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I have a general algebra textbook that uses category theory as a foundation, and although I haven't gotten very far into it, my impression is that either catego

I have a general algebra textbook that uses category theory as a foundation, and although I haven't gotten very far into it, my impression is that either category theory or ZFC can be used to derive the other one. As a result, it doesn't matter a whole lot where you start since you get both eventually anyway. If category theory adds tools that make it easier to prove theorems, that's great and useful, but no one really cares if it was foundational or derived.

dwohnitmok 7y ago
This is one main reason why the larger mathematical community does not closely follow these foundational discussions (there are foundations that are strict extensions of ZFC but these extensions are usually conservative). The usual response is that it would be nice if the formal foundations of mathematics corresponded more closely to the informal intuitions of mathematicians, which is arguably not true of set theoretic foundations and perhaps is more true of category theory. Whether or not you agree about the correspondence or whether even if you agree you find it at all convincing is up to you.