It's amazing how many people really don't get mathematics. Just as the popular image of science involves test tubes of bubbling stuff and cries of "eureka", most folks seem to think that maths is what happens when Russell Crowe attacks a poor innocent blackboard with greek graffiti.
Nothing could be further from the truth. In actual fact, maths is a climbing frame.
Bear with me here.
Maths proceeds as follows. First, find a bunch of factoids about some area of the mathematical world. Then "climb the frame" by placing them into some sort of abstract context. Then explore the internal workings of that paradigm*. Then climb back down from the general rules to the specific.
If you've done this right, you may find that you've descended to Earth somewhere quite different from the point where your feet left the ground. That's mathematics.
* Philosophy-of-science geeks will note that the process I'm describing is very similar to Kuhnian paradigm formation. The two are very similar, but there are some differences - maybe I'll do a post on it.