I am currently working on a nice little proof. I was initially very confident that I could find the proof — mostly because I already gave a different proof for the same observation, albeit with completely different tools.
This time around the proof will probably end up as a series of stronger and stronger lemmas. Already 3 weeks ago I thought I had shown the strongest intermediary lemma I have formulated so far. Unfortunately, Andreas shot it down, then I found a different proof, and Andreas shot it down again, then I found another proof, and Francois shot it down. As painful as this sounds (and really, really is), I am so lucky to have such colleagues.
Finally, I think I have a found a proof for the intermediary lemma that will live (well, maybe I should wait until Francois sees it later today…). The creepiest thing about this proof, however, is Andreas’s precognition.
When I showed him the failed second attempt (which failed pretty much at line 1…), we discussed the phenomena involved and Andreas made two comments about the problem itself. The first was that due to the setting, an indirect proof seemed to him to be the way to go; second, he gave a very simple example, a special case of the problem that should turn up in some general form. And as you might expect, these two predictions came true — in almost every respect.
The first time I ever heard of such precognition was from a student of Stevo Todorcevic when I was just starting out on my PhD — and it scared the hell out of me to hear that he predicted a complication that the student only found after 3 months of work on that problem. I call this phenomenon ‘supervisor precognition’. It’s not like supervisors in mathematics always have an idea for an actual proof of a problem, usually not even for a strategy. However, supervisors often have a small but brilliant insight into the situation as such and might spot some critical properties far ahead, long before the student actually gets there.
I know that this strength has as much to do with experience as it does with mathematical talent, but it is both annoying and wonderful. Annoying, since I would like to pretend that I could be as productive without these amazing insights from other people. For Hikaru no Go fans, it feels like Akira playing Sai in the first game — it’s a move from far above, so to speak. To use a metaphor that I don’t really like, if we fight through a jungle to get on a mountain top, this precognition maybe is the equivalent of a greater height, allowing to see not the exact path ahead, but a few major obstacles ahead.
I think this precognition is perhaps the most important strength of a good supervisor. On some level, this precognition needs to be present to guide students to their own, independent research. Students should look out for signs of it (and ask around who’s got it) and researchers should try to develop it (if anyone can tell me how, please tell!). Now if only my proof was finished…