Subhotosh,But a+b = b+a is not only restricted to natural numbers.
As I read the article, Peano arithmetic proves commutative property for addition of natural numbers only. Does not include fractions, vectors, matrices ....
So - again I am not a bonafied mathematician - Peano postulate supports commutation properties in natural number field. But does it prove it?
Yes my recollection is that there is a proof in the natural numbers based on the Peano Postulates, and that seems to be what the article that I cited says and what is implied by other articles that I glanced at.
And then if my recollection is not playing me false, we can define integers as sets of ordered pairs of natural numbers and prove it all over again for those sets because of what we proved for natural numbers. What we get out of that is the additive inverse and the additive identity. And then we can repeat that process for rational numbers defined as sets of ordered pairs of integers. What we get out of that is the multiplicative identity and the multiplicative inverse. However, my course on fundamentals did not extend to real numbers (historians never need to deal with irrationals). I personally have to take on faith that mathematicians have proved the fundamental laws of arithmetic for real numbers because I never studied analysis.
At some point, I sort of remember that the extension continues by using isomorphisms and maybe ring theory. We are getting way above my pay grade. I could not prove an isomorphism today if my life depended on it.
I do not claim to have learned much mathematics (although a lot more than most historians bother with), and I have forgotten a lot of what I did learn (all you really need to know in a bank is some arithmetic and to keep your hands off the cash). I just have experience tutoring kids in algebra, pre-calculus, and a few other very basic subjects. But I am 99% confident that all of arithmetic has been axiomatized at a level below the commutativity and associativity of addition and multiplication and the distribution of multiplication over addition.
Of course, ultimately we do have to rest on unproved axioms.