Say I am explaining to a kid, A+B is the same as B+A for natural numbers.
The kid asks: why?
Well, it’s an axiom. It’s called commutativity (which is not even true for most groups).
How do I “prove” the axioms?
I can say, look, there are 3 pebbles in my right hand and 4 pebbles in my left hand. It’s pretty intuitive that the total is 7 whether I added the left hand first or the right hand first.
Well, I answer that on any exam and I’ll get an F for sure.
There is something about axioms. They can’t be proven and yet they are more true than conjectures or even theorems.
In what sense are axioms true then?
Is this just intuition? We simply define natural numbers as things that fit these axioms. If it’s not true then well, they’re not natural numbers. That may make sense. What do mathematicians think? Is the fact that the number of pebbles in my hand follows the rules of natural numbers “science” instead of “math”? Looks like it’s more obvious than that.
It looks to me, truth for axioms, theorems, and science are all truth in a different sense, isn’t it? We use just one word to describe them, true. I feel like I am missing something here.
You only need to “prove” an axiom when using it to model a real-world problem. In general, mathematicians just say “these are my assumptions (axioms), this is what I can prove with them” – they often don’t care whether it models a real-world problem or not.
When using math to model real-world problems, it’s up to you to show that the axioms actually hold. The idea is that, if the axioms are true for the real-world problem, and all the logical steps taken are sound, then the conclusions (theorems etc.) should also be true in your real-world problem.
In your case, I think your example is actually a convincing “proof” that your axiom (commutativity of addition over natural numbers) holds for your real-world problem of counting stones: if I pick up any number of stones in my left- and right-hands, it doesn’t matter whether I count the left or right first, I’ll get the same result either way. You can verify this experimentally, or use your intuition. As long as you agree that the axioms of the model fit your problem, you should agree with the conclusions as well (assuming you agree with the proofs, of course).
Of course, this is not a proof of the axioms, and it’s entirely possible for someone to disagree. In that case, they don’t believe that the natural numbers are valid model for counting stones, and they’ll have to look for a different model instead.