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Anonymous at Sat, 7 Sep 2024 21:57:50 UTC No. 16366630
Is pure math just understanding the hierarchy of spaces (topological, vector, measure, metric, etc) and what kind of transformations are possible btn them in the same way science is about categorizing observable phenomena? I have a hard time thinking about learning random theorems in random fields and feel more comfy finding out what kind of things I can do in a measure space and not a vector space, like why are norms important instead of metrics, etc. These kinds of questions are more interesting to me than mastering the proof of the heine-borel theorem for instance. Are there books that take this approach to teaching math or do you need to suffer through 4 yrs of undergrad material to appreciate the answers?
Anonymous at Sun, 8 Sep 2024 00:58:45 UTC No. 16367055
>>16366630
You are describing a mildly retarded version of the ethos behind category theory. And no there's a lot more. But this is an important thing to be able to see regardless.
Anonymous at Sun, 8 Sep 2024 01:39:17 UTC No. 16367220
You don't care about groups (algebraic structures)?
Mathematical concepts by Jost should be a good start
Anonymous at Sun, 8 Sep 2024 07:47:57 UTC No. 16367607
>>16367220
don't they live in vector spaces?
Anonymous at Sun, 8 Sep 2024 08:32:42 UTC No. 16367663
>>16367607
Not all groups can be described as vector spaces. One of the most intriguing applications of groups is elliptic curves where you can encode operations on a 2d projection of an elliptic curve. It's used in encryption, and to prove theorems about specific groups using the theory of elliptic curves.
Anonymous at Sun, 8 Sep 2024 08:34:46 UTC No. 16367670
>>16366630
Well that's the beautiful platonic side of math you find in algebraic, analytic and topological stuff. But there also exists the unstructured horror of fields like combinatorics, numerical optimization and statistics.
Anonymous at Sun, 8 Sep 2024 08:36:44 UTC No. 16367673
>>16367670
I love combinatorics solely for generating functions. Actually insane how many types they are and how they are used in proving really though conjectures.
Cult of Passion at Sun, 8 Sep 2024 08:39:53 UTC No. 16367680
>>16367673
Its so simple and I found that made it super universally applicable. Rudimentary, axiomatic, easily missed because its so "simple", like Arithmetic or Number Theory.
Anonymous at Sun, 8 Sep 2024 08:45:24 UTC No. 16367684
I should have studied math, I want to know enough to have these abstract thoughts about it.
I don't think I'll ever do another undergrad, what do?
Anonymous at Sun, 8 Sep 2024 08:59:40 UTC No. 16367702
>>16367684
You definitely have to like it. It's really hard to learn math without enjoying it to an extent, especially when you venture out of highschool math and calculus.
For any advanced math subject, you'll need to know linear algebra and proof techniques structure + formal logic (which are arguably easier than calculus). And some you'll also need to know said calculus. After covering all these bases, you can start appreciating more advanced subjects, it just becomes a question of what interests you and what problems motivate you to explore a field.
sage at Sun, 8 Sep 2024 11:09:23 UTC No. 16367853
>>16366630
>retarded thread pic
Come back to /pol/, they need you.
Anonymous at Mon, 9 Sep 2024 06:55:09 UTC No. 16369322
>>16367663
don't curves live in vector spaces? It's hard to imagine a curve without a notion of vector distance.
Anonymous at Mon, 9 Sep 2024 07:00:46 UTC No. 16369325
>>16366630
>science is about categorizing observable phenomena
This board is worse than /b/
Anonymous at Mon, 9 Sep 2024 07:12:06 UTC No. 16369337
>>16369322
What about curves in projective space?
Anonymous at Mon, 9 Sep 2024 07:35:14 UTC No. 16369352
>>16369325
lmao, what's wrong about that, even within physics, what you observe is categorized btn quantum mechanics and classical mechanics
Anonymous at Mon, 9 Sep 2024 11:12:49 UTC No. 16369499
I'd like to think about it as finding a sense of size and distance of sets as your notion of abstractness of space increases. So each space has its own weird sense of distance and the more grounded you are about them, the more you start to understand math.