On a very abstract level, I don't really think it makes a difference. If you start from one sub-domain as a primary foundation in mathematics, you eventually wind up realizing you can define that sub-domain easily, using different sub-domains of math. Realizing that is I think, where category theory sprang from. That last bit, I could be totally wrong on - I tried to teach myself category theory and wound up building webs using abstract algebra and formal logic.

That's obvious if you program, though (to me). What you can do in one language, you can do faster, slower, more concise, more verbose, more symbolic, more literal, and so on, in another. You can build constructions and you can break them apart, depending on what layer of abstraction you can get to. You can call it by it's specific name and functionality, in how the part interacts in relation to the parts around it, or you can call it objects and morphisms.

I might be totally wrong, again, or using the wrong words and confusing everyone including myself (which is why I like to study formal languages, too). But I think that no matter where you start from, you will eventually find yourself in a place where you are scratching your head, confused about what you know, what you don't know, what everyone knows, and what everyone doesn't know.

Oh, and it's probably totally obvious if write proofs, believe in induction, and get very philosophical.