Math is an agreement to explore properties of a well defined set of objects and their relationships using logic.
Philosophy is when you broaden the set of objects to include the human experience but you are still bound by logic, unlike say, poetry (or everyday chatter) where you can use language to deliberately (or often unwittingly) defy logic for emotional effect.
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.
You have to define what a "whole number" is first, and that is a philosophical question. You may think it's obvious [1] what a "whole number" is and how it behaves under addition, but to make it formal you have to write it down. Pretty philosophical if you ask me.
[1] everyone though that, until a few smart logicians showed up in XXth century and proved everyone else wrong.
The effect of a brick falling on someone’s head is not a philosophical question, and neither is one about the common that clearly exists between, say, three horses and three apples you want treat these horses with.
