Math is different in that one feels there should be something close to "absolute truth." That is, given a set of axioms, it is possible to verify with complete certainty whether a proof follows from those axioms or not. Though when proofs become so long and abstruse that only a handful of people can even read them, perhaps that's no longer true. In other fields, it's more clear that nothing is known with 100% certainty. That differences of opinion or experimental error could always lead to uncertainty.

Unfortunately, math doesn't really permit this type of truth: if your axioms are strong enough to prove general statements about arithmetic, there is no effective procedure to determine whether an arbitrary proof follows from those axioms.

Did you mean to write “there is no effective procedure to determine whether an arbitrary formula follows from those axioms?”

A proof is exactly how we demonstrate that a formula follows from the axioms.

Still, given a set of axioms, statements will fall into one of three categories. 1) Provably True, 2) Provably False, 3) Undecideable

Claims that a statement is in category 1 are fully verifiable (by providing the proof). The same goes with claims that a statement is in category 2.

Yup, not picking on mathematics at all :-)