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Below are some topic suggestions for an essay for 80-110 Nature of Mathematical Reasoning. Feel free to suggest an own topic that is related to the material presented in class. Any connections between mathematics and philosophy, psychology, sociology, or history are worth considering.
The essay should be 5-10 pages long and include a short presentation of technical/mathematical material. Make sure to also discuss the connection of the topic to the material presented in class.
The paper topic and an outline should be discussed with the instructor no later than two weeks before classes finish. The essay should be handed in one week before classes finish.
The topics marked with a are the ones I find most interesting myself, but that shouldn't affect your decision too much!
Exposition of the development of number systems that eventually led to the introduction of a symbol for zero (Babylonian, Indian, Egyptian mathematics).
Literature: Kaplan, Ifrah.
Discuss the mathematics needed to organize the construction of the great pyramids in Egypt. This includes the design of the building itself, the calculations of the amounts of material needed, the food for the workers, and many other things.
In the course we've touched upon the history of mathematics only in the Western hemisphere, but considerable developments took place also in India and China. Present these developments.
Literature: Struik, Kline
Find out how Thales proved the theorem that every triangle inscribed in a semicircle with one side at the base of the semicircle is a right triangle.
Describe the history of the five platonic solids (regular polyhedra): How did the Greeks know there are only five of them? What role did they play in Greek or other thought (e.g., in Kepler's astronomy)?
Critically discuss Euclid's argument for the Pythagorean theorem.
What happened to the idea of axiomatizing a field of knowledge in the time after Euclid? Find out, and write a short history about it.
Find out and describe how the meaning and use of the term axiom changed from the ancient Greeks to modern times.
Find out and describe how the meaning and use of the term hypothesis changed from the ancient Greeks to modern times.
Find out and describe how the meaning and use of the term model changed from the ancient Greeks to modern times.
Explain and compare the basic calculating algorithms with Roman numerals. Maybe you can invent a new, improved algorithm?
Describe how it works, and when it was used.
Summarize the development of the Arabic numerals in Europe.
Exposition and discussion of the development of various methods for calculating the value of pi in the course of history up to modern times.
How did the view originate that people regarded the Earth to be flat, despite the fact that already ancient civilizations knew that it is a sphere?
Find out how some mathematicians died, and write about it (e.g. the discoverer of the irrational numbers was allegedly drowned, Galois died in a duel).
Discuss the connection between mathematics and music. The Pythagoreans could be a starting point, but also symmetries in Bach's music or repetitiveness in contemporary music could be discussed.
Clifton Fadiman, Fantasia Mathematica and The mathematical magpie are nice collections of excerpts from the literature and poetry that deal with mathematics.
Write about how mathematics is presented in the literature and how this actually reflects the nature of mathematical reasoning as presented in the 80-110 course.
Discuss the importance of Galileo's familiarity with mathematics for his works on physics.
Literature: Dana Sobel, Langford.
Riemann's lecture is regarded as a fundamental paper in systematizing non-Euclidean geometries. Write a summary of the talk. [Remarks: I haven't read the paper myself yet, so I don't know how technical/difficult it is, but it definitely is interesting.]
Exposition of the history of non-Euclidean geometries (18-19th century).
Discussion of the emergence of set theory in the late 19th century and the paradoxes of set theory that were discovered later.
Discussion of various kinds of paradoxes, using Quine's article `The ways of paradox'.
Describe different logics, e.g, classical, intuitionistic, modal, temporal What are the differences, and which is the right one?
Write a crystal-clear exposition of a proof presented in class (or another of similar difficulty). Explain what it proves, when it was proved for the first time, why it is important, etc.
Discuss the difference of direct and indirect proofs. What are reasons one could be against using proofs by contradiction in mathematics?
This gives you the opportunity to see what current philosophers of mathematics talk about!
Stewart Shapiro presents an overview on questions and positions in philosophy of mathematics in chapter 2 of his introduction to philosophy of mathematics Thinking about mathematics (1999). Present and contrast the positions in a concise fashion.
What could they be, and how do we know about them? Beware: you could probably spend your life thinking about this question, and write a PhD about it...
On the use of machines in mathematics: From Leibniz' dream of a "calculus ratiocinator", to Hilbert's program, and finally to the use of computers in finding proofs. Expectations, achievements and limitations.
Pick a famous female mathematician, like Hypatia (Greek) or Emmy Noether (20th century), and write about her life and her mathematical achievements.
Read Andrea Nye's `Words of Power' and discuss it.
Discuss the problems and difficulties that can arise when learning mathematics. If you have own experiences of this sort, you can write about those.