Tentative Syllabus

TENTATIVE SYLLABUS MATH412 HISTORY OF MATHEMATICS Fall 2015

Instructor Contact Information:

Dr. Charles Groetsch, Citadel Distinguished Professor of Mathematical Science, 321 Thompson. (voice) 843-953-5036, (fax) 843-953-7391, Dept. Office (843-953-5048) email: [email protected]

Text: D. Burton, History of Mathematics; An Introduction, and selected readings.

Course Goals:

Develop the principal themes of basic mathematics (arithmetic, geometry, algebra, and calculus) in an historical context. Provide students with a deeper understanding of basic mathematics and instill within them the confidence to investigate mathematical ideas independently; develop an appreciation of the mathematical contributions of various cultures and of the interplay between history, culture and mathematics.

Objectives: 1. Investigate the origins of numerical and geometrical systems in several ancient cultures. 2. Develop facility of working in various numeration systems and a critical appreciation of different systems. 3. Study the development of axiomatics and the idea of proof in the ancient Hellenist world. 4. Solve mathematical problems using the historical systems and techniques; contrast this with modern methods. 5. Assess and evaluate the mathematical development of various cultures. 6. Study, from a historical viewpoint, the fundamental theorems in each of the following four broad areas of basic mathematics: arithmetic, geometry, algebra, calculus. 7. Investigate the origins of the current mathematical world view of the physical universe 8. Complete a substantial writing project on an aspect of the history of mathematics.

Assessment:

Homework problems will be assigned regularly and must be neatly written and handed in on time. There will be a one-hour in-class closed book test, a required paper, and a comprehensive two-hour final exam. The homeworks will count for 20% of the course grade. The one hour test will count for 25%; the paper will count for 20%, and the final exam will count for 35% of the course grade.

Topical Coverage:

The emphasis will be on intellectual history and the organic development of mathematical systems. Topics include ancient mathematical systems (Egyptian and Mesopotamian); Greek geometry and number theory; the beginnings of algebra in the Islamic world; renaissance algebra; the mathematization of the natural world; and the beginnings of calculus and analysis and more modern topics as time allows.

History of Math Paper:

A 6- 9 page paper is a course requirement. Some suggested topics are listed below, or you may choose your own topic, with the instructor’s approval. Each paper must include each of the following:

1) A narration, perhaps biographical details on a specific mathematician, or background on a school of mathematics or a whole culture.

2) Some mathematics, for example, proofs of some specific theorems and illustrations of their applications and consequences, examples, diagrams, etc.

3) A bibliography containing citations from the paper itself and a guide to further reading.

The paper will be graded on (a) evidence of mastery of the mathematics involved, (b) clarity and originality of the exposition, and (c) indication of how the paper relates to other academic work, or for prospective teachers, plans for using the topic in their classroom.

A short proposal is due by 2 October.

Here is an example of what a proposal might look like:

Title: Irrartionality of e and pi.

My paper will explore the historical development of the famous mathematical constants e and pi; give proofs from the literature that these constants are irrational, and suggest topics for further exploration.

1) The history of pi and e a) pi in various historical cultures b) Archimedes and pi c) approximations to pi d) pi today e) the emergence of e – the work of Euler

2) Some mathematical demonstrations a) A proof that pi is irrational b) A proof that e is irrational c) A remarkable identity involving pi and e

3) Bibliography and Guide to Further Reading

T. Apostle, Advanced Calculus, publisher, date Beekman, A History of pi, publisher, date W. Dunham, Euler: The Master of Us All, publisher, date E. Moar, e the Story of a Number, publisher, date Etc.

Some possible topics for your paper (or suggest your own):

Edmund Halley’s gunnery rule

The Hardy-Littlewood collaboration

Ramanujan

History of iterative methods for solving equations

L’Hospital, Bernoulli and the first calculus text

Maria Agnesi: her life and work

Count Wronsky and the Wronskian

Figurate numbers in Greek mathematics

Newton vs. Leibniz

Marin Mersenne and his connections

Leonardo of Pisa and the Italian renaissance of mathematics

Early function concepts in Tartaglia and Galilei Karl Weierstrass and polynomial approximation

Vieta and the birth of algebra

Eratosthenes and other measurers of the earth

Evangelista Torricelli and his work

The Bernoullis: a mathematical dynasty

Cauchy and the rigorous development of calculus

The birth of probability theory