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Indeed, I've been told countless times I don't "need" to learn some branch of math I was interested in (by mathematicians no less).

Also, I've been told I "can't avoid" some branch of algebra (I've never touched it since).

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Those aren't necessarily incompatible. Nobody can avoid linear algebra, but almost nobody actually needs to use or know linear algebra. Heavily symbolic fields like math are probably not great examples though - different methods can be used to reach the same results because it's all fundamentally the same, just differing representations.

didn't godel prove something like the conclusions you can reach depend on your axioms?

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