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Comment by dahart | Hacker News Reader

I've always liked that line, but it's clearly being cheeky and there are times it's applicable and times it's not.

All arithmetic is math, but not all math is arithmetic.

Except that if you dig deep enough, all math is arithmetic under the hood. Algebra is doing the arithmetic on variables before they're numbers. Calculus is just useful shorthand for compound arithmetic; the derivative is a divide and the integral is a sum. Exponentiation is repeated multiplication, which is repeated addition. Linear algebra is arithmetic on blocks of numbers.

So we abstract over the individual arithmetic operations for our own convenience, there are too many of them. But it's all still fancy arithmetic.

Maybe we should call math "fancy arithmetic" to help all the people who are intimidated by math...

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Actually, the opposite is true. If you dig deep enough, it turns out that arithmetic is far more complicated than most other areas.

Then maybe instead of calling math fancy arithmetic, we should call arithmetic "hard math"? :P

Seems like you agree more than disagree; math and arithmetic aren't easily separable, if you're being serious instead of cheeky?

Well, I was trying to make two points.

1) Arithmetic (number theory) is far harder than most people realise, to the point that its considered one of the most complicated mathematical disciplines.

2) Most subject areas in math bear little to no resemblance to arithmetic, so it doesn't make sense to call all math 'Fancy Arithmetic'.

Sure, understood and I agree. I was also trying to make one point: all math is built on arithmetic operations if you dig down to first principles, so it does make some sense to call all math 'fancy arithmetic'. There is no math at all without the concept of addition sitting somewhere under it.

But I'm not actually proposing that, just sharing a cheeky point of view to contrast @taneq's boss, so don't take me too seriously. ;)

> 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

What is interesting is how the same approach works for analogous structures like lists - because of the monoid abstraction.

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?

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