This schedule will be periodically updated as we go along. It is subject to change. For the syllabus, click here. For course reserves, use math432fa17, passwd ivan.
I. Fermat’s Enigma, or Math’s Greatest Hits
Simon Singh’s book provides a historical narrative incorporating several engaging episodes from mathematics. We will pay attention to the audience(s) he aims at, particularly as topics become increasingly esoteric. Your first essay will be based on this book.
Wed 8/30 (1) Introduction (2) sample discussion (3) fill out questionnaire w/ essay (4) beginning of BBC film, Fermat’s Last Theorem.
HW for 9/6: (1) Watch rest of BBC film (2) Read Chapters 1-2 of Singh (3) MC1 due on BBd
Wed 9/6 (1) Set up groups, introductions (2) Discuss movie and Chapters 1-2 of Singh (3) Answer MC2 in class
HW for 9/11: (1) Read Chapters 3-4 of Singh (try to read the rest as well) (2) MC3 due on BBd (3) Rewrite opening passage of questionnaire essay
Mon 9/11 (1) Quiz MC3ALT (2)Discuss Chapters 3-4 of Singh (3) Mathematics map (4) Topics for Essay 1
II. Nothing was a problem for the Greeks.
We look into the early development of mathematics and its foundations. Much is owed to the Greeks, such as the concept of proof. But some ideas, like the crucial step of defining zero (nothing) to be a number, were introduced and honed by non-Western cultures. Ascribing credit can still be a matter of controversy. Your second essay will be based on this material.
HW for 9/13: (1) Read Chapter 1 of Boyer & Merzbach (2) Chapter II of Kline along with pages 27-31 from Chapter III (3) HW on BBd, explained in class. (bring typed paper copy to class, and also submit on BBd)
Wed 9/13 (1) Discuss Prehistoric Math, Pythagoras and Greeks (2) Discuss Essay 1, finalize topic and rubrics.
HW for 9/18: (1) Draft of Essay 1 due. (2) Pages 1-36 of Lost Discoveries. Also, skim rest of Chapter 2, reading India and Zero more carefully, along with notes 26, 54 and 163 of Chapter 2. (4) Reading assignment 5 on BBd
Mon 9/18 (1) Discussion of contributions by non-Western cultures (2) Controversies about discovery of Pythagoras Theorem and Zero (3) Recent evidence for discovery of zero (article and video) (4) Bollywood film clip on Vedic mathematics (see instants 20:27 and 1:22:25)
HW for 9/20: (1) Give feedback on Essay 1 (turn in to me and to essay-writer) (2) Dig up more on: What is Vedic math? Who should get credit for Pythagoras Thm? For discovery of zero? (3) Look for skeptical reviews of Teresi’s book. (4) Additional References: Chapter 4 and 5 of Kaplan, Chapter 3 of Zero: The Biography of a Dangerous Idea by Charles Seife (course reserve) (a possible reference for essay material: Chapter 5 of Burton along with Sec 6.1 and 6.2 upto Hindu-Arabic numerals).
Wed 9/20 (1) More discussion of Kline’s outlook vs Teresi’s (2) Who should get credit for various discoveries? (3) Oral feedback on Essay 1 (4) RA6 in class
HW for 9/25: (1) Chapter 1 of “Is God a Mathematician?” by Mario Livio. Also, pages 33-40 (course reserve). (A version of this book is archived at this site.) (2) AMS article by Edwards (3) Negative review of Teresi (4) RA6 on BBd
Mon 9/25 (1) Discuss Plato. (2) Is mathematics invented or discovered? (3) Continue discussion from Wed on Teresi. (4) Essay 1 draft returned, discuss
HW for 9/27: (1) “Journey Through Genius” by Dunham page 27-53 (skim pages 40-47) (2) Read about Euclid’s contributions in other texts (3) MC7 on BBd
Wed 9/27 (1) Discuss Euclid’s contributions. (2) Film clip from “Lincoln.” Starting at 1:40 (3) Discussion of topics for Essay 2 (4) Questions for Prof. Tatarewicz
III. Harnessing Infinity
With the Renaissance (roughly the 14th through 17th centuries), Western mathematics started advancing again. One of the key developments, calculus, depended on understanding (and exploiting) the infinite and the infinitesimal – something Archimedes already had experimented with hundreds of years earlier. Cantor put infinity on firmer ground with his study of cardinality.
HW for 10/2: Revised version of Essay 1 due.
Mon 10/2 (NOTE: use picture of cannonball falling straight down) Guest lecture by Professor Joseph Tatarewicz, Dept of History, UMBC, on historical and philosophical reasons that focussed attention on infinity. Tentative abstract.
HW for 10/4: (1) Read about Galileo in Burton (Chapter 8, up to Page 347) (2) New Yorker piece by Gopnik
Wed 10/4 (1) Discuss Professor Tatarewicz’s lecture and Galileo. (2) Finalize task for Essay 2.
HW for 10/9: (1) Read (optional) about the invention of calculus in a reference like Burton, including the contributions of Newton and Leibniz. (2) Read summaries given here (skim) and here (read). Also, you may find interesting the succinct account of the Newton-Leibniz controversy in Dunham, page 187-190. (3) Read Wall Street Journal article, Calculus is so last century (4) MC8 on Bbd
Mon 10/9 (1) Discuss readings on Calculus, including relevance of subject.
HW for 10/11: Watch film on Archimedes and calculus: “Infinite Secrets” on YouTube (2) MC9 on BBd
Wed 10/11 (1) Discuss film (2) Peer Review Exercise: I will hand out a made-up essay which we will review in class
HW for 10/16: (1) Draft 1 of Essay 2 due (2) Read about criticism about calculus and Cantor’s work on infinity in Dunham, page mid-248 to 266.
Mon 10/16 (1) Discuss “Infinite Secrets.” (2) Discuss criticism of calculus and Cantor excerpt (3) Possible group RA in class on Cantor
HW for 10/18: (1) Give feedback on Essay 2 (turn in to essay-writer) (2) Read more about Cantor in Dunham, Chapter 12 (3) Read short story by Borges, The library of Babel
Wed 10/18 (1) Feedback on Essay 2 (2) Discussion on Cantor (cont) and Borges story
IV. How to surprise a Mathematician
Starting in the nineteenth century, mathematics was rocked by more surprises beyond the multiplicity of infinity. Euclid’s parallel postulate could be replaced by a contradicting alternative and still lead to consistent geometries. Russell’s paradox showed sets could not be defined naively. Godel’s incompleteness theorem showed that Hilbert’s dream of finding a complete and consistent set of axioms for all of mathematics was impossible. The rise of computing in the twentieth century resulted in great advances and drastic changes.
HW for 10/23: (1) Read about non-Euclidean geometry in Dunham, pages 34-36, 53-60, 245-248 (I also recommend you skim Burton, pages 581-599) (2) Read this more user-friendly article on spherical and hyperbolic geometry (3) RA 11
Mon 10/23 (1) My favorite joke on surprising a mathematician (2) Discuss non-Euclidean geometry.
HW for 10/25: (1) Draft 2 of Essay 2 due (turn in to me) (2) Read excerpt from Logicomix (course reserves) (3) Read pages 690-696 (mid-page, until “antidote to the paradoxes”) of Burton.
Wed 10/25 (1) Discussion of General Relativity (2) Discussion of Russell’s paradox and Zermelo’s axiom to remove it. (Also Cantor’s paradox.) (3) The Cantor set.
HW for 10/30: (1) Draft 2 returned with my comments. (2) Read page 621 of Burton on Hilbert’s axioms in geometry (3) Also in Burton: Kronecker (page 673-mid 675, read), continuum hypothesis (Page 689, read), Hilbert and Brouwer (pages 701-707, read to get general gist). (4) Reread pages 134 to 144 of Fermat’s Enigma (5) RA 12
Mon 10/30 (1) Zermelo’s Axiom of choice and consequences like Banach-Tarski paradox, or all vector spaces having bases (2) Discussion of constructionist approach (3) construction of naturals, integers, rationals and Dedekind cuts for reals (4) Godel’s theorems
HW for 11/1: (1) Read opinion pieces on mathematics outreach: “How to fall in love with math” and “The importance of recreational math” (2) Optional, if interested: Explanation of Banach-Tarski paradox
Wed 11/1 (1) The playing card project (class task and discussion) (2) More on editing an essay (3) Discussion of ideas for projects.
V. Why we are the way we are
In these readings, we will look at how the culture of mathematics has evolved. One of the earliest, fundamental changes was the insistence on using proofs to justify theorems. But other entrenched attitudes have also been evolving: dividing mathematics into “pure” and “applied,” or thinking of certain parts as “beautiful” and others as “ugly.” Perhaps the greatest transformations have come in the relation of mathematics to other fields, and in the people, such as women and minorities, who make it their profession.
HW for 11/6: Final version of Essay 2 due. Submit paper copy and on Blackboard.
Mon 11/6 Guest lecture by Professor Anne Rubin, Department of History
HW for 11/8: (1) film on Ramanujan (2) Chapter 5 of The Man Who Knew Infinity (book by Robert Kanigel) (3) Watch excerpts from The Man Who Knew Infinity (movie) (4) Optional, if you have time: read foreword by C.P. Snow in A Mathematician’s Apology. (5) RA 13 (6) Problem on prime numbers
Wed 11/8 (1) Discussion of project ideas (2) Discussion of Ramanujan, importance of proofs (further reading: Chapter 6, section 4 of book by Kanigel, and this article by the creator of Mathematica. Also, automatic theorem proving of Ramanujan’s results. See also this article on questions about whether computers should do automatic proofs.)
HW for 11/13: (1) Final version of essay 2 returned (2) Read sections 1 – 18 of A Mathematician’s Apology. (Use link from 11/18, or this link which may be easier to read.)
Mon 11/13 (1) Discussion of project ideas (2) Discussion of Hardy’s “apology” regarding his comments on teaching and exposition, and on age. (3) RA 14 done in class.
HW for 11/15: (1) Read remaining sections 19-29 of A Mathematician’s Apology. (2) Yet another op-ed by me, this one on ageism in math (the last one, I promise)
Wed 11/15 (1) Project proposals due. (2) Discussion of Hardy’s “apology” regarding his comments on usefulness of math. How applied and pure math have interacted (3) Solution of Ramanujan puzzle if we have time (4) RA 15 will be done in class, but individually – it will be a writing exercise, geared towards assessing how well you have absorbed Sections 1-29 of Hardy.(Write something positive about him.)
HW for 11/20: Topic: Tracing the evolution of math education. Changing perceptions on mathematics’ place in society. (1) Read “The Two Cultures” by C. P. Snow up to page 22 (2) Read article on history of US math education by J. Furr (3) Read Op-ed by Andrew Hacker in NYT. (4) Read Short review on “Does Mathematical Study Develop Logical Thinking?” in AMS Notices. (5) Additional optional resource: “Historical outline:1920 to 1980” section in article by Klein. (6) RA 16
Mon 11/20 (1) Short presentation on history of math education (2) Discussion (3) Experiment based on “Does Mathematical Study Develop Logical Thinking?”
Wed 11/22 No class.
HW for 11/27 Topic: The changing face of mathematicians. Women in mathematics. Why is representation important? Reading assignments: (1) Article on Sonya Kovalevsky (OK to skim mathematics) (2) Article on Emily Noether (3) Article from National Geographic on women in STEM (4) Article from Popular Mechanics on women in “Hidden Figures.” (5) If you can, watch the movie Hidden Figures. If not, you can watch an excerpt on Euler’s Method from the movie, followed by a medley of more excerpts that will play in succession and yet some more excerpts, including a preview or two here. (6) RA 17
Mon 11/27 Guest Lecture by Professor Kathleen Hoffman, Department of Math and Stat, UMBC
HW for 11/29 Topic: From experiment to proof back to experiment. Tracing how the computer has changed math. Reading assignments: (1) Look up Euler’s Method for solving ODEs and read this article (2) The computer: ruin of science and threat to mankind (essay #41, skim sections if they seem to get repetitive) (3) Notes on Tom Stoppard’s play, Arcadia. (Read all sections. Actual play can be read here or watched here if you are so inclined – starting at time 2:40.) (4) Submit writing part of class project if you want me to give comments. (No late work accepted after end of class.)
Wed 11/29 (1) Discussion of last question from 11/27 (2) Presentation on the topic of scientific computation (3) Project progress reports (4) Setting up of writing group
HW for 12/4 What is mathematics? A chance to reflect upon and synthesize what we have learned through its history. (1) RA 18 Submit through Bbd: the best quote you can find that answers this question. Can be a sentence, or up to a short paragraph. Also in this RA, write a single paragraph on why you feel this quote encapsulates the answer. Bring a hard copy to class. (2) Submit preferred date and estimated duration of project presentation
Mon 12/4 (1) Writing part of project returned (2) Discussion and feedback on project progress (3) Discussion of topic and quotes (4) Short essay to be written in class
HW for 12/6 Submit preferred date and estimated duration of project presentation if you haven’t already.
Mon 12/6 Student presentations for project
HW for 12/11 Final version of project writing assignment due.
Mon 12/11 Student presentations for project
FINAL EXAM SLOT 12/20 1 to 3 pm, in usual classroom
Wed 12/20 Student presentations for project, attendance mandatory