Paper ideas

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 thinking?**"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 this chapter 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 alternatives?**Ethnomathematics****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 differently.**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 Lakatos.**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.)