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Perhaps this is a biased forum for this "downer" question, but I've never figured this out. At what stage is it safe to share mathematical ideas? Many have told me that there is serious danger of ideas being stolen if shared in a premature form, but I've found that sharing ideas often moves things along much more quickly than holding them back.

This is definitely not the best question for MO, but I think some expert advice on the topic may be helpful to young mathematicians.

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This assumes that ideas can be owned and stolen, rather than, say, borrowed. – Thierry Zell Oct 27 '10 at 4:00
...or built/expanded/improved upon, for that matter. – J. M. Oct 27 '10 at 4:07
Well, one reason I might have for not sharing my ideas is that so many of them are so silly... – Minhyong Kim Oct 27 '10 at 4:13
shrug I share pretty much all my ideas, silly or not, with anyone who asks. In fact this is one of the reasons I've never worked for GCHQ (UK government crypto people---analogue of NSA)---I can't face the idea of having to work on a secret problem that I then can't talk about with other people. Sharing is exactly what supervising PhD students is about. – Kevin Buzzard Oct 27 '10 at 10:51
Come to think of it, I do like listening to other people's ideas, like Kevin's. – Minhyong Kim Oct 27 '10 at 15:40

Well, a personal anecdote, I worked with a famous guy for some years. His basic strategy was to mail, on paper, anything publishable in a draft to one or two dozen parties that might be presumed to be interested. If nobody replied inside a month he submitted it somewhere. The point, in my mind, was that if one other person sees your stuff early you may get robbed, but if 20 see it early they are all witnesses. Later they came up with the arXiv.

The other, well-known side, is that if you share your stuff with the top expert in the field, that person may send you back a note saying "that was fun, here is the answer" and promptly forget all about it. You have not been cheated but there is still a problem.

EDIT There seem to be mixed impressions of what I meant in the preceding paragraph, and for whom the situation would remain a problem, so maybe I had better describe my own experience again. I have told this story many times, with names, and I think the story only does people credit, but I think on MO I ought to stick to anonymity. Email me if you want more detail. In graduate school I was working on minimal submanifolds. My adviser came up with a fairly specific problem, suggested I work on it, and asked one or two guys in the same department if they thought it was new, which they did. It still took me some time but I was getting there. My adviser was away somewhere giving a talk, and, once again, mentioned the problem to a guy. The difference was that this guy is a leading light in similar problems, went home, solved my dissertation problem in one evening on some 30 pages of notes, and put those in a drawer and forgot all about it. But at some point he happened to mention to my adviser that it was a good problem, he had completely solved it. My adviser mentioned this to me, and I was terrified. How could I submit this as a dissertation if this other guy solved it already? At some point I contacted him, he said, don't worry, it's your problem, I don't need it, you just finish it up and it's your dissertation. Finally, after I finished, I did ask to see his notes, he found them eventually and sent me copies, but even between him and his adviser at the 1992 Park City summer program no sense could be made of the notes by anyone concerned.

So I suppose I would say, along with Kevin's comment, that the nature and severity of the "problem" when the world champion in your area solves your problem in an afternoon (but has not the slightest intent to publish, ever) depends on your position and how much you need this as a publication/dissertation and how critical it may be that the work be perceived as your own and original. I may have misunderstood my position in graduate school, and everybody behaved well in my opinion, but it was certainly scary. I think I do see Kevin's point that, as a journal referee, he is often confronted by work that is already known, "in the air" as they say, or where the most likely techniques are pretty obvious as soon as the statement of the theorem is read, but he will still accept it for many journals.

I think it is fair to say people pick and choose what of their stuff to put on MO. This is probably healthy. We should struggle rather than getting handed everything.

Given that this question is Country and Western, I am taking this opportunity to point out that I went to high school with Paul Ginsparg, founder of the arXiv. He was a year older. It is a good bet that he is still a year older. Also Natalie Portman.

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Oooh. Do people really sap off expert answers like that? That's slimy and must feel very unpleasant! I was under the impression that we are not inclined to share ideas because it is natural to want to solve things ourselves... – Jon Bannon Oct 27 '10 at 10:19
@Will: I don't see the problem in your second para. I see papers on the arxiv, and published papers, where the main theorem is either something that I knew already, or something that I didn't know but the moment I see the statement of the theorem I think "this technique probably works" and then read the proof and see that the author used this technique. If I am sent such a paper to referee, I reject it if it's a strong journal but might easily recommend publication if it is not. Where's the problem with that? – Kevin Buzzard Oct 27 '10 at 10:43
Well, the problem is that one may not send it at all, after having received a solution from another person... – Andrea Ferretti Oct 27 '10 at 14:04
Andrea, yes, that's about the size of it. I described such an episode in two extra paragraphs in an edit, the "paper" was my dissertation, things could have worked out badly, at least that is how it seemed to me at the time. – Will Jagy Oct 28 '10 at 16:50

While I feel lame writing "actually, you should read about this on my blog"...

Actually, you should read about this on my blog. There's quite a serious an interesting discussion in the comments section.

I would say the one sentence summation is "it's never safe to share your ideas, but it's never safe to not share them either." This simply isn't a question with a right or wrong answer; I think it's better to err on the side of openness, but it is not without pitfalls.

One point I would add to the discussion there is just how darn hard it is to steal ideas; it's hard enough to figure out what's going on in a paper when it's written up. Generally, the person who came up with an idea has enough of a head start that it would be more work than it was worth to try to steal their ideas.

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This is the kind of thing I think it is good for young people to hard it is to steal ideas! I think the payoff from hearing people's perspective on things, and finding out if you're being dumb about something early on outweighs some other evils. – Jon Bannon Oct 27 '10 at 10:16
I agree with Ben. In fact, the other side of the coin is equally valid. Making your ideas clear enough to share them is hard! Perhaps I should say ' making them clear enough to be stolen' is hard! – Tim Porter Oct 27 '10 at 15:56
A paper can be stolen. An idea would be more difficult, because the idea does not come in isolation but is connected to the mathematician's experience, strengths and vision. If I have a good idea, and someone takes it, runs with it to achieve much more than I could have, was it really my idea to begin with? (This is also an argument against the worst excesses of patents, where one sees vague concepts being patented instead of actual processes.) – Thierry Zell Oct 27 '10 at 19:06
Ideas can't be stolen, but credit can (I think one of the good points of the discussion linked above is that this can happen even if you do publish in a timely manner; people get confused easily about who did what first), which is really what people are talking about here. You can build on and repurpose other peoples' ideas in a way which is ethical and gives them fair credit, or you can do that in ways that do not. I think most people are striving for the former, but it can be a tough line to walk. – Ben Webster Oct 27 '10 at 20:39

My advisor gave me advice which I have found very useful (and which is largely in line with Ben Webster's response):

"By talking to other mathematicians about your work, you will generally gain far, far more than you will lose."

He saw isolation as one of the biggest risks for a young mathematician. I have followed his advice, and nearly all of my papers have benefitted from the suggestions and insights of other mathematicians.

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While I broadly agree, the devil is in the details; specifically, which details you share with whom. – Yemon Choi Oct 28 '10 at 16:23

I can think of one situation where sharing your ideas can be dangerous, and it is when you have formulated and proved a conjecture that is hard to synthesize but easy to prove. If you haven't written and submitted a paper, sharing the idea may make you vulnerable to people who write quickly. This happened to a colleague of mine - the other person's paper only acknowledged that she had formulated the conjecture, but not that she had proved it.

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Did this story have a happy ending? – J. M. Oct 27 '10 at 13:47
Though I am generally in favor of speaking openly about ideas, I agree that this is a case where sharing an idea can be risky. I know of at least one similar story, and it was partially caused by miscommunication about the state of the conjecture. This story had a happy ending, since everyone involved was friendly and honest about it, but it could have easily turned into a bigger problem. – Daniel Erman Oct 27 '10 at 14:46
Regarding the ending: My information is one or two years old, so things may have happened since then. The last thing I heard is that the problem is unresolved, she does not have tenure (contra the other party) and therefore feels uncomfortable raising the issue in a professional setting, and she no longer talks to said professor about her work. – S. Carnahan Oct 27 '10 at 15:05
It seems to me that a small part of the bad behavior here is on the part of the editor who managed that paper. I have several times had editors send me papers which improved on or corrected my ideas, with the editor asking me to confirm that my work was fairly represented. Is this not standard protocol? – David Speyer Oct 27 '10 at 18:01
That's never happened to me.... – Ben Webster Oct 27 '10 at 21:07

One thing that does not seem to have been emphasized in the other answers so far is that it matters quite a lot who you share your ideas with. There are some people that should be avoided.

In graduate school I solved a problem that had been circulating in the department for a while. Before I had had a chance to write it up formally, someone (more senior than myself) who had tried unsuccessfully to solve it asked me to explain my solution to him. As I started to explain my solution, he would repeatedly interrupt and say, "Oh I see now...this is what you do." Then he would start talking and writing on the blackboard. When he got stuck (because he didn't really see how to solve it), he would then stop and invite me to continue. The scenario would repeat. He would also criticize my presentation. The whole time, he acted as if he didn't believe that I had solved the problem, and seemed to be trying to get me to give him my ideas while behaving in a way that would allow him to say afterwards that I had had some good ideas but had not really solved the problem and that he was the one who had really solved it. I don't know if he was doing this intentionally or whether this was just his personality, but needless to say, it was a very unpleasant experience and I avoided sharing my ideas with him from then on.

While in general I am a fan of sharing ideas, I think one should be aware that there are people out there who (consciously or unconsciously) do steal ideas. They are rare (the above person is the only one that I explicitly try to avoid sharing ideas with) but they do exist.

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Your example may be a little extreme but I think on a smaller scale this happens all the time. It is really hard to be objective about why you understood something. Once ABC mentions to you that XYZ is true, then you first don't understand it, then you realize that of course XYZ is true. Then you write up a proof of why XYZ is true, as a Lemma in your paper, of course with your own reasoning; and by the time you post the paper you have completely forgotten that XYZ would never have occurred to you if ABC hadn't told you about it. – Arend Bayer Oct 29 '10 at 16:22
@AByer: What you say is true, but what I think doesn't happen all the time is to ask someone who has announced the proof of a precise conjecture for a private preview, and then aggressively and repeatedly interrupt the presentation in order to cast doubt at every opportunity. – Timothy Chow Oct 29 '10 at 22:00
Never seen first-hand, but I've heard from at least three junior people being offered to discuss their work with a more senior person, and then felt pressured into giving the senior person co-author credit. It's not pretty, didn't work, and causes a lot of unpleasantness for all involved. But I cannot imagine that one can take this kind of strategy very far: word gets around. – Thierry Zell Nov 5 '10 at 4:32

Thus far, the responses about the negative aspects of sharing only discuss idea "theft." This would seem to apply mostly when the ideas being shared are already publishable, near publishable, or at least very likely to lead to something publishable. The biggest danger I have ever experienced with sharing occurs earlier in the process and has nothing to do with theft. It is to enter the following pattern:

(1) I share an idea or problem with somebody before I have really given myself time to marinate in it. (2) The other person introduces their own ideas--- and is so persuasive, or detailed, or optimistic about their own views, that I get caught up in them. (3) I lose the thread of my original idea and/or spend a long time fruitlessly trying to attack my own problem with somebody else's way of thinking.

I'm not saying that my way is always better or leads to the right thing, and that the only thing stopping me from publishing great results is distractions from lesser minds. I am saying that sometimes you should not share your ideas until you are either well and truly stuck, or far enough along that you could give a coherent (if detail-free) seminar talk about them. You need to know enough about your own ideas that you can put them down for a minute and compare them with someone else's without information loss. When this is, exactly, changes from person to person and problem to problem. But if you share too soon, you run a risk of spending a lot of mathematician-hours checking the details of somebody else's half-baked idea before you have even checked your own. I think grad students are generally more in danger of this than they are of idea theft. (It is easy to unconsciously take an established person's educated guess for "expert wisdom", even when it is not intended as such.)

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Always, the knowledge should be shared. The individual credit is not important, the important thing is that the mathematics grows up, independently who did a particular contribution.

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While this is certainly the "morally right" idea, in practice it is important to get credit for work, perhaps even more so in the beginning of one's career. In particular, it cannot be denied that nowadays people are being judged by the number of publications and citations. Even though I guess most mathematicians are quite unhappy about this, it would be naive to have this idealistic view of science. – Pieter Naaijkens Oct 28 '10 at 10:53
My comment should not be interpreted as an advice against sharing, by the way. – Pieter Naaijkens Oct 28 '10 at 10:54
If it's about getting math done, I can't theoretically...why not sharing helps!! – Jon Bannon Oct 28 '10 at 11:23
@Jon, If it were strictly about "getting math done" then we could simply let the automated theorem provers loose -- a computer should be able to produce a far greater volume of new mathematics than all the mathematicians who have ever been alive just by repeated applications of logical rules starting from the axioms. – JBL Oct 28 '10 at 13:19
@Jon, if it were just about getting math done then I might as well apply to be an amanuensis for someone brilliant. I do find this slightly Panglossian attitude a bit odd. I also think there is a risk of encouraging sharing at the expense of actually seeing one's ideas through to proofs (my usual cry of Stone Soup applies here, and I disagree with Wikipedia and others who think the moral is "co-operation" is good, for me the moral is called "cadging a free-meal") – Yemon Choi Oct 28 '10 at 16:21

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