First of all: there is a philosophy of mathematics tag in this SE, so this could be the proper site for my question. I do not know how to evaluate this.
When we assume an axiom to prove something, why can this be done? And by this I mean exactly the assumption of an axiom. What is my ``axiom´´ that allows me to assume axioms? Would that be an axiom of mathematics as any other or would that be a principle, as in the Principle of Identity, which I interprete as an axiom of logic and different than the axioms of a mathematical theory. But then my questions start: would not that make the whole use of logic paradoxical? Why would an axiom of logic be valid per se but an axiom of mathematics not? Where the differences from axioms of logic and axioms and mathematics start?
I know there are a lot of questions and this whole discussion may be too vague, but, come on, I am under the tag of philosophy. If there is a good answer about the title question, I would accept it. Thanks for reading.