Assuming that something is true for constructive reasoning purposes is not "faith-based", it is probabilistic. I don't have "faith" that the sun will rise tomorrow but I assign it a high probability based on priors for the purposes of practical reasoning. We provisionally accept certain axioms because of their high utility but the axioms may be discarded at any time if the utility or correctness is in doubt. Non-axiomatic reasoning systems are a real thing, we use them in computer science.

This entire discussion is predicated on a minimalist set of mathematical axioms being valid, but there isn't even agreement on what that set of axioms is (though it may seem so to a layman). We still manage to get on in the world because we can do highly effective utilitarian reasoning without that being resolved.