[mikorym]

> And if the axioms do not hold, the theory is bunk.

You assume the axioms hold. That is what an axiom is: an assumption.

> Mathematics is precise and sound; it's not gospel.

We do think that mathematics is precise. We don't really know if it is sound (unless I am missing something). For example, what is a set? It is not a "collection of things". Rather, it is an object in some mathematical setting.

The topic of whether mathematics is "correct" is something else. We can all go out and build a DIY logical machine from scratch and see for ourselves that the mathematics we use give the results that we expect. Applied mathematics in this sense concerns itself with mathematics that suitably describe real situations.

[drilldrive]

>We don't really know if [mathematics] is sound. My apologies, I was hoping to be concise. I meant to say that mathematics is sound relative to the assumptions, which is exactly correct. But if the assumptions do not hold in reality (and they never quite do), then the theory as a whole is slightly off. I mean to say that mathematics is never 'correct' with reference to reality, but of course is always 'correct' with reference to the axioms.