Math/Hist 388: History of Mathematics

Fall 2012, Section 1
MWF 8:30-9:20 am
JEB 026
Prof. Ely
314 Brink Hall
208-885-6740

Office Hours: M 9:30-11:30,
F 9:30-10:30
or by appointment
Required Book: A History of Mathematics: An introduction. (3rd Edition) Katz, V. 2009.
What kinds of things do we study in a history of mathematics class?
  • How did the mathematical ideas that we use develop through time and place?
  • What cultural norms, social institutions, and personal beliefs contributed to their development?
  • What is mathematics? And is it discovered or invented?
  • How do we figure out the mathematical practices and ideas of cultures with different languages and notations?
What do we actually do in a history of mathematics class?
  • Reading (lots) — We will be examining some primary sources, as well as reading articles and book chapters by historians. Figuring out what some of these people are saying and doing is not easy, so it will be important to explore these texts together in class.
  • Writing (quite a bit) — Writing responses to the readings will help you organize your ideas about the readings. But the most productive writing will be for your research project, where you have to refine and develop sophisticated answers to a complex question of your own devising.
  • ‘Rithmetic (some) — There is no better way to learn the mathematical practices of another society than to tackle the same questions that they tackled, using the same notations and methods.

Course Schedule

Part I — Deciphering SourcesAncient Mesopotamia & Egypt

Week 1
Response Reading:
Robson, E. Words and pictures: New light on Plimpton 322. (found on our online site, see p. 6-7 of the syllabus)
Textbook Reading:Ch 1 / Mon, Aug 20Introduction to course
Old Babylonian notation
Wed, Aug 22Old Babylonian mathematics
Fri, Aug 24Babylonian and Egyptian mathematics
Response Reading: Robson (about Plimpton 322)
Response Questions:
1. What are three competing accounts of how the Pythagorean triples on the Plimpton 322 tablet were generated? Which of these accounts does Robson think is correct, and why?
2. What point is Robson making about how research on the history of mathematics should be conducted?

Part II — GeometryAncient Greece

Week 2
Response Reading:
Plato. Excerpts from Meno.
Textbook Reading: Ch 2 / Mon, Aug 27Intro to Greece (500-200bc), Pythagoreans and incommensurability
Wed, Aug 29Zeno’s paradoxes and the responses of Socrates and Aristotle
Math Problem: Babylonian and Egyptian (pp 28-30 pick one: #10, 19, 20, or 27)
Fri, Aug 31Logic in Plato and Aristotle
Response Reading: Meno
Response Questions: Choose 2-3 of these to focus on.
1. What is Meno's question about how knowledge is gained, and what is Socrates' answer?
2. How does Socrates' interview with the slave boy support his position about the nature of learning?
3. Why does he choose to discuss geometry with a slave boy in order to prove his point, rather than choosing to discuss, say, politics with, say, Meno?
4. What do you think of Socrates' ideas here? How do you think they can give us insight into how the Greeks think about math (and how we think about math)?
Week 3
Response Reading:
Same as textbook reading.
Textbook Reading: Ch 3 / Mon, Sep 3Labor Day — No class!
Wed, Sep 5Geometric problem solving in Plato’s time: three tough problems
Math Problem: Pythagorean (pp 47-8 pick one: #10 or 13)
Fri, Sep 7Euclid and paradigmatic Greek practice
Response Reading: Katz (textbook) Ch 3
Response Questions: Attend to 3 of these.
1.What is the difference between postulates, common notions, propositions, and definitions in Euclid’s Elements?
2. What is the general goal of Book I of Euclid’s Elements, and how is Euclid’s Postulate 5 (the “Parallel Postulate”) used in accomplishing this goal?
3. What is the Euclidean algorithm? How does it work? What does it have to do with commensurability?
4. What is the method of exhaustion? How is it used in the Elements?
Week 4
Textbook Reading: Ch 4 / Mon, Sep 10Euclid and paradigmatic Greek practice, cont’d
Wed, Sep 12Archimedes
Math Problem: Euclid (pp 90-92, pick one: #6, 10, or 38)
Fri, Sep 14Analysis and Synthesis in Archimedes
Response Readings: Pappus AND Archimedes
Response Questions: Pappus describes the difference between synthesis and analysis in Greek mathematics. Carefully read and understand Propositions 1 and 7 of Archimedes' On the Sphere and Cylinder and Proposition 1 ofMeasurement of a Circle.
1. In your own words, what is the difference between analysis and synthesis for Archimedes? Why does he use both in each of his propositions, and what role does each play?
2. Please read The Sand Reckonertoo (you don’t have to grind through every detail!). What reflections do you have about it?
Week 5
Textbook Reading: Ch 5
Ch 6 / Mon, Sep 17Apollonius and conic sections, Ptolemy and astronomy
Wed, Sep 19Intro to Rome (200bc-400ad), decline of Greek math,
Diophantus and algebraic thinking in late antiquity
Math Problem: Archimedes or Apollonius (pp 127-131, pick one: #8 or 29)

Part III — AlgebraIslamic and European Middle Ages and Renaissance

Week 5 cont’d
Textbook Reading: Ch 8 / Fri, Sep 21Mathematics of India, the Hindu-Arabic number system
Library assignment due (tentative)
(No response reading)
Week 6
Textbook Reading:
Ch 9 / Mon, Sep 24Intro to Medieval Islamic mathematics (800-1400), the development of algebra
Wed, Sep 26Medieval Islamic algebra, trigonometry, and astronomy
Math Problem: Diophantus (p 114 Do both #7 & 24, or do #12)
Response Readings: al-Khayyami AND Katz pp 265-292)
Response Questions: Al-Khayyami's work is about solving 3rd-order equations. The excerpt you have is where he lays the groundwork for this. On pp. 208-210, his purpose is to show how to solve the simplest case: x^3 = c. When you read this, pay attention to what he shows in each of the parts that say "demonstration," and try to understand how they fit into solving x^3 = c.
1. Does al-Khayyami pay attention to homogeneity or not? What evidence in the excerpt that supports your answer?
2. Does al-Khayyami use analysis? synthesis? both? What evidence in the excerpt that supports your answer?
Parts from the Katz book will help you understand the following two questions:
3. How does al-Khayyami use the work of al-Khwarizmi? What evidence in the excerpt supports your answer?
4. Later on in the book, al-Khayyami solves a variety of cubic (i.e. 3rd-order) equations. A later mathematician, al-Din, filled in a gap in his work on this. What is it, and why is it important?
Fri, Sep 28The advent of the university in high medieval Europe (c1200-1400), the transmission of Greek and Islamic documents, Leonardo of Pisa
(No response reading)
Week 7
Textbook Reading: Ch 10, 12 / Mon, Oct 1More Medieval mathematics,kinematics of Nicole Oresme
Wed, Oct 3Algebra in the Renaissance (1400-1600)
Math Problem: Islamic (pp318-9, pick one: #2 or 18)
Fri, Oct 5Algebra in the Renaissance, cont’d
Response Reading: Vieta
Response Questions: This reading is highly truncated and short, which makes it even more difficult to understand. I’ll try to find a better version to post for you to read. Either way, do your best to answer these questions, and we’ll go through it and its importance together carefully in class.
1. What are the differences between what Vieta specifies as zetetic, poristic, and exegetic analysis?
2. What is the Law of Homogeneity, and what are species for Viete?
3. In Chapter V, Vieta represents certain mathematical things with letters. Which things, which letters, and why is this important?

Part IV — Analytic Geometry & CalculusThe Enlightenment

Week 8
Textbook Reading: Ch 13
Ch 14.1-14.2 / Mon, Oct 8Exam I
Wed, Oct 10The Scientific Revolution and inductive reasoning: Copernicus, Kepler, Galileo Math Problem: Cardano (pp. 418-420, pick one: #32 or 38)
Fri, Oct 12Analytic Geometry and deductive reasoning: Descartes and Fermat
Response Reading: Descartes
Response Questions:
1. In the excerpt from the Regulae, what is Descartes' complaint about the "ancient" philosophers and mathematicians?
2. In the Discourse on the Method excerpt, Descartes outlines his approach to philosophical reasoning. In the Geometrie excerpt, he outlines his approach to geometry. How are these approaches similar to one another? (In other words, why does it make sense that the same guy came up with both of them?) How do his programs of philosophy and mathematics differ from the programs of the "ancients" that he griped about in the Regulae?
(Hint: In order to understand his approach to geometry, think about the first sentence of the Geometrie excerpt. How does this relate to graphing something on a Cartesian plane?)
Week 9
Textbook Reading: Ch 15.2
Ch 16.1 / Mon, Oct 15Indivisibles in the 17th Century: Galileo, Cavalieri, Torricelli
Response Reading: Galileo
Response Questions: Galileo gets in trouble with this work because he is attacking the "Peripatetic" philosophers (philosophers from the medieval tradition that is based on Aristotle's works, identified with the Catholic church). It's not just because he's attacking the Aristotelian ideas, it's also because he makes the Peripatetic guy (Simplicio) sound really dopey.
1. What is Salviati's answer for why the inner and outer circles can end up going the same distance after one revolution, even though their circumferences are different?
2. What are Simplicio's two difficulties with this answer? How does Salviati answer these with the soupdish example?
3. Do Galileo's ideas here challenge the Aristotelian notion of absolute vs. potential infinity? Why or why not?
Wed, Oct 17Approaching calculus: tangents, areas, logarithms
Math Problem: Analytic geometry (pp 501-504, pick one: #20 or 21)
Fri, Oct 19Isaac Newton
Due: Research project topic and preliminary bibliography
Week 10
Textbook Reading: Ch 15.3
Ch 16.2
Ch 17.1.1-17.1.2 / Mon, Oct 22Gottfried Wilhelm Leibniz
Response Reading: Mancosu (on Leibniz’ infinitesimals)
Response Questions:
1. a. What are Rolle's objections to Leibniz' infinitesimal system?
b. How does Varignon answer these objections? And how are these answers different from Leibniz's answers?
2. What are Nieuwentijt's objections, and what was his counterproposal?
Wed, Oct 24The Fundamental Theorem of Calculus—Barrow, Newton, and Leibniz
Math Problem: Calculus (pick one: pp 539-541 #9, 11, or pp 579 #16, 18, or 25)
Fri, Oct 26The 18th Century—the “Age of Reason” in continental Europe, the uses of (and objections to) the techniques of infinitesimal calculus, Leonard Euler
Week 11
Textbook Reading:
Ch 17.1.3-17.4.4 / Mon, Oct 29Leonard Euler (cont’d)
Wed, Oct 31Differing development of British and Continental mathematics in the 18th Century
Math Problem: Applications of Inf’l Calculus (pp 636-639, pick one: #1, 3, 6, or 7)
Response Reading: Grabiner (on Lagrange and Maclaurin)
Response Question: What is the difference between Maclaurin and Lagrange's uses of calculus? Precisely what is the root of this difference?
Fri, Nov 2No class or office hours – I’m at a conference.

Part V — Foundations of Mathematics The 19th and 20th Centuries

Week 12
Textbook Reading: Ch 22.1-22.2
Ch 20.1-20.2
Ch 24.2 / Mon, Nov 5The Crisis of Analysis in the early 19th Century: Fourier and Cauchy
Note: office hours cancelled today
Wed, Nov 7Non-Euclidean Geometry
Fri, Nov 9Cantor’s infinities
Due: Research project question
(No response reading until Monday)
Week 13
Textbook Reading: Ch 25.1 & 25.6 / Mon, Nov 12The philosophy of mathematics: the “isms” of the early 20th Century
Response Reading: Davis & Hersh
Response questions:
1. According to David & Hersh, what is the mathematical philosophy of "platonism", and why is it so common among mathematicians?
2. Choose two of the following: logicism, formalism, or constructivism (i.e. intuitionism). For each ism, how does it challenge the view of platonism?
Wed, Nov 14The foundations of mathematical logic: Frege, Zermelo, and Russell
Math Problem: pp 706-707 #2 or 3, or pp 813-815 #7 or 9, or pp 926-928 #1
Fri, Nov 16Hilbert’s formalist questions, Gödel’s answers
(No response reading)
Nov 19-23 / NO CLASS THIS WEEK — HAPPY THANKSGIVING!!!!

Part VI — Mathematics in Other Civilizations China, The Americas, Oceania

Week 14
Textbook Reading: Ch 7
Ch 11 / Mon, Nov 26Mathematics in Ancient and Medieval China
Wed, Nov 28Mathematics in Ancient and Medieval China, discuss research project review process
Math Problem: Chinese (pp 226, pick one: #1, 2, 4, or 7)
Fri, Nov 30Mathematics in pre-Columbian America: Maya & Inca
(No reading due)
Week 15 / Mon, Dec 3Mathematics of Oceania, Australia, New Zealand
Due: First draft of research project
Wed, Dec 5Reviewing research projects
Due: Peer reviews of research projects
Fri, Dec 7Fun topic (to be determined)
Finals Week / Tues, Dec 11Exam II (7:30-9:30am)
Fri, Dec 14Due: Final draft of research project

Grading Summary

Duedate: Points:
  • Response Readingsusually weekly (see below)100
  • Math Problemsusually weekly (see below)36
  • Exam I Mon, Oct 850
  • Exam II Tues, Dec 11 (7:30-9:30am)50
  • Research Project:125
oLibrary Assignment (tentative)Fri, Sep 21 5
oTopic and Preliminary BibliographyFri, Oct 19 10
oResearch Question Fri, Nov 9 10
oFirst Draft Mon, Dec 3 60
oPeer Review Wed, Dec 5 10
oFinal Draft Fri, Dec 14 30
o
Total: 361 points
Course Grades:90-100%: A 80-90%: B 70-80%: C 60-70%: D 0-60%: F

Response Readings(100 points)

One goal of this class is for you to get a good taste of what it is like to read and analyze historical literature. So every week we will read an article, book chapter, or primary source excerpt, and then discuss it in class. Usually these class discussions about the readings will occur on Fridays.

In order to be ready for these discussions, you must read the week’s Response Reading and—you guessed it—write a response to it. A response should not just be your random musings; it should address the response questions that are included in the day-by-day schedule above. These questions are meant to help you focus on several important elements of the reading. Your response is ultimately for your later benefit—it is a summary of the meaning and importance of the reading for when you look back at the reading later and try to remember what it was about.

There are 11 responses during the semester. Each is worth 10 points, and these points include your attendance on the day of the class discussion as well. Your lowest grade of the 11 responses will be dropped. You may not turn these in late—their purpose is to prepare you for the class discussion.

How to write a reading response:

  • Answer the reading questions, or at least give it a good try. Start your answers with the question itself, or at least with some statement like “The difference between analysis and synthesis for Archimedes is…”. That way the reader knows what you’re talking about.
  • Write down questions that arise as you read. You don’t have to fully understand the reading—that’s what the class discussion is for! So write down “What the heck does Robson mean on page 25?” or “Why does Viete make such a big deal about the Law of Homogeneity?”
  • A response should be to the point. Maybe it’s one page. If you can get it said more succinctly, that’s fine.
  • Feel free to add your own opinions or reflections at the end of your response.

Instructions for obtaining the Response Readings:

1. Go to db.lib.uidaho.edu/ereserve/show_course.php?pointer=3554 (sorry about the awful url)

2. Username:reserve

Password:Fd73gb

3. When you click on the appropriate reading, it should pull up a pdf of the document. Print it and read it!

Math Problems(36 points)

All this reading and writing—wouldn’t it be nice to do a little math too? As I said earlier, the best way to learn the mathematics of another culture is to work their problems using their notations and methods.

Every week (roughly) you will also have to do one of the math problems specified on the class schedule. To get full credit, you must show your work and carefully explain what you are doing. These problems are specified on the syllabus in the schedule. These are usually due in class on Wednesdays. There are 13 problems in the semester, and each is worth 3 points. A day late with these is okay, but keep it in the same week. I will drop the lowest one.

Exams I & II (50 points each)

Friday, Oct 8in class

Tuesday, Dec 11 7:30-9:30am

For each exam I will hand out study guide ahead of time containing some short answer questions and essay questions. From among these I will choose 5 short-answer questions and 2 essay questions to go on the test. You will have to answer 4 of the short answer questions and 1 of the essay questions.

Research Project(125 points total)

The most important assignment of the semester is the research project. This is where you get a chance to deeply investigate an interesting question in the history of mathematics. The crucial part of this project is that it is directed by an interesting and tractable question that you care about. On the next page, I talk more about what makes an interesting research question, rather than just a research topic (your paper will have both).

Don’t worry, this project will not blind-side you. You will receive support, guidance, and feedback on your project at every step of the way. A guided day in the library will help you find resources to the paper. You will get feedback from your peers and me about your research question. And at the end of the semester you will get substantive feedback on your project rough draft from your peers and from me. Your final project grade will depend on how well you use this feedback in writing your final draft.

Here are the details about the research project and its requirements and due-dates:

Library Assignment (Due Friday, Sept 21) (5 points)

The idea of this assignment is to force you to familiarize yourself with some of the resources that are available at the UI library, to prepare you for doing your bibliography and research.

Topic and Preliminary Bibliography (Due Friday, Oct 19)(10 points)

The topic of your research project may be anything that you are interested in related to the history of mathematics. When choosing a topic, don’t be restricted just to topics that we are specifically focusing on in class! Feel free to make your project about something like the influence of computers on mathematics, or the life of Evariste Galois, or Sophie Germain’s correspondence with Gauss, or the mathematics of sub-Saharan Africa, or any number of other topics that we won’t specifically address in course lectures and discussions.

The topics and sources assignment will be a single page. Include your topic and a preliminary bibliography with at least three sources. The preliminary bibliography must include:

  • A primary source. For example, if your topic is Euler’s physics, you could cite the webpage homepages.wmich.edu/~mcgrew/euler.htm, which contains an full-text English translation of Euler’s letters to a German princess. Another example: if your topic is Laplace, you could cite his work “A Philosophical Essay on Probabilities,” found in Stephen Hawking’s anthology God Created the Integers.
  • An article from a scholarly journal. Such periodicals might include Historia Mathematica, Archive for History of Exact Sciences, Isis, Convergence, etc. Information on locating such articles is included below.
  • A book. Don’t forget to use Interlibrary Loan if you find the UI Library’s holdings to be inadequate!

Searching for bibliographic sources: The research tools that I think will be most important for you include: