The question is about comparing axioms with belief and religion; so my question is kind of off-topic, but I have to ask: How about philosophy? How does proving a philosophical theorem start? Is it by an axiom or something else? To put it another way, what is the philosophical equivalent of scientific method?
A statement about facts belongs to the natural sciences. A statement about numbers (or dually: sets) belongs to math. A statement about statements, belongs to philosophy. Since the scientific method requires testability against facts, while (in this definition) statements can impossibly be facts in reality, it is not possible to apply the scientific method in philosophy. The same holds true for programming. Any statement about programming languages or programs is either mathematical or philosophical, but never scientific.
