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I have no problem with the fact that, at bottom, all reasoning, including from assumed axioms in formal systems, requires some element of faith.

Not necessarily. The Axiom of Choice is a great example.

Um...are you arguing with me or agreeing with me? I agree that if you are going to offer formal proofs, you have to reveal what are the assumptions of your formal system--which you accept on faith, since you don't prove axioms, and that would include the Axiom of Choice.

You don't have to "believe" it, you just have to say whether or not you assume it.

What would be the point of assuming Axioms you don't know are true? There are important meta-mathematicians that have touted formal systems entirely divorced from domains of discourse, real or ideal, but you have been touting mathematics as the uber-cover of Everything, so I would not have thought you would be going in this direction.

Surely it depends on what you want to know. You don't have some natural world as a crutch. Theorems proven with or without the AoC are simply done that way. Your system depends on assumptions. Without the AoC, you are constrained and can only gather certain conclusions. With it, you can get a lot more, but at the cost of an assumption.

A lot of scientists do not realize exactly how many things they assume when they use some ancient model. It's essential to understanding when it fails to predict something.

When you strip the particulars from any problem, you get a much better perspective on the one you care about. You can see how it relates to similar problems and how that was attacked. You can then see that biology is chemistry is physics is engineering in a very broad sense. It's the same issues and the same problems over and over again. But this involves dealing in the abstract.