I think any high level thought or movement is intuitive and approximate and not completely trustworthy, including high level thoughts about mathematics.
You find things by looking across long distances, but constructive proof steps only cross short distances. Nothing new is actually found by applying simple rules. Mathematical proofs don’t represent a way of thinking, they’re artifacts produced after the thought has been done and the realization has been had, they only exist to validate and to discipline (train) the higher-level heuristics you really use when you’re navigating the overarching space of mathematics.
I’m not a mathematician, but if someone had told me this when I started undergrad, much more likely I would’ve been better at it and I would’ve ended up being a mathematician in that timeline.
I think any high level thought or movement is intuitive and approximate and not completely trustworthy, including high level thoughts about mathematics.
You find things by looking across long distances, but constructive proof steps only cross short distances. Nothing new is actually found by applying simple rules. Mathematical proofs don’t represent a way of thinking, they’re artifacts produced after the thought has been done and the realization has been had, they only exist to validate and to discipline (train) the higher-level heuristics you really use when you’re navigating the overarching space of mathematics.
I’m not a mathematician, but if someone had told me this when I started undergrad, much more likely I would’ve been better at it and I would’ve ended up being a mathematician in that timeline.
This is consistent with what I’ve heard/read elsewhere, yeah.