Mathematics is the exploration of the a priori. Historically some axioms have been deemed more real than others. I think Gauss rejected non-eulidean geometry for instance. But this point of view has changed. With abstract algebra, and I believe also computer science, modern mathematics is about exploring connections between structures such as they emerge from stipulated axioms and rules of inference. It is science in the sense that ideas and hypotheses can be tested experimentally. But a proof requires more than non-falsification. Then there is the complication of potentially irreducible computation problems, where essentially a kind of mining of the computational space is the only way forward. This is the new kind of science Stephen Wolfram speaks of.
