I think math could be defined as the study of axioms, and theorems you can derive from those axioms.
For example, arithmetic. The axioms are stuff like "zero exists" and "if $a=b$ and $b=c$ then $a=c$" and "every natural number has a number after it" and "$a+1$ means the number after $a$" and "$a+(b+c)=(a+b)+c$", etc.
And using the axioms, you can prove that $1+1=2$:
$a+1$ means the number after $a$.
Thus, $1+1$ means the number after $1$, which is $2$.
And, building off of that, you can prove that $2+2=4$:
$2+2=2+(1+1)$, by the previous theorem.
That equals $(2+1)+1$, since we can move parentheses around.
Since "$a+1$" means the number after $a$, we have:
And geometry is what we get with Euclid's axioms. (Technically, there is non-Euclidean geometry, too...) etc.