To add to micromass' answers, I would say that mathematics simply relies on the laws of logic. And in order to use any mathematics, you certainly need to prove that all your statements are logical!
It would be like using Taylor's theorem on some polynomial, and when some one asks why you can do that, you'd just say, "well the book says so." Well, why does Taylor's theorem work in the first place? "Higher-level mathematics" proves it.