Are the basic properties of algebra just axioms? I was reading online and found resources that claim that the following properties are just axioms:


*

*Commutative Property of Addition and Multiplication

*Associative Property of Addition and Multiplication


I was wondering if it is true that these properties are just axioms (i.e. they do not have proofs). Specifically, if they are axioms, I am curious as to how they were "discovered." For instance, $a * (b * c) = (a * b) * c$ has been taught to me from a very young age, so it is almost habitual. How were mathematicians of the past able to reason that this property (and the other properties) is true with complete certainty? Thank you!
**Please note that I am not looking for examples of these properties being true. I understand that 5 * (3 * 4) = (5 * 3) * 4. I am wondering how the logic behind the generalization of these properties came about.
 A: One might argue that they were discovered empirically before they were formalized as axioms. For example, you can visualize the commutativity of multiplication (that is, that $a*b = b*a$) by stacking $a$ many objects in $b$ many rows, and then by stacking $b$ many objects in $a$ many rows and noticing that you will have the same number of objects (and you can do a similar sort of thought experiment to conclude the axiom you stated must be true). Undoubtedly people noticed this property before deciding on the need to formalize mathematics axiomatically.
On the other hand, how were mathematicians able to reason that this is true with complete certainty? Well, they weren't and aren't able to, since you can't prove an axiom. For example, one of Euclid's original axioms of geometry is that given two points, you can draw a line segment between them. There is no way to prove this statement without assuming something else - but it seems very obviously true! Axioms are what works as a good starting point for developing a theory about a given subject, and usually there is a good intuitive basis for why they were chosen.
A: Yes, the properties you mention are axioms. Specifically, they are part of the axioms for defining a commutative ring. It turns out that these axioms are both natural to consider, and also provide a lot of mileage in what they provide. However, there are several areas of math which examine precisely what "goes wrong" and what "stays the same" when commutativity and associativity are not required. See, for instance, division algebras or non-associative algebras
