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.