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I've tended toward "invention" ever since reading Hoffstadter's Godel Escher Bach and trying to wrap my brain around Godel, who essentially proves that mathematics by definition cannot be both consistent and complete; any mathematical system which attains a certain level of "completeness", i.e., it is a powerful enough tool to describe pretty much anything, becomes inconsistent in that perfectly valid expressions can be formed which are paradoxes, both true and not true. And conversely, any system not plagued by this problem is simply too limited (not powerful enough) to be comprehensive.
To me, this sounds like mathematics is fatally flawed in much the same way as Newtonian physics, which turned out to be an over-simplification once we attained the ability to measure things which are very small or very fast. It seems to me if mathematics were an inherent part of "the way things are", it should work perfectly without paradoxes and inconsistencies. The fact that it doesn't work that way suggests to me that it is an invention of the human mind, and like many inventions is not so much perfect as it is a convenient tool.
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