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.
Best Answer
Roughly speaking an axiom is something taken to be true without proof. Many mathematicians have worked to minimize the number of axioms needed define all of the mathematical operations your are familiar with. However, the modern set of axioms used to define mathematical operations are rather tricky and take a fair amount of background knowledge to understand.
So to answer your question, no addition/subtraction/multiplication/division are not axioms, but rather definitions.
But to make matters tricky, you have to define these operations for different types of numbers. For example, the natural numbers 1, 2, 3, etc. The integers -3, -2, -1, 0, 1, 2, 3 etc. Fractions 1/2, 1/3, 4/5, etc. Real numbers pi, e, etc.
As for why 3 times 3 is nine, it depends on the precise way it is defined. But if, for instance, we define multiplication for the natural numbers as repeated addition, then 3 x 3 is defined to be 3 + 3 + 3 and 3 x N = 3 + 3 + 3 + ... + 3 + 3 (where there are N threes). If addition has already been defined, then one follows the previously defined rules to calculated these sums.