[jveld]

Here's how I explain "pure math" to people, based on my experience taking a highly theory-oriented linear algebra course and thinking that "this stuff couldn't possibly be useful." Boy, was I wrong...

I think of (pure) mathematics as exploring the structures generated by simple rules. You start with some system of axioms, maybe those of group theory or linear algebra, and you see where it takes you. Often, there are richer, but closely related structures available by adding additional axioms or constraints. For example, add commutativity to group theory and you get abelian groups. Add metrics to vector spaces and you get topology (sorta).

This is useful because the world is full of complex systems that emerge from simple rules. Therefore, when we observe that some system in the real world displays the characteristics of a known mathematical structure, we inherit a bunch of free knowledge about that system.

In practice, most math falls on some spectrum between the above definition and "applied math". Historically speaking, it's a pretty modern idea (although its pedigree begins with Euclid). Before the mid-nineteenth century, mathematicians had indeed been chasing puzzles like "how to find the roots of polynomials" and "can you square a circle using constructions?" for several centuries. And puzzles are certainly not dead, as the millenium prize clearly shows. Number theory also doesn't play nice with this definition. I intentionally ignored that - if I start thinking about the ontology of numbers I risk losing quite a bit of sleep :p.

[tikhonj]

I think number theory fits perfectly: the definition of the natural numbers is dead simple and, well, eminently natural but results in an intricate structure we're nowhere near understanding.

[jveld]

good point - for some reason it hadn't clicked that number theory only involves the integers and not, ya know, complex numbers.