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Does it explain why ab = ba?

Timothy Gowers has a few thoughts on this:

https://www.dpmms.cam.ac.uk/~wtg10/commutative.html

That can bring difficulties, as far as I know the commutative property is a fundamental property in many algebra systems. So it's one of the few assumptions from where everything else builds up, not sure how would you go about explaining that.

You can show a few pictures of examples of how it works in real life, but axioms are the basis from which you build everything else. You just accept them as being useful and prove the rest.

ab=ba is not an axiom for integers (where multiplication is defined as repeated addition), or rationals (where multiplication is defined in terms of integer multiplication). I suspect that it is similarly not an axiom in the real numbers (and, by extension, the complex numbers), but it has been a while since I looked of how the reals were defined.

If we consider fields, then the field operation commonanly referred to as multiplication, is defined to be commutative.

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