# In what sense are math axioms true?

Say I am explaining to a kid, $$A+BA +B$$ is the same as $$B+AB+A$$ for natural numbers.

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 $$33$$ pebbles in my right hand and $$44$$ pebbles in my left hand. It’s pretty intuitive that the total is $$77$$ 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.