Are mathematical operations axioms?

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.

• They are defined by axioms; see e.g. Group theory: the axioms defines the behaviour of composition operation: $\circ$. – Mauro ALLEGRANZA May 13 at 6:48
• The same for $+$ and $\times$ for arithmetic, as defined by Peano axioms – Mauro ALLEGRANZA May 13 at 6:49
• The way the numerical operations are defined gives us a "procedure" to compute the result of every operations: thus, we compute $3 \times 3$ and we find $9$ (exactly as at school...). – Mauro ALLEGRANZA May 13 at 13:33
• Expressed in a more "mathematical" form: we apply the axioms and the definitions. $3 \times 3= 3+ (3 \times 2)= 3 +(3 + (3 \times 1))= 3 + (3+(3 + (3 \times 0)))$. – Mauro ALLEGRANZA May 13 at 13:36
• Now apply the axioms for sum. – Mauro ALLEGRANZA May 13 at 13:36

There are two different ways to discuss binary operations like addition and multiplication. The first is as a function from $$f:(X,X)\rightarrow X$$ so for example if $$X=\mathbb{N}$$ then $$f(n,m)= n+m$$. Binary operations are defined along these lines and you can construct them rather than axiomize them.

That being said, in higher algebra it's typical to treat binary operations with certain properties as axioms. For example in the definition of a group $$G$$ we can express closure as the property that for a binary operation $$*$$ then if $$g*h \in G$$ we say the binary operation is closed which is a subtle and powerful property. Addition and multiplication in the rational numbers are a simple example of this.

We might also want to impose further conditions like there exists an identity element $$e$$ such that for all $$g \in G$$ then $$e*g=g*e=g$$. If the binary operation is addition then $$e$$ would be $$0$$.

Perhaps the most powerful property a binary operation can have is associativity, that is $$(g*h)*j=g*(h*j)$$ which should also be familiar from addition and multiplication. In fact, because associativity fails for subtraction we just get rid of it and add negatives. That for all $$g$$ there exists a $$g^{-1}$$ such that $$g^{-1}*g=e$$ so a simple example for addition would be $$4 + (-4)=0$$. We just add negatives to avoid subtracting because then we can always use associativity. It's similar with division and multiplication, we just don't divide we assume all the reciprocals exist.

So we start with some definitions and decorate them with axioms to derive results. Most mathematical objects will come this way, with some underlying set and then a bunch of assumptions about what you can do with that set. We've packed a lot of detail into the statement "$$G$$ is a group" and it becomes useful to keep the detail hidden and just assume that $$G$$ is a group once the material is understood.

• Well, I understand that there are properties for certain groups of numbers. I don't understand something else, for example, if we work with natural numbers, then their product should also give a natural number 3*3=9, but why can't the answer be 10? Are there any laws that regulate the numerical result, because 10 is also a natural number? – Honza Prochazka May 13 at 13:19

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.