Yesterday, as I was walking to my office in the morning, I planned to write a post in which I was going to say that Polymath9 had basically been a failure, though not a failure I minded about, since it hadn’t had any significant negative consequences. Part of the reason I wanted to say that was that for a few weeks I’ve been thinking about other things, and it seems better to “close” a project publicly than to leave it in a strange limbo.
When I got to my office, those other things I’ve been thinking about (the project with Mohan Ganesalingam on theorem proving) commanded my attention and the post didn’t get written. And then in the evening, with impeccable timing, Pavel Pudlak sent me an email with an observation that shows that one of the statements that I was hoping was false is in fact true: every subset of can be Ramsey lifted to a very simple subset of a not much larger set. (If you have forgotten these definitions, or never read them in the first place, I’ll recap them in a moment.)
How much of a disaster is this? Well, it’s never a disaster to learn that a statement you wanted to go one way in fact goes the other way. It may be disappointing, but it’s much better to know the truth than to waste time chasing a fantasy. Also, there can be far more to it than that. The effect of discovering that your hopes are dashed is often that you readjust your hopes. If you had a subgoal that you now realize is unachievable, but you still believe that the main goal might be achievable, then your options have been narrowed down in a potentially useful way.
Is that the case here? I’ll offer a few preliminary thoughts on that question and see whether they lead to an interesting discussion. If they don’t, that’s fine — my general attitude is that I’m happy to think about all this on my own, but that I’d be even happier to discuss it with other people. The subtitle of this post is supposed to reflect the fact that I have gained something from making my ideas public, in that Pavel’s observation, though simple enough to understand, is one that I might have taken a long, or even infinite, time to make if I had worked entirely privately. So he has potentially saved me a lot of time, and that is one of the main points of mathematics done in the open.