My partner asked me an interesting question yesterday: are there any Buddhist monks who do math as a meditative craft? As far as I know, the answer is no. In fact, the only mathematician monks or “monks” I can think of are the Pythagoreans and the mathematicians in Neil Stephenson’s book Anathem, but the question is still interesting, and it got me thinking.
Over the last few weeks, I’ve started to really miss doing math. That’s been a surprising and gratifying feeling, since my attitude towards math post-grad school has generally been something along the lines of “I should keep studying math, but it’s stressful and I’m so bad at math and I never get anywhere anyway, and besides, I’m busy.” Now and then, I’ve pulled out a book on something or other and started reading, but never got past the first few pages. I suppose that’s not very surprising, since I always found reading math to be one of the most frustrating and unproductive activities I undertook as a grad student.
So this time around, I decided to do something totally different and just pick interesting and challenging problems to work on without trying to sit down and study new areas of math. I’ve been doing problems from the International Math Olympiad (and some of the national Olympiads used to pick teams from various countries), which is for high school students, and the Putnam competition, which is for undergrads. The IMO doesn’t (officially) require any math beyond pre-calculus (the Putnam assumes more: advanced calculus, linear algebra, real and complex analysis, etc, but then, it’s for college math majors), so in theory, it should be accessible to lots of people, but it’s really, really, really hard. Actually solving the problems often requires a lot of math that’s not part of the usual high school/college sequence, as well as a great deal of ingenuity. The thing about this working on these problems is, even though in the back of my head I can hear the voice of one particularly obnoxious former professor telling me that these problems are too elementary and I should be doing serious math, I’m having a blast (Said professor actually told me this, though it was when he saw me reading an Abstract Algebra book typically used for Master’s or PhD level Algebra classes, that he deemed to be too elementary.) Other than a few days last summer working on some interesting problems at a Math Circle workshop, it’s the first time in years that I can recall genuinely enjoying sitting down with a notebook and pen and thinking.
So anyway, all of this led me to start thinking about the difference between this experience of doing math and the experience I had in grad school, where I mostly felt constantly panicked, because I couldn’t force myself to have insights fast enough. Now that the idea has been pointed out to me, I think that the difference between those two experiences is precisely the difference between math as a practice and math as a performance. When (Buddhist) meditators speak of “having a practice,” what they typically mean is that they sit down and meditate every day (or nearly every day). Even though you will inevitably “get somewhere” if you meditate every day (you’ll probably be calmer, a little slower to react angrily, etc), the point of meditating is just to meditate.
This is how working on these math problems for the last few weeks has felt. I just sit down and do math. Obviously, I have a goal, since I want to solve the problems, but there’s no timeline, no rewards for solving the problem (except satisfaction), no threats, no anything. I can drop a problem that I realize is too hard, or I just don’t find interesting. I can read the answer if I’m at a loss or interested but don’t want to take the time to solve it. I can peek at the answer if I need a place to start. I don’t have to waste hours writing up solutions in formats that will please my professor. I can pick problems that involve math I understand, or math that is just barely beyond my understanding, or math I once understood but have forgotten. And as long as I play with some problems every day, or most days, or occasionally, I’m successfully “doing math.”
This is not at all how I felt when I was a graduate student, where I felt that my job was to perform constantly, and to improve my performance at a rate much faster than came naturally to me. And just as you can’t get better results by worrying about how much your meditation is progressing, you can’t have mathematical insights faster by trying to have them. You can’t hurry insight by threatening it with a bad grade. All you can do is take potentially talented but late-blooming young mathematicians like me, and make them feel stupid and afraid of math.