Are mathematical operations axioms?
I will give an example of multiplication, but this also applies to division, subtraction and addition. Idea of multiplication was invented by people to increase/decrease something exactly N times. For example: I want to increase the number 3 three times, the answer of course is 9, but what is the confirmation of this?
Logically, I understand that if I want to increase something three times, it must be three times larger than original, and this is an axiom or it is just an abstract operation to get a product that must be exactly N times larger? What proof that the answer should be exactly this, pure logic? I don't ask about axioms of properties like associative, commutative...
I am not an expert in mathematics, my level of knowledge is high school.
Thank for you answer.