Math 420
Extended Essay #4
Due: Wednesday, May 4 (in class), but please see note below.
For the last extended essay, you have a choice of several
possibilities
- The discovery of non-Euclidean geometry, combined with the move towards
rigor in mathematics resulted in some stresses in mathematics that continue
today. Consider the history of the parallel postulate and its
resolution in the work of Lobachevsky and Bolyai and discuss what the
reaction to these ideas were. Why did Gauss not publish? Why was
this work such a shock to nineteenth century mathematics?
- The crisis continued: The move to increased rigor in mathematics
generated a problem. What constitutes a proof in mathematics?
What things are permitted, and what are not? Look at some approaches
and their history (how did they come about? A good place to start is
the paper from Mathematics Magazine which can be found
at this location.
- More woes. The move to axiomatize mathematics generated still
further problems. One part of the effort would be to figure out an
algorithm for proof. Discuss the life and work of Kurt Gődel
and dissect his famous theorem on axiomatic systems. Discuss what this
means for mathematics.
- Finally, if none of this appeals to you,
pick a mathematician who interests you, and tell that person's story.
Our textbook, Wikipedia, and the St. Andrews website is a good place to
start, but try and look for further information.
As always, essays should be written as an
academic paper (citations, bibliography and the like) and should be about five
pages or so. If you are already looking at one of these topics for your
term paper, please consider tackling something else for this essay.
I am a bit behind in getting this to you.
Because of that, this essay will be accepted without penalty until the time of
our final exam (but please remember that the term paper is also due at that
time).
Any questions? Please ask!
-Bob