There is a debate in mathematics and the philosophy of mathematics as to what the best foundation for mathematics should be, or even if their should be one. The usual suggestions are those mentioned in the subject line thought they are certainly not the only options. I am merely curious what people (especially mathematicians) have to say about this.
I think I'd be of the persuasion that says no particular theory should be considered "foundational" to mathematics. In my opinion, they are just different faces of the same multifaceted coin (maybe there's a theorem involving a coin with more than two sides in there somewhere?).
Really, I can't say, though. I'd certainly be interested to know what you think. It's not a topic I've previously put a great deal of thought into, so if you have insight... well, please share!
I agree. I do not think there should be a foundation in this sense, though many people do. I do however think that metaphysical and epistemological foundations are needed. Perhaps I'll write a hub on mathematical foundations one of these days, it's something I've researched a fair amount, though the details escape me at the moment.