[broadwaylamb]

I think the issue here is that if mathematics is talking about anything at all, then the search for the first cause tells us what that thing is. This is a philosophical issue separate from the sociological activity of doing mathematics. The latter doesn't need an all-encompassing foundation to continue to do useful and interesting mathematics, so pragmatics shrug the issue off. On the other hand, even if mathematics is purely about "structure", then we should still be able to organize our knowledge in a way that provides a single foundation (even if that means showing that the different formulations are logically equivalent).

[jessaustin]

One suspects that if the search for "foundations" were made rigorous enough to analyze in any meaningful way, someone like Gödel would come along with some sort of impossibility theorem. In other words, if we were to really examine the idea that any particular foundation could be the best/most natural/simplest/whatever, we'd see that it is folly.