> In the empirical sciences you have to both check the argument and also test the conclusion against observations

That isn't true, you just test new axioms but most stuff we do in empirical sciences don't require new axioms.

The only difference between material sciences and math is that in math you don't test axioms while in empirical sciences you do.

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.