I'm a graduate student in computer science but I have an MA in math, so I think I can offer some perspective on the question of what mathematicians do all day.  I've frequently joked with friends and colleagues that if I had to explain what I do to my early school-age nephew, I'd tell him "I count, color, draw pictures, and talk to people."  And, really, in terms of the actual activities I do, that's accurate.<p>But, the point of all these activities is to solve problems.  It's been famously attributed to Erdos that "a mathematician is a device for turning coffee into theorems."  In this case, of course, theorems are the problems we set ourselves to solve, often without any care or concern whether they reflect anything in the real world.<p>This then brings me to something one of my professors once said: "mathematicians today are doing the science of 200 years from now."  I think it's more accurate to say we're building the tools necessary for the science of 200 years from now, but then his little quip loses something in pithiness.  Nonetheless, when I think about all the mathematics I've studied and the ways in which it's been applied, I think I can agree with the sentiment behind it.<p>Finally, though it was Newton (who would probably consider himself primarily a physicist rather than a mathematician) who said he stood upon the shoulders of giants, I'd like to mention that mathematicians owe perhaps the greatest debt of all scientists to those who came before them.  I was taught things in undergrad math courses that were research-level mathematics only 100 years ago, and I've done homework exercises in grad school that were originally the basis for PhD theses some 70-ish years ago.