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.
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