Insight and Invention in Geometry
A joint seminar in Mathematics and Cognitive Science
Professor Douglas Hofstadter
Q700 / M490 and M590
Tuesday / Thursday 11:15 a.m. - 12:30 p.m.
Rawles Hall 104
This seminar is about how creative and insightful mathematical thinking works, using geometry as the arena in which to look at the phenomena. With the humble triangle as our springboard, and inspired by the vast variety of triangle centers and the unexpected wealth of relationships tying them together, we will explore Euclidean geometry. This will lead us to projective geometry, and we may dip at least a bit into non-Euclidean geometries of various sorts.
All this geometrizing will be done with an eye on the nature of mathematical understanding and imagery. We will focus on the ways in which intuitive, imagistic understanding of ideas, always striving for greater simplicity, facilitates the invention of new ideas through non-obvious analogical leaps. The key role played, in such leaps, by a subjective esthetic drive, and the variety of qualities that make up this subjective drive, will constitute a companion focus of our attention. Finally, the relationship between the arts of mathematical concept-invention and conjecturing and the art of theorem-proving will be scrutinized.
This course can profitably be taken by both graduates and undergraduates, whether they are in mathematics or not. It has no advanced prerequisites other than an enjoyment of math. On the other hand, it does require students to get directly involved in the ideas of geometry and to carry out creative mathematical explorations. Students whose mathematical background is more advanced will naturally gravitate towards the more abstract ideas, while those whose background is more elementary will tend to stay at the more concrete end of the spectrum. But it is equally possible for new insights into triangles, into mathematical thinking, and into the nature of creativity to pop up at any point along the concreteness/abstraction spectrum.
The seminar will make use of the computer program Geometer’s Sketchpad, which brings visual experience directly into the forefront. From the very start of the course, students will be encouraged to make personal explorations and discoveries, and in the second half of the semester, they will write a paper of five to ten pages describing both their geometrical discoveries and the cognitive pathways (both fruitful and fruitless) that they followed in making them.
Coxeter and Greitzer. Geometry Revisited (MAA). David Wells. The Penguin Dictionary of Curious and Interesting Geometry (Penguin).
There will also be various handouts, some written by myself, others taken from various books and articles.