now... now I'm getting nervous!

Now, taking into account that the person who asked me to referee this paper told me: be selective, we accept only 20% of submitted papers,

what should you do in these cases? Reject? Ask for a revision? Not getting nervous and try to see if the rest of the piece is good (I'm quite a good guy and I'm doing that at the moment)?

Of course I will talk with the editor, but I also would like to know more opinions that might be helpful in future.

Update: Thank you very much to everybody for the numerous and helpful comments.

Valerio

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Nice question, but be careful: you're supposed to be anonymous. The author might recognize that you're talking in his paper if he reads mathoverflow. –  Joël Nov 30 '11 at 23:21
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I don't think that author can recognize me by these infos, anyway, I think there is a way to put questions on MO anonymously, but I don't know how to do.. –  Valerio Capraro Nov 30 '11 at 23:29
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Re 3, citing an entire book without specifying the precise location is common, though it shouldn't be. –  Tom Leinster Nov 30 '11 at 23:40
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@Valerio: just regarding the 'I don't know how to do'. I never asked anything, but I think just logging out/not being logged in before clicking on 'ask question' would basically work. –  quid Dec 1 '11 at 1:57
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@Valerio: If I were you, I wrote a report as you explained: Actually you have nicley categorised paper flaws into 4 levels according to yourself). Flow your own way to write a report; At the end of your report you should write your final decision of course. This must be well-motivated from your list of comments/corrections. –  Alireza Abdollahi Dec 1 '11 at 8:53
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8 Answers 8

My suggestion would be to

a. See if the claimed result is interesting. If not, reject.

b. If the claimed result is clearly interesting, be honest and say that the paper is clearly not publishable as is (and give your points 1-4, though really already 1 is sufficient), and should be revised before refereeing (this is often a box you can check if use one of the semi-automated review submission systems), but you would be willing to look at it once it is so revised.

c. Only get nervous over your own papers.

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I would add: trust the editor. Write a report that summarizes all your impressions, and let the editor decide what this means for the future of that paper. –  Thierry Zell Dec 1 '11 at 4:57
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@Thierry: I don't think I agree at all. With the current situation in many journals (waiting lists etc.) many editors are not as thorough as they should be, and reject papers even if a referee suggests to forward a paper for revision (if the referee is not enthusiastic enough). So, if result is clearly interesting, it is worth emphasizing it strongly enough to make sure it's easy for the editor to spot it! –  Vladimir Dotsenko Dec 1 '11 at 10:32
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@Vladimir: ultimately, the editor knows better than the referee what the journal wants or does not want; if some journals don't even want to bother with papers that will require too many revisions, it's regrettable and I'm not sure it's the right attitude to have, but maybe it's because there are simply too many math papers out there. –  Thierry Zell Dec 1 '11 at 23:08
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...and that ("too many maths papers out there") is precisely why it's the first responsibility of the referee to make it crystal clear if the paper is interesting or not in the first place. In my opinion, the key duty of the editor is not to decide if the paper is interesting but to be capable of picking a qualified referee for that. If a referee spends all their time on listing improvements/corrections but is not crystal clear whether the paper is interesting or not, it's a bad referee, no? –  Vladimir Dotsenko Dec 2 '11 at 8:43
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I agree with this, but I'd just like to add that, in this situation, don't ONLY reject on grounds of interest. Now that you've done the work to read the paper, please do put in the report the (major) problems you found. If you don't, the author is likely just to resubmit their paper elsewhere, and some other poor referee will have to find the mistakes. –  Matthew Daws Dec 16 '11 at 13:32
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Others have already answered your question well, but I want to add something about #4, the mathematically more serious mistake. Since by your own admission you are inexperienced, I can see why you spent three evenings trying to repair the proof, but I personally would never spend that much time in such a situation. In my report, I would simply say that the proof is not acceptable in its current form because claim X is insufficiently justified. (I would also keep going through the proof under the tentative assumption that claim X is correct, to see if there are any other issues later on in the proof.) Say that the author must either provide more justification of X or find a way around it or retract the theorem. In one case where I did this, the author did indeed retract the theorem (it was not the main result) because he was unable to repair the proof.

Proofs are supposed to be verifiable by competent readers with only modest effort. If you're qualified enough to referee the paper then you should not be working so hard to understand the proof.

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Absolutely! But let me say-- if you have only had one case of the theorem being retracted, you are refereeing an order of magnitude better class of papers than I am. Which is perhaps to say, there is a difference between a proof which is not well justified, and a proof which just cannot be right (and I think the OP initially gave the game away a little, and I suspect the latter is the case here). –  Matthew Daws Dec 16 '11 at 13:37
    
@Matthew: Those types of papers typically go in my reject pile pretty quickly. I'm talking about a good paper with a mistaken side result. –  Timothy Chow Dec 16 '11 at 16:13
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3-4. Do not hesitate to ask the author (via the editor) what he had in mind.

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This is the kindest course of action. On the other hand, it will take up more of your time. So, as an easy criterion, ask yourself first "If all of these problems are solved, would I definitely accept the paper?" If the answer is "no" then just save everybody's time and reject now. –  Sam Nead Dec 1 '11 at 10:37
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It is the author's job to be comprehensible, clear, and correct. If the paper falls short in that regard, you are doing the author and the editor a favor by asking for clarifications.

If you are feeling at all unsure, you could always say things like "I'm afraid I got a bit confused by Y", "I was not able to see how the proof of Lemma 1 handled infinite sets, would you please elaborate", "Would you please say where in [X] result Z can be found?", etc.

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I like the comment of Francis Crick in `What mad pursuit' that one should always start a referee report with praise, but of course not hold back on the detailed criticism.

I have found that a referee can also help the author by asking for clarification in the Introduction of the aims of the paper, what does it accomplish in relation to previous work in the area, and what questions are still open. The author may also not have got the title right in relation to the expectations of readers: some of my early papers had lousy titles which not not convey easily what was in the paper.

It is useful to know that the referee is not responsible for the correctness of the paper: that is the responsibility of the author. Of course, if the referee has doubts on the correctness, these must be raised.

I am unhappy about remarks such as "there are doubts about the interest of this area", since some papers are of interest precisely because they go against the so-called `mainstream', which, to mix a metaphor, flaps about like a sail in a gusty wind.

The job of referee is very important, and I wish you luck and interest in this.

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Point (3): I would ask the author to be more specific, unless the results are basic stuff. People have asked me in the past to add a theorem number or at least a section number.

Point (2) and (4): that's exactly what goes in your referee report. Point (2) is a minor change, point (4) is a major change. List them all and send them back in a nice list.

Point (1): I would point out briefly that the English needs a review; I think proofreading the English should be a job for mr. Springer or mr. Elsevier after acceptance.

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While I agree that it's not really the responsibility of the referee to fix the English, I know from experience that in most cases, if I don't fix the English, nobody else will (or at least nobody else will do as good a job). The other mathematicians in the supply chain will think it's not their job, and the non-mathematicians will be hampered by their lack of knowledge of mathematics. So I usually try to fix the English myself. –  Timothy Chow Dec 1 '11 at 15:35
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You are right that this aspect is often overlooked. I am not an editor in any journal (too early in my career for that), but if I were one I would insist on the journal's production team doing it. Proofreading and typesetting is more or less the only things journal do in-house to justify the big money they cash in, so in my humble opinion they should at least do it properly. –  Federico Poloni Dec 1 '11 at 22:44
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Yes, Mr Springer and Mr Elsevier should employ competent proof editors in the fields they publish in. –  David Roberts Dec 2 '11 at 0:27
    
My limited experience tells me that the cheaper the journal is, the most competent the proof editors are. Probably, both are correlated to the academic/non-profit publishing. –  Benoît Kloeckner Dec 16 '11 at 14:19
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This is really a comment but I don't have enough reputation points for that. It is a little more specific than Joël's.

It's a small world. How many papers are out there where the author defines some type of supremum of a set of elements in a group? Perhaps you have disguised the actual mathematical structure.

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Many people believe that the single-blind refereeing process is stupid and just slows things down. I am not sure whether I agree or disagree. –  Igor Rivin Dec 1 '11 at 13:26
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Even if the single-blind refereeing process is stupid, accepting a refereeing request under a single-blind system with the intent of breaching the anonymity seems to be a recipe for getting people angry without making any real progress towards reforming the system. –  Timothy Chow Dec 1 '11 at 15:40
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It's a matter of (alleged) carelessness here; I wouldn't assume any intent of breaching the anonymity on the side of the OP. –  darij grinberg Dec 1 '11 at 17:29
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You can do the job of a referee, and you can add to it if you wish the job of a copyeditor/editor. I think the criticisms you made all address the readability of the paper; if the author cares, he/she will incorporate your remarks. The ones that matter to the publisher are the referee remarks, and you should ask your advisor/community how to present those. As mentioned by others, the paper should be correct and interesting (and readable). Be clear about what is needed to achieve those goals. After that , add what reasonable efforts you believe the author should do to make the paper more readable.

Gerhard "Ask Me About System Design" Paseman, 2011.11.30

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