[Doctor Stochastic]

When people use math and other sciences to make predictions about the real world the axioms become matters of faith since you cannot actually choose what is real.

No. People have choice, not faith, in which axiom systems to use in a particular situation. Should an axiom system not yield the desired results (with respect to physics, for example), another axiom system will be invented. Note that only five axioms suffice to give the integers (and prove that such axioms cannot be proved consistent.) A different set is needed for geometry (and these can be used to prove that geometry is consistent). Other axioms are needed for complex numbers, etc. Groups need only four axioms but it's surprising how rich the theory is. (Semi-groups and quasi-groups need only three.)

[unlearner]

"No. People have choice, not faith, in which axiom systems to use"

As I am pointing out elsewhere, and at the risk of being redundant:

We base real-world decisions on the reliability of a loose correspondence between mathematical theories and the real world. That is a "leap of faith". So is the idea that evolutionary theory represents the actual history of life on earth. And I contend that lacking any clear logical proof back to foundational axioms, the theory of the origins of species is itself axiomatic and thus must be accepted on the basis of faith if one is to believe that the theory roughly corresponds to actual events.