As my old boss once said, "never confuse mathematics with mere arithmetic."
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...
Seems like you agree more than disagree; math and arithmetic aren't easily separable, if you're being serious instead of cheeky?
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'.
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 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.
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?
I mean, there is theory & practice in every field, technique and discovery. A medical scientist studying the development of aberrant cells in a petry dish has a pretty different job to a pediatrician or radiologist. They generally seem to coexist happily under the "medicine" label. For first year undergraduates, they would probably need to take most of the same classes.
I love talking about both subjects I went to university for actually and both can be difficult to get people interested in. There are massive misconceptions about both.
- They see a shopping bill or a phone number and ask you if you can add them.
- They see random equations, often in the background, and ask you if you understand what it means.
- arithmetics (a.k.a. Number Theory) is a vast branch of modern pure mathematics;
- numeric calculations, i.e. how to do them efficiently, is a very important part of the (applied) mathematics.