Math 108: History of Mathematics Spring, 2000

Prof. J. Grabiner

Office phone: 73160; Secretary: 73061; email jgrabiner@pitzer.edu

Office Fletcher 222; Tentative Office Hours MKF 11-12, Th 3:15-4:15

Required Books:

Victor J. Katz, A History of Mathematics: An Introduction (2d ed., Addison-Wesley, 1998)

  • (A copy of the first edition {Harper-Collins, 19931 is on reserve at Pomona Science, but the page numbers are different. The 2d edition is better. This is the best one-volume history in existence, and it is worth buying and keeping.)
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    Plato, Meno (Bobbs-Merrill pb)

    Rene Descartes, Discourse on Method (Bobbs-Merrill pb)

    Jacques Hadamard, Psychology of Invention in the Mathematical Field (Dover pb)

    Week Topic and Assignments (Lectures will focus on specific topics, and will reflect the instructor's view of what is most important; Katz will provide an overview and some details.)

    Jan. 19 Introduction to the course

    Jan. 24-26

    Mathematics in Egypt and Babylon; Mathematics in Africa, the Americas, & the Pacific.

  • Katz, 1-45, 332-341.

    Recommended: H. Frankfort, Before Philosophy; O Neugebauer, Exact Sciences in Antiguity

    M. Ascher, Ethnomathematics; Carl Boyer and Uta Merzbach, A

    History of Mathematics; B. L. van der Waerden, Science

    Awakening; A. Aaboe, Episodes from Early Mathematics.

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    • Jan.31-Feb.2 From Pythagoras to Euclid.

  • Katz, 46-58

    Plato, Meno (all) for Wed. discussion

    Recommended: W. Knorr, The Evolution of the Euclidean Elements; W. Knorr, The Ancient Tradition of Geometric Problems; David Fowler, The Mathematics of Plato's Academy.

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    • Feb. 7-9 Euclid and His Time

  • Katz, 58-101;

    Recommended: Aaboe, op. cit., and (superb!) Knorr, The Evolution of the Euclidean Elements; T. L. Heath, ed., The Thirteen Books of Euclid's Elements (Euclid, in full and in English).

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    • Feb. 14-16. Archimedes

  • Katz, 102-134. (See more assigned reading 5 lines down)

    Recommended: T. L. Heath, ed., The Works of Archimedes; E.J. Dijksterhuis, Archimedes (2d edition has updated bibliography)

    Apollonius, Diophantus, Hellenistic Mathematics Katz, 135-191.

    Recommended: J. Klein, Greek Mathematical Thought and the Origin of Algebra

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    • Feb. 21-23. China and India; Mathematics in the Islamic World

  • Katz, 192-237; 238-287.
  • Recommended: George Gheverghese Joseph, The Crest of the Peacock: NonEuropean Roots of Mathematics; Frank J. Swetz and T. Kao, Was Pythagoras Chinese? Right Triangle Theory in China; J. L. Berggren, Episodes in the Mathematics of Medieval Islam - N. L. Rabinovitz, Probability and statistical inference in Ancient and Medieval Jewish Literature
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    • Feb.28-Mar. 1 The Latin Middle Ages; The Renaissance

  • Katz, 288-326; 327-331; 342-384. Start on 385-430 as well.
  • Recommended: M. Mahoney, "Mathematics," and J. Murdoch and E.Sylla, "The Science of Motion," in D. Lindberg, ed., Science in the Middle Ages O. Ore, Cardan: The Gambling Scholar; J. Klein, Greek Mathematical Thought and the origin of Algebra; Judith Grabiner, "Mathematics," in P. F. Grendler, ed., Encyclopedia of the Renaissance (6 vols., 1999), vol. 4, 66-72.
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    • Mar. 6 Mathematics and early 17th-century science

  • Katz, pp. 385-430.

    • Mar. 8 midterm Examination. Study questions will be provided.

    This is a short-answer examination.

    March 13-17 Spring Break

    March 20-22 Analytic Geometry

  • Katz, 431-448 (for Monday's lecture). Start on 448-503.

    Descartes, Discourse on Method, all. (For discussion Wednesday)

    Recommended: M. Mahoney, The Mathematical Career of Pierre de Fermat; C. Boyer, History of Analytic Geometry

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    • Mar. 27-29. 17th-century mathematics before the calculus; Newton and Leibniz

  • Katz, 448-467; 468-503. Then, 503-543.

    Recommended: I. Hacking, The Emergence of Probability;C. Boyer, History of the Calculus (covers antiquity - 19th c.) M.Baron, Origins of the infinitesimal Calculus (focus on 17th c.) A.R. Hall, Philosophers at War: The Newton-Leibniz Controversy

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    • April 3-5 Mathematics and Physics in the Scientific Revolution; 18th-century mathematics

  • Katz, 544-595, 596-648.

    Recommended: I. B. Cohen, The Newtonian Revolution; T. L. Hankins, Science and the Enlightenment; M. R. Kline, Mathematical Thought from Ancient to Modern Times (relevant chapters); H. Goldstine, History of Numerical Analysis from the 16th through the 19th century

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    • April 10-12 Rigorization of the calculus.

  • Katz, 704-737. Handouts: translations from Cauchy.

    Recommended: J. V. Grabiner, The Origins of Cauchy's Rigorous Calculus

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    • April 17 Mathematical creation (class discussion Mon.); based on Hadamard, Psychology of invention in the Mathematical Field April 19 Study Day: no class. Instructor will be available in office.

    April 24-26 Women in Mathematics: Student Reports in Class..

  • Details and suggested sources will be forthcoming. Each student will have k minutes where

    k + 1 = total time

  • number of students
    •

    • for the in-class report.

    A (roughly) 3-page written version, with title and bibliography, will be due on Monday, May 1.

    Recommended: J-.Alexanderson & D. Albers, Mathematical People: More Mathematical People; Claudia Henrion, Women in Mathematics (her bibliography is especially worth consulting); C. C. Gillispie, ed., Dictionary of Scientific Biographv, 16 vols., the place to start for reliable biographies of any deceased scientific figure; Ann H. Koblitz, Sofia Kovalevskaya: Scientist, Writer, RevolutionarV; Constance Reid, Julia Robinson.

    May 1-3 Modern Mathematics; Bringing it All Together Katz, (skim) chapters 15, 16, 17, 18. Choose one section based on your own mathematical background; problem and questions due Friday May 5. Seniors: Yours is due Monday. Senior- finals TBA.

  • Recommended: T. Hawkins, Lebesgue's Theory of Integration; J.L. Richards, Mathematical Visions: Non-Euclidean Geometry in Britain; G. Moore, Zermelo's Axiom of Choice; Constance Reid, Hilbert
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    • Written assignments and grading:

    (i) Weekly assignments. Every Monday (except Jan. 24, Mar. 20, April 24, and (for non-seniors) May 1 (though non-seniors will do one for May 5 instead), you will hand in both of the following:

    (a) the solution of any problem in the previous week's chapter, from Katz (your choice; it's most valuable if you pick the hardest one you can reasonably do);

    (b) your answer to two of the discussion questions in the previous week's chapter(s) from Katz (again, your choice; pick something that interests you, but do NOT write two questions on virtually the same topic). one page, double-spaced, should be enough for each question.

    For both (a) and (b), GIVE THE PAGE AND QUESTION NTJMBER.!

    Please type (b); make (a) as legible as possible.

    (ii) Midterm Wednesday, March 8. It will be short-answer, and study suggestions will be provided ahead of time.

    (iii) Final Examination (short answer plus essay): Friday, May 1.2, 8 AM. Study suggestions will be provided for this as well; senior will make separate arrangements.

    (iv) Grading: Weekly assignments 35%, midterm 20%, report. 15%, final exam 30%.

    Late work (without compelling reason) docked 10% per CALENDAR day.