"This doesn't cause religious disputes, and indeed mathematicians are happy to jump from one set of constraints to another."
For me, that is the difference between an axiom and a belief. People with beliefs think their beliefs to be better than those of others. With axioms, that is not so. In fact, mathematicians like to doubt their axioms "do we really need the parallel postulate" was, for centuries, a subject of thought"
Actually for a while the parallel postulate was a matter of religious zeal. The studies of consistent geometries that had different behaviors for "parallel" lines was, for some reason, profoundly disturbing to some christians who tried to prove that the parallel postulate was not needed (i.e. That non-euclidean geometry was inconsistent). This was a matter of amusement for mathematicians, even though many mathematicians were a bit disturbed by non-euclidean geometry.
Lewis Carrol, who was a mathematician and some kind of Anglican, couldn't stomach complex numbers. So religion can become confused with axioms, but it tends to start with religion, not axioms.
For me, that is the difference between an axiom and a belief. People with beliefs think their beliefs to be better than those of others. With axioms, that is not so. In fact, mathematicians like to doubt their axioms "do we really need the parallel postulate" was, for centuries, a subject of thought"