[pflats]

I'm going to be rather dismissive in my reply, and for that, I apologize, because I'm not quite sure how else to respond.

This is more or less a non-issue. Thanks to mathematicians building on Euclid for the last 2300 years, we have a system of mathematics built on a few basic principles (that you would not disagree with) and deductive reasoning. If you take a theorem that is accepted as proven, you can almost definitely follow an immense chain of logic back to the fundamentals. It will take you a ridiculous amount of time to do so, but it is possible.

If you're referring to specific debates in the math community (e.g. "I feel that the general math community accepting the axiom of choice was a bad idea") then that's worth being specific about in your post.

[AimHere]

>we have a system of mathematics built on a few basic principles (that you would not disagree with) and deductive reasoning

I think it's even better than that; mathematicians don't necessarily care whether the reader 'agrees' with the axioms, or whether they're in any sense 'true' or 'false'. Mathematics is always of the form 'if these axioms are true, this theorem follows from it'.

The real world and the notions which people consider to be self-evidently true is just some messy slimy gooey gunk best left to psychoanalysts and theoretical physicists and sewer workers and the like.

[pflats]

Oh, I agree, that's the best part. Pick your rules: oh, you picked those seven? You've got a ring; here are your math rules!

In that specific case, though, I figured I'd point out that the basic rules of math aren't usually things people squabble over. (Although I do enjoy a little mathematical philosophy from time to time.)

[pseut]

Even better is when, to everyone's surprise, those seven turn out to be relevant for describing real world phenomena.