Many young, and not so young, mathematicians struggle with how to spend their time. Perhaps this is due to the 90%-10% rule for mathematical insight: 90 pages of work yield only 10 pages of useful ideas. A venerable mathematician once described his career to me as constantly stumbling around in the dark. Of course, this struggle is largely personal…perhaps it is an evolutionary struggle to find one’s own way to find something relevant to contribute to mathematics as literature. Such an evolutionary struggle necessitates the artist’s turmoil and requires an acknowledgement of “l’importance d’être seul”. Even when I have successfully done some reasonable mathematics, it behooves me to look for a more efficient way to proceed. This sort of searching turns up various nuggets of advice on how to do mathematics, like the following (paraphrased) ones:
Read a paper of a master for a year, and you will get something– advice given to young mathematicians by Israel Gelfand at Harvard during his 90th birthday conference on the unity of mathematics.
Try to imagine a proof, no matter how vague-Gowers’s online essay on the philosophy of mathematics and our relationship to formalism
I try to colorize it in my mind, to try to see what it’s really getting at, rather than simply what it says-Thurston’s online response to Ashley Reiter on reading mathematics.
Also recall Grothendieck’s images of a softening nut and the rising sea…online interviews/documents with suggestions of Atiyah, Singer, Connes, Gromov…
There are many other such nuggets, many of which can be found on posts on this site. These all provide very inspiring images, but I would like to be more blunt:
What is your method? How do you go
about doing mathematics on a day to
The method is “keep trying”. Try reading the paper again, try reading a different paper, try stopping reading papers and just work it out yourself. Try talking to other people, try struggling on your own, then try talking to people again. Try a different problem, then try going back to the same problem. Try the same method again and again, then try different methods, then go back to the first method and try it one more time. Try just writing out some calculations even if you don’t see them going anywhere, try a stupid example, try drawing some pictures. Finally try taking a break, and then also try not taking a break. Try everything once, then try it again!
But enough of that, here’s some more practical advice from my own experience (I’m a postdoc by the way).
1) For a hard problem, work intensely on it for a few weeks or months, until you feel like you’ve completely hit a wall. During this period you want to be producing as many calculations, lemmas, pictures, examples, etc as possible. They don’t all have to be super-relevant — your goal is maximum output of material. Then when you finally feel like you’re going insane, take a deep breath and move on to something else. After a break of a few weeks or months, which may be spent either on work of some other kind or even on vacation, come back to the problem and see if the fresh perspective helps. It may take multiple cycles like this for a really hard problem.
2) Look for a variation of your problem which is easier or which your methods can better handle. Even if the variation is not as strong of a result as what you are ultimately aiming for, it may still “count” and end up being a nicer paper than you realize at first. It’s good for your morale to register progress in this way, and if you can get a publication it’s good for your CV too. Here is where advisors and mentors can help a lot — they can help you identify which variation is meaningful, doable, and “interesting”.
3) Remember that almost nothing ever works. One of the worst feelings to me is when I find a mistake in something I thought I had figured out, and all of a sudden weeks or months of progress begin to unravel. This can be extremely depressing. Try not to take it too hard, and over time you can rebuild and recover. Try also to appreciate the progress you are making, even if it is much less than you wish it was.