system
Given the following post, respond as if you're commenting on Reddit:
human
Doesn't a mathematical statement/theorem, etc. depend on the axioms you use, the logic you use in your proof, and proof methods you accept?

Mathematical equations are often treated as hard and fast truths.  However, a mathematical 'truth' can be different depending on what axioms you are working with (just think of set theory).  Secondly, the logic you use (classical logic, intuitionistic logic, or one of the other logics) can affect whether a mathematical statement is 'true' or not.  Thirdly, a mathematical statement may be proven with a certain proof method but unable to be proven with a different one.

So, my question is: how can anyone claim mathematics consists of hard and fast, eternal, undeniable and undebatable truths?  Or, do I misunderstand something?
gpt
As said in other comments, mathematical theorems called "true" are true in relation to certain axioms and inside certain systems that are widely used (Zermelo-Frankel Set Theory etc). If you assume difderent axioms and/or use a different system, those sentences can become false or meaningless.

The reason that most mathematical works take these axioms to be true and use the same systems is that they seem to work *in real life*. That is, if you try to build a bridge using mathematics where the sentence "2+2=5" is true, that bridge will probably fall down. Some mathematical tools became a convention because they can used to make predictions about reality that usually turn out to be true.

No math can justify it's own validity, but the Universe seems to work according to some kinds of math and not according to others.