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Mathematical axioms have nothing to do with the axioms of the scientific method (and the example you provide does have proofs, mathematical proofs). You can change mathematical axioms all you want, devise perfectly functional mathematics and it changes nothing. There is no inherent connection between mathematical axioms and the acutal observed universe around us.
The reason we use the mathematical axioms that are most commonly taught is that they have proven best as providing descriptive and predictive power for the universe around us. If it turns out that "parallel lines intersect at Trump Tower but at not other point" is a better geometric axiom for describing our universe it would soon supplant the traditional Euclidean axiom.
If you want to call that faith, I can't stop you but I think it is a perversion of the word that removes all meaning.
There is really only one axiom of the scientific method: that the fundamental properties of the universe are consistent across space and time. Yes, this, I suppose, requires an unprovable faith. But to assume a different axiom is to render all observation of our universe pointless. But again, it is an axiom that is supported by observable evidence.
Of course, pretty much all religious thinking is a rejection of that axiom and that is why I hold religion and scient to be unreconcilable.
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