Great Problems of Mathematics: A Course Based on Original Sources

Stimulating problems are at the heart of many great advances in mathematics. In fact, whole subjects owe their existence to a single problem which resisted solution. Nevertheless, we tend to present only polished theories, devoid of both the motivating problems and the long road to their solution. As a consequence, we deprive our students of both an example of the process by which mathematics is created and of the central problems which fueled its development. A more motivating approach could, for example, begin a discussion of infinite sets with Galileo’s observation that there are as many integers as there are perfect squares. This observation seems as paradoxical to today’s students as it did to Galileo. Its ingeniously simple resolution (through a better definition of “size”) is a tremendous educational experience, an example of the kind of education which the German logician Heinrich Scholz characterized as “that which remains after we have forgotten everything we learned”. We have designed a lower division honors course aimed at giving students the “big picture”. In the course we examine the evolution of selected great problems from five mathematical subjects. Crucial to achieving this goal is the use of original sources to demonstrate the fundamental ideas developed