Instructor: Professor William C. Summers, M.D., Ph.D.
Office Hours: By appointment
email to instructor: (click here)
Thursday 130-320 pm Room: TBA
SEMINAR ABSTRACTS: Click here for Abstracts of Student Seminar Presentations (to be read prior to class!).
SCHEDULE OF SEMINAR PRESENTATIONS
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 Docuprint (27 Whitney Ave). The following text should be purchased for the course and is available at Labyrinth Books: Kline, Morris. Mathematics: The Loss of Certainty. Oxford Univ. Press, 1982 (PB) (Publisher’s price: $19.95)
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.
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.
Sept 2: Introduction: scope and emphasis; notation; pure/applied tension; motivations.
Sept 9: Search for certainty: early Greek mathematics and Aristotle's logic. Reading: *(1) Aristotle: "Prior Analytics, Book I. Parts 1-6" (pp. 39-47). (2) Kline, Chapter 1 "The genesis of mathematical truths." (pp. 9-30).
Sept 16: Search for certainty: proofs and Euclid. Reading: *(1) Heath, "Euclid, Book I: Definitions, Postulates, Propositions 1 and 2." pp. 153-158; 241-246. *(2) Boyer, Ch. 7, "Euclid of Alexandria" (pp. 100-119).
Sept 23: Search for certainty: Probability two kinds Reading: *Hacking, Ch. 3 "Opinion"; Ch. 16, "The art of conjecturing"; Ch. 17, "The first limit theorem."
Sept 30: 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 7: 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 14: 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 21: 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 28: 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 4: NO CLASS
Nov 11: 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 18: 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.
Dec 2: Computation, computers, and computability: What is information? Turing and his machine. Reading: *Turing, A.M. "Can a machine think?" in "The World of Mathematics," ed. James R. Newman. pp. 2099-2123; *Shannon, Claude.
"A mathematical theory of communication" Bell Labs Technical Journal, 1948.
Final Paper Due: 4:45 PM, 6 Dec 2010; hand in at mail box in room 240 HGS
By 24 Sept 10: Selection of topics: consult with instructor on sources and research strategy
6 Dec 2010: Final paper due (3500-5000 words).
Possible topics for papers (definitely not limiting):