Quote by HallsofIvy
...there is not just one "mathematics" there are many different "mathematical theories" (group theory, linear algebra, Euclidean geometry, etc. are all different "mathematical theories"). Mathematical theories are "templates"

That's why I believe that mathematics, taken as a whole, may be a larger (or perhaps more varied) invented structure than discovered reality itself. There seem to be parts of mathematics which don't correspond to parts of reality, spare mathematics, as it were, from a physicists point of view. Perhaps Hamilton's Quaternions are an example? But then what look like spare parts suddenly turn out to be very useful  Tensor analysis for General Relativity for example. One never can tell!
As to the original question:
Quote by nightflyer
: Why is mathematics  a language developed through reasoning  so effective when it comes to describing the empirical world?

, I've set out my ten cents worth in another concurrent thread: (
http://www.physicsforums.com/showthread.php?p=1611545)