Cognitive Science | Seminar in Cognitive Science
Q700 | 26805 | Hofstadter

2:30-3:45, TR, CRCC, 510 N. Fess
Above section meets with Math M590

TOPIC:  Mathematical Thinking in Geometry

The goal of this seminar is to observe how creative and insightful
mathematical thinking works, using geometry as the arena in which to
look at the phenomena.  Using the humble triangle as our
springboard, we will first explore Euclidean geometry in some depth
(inspired largely by the vast variety of so-called “triangle
centers” and the unexpected wealth of relationships tying them
together), and this exploration will naturally lead us into some
contact with projective geometry and will even have us dip at least
a little bit into non-Euclidean geometries of various sorts.
All of this geometrizing will be done with an eye constantly
kept on the nature of mathematical understanding and imagery,
relentlessly focusing on the various ways in which deep intuitive,
imagistic understanding of ideas, always pushing for greater
simplicity, facilitates the invention of new ideas through non-
obvious analogical leaps.  The key role played, in such leaps, by
some kind of esthetic drive, and the variety of qualities that go
into this drive, will constitute a companion focus of our attention.
Finally, the relationship between free-floating mathematical
invention and the more grounded activity of proving of theorems will
be scrutinized.
This course can profitably be taken by both graduates and
undergraduates, both in mathematics and not in mathematics.  It does
not have any advanced prerequisites (mostly just an enjoyment of
math -- not even a prior calculus course!), but on the other hand it
does require students to get their hands dirty in carrying out
creative mathematical explorations.  Those whose mathematical
background is more advanced will naturally gravitate towards the
more abstract, whereas those whose background in more elementary
will tend to stay at the more concrete end of the spectrum. But
thatıs fine -- it is equally possible for new insights into
triangles, into mathematical thinking, and into the nature of
creativity in general to pop up at any point along the
concreteness/abstraction spectrum.
The course will make use of the elegant computer program
Geometerıs 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.


David Wells, "The Penguin Dictionary of Curious and Interesting
Geometry" (Penguin)

Coxeter and Greitzer, "Geometry Revisited" (MAA)

There will also be various handouts, some written by myself, others
taken from various books and articles.

For additional information please contact Helga Keller