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Vladimir_Nesov comments on What mathematics to learn - Less Wrong
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I don't believe specialization on the level you imply is sustainable. You'd get lost even on upper undergraduate level if all topics outside most of your recommended sets are completely left out. The mathematical maturity that allows you to imagine limiting your study to just a few topics came from having studied the others.
I probably didn't make this clear, but I do agree with you. If you want to study mathematics you need to study lots of areas which is why they have general requirements like that for getting degrees in mathematics.
But if you are an undergraduate looking for a particular subject to get you in the "mathematics groove" I think (1) is a good recommendation for independent study alongside classes, (3) is really a way to lay a foundation for mathematical maturity; certainly a class on proofs and logic would be necessary before almost any of my recommendations.
Again: I didn't mean my list to be a list of "only study these things" so much as "if you want more math, these might be good places to look."