Sidoli, Nathan Camillo
Fall, 2021
Office hours: Thursday, 4th and 5th

SILS, 11, 1409
03-5286-1738
[email protected]

Announcements

I will put announcements about the class in this space. Please check here periodically as the term progresses.

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 pack 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: Henri Poincaré, Mathematical Creation, from Science and Method (1908), reprinted as Mathematical Creation, Reflections: Journal of Science Education, 5 (2000), 85-94. (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

    Assignment 20%
    Midterm exam 40%
    Final exam 40%

    General Format

    The class meets once a week for a lecture. Students are expected to attend the lectures, complete one assignment and write a midterm and final exam.

    Assignment: A graphic image, or new media document

    Read through the first book of Euclid’s Elements. (There are many versions available. The Japanese version is: エウクレイデス『エウクレイデス全集』第1巻 原論 I-VI、斎藤 憲 訳・解説、三浦 伸夫 解説、東京大学出版会2008年。If you cannot find a printed version, you may use the online version by D. E. Joyce. There is another online version of the Elements provided by the Clay Mathematics Institute, which includes the Greek text and images of the oldest manuscript. See also the fascinating 1847 edition by Oliver Byrne, which gives visual arguments for the geometry of the first six books.) Develop some graphic image or audio-visual presentation that gives the viewer an impression of the logical structure of the text. The data for the logical structure of the first book can be found on D. E. Joyce’s website. You may focus on either the constructions or the theorems, or you may develop a single presentation to convey both.

    Further Readings and Materials for Assignment
  • Euclid’s Elements: For a file of the full text in both Greek and English, see Fitzpatrick, R., Euclids’s Elements of Geometry.
  • Constructive Geometry in the Elements: Beeson, M., 2009, Constructive Geometry, and Beeson, M., 2009, The Parallel Postulate in Constructive Geometry.
  • Euclid: The Game: There is an online game that helps you develop a sense of how constructions are used by Euclid. See Euclid: The Game.
  • Exams

    There will be two take-home exams, a midterm and a final. The exam sheets will distributed at the end of the class before the exam. You will have 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. (Please note that students enrolled in the GSFSE will have different exams from everyone else.)

    Classroom Etiquette

    Please follow basic rules of decorum – do not sleep, eat, or carry on individual conversations in class. Finally, do no 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 behind them.)

    Discussion Topics, Readings and Assignments

    Week 1: Sep 28

    Introduction to mathematical reasoning

  • No reading.
  • Lecture notes: Introduction to proofs.
  • Introduction

    Week 2: Oct 5

    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 12

    The birth of demonstrative mathematics: Hippocrates

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

    Week 4: Oct 19

    Euclid I: 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.)
  • Euclid’s Elements Book I

    Week 5: Oct 26

    Euclid II: Number theory (short movie)

  • Reading: Dunham, chap. 3.
  • Lecture notes: The Euclidean Algorithm.
  • Movie: You can watch the documentary on YouTube (Part 1, Part 2).
  • Background reading for Greek mathematics: Netz, R., Greek Mathematicians: A Group Picture.
  • Euclid’s number theory

    Week 6: Nov 2 (Assignment due)

    Archimedes: Geometric problem solving (scenes from BBC documentary)

  • Reading: Dunham, chap. 4.
  • Website: The Archimedes’ Palimpsest Project.
  • Archimedes

    Week 7: Nov 9 (Midterm exam sheets distributed)

    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 8: Nov 16 (Midterm due)

    Midterm exam: Discussion of the questions and answers

  • No reading.
  • Holiday: Nov 23

    No Class

  • No Reading.
  • Week 9: Nov 30

    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 7

    Euler and number theory

  • Reading: Dunham, chap. 10.
  • Video: There is a Numberphile video dealing with Fermat‘s Little Theorem on YouTube (Liar Numbers - Numberphile).
  • Euler

    Week 11: Dec 14

    Non-Euclidean geometry, I

  • Reading: Poincaré’s essay on Mathematical Creation; Source pack 4, Non-Euclidean geometry.
  • Lecture notes: Sacccheri’s proof of the parallel postulate.
  • Movie: You can watch a documentary on this subject on YouTube (Part 1, Part 2).
  • Supplementary material: Tibor Marcinek’ss Geogebra website of the Poincaré Disk.
  • Non-Euclidean geometry

    Week 12: Dec 21

    Non-Euclidean Geometry II; and Cantor and non-denumerability

  • Reading: Dunham, chap. 11.
  • Cantor on the continuum

    Holiday: Dec 28

    No Class

  • No Reading.
  • Holiday: Jan 4

    No Class

  • No Reading.
  • Week 13: Jan 11

    Cantor and the transfinite realm

  • Reading: Dunham, chap. 12.
  • Supplementary material: see an article from Quanta on recent mathematical work on the continuum hypothesis: How Many Numbers Exist?...
  • Cantor on transfinite numbers

    Week 14: Jan 18 (Final exam sheets distributed)

    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.
  • Turing and his machine

    Week 15: Jan 25 (Final exam)

    Final exam: Discussion of questions and answers.

  • No Reading.