Daniel Zwick

Calculus 16300, Section 23

Miklos Abert

5/16/05

I once believed that mathematics is simply the study of numbers. To an extent, this appears to be true, and high school mathematics certainly gives this impression, as most of the work done involves computation, with a few proofs thrown in along the way. Now, it is clear to me that math is not what I once thought it was. In fact, much logical reasoning seems to be involved, and among the vast array of topics covered in the field of math, many are quite abstract.

For instance, this year we learned topology, which totally took me by surprise. One can certainly see the involvement of familiar mathematical concepts, such as distance and continuous functions, yet the manner in which these familiar concepts are defined do not seem intuitive to most students. We learned that the notion of continuous functions, in topology, involving the fact that the preimages of open sets are open, is actually the fundamental definition of continuity. Yet this is quite a difficult idea to wrap one’s head around when, throughout one’s life, one has been taught that a function is continuous if, for all epsilon greater than zero…well, you know the mantra! However, this is only difficult on first glance—in proving that this fundamental notion of continuity is true, it became much clearer that these two definitions do, in fact, agree. Even the latter, familiar definition was reformulated, using the notion of a distance function, when we discussed topology, though this was perhaps, more easily understood.

From this year, it became clear that in order for one to truly ‘succeed’ in mathematics, or perhaps even simply to survive, and certainly in order to enjoy math, one must be willing to put in the proper amount of time. The proof-based nature of this course does not allow one to simply sit back in class and perform some easy calculation to solve a given problem; proofs are often quite difficult, but it appears as though proofs are simply fundamental components of what we call math. In order to ease the difficulty one often encounters in proofs, one must be attentive at all times, as in order to prove a given idea, one often must utilize information learned earlier on.