History of Mathematics

Course Description

For most of western history, an understanding of mathematical proof was considered essential to a well-rounded education. There is the common story that Abraham Lincoln always carried with him a copy of Euclid’s Elements. Cicero tells us that he sought out the grave of Archimedes, in order to pay tribute to one of the great icons of the intellectual culture he wished to emulate. Neither of these men had any advanced training in mathematics; they were statesmen and orators. They read mathematics because it can compel with a force far beyond that of great speeches. Mathematics is capable of convincing us of things we can all agree upon, and it does this through rigorous proof.

This course explores the origins of various types of deductive argumentation by reading mathematical proofs in primary sources along with scholarly commentary. We will look at the sources of different approaches to mathematical demonstrations in a number of diverse cultures and periods. We focus on the continuous tradition that began in Greece, was adopted by Arabic and Hebrew scholars, and was further cultivated in the Latin west. Finally, we will confront contemporary challenges to the idea of proof such as the delicate relationship between rigor and intelligibility and the possibility of computer generated proofs.

This is not a mathematics course. The texts will demand careful attention to the reasoning but they will not require special mathematical training beyond what you learned in High School. Although it is important to understand the mathematical arguments, we will also pay attention to the text’s historical and philosophical aspects. Each time we meet, we will work through a proof or two on the board, taking our time until everyone agrees that the demonstration is convincing. We will then discuss the proof using contemporary scholarship as our point of departure.

Required Texts

Dunham, W., 1991, Journey Through Genius: The Great Theorems of Mathematics (Penguin: New York). (To be purchased from the Co-op.)

W.ダンハム『数学の知性 : 天才と定理でたどる数学史』中村由子訳、現代数学社、1982年。(In the library.)

Bos, H., 2001, excerpt from Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction (Springer: New York). (See class list, below.)

Copeland, B.J., and Proudfoot, D., 2006, Turing and the Computer, from Alan Turing’s Automatic Computing Engine. (See class list, below.)

Rowe, D., 2006, Euclidean Geometry and Physical Space, Mathematical Intelligencer, 28, 51-59. (See class list, below.)

Source pack 1: Katz, V., Imhousen, A., Robson, E., Dauben, J.W., Plofker, K., and J.L. Berggren, 2007, The Mathematics of Egypt, Mesopotamia, China, India and Islam (Princeton University Press: Princeton). (See class list, below.)

Source packs 2-4: Flauvel, J. and J. Gray, 1987, The History of Mathematics: A Reader (The Open University and Pelgrave Macmillian: Hampshire). (See class list, below.)

Source pack 5: Nicholas Lobachevskii, Excerpts from his Theory of Parallels (See class list, below.)

Suggested Reading

General background reading I : Crowe, M., 1988, Ten Misconceptions about Mathematics and its History, in W. Aspray and P. Kitcher, eds., History and Philosophy of Modern Mathematics, Minneapolis, 260-675.

General background reading II : Lakatos, I., 1978, What does a mathematical proof prove?, in eds. J. Worral and G. Lurrie, Imre Lakatos: Mathematics, Science and Epistemology, Cambridge, 61-69.

Background reading (Babylonian): Robson, E., 2008, Mathematics in Ancient Iraq: A Social History (Princeton University Press: Princeton). (A selection.) (See class list, below.)

Background reading (Greece): Netz, R., 2008, Greek Mathematicians: A Group Picture, in C.J. Tuplin and T.E. Rihil, eds., Science and Mathematics in Ancient Greece, Oxford, 196-216. (See class list, below.)

Background reading (Descartes): Macbeth, D., 2004, Viète, Descartes, and the Emergence of Modern Mathematics, Graduate Faculty Philosophy Journal, 25, 196-216. (See class list, below.)

Grading

There are two groups of students enrolled in this course: (1) students from the Main Waseda campus, and (2) students from the Nishi-Waseda Campus. The content of the assignment and exams will be different for the two groups, but percentage of the overall grade will be the same.

Active participation (in-class discussion)	20%
Assignment	20%
Midterm exam (take-home)	30%
Final exam (take-home)	30%

Exams

There will be two take-home exams, a midterm and a final. The exam questions will be distributed online at least a week before the exam papers are due. You will have at least one week to answer all of the questions. Please submit your exam electronically to Moodle before class AND bring a copy of your answers to class on the day of the exam. We will spend the exam period working through the answers that everyone gave. A general statement of the grading standards will be distributed before the exam.

General Format

The class meets once a week for a lecture. Students are expected to attend the lectures, engage in class discussions, and write a midterm and final exam. After each lecture, there will be a short discussion period for student participation based on questions that I will post in class.

Classroom Etiquette

Please follow basic rules of decorum – do not sleep, eat, or carry on individual conversations in class. Finally, DO NOT use mobile phones, smart phones, or laptops in class. (Unfortunately, a large percentage of students use their laptops to do unrelated things during class, and this distracts both them and everyone aroud and behind them.) I will be very strict about enforcing the rule about devices and laptops, so if you feel that you must use devices, I encourage you to enroll in a different class.

Assignment: A graphic image, or new media document

There will be one assignment, the details of which will be distributed distributed at the end of the class before the exam.

Further Materials and Readings for Assignment

Online version of the Elements, which includes information about all of the logical dependencies (this is the data you need for the assignment).

For a PDF file of the full text in both Greek and English, see Fitzpatrick, R., Euclids’s Elements of Geometry.

Web applet for Euclidean constructions: Euclid: The Game. (An online “game” that helps one develop a sense of how constructions are used by Euclid.)

Armann, B., 1999, Euclid: The Creation of Mathematics, Springer, New York. (Not available for download.)

Constructive Geometry in the Elements: Beeson, M., 2009, Constructive Geometry, and Beeson, M., 2009, The Parallel Postulate in Constructive Geometry.

Hartschorne, R., 2000, Geometry: Euclid and Beyond, Springer, New York. (Not available for download.)

Mueller, I., 1981, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements, MIT Press, Cambridge, MA. (Reprinted by Dover, 2006.) (Not available for download.)

Discussion Topics, Readings and Assignments

Week 1: Oct 8

Introduction to mathematical reasoning

No reading.

Lecture notes: Introduction to proofs.

Introduction

Week 2: Oct 15

Babylonian and Egyptian mathematics

Reading: Source pack 1, Scribal mathematics.

Background reading for Babylonian mathematics: Robson, Mathematics in Ancient Iraq, chap. 1.

Mathematics in Ancient Egypt and Mesopotamia

Week 3: Oct 22

The birth of demonstrative mathematics: Hippocrates

Reading: Dunham, chap. 1; Source pack 2, Early Greek mathematics.

Early Greek mathematics

Week 4: Oct 29

Euclid: Foundations of geometry

Reading: Dunham, chap. 2.

Supplemental reading: Euclid’s Elements, Book I.

Online sources for Elements: D. E. Joyce’s online version of Euclid’s Elements. Oliver Byrne’s 1847 edition of Euclid’s Elements, using visual arguments.

In order to get a sense of how constructions function in Euclid's Elements, see Euclid: The Game. (It sometimes takes a long time to load.)

Documentary on ancient mathematics: You can watch the documentary on YouTube (Part 1, Part 2).

Euclid’s Elements Book I

Week 5: Nov 5 (Assignment due, direct submission)

Archimedes: Geometric problem solving (scenes from BBC documentary)

Reading: Dunham, chap. 4.

Website: The Archimedes’ Palimpsest Project.

Archimedes

Week 6: Nov 12 (Midterm exam sheets distributed online)

Descartes and symbolic algebra (short movie)

Reading: Bos, H., Descartes’ solution to Pappus’ locus problem; Source pack 3, Descartes’ solution to the locus problem.

Movie: You can watch the documentary on YouTube (Part 1, Part 2).

Lecture notes: The Pappus Problem.

Background reading for Descartes: Macbeth, D., Viète, Descartes, and the Emergence of Modern Mathematics, Gaukroger, S., The Nature of Abstract Reasoning, Isaac Newton's solution to the Pappus locus problem: An extract from the Principa.

Descartes

Week 7: Nov 19 (Midterm due, online submission, in-class discussion)

Midterm exam: Discussion of the questions and answers

No reading.

Week 8: Nov 26

Perfect induction and probability theory

Reading: Devlin, The Unfinished Game, Chaps. 2, 3, 4, 5, 6 (especially Chaps. 2, 4, 6).

Lecture notes: The Arithmetic Table.

Supplemental readings: Pengelley’ notes on Pascal, including excerpts from the Treatise on the Arithmetic Triangle; the 1654 correspondence between Pascal and Fermat on probability theory.

Perfect Induction and Early Probability Theory

Week 9: Dec 3

Newton, Leibniz and the calculus (short movie)

Reading: Dunham, chap. 7.

Movie: You can watch the documentary on YouTube (Part 1, Part 2).

Newton

Week 10: Dec 10

Non-Euclidean geometry, I

Reading: Rowe, D., Euclidean Geometry and Physical Space; Source pack 4, Non-Euclidean geometry; Lobachevskii, excerpt from his Theory of Parallels.

Lecture notes: Sacccheri’s “proof” of the parallel postulate.

Movie: You can watch a documentary on this subject on YouTube (Part 1, Part 2).

Non-Euclidean geometry (Part 1)

Week 11: Dec 17

Non-Euclidean Geometry II

Reading: Gray, J., a chapter on Riemann: Geometry and Physics, from his Worlds out of Nothing (including translated excerpts from Riemann’s lecture on the hypotheses of geometry); Hartshorn, Sections 37 and 39 from his Geometry: Euclid and Beyond.

Lecture notes: Poincaré’s Disk.

Supplementary material: Tibor Marcinek’s Geogebra website of the Poincaré Disk.

Non-Euclidean geometry (Part 2)

Holiday: Dec 24

No Class

No Reading.

Holiday: Dec 31

No Class

No Reading.

Week 12: Jan 7

Cantor on non-denumerability and the transfinite realm

Reading: Dunham, chaps. 11 and 12.

Supplementary material: see an article from Quanta on recent mathematical work on the continuum hypothesis: How Many Numbers Exist?...

Cantor, non-denumerability, and the transfinite realm

Week 13: Jan 14 (Final exam sheets distributed online)

Turing and computing

Reading: B.J. Copeland and D. Proudfoot, Turing and the computer.

Supplementary material: Computerphile videos on Turing’s halting problem: Turing & The Halting Problem, and Halting Problem in Python. YouTube video on the Enigma machine. YouTube video using catagory theory to describe a general diagonal argument.

Turing and his machine

Week 14: Jan 21 (Final exam, online submission, in-class discussion)

Final exam: Discussion of questions and answers.

No Reading.