Note: these posts are copied over from the ‘mathbucket’ section of my old tumblr blog and I haven’t put much effort into this, so there is likely to be context or formatting missing.
Actually programming is doing one good thing for me, by forcing me to engage with a more algorithmic/symbolic mode of thinking that I’ve kind of ignored as much as I can in the last few years. I find it frustrating a lot of the time, partly because it’s not particularly how my brain works but also because normally the structural component is the the main thing, there is no there there. (Or what’s there is incredibly intrinsically uninteresting to me, like parsing some file or whatever).
At the other end of the spectrum is differential geometry, which I have this kind of doomed love for despite not being especially good at it. I love it because the questions are so tangible – ‘how does this surface curve?’ – and the particular methods you use are correspondingly less important if you have the right intuition for the tangible problem. I mean they are still important, there are definitely more and less elegant ways of doing things, but structure is at least somewhat downplayed compared with the actual thing you want to know about, which is how this surface curves.
I mean I found a differential geometry book by Serge Lang in the library once, I don’t think I dreamt it, and it was a proper Bourbaki-style algebraist’s version of differential geometry. No pictures and everything was done with some kind of quadratic form iirc. After that I was kind of convinced that you can build differential geometry out of anything you have to hand.