There are no axioms in philosophy, in math 1+1 = 2. That's an axiom.
Within philosophy however, there's not one type of philosophy that is right, it's subjective so no.
That's not an axiom though. But (depending on your formal system of choice - for example Peano arithmetic) you can usually prove this using your axioms.
What about "a+b = b+a" though? You can prove this in usual peano arithmetic, but there are (well-defined and studied by mathematicians) models, where you can prove that 1+1=2 but not that "a+b = b+a" (for arbitrary a and b). You could say that the truth of this statement is subjective. [1]
A more famous example is an axiom of choice - you can decide that you use it or not, and you'll do a mathematic in a bit different universe depending on that. So I'd say there is more than "right" way of doing math.
1 + 1 = 2 is a definition of 2, as the successor of 1. It is not a particularly fundamental one, as we can do arithmetic without it, just not in any base > 2.
Definitions are axioms, and they are inescapable in philosophy. Rationality cannot get you any further than what follows from your axioms.
