History of Mathematics--Term Paper Topics

No papers will be permitted which relate
to John Nash, Fermat's Last Theorem, or the solutions of equations using
radicals.

**Historical emphasis**

1. Post-1965 discoveries about the history of pre-17th Century Chinese mathematics.

2. Same as 1. but for Egyptian mathematics.

3. Same as 1. but for Babylonian mathematics.

4. Same as 1. but for Greek mathematics.

5. Same as 1. but for Indian mathematics.

6. Same as 1. but for Arab mathematics.

7. Same as 1. but for another culture or civilization.

8. The interaction of navigation with the development of mathematics.

9. The interaction of astronomy with the development of mathematics.

10. The interaction of physics with the development of mathematics.

11. The interaction of commercial activity with the development of mathematics.

**Philosophical emphasis**

1. Bishop Berkeley’s philosophy and his critique of the calculus and the responses to his critique.

2. The philosophy of Ludwig Wittgenstein, and, in particular, his views on the foundations of mathematics and his critique of Bertrand Russell.

3. Discuss current controversies concerning the death of proof and/or the stagnation of science.

**Biographical**

1. The life of Archimedes, his principal mathematical AND scientific accomplishments, his predecessors, and his influence on the future of mathematics and science.

2. The life of Galileo, his principal mathematical AND scientific accomplishments, his predecessors, and his influence on the future of mathematics and science.

3. The life of Kepler, his principal mathematical AND scientific accomplishments, his predecessors, and his influence on the future of mathematics and science.

4. The life of Euler, his principal mathematical accomplishments, his predecessors, and his influence on the future of mathematics.

5. The life and philosophy of Pascal, his principal accomplishments, and his influence on the future of mathematics and science.

6. The Bernoulli family, their relationships with one another and with other mathematicians, their accomplishments and their influence.

7. The life of Gauss, his principal mathematical accomplishments, his predecessors, and his influence on subsequent mathematics.

8. The life of Lobachevsky, his principal mathematical accomplishments, his predecessors, and his influence on subsequent mathematics.

9. The life of Sophie Germaine, her principal mathematical accomplishments, her gender-related difficulties, and her influence on subsequent mathematics.

10. The life of Riemann, his principal mathematical accomplishments, his predecessors, and his influence on subsequent mathematics and science.

11. The life of Sonia Kovelevskaya, her principal mathematical accomplishments, her gender-related difficulties, and her influence onsubsequent mathematics.

12. The life of Hilbert, his principal mathematical accomplishments, his predecessors, and his influence on subsequent mathematics.

13. The life of Emmy Noether, her principal mathematical accomplishments, her gender-related difficulties, and her influence on subsequent mathematics.

14. The life of Alan Turing, his principal accomplishments, his predecessors, and his influence on subsequent mathematics.

15. The life of Julia Robinson, her principal mathematical accomplishments, her gender-related difficulties, and her influence on subsequent mathematics.

16. The life of John Conway, his principal
accomplishments, his collaborators, and his influence on contemporary mathematics.
Caution: The source materials for Conway will be contemporary, because
he is a currently active mathematician.

**Thematic**

1. The evolution of the concept of number, from the Greek view that the only numbers are the counting numbers, through the acceptance of rationals, irrationals, complex numbers.

2. The evolution of modern algebraic notation from the work of al-Khwarizmi through Descartes.

3. The evolution of the concept of function. When did the idea first appear? What did Newton think a function was? Fourier? Continue through the 19th Century.

4. The history of the sphere-packing problem first posed by Kepler. Discuss Newton’s argument through Gauss’ work on lattice packing.

5. Discuss the evolution of the concept of "higher-dimensional spaces" and "vectors" in the work of Argand, Hamilton, Grassmann, ...

6. History of Statistics, from its beginning through its development in India and elsewhere in the first half of the 20th Century.

7. History of Combinatorics, with a discussion of the place of the work of MacMahon.

8. History and controversy pertaining to the Data Encryption Standard (DES).

9. History and controversy pertaining to Pretty Good Privacy (PGP encryption).

10. History of the four-color problem from Heawood to its solution in the 20th Century.

11. Trace the history of mathematical induction. Be sure to remark on the place of the Peano postulates in this history.

12. Trace the history of place-value notation.

**Mathematical emphasis**

1. History of the traveling salesman problem.

2. Prove Propositions 13 through 17 of Euclid, Book 13. These are about the five Platonic solids.

3. What should be the rules regarding the proper use of a straightedge with two fixed marks in geometric constructions? (Your results must make it possible to trisect angles by its use.) How do you use this marked straight-edge to trisect angles? Try to develop a theory of what numbers are "constructible" using a compass and marked straight-edge.

4. Discuss the following problem. If you can trisect an angle using geometric tools, can you use the same tools to solve any cubic equation? Some cubic equations?

5. Develop the Law of Cosines along the lines of the geometric proof of the Pythagorean Theorem (see Propositions 12 and 13 of Euclid, Book 2)

6. Develop the idea of continued fractions, at least so far as to be able to get the first two fractional approximations for pi (the first is 22/7).

7. Discuss "surreal numbers" and combinatorial games, a subject that has blossomed since 1970.