To me, the morally correct proof of commutativity of addition is the Eckmann-Hilton argument, aka you pick the right side up, move it to the left, and put it back down. Exercise to the reader as to how to actually do this on the integers without defining addition the usual way.
At least, when I tried to adapt it it seemed like it worked—I’ve not actually seen someone else do it, despite how obvious it is to try applying it to integer addition.
This also lets you prove associativity from the interchange law for whatever two additions you came up with in the exercise.
With the negative and rational numbers, personally I would say “well, you should be able to split 1 into two parts that add to it; now, what would that require?” and likewise for the negative numbers. For exponentiation by a rational, it’s “I should be able to split x into two parts that multiply to it!”
The ring theoretic acknowledgement of the fact that multiplication is morally repeated addition is that you have a ring homomorphism from the integers to any ring, though the problem is that if 1+1...+1 = 0 then it’s not injective; but the integers mod n will be for some n. Likewise, the rationals embed in any field where 1+1...+1 ≠ 0. Though this fact does get mentioned in books and classes, it should perhaps be moved up as one of the first exercises.
What I want to know, though, is what your problem with the algebraics are! What else are you to do besides roots of polynomials? Though, I will say that I wish I had realized much earlier that rational functions really are ‘everything algebra can do’. For roots, a root of a rational function must be a root of the numerator, so you can just look at polynomial roots.
There are alternate constructions of the reals. I think the Eudoxus reals are philosophically interesting—they are based on the integers, and use a intuitive idea of counting how multiples line up. The nonstandard analysis stuff is supposedly more intuitive, but I don’t know if their version of the reals is too.
The reason why classes and books don’t talk about hyperoperators is that, aside from the Ackermann function, I’ve never seen any beyond exponents actually come up. Unless you meant that elementary school teachers weren’t telling you that multiplication was repeated addition, in which case I’m mildly horrified.
In case you want an example of unnecessary assumptions, I’ve found that most don’t know that you don’t have to assume that ring or module/vector addition is commutative—it’s derivable from the other axioms.
To me, the morally correct proof of commutativity of addition is the Eckmann-Hilton argument, aka you pick the right side up, move it to the left, and put it back down. Exercise to the reader as to how to actually do this on the integers without defining addition the usual way.
At least, when I tried to adapt it it seemed like it worked—I’ve not actually seen someone else do it, despite how obvious it is to try applying it to integer addition.
This also lets you prove associativity from the interchange law for whatever two additions you came up with in the exercise.
With the negative and rational numbers, personally I would say “well, you should be able to split 1 into two parts that add to it; now, what would that require?” and likewise for the negative numbers. For exponentiation by a rational, it’s “I should be able to split x into two parts that multiply to it!”
The ring theoretic acknowledgement of the fact that multiplication is morally repeated addition is that you have a ring homomorphism from the integers to any ring, though the problem is that if 1+1...+1 = 0 then it’s not injective; but the integers mod n will be for some n. Likewise, the rationals embed in any field where 1+1...+1 ≠ 0. Though this fact does get mentioned in books and classes, it should perhaps be moved up as one of the first exercises.
What I want to know, though, is what your problem with the algebraics are! What else are you to do besides roots of polynomials? Though, I will say that I wish I had realized much earlier that rational functions really are ‘everything algebra can do’. For roots, a root of a rational function must be a root of the numerator, so you can just look at polynomial roots.
There are alternate constructions of the reals. I think the Eudoxus reals are philosophically interesting—they are based on the integers, and use a intuitive idea of counting how multiples line up. The nonstandard analysis stuff is supposedly more intuitive, but I don’t know if their version of the reals is too.
The reason why classes and books don’t talk about hyperoperators is that, aside from the Ackermann function, I’ve never seen any beyond exponents actually come up. Unless you meant that elementary school teachers weren’t telling you that multiplication was repeated addition, in which case I’m mildly horrified.
In case you want an example of unnecessary assumptions, I’ve found that most don’t know that you don’t have to assume that ring or module/vector addition is commutative—it’s derivable from the other axioms.