Instructor: Professor William C. Summers, M.D., Ph.D.
Office Hours: By appointment
email to instructor: (click here)
Wednesday 130-320 pm Room: TBA
Thursday 130-320 pm; Room: BASS LIBRARY L70
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).
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.
Jan 17: Introduction: scope and emphasis; notation; pure/applied tension; motivations.
Jan 24: 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).
Jan 31: 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).
Feb 7: Search for certainty: Probability two kinds Reading: *Hacking, Ch. 3 "Opinion"; Ch. 16, "The art of conjecturing"; Ch. 17, "The first limit theorem."
Feb 14: 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).
Feb 21: 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).
Feb 28: Search for certainty: Frege, Russell and Whitehead, Gödel. Reading: Kline, Ch. 10, "Logicism versus intuitionism." pp. 216-244. Ch. 12, "Disasters" pp. 258-277.
Mar 6: 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.
Mar 27: 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.
Apr 3: 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.
Apr 10: 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.
Apr 17: 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.
Apr 24: Computation, computers, and computability: What is information? Reading: *Shannon, Claude. "A mathematical theory of communication" Bell Labs Technical Journal, 1948. Part I, pp. 1-19.
Final Paper Due: 5:00 PM, 1 May 08; hand in at drop-box outside room 315 WLH
Schedule:
By 31 Jan 08: Selection of topics: consult with instructor on sources and research strategy
Weeks of 31 Jan - 27 Mar 08: 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.
25 Mar 08: 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 27 Mar 08 and thereafter.
1 May 08: Final paper due (3500-5000 words).