>>12844417>makes me really appreciate how much work our precedessors put into fine tuning the subjectThis is something I definitely didn't realize when I first started learning the "big boy" (advanced undergraduate) topics like topology, algebra, rigorous analysis etc.
Everything is so carefully put together, sitting at just the right level of abstraction. The definition of a topological space is such a remarkably elegant way of formulating everything.
So when you're learning topology starting right from this highly evolved definition of a topology, in way you're kind of working "backwards". You solve all these toy problems that do a good job of illustrating why these definitions all exist. You're not really doing the messy work of "real math", hacking through the weeds, figuring out how things work. Instead you're being shown this remarkably powerful theory, and you're being initiated into it, and you won't have an appreciation for how this all came about, how all these topological definitions (like say compactness) extract the ESSENCE of all these important concrete mathematical objects that were studied historically, and motivated all these crucial definitions.
Yeah, just something I find a bit interesting to think about. Learning from textbooks, everything is so damn clean. Even the "messy" topics like analysis are clean as fuck coming out of Rudin. Of course, this isn't a criticism or anything. Theory should be presented in a clean, elegant and pedagogically sound manner.
I've become a lot more interested in the history of mathematics recently. It's cool to see the transition from "concrete" to "abstract". Like going back to Galois, the modern notion of a group didn't exist when he formulated his ideas. He just worked directly with permutation groups, which of course makes perfect sense, them being the prototypical groups.
