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> I was also trying to make one point: all math is built on arithmetic operations if you dig down to first principles, I am a newb at this stuff, but I believe

> I was also trying to make one point: all math is built on arithmetic operations if you dig down to first principles, I am a newb at this stuff, but I believe (basic) number theory actually builds on recursive structures, and then uses induction to make statements about behavior. https://en.wikipedia.org/wiki/Peano_axioms#Addition https://en.wikipedia.org/wiki/Peano_axioms#Addition What is interesting is how the same approach works for analogous structures like lists - because of the monoid abstraction.

dahart 9y ago
This is interesting! I'm not sure, but I think this may reinforce what I said, not contradict it. I'm suggesting that math operations we know all build on top of addition, not that addition is necessarily the atom. Addition can build on top of something else, and still sit under all higher level math. But that said, I didn't learn addition in terms of monoids, and I don't know anyone else who did. This might be a grand unified theory of math under which addition fits, but physical addition has no recursive monoid analogue. You can add two weights together to get a measurable sum, and it does not depend on a recursive structure that uses induction. You just put two separate things side by side on the scale. Just because we can explain addition using monoids and induction does not mean that addition is made of monoids and induction. Also, I probably wouldn't use Peano theory as the starting place to teach addition. As a pedagogical tool, this is advanced math, and I guess would be a bigger impediment to learning than Greek symbols, for beginners. Right?