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Followup: working in secret July 19, 2008

Posted by Ben Webster in Uncategorized.
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I got a bit behind on responding to comments to my post on Li’s preprint, so I thought I would just start a new thread.

Now, I don’t want to concentrate too much on the particular’s of Li’s case, since I don’t know Li or too many of the specifics.  I’ll just clarify that I don’t think Li is a crackpot, or did I say in my past post that I though that.  But that’s exactly why I feel like releasing his preprint the way he did was a mistake.  Certainly, there’s a bit of my taste for facetious exaggeration (I’m sure my readers have noticed this tendency in the past) in describing the unfortunate aspects of that as “crackpotesque,” but I think that also helps convey what about it I thought was a bad decision.

I’m more interested in expanding on some of the comments in that thread.  Gil Kalai said:

There are two extreme ways to practice math (with many altenatives in between.) One way is to work secretly on a big problem, to tell nobody or very few people about it, to discuss with nobody the techniques you are using, and then after many years to astonish the world with a preprint or a lecture) presenting the solution. The other extreme way is to work while at any time discussing your thoughts and ideas with everbody (perhaps also on blogs), write papers with partial progress and conjectures etc.

The advantage of the first avenue is not just the fear that somebody will use your ideas but also that it helps the researcher to stay concentrated, and avoid outside preasure and distractions of various types. A clear disadvantage of the first avenue is that feedbacks from others can be useful at intermediate stages of the process towards a mathematical discovery.

I’m curious: does anyone out there think that Gil’s “first avenue” sounds like a good idea?  It sounds crazy to me.  Maybe I lack the self-confidence to think I would succeed at it (not something I’m regularly accused of), but it seems like asking for trouble, both in terms of actually getting the math done and in terms of one’s career.  Obviously, there are dangers in revealing your ideas and results to other people.  I think outright theft is relatively rare, but someone “eating your lunch,” implementing something you had hoped to do before you have a chance, is a very serious concern.

But I think people’s cognitive biases cause them to be too sensitive to this possibility, while forgetting about the upside, because the danger of having one’s work stolen is so obvious and painful, and the dangers of secrecy are much less obvious.  It’s important to remember though, a co-authored paper which actually happens is much better than a solo one which never does, or even which happens a few years down the road (at least for those of us worried about jobs).  Not to mention the very real possibility that people will independently come up with the results you wanted.  As Greg said:

What is true is that you’re much more likely to lose credit by being secretive than by being open.

I find it interesting that Gil mentions that

The first avenue, had spectacular successes in the last few decades…

I assume he’s referring to Wiles’s proof of Fermat and Perelman’s proof of Poincare (are there other examples I’m missing?), which were certainly both spectacular, but they were somewhat qualified as successes.  I mean, Wiles’ worked mostly in secret for years, announced a false proof, and then fixed it after getting input from other people.  It’s likely that he would have discovered his mistake much earlier if he had been talking to a larger circle of people; one could argue that the level of secrecy he maintained might well have cost him the Fields Medal, which he would have been a shoo-in for if his initial proof had been correct (he had turned 40 by the time he fixed it).  [EDIT: I seem to have just been wrong about this.  I read one of my sources as suggesting he was eligible for the 1994 Prize, but the Wikipedia page seems to say no.]

In the case of Perelman, well, what can one say?  One could argue that his behavior has seriously compromised his standing in the mathematical community, but I suppose he doesn’t really care.  Certainly his work would have been a lot more digestable if he had consulted with people more, and seems to be sufficiently complicated that it would have been awfully hard for someone to absorb quickly enough to have scooped him.

Not to mention that this style is much less suited to what most mathematicians do than, say, Wiles’s situation.  When it’s very important for you to prove a particular result which many other people would like to prove, secrecy makes a certain amount of sense, but when you want to establish a reputation as an effective and interesting researcher, having other people build on your research is the best thing that could happen (even if they’re solving problems you had hoped to do yourself), given that it happens pretty rarely.  You’ll be a much better mathematician if you think of people interested in the same problems as potential coauthors rather than potential rivals.

As both Greg and Terry mentioned, having coauthors is a very good thing on a lot of levels.  Obviously, they can stress you out from time to time, but as a general rule, they will lead to you doing better and more mathematics, and writing better and more papers.  On some level, the most important thing to remember is that like trade, talking to other people about mathematics is very much a positive sum game, since the other person is likely to have some small piece of knowledge that you are lacking, or to bring some skill or temperment to a collaboration.  I’m by nature a very impetuous, big-picture-oriented research, and having someone who focuses more on details and makes me come back to earth is a large boon. There’s some risk that like the gains from trade, gains from mathematical discussion will be inequitably distributed, but they’re still indisputably there.

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Comments

1. Alonzo Moseley - July 19, 2008

While it may be better for most people to work with collaborators, I think a certain type of person can’t function in this traditional way. A lot of the more outgoing mathematical leaders are also highly competitive, and a sensitive/loner type might not be able to handle this cutthroat environment. (Yes, I know people like to deny that the math world is like that.)

In Perelman’s case, his first introduction to the university level math world was being accepted under a strict anti-Semitic quota. And he had to show up and see these people every day. Maybe some people liked to justify the policy. One would imagine there was all sorts of unpleasantness unrelated to this issue as well; it’s hard to believe math departments are going to be nasty in only one way. Who knows what else he had to put up with as he progressed further along in his career.

Anyhow, I imagine a sensitive/loner type might have a lot of trouble handling the real world of math research at the top levels. Suppose Perelman discussed his early efforts on Poincare. Can you imagine the feeding frenzy that would have occurred once people realized he was on to something? It would have been intolerable for someone like him. Not to mention the issue of someone beating him to the punch and then deciding that Perelman didn’t really know how to get to the end, and besides Hamilton developed the whole program anyhow, and so…

I think one should not be critical of people with unusual psychological makeups, and just be glad that he was able to accomplish what he did, and was willing to show the world what he had done. I’m sure he was doing the best he could under very difficult personal circumstances.

2. Nathan Dunfield - July 19, 2008

While it may be better for most people to work with collaborators, I think a certain type of person can’t function in this traditional way.

A minor point, but historically the “traditional way” was for mathematicians to work alone, if perhaps not in secret; collaboration has increased markedly in past 30 years, perhaps as a consequence of things like email. There was a Notices article a while back that documented the increasing proportion of mulit-author papers in MathSciNet, for instance.

3. Jason Starr - July 19, 2008

I think where one’s style falls between your two extremes very much depends on the type of mathematician. Most of us (myself included, obviously) are elaborating on programs developed by others and occasionally pushing things forward with our own insights. For this type of mathematics, openness and collaboration are crucial.

But there are mathematicians who advance not just a program, but entire fields with mind-blowing new ideas. I think for these people collaboration is more about “team-building” than about anything else. While I completely agree with your attitude for the former mathematicians, the latter (much rarer) mathematicians should obviously do mathematics however they like. After all, they are the ones who do most of the important mathematics. Also, based on my few experiences with these latter mathematicians, I doubt they particularly care how the rest of us view their decisions about openness or secrecy.

4. Emmanuel Kowalski - July 19, 2008

I think one should be careful to take into account the (significant) difference between working “non-secretely” in the sense of talking to colleagues about what one is doing, and actually collaborating for a joint work. Many mathematicians may do a lot of the first without having many joint papers (for instance, Lafforgue’s work).

As for other successes of the lonely researcher school, I’m sure there are many, if one looks a bit carefully. I’ve been told that my colleague Demetrios Christodoulou did not really tell many people about his recent work, before putting it on arXiv, and it seems to be an extremely remarkable, 594 pages long, piece of work (if I understand right, it shows that general relativity permits the creation of black holes even in the absence of matter; unfortunately I don’t know much about the subject…)

5. Scott Carnahan - July 19, 2008

In the case of Perelman, well, what can one say? One could argue that his behavior has seriously compromised his standing in the mathematical community, but I suppose he doesn’t really care.

I think the opposite is true. There are certain members of the mathematical community whose behavior has turned him away from mathematics. Given the way his results were treated after he had written them up, I think he was justified in not publicizing his techniques until he had finished writing them up to his own satisfaction. If you work in a field that has predatory mathematicians, you can either change fields, or employ some secrecy.

6. anon - July 19, 2008

(are there other examples I’m missing?)

If my general understanding is correct, Taubes’ solution to the (3-dimensional) Weinstein conjecture might tend to fit in Gil’s first category. Interestingly, in this case I think people’s general expectation had been that the conjecture was going to be cracked by symplectic field theory, which had a sizable number of people talking to each other a lot and writing lots of joint papers about it with partial results and so on. So in some sense there was a competition between (moderate versions of) the first and second avenues here. And the first won. Of course, you could argue that this had more to do with Taubes’ methods just being more suitable for this particular question than SFT, and it’s certainly true that SFT has led to plenty of other interesting results.

If you’re a young postdoc looking to maximize your publication count and to make sure as many people as possible know your name, then it’s clear which of these avenues is most appropriate for you. But with at least some more senior people, it is simply the case that they know more about certain topics (the analysis of the Seiberg-Witten equations in Taubes’ case) than anyone else in the world (well, perhaps except for two people whom he could meet for lunch whenever he wanted), and it sometimes pays off greatly for them to slowly mine that expertise as extensively as they can.

7. Vishal Lama - July 19, 2008

In the case of Perelman, well, what can one say? One could argue that his behavior has seriously compromised his standing in the mathematical community, but I suppose he doesn’t really care.

Well, isn’t it true that a bunch of highly-gifted mathematicians (we all know who) actually tried to publish their version of Perelman’s proof giving almost no credit to Perelman and that they soon withdrew their paper only after there was an “outcry” (for very obvious reasons) over it? To put it bluntly, it was a case of “poaching” of ideas and personally I feel that is much worse than some mathematician, such as Li, submitting a false proof and later retracting it! Anyway, it is hard to see how Perelman compromised his standing in the mathematical community given that the latter actually failed him and not the other way around.

8. Anonymous - July 19, 2008


Given the way his results were treated after he had written them up, I think he was justified in not publicizing his techniques until he had finished writing them up to his own satisfaction.

I fully agree with what you’ve written, but I think this is a different issue. Perelman was treated poorly by the Yau school, but he also behaved poorly himself after he announced his proof. The difficulty is that he wrote up his results to his own satisfaction but to nobody else’s. In particular, he wrote just barely enough to convince everyone (eventually) that he had a full proof in his head, while providing minimal explanation and guidance for the rest of the community. In the process, he wasted several person-years of effort to reconstruct what he already knew and didn’t want to explain.

Part of the honor and respect given to solvers of famous problems is a recognition of their contribution to the mathematical community. When someone like Perelman contributes as little as he can get away with while still getting credit for the result, it’s no wonder some people become angry with him.

For example, I’m not sure he should have been offered a Fields medal. It’s tantamount to saying that if your work is sufficiently brilliant, then you’re excused from fully documenting or explaining it. This doesn’t seem like a good message for the IMU to be endorsing. It’s certainly true as a practical matter that you can get away with a lot more if many people are eager to decode your manuscript, but I feel that the community should resist this phenomenon rather than give in to it.

9. Will Orrick - July 19, 2008

Wiles has plainly stated that his decision to work secretly was a deliberate one, it’s not so clear in the case of Perelman. Here’s a quote from Nasser’s article:

‘In 1995, Hamilton published a paper in which he discussed a few of his ideas for completing a proof of the Poincaré. Reading the paper, Perelman realized that Hamilton had made no progress on overcoming his obstacles—the necks and the cigars. “I hadn’t seen any evidence of progress after early 1992,” Perelman told us. “Maybe he got stuck even earlier.” However, Perelman thought he saw a way around the impasse. In 1996, he wrote Hamilton a long letter outlining his notion, in the hope of collaborating. “He did not answer,” Perelman said. “So I decided to work alone.”’

A second quote from the same article, regarding possible concerns that the posted proof might turn out to be wrong: ‘“My reasoning was: if I made an error and someone used my work to construct a correct proof I would be pleased,” he said. “I never set out to be the sole solver of the Poincaré.”’

10. Emmanuel Kowalski - July 19, 2008

Once time is taken into account, Perelman’s attitude has actually probably benefited mathematics (and in particular his field). It is true that, for a few years, the status of the proof was very unclear, for non-experts in particular. But now, we have two complete book-length treatments, starting from a graduate student level to reach a full proof of the Poincare conjecture (not to mention T. Tao’s rather amazing run-through on his blog). Because of this and various summer schools, etc, there are certainly orders of magnitude more young mathematicians who understand the proof than would have been the case if his original papers had been forbidding but acceptable Annals fare, readable only by a select few, and no particular incentive for anyone else to immediately rewrite the whole in book form.

Contrast with Wiles’s proof, which was published fairly quickly once he found the way to correct the mistake in the first version, and which was confirmed almost immediately by the most respected experts: there is still no really easily accessible way to learn about the proof, and probably less people who understand the details than for Perelman (to be fair, one may probably argue that, in terms of the amount of background needed, Fermat’s Great Theorem is indeed less accessible than Perelman’s work and not readily reducible to a single book).

Also there are very interesting remarks along similar lines in Thurston’s paper on proof and progress in mathematics, in particular Section 6 about his experiences with working first on foliations, then on hyperbolic 3-manifolds, and the different fortune of the two fields and the community around them after his work.

11. Anonymous - July 19, 2008

Once time is taken into account, Perelman’s attitude has actually probably benefited mathematics (and in particular his field).

I strongly disagree that Perelman’s approach was a net benefit to mathematics. I’d bet that all the summer schools, Terry’s blog posts, and at least one of the two book-length treatments would have occurred even if Perelman had provided more details. However, they would have been even better, since they could have taken into account more of Perelman’s insights, and less time would have been wasted.

I agree that the comparison with FLT is tricky, because of the massive amount of background required for Wiles’s proof. There certainly were attempts to make it as accessible as possible. (For example, there was a fantastic instructional conference at BU in 1995, which led to a volume of survey papers that outline the proof and explain various parts in more detail.) However, if you want a complete understanding, with no black boxes, then it really requires a fairly massive amount of background.

Thurston’s comments are pretty interesting, but the situation feels fundamentally different. Thurston spent a lot of time explaining foundational material and ideas, with the goal of increasing the community’s understanding of his techniques. By contrast, after posting sketchy technical preprints on the arXiv, Perelman spent a negligible amount of time lecturing and answering questions, and none at all in producing a better write-up. He doesn’t have the excuse that he was engaged in a Thurston-style community education project.

The fact that, nevertheless, the community managed to understand Perelman’s proof isn’t because he had a wise strategy to encourage that outcome, but rather because the result is fantastic and there was bound to be intense interest in it. If he had actively participated in the process, we would have understood more sooner.

So I wish Perelman had been more like Thurston, and at the very least he ought to have been like Wiles.

12. Alonzo Moseley - July 19, 2008

I think that one cannot separate Perelman’s working alone with his decision to withdraw from the math world and go home to Russia. He made this as a personal, philosophical decision. If he had chosen a different life philosophy, that would have meant joining a math department and caring at least a small amount about things like his reputation, getting tenure at a good place, and so on. I don’t believe he would have succeeded in solving the geometrization conjecture in this fashion, even if he was very idealistic for a university mathematician. The risk is simply too great for a young person, to work on something of this magnitude. Not to mention the distractions from the rest of the job, dealings with “competitive” colleagues, and so on.

Look at analytic number theory, another field with a large conjecture looming. Many of the top young minds do work on the Riemann hypothesis. They also produce many papers, advise students, go to conferences, and presumably, spend a lot of time comparing themselves to their rivals and worrying about beating their opponents to the next result. I think any normal human being will get corrupted by this, even an idealistic genius like Perelman. At the very least it will interfere with his concentration.

People say, well, why didn’t he at least write it up nicely afterwards for public consumption? My personal opinion is that he is ensconced in a certain monastic mentality, and would not be able to suddenly become “normal” for a mathematician and do things like that. I really think he wouldn’t be able to force himself to write a Tian/Morgan style book. Think about his lifestyle, his philosophy.

13. Emmanuel Kowalski - July 19, 2008

By the way, I would also agree that it would be better if Fields medals were given to mathematicians who are not only brilliant but also do the work of writing things up to the right standards of exposition. But I don’t think that Perelman is such a bad example here, since after all, it seems that he had said beforehand that he would refuse the medal (although for other reasons maybe), and the IMU is responsible for still going through the offer.

14. Vishal Lama - July 19, 2008

By the way, does anyone doubt that Hollywood will make a movie on Perelman, say, in 10 years? I would guess that Sylvia Nasser would first write a book on Perelman (just as she wrote one on Nash), and then follow it up with a movie that would of course be full of drama. We already have all the ingredients necessary for such a movie!

15. Will Orrick - July 19, 2008

one could argue that the level of secrecy he maintained might well have cost him the Fields Medal, which he would have been a shoo-in for if his initial proof had been correct (he had turned 40 by the time he fixed it).

Wiles was out of luck anyway, since he turned 40 in 1993, the year in which he announced his first proof, and the year prior to the ICM. To be eligible for the 1994 Fields Medal, he would have to have turned 40 on or after 1 January 1994.

16. Will Orrick - July 19, 2008

…of course, mathematicians may still exist who would regard proving Fermat’s Last Theorem as adequate recompense for missing out on the Fields Medal.

17. Alonzo Moseley - July 19, 2008

Vishal:

And of course, if the Nash movie is any indication, there will be various embellishments to the story too… The young enthusiastic Grisha Perelman, in love with the beautiful Svetlana… suddenly finds his world turned upside-down. His collaborator, who he thought he could trust.. betrays him and runs off with Svetlana! Grisha, despondent, decides to withdraw from the world… and do the one thing that can keep him from thinking about Svetlana, his collaborator, and this shocking betrayal..

Camera zooms in on Grisha..

I vill not stop verking on zis… until I hev solved… Poincare!!!

Coming soon to a theater near you!!

18. Anonymous - July 19, 2008

…of course, mathematicians may still exist who would regard proving Fermat’s Last Theorem as adequate recompense for missing out on the Fields Medal.

Exactly. I actually wish Wiles had been eligible for a Fields medal, not because he needed one, but rather because his fame would have brought honor to the medal. It’s the same situation as Serre and the Abel prize: no well-informed person’s opinion of Serre went up because he got the prize, but lots of people’s opinions of the Abel prize went up.

I think any normal human being will get corrupted by this, even an idealistic genius like Perelman.

I’m not sure I’d characterize it as corruption, or even distraction. The monastic ideal suggests that thoughts produced in isolation are purer and more valuable than anything that arises through interaction with others. That may be true for some people, like Perelman, but I don’t think it’s true in general. Stimulation, inspiration, and feedback are really important for most people. You could even view this as one of Perelman’s weaknesses: he’s awfully smart, but perhaps he lacks the ability to interact successfully with other researchers. Who knows how much more he could have accomplished if he had that ability? I don’t mean to demean him and his amazing accomplishment, but I also don’t think we need to buy in to his philosophy of research. It may work well for him, but I don’t think it’s a superior philosophy that everybody could profitably apply, if only we shared his noble lack of concern for status and rewards.

19. Alonzo Moseley - July 19, 2008

I didn’t mean to suggest this would work for everyone, or even many people. I do think though this was the best for someone with his unusual personality. And while more interaction in general makes someone a better mathematician, I do believe that solving the geometrization conjecture using the Ricci flow, something that took Perelman several years of concentrated effort without distractions, would not have been achievable in a more traditional academic path.

20. Anonymous - July 19, 2008

And while more interaction in general makes someone a better mathematician, I do believe that solving the geometrization conjecture using the Ricci flow, something that took Perelman several years of concentrated effort without distractions, would not have been achievable in a more traditional academic path.

It’s difficult to say. Wiles shows the opposite phenomenon: he successfully followed a traditional academic career path, and then he used the freedom of tenure to focus for seven years on a difficult problem. Overall, there’s certainly pressure pre-tenure, but post-tenure one is probably in a better position to focus than in just about any other job. The one way to do even better is not to care about having a job at all, which requires being either rich or eccentric.

Speaking of being rich, are there any notable mathematicians who don’t hold any job because they are independently wealthy? I know of a number of mathematicians who are rich enough that they don’t need to work, but they either work anyway or no longer publish papers. Centuries ago, there were plenty of examples, but I can’t think of any today.

21. Hany - July 19, 2008

I agree with Alonzo; every single mind has its peculiarities, especially those who solve difficult and complicated problems. If Perelman was a different person, he might not have succeeded in proving the geometrization conjecture.
I do not actually understand why there should be a ‘good’ way to do mathematics. Everyone works according to his temperament, his capacities and what external conditions permit him. Variety of styles is normal and healthy.

22. estraven - July 20, 2008

@emmanuel kowalski:
I think there should be prizes for expositions (actually, there are already: maybe there should be more). But the Fields medal is a research prize, and it should stay such. On the other hand, maybe we should fix the quirk that makes birthdate modulo 4 relevant.

23. burnt too - July 20, 2008

There are many kinds of people, and many ways of conducting you life, and all this variety exists within mathematics too.

For example there is a type of mathematician who discusses his ideas with others freely and openly, and is happy to share credit (even as a gift) to others. Such a person is a target for ruthless exploitation, and it is hardly surprising when he says enough is enough, and switches to working alone and in secret.

Ben Webster, you have been very naive about this in these two threads.

24. Ben Webster - July 20, 2008

burnt too-

I may well be naive. I certainly won’t pretend to have been in this game as long as many people.

But any discussion of the dangers of sharing one’s ideas must be tempered with considering the dangers of not sharing them. I feel as though the cognitive biases of humans are pretty heavily tilted toward the former rather than the latter.

If you share my penchant for analogies (I love them almost as much as facetious exaggeration), one might say that outright thievery is like terrorist attack, and becoming isolated from not sharing ideas enough is like radon seeping into your basement. Noting that radon gas actually kills many more people than terrorism is not ignoring the danger of terrorism, but rather pointing out that one has to keep things in perspective.

25. Alonzo Moseley - July 20, 2008

Looking at Webster’s webpage, it’s like he’s the poster child for someone for whom everything has gone right. Given the pyramidal structure of the academic pipeline, I think this is the exception unfortunately rather than the rule. Only a small percentage can make it, and sometimes people really do get screwed. Everyone who spends more than a few years in the academic world hears the stories about the advisor who gives the brilliant student an unsolvable problem, the postdoctoral mentor who decides that his mentee really doesn’t know what he’s doing and doesn’t deserve to be a coauthor for what they’re talking about, or the new PhD with who interacted extensively with many supportive people who wrote excellent references, whose exciting new invariant turns out to be equal to zero on all manifolds.

Since we’re in the analogy game… suppose you have someone who has no problem walking through the worst neighborhoods at all hours. He says, I’ve never been mugged ever, people are just paranoid. Well maybe he’s right, but maybe he’s just lucky. Also I think some people are better judges of characters than others, and sort of instinctively know who to talk to and who not to talk to. You see these God-awful professors you’d think couldn’t get a student if their lives depended on it, but then you have some student deciding to work with him because “he’s the only one who does noncommutative categorical complex analysis here”.

26. Anonymous - July 20, 2008

I agree with Ben: if you get scooped, it’s easy to see the damage, while nobody notices the great theorems they could have proved if they had just talked with the right people.

27. Ben Webster - July 20, 2008

Alonso-

I’ll just note, street crime is also an excellent example of something which people tend to wildly overjudge the danger of. Children are at worse risk of dying young in the richest of suburbs than in the worst crime infested ghetto, because they travel in cars so much more in the suburbs. So remember, everything in life has trade offs.

As for my personal experience, you’re right that my view of things has colored my perspective on this a lot, not only because I have not had my results stolen from me, but because I’ve found talking to people, and collaborating them to be incredibly fruitful. There is a lot of research I never would have done if I hadn’t revealed to various people at various points what I was hoping to prove, and asking their input on it. So I’m emphasizing the upside of openness because I know it’s there, and I think that’s important for people to remember. Even if one of my results was stolen from me now, I would be upset, but I would still probably think that a certain level of openness about this stuff had been a net positive to my career.

28. Jason Starr - July 20, 2008

I think it is silly to frame this discussion in terms of a “paranoia” vs. “security” analogy. I think few mathematicians who work in private do so because of fear of theft: they do it so as not to embarrass themselves at a premature stage of a project, they do it because they don’t think collaborators will improve a project, and often they do it because they have found no one interested in collaborating on that particular project. Personally I share Ben’s sentiment that the best thing for mathematics is for everybody to communicate openly and often without too much regard who ends up getting “primary credit” for the final result. But I think most mathematicians consider this decision as a matter of style, and not as a “moral issue” in either direction (which is a bit how some of the comments sound).

29. burnt too - July 21, 2008

When estimating probabilities it is useful to bear in mind that

P(A|A)=1

30. Gil Kalai - July 21, 2008

“I’m curious: does anyone out there think that Gil’s “first avenue” sounds like a good idea? It sounds crazy to me.”

It is probably a good idea for some people and not so good for others, and it is important, in my opinion, that we will have openness for various ways of practicing mathematics.

There is a little point here worth mentioning. On the scale between the first avenue (of the loners) and the second avenue (the collaborators) we the flamboyant bloggers represent probably some extreme community in the far-end of the second direction. So polling about it in a blog may lead to some bias.

(Of course, it is very flattering, Ben, that you quote me so generously)

“Personally I share Ben’s sentiment that the best thing for mathematics is for everybody to communicate openly and often without too much regard who ends up getting ‘primary credit’ for the final result.”

This is a complicated issue and it is not unique for mathematics. Complete openess and transparacy is a nice idea. I am not sure it works. And it has some disadvantages. My heart certainly tends to the open avenue.

“What is true is that you’re much more likely to lose credit by being secretive than by being open.”

I dont think credit is the main issue. The main issue is being able to stay concentrated and complete your task. For that sometimes for some people loneliness and solitude and secracy can be an advantage. (And also I do not see the evidence for Greg’s factual statement.)

31. Greg Kuperberg - July 21, 2008

Okay, this discussion has swept together various different issues. In particular working in secret is really not the same thing as publishing alone. I have no particular habit of working in secret, in fact I’m afraid that I would lose focus if I did. But somehow, even though I never consciously strived for it, most of my papers are solely authored. If the question is how best to contribute to mathematics, then I don’t see anything wrong with having coauthors or not having coauthors. To an extent, it’s a matter of convenience and agreements rather than the research itself. If you collaborate in the generalized sense — as almost all mathematicians do — then you can often choose to divide your ideas into separately authored papers, or choose to combine them into a joint paper, as you prefer.

Gil Kalai: I do not see the evidence for Greg’s factual statement.

I know of many results in mathematics that are accidentally secret, just because the authors procrastinated with writing them up. They often do lose credit that way. I have rarely seen people lose credit expressly by sharing their ideas. I have more often seen people filch credit from published papers, by adding window dressing to someone else’s original ideas. But usually they can still do that whenever the original ideas are published.

Perelman was treated poorly by the Yau school, but he also behaved poorly himself after he announced his proof. The difficulty is that he wrote up his results to his own satisfaction but to nobody else’s.

The only really tragic thing that Perelman did was to quit mathematics in protest. That was an unaccountable overreaction. Also, as I understand it, Perelman meant to protest the entire mathematical community, not Yau specifically.

As for writing up his results “to his own satisfaction” and not to “ours”, I don’t think that we’re in a position to blame him. That’s like blaming Schubert for not finishing his 8th symphony. I.e., maybe Perelman could have written more, but he only deserves praise for what he did write.

On the other hand, even though he was entitled to do his work in any style he wanted to, and he was entitled to write it up any way he wanted to, that does not necessarily mean that it’s a good idea to follow his example. Especially on the first point, and to get back to my first remark, I’d be afraid to work in isolation. I know very few people who can work in isolation without frittering away their time; certainly I can’t.

32. burnt too - July 21, 2008

If you share your ideas you are definitely in danger of being completely deprived of credit. And you just face a backlash if you make a squeak of protest.

The ability of people to get away with unethical conduct is significantly enhanced by the fact that others dismiss such unethical conduct as a rare abberation rather than a dominant factor in the research environment.

The risk of theft of ideas is definitely the number one factor by a wide margin, for me choosing to work alone and in secret.

Just read comment #4 by Chip Neville, in

http://sbseminar.wordpress.com/2008/07/03/request-lis-preprint-or-on-not-being-a-crackpot/#comments

to see how bad things can be.

33. Greg Kuperberg - July 21, 2008

It’s hard to know what to make of Chip Neville’s account. I did find a distinguished senior mathematician who fits his description. I don’t know if he is referring to that person or not. If he is, then it’s hard to believe that what happened was worse than an honest misunderstanding. Now, I don’t want to make light of honest misunderstandings, because even among high-minded professionals, they can be pretty bad. But saying that you shouldn’t collaborate because you could lose credit is like saying that you shouldn’t date because you could be jilted.

Gil gives a good reason that you might temporarily collaborate less rather than more. Namely, if you want to concentrate on your ideas and finish a good result. That makes a lot of sense. Maybe it’s why some of my papers are solely authored when they could have been joint work (although I don’t have any specific people in mind). Sometimes I just felt confident that I could complete the result.

But this is not the same as circling the wagons around your result so that no one else steals it. I have heard people talk this way: “I have this idea for this major open problem, but I’m not finished and it’s highly confidential, so tell no one.” It is usually a way to take much of the fun out of math for dubious gain, sometimes for negative gain. And often the whole thing is a charade, because the Valuable Secret Idea isn’t all that secret or all that valuable.

34. Alonzo Moseley - July 21, 2008

I think the analogy with dating is actually quite valid. Some people who have been burned in relationships just prefer to remain single, and it’s not right to demand that they do otherwise without knowing their situation.

And it might be more than an analogy. Looking at the websites of the pro-collaborationists on this thread, many are in families full of mathematicians. So these are examples of people who can make good relationships with mathematicians. But anyone who’s been in a math department sees a lot of the opposite, many screwed up marriages/divorces etc. And just like in a relationship where normally no one thinks they are the one doing the burning, a lot of
these cases of academic disputes come from some egomaniac being convinced he is doing nothing wrong, it really does obviously all follow from his 1983 paper which he published with someone else he’s not on speaking terms with. The most absurd example I’ve personally experienced was when a referee for one of my papers decided that it all followed directly from his work, except one formula. He decided that this (not very short) formula was in fact true, that he proved it but that my proof was not valid. He said I just conjectured it correctly. Problem was, this formula was in fact not remotely valid, I made a mistake in it. Okay, so this isn’t my best paper ever… but this is a classic example of this phenomenon.

Anyhow I think that one should just be tolerant of people who have had bad experiences and prefer to go it alone. For them this might be the right option because collaborating is too painful or otherwise difficult for them, and one shouldn’t be overly judgemental.

35. Alonzo Moseley - July 21, 2008

By the way, I just want to make clear, that my paper did not remotely follow from his work. I promise I am not just some idiot :)

36. Matthew Emerton - July 21, 2008

This comment addresses a side issue raised in some of the previous
comments — the accessibility of the proof of FLT. As was noted
in post 11, many of the details of the proof were written up in
“the BU volume” soon after the papers of Wiles and Taylor-Wiles
appeared; the contributors to this volume included most of the
experts in the field at the time. This book has served as a basic
reference for many younger researchers in the field (myself included) who grew up mathematically in a post-FLT world.

Also, the Taylor-Wiles method (as it is known) for proving modularity (of elliptic curves, or more generally Galois representations) is now very well understood by researchers in
algebraic number theory, has been generalized in many directions,
and is to some extent a “standard method”. It may not be
widely known outside a circle of expert number theorists (in
comparison say to Ricci flow, the idea of which I would guess
is at least vaguely understood by many more mathematicians), but this is I think generally true of algebraic number theory. For example, class field theory has been a well-established topic
of algebraic number theory for more than one hundred years,
and completely proved in its modern form for more than
fifty years, but — at least in the number field case, as opposed
to the function field case (which admits a geometric interpretation
that is probably more widely known) — is little known outside
the circle of specialists in algebraic number theory. (Since
class field theory and its cohomological consequences are among
the key underpinnings of the Taylor-Wiles method, this incidentally creates one barrier to a wider understanding of the latter method.)

In any case, I just wanted to make the point that Wiles’ work has been thoroughly digested and expanded upon, and that the BU book serves as a crucial primer for students in the field, written by the experts.

37. Daniel - July 21, 2008

For me, the problem with collaboration is an ego problem. Collaborating with somebody below me on the food chain is hard because I don’t want my incompetence to be obvious to them, and collaboration with somebody above is hard because I feel anything I say to them would be either trivial or wrong.
Some people are afraid of having their ideas stolen, it seems. I don’t worry about that much… I worry more about my ideas being revealed to the world (i.e. my potential collaborator) as being total nonsense. And at least 95% of the time that is indeed what they are.

38. Emmanuel Kowalski - July 22, 2008

I must admit I had forgotten about the BU volume when doing the comparison between the current state of accessibility of the results of Wiles and Perelman (though I have it on my shelf…), which leaves the only issue (more or less unavoidable at the moment) of the amount of background needed.

The mention of Class Field Theory seems highly pertinent as an example of a theory that is, in a sense, classical and well-understood, yet quite hard to get into — even for number theorists.

(Incidentally, when it comes to another classical diophantine question, the Catalan Problem, there are, or will soon be, books giving the complete proof “from scratch”).

39. Anonymous - July 22, 2008

Lest people think that Moseley and “burnt too” are representative, I’ve been around a while. I have seen a handful of ethical slip-ups but I’ve seen dozens, maybe hundreds, of occasions where mathematicians were generous and helpful. Ultimately, whether one collaborates or not, is a matter of choice and people should do what they feel comfortable doing, but young people should not be discouraged from interacting by a couple of people with an ax to grind.

Posting anonymously so that Moseley does not scrutinize my cv for reasons why my opinion might be unreliable.

40. David Speyer - July 22, 2008

Daniel,

Obviously, you should work in whatever way works and feels comfortable for you, but I find your response odd. I have run into mathematicians that I feel too dumb, too smart or too different from to collaborate with. My general standard, though, is that I can collaborate with anyone with whom I can have a mathematical conversation and feel that we are both contributing. (Obviously, we will only actually wind up collaborating if we have common interests, are both interested in taking on a new project, and get lucky.) Since I talk a lot of math when socializing at conferences, I feel confident in saying that most mathematicians (at least in the fields I work in) pass this test. Is your test more stringent, or is that not your experience?

41. Greg Kuperberg - July 22, 2008

I think the analogy with dating is actually quite valid. Some people who have been burned in relationships just prefer to remain single, and it’s not right to demand that they do otherwise without knowing their situation.

I completely agree with you here, but I interpret this thread to be more about advice than demands.

42. Anonymous - July 22, 2008

For me, the problem with collaboration is an ego problem. Collaborating with somebody below me on the food chain is hard because I don’t want my incompetence to be obvious to them, and collaboration with somebody above is hard because I feel anything I say to them would be either trivial or wrong.

Regarding the first issue, I think feeling incompetent is fairly common. Specifically, when you write a paper, you know exactly what led to it, but when you read other people’s papers, they look much deeper and more mysterious. Furthermore, it’s easy to assume everyone else must already understand everything you understand (while every day you see clear evidence that the opposite isn’t true). The good news is that you most likely don’t have anything to hide, and furthermore some people who intimidate you probably also feel incompetent.

For the second issue, hard work is the typical way to make up for any shortcomings as a collaborator. If you’re collaborating with someone who has a lot more experience, and perhaps even more talent, then it can be hard to interact on an equal footing. However, if you devote more time and effort to the collaboration, then you can often hold your own even in cases you never would have expected.

43. Anonymous - July 22, 2008

I completely agree with you here, but I interpret this thread to be more about advice than demands.

Exactly. I certainly don’t want to come across as advocating that everybody must collaborate or share ideas freely – people are and of course should be free to do whatever seems best to them, and I’m sure the best course of action varies between different people. My one concern is that I don’t want young mathematicians who read this exchange to develop unnecessary fears about the dangers of not guarding their ideas carefully enough. The worst-case scenario is bad, but it is also rare.

To return to the relationship analogy, it feels like advising young people that they should never experience romance, out of fear that the relationship may become abusive. I don’t think such advice would make sense, even though abusive relationships are a real risk and everyone should be told about warning signs.

44. Alonzo Moseley - July 22, 2008

I should clarify, I was referring to the original post and some of the comments which were critical of Perelman for example without having any idea what his situation is. I realize it would be very bad for mathematics if young people were generally advised to go into hiding.

45. burnt too - July 22, 2008

I need to reiterate

“There are many kinds of people, and many ways of conducting you life, and all this variety exists within mathematics too.

For example there is a type of mathematician who discusses his ideas with others freely and openly, and is happy to share credit (even as a gift) to others. Such a person is a target for ruthless exploitation, and it is hardly surprising when he says enough is enough, and switches to working alone and in secret. ”

“If you share your ideas you are definitely in danger of being completely deprived of credit. And you just face a backlash if you make a squeak of protest.

The ability of people to get away with unethical conduct is significantly enhanced by the fact that others dismiss such unethical conduct as a rare abberation rather than a dominant factor in the research environment.

The risk of theft of ideas is definitely the number one factor by a wide margin, for me choosing to work alone and in secret.

Just read comment #4 by Chip Neville, in http://sbseminar.wordpress.com/2008/07/03/request-lis-preprint-or-on-not-being-a-crackpot/#comments to see how bad things can be.”

I find it very disturbing that some people are so dismissive of the genuine grievances of victims of unethical practices in academia. It send a chilling message to victims that they should keep their mouths shut, because they’ll encounter a backlash if they speak up. This in turn further empowers the perpetrators.

46. Greg Kuperberg - July 22, 2008

I don’t see that anyone here wants to dismiss unethical practices in academia. I certainly don’t. On the contrary, I agree with Louis Brandeis: “Sunshine is the best disinfectant.” But in order to take grievances seriously, you have to know what actually happened. If you don’t have the facts, then the grievance itself could empower perpetrators instead of correcting them. (That is what apparently happened in the famous David Baltimore case, for example.)

Chip said that 30 years ago, a more senior mathematician who he didn’t want to name filched his ideas. For all I know, his grievance is very important. But he didn’t say enough to inform any useful reform or advice.

47. Scott Carter - July 22, 2008

I thought the movie Life after Poincare was a big budget production;-)

48. burnt too - July 22, 2008

You (Greg Kuperberg) now reasonably acknowledge (#46) that you don’t have enough information. But earlier you said (#33) “it’s hard to believe that what happened was worse than an honest misunderstanding.” But this is exactly the kind of conclusion that you just didn’t have grounds to draw. I think this exemplifies a very common systematic bias that leads to discounting or dismissing victims’ accounts.

Some other things to consider. For a person to be able to swipe someones deep mathematical results (or crucial ideas) and credibly present them as their own, that person has to be an expert, presumably someone who was conceivably capable of finding the results themselves, even if in fact they didn’t. Also some people may feel entitled or annointed to be the one to prove a certain result, and they retain that feeling even after someone else proves it, and they move to stake their claim to it anyway, often with the support of others. So your “honest misunderstanding” conclusion really was completely groundless (not to mention that it bluntly contradicts Chip Neville’s account).

49. Greg Kuperberg - July 22, 2008

I don’t know that my remark about honest misunderstandings was groundless. But okay, I concede that I was more correct the second time.

50. Anonymous - July 22, 2008

The ability of people to get away with unethical conduct is significantly enhanced by the fact that others dismiss such unethical conduct as a rare abberation rather than a dominant factor in the research environment.

I believe severe misbehavior is rare, but not everybody agrees. Does anybody know of any actual statistics? The AMS Committee on Professional Ethics (COPE) investigates allegations of unethical conduct, and they keep archives, but the archives are secret. I wish they would at least release some anonymous statistics. (How often are complaints raised about different sorts of behavior, what are the outcomes of the cases, etc.?)

Surely someone must have tried to gather statistics. If not, then this would be well worth trying. I’ve seen surveys about scientific misconduct, but they are often poorly designed, and in any case they typically address very different sorts of misconduct than I expect occur in mathematics. For example, I imagine there is very little research fraud in mathematics (in the sense of knowingly attempting to publish incorrect proofs), while on the other hand some authorship practices that might be considered normal in, say, biology would be considered unethical in mathematics.

I’d also encourage anyone who feels wronged to pursue their case, for example through COPE. Regardless of how common they are, unethical mathematicians certainly exist, and they should not be allowed to get away with their behavior.

I find it very disturbing that some people are so dismissive of the genuine grievances of victims of unethical practices in academia. It send a chilling message to victims that they should keep their mouths shut, because they’ll encounter a backlash if they speak up. This in turn further empowers the perpetrators.

I’m sorry if I seemed dismissive. I don’t mean to dismiss genuine grievances. My impression is that only a very small fraction of mathematicians feel aggrieved, but I have no informed opinion as to the fraction of aggrieved mathematicians who have actually suffered a real injustice. I don’t mean to suggest that this fraction is small; I simply have no data or experience with which to estimate it.

51. Daniel Moskovich - July 22, 2008

I think the line between ethical and unethical practices in mathematics is very fine and often seems quite subjective. Vagn Lundsgaard Hansen’s “Good conduct in the sciences” (Proceedings of the International RILEM Conference on Volume Changes in hardening Concrete: Testing and Mitigation (ISBN: 2-35158-004-4) , pages: 1-10pages: 418, 2006, RILEM Publications S.A.R.L., Bagneux, France) seems a good place to start. COPE doesn’t really seem to define what consists an unethical practice, and it can be quite unclear.
Some fairly common stories it seems:
Sample case 1: X talks about his ideas to Y at a conference.Y turns it into a paper with his friend Z, perhaps acknowledging “useful conversations” with X. Ethical or unethical? I’m not at all sure. We don’t know whether X could have written up the results and completed them. We can’t hold up mathematics waiting for X to publish…
Sample case 2: X and Y are doing joint work. Y completes the next step with Z and they publish jointly, leaving out X. Ethical or unethical? X worked developing the idea, and feels that he/she did all the hard work while Y ran off with Z to skim off the cream. I’m not at all sure.
Sample case 3: X writes a paper which is completely unreadable. Y gets fed up trying to decode it and reproves the result, perhaps with insufficient acknowlegement, and it becomes known as “Y’s theorem”. X feels his ideas are stolen. Ethical or unethical? I have no idea…
Sample case 4: X and Y talk at a conference, and the conversation leads to an idea leading to a great breakthrough by Y. But ideas are not concrete objects, and Y sees it as basically his/her idea, while X sees it as Y stealing his/her idea…
Sample case 5: X and Y are making little progress in joint work because the method is bad. So Y teams up with Z, tries something new, and solves the problem. X feels cheated.
Sample case 6: X told Y something. Y didn’t understand it and/or forgot it. Later Y rediscovered it and published it. X says “I told you this proof at a conference in Sicily in 1978″.
What I’m trying to point out is that ethics in mathematics is often not well-defined. It can be unclear what is ethical and what is unethical, because of “firstness”, naming of theorems and objects after people, etc. Maybe we need an easy set of universally agreed-upon guidelines, so that we know when we are acting unethically…
Issues of credit seem to really poison some relationships between mathematicians, and I’m sure both parties believe they acted ethically in a majority of cases (people usually believe they are right). And without universal guidelines, who are we to judge otherwise?
Let me point out one place where this hurts- I am often afraid to reveal incomplete ideas at conferences, not because I’m afraid they’ll be stolen directly, but because I’m afraid somebody will tell me (or worse yet, hint to me) how to complete them, and then who am I to say that the whole idea wasn’t theirs? It isn’t clear what ownership can be claimed for partial ideas.
Something else I wish existed- a service where you sent introductions to papers, explaining in detail who gave which input for what, and they would give guidance on how best to acknowledge other people’s ideas in the given context. I often feel really bad about the standard “thanks to X and Y for useful conversations” when X might have essentially proved my Lemma 3.1, 4.5, and 7.4, while Y is just a big name who made encouraging noises…

52. Anonymous - July 22, 2008

Something else I wish existed- a service where you sent introductions to papers, explaining in detail who gave which input for what, and they would give guidance on how best to acknowledge other people’s ideas in the given context. I often feel really bad about the standard “thanks to X and Y for useful conversations” when X might have essentially proved my Lemma 3.1, 4.5, and 7.4, while Y is just a big name who made encouraging noises…

I don’t know how much one could say about any given case without knowing a tremendous amount of detail, so I suspect such a service wouldn’t be feasible. However, there are a few generic suggestions. One is to offer credit for specific results. For example, if X suggested the proofs of Lemmas 3.1, 4.5, and 7.4, you can actually thank X specifically for that in the acknowledgements (as well as in the body of the paper near each lemma statement). This is a good way to give special recognition to important contributors who don’t rise to the level of full coauthors. A second, related suggestion is to describe general contributions. For example, instead of thanking several people for helpful conversations, you can thank one person for suggesting useful references, one for providing feedback and corrections on the manuscript, one for serving as a sounding board for ideas, etc.

53. Anonymous - July 23, 2008

Maybe we need an easy set of universally agreed-upon guidelines, so that we know when we are acting unethically…
Issues of credit seem to really poison some relationships between mathematicians, and I’m sure both parties believe they acted ethically in a majority of cases (people usually believe they are right). And without universal guidelines, who are we to judge otherwise?

It would definitely be nice to have more guidelines, although I think it would be difficult to formulate universally agreed-upon guidelines that were detailed enough to resolve most real-life issues. Despite this, my guess is that for most actual disputes, there would be a pretty clear community consensus as to who was at fault, given enough detailed information (which might not be reliably available). Organizations like COPE could uncover such a consensus even in the absence of universal guidelines.

Among your sample cases, 3 and 6 seem relatively straightforward to me, although the facts of the cases may be in dispute. The others largely come down to questions of coauthorship and collaboration. Here are two guidelines I think are fairly common; I’d be curious to know whether other mathematicians agree that they capture community norms.

The first is about collaboration, and says that once two people have started collaborating on a specific topic, neither one can withdraw unilaterally from the collaboration. The other party can reasonably expect to be a coauthor on the next publication on this topic (at which point the collaboration can be ended by either side). Of course, the tricky case is when one collaborator doesn’t live up to expectations, for example by putting in very little work. My understanding is that this possibility is an incentive to choose collaborators carefully, and that the threat of cutting off future collaboration is supposed to discourage laziness. Another issue is that it can be difficult to distinguish between isolated conversations and the beginning of a collaboration. I don’t know what to say about that, except that if it matters to you whether you are officially collaborating, then you should discuss the issue with your hypothetical collaborator.

The second guideline is about coauthorship. It says that if you have any uncertainty as to whether someone should be listed as a coauthor, you should defer to them on this decision. On the other hand, if someone asks you whether you should be a coauthor and you aren’t certain, you should decline. This process obviously assumes a certain amount of good faith on the part of everyone involved, so it may not be universally applicable. Like the first guideline, it has the property that nobody can take advantage of the same coauthor more than once, which limits abuse. However, in principle it allows a corrupt mathematician to take advantage of lots of different people, each once. One advantage, though, is that it (I think correctly) judges denying someone a well-deserved coauthorship to be a greater injustice than giving someone an undeserved coauthorship.

How closely do most mathematicians adhere to these principles? And, regardless of whether they describe what people actually do, do they at least describe how they think other people ought to behave? I’ve heard both guidelines advocated by senior mathematicians, but I don’t know how idealistic or eccentric they were.

54. Daniel Moskovich - July 23, 2008

Anonymous- regarding guideline 1, I’m not sure I agree. I think a coauthor should have to have done at least some of the research AND some of the writing. Nobody should be a coauthor just because a collaboration was started… Also, collaborations which are started may often be abandoned, just as people break up- consider sample case 5.
It’s a very touchy situation, which I wish there were clear rules how to handle. I think that without a contract, a collaboration is like some form of dating, and until the proverbial ring is on the finger there is no commitment to have children together. Whether or not Y should be a coauthor depends on whether Y in fact did some of the work (must be defined) and some of the writing up.
How specific do you have to be to be “collaborating”? If you use somebody as a “sounding board”, give them your proof as you find it in real time, etc., is that person a collaborator and coauthor?
Another real-life issue- let’s say X collaborates with Y. X tries procedure A and Y tries B. Procedure A works and gives a complete proof, but Y is still stuck with B. Is the paper joint? What is the ethical thing to do? 100% of the work that actually gets published was done by X…

55. James - July 23, 2008

I agree with Anonymous #52 that acknowledgments are better when specific. The only possible exception I can think of is when you’re thanking very famous mathematician X for their interest and encouragement. Then it might read like “X thinks what I’m doing is interesting”. So then you might rather keep things vague and leave open the possibility that X actually helped rather than risk looking like an insecure name dropper.

56. Alonzo Moseley - July 23, 2008

Watching #53 and #54 debate, I see two reasonable people trying to understand the reasonable way to go about things. Really, I think
such people would rarely end out in such disputes, with or without a kind of honor code like #53 is suggesting. The problem in my mind is that the egomaniacs are unlikely to abide by any reasonable set of rules and then there is no means of enforcement. If you meet such people, you realize that even if one of these jack***** will agree to say not end collaborations once they have started, you could never trust them to keep the agreement.

My personal point of view is just to make sure your collaborator is ok and that you have grounds for this particular collaboration, and then make a set of rules for that specific collaboration. Each situation is different.

57. Charles - July 23, 2008

Daniel, I actually got into that precise situation with some stuff I did two summers ago, and I know that my decision in that case (there were other factors too) was that my name doesn’t go on the paper.

Regarding collaboration, there’s already a code of conduct out there, though I don’t know how many people follow the Hardy-Littlewood axioms of collaboration http://www.math.ufl.edu/misc/hlrules.html

58. Scott Carter - July 23, 2008

Story: A mathematician, Nameless, asked to visit me about his groundbreaking result. As it happened at his proposed time of visit there was a mini-
conference on the same subject. I said, “Sure, come down, tell us about your stuff.” His talk was a disaster. The world experts on the subject were present, and he would not reveal details because the manuscript had not been finished.

Clearly, he had been burnt in the past with someone stealing essential ideas. I believe that someone became a better known mathematician, and Nameless
became lesser known.

Personally, I don’t believe Nameless became lesser known because his result got pinched. He became lesser known because he feared getting pinched. His paranoia made him difficult to work with.

Once the paper that Nameless didn’t talk about was finished, it turned out to be a bit of a dud. Or more precisely, it is an extremely difficult paper to read because the essential idea was not explained verbally to us. As it is, I *think* I know what the main idea is, how it relates to existing literature, and what the main conjecture with respect to this work is. My own desire to study these ideas was severely diminished by Nameless’s attitude at the conference.

As people think about ideas being or not being shared, they might consider how big an idea is it. If you could not get it to work, is it because it is wrong or because you are less competent than the person with whom you are willing to share it. Is it a good idea, but you don’t have the time to develop it. Is your skill in taking an idea an developing special cases or intricate details. To what extent is the goal of the game to develop mathematics and to what extent is the goal of the game to develop your own reputation and/or career?

Arnold’s principle indicates that you probably won’t get credit for your idea anyway. You might get lucky and get credit for something else.

Pinching other people’s ideas is not a good thing to do. When in doubt, ask if it is OK to write it up with an acknowledgement, or put the other person’s name on it and see if it is OK like that.

59. Matthew Emerton - July 23, 2008

Regarding collaboration, and in particular the the final question asked by Daniel (#54 above), I think that one should adhere to axiom 4 of Hardy and Littlewood: once two (or more) people are working together seriously on a problem, the work is joint, regardless of who actually proves the final results. It is impossible to predict in advance how the lines of research will develop, or whether method A or method B will succeed. (Otherwise, none of the collaborators would be pursuing these methods in the first place.) Of course, a collaborator can always opt out of a collaboration, or decline to have their name placed on the paper — but that should be their decision; it shouldn’t be forced on them by the other members of the collaboration. (In particular, I agree with the guidelines suggested by #53.)

On a related note, I would suggest that if one begins to discuss ideas about a specific problem seriously with another mathematician, it is good to decide at the very start whether or not you are officially working together (with the goal of writing a joint paper, say). This can avoid unpleasantness or misunderstandings later.

60. Matthew Emerton - July 23, 2008

Just to clarify: when I wrote “a collaborator can always opt out”, I meant that they could opt to not have their name attached to the results of the collaboration. I didn’t mean that they can opt out and take the results with them. (In practice, this typically means that they can decide to not have their name on the paper; they can’t go off and publish the results produced during the period of collaboration under their name alone.)

61. Ben Webster - July 23, 2008

I’ll just note, since it’s taken on a life of its own in the comment thread: my intent in the post was not to criticize Perelman (or Wiles, for that matter), but to point out that I would not recommend his path to anyone else, and that mathematics would almost certainly have benefited had he found other mathematicians to communicate to about this stuff. Whether the mathematics community “deserves” his help is another question entirely, but one I am less concerned about (given that I don’t feel any need to judge his ethics). I am a member of the mathematics community, and I am selfish on its behalf.

62. Ben Webster - July 23, 2008

once two (or more) people are working together seriously on a problem, the work is joint…

right, though this creates really sticky questions about what “a problem” is. If I write a paper with Y, and then come up with a generalization of this paper after it’s done and Y and I aren’t in regular contact, does that count?

63. Jonathan Vos Post - July 23, 2008

Have a mixed portfolio of collaborators. In school, and at home, and in social groups, we learn how to do collaborative work. In university, we have a chance beyond study groups, namely in working with one or more mentor professors. There is a network effect, where successful collaboration links you into the community of Mathematicians, and leads to more collaborators. Erdos number, and all that.

I say so, despite having the Chairman of my Department in grad school pay me a flat fee for my coauthorship of a textbook that, in later editions earned him roughly $300,000. He’d given me sole authorship of the Teacher’s Guide, and then made trivial changes to it and removed my name from later editions. I say so, despite the lunatics at Rockwell International who ran amok plagiarizing and forcing out at Caltech and MIT graduates from their department, who had the clever tactic of pre-emptively accusing their victims of being the plagiarists.

The first case is an anomaly of a departmental dictator-Chairman and exploitable grad students. The second case is an extreme version of what is Standard Operating Procedure in the corporate world.

I suspect that statistics would support my argument that the more collaborators, the better. The two cases considered by Gil Kalai and Ben Webster are akin to a choice of a one-shot rifle with telescopic sight and a shotgun. It depends on what you’re hunting, or protecting yourself against.

As to which mathematicians are wealthy enough to rise above the fray, Jim Simons. Witten invented the framework for understanding the Jones polynomial using Chern-Simons theory. This had far reaching implications on low-dimensional topology and led to quantum invariants such as the
Witten-Reshetikhin-Turaev invariants. But the point is, that Simons is, having switched out of math, the James Harris “Jim” Simons trader, and philanthropist.

In 1982, Simons founded Renaissance Technologies Corporation, a private investment firm based in New York with over $30 billion under management; Simons is still at the helm, as president, of what is now one of the world’s most successful hedge funds.

Simons earned an $2.8 billion in 2007, $1.7 billion in 2006, $1.5 billion in 2005, the largest compensation among hedge fund managers that year, and a mere $670 million in 2004. With an estimated current net worth of around $5.5 billion, he is ranked by Forbes as the 57th-richest person in America. He was named by the Financial Times in 2006 as the ‘smartest’ billionaire.

I’d like to see a movie about Witten. Edward Witten is that American theoretical physicist at the Institute for Advanced Study who, in 1990, and this doesn’t usually happen to Physicists, was awarded the top prize in Math, namely the Fields Medal. In 1995, he suggested the existence of M-theory at a conference at USC, and used M-theory to explain a number of previously observed dualities, sparking a flurry of new research in string theory called the second superstring revolution.

But he’s more colorful than Hawking, if the Press would only wake up. After all, he got his bachelor’s degree in History (with a minor in Linguistics) from Brandeis, planned to become a political journalist, and published articles in The New Republic and The Nation. Then he worked for George McGovern’s presidential campaign. Then, he
went to U. Wisconsin-Madison for one semester as an Economics graduate student before dropping out, returning to academia, switching to Applied Math at Princeton before shifting departments again and getting his Ph.D. in physics in 1976 under David Gross, the Nobel laureate in Physics in 2004. Gross would play the banjo music for the movie, too.

Data on Witten and Simons from wikipedia.

64. Emmanuel Kowalski - July 23, 2008

I think there is another danger in not communicating enough, which has not been mentioned explicitly, but is very real: by working alone or secretly, the chances increase of spending time and energy on reproving something already known and published. I don’t know if this has been studied in comparison with fraud problems, but I would guess that tit happens rather more frequent, especially for younger people, and up to the “middle-range” of difficulty. (The guess is partly based on having heard directly, concerning people of my generation who I know personally, from many more instances of this, compared with a single case of theft).

On the other hand, compared with the outcome of a fraud case, such a situation is not necessarily so disastrous: the learning involved in rediscovering something may be one of the best ways to understand it really deeply, and it may lead naturally to new insights which would have not come to mind by directly quoting or reading the literature. But the loss of time in particular may be a real problem at the beginning of a career. Also, there is quite often a sufficient difference between two approaches that there would be something to save of the work.

A famous instance of rediscovery was Turing’s proof of a version of the Central Limit Theorem, as he was unaware of Lindeberg’s earlier work.

65. Anonymous - July 23, 2008

How specific do you have to be to be “collaborating”? If you use somebody as a “sounding board”, give them your proof as you find it in real time, etc., is that person a collaborator and coauthor?

Yeah, that’s definitely tricky. I think the place where it comes up most dramatically is graduate school, in the advisor-advisee relationship, and secondarily with postdoctoral mentors.

Graduate advising practices seem to vary quite a bit. My approach is to avoid coauthoring papers with students on their primary thesis project (although I sometimes do it on side projects). In particular, I avoid any serious thinking about student projects while I’m not meeting with them: I may do a little idle brainstorming or search for references, but I never get out scratch paper. During meetings, I feel much more free to suggest ideas, try calculations, etc., but with the understanding that no contribution I make during a meeting would give me coauthorship. During early meetings, my contributions often matter quite a bit, but as the student develops more experience, they matter less and less, and eventually they are negligible compared to the work the student is doing outside of our meetings (which justifies having the student be the sole author).

Postdocs are a little different. I think it’s important to offer them a safe sounding board for ideas, without the need for coauthorship, but I also feel much more free to collaborate with them.


Another real-life issue- let’s say X collaborates with Y. X tries procedure A and Y tries B. Procedure A works and gives a complete proof, but Y is still stuck with B. Is the paper joint? What is the ethical thing to do? 100% of the work that actually gets published was done by X…

My take is that this is precisely the situation the guideline is intended for, and that X and Y should coauthor a paper (assuming, let’s say, that Y genuinely worked hard on approach B). The idea is that nobody could have predicted in advance which approach would work: if X was certain B was hopeless, then the collaboration should never have been started. Y was making a valuable contribution by exploring B, even though it didn’t happen to work out. In practice, hopefully X would keep Y up to date on the progress of A, and Y would start contributing to that approach once it became clear that it was superior.

There are certainly times when it might be best for Y to withdraw willingly from the collaboration. For example, if X’s solution is exceptionally brilliant (and Y recognizes that it is far more impressive than his/her work on B), or if approach A turned out to require sophisticated background and techniques Y was unable to apply. However, I feel that it’s up to Y to withdraw, and that X should not try to force the issue.

Leaving the decision up to Y allows one form of abuse (by lazy coauthors), but it defends against another form. X could work secretly on A, without keeping Y up to date on progress, with the hope of writing a singly-authored paper if A works out, and if not then falling back on a jointly-authored paper on B with Y. Using the collaboration with Y as an insurance policy is definitely unethical.

The problem in my mind is that the egomaniacs are unlikely to abide by any reasonable set of rules and then there is no means of enforcement.

Definitely, just articulating the community’s norms won’t stop someone from deliberately breaking them. However, I hope it might encourage people who have been mistreated to complain more effectively. If you are deciding whether to ignore a situation or try to get justice, it could be helpful to know how different people view your type of situation. If it’s generally considered a gray area, then it’s probably best to forget about it. If everyone agrees on the underlying principles, then you might at the very least be able to shame the transgressor.

66. Double follow-up: pass the popcorn « Secret Blogging Seminar - July 23, 2008

[...] followup posts going), we finally got a truly exciting and original idea:  when is somebody gonna make a movie about Ed Witten?  If string theory is confirmed at CERN, it would make a great ending (and if it [...]

67. David Speyer - July 23, 2008

Matthew Emerton, #59, writes “once two (or more) people are working together seriously on a problem, the work is joint, regardless of who actually proves the final results. ”

I think this needs some caveats. When Nick Proudfoot and I wrote our paper “A Broken Circuit Ring“, one of our hopes was that we could use the methods of that paper to resolve a certain open question about the h-vectors of broken circuit complexes. (See section 5 of Schwartz and Remark 9 in our paper.)

Eventually, Nick and I convinced ourselves that this approach could not work. If Nick now finds some unrelated approach and answers Schwartz’s question, I don’t think he is obliged to do anything other than credit my previous discussions with him.

There are several problems against which I pit every new technique I learn. (A suggestion of Feynmann’s.) If I am collaborating with other mathematicians, I will certainly discuss with them whether our methods could solve one of those problems. However, and I try to be open about this, that doesn’t mean that I will involve them in any other work that I do attacking that problem.

68. David Speyer - July 23, 2008

A shorter way to state the above might be: there is an obligation to include one’s past collaborators in future work on the same topic. However, a topic may be a problem, a technique, a particular mathematical object, or some combination of the above.

My own (not very long) experience has been that I do not feel that I have ever been taken advantage of (and don’t think I have taken advantage of anyone else). I have had to carefully hash out which areas of research were and were not areas of collaboration, and often these discussions happened later than they should have. In my mind, a useful topic of conversation might be “what topics should be discussed with a mathematical partner before entering into collaboration?”.

69. Gil - July 23, 2008

The question about discussing incomplete ideas and projects, or being secret or shy about them is a delicate issue. I do not thing there is a recipe and certainly not a recipe good for everybody. Mathematicians are usually thoughtful people and this is an another issue that deserves careful thinking on a case-by-case basis. And, as people noted there are advantages and disadvantages for both extreme avenues.

The issue of being completely open or keeping things secret regarding ideas and partial progress is separate from the issue of collaboration. Also when you collaborate there are various cases when it is reasonable not to make partial progress and ideas public.

As a matter of fact, One should be even more careful about these matters in case of a collaboration. When you want to report partial progress and general ideas of a joint project it is wise to coordinate it with your collaborators.

70. Daniel Moskovich - July 23, 2008

I think there’s another reason we need a clear code of ethics- so that we ourselves can avoid acting unethically. There are many cases I’ve been in myself when I don’t know what the ethical thing to do is, and a code as well as accepted guidelines would help. Also so that you realize when somebody else acted unethically towards you, and when they were just not nice but not necessarily unethical (possible example: writing a bulk e-mail stating “all results in paper A are wrong.” (true story), and being right about it).
Concerning Scott’s story, if Nameless (N for short) is whom I think he might be, it’s complicated and interesting. First, N had indeed had his ideas brazenly stolen in the past. Secondly, N had just proved a conjecture a lot of people were interested in. But the proof had many ad-hoc steps. As a perfectionist he wanted to do these steps properly and methodically (I definitely understand that). But that might take many years- indeed, one of the steps in particular still uses an unsatisfying ad-hoc argument. Meanwhile, as I understand it, for the sake of mathematics X wanted the proof out and was publishing parts of it, using the ad-hoc arguments, as “forced” joint papers.
X had the interests of mathematics in mind, and one can’t call his actions unethical per se… but the situation really stressed-out N, who doesn’t get sufficient credit I don’t think.
It’s hard to say how N should have acted- people have different personalities- but I really hope for the sake of mathematics that that particular proof gets rewritten as a short monograph or something, with all the steps as they should be, so as to make it understandable and readable to somebody who was not part of the in-group.
Indeed, how should one handle such a situation? Let’s say you found an exciting new invariant, but the construction uses silly ad-hoc arguments, and it’s hard to calculate. You want to withhold publication until you do the work properly, with the “right” arguments, in a way which is useful to the rest of mathematics.
Do you put the intermediate construction on your homepage? That would be the option I would be tempted to take. Do you announce your invariant but not publish for a long time? (maybe including a formula in the announcement?) Many people have taken that route. And what do you do if X writes up part of the construction independently before you make any public announcement, and wants a joint paper?

71. Gil - November 16, 2008

Michael Nielsen deals with related issues on his blog and book. See for example:

http://michaelnielsen.org/blog/?p=485

72. Mathematics, Science, and Blogs « Combinatorics and more - January 28, 2009

[...] Mathematics, Science, and Blogs By Gil Kalai Michael Nielsen wrote a lovely assay entitled “Doing science online” about  mathematics, science,  and blogs. Michael’s primary example is a post over Terry Tao’s blog about the Navier-Stokes equation and he suggests blogs as a way of scaling up scientific conversation. Michael is writing a book called “The Future of Science.” He is a strong advocate of doing science in the open, and regard these changes as truly revolutionary.  (The term “Science 2.0″ is mentioned in the remarks.)  Michael’s post triggered Tim Gowers to present his thoughts about massive collaboration in mathematics. This is a very interesting post and there are interesting follow-up remarks.  Ben Webster in “Secret blogging seminar”  mentioned in this context earlier interesting related posts about “Working in secret“.   [...]

73. Successful Researcher - October 29, 2009

An interesting discussion on scooping in (theoretical) physics and the role of arXiv can be found here.

74. Lark Speyer - February 16, 2012

Not to over-emphasize my own personal favorite bizarre research news of the month, but the Andrulis incident seems another interesting example of someone who (presumably) hoped to blow the world away after decades of work in secret. The latter comments in both the Andrulis-related threads on Retraction Watch devolve into long back-and-forths on both technical and ad hominem levels between a defender and a debunker.

And then there’s a recently published book on introversion and creativity, Quiet, also arguing for the ‘lone genius’ and against ‘collaboration.’


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