Change, Probability, and Numbers

Fernando Q. Gouvêa

These are the slides from a talk about Change, Probability, and Numbers, a course for non-majors on eighteenth-century mathematics which was part of Colby’s “Integrated Semester” program. I have appended by course syllabus and a description of the contents of the “coursepack” prepared by Colby Custom Publishing.

Slide 1

The Integrated Semester Program

• Four (or three) professors from different departments teach about the same time period (or theme).

• Students take the courses as a block.

• Lots of incentive to build bridges between courses.

• Open to all students, but emphasis is on first-year students.

Slide 2

“Enlightenment and Revolution”

• History course: focusing on Enlightenment and Revolution in France.

• Literature course: focusing on French literature of the second half of the eighteenth century.

• Music course: focusing especially on Vienna and Mozart.

• Mathematics course: focusing on mid-to-late eighteenth century, but with a broader scope than all the others, both in period and in location.

Slide 3

Constraints on the Mathematics Course

• Should be accessible to first-year students, most likely students who would not be planning to major in science or mathematics. In particular, the students could not be expected to know any Calculus.

• Should do at least a little to fill in the picture of science during this period.

• Should be a mathematics course (fulfilling a “quantitative reasoning” requirement), rather than a history of mathematics course.

Slide 4

Choices

• Start with a survey of science and mathematics in the eighteenth century. Special focus on mathematics (pure and “mixed”), which at the time included theoretical physics. Some attention to experimental science.

• Choose a few mathematical topics that were of interest in the eighteenth century, and discuss them in some depth.

Slide 5

Historical Survey

• Text: T. L. Hankins, Science in the Enlightenment.

• Focus mostly on pure and “mixed” mathematics, paying attention to interactions with the rest of the culture.

• Complements: invite other professors to talk about related issues (“Newtonianism” in English poetry, science in eighteenth-century America, etc.).

Slide 6

Topics in the Historical Survey

• Overview: organization of knowledge, science/mathematics and the cultured person, experimental science, the scientist as cultural hero (Newton, Franklin, reading from Voltaire on Newton).

• Newtonian mechanics: the work of Newton, its reception on the continent, development of “rational mechanics,” variational principles, the vibrating string. (Article by V. F. Rickey on Newton’s life and work, discussion of Euler, Maupertuis, Daniel Bernoulli, d’Alembert, Lagrange)

Slide 7

Topics in the Historical Survey

• Newtonian cosmology: the problem of the shape of the Earth, conflict with Cartesian thought, the motion of the moon, the return of Halley’s comet, celestial mechanics, cultural impacts. (Discussion of Maupertuis, Euler, Clairaut, Laplace)

• “Political” mathematics: Borda and Condorcet on voting, the debate on smallpox inoculation, mathematics and science and the French Revolution. “Natural history” and the reaction against mathematical science (Buffon, Diderot, Rousseau).

Slide 8

Supplementary Readings

We provided a number of readings in a “coursepack” created by the Colby Bookstore’s custom publishing service. This included both primary sources and some secondary sources on the period in question.

Slide 9

Mathematical Topics

1. The “metaphysics of the calculus:” what is a derivative? (Readings: Newton and Leibniz on derivatives, Berkeley, d’Alembert on differentials and limits, J. Grabiner on the history of the derivative, modern summary of calculus by R. Hersh.)

2. Probability theory and its applications. (Readings: Pascal and Fermat on the problem of points, extracts from Laplace’s “Introductory Essay,” L. Daston on expectation, risk, St. Petersburg paradox, smallpox inoculation, etc.)

3. Number Theory. (Readings: some of Fermat’s letters, Pascal’s dismissal of Fermat, Euler’s number theory, Euler’s first proof of “Fermat’s Little Theorem.”)

Slide 10

Last-minute changes

• Decided to start with a “what is mathematics” discussion, using Bullock’s “Literacy in the Language of Mathematics” as a theoretical discussion and Euler’s article on the problem of the Königsberg bridges as a sample.

• Put a lot more emphasis on Borda and Condorcet’s voting theory than I had planned originally, responding to student interest.

• Added readings not originally planned. (Coursepack will be better next time!)

• Showed a few videos, including a tape of the Nova program on “The Proof” of Fermat’s Last Theorem.

Slide 11

Assignments

• Lots of reading to prepare for class discussion.

• Weekly reaction papers.

• Your mathematicians: students wrote short biographies and bibliographies of mathematicians of the 18th century. (Later collected and distributed to the whole class.)

• Periodicals: students located, read, and summarized four articles from the periodical literature about science and mathematics in the eighteenth century.

• Term paper: a major paper on a topic related to the class. Topics included: Bernoulli’s theory of the effects of smallpox inoculation, Euler’s algebra textbook, Émilie de Châtelet, scientific and popular views of comets, etc.

Slide 12

How did it go?

• Students were generally interested and participated in class. Discussions had to be carefully planned.

• The reaction papers were very useful, both to keep students reading, and to provide some feedback about how much they were getting out of their reading.

• A lot of help with writing was necessary.

• Semester went by too fast!

Slide 13

What didn’t work well

• Newton readings were too difficult (and there were too many of them).

• The issue of the meaning of the derivative was interesting but never really became important to the students. “Interesting,” but not much to discuss.

• Number theory came late in the course and we didn’t have enough time to do it justice. Students were fascinated by numerical checks that Fermat’s claims were correct, but less fascinated by Euler’s proofs.

• We kept running into the problem of the students’ lack of mathematical knowledge.

Slide 14

What worked best

• Euler on the Königsberg bridges.

• Borda and Condorcet on how to count votes.

• Probability theory, especially when applied to concrete issues.

• Students got a very clear sense that mathematics and science were of general interest during the Enlightenment.

• Students got a very clear view of mathematics as a human activity connected with other intellectual pursuits.

Slide 15

Other comments

• Being part of the integrated semester forced us away from an “internalist” approach to history.

• There could have been a lot more connections with the other courses.

• There were social and psychological issues connected with having a small group of students all taking the same classes.

• Developing students’ writing ability turned out to be a major issue that required a lot of effort and attention.

Slide 16

For more information

Most of the course materials are on the web, at http://www.colby.edu/personal/fqgouvea/177/index.html

A small number of handouts are available here, or email me at fqgouvea@colby.edu.