[KayEyeDoubleDee]

So how come we first "invent" an area of mathematics and only subsequently "discover" that it describers Nature

I would argue that historically this is not the order of these two events. It is only recently that a few obscure areas of mathematics have been shown to have physical "manifestations". For instance, I read recently about an example involving the Banach-Tarski paradox.

[TopQuark]

Really? From Euclidean geometry to Rimannian. How about the classical and variational calculi?

I am not sure why you refer to Banach-Tarski. These things, as far as I understand stem from the Axiom of Choice and should not be THAT surprising: once you accept it, you are stuck, for instance, with non-measurable sets.