Roman Kossak - Mathematical Logic

Yet although most members of our species have some capacity for geometrical and numerical concepts, very few human beings have much capacity for doing or even understanding mathematics. This is often held to be the consequence of bad education, but although education can surely help, there is no evidence that vast disparity of talent, or even interest in, mathematics, is a result of bad pedagogyA paradox: we are the mathematical animal. A few of us have made astonishing mathematical discoveries, a few more of us can understand them. Mathematical applications have turned out to be a key to unlock and discipline nature to conform to some of our wishes. But the subject repels most human beings.

The professor smiled. He liked my answers, and I was happy with them too. It seemed I had understood something and that this understanding was somehow complete. The truth was simple, and once you grasped it, there was nothing more to it, just plain simple truth. Perhaps such rare satisfying moments were the reason I decided to study mathematics.

In my freshmen year at the University of Warsaw, I took Analysis, Abstract Algebra, Topology, Mathematical Logic, and Introduction to Computer Science. I was completely unprepared for the level of abstraction of these courses. I thought that we would just continue with the kind of mathematics we studied in high school. I expected more of the same but more complicated. Perhaps some advanced formulas for solving algebraic and trigonometric equations and more elaborate constructions in plane and three-dimensional geometry. Instead, the course in analysis began with defining real numbers. Something we took for granted and we thought we knew well now needed a definition! My answers at the blackboard that I had been so proud of turned out to be rather naive. The truth was not so plain and simple after all. And to make matters worse, the course followed with a proof of existence and uniqueness of the exponential function f ( x )= ex , and in the process, we had to learn about complex numbers and the fundamental theorem of algebra. Instead of advanced applications of mathematics we have already learned, we were going back, asking more and more fundamental questions. It was rather unexpected. The algebra and topology courses were even harder. Instead of the already familiar planes, spheres, cones, and cylinders, now we were exposed to general algebraic systems and topological spaces. Instead of concrete objects one could try to visualize, we studied infinite spaces with surprising general properties expressed in terms of algebraic systems that were associated with them. I liked the prospect of learning all that, and in particular the promise that in the end, after all this high-level abstract stuff got sorted out, there would be a return to more down-to-earth applications. But what was even more attractive was the attempt to get to the bottom of things, to understand completely what this elaborate edifice of mathematics was founded upon. I decided to specialize in mathematical logic.