It's both intuitively obvious geometrically, and algebraically is conceptually tricky to grasp. Or, more precisely it's trivial to decide that arithmetic operations are commutative or not, but tricky to decide correctly for all operations a child knows.

I also think you define integers as successors. Z, S(Z), S(S(Z)) etc. you can prove that a * b = b * a, given that you have defined the + function and * functions.