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