[griffzhowl]

Empirical science uses both deductive logic to make predictions, and observations to check those predictions. I'm not saying that's all it involves. Not sure which part of that you disagree with

And a lot of what goes on in foundations of mathematics could be described as "testing the axioms", i.e. identifying which theorems require which axioms, what are the consequences of removing, adding, or modifying axioms, etc.