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I am a mathematician. I recently formulated a conjecture, and was working toward proving it. Another mathematician I know only via email also conjectured the same around the same time.

I now managed to prove the conjecture. I think it is an important result.

However, my paper on this is not ready to be put on the arXiv, because there are natural questions related to this conjecture that need to be solved first (otherwise the paper will look incomplete). I know I can solve these questions, but it will take me time to do that.

From a recent preprint this person put on the arXiv recently, I am convinced that he is also not too far from proving this result (my solution did not use his results and was not inspired by them). Thus, I am worried that he will also obtain the same result.

I was wondering if it would be ethical for me to "block him" and establish priority by emailing him my solution of the conjecture?

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It seems to me that there is some amount of "having your cake and eating it too" in your question.

First of all, I think private mathematical correspondence, including sending preprints, should only be "friendly" (or "collaborative"); it should not be "unfriendly" (or "competitive"). If someone emails me a paper, I don't in general have an obligation to read it. If you think I am about to prove something very soon and you send me something with the aim of taking away the independent nature of my proof, that seems quite unfriendly. The correspondence doesn't advance the field in any way; it only seeks to advance you at my expense. As others have said, if the natural state of affairs is that both you and I will arrive at proofs of the same result independently in roughly the same time frame -- great, we will both get the credit, and it will not be divided in half in any clear way. I don't see why you need to perform machinations to try to prevent that.

Also, you are trying to establish priority in a rather strange way when there are two much more standard ways: submit to a journal and/or post on the arxiv. Regarding this, you say:

My paper on this is however not ready to be put on the arxiv, because there are natural questions related to this conjecture that need to be solved first (otherwise the paper will look incomplete).

But you also say:

I now managed to prove the conjecture. I think it is an important result.

If the conjecture is important, then a proof of it should make for a very nice paper. Most papers "look incomplete" in the sense that they leave something natural undone. And most of us do not get to work out all the consequences of our results in the same paper as those results are first published. Doing too much in one paper is risky for several reasons, one of which is that the longer you sit on a given result, the larger the chance that someone else will do it. You seem to be trying to circumvent this natural tension with your private correspondence. While not unethical per se, I don't expect it will be well received.

Given the situation and the concerns, I suggest that you do upload to the arxiv a short note that proves your conjecture and that does not have all the bells and whistles you would like to eventually have. You say that it is two pages long, so that I would expect that the relevant mathematical community will be able to quickly ascertain whether it is a correct proof of the conjecture. Because of this you don't have to submit right away or even submit the version of the paper you put on the arxiv. You can take the time to make the additions that you want. Of course, other members of the community can respond to your arxiv preprint and build on it as well. That's what's fair, it seems to me.

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    +1: I like the point emphasising that arXiv permits early claim of a proof at the cost that competitors now have equal standing as to where develop things further. Trade-off well emphasised. – Captain Emacs Feb 22 '18 at 10:50
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    "It is amazing what you can accomplish if you do not care who gets the credit." ~ Harry S Truman. – emory Feb 22 '18 at 11:44
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    "It is amazing what you can accomplish if you do not care who gets the credit." While this is a nice quote the wikiquote article for Harry S. Truman has it in the misattributed section. – ValarDohaeris Feb 23 '18 at 13:04
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    @ValarDohaeris: isn't that the point? – Kevin Feb 23 '18 at 15:26
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    @ValarDohaeris "The problem with quotes on the Internet is that it is hard to verify their authenticity." ~ Abraham Lincoln – user78960 Feb 23 '18 at 18:32
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You are asking the wrong question (or to be precise, a loaded question). It is ethical to send anyone emails containing mathematical ideas (with some obvious exceptions, such as if you expect the recipient to use your ideas to commit a crime), so that’s not the issue. The problem with your plan is that the recipient has no obligation to read your email (and indeed if I were in his shoes I would not read it, certainly not past the point where you claim you have a proof of the result I am trying to prove but for some lame-sounding reason are not yet ready to publish it), so your perception that the email ties his hands somehow, establishes priority, or blocks him from publishing his own proof and claiming to be the first to do so, is simply incorrect.

There is on the other hand a correct and sure way to establish priority, and that is to post publicly a timestamped proof of the result (the arXiv would be the canonical place to do it). Do not worry about the paper being incomplete; as long as the proof is complete, and correct, you have established priority and are free to later add other things or make other improvements to the paper by posting a new version or a completely new preprint.

As for the collaboration idea suggested by some people, of course you are free to propose a collaboration, but since you already proved the result, I don’t see what purpose that serves other than as an act of charity, and in fact it would be a bit dishonest to pretend you need the other researcher’s “help” to prove something you already proved. It might make sense in a situation in which you have both proved the result around the same time and found substantially the same proof and decide to merge your papers into one joint publication (I have seen this done on several occasions), or if you both have different proofs and decide to publish them both in one paper. But neither of those scenarios sound like the situation you are describing.

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    @Worried how would that change anything? – Dan Romik Feb 21 '18 at 16:24
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    @Worried Your email to him doesn't prove that you had the idea first. For all you know, he could be sitting on the same idea you are, for the same reasons. – user37208 Feb 21 '18 at 16:38
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    @Worried okay, that makes sense. But as I said he doesn’t have to read the email, and as user37208 points out, even if he does read it he could claim, even somewhat believably given how short the proof is, that he already knew the proof before reading it. So this information doesn’t change anything about my answer. There’s only one sure way to establish priority, and that is to make the proof public. – Dan Romik Feb 21 '18 at 16:40
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    I don't think there's a way to prevent a person from claiming that they independently achieved the same result as you. Even publication order is a sketchy concept; imagine that your competitor has already published somewhere a partial proof. – Alexey B. Feb 21 '18 at 19:15
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    @Azor-Ahai many mathematical ideas can be used to commit crimes, for example an efficient method for factoring integers could facilitate breaking RSA encryption, with lots of criminal applications. – Dan Romik Feb 21 '18 at 19:42
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Consider this:

Either you're faster anyways or you want to use his ethics to publish a result he would have obtained faster.

Can something that at best accomplishes nothing and at worst steals merits be ethical?

  • I like how you've reduced this to simple terms. But I'd go more with Pete's take/answer: it's not so much unethical as it is unfriendly. – zibadawa timmy Feb 21 '18 at 22:57
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    @zibadawatimmy he says “it only seeks to advance you at my expense.“ - that's unethical, no matter what he calls it, though I think you understood Petes answer wrong, his unfriendly doesn't exclude unethical. – DonQuiKong Feb 22 '18 at 5:17
  • This should be the top answer. It's clearly unethical—there's no dancing around it. – AmagicalFishy Feb 23 '18 at 0:38
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Priority doesn't mean the idea entered your head first; it means you published it first. And in fact, sending a preprint doesn't even establish that you had the idea first. For all you know, your competitor may be sitting on the same idea as you, for the same reasons (as I said in a comment).

In terms of advice, you should ask yourself how important this theorem really is. If it's something that will make a splash, you should post a short preprint to arXiv as soon as you can. But if it's merely a good result of the level one would routinely see in a solid-but-not-elite journal, you should solve the related questions you mentioned, and submit everything to a solid-but-not-elite journal. If your competitor publishes a paper with some overlap around the same time, that's not the end of the world. You'll both get credit, even if he beats you to the punch by a month.

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You and a competitor are both racing to solve the same problem, you are both aware of each other, and the "winner" is the first to make their result public.

You're almost ready to make your result public and your competitor might be similarly ready. If you reveal your position to your competitor, then they might make their result public immediately. (They might even be able to do so ethically, e.g., perhaps they receive your mail, largely ignore it, and publish.) Thus, rather than "block him," you'll actually tip him off. Given that emailing needn't constitute a "block," I see no ethical dilemma.

If you want to establish priority, then publish what you have, perhaps after removing any discussion of what might follow from your results (that can be added later). There is certainly no ethical issue of publishing a finished result.

You should also consider collaborating with your competitor. You can do so before or after establishing priority. Ultimately, you have both independently proved a result, and you can both benefit from working together, in particular, you can check each other's work and you can both contribute different ideas. (Although you've proved the same result, it is unlikely that you both think about the result in the same way.) If you don't collaborate, then you might both want to include a statement that acknowledges that the other proved the result independently.

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    Thank you, but this actually doesn't answer the question I asked at all. – Worried being scooped Feb 21 '18 at 15:06
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    Are you a mathematician? While this somewhat ruthless mindset may be common or necessary in some fields, I don't find it particularly common in mathematics. Most topics attract a small number of researchers. Cooperation and (potentially) multiple proofs of a result are more highly valued than successfully posting "first!" in my experience. – zibadawa timmy Feb 21 '18 at 15:08
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    In math, you can publish your proof a few months after someone else, and (assuming it's sufficiently different and interesting) it will be generally be considered that you both proved the theorem "around the same time". There's a limit to that, of course, though I wouldn't be able to determine a precise threshold. – user9646 Feb 21 '18 at 15:20
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    I agree with both zibadawa timmy and Najib Idrissi: Being first isn't crucial. Actually, if you're first, yet your competitor writes a better paper, then your work will be forgotten, your competitor's remembered. – user2768 Feb 21 '18 at 15:38
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    @DanRomik The worry isn't what I was referring to. It's the "tip your hand and he'll bite it; so don't do that" bit. I don't know very many mathematicians that behave that way, making an attitude of "it's every man for himself, trust no one" rather needlessly...ruthless. Also note this post has been edited like 8 times since I first made that comment. – zibadawa timmy Feb 21 '18 at 17:38
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The situation you describe makes little sense.

First, "establishing priority" means proving to the world at large that you were first. Mailing something to your rival proves nothing to anybody except them.

Second, you claim that proving this conjecture is "an important result" but that it's not enough for a paper on its own. That's a fairly direct contradiction. Further, the fact that it can be proven in two pages and that two different people have independently done or almost done this at the same time suggests that it really isn't that big a deal.

If you're really that bothered about it, put a note containing your proof on ArXiv and then flesh it out into the full paper. But, really, it sounds like you should just write your paper, upload it to ArXiv and submit it as normal. If it's as important as you say it is, it's unlikely that there'll be an issue with journals accepting more than one paper on the subject, especially if the proof techniques are different. (And if you've both independently come up with the same short proof, that really does suggest that the conjecture is in the "obvious, really, when you think about it a bit" category of not very important conjectures.)

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    The sentence starting "Further..." in the second paragraph seems reasonable, but note the following counterexample: arxiv.org/abs/1605.09223 (read the comments), which later appeared as annals.math.princeton.edu/2017/185-1/p08 – Spiny Feb 22 '18 at 14:18
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    @Spiny Sure, and I'm sure there are plenty of other examples. Hence "suggests" rather than "conclusively proves". – David Richerby Feb 22 '18 at 15:04
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    Yes, I wasn't trying to contradict you; just pointing out a curious and interesting (to me!) occurrence which seems relevant. – Spiny Feb 22 '18 at 15:33
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I believe such an email would be inappropriate, impolite, rude.

IMO, you shouldn't try to force him to stop working on the conjecture this way, because :

  • You don't know his reaction, he might take it very badly.
  • You don't know how far he his with his work, he might as well be finished and about to publish ?
  • You should see the potential positive outcome in this rivalry instead of the danger. Could he enrich with a second alternate proof ? If he proves it in the same way, doesn't it give credit to it ?
  • In the other way around, how would you feel if he would email you a preprint ?

For those reasons, I wouldn't write an email at all and follow @pete-l-clark advise.

However, if you're keen to write to him, here are the ideas I would put in such an email :

I heard you work on this conjecture as well.

I wanted to let you know that I found the proof of the conjecture, and that I'm about to publish.

I would appreciate if you could share your progress with me as well.

OR

Later on - once a publication has been made -, a discussion about our work would be interesting.

(Though, note that one could also take this kind of email as an attempt to put pressure on the receiver, but that shouldn't be your intention IMO.)

  • Sending a relevant preprint to a colleague is "inappropriate, impolite, rude"? really? – Worried being scooped Feb 22 '18 at 7:53
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    @Worriedbeingscooped because your question makes it very clear that you don't actually want to "send a relevant preprint to a colleague", you just want to mark your territory and prevent them publishing before you. Sending somebody a preprint because you believe they would be interested in your work is not rude. Sending somebody a preprint with the intention of sabotaging their work for your own personal benefit is a very different matter. – Chris H Feb 22 '18 at 8:38
  • Regarding the intention, even if I put it on the arxiv, I would send him a notification, because it is a work he should see. – Worried being scooped Feb 22 '18 at 8:39
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    @Worriedbeingscooped I would find it very impolite if I would receive such an email. To answer your question directly, I don't find it ethical. – Mago Feb 22 '18 at 10:40
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    @einpoklum Yes, those are just raw ideas, formulation and wordings should be changed. Also, as I wrote, in this situation I wouldn't actually send the email. – Mago Feb 22 '18 at 10:42
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The simple solution is to put the science as a higher priority than recognition and then establish an agreement to work together with both names on the paper. A "failure to work with others" is probably one of the most common reasons why we have yet to cure cancer, send people to Mars, have flying cars (could you imagine people cutting each other off while traveling an air-highway?), etc.

  • I was about to post essentially the same answer. I recommend you look up the history of the HOMFLY polynomial:" Its name is an acronym for the last names of its co-discoverers: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter." – Ethan Bolker Feb 23 '18 at 22:09
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One possibility is to publish the proven conjecture to the arXiv by itself to establish priority. Leave the fleshing out of details and answering of the other questions to the paper you'll be submitting to a journal.

My paper on this is however not ready to be put on the arxiv

I don't know if there is such a thing. Okay, papers consisting of only a few words are not ready for the arXiv, but that's not what you have here. Since you presumably will be publishing your result in a journal, the completeness of what you put on the arXiv is not of paramount importance. Since the arXiv is not peer reviewed and often has things put on it such as short essays commenting on practices in the field and joke/April Fools paper (at least in my field, astronomy), a short paper focused just on one result not fully fleshed out would fit in quite naturally.

Another possible alternative is something that just started in my field: American Astronomical Society Research Notes. These are intended to be very short summaries with very fast turnaround times (I believe they are not peer reviewed) of summaries of works in progress, results that need to get out quickly (e.g., observation of a transient phenomenon), or negative results that aren't worth writing a full-fledged paper for. I don't know if mathematics has something similar.

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    The similar thing in mathematics is the ArXiv. There's never any point-your-telescope-this-way-quick!! news in mathematics, and the rest is exactly what ArXiv is good at. – David Richerby Feb 23 '18 at 9:50

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