Do you believe in god or math?

[CliveStaples]

Blaaaaagggghhhhh. Toothless subjectivity monster strikes again.

But this 1 + 1 = 3 business...

Given how we've defined the natural numbers and the addition of numbers and all that stuffs, statements like...
"Given what we defined '1', '2', '+', and '=' to mean, 1 + 1 = 2 is true"
are tautologies. Which is to say, they are absolutely true, but completely devoid of content. The notion that our numbers can exist as a wholly self-contained system based on axioms is certainly valid. But the numbers aren't going to mean anything until we prescribe some way of relating them to the world in front of us, right?

That's what makes it such a powerful system. Once you know that something has the structural properties of this meaningless number/arithmetic system, you know how to work in it.

Because (perfectly rigorous*) statements like "This list of axioms implies that theorem" are tautologies, they are absolutely true, but have nothing to do with our world. When said math is situated in regard to the real world, it may be false, but it has something to do with our world. Our world appears to play (somewhat) nicely with our mathematics, but this sort of tension isn't going to ever go away.
(*yes, I know that's another idealization. But poo-poo-ing the rigor of math within this conversation is like a black hole calling the sun dark)
If you agree with that last paragraph, I don't think there's anything more to discuss. If you disagree with that last paragraph, I'm not very interested in continuing this discussion.

It's not true that "they have nothing to do with our world". It depends on whether that axiom holds in our world; if it does, then the theorems hold, and for the same reason that they did in the abstract/"meaningless" setting: because it can't be otherwise, given the axioms.