Cognitive Science | Insight and Invention in Geometry
Q700 | 27979 | D. Hofstadter


Insight and Invention in Geometry

Professor Douglas Hofstadter

A joint seminar in Mathematics and Cognitive Science
Fall 2010

M490–27622 and M590–27623
Q450–27972 and Q700–27979

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.

References

H. S. M. Coxeter and S. L. Greitzer. Geometry Revisited (MAA #19).
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.
For additional information please contact htkeller@indiana.edu / Tel.
812.855.6965