When I was in college and grad school, I did a lot of formal logic. I took logic classes both on their own and as part of linguistic semantics. Logic is basically mathematics, even though it uses letters instead of numbers. Two things were really hard about it: first, understanding principles of a logic and its manipulation enough to do exercises, to express things within the system, and to understand proofs about the system; second, to stop trying to understand in order to learn to do exercises and proofs. The latter version is like learning to type; you have to just keep doing it until it works right- then you can. Unlike typing, then you can perhaps start to understand what you’re doing. you do any logic or mathematics, you know that proof by induction feels like it makes no sense if you try to understand it before learning it. That’s a pretty hard thing for me, because I tend to learn by having a conceptual framework, knowing what we’re getting at in the general sense before I look at the details. To do anything else is generally just confusing, or in the case of logic classes, it was a leap of faith.
I was thinking about this today because I just heard about a mathematician in the news, Grigori Perelman, whose work on the Poincaré Conjecture has earned him some fame and awards, although he has opted to stay at home rather than accept them. I have gotten to know quite a few mathematicians (and logicians). What strikes me is that the career of the theoretical mathematician involves quite a bit of thinking- and then waiting until you come up with a way of proving something. Think, wait, think… I don’t know if I really liked doing logic, but I did benefit in some sense from its structure; it gave me some way of trying to squeeze and shape my thinking into something a bit more focused and a bit quieter. It’s not exactly meditation, but whatever works is just fine. I guess that’s not enough for me. Perhaps that’s because I get distracted by other interesting things, or perhaps it’s because I find other things more interesting or important. I do know that my doctoral research advisor’s stated motive for his work in very abstract logic did not hold for me; he told me that he did this because “it keeps me off the street.” What he did was describe logics that described other logics. An outlet for meta-meta thinking?
Funny thing is, while logic itself embodies structure, that doesn’t mean that the time spent working on a dissertation is structured. In hindsight, that’s why research didn’t end up being the thing that kept me off the street. Thinking about structure doesn’t provide one with structured days.