This really deserves a longer and better answer, but I'm struggling to explain. It's a pretty deep conceptual thing, but I'll try.
As a completely uneducated simpleton, it seems bizarre to me
that addition and multiplication are considered "different"
in the deeper explorations of math and number theory.
It seems like multiplication is just an extension of addition.
You're not alone in this, but as you go on in advanced math you find more and more that multiplication is not really repeated addition, it just happens to coincide with repeated addition when that makes sense. The problem/opportunity is that multiplication still makes sense when repeated addition doesn't.
It might be easier to think of this with regard powers. People teach that A^5 is just AxAxAxAxA. You then deduce that A^a x A^b = A^(a+b). From that you start to assign meanings to things like A^0. And A^(-1). But what does it mean to multiply together -1 copies of a number? That doesn't make sense!
And what about A^{\pi} ? How can you have a transcendental number of things multiplied together? It doesn't make sense!
As you get deeper into math you need different definitions of powers, and of multiplication, and you find they they coincide with repeated multiplication and repeated addition, they may have originated with those ideas, but that's not really the best way to think about them, and it's not, in some sense, what they "are".
A poor analogy might be this. To an outsider, Smalltalk and Haskell will kind of look the same. They're programming languages, they do the same things. But they are really very different animals. So multiplication is really a very different animal from repeated addition.
It might be easier to think of this with regard powers. People teach that A^5 is just AxAxAxAxA. You then deduce that A^a x A^b = A^(a+b). From that you start to assign meanings to things like A^0. And A^(-1). But what does it mean to multiply together -1 copies of a number? That doesn't make sense!
And what about A^{\pi} ? How can you have a transcendental number of things multiplied together? It doesn't make sense!
As you get deeper into math you need different definitions of powers, and of multiplication, and you find they they coincide with repeated multiplication and repeated addition, they may have originated with those ideas, but that's not really the best way to think about them, and it's not, in some sense, what they "are".
A poor analogy might be this. To an outsider, Smalltalk and Haskell will kind of look the same. They're programming languages, they do the same things. But they are really very different animals. So multiplication is really a very different animal from repeated addition.