Massachusetts Institute of Technology

Department of Mathematics

18.995 Cultural History of Mathematics

Fall 2009

Is mathematics a purely intellectual exercise isolated from social influences? Does the development of mathematics follow its inner logic, or is it subject to the pressures and biases of the time? Do mathematicians think differently if they live in different cultures? This discussion-based seminar will address these questions while exploring patterns of mathematical practice in different historical contexts: Greek antiquity, the Middle Ages, the Scientific Revolution, the Age of Enlightenment, the Romantic era, Victorian culture, and the tumultuous twentieth century. The students in the class will read and discuss short articles by professional historians. The readings link mathematical innovations with contemporary developments in politics, literature, philosophy, and theology. This seminar presents an opportunity to gain deep cultural understanding of some basic mathematical concepts and methods of reasoning.

Discussions

Each week’s readings must be read prior to recitation section. Active participation in the discussions is expected.

Weekly Reading Response

A short reading response (1 page; roughly 300 words) must be submitted via the Stellar class site by 9 am every Monday. PDF format is preferred, but PS, DOC, and TXT are also allowed. A few tentative questions will be provided to stimulate your thoughts, but you are encouraged to raise your own questions.

* avoid generalities, be specific;

* define your personal stance towards issues raised in the readings;

* focus on the points where you disagree, or where you can push the argument further;

* cite examples from your personal experience or from other literature;

* ask provocative questions, even if you do not know the answers.

Your reading response will be made accessible online to other students in the class after the deadline. It may be part of discussion in class. Reading responses will be graded as Check+ / Check / Check–

All the responses must be submitted to complete the course.By the instructor’s permission, graduate students may submit a 12-page final paper instead of the weekly responses.

Grading

*Final grades will be based on:
*Attendance (20%)

Participation in discussions (40%)

Weekly writing or Final paper (40%)

Readings

All required readings are available on the Stellar class site. The recommended readings are on reserve at MIT libraries. The main recommended textbook:

Ronald Calinger, *A Contextual History of Mathematics to Euler* (Upper
Saddle River, NJ: Prentice Hall, 1999).

Primary sources (original writings by mathematicians) are not required, but if you wish to consult them, a collection of primary sources is also put on reserve:

Ronald Calinger, ed., *Classics of Mathematics* (Englewood Cliffs, NJ:
Prentice Hall, 1994).

Topics and readings

09/14 (1) __Introduction__

09/21 (2) __Greek Mathematics and the Idea of Proof__

G.E.R.
Lloyd, “The Conception and Practice of Proof,” chapter 3 in *Demystifying
Mentalities* (Cambridge, UK: Cambridge University Press, 1990), pp.
73-97, 105-111, 120-122.

Recommended:

Calinger, *A Contextual History of Mathematics*, chapters 3, 4, 8.

Raviel Netz, *The Shaping of Deduction in Greek Mathematics: A Study in
Cognitive History* (Cambridge, UK: Cambridge University Press, 1999).

09/28 (3) __Medieval Mathematics: Theology and Commerce__

Mark Thakkar, "Mathematics in Fourteenth-Century Theology," in *The
Oxford Handbook of the History of Mathematics*, eds. Eleanor Robson and
Jacqueline Stedall (Oxford, UK: Oxford University Press, 2009), pp. 619-638.

Joel Kaye, *Economy and Nature in the Fourteenth Century: Money, Market
Exchange, and the Emergence of Scientific Thought* (Cambridge, UK:
Cambridge University Press, 2004), pp. 200-220.

Recommended:

Calinger, *A Contextual History of Mathematics*, chapter 12.

10/5 (4) __Early Modern Mathematics as a Voyage of Exploration__

Amir R. Alexander, "Exploration Mathematics: The Rhetoric of Discovery
and the Rise of Infinitesimal Methods," *Configurations*, vol. 9, no. 1
(Winter 2001): 1-36.

Recommended:

Calinger, *A Contextual History of Mathematics*, chapter 14.

10/13 **Tues** (5) __Mathematics as Cultivation of Virtue in the
Scientific Revolution__

Matthew L. Jones, "Descartes’s Geometry as Spiritual Exercise," *
Critical Inquiry*, vol. 27, no. 4 (Summer 2001): 40-71.

Recommended:

Calinger, *A Contextual History of Mathematics*, chapter 15.

Matthew L. Jones, *The Good Life in the Scientific Revolution:
Descartes, Pascal, Leibniz, and the Cultivation of Virtue* (Chicago:
University of Chicago Press, 2006).

10/19 (6) __The Scientific Revolution: The Mathematization of Nature?__

Steven Shapin, "Certainty and Civility: Mathematics and Boyle’s
Experimental Conversation," chapter 7 in *A Social History of Truth*
(Chicago: University of Chicago Press, 1994), pp. 310-354 (skim pp.
338-348).

Recommended:

Calinger, *A Contextual History of Mathematics*, chapters 16 and 17.

10/26 (7) __Mathematics as Rational Calculation in the Enlightenment__

Lorraine Daston, "Enlightenment Calculations," *Critical Inquiry* 21
(Autumn 1994): 182-202.

Lorraine Daston, "Fitting Numbers to the World: The Case of Probability
Theory," in *History and Philosophy of Modern Mathematics*, eds.
William Aspray and Philip Kitcher (Minneapolis, MN: University of Minnesota
Press, 1988), pp. 221-237.

Recommended:

Lorraine Daston, *Classical Probability in the Enlightenment*
(Princeton, NJ: Princeton University Press, 1995).

11/02 (8) __Romantic Mathematics: Beauty, Rigor, and the Genius__

Amir R. Alexander, "Tragic Mathematics: Romantic Narratives and the
Refounding of Mathematics in the Early Nineteenth Century," *Isis* 97
(2006): 714-726.

Judith V. Grabiner, “Who Gave You the Epsilon? Cauchy and the Origins of
Rigorous Calculus,” *The American Mathematical Monthly *90 (1983):
185-194.

Recommended:

Judith
Grabiner, *The Origins of Cauchy’s Rigorous Calculus* (New York: Dover,
2005).

11/09 (9) __Oral and Written Cultures in Nineteenth-Century Mathematics__

Andrew Warwick, "A Mathematical World on Paper: Written Examinations in
Early 19th Century Cambridge," *Studies in the History and Philosophy of
Modern Physics*, vol. 29, no. 3 (1998): 295-319.

David E. Rowe, "Making Mathematics in an Oral Culture: Göttingen in the
Era of Klein and Hilbert," *Science in Context* 17 (2004): 85-129.

Recommended:

*
*
David E. Rowe, “Klein, Hilbert, and the Göttingen Mathematical Tradition,”
*Osiris*, 2nd series, 5 (1989): 186-213.

Andrew Warwick, *Masters of Theory: Cambridge and the Rise of
Mathematical Physics* (Chicago: University of Chicago Press, 2003).

11/16 (10) __Mathematics and the Arts: The Rise of Modernism__

Jeremy Gray, "Modernism in Mathematics," in *The Oxford Handbook of the
History of Mathematics*, eds. Eleanor Robson and Jacqueline Stedall
(Oxford, UK: Oxford University Press, 2009), pp. 663-683.

Recommended:

Jeremy J. Gray, *Plato’s Ghost: The Modernist Transformation of
Mathematics* (Princeton, NJ: Princeton University Press, 2008).

Leo Corry, "How Useful is the Term ‘Modernism’ for Understanding the
History of Early Twentieth-Century Mathematics?" in *Modernism in the
Sciences, ca. 1900-1940*, eds. Moritz Epple and Falk Mueller (Berlin,
Akademie, forthcoming).

Linda Dalrymple Henderson, *The Fourth Dimension and Non-Euclidean
Geometry in Modern Art* (Princeton, N.J.: Princeton University Press,
1983).

Joan L.
Richards, *Mathematical Visions: The Pursuit of Geometry in Victorian *
*
England *
(Boston:
Academic Press, 1988).

11/23 (11) __Mathematics and Theology: Views of the Infinite__

Joseph W. Dauben, "Georg Cantor and Pope Leo XIII: Mathematics, Theology,
and the Infinite," *Journal of the History of Ideas*, vol. 38, no. 1.
(1977): 85-108.

Loren Graham and Jean-Michel Kantor, "A Comparison of Two Cultural
Approaches to Mathematics: France and Russia, 1890-1930," *Isis* 97
(2006): 56-74.

Recommended:

Joseph W.
Dauben, *Georg Cantor: His Mathematics and Philosophy of the Infinite*
(Princeton, NJ: Princeton University Press, 1990).

Loren Graham and Jean-Michel Kantor, *Naming Infinity: A True Story of
Religious Mysticism and Mathematical Creativity *(Cambridge, MA: Belknap
Press, 2009).

11/30 (12) __Mathematics and Philosophy: Bourbaki and Structuralism__

David Aubin, "The Withering Immortality of Nicolas Bourbaki: A Cultural
Connection at the Confluence of Mathematics, Structuralism, and the Oulipo
in France," *Science in Context*, vol. 10, no. 2 (1997): 297-342.

Recommended:

Leo Corry, "Nicolas Bourbaki: Theory of Structures," chapter 7 in *
Modern Algebra and the Rise of Mathematical Structures* (Basel:
Birkhäuser, 1996).

Liliane Beaulieu, “Bourbaki’s Art of Memory,” *Osiris*, 2nd series,
14 (1999): 219-251.

12/07 (13) __Doing Mathematics on/with/by a Computer__

Donald MacKenzie, "Slaying the Kraken: The Sociohistory of a Mathematical
Proof," *Social Studies of Science *29, no. 1 (February 1990): 7-60.

Recommended:

Donald MacKenzie, *Mechanizing Proof: Computing, Risk, and Trust *
(Cambridge, MA: MIT Press, 2001).

Slava Gerovitch, *From Newspeak to Cyberspeak: A
History of Soviet Cybernetics *(Cambridge, MA: MIT Press, 2002).