I recently posted a short (6 page) note on arXiv, and have more or less decided that I should not submit it to a journal. I could have tacked it onto the end of a previous paper, but I thought it would be somewhat incongruous -- it is an interesting consequence of the key lemma unrelated to the main result. I really liked the concision of the paper and didn't want to spoil it.

This brings to my mind several philosophical and/or ethical questions about the culture of publishing that I find interesting, particularly at this point in my life since I am nearing the point of seeking a permanent position.

  1. Two things are clear: It is to one's advantage, especially in the early career stage, to have many publications. It is also to one's advantage to have a strong publications. So take a result that is not very difficult to prove, but is interesting mostly because I think it may be useful as a stepping stone to another as-yet-unknown result. Given that I have already put it on arXiv, is "because I think it could be published" a good enough reason to publish it?

  2. One good reason to submit a paper to a journal is to have it in the refereed scientific record. I was tempted to claim the result at the end of the previous paper without proof as a remark, but in the end thought better of making such a claim without giving a proof. How bad is it to make such a claim if the proof (or even the truth) is not obvious, but can be proven as a reasonably straightforward generalization of someone else's proof of a different result?

  3. This is a more pragmatic and frank question. Should young researchers be careful to avoid the impression of splitting their work into MPUs (minimum publishable units)? In this instance my reasons for not including the result in another paper are somewhat complex and non-obvious.

  4. As a counterpoint to 3, how should one balance the purity/linearity of the ideas/results in a paper, versus including as many related results as possible?

I realize that these are very soft and subjective questions, but I am interested to hear opinions on the matter, even if there is no universally true answer.

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    $\begingroup$ Could you possibly mark your question as Community Wiki? $\endgroup$ – Andrey Rekalo Oct 10 '10 at 22:45
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    $\begingroup$ Can whoever made the vote to close explain it? I think if the question were made CW it would be fine. Many similar soft questions about career advice have stayed open and been quite popular. $\endgroup$ – Qiaochu Yuan Oct 10 '10 at 23:18
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    $\begingroup$ Community wiki activated. $\endgroup$ – Andrew D. King Oct 10 '10 at 23:31
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    $\begingroup$ Andrew, you haven't actually explained, why you don't want to publish the result. $\endgroup$ – Alex B. Oct 11 '10 at 0:35
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    $\begingroup$ There are a few reasons. The result seems a bit small and strange on its own, and I'm not sure whether or not others would find it as interesting as I do without further evidence. I feel like I would rather spend more time mulling over the implications to bigger questions to see if such an application comes to me. I put it on arXiv because I am happy for others to do the same, I do think it's interesting, and I claimed some time ago to several people that I knew how to prove it.... $\endgroup$ – Andrew D. King Oct 11 '10 at 1:00

This is a question of interest to most mathematicians who are research active and not slowly but surely knocking off important problems in their field at the rate of one per paper. (I think I could have ended the previous sentence at the word "active" without much affecting the meaning!)

I think the answer is ultimately quite personal: you are free to set your own standards as to how much of your work to publish. I myself understand the psychology both ways: on the one hand, math is usually long, hard work and when you finish off something, you want to record that accomplishment and receive some kind of "credit" for it. On the other hand, we want to display the best of what we have done, not the entirety. This position is well understood in the artistic and literary world: e.g. some authors spend years on works that they deem not ready to be released. Sometimes they literally destroy or throw away their work, and when they don't, their executors are faced with difficult ethical issues. (This is roaming off-topic, but I highly recommend Milan Kundera's book-length essay Testaments Betrayed, especially the part where he details the history of how after Kafka's death, his close friend Max Brod disobeyed Kafka's instructions and published a large amount of work that Kafka had specifically requested be destroyed. If Brod had done what he was told to do, the greater part of Kafka's Oeuvres -- e.g. The Trial, The Castle, Amerika -- would simply not exist to us. What does Kundera think of Brod's decision? He condemns it in the strongest possible terms!)

Another consideration is that publication of work is an effort in and of itself, to the extent that I would not say that anyone has a duty to do so, even after releasing it in some preprint form, as on the arxiv. A substandard work can be especially hard to publish in a "reasonable" journal. I have a friend who wrote a short note outlining the beginnings of a possible approach to a famous conjecture. She has high standards as to which journals are "reasonable", and rather than compromise much on this she determinedly resubmitted her paper time after time. And it worked -- eventually it got published somewhere pretty good, but I think she had four rejections first. I myself would probably not have the fortitude to resubmit a paper time after time to journals of roughly similar quality.

As you say, though, one advantage of formal publication is that the paper gets formal refereeing. Of course, the quality of this varies among journals, editors, referees and fields, but speaking as a number theorist / arithmetic geometer, most of my papers have gotten quite close readings (and required some revisions), to the extent that I have gained significant confidence in my work by going through this process. I have one paper -- my best paper, in fact! -- which I have rather mysteriously been unable to publish. It is nevertheless one of my most widely cited works, including by me (I have had little trouble publishing other, lesser papers which build on it), and it is a minor but nagging worry that a lot of people are using this work which has never received a referee's imprimatur. I will try again some day, but like I said, the battle takes something out of you.

Finally, you ask about how it looks for your career, which is a perfectly reasonable question to ask. I think young mathematicians might get the wrong idea: informal mathematical culture spends a lot of time sniping at people who publish "too many papers", especially those which seem similar to each other or are of uneven quality. Some wag (Rota?) once said that every mathematician judges herself by her best paper and judges every other mathematician by dividing his worst paper by the total number of papers he has published. But of course this is silly: we say this at dinner and over drinks, for whatever reasons (I think sour grapes must be a large part of it), but I have heard much, much less of this kind of talk when it comes to hiring and promotion discussions. On the contrary, very good mathematicians who have too few papers often get in a bit of trouble. As long as you are not "self plagiarizing" -- i.e., publishing the same results over and over again without admission -- I say that keeping an eye on the Least Publishable Unit is reasonable. Note that most journals also like shorter papers and sometimes themselves recommend splitting of content.

So, in summary, please do what you want! In your case, I see that you have on the order of ten other papers, so one more short paper which is in content not up there with your best work (I am going entirely on your description; I don't know enough about your area to judge the quality and haven't tried) is probably not going to make a big difference in your career. But it's not going to hurt it either: don't worry about that. So if in your heart you want this work to be published, go for it. If you can live without it, try that for a while and see how you feel later.

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    $\begingroup$ +1 for recommending Kundera, off-topic though it was :-) $\endgroup$ – Alex B. Oct 11 '10 at 2:22
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    $\begingroup$ Pete Clark, excellent comparison of mathematics to art. A proof is not merely a recitation of truths; it is also an elaborate construction that leads the observer/reader down the path, showing how obvious the path is when the proper signs are visible or are made visible. That is the artistry of a mathematical proof, and the reason for why different proofs of the same underlying mathematical theorem can have different levels of satisfaction or acceptability for different readers. $\endgroup$ – sleepless in beantown Oct 11 '10 at 2:42
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    $\begingroup$ informal mathematical culture spends a lot of time sniping at people who publish "too many papers" Really? This seems to be a piece of mathematics culture I have completely missed. I don't think I've ever heard this complaint leveled at anyone (on the flip side, I've heard plenty of complaining about people who don't write up results and let them become folkloric). $\endgroup$ – Ben Webster Oct 11 '10 at 5:53
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    $\begingroup$ 1. Ben: Sure, have you not heard anyone described as having "100 papers and no theorems", or anything like it? I wouldn't say I approve, but it's probably not so good if the expected value of one of your papers, judged by the people who are interested in your more important work, is less than their threshold for reading a paper. There are some people to whom this applies. 2. Pete: If you're interested in the Kafka story, you should read the recent NY Times Magazine article by Elif Batuman. $\endgroup$ – JBorger Oct 11 '10 at 6:18
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    $\begingroup$ @James: thanks very much. I have decided to read the article now instead of further discussing how prevalent this negative aspect of mathematical culture may or may not be. If anyone wants to join me: nytimes.com/2010/09/26/magazine/26kafka-t.html $\endgroup$ – Pete L. Clark Oct 11 '10 at 6:24

My philosophy is: each paper should have one result (or a very closely connected collection of results) in it. That is, each paper should tell one story. This is based on past experience: when I put two different (although connected) results in one paper, everybody learned about the most important one, and nobody realized that the paper also contained the second.

For a more famous example, Alexander's paper on the Alexander polynomial contained a bunch of different results, including the skein relation (all connected, of course, because they were all about the Alexander polynomial). Everybody somehow forgot about the skein relation result, and did not realize that this result was also in the paper until after Conway had rediscovered it.

Would it have helped if the skein relation had been published in a different paper? I can't tell ... maybe everyone would have just forgotten entirely about the second paper. And I also can't say for sure why the skein relation result was overlooked. But my theory is that everybody just remembers one story associated with each paper, and that you shouldn't bother putting in anything that's not in that story.

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    $\begingroup$ I agree, which is why in my example I omitted my lonely orphaned result from the paper. But it's not always obvious whether a result is connected enough (or for that matter, interesting enough, brief enough, etc.) to be included. But obviously one should be careful: If the result is really interesting, maybe it should be in a separate paper. There are lots of examples of important results that get buried in the back of a paper, where their importance is not pointed out as vigorously as it should be. $\endgroup$ – Andrew D. King Oct 11 '10 at 14:08
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    $\begingroup$ One example that jumps out at me is the NP-hardness of fractional colouring, which appears at the very end of a very important paper of Grötschel, Lovász, and Shrijver. In fact, maybe examples of this would be an interesting topic for another community wiki. $\endgroup$ – Andrew D. King Oct 11 '10 at 14:10
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    $\begingroup$ I like to put Peter's comment into a slightly different context: the mathematical record of all published papers is a story. A paper is in some sense just one point in the plot of that story. So a paper has to include something big enough to get noticed, but since it's always remembered as just one plot point, you should try to stick to having a single paper containing a single principal idea, as that's how papers tend to be remembered. If your paper doesn't contain a rememberable result, it is forgotten. If it contains more than one, likely only one will be remembered. $\endgroup$ – Ryan Budney Oct 11 '10 at 23:09

My answer is similar to that of Pete Clark's.

Ethically, as long as what you write is sincere, you can arrange your results in whatever way you want in papers, and you can publish them in journals or not as you please. If you look at the way that many strong mathematicians write papers, some of them simply break every rule that you can imagine as to what "should count" as a paper. They can get away with it because they are trying to do something hard. For instance, Perelman's papers are extremely telegraphic and break a lot of rules. As it happens, they aren't published in journals, but no one cares, other than that I'm sure that many journals would love to have published these papers.

Unfortunately, if you need a job, the letter of publishing ethics is more important than the spirit, unless you game the system to some blatant extreme. It is just the mechanics of job searches and not the fault of any one search committee. You get almost no penalty for publishing small results; on the contrary it is risky not to expand a good result into several papers. In any case, many published results are routine extensions of known mathematics, dressed up by including every little detail of "the proof". Up to a point, even that is ethically fine. Unfortunately, it really is true in both hiring and promotion that quality is controversial, while quantity is an objective standard. (Even though quantity is also a highly distorted standard that is not measured in a consistent way.)

Mathematics is rigorous (ideally), but mathematics journals and the mathematics profession are not. I don't think that it works to impose axioms on the latter two. It's not that I don't care or that I believe that there are no standards, but I would rather see common sense than strict conventions.

  • $\begingroup$ What do you mean by telegraphic? $\endgroup$ – JBorger Oct 11 '10 at 22:52
  • $\begingroup$ I mean overly abbreviated. One of the definitions of "telegraphic" in Wiktionary is "brief or concise, especially resembling a telegram with clipped syntax". At a deeper level, Perelman's ideas had a great deal of credibility; if not for that fact, his proof as written would have been called incomplete. Actually, any proof written for a human audience is like this, the only question is to what degree. A formalized proof that can be checked by a software proof assistant is very different from a publishable paper. $\endgroup$ – Greg Kuperberg Oct 12 '10 at 7:00

In terms of what's best career-wise, I think it would depend a lot on where you end up. As far as what's best for the community, obviously many redundant papers are not ideal, but it would make sense to publish your result separately if both hold:

  1. it does not quite fit in with the rest of the other paper (as you already mentioned);
  2. you find it interesting enough that there is reason to believe it might be used later.

We like to complain about the over-abundance of papers in our fields, but, too often, very useful results end up in footnotes of almost unrelated papers, which makes it awkward for reference and dissemination. And there's nothing wrong with concise papers.


Following Yemon's suggestion, I'm turning a comment into an answer:

I would say, if you think that the result is interesting enough to be put on the arXiv, then it should be interesting enough to appear on paper. Besides, the opinion of the referee, whose main job will be to assess exactly that - the interest of the result - will provide a nice check for your own judgement. If it's just you who finds the result interesting, then it won't get published anyway. If on the other hand the referee agrees with you, then you will be glad that you submitted, since it indicates that the community is interested after all.

As for strategic career-planning, I'm not in the position to give advice on that, but it would seem very strange to me if an extra publication turned out to be detrimental to the career.

  • $\begingroup$ Alex, great answer! $\endgroup$ – Hailong Dao Oct 11 '10 at 1:55
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    $\begingroup$ The trouble comes when you try to find an outlet for that paper. IMHO there is lots of folklore that could do with being written up, and might find a home on the arXiv, but which is unlikely to be accepted by any journal which has reputation to preen and backlogs to prune $\endgroup$ – Yemon Choi Oct 11 '10 at 5:31
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    $\begingroup$ But folklore and new results are essentially different: "folklore" means that lots of people have thought about the result and everyone is convinced beyond any reasonable doubt that the result is true. If you think that a new result is worth disseminating, then you should also agree that this result is worth being scrutinised by someone dedicated to picking holes in it. I would leave it to the journal to worry, whether they want to publish it. If I think that the result is interesting, then I will submit it. If the journal disagrees, then that gives me valuable feedback. $\endgroup$ – Alex B. Oct 11 '10 at 7:05
  • $\begingroup$ We seem to have different definitions of "folklore" - yours may well be the right one. What I was thinking of was results which are surely not new observations, and which you want to use, but which haven't been written down anywhere, and which are not immediately obvious. I admit to bias here, having recently written a note which is new in the sense that it had calculations I couldn't find in the literature, and which were needed for another paper, but which didn't contain anything surprising. $\endgroup$ – Yemon Choi Oct 11 '10 at 7:32
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    $\begingroup$ I see. I think I would prefer such calculations to be included in the paper where they are used. If they are too ugly to be read by mortals, then maybe put them into the appendix. My thinking is that whoever reads the paper shouldn't have to take on trust calculations on the arXiv that noone will ever peer-review. If the calculations are re-usable, that's a bonus. Then people will refer to your paper because there is something genuinely useful for them in it. $\endgroup$ – Alex B. Oct 11 '10 at 7:41

I strongly feel that a nice concise result that can stand on its own should be and is worthy of publication.

Anchor it with a prologue consisting of some of the context of your paper where you felt that your key lemma leads to this result. Perhaps add some interesting consequences that you can be certain of, and some conjectures which you can intuit but not yet prove, and would be worthy of investigation, in your estimation.

Don't worry about someone else thinking that you are breaking apart pieces that belong together into "Minimum publishable units". It's wonderful if one paper can consist of an astounding result with many sequelae; but it's just as wonderful to have a paper with a good clear proof and result, with sequelae sequestered into following papers which build upon that result. Perhaps this short paper should be published in a different journal, a different category of mathematics, a journal with a different focus?

If you felt that the pieces belonged together, you could have included it in your prior paper. You obviously feel that this result is a consequence of your paper's key lemma, but pointing in a different direction which did not flow along with or fit in with the construction or design or architecture of your prior paper. My opinion is to go ahead and fill out this new result with more implications and more context, and attempt to publish it as a separate piece.

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    $\begingroup$ + for the phrase "sequelae sequestered"! Even though it relies on a secondary meaning of "sequela" ... $\endgroup$ – Joseph O'Rourke Oct 11 '10 at 1:33
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    $\begingroup$ I agree with Joe. You shouldn't use the length as a definitive criterion. I've read 50 page papers which ultimately had nothing much to say, and 2 page papers that were inspiring. As you said, if you had thought of the result when you were writing the previous paper you would have included it. But you didn't see it until later. So you've already decided that it's worthwhile to publish. It's easy to fall into the trap (I speak from my own experience) that you might be able to follow the path opened up by the new result to something more significant. That's ultimately self-defeating. $\endgroup$ – Victor Miller Oct 11 '10 at 16:18

Deans can count.

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    $\begingroup$ Deans can count, ... but they cannot read. $\endgroup$ – Steven Heston Oct 11 '10 at 20:28

I would say that concerning the overwhelmingly large number of mathematical papers that appears, the main question (for the mathematical community, your career is something else) is: will the publishing of your paper add more signal (being MathReviewed, if someone needs the result, she or he is likely to find it) or more noise? For example, concerning a slightly different question (when should one split a paper in two?), I tend to think that if splitting does improves the probability of finding any of the results by a MR search, then splitting is actually good.


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