[sweetjazz]

It's too complex to be one or the other.

Sense in which mathematics is invented: People get to choose the axioms that they are interested in; what mathematicians study is the invention of human minds.

Sense in which mathematics is discovered: Once the basic axioms of study are agreed upon in a field, the mathematical works consists in deducing what follows. What matehmaticians do is discovery of logical consequences.

Since most of the mathematicians working at any given point fall into the latter, I would tend to describe mathematics more as a process of discovery then of invention. But leaving out invention altogether would miss out some of the key developments in mathematical thought.

The standard "invention" example is the discovery of a non-Euclidean geometry, one that satisfied all the axioms of Euclid except the parallel postulate. On the one hand, a realist could describe this as the discovery of a logical system which has always been in some form of abstract existence. Yet, it is hard to deny that what took place was clearly an inventive process. The mathematicians involved "found" a model for an axiomatic system that was being sought because of its own intrinsic interest.

In the end, invention and discovery are very similar. Can we not describe Edison's invention of the light bulb as the discovery that certain electromagnetic principles can be used to build a physical device that illuminates nearby objects? Can we not describe Maxwell's discovery of the laws of electromagnetic radiation as the invention of a mathematical formulation that accurates predicts electromagnetic phenomena?

I think that the main difference in the two words is the connotation of the process involved in obtaining the result. And as far as mathematics goes, both connotations convey aspects of what really goes in the creation of new mathematics.