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.
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] https://en.wikipedia.org/wiki/Robinson_arithmetic