CSPC 360a (TD)

Introduction to the History of Mathematics: Certainty, Uncertainty and the Infinite

Instructor: Professor William C. Summers, M.D., Ph.D.
Office Hours: By appointment
email to instructor: (click here)

Fall Term-2003

Thursday 7-9 pm; Room LC 105 (Note Room Change)

SEMINAR ABSTRACTS: Click here for Abstracts of Student Seminar Presentations (to be read prior to class!).

Course Description

This seminar course will consider the history of several mathematical topics from antiquity until the present time. The course will NOT be a mathematics course, but rather will treat mathematics as examples of intellectual problems rather than technical accomplishments. The historical method will be applied to illuminate the role that these mathematical concepts played in various time periods both from the point of view of general intellectual history and of the disciplinary history of mathematics. In order to maintain the focus and allow in depth study, only three topics will be considered: (1) the search for certainty, i.e., the development of the axiomatic method of deductive proofs from fundamental assumptions, and its corollary, probabilistic inference; (2) the puzzles and paradoxes related to the concepts of infinity and infinitesimals, including such topics as the origins of the calculus; and (3) the mathematics of computability, the basic concepts of information theory, computational procedures and artificial intelligence.

The course will be accessible to anyone with a prior exposure to some calculus. The readings will be in both original sources, e.g., Euclid, Aristotle, Newton, and secondary interpretive sources. The seminar will examine some of the primary sources in detail, so that the seminar participants can gain experience in first-hand close reading of complex technical material in a cooperative and mutually reinforcing environment.

A final paper (~ 15 pages) will be required on some mathematical topic (broadly conceived), not necessarily related to one of the main themes in the seminar. For the midterm evaluation, there will be a short take home essay assignment (3-4 pages).

Texts and Papers: There will be a course packet of the readings marked (*) available at York Copy. The following text should be purchased for the course and is available at Book Haven:

Kline, Morris. Mathematics: The Loss of Certainty. Oxford Univ. Press, 1982 (PB).

Additional Bibliography

Aristotle (The Complete Works of), Edited by J. Barnes. Princeton Univ. Press, 1984.
Boyer, Carl B. A History of Mathematics, 2nd Ed. Revised by Uta C. Merzbach. J.Wiley, 1991.
Dauben, Joseph, Georg Cantor Harvard Univ. Press. 1979.
Hacking, Ian. The Emergence of Probability. Cambridge Univ. Press, 1975.
Hardy, G.H. A Mathematician's Apology. Cambridge Univ. Press, 1967.
Heath, Thomas L. Euclid's Elements (3 vols. Second Edition) Dover, 1956.
Lindberg, David C. The Beginnings of Western Science. U. Chic. Press, 1992.
Newman, James R. The World of Mathematics, 4 vols. Simon and Schuster, 1956.
Newton, I. Philosophiae Naturalis Principia Mathematica trans. by A. Motte and edited by F. Cajori. U. Calif Press, 1960.

Syllabus

Sept 4: Introduction: scope and emphasis; notation; pure/applied tension; motivations.

Sept 11: Search for certainty: early Greek mathematics and Aristotle's logic. Reading: *(1) Aristotle: "Prior Analytics, Book I." (pp. 39-84). (2) Kline, Chapter 1 "The genesis of mathematical truths." (p p. 9-30).

Sept 18: Search for certainty: proofs and Euclid. Reading: *(1) Heath, "Euclid, Book I: Definitions, Postulates, Propositions 1 and 2." pp. 143-246. *(2) Boyer, Ch. 7, "Euclid of Alexandria" (pp. 100-119).

Sept 25: Search for certainty: Probability two kinds Reading: *Hacking, Ch. 3 "Opinion"; Ch. 16, "The art of conjecturing"; Ch. 17, "The first limit theorem."

Oct 2: Search for certainty: Postulate 5 as Trojan horse. Reading: *(1) Heath, "Euclid, Book I: Notes on Postulate 5." pp. 201-220. (2) Kline, Chapter 4 "The first debacle: The withering of truth" (pp. 69-99).

Oct 9: Search for certainty: 19th century foundationalism. Reading: Kline: Ch. 8: "The illogical development: At the gates of paradise"; Ch. 9: "Paradise barred: A new crisis of reason." (pp. 172-215).

Oct 16: Search for certainty: Frege, Russell and Whitehead, Gödel. Reading: Kline, Ch. 10, "Logicism versus intuitionism." pp. 216-244. Ch. 12, "Disasters" pp. 258-277.

Oct 23: Infinity: Zeno, Aristotle and Euclid avoid it. Reading: *(1) Boyer, Ch. 5 "The heroic age." pp. 62-81; *(2) Aristotle, "Physics, Book III, Parts 4-8," pp. 345-354.

Oct 30: Infinity: Medieval religion embraces it. Reading: *(1) Lindberg, David C. Ch. 11, "The medieval cosmos" pp. 245-280 in "The Beginnings of Western Science." U. Chic. Press, 1992. *(2) Boyer, Ch. 14, "Europe in the middle ages." pp. 246- 268.

Nov 6: Infinity: Newton confronts it. Reading: *(1) Newton, Book I: The motion of bodies, Section I: The method of first and last ratio of quantities. pp. 29-39. *(2) Boyer, Ch. 19, Newton and Leibnitz pp. 391-414.

Nov 13: Infinity: Cantor and Russell deal with it. Reading: *(1) Dauben, Joseph, Ch. 6, "Cantor's philosophy of the infinite." in "Georg Cantor" Harvard U. Press 1979. *(2) Hahn, Hans "Infinity" in "The World of Mathematics," ed. James R. Newman. pp. 1593- 1611.

Nov 20: No Class.

Dec 4: Computation, computers, and computability: Turing and his machine. Reading: *Turing, A.M. "Can a machine think?" in "The World of Mathematics," ed. James R. Newman. pp. 2099-2123.

Dec 9: Computation, computers, and computability: What is information? Reading: *Shannon, Claude. "A mathematical theory of communication" Bell Labs Technical Journal, 1948.

Term Papers and Deadlines

Final Paper Due: 5:00 PM, 12 Dec 03; hand in at 332 Bass

Term Paper Information:

Schedule:

By 26 Sept 03: Selection of topics: consult with instructor on sources and research strategy

Weeks of 29 Sept - 3 Nov 03: Conduct research on the topic of final paper: Individual meetings with instructor to discuss final paper: the aims of the paper; the use of research sources; methods of citation; the general scope of the paper.

3 Nov 03: Preliminary draft (1000 words) due. This draft will be posted on the web and will be the basis for seminar presentations scheduled for weeks of 3 Nov 03 and thereafter.

12 Dec 03: Final paper due (3500-5000 words).

Possible topics for papers (definitely not limiting):

Biographical study of one or more mathematicians (e.g., Fermat, Turing, Diophantus, Cardano, Bernoulli family, Lvov school, Bourbaki; Cayley)

Famous proofs: (e.g., Pythagorean theorem; trisection of angle; squaring the circle; duplication of the cube)

Mathematical puzzles

Computational theories and mechanical computing (e.g., calculating machines; slide rule; computers)

Mathematics in art (e.g., perspective geometry; cubism; numerology)

Number mysticism, divination

Axiom-of-choice debates

Math anxiety in girls, the pedagogical debates (e.g., psychological vs physiological)

Foundations of calculus (e.g., definitions of integral, continuity problems)

Mathematics of gambling

History of logarithms

Mathematical notations (history of symbols and representations)

Non-Western origins of mathematics

Arabic development of algebra

Axiomatic attempts at arithmetic (e.g, work of Peano and Dedekind)

Mathematics and music (e.g., well-tempered scale, serialism, harmony and dissonance)

This page last updated: 9 September 2003

email to William C. Summers: (click here)