[ColinWright]

Arithmetic is to mathematics as planing a piece of wood is to building a piece of furniture, or as typing is to programming. It's a critical, underlying skill, but it's not the whole thing.

The question of what mathematics might actually be is like the question of what pornography is. I can't define it, but I know it when I see it.

[beejiu]

From my experience of professional mathematicians at university, arithmetic was usually their worst skill. They'd quite often struggle (for a short moment) to multiply, say, 12 and 16 together. The public perception of a mathematician is somebody who remembers Pi to a 1,000 digits. A real mathematician doesn't care less.

[lostcolony]

This. So much this. Arithmetic is bookkeeping, it's turning an elegant abstract process into an ugly solved instance through rote mechanical labor.

In most of my college math classes, I largely 'got' it, but what killed me, and many other students, was the arithmetic. We could take multiple integrals, but dammit, 2*3 /= 5. Thank God for partial credit.

[barrkel]

I see mathematics as the act of using symbols and rules for manipulating those symbols to generate statements that are true if the axioms are true. (The statements are generally more interesting if there is a correspondence with real life somewhere. But there doesn't need to be one. Abstract nonsense and all that.)

That's a fairly vague and and non-rigorous statement. Making it rigorous and specific could possibly involve mathematics, if desired.

So in this respect I see arithmetic is a bit like an instance (in the OO sense) of the act of mathematics, but mathematics is a class of thinking and, especially, expressing. In particular, the fact that most arithmetic has a fixed set of rules and you don't generally invent new rules consistent with some meta-rules - the algorithm being performed could be trivially done by a computer - suggests to me that it doesn't really require much mathematical thinking.

When I was younger (mid-teens, and aspiring programmer), adults used to ask me if mathematics was important for programming. I would reply that it is not, that it is very rare for complex arithmetic or calculus etc. to be useful in most programs. That statement was true for both how I and those I was talking to saw maths at the time. But now, looking back, I think programming does actually use some mathematical thinking - real mathematical thinking - albeit not requiring anything like the same level of rigour. You invent your symbols, and compose them to solve the problem, and try to ensure invariants are preserved, and convince yourself that all cases are handled and the result won't have holes in the proof - i.e. bugs.

[threatofrain]

In your description, mathematics is the pure study of theoretical computing machines. However, I think that the act behind mathematics is what people are interested in -- general or capricious analogy-making ability. That's what we want to develop in young people.

We want the thing or process that operates the math, including novel operations for novel situations, and I believe that analogy-making is that thing.

And this is may be why people have an emotional resistance to arithmetics in school. Because arithmetics is an instance of math, but what people are practicing is a particular instance of operation, rather than the skill of operation.