For those on the reading/writing track, there's a five-page paper due
by the end of the term. Five pages is really not a long paper, so
you'll need to be very efficient in making a point. The principle is
the same as for reading responses: Don't Summarize.
Reading biographies is particularly dangerous because you will be
tempted to recap rather than make an argument.
Here are a collection of ideas of topics and sources.
Expand on a reading response: you do not have to read
anything new for your paper---you can just take any of the readings you
have already done and write a more extended response to it.
History of dimension: The Oxford English Dictionary seems to
say that the first use of the word "dimension" in English is from
translations of Euclid's Elements. By the late nineteenth century,
dimension was a concept well enough understood to be the subject of an
interesting political/social satire, E.A.Abbott's classic
Flatland. What is the early history of fourth-dimensional
"In the air": Why is it that over and over again, a question has been
unresolved for hundreds of years, and then several parties independently address it
at around the same time? For instance, non-Euclidean geometry?
Omar Khayyam had some ideas that were followed up by Girolamo Saccheri, but didn't seem to
be going anywhere. Then a hundred years later, four different mathematicians hit on
a new interpretation of these developments, and non-Euclidean geometry exploded into
existence. It's lazy to say that the new ideas were just "in the air."
What happened in the hundred or so years
following Saccheri's work that paved the way for non-Euclidean geometry?
Sources include Kant and Raymond Wilder's (modern) book, Mathematics
as a cultural system.
Math Topic X: Focus on a mathematical topic, like
hypercomplex numbers (Stillwell Chapter 20) or probability or
representation theory, and look into its history, relating it to the
themes of the course.
Georg Cantor: We've read a bit about Cantor, his estrangement from the
mathematical profession, his mental breakdowns, and unusual relationship with the Church.
Read about the philosophical content of his work and its reception in
from Dauben's biography of Cantor, and make a connection with other material
from the course or elsewhere.
Public math contests: there are several famous math duels in
history, especially the one between Cardano and Tartaglia over the
cubic equation and the priority for its solution. What's the history of
this social ritual and how did it give way to more modern ways of
adjudicating priority disputes?
The New Math: this reform movement has a bad rap, but is it
really pedagogically different from standard curricula, and what
historical and philosophical perspectives does it reflect? Source:
texts from the sixties.
Psychology of mathematical invention:
sources include Piaget, Hadamard, Poincare
Maps and projections: different maps of the world have
tradeoffs in terms of their accuracies and distortions; this is
necessary when describing a mainly spherical globe on a flat plane.
Mercator's projection (sixteenth century) has a long pedigree and is
still found in schools. What cultural values are reflected in
Mercator's projection? What are some mathematically viable
History of math departments/History of journals:
focus on one of the institutions that has become essential to the
transmission of mathematical ideas. Look at its early history and try
to find some important feature, now taken for granted as an obvious
feature of the institution, that could have been chosen or designed
The "one, two, three, many" meme: there is a story
frequently told in the literature that such-and-such a culture only has
number words that count up to three, and anything after that is simply
designated as "many."
Proofs and Refutations: find another example in mathematical
history that illustrates (or otherwise illuminates) the development of
mathematics by the method of Proofs and Refutations described by
Big questions: In what way does 2+2=4?
What does a proof do? Is math a kind of science?
What is math, anyway?
(I totally encourage you to tackle questions like these.)
Student Paper Contest in the History of Mathematics
I'll help you beef up your paper for submission to the student paper
contest of the MAA (Mathematical Association of America) if you are
interested. Their deadline is March 31.