Understanding how mathematics is made, and how it interacts with everything else, helps us understand what it is and what it's good for. The theme of these five lectures is Mathematics and the World, with world understood fairly broadly. The first three lectures ask general questions about the relationship between mathematics and the sciences; the last two, focusing on key individuals, address the relationship between mathematics and the psychology and philosophy of discovery. All these topics place mathematics in historical context.

Lecture 1: Why is mathematics the language of science? A historical view

We take for granted that the physical sciences are mathematical, and expect that the biological and social sciences will also use mathematics. We expect mathematics not only to provide precise description, but also to be the instrument of new discoveries. We'll see that the triumph of this view was not inevitable, and owes as much to religion and philosophy as it does to scientific practice. Key figures range from Pythagoras and Plato through the Islamic world to Galileo, Kepler, and Newton.

Readings: (1) Requirements for the physics major at Kansas State (see catalogue) (2) Victor Katz, A History of Mathematics, 2d. edition, sections 2.1-2.2, 10.3.2-10.3.4, 10.5.1, 12.5.6, which are pp. 47-54, 402-416, 421-425, 516-519.

Recommended reading: I. B. Cohen, Birth of a New Physics

Lecture 2: How did we come to live in a non-Euclidean world?

Euclidean geometry, with the theory of parallels learned in high school, appears to be the real geometry of real space. If so, how did the counterintuitive non-Euclidean geometry ever even occur to people? How did people come to recognize that physical space is best modeled in a non-Euclidean way? Key figures include mathematicians like Euclid, Omar Khayyam, Bolyai, Lobachevsky, and Gauss; philosophers like Plato, Aristotle, and Kant, and physicists like Helmholtz and Einstein.

Reading: Euclid's Elements: Postulates 1-5; Theorems I:27-I:32. George Gamow, One, Two, Three...Infinity, chapter in Space, Time, and Einstein, sections 3 and 4 (pp. 103-112 in my edition). Katz, sections 2.4.1, 2.4.2, 7.4.2, 14.3.2-14.3.3, 17.2 (especially 17.2.4), which are pp. 58-62, 65-66, 269-271, 624-630, 772-785.

Recommended reading: Jeremy Gray, Ideas of Space.

Lecture 3: A new way of thinking: statistics, science, and society

If mathematics is certain and the mathematization of science is intended to achieve certainty, what made scientists apply statistics to nature? The focus is on the nineteenth century, with attention being paid to Lagrange, Adolphe Quetelet, Auguste Comte, and James Clerk Maxwell.

Reading: Katz, section 16.5, especially 16.5.3, which is pp. 753-760, esp. 758-760.

Recommended reading: Darrell Huff, How to Lie with Statistics; Theodore Porter, The Rise of Statistical Thinking

Lecture 4: Descartes and Problem-Solving

Everybody has heard that Descartes (along with his contemporary Fermat) invented analytic geometry. But this was not what it seems, since Descartes didn't even have Cartesian coordinates. We will see what he did do: pioneer a new approach to solving problems, one which transformed mathematics -- and which has much to teach us.

Reading: Katz, sections 5.3, 9.4 Introduction and 9.4.1, 11.1.2, 12.1.2, which is pp. 182-187, 367-372, 436-442, 472-473.

Recommended reading: Rene Descartes, Discourse on Method; Meditations

Lecture 5: The psychology of discovery: is the calculus algebra or geometry?

Different ways of thinking have, historically, led to important discoveries. We look at contrasting ways of thinking about mathematics, illustrate these with respect to the differential calculus in the eighteenth century, and explore the implications of this contrast for understanding how mathematics develops and how it is taught. The key figures will be Colin Maclaurin and Joseph-Louis Lagrange.

Reading: Jacques Hadamard, Psychology of Invention in the Mathematical Field, chapter VII: Different Kinds of Mathematical Minds.

Recommended reading: Katz, sections 13.2.2, 13.5.4, which is pp. 562-565. The rest of Hadamard's book, especially chapter I, General Views.

Required Texts:

A History of Mathematics: An Introduction, by Victor J. Katz, second edition, Addison Wesley Publishing Company, February 1998

A Source Book in Mathematics, 1200-1800, by Dirk J. Struik, reprinted edition, Princeton University Press, August 1986

Problems: To be assigned