Are you familiar with Russell's teapot? The burden of proof lies upon the person making an assertion or claim.
This is not to say that mathematics is wrong, of course, just that since it is not internally consistent -- and since nothing is -- everything becomes an object of belief at some level. Therefore relying on notions like "the burden of proof lies upon the person making an assertion or claim" is turtles all the way down -- NOTHING is provable and EVERYTHING is a belief.
If that's the case, then it's tautological. In practice, "provability" is partially quantitative, not starkly Boolean (True/False)
I'm not sure Russell's Teapot concept was troubled by his later failures to create an internally consistent body of mathematics:
> Russel's teapot was mathematics as he could not meet the burden of proof at the axiomatic level, yet we still believe it to be true.
You don't need to meet a BoP at an axiomatic level, by definition of what an axiom is. We still accept the analogy of the teapot, because the axioms are accepted by standard. If this acceptance is "faith", it is still far from religious faith, whose "axioms" are not reduced/irreducible, but complex, dependent (on reason) and ad-hoc.
Giant enthymematic jump right there.