Symbol and Meaning

I made an abbreviated effort to express my current epistemology of mathematics, in which symbols take on a primary role, to one of my fellow grad students. His response was, "But they're more just than symbols."

Just symbols? I can see where he's coming from. Anyone who's tried to explain mathematics to someone else knows the frustration of trying to express concepts as more than mere symbolic manipulation. As grad students we're expected to teach mathematics to students who are often pretty uninterested in math, and who have often been taught their whole lives that math is nothing but symbols which have no connection to the real world. Trying to convince these students that there is real meaning behind these symbols can be exceedingly frustrating, especially if they care about their grades but not about the material. Students who try merely to memorize symbolic manipulations without understanding any of the concepts will do poorly in a college calculus class, and the worst part is they might not ever know why.