Considering Peano axioms we'll define addition, multiplication and exponentiation operations.
We can then prove that addition and multiplication operations are commutative and associative. The proof for addition operation is pretty straightforward but for for multiplication it a bit more difficult. For example, to prove associativity, we have to prove distributivity first.
Moreover, for the definition itself it is not obvious that commutativity should take place, because the definition is not symmetric.
$a \cdot 0 := 0$
$a \cdot S(b) := a \cdot b + a$
Here $S(b)$ is the "next" function from Peano axioms
This definition is not symmetric with respect to $a$ and $b$, so commutivity is not obvious.
For exponentiation operation neither commutivity nor associativity take place. And I am very curious about the logical explanation why that happens
NOTE: The explanation I am looking for is somehow similar to a Abel–Ruffini theorem which explains WHY there is no an algebraic solution for a general polynomial equation