All pure math can be deducible from axioms doesn't mean that all math can be deducible from a single set of axioms. Rather, it means that for each mathematical proposition there is a set of axioms from which one can deduce it.
My main point is I disagree with the view of mathematics as nothing more than some axiomatic program-- in 1903 many were hopeful that a system (like Russell's formal logic in Principia) could simply generate the truths of mathematics. Gödel shattered that dream.
I just don't see how this follows from Godel. It gives us a more expansive view of math, but I don't see how any fundamental understanding is overturned. I don't see how this takes away from the connection between axioms and theorems. The characterization of math as discovering the logical consequences of axioms is just as true.