If Math is Based on Unprovable Axioms and can't be Proven - Why Does Math "Work" so Well?
I have heard that both Mathematics and Religions are based on "axioms" that are by definition unprovable. I have also heard some people argue that Mathematics, like Religion - only provides us with a logical framework to interpret the world with:
If the above is true - why has mathematics (a system of beliefs based on unprovable axioms) proved to be so effective in understanding, analyzing and predicting the behavior of systems and phenomena in the world; whereas the same can not be said about religion (also a system of beliefs based on unprovable axioms)?
In the field of Probability, Kolmogorov's Axioms make fundamentally unprovable statements such as "If A is a subset of, or equal to B, then the probability of A is less than, or equal to the probability of B." These Axioms are regarded as being established, accepted, and self-evidently true - and also serve as a premise or starting point for further reasoning and arguments. In Zermelo-Fraenkel Set Theory, the "Axiom of Union" states that "for any set of sets F there is a set A containing every element that is a member of some member of F" - the legitimacy and the validity of this axiom is generally not up for debate, it is generally considered as "infallible" regardless of your attitudes towards it. The same way, ancient religions made similar unprovable statements that were accepted at the time and also considered as starting points for further reasonings and arguments, such as "an afterlife self-evidently exists and the ratio of your soul's weight to the weight of a feather will decide your position in this afterlife". These statements were also considered to be infallible, regardless of attitudes held towards them.
Yet as we all know, Kolmogorov's Axioms along with other collections of axioms, characterize a system of beliefs (i.e. mathematics) that has been shown to be very useful in consistently and accurately understanding, analyzing and predicting the behavior of systems and phenomena in the world (e.g. science, engineering, technology) - so much so, that mathematics almost appear to be "objectively true" and pass off as unquestioned and "factual knowledge" in our everyday life. Is this (i.e. the power, achievements and the reputation of mathematics) attributable to the fact that perhaps our world in it's su generis form might actually have an underlying mathematical structure and we have correctly identified parts of this underlying mathematical structure and created a set of axioms that is consistent with this mathematical structure? Or is mathematics at it's fundamental level no different than mystical rain dance rituals based on unprovable commandments and tautologies, and it's plethora of successful implementations over the past thousands of years merely an inconsequential result? Are deluges and rain floods understandable through man's turbulent relationship with some God - or are they understandable through atmospheric weather patterns modelled by systems of differential equations?
Even though Godel's Incompleteness Theorem exposed jarring flaws in mathematics as a whole, revealing that "a complete and consistent set of axioms for all mathematics is impossible", - why has mathematics, albeit based on a unprovable axioms, managed to evade the eventual pitfalls of other axiomatic based systems, and managed to work so "well" over the years?
