IN207 Honors Seminar in Mathematics
INFINITY
Guide: James V. Rauff, Professor of Mathematics Office: Shilling Hall 203J Office phone: 424-6249 Office hours: 2-3 MTWTh & by appointment
Textbook: To Infinity and Beyond by Eli Maor (Princeton, 1991) ISBN 0-691-02511-8
Readings: Additional readings by a variety of authors will be handed out or assigned for online reading.
Course description: In this seminar we undertake an exploration of the mathematics of infinity Our main guide to the infinite will be mathematics, but we will also be assisted by anthropology, art, linguistics, literature, philosophy, mythology and physics.
Requirements: The ideas that you will encounter in this course are dense, deep, and complex. It is important that you attend carefully to the reading and writing assignments. It is also important that you attend class and participate in the discussions.
- You will be given reading assignments for every class
period and writing assignments almost weekly. Writing assignments must be typed
(double-spaced), but extensive mathematical notation may be
handwritten. Assignments will be posted in the Daily Planner shown
below.
- You will also be asked to work on in-class cooperative
explorations and problems.
- You will be expected to contribute to discussions in
class. I prefer spontaneous and engaged discussion, but reserve the
right to query individuals directly.
- You also will prepare a term project on some aspect of
infinity (history, theory, experiment, personality, mystical, etc.)
This project can be a traditional term paper, a multimedia presentation, a
one-act play, a computer program, a musical composition, a work of
visual art or whatever moves you and is approved by me. Group
projects involving 2 or 3 people are encouraged. The nature of the project
is up to you, but it must include a written explanation of the aspects or
concepts of infinity addressed in the work and your one-page
self-assessment of the project. All projects will be presented to the
seminar in December. Your presentation should be about 10 minutes in
length. You should discuss your ideas for your project with me
before November 3. A brief list of example topics can be
found by clicking here. You are not limited
to these choices!
Your grade will be based upon the quality of your oral and written work. Written assignments and in-class activities will count for 75% of your grade and the final project 25%. My grading scale is unimaginative: 93% + is an A, 83% a B, 73% a C, etc. Late written assignments will be penalized.
Keys to Success: These are obvious, but worth reiterating. Attend class. Do all the reading and writing assignments promptly. Participate in class discussions. Ask questions. Ask more questions.
Schedule of Assignments: Reading and writing assignments are listed on the day they are due.
LAST UPDATED ON November 10, 2009
|
Date |
Reading |
Writing/Events |
Links |
|
Aug. 25 (T) |
none |
Introductions |
|
|
Aug. 27 (Th) |
Maor: pp.14-16 & Borges: The
Library of Babel |
Questions for
discussion: 1.
Suppose you put one penny
on the upper left-hand square of a chessboard. Then put 2 pennies on the next
square and 4 on the next and so on, doubling as you go. How many pennies are
on the 30th square? How
high is the stack? How much does the
stack weigh? Same questions for 50th
square? 64th square? 2.
How many books are in the
Library of Babel? 3.
Are the seven Harry Potter
novels in the Library of Babel? 4.
Can 410 pages of repeated
MCV correspond to any language? 5.
Is it possible that every
string of the 25 symbols of the Library's language that you can imagine has a
terrible meaning in some secret language? 6.
Is it illogical to think
that the world is infinite? |
|
|
Sept. 1 (T) |
Maor:
pp.2-5 Maor:
pp.29-33 Maor:
pp.40-43 |
Questions to ponder (not to be handed in): 1. Calculate these infinite sums: (a)
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … (b)
1/3 + 1/9 + 1/27 + 1/81 + … (c)
2/5 + 4/25 + 8/125 + 16/625 + … (d)
1/3 + 2/9 + 4/27 + 8/81 + 16/243 + … 2.
Show how the Grandi series, 1 + 1 – 1 + 1 – 1 + 1 – …, can be used to
argue that 2 = 5. |
|
|
Sept. 3 (Th) |
Maor:
pp.54-60 |
Hand in your
responses to these: Show that each of these sets is
denumerable by exhibiting a 1:1 match between the set and the set of natural
numbers. a. The set of odd natural numbers b. The set of natural numbers that are multiples of 5 c. The set of integers d. The set of perfect squares e. The set of finite sequences of the letters B and A. f.
The set of rational numbers |
|
|
Sept. 8 (T) |
Maor:
pp. 61-66 |
Hand in your
responses to these: 1. Define a "chuckle" to be any finite sequence of the
words "HA" and/or "HO". For example,
HAHAHAHAHAHOHOHA and HOHOHAHO are
chuckles. Prove that the set of all chuckles is denumerable. 2. Define a "laugh" to be a denumerable sequence of
the words "HA" and/or "HO". For example, HAHAHAHAHA....
and HAHOHAHOHAHOHAHO... and HAHAHOHAHOHOHAHOHA ... are
all laughs. Prove that the set of laughs is not denumerable. |
|
|
Sept. 10 (Th) |
Maor:
pp.253-254 |
|
|
|
Sept. 15 (T) |
Maor:
pp. 76-82 |
Hand in your responses to these: 1.
Calculate. a. b. c. 2.
True or false? a. b. 3. Use the set definition of addition to prove 4. Use the set definition of multiplication to prove 5. Use the set definition of exponentiation to prove |
|
|
Sept. 17 (Th) |
Maor:
136-148 |
For discussion: 1. What is your favorite example of the
concept of infinity in (a) literature
(b) cinema (c) music (d) poetry?
2.
Do these view infinity positively or negatively? 3. How do these concepts compare with the
mathematical notion of infinity? 4. Is blue your color for infinity? |
|
|
Sept. 22 (T) |
Maor:
Chapter 20 |
|
http://www.protozone.net/ASHOCK/AJBord.html http://www.scienceu.com/geometry/handson/kali/kali.html |
|
Sept. 24(Th) |
Maor:
Chapter 21 |
Hand in your responses to these:
a.
b.
c. d. e. 2. Using this figure
as the base pattern, construct each of the seven types of strip symmetries.
|
|
|
Sept. 29 (T) |
2.
Crump – The Anthropology of Numbers, Chapter 7: Time |
For
discussion: 1.
How do people get the idea of time? 2.
What concepts are involved in the measurement of time? 3.
What is a synodic month? 4.
What is a solar year? 5.
What is Crump's distinction between nature and cosmos? 6.
What is chaos? 7.
What are some socially constructed time units in your culture? 8.
Can socially constructed time be infinite? Explain. 9. How would one calculate using Ireneo
Funes' numeration system? 10. Ireneo remembered every leaf and every
time he perceived that leaf. He would also remember every time that he
remembered that he perceived that leaf. He would also remember every
time that he remembered that he remembered that he perceived that leaf. What conclusions/paradoxes result from
this line of reasoning? |
|
|
Oct. 1(Th) |
Hand in your
responses to these: 1. How do
people get the idea of time? 2. What
concepts are involved in the measurement of time? 3. What
are some socially constructed time units in your culture? Which of these are linked to natural
cycles? 4. Can
socially constructed time be infinite? Explain. 5. Ireneo remembered every leaf and every time
he perceived that leaf. He would also remember every time that he remembered
that he perceived that leaf. He would also remember every time that he
remembered that he remembered that he perceived that leaf. What conclusions/paradoxes result from
this line of reasoning? |
|
|
|
Oct. 6 (T) |
1.Harris
– Mathematics in a Cultural Context pp.53-73 2.Jakamarra-Marlujarrakurlu |
For
discussion: 1. What
is the Dreaming? 2.
What is social time? 3. Do
we experience time as duration? Why? 4. What
are some natural units of time? 5.
When does a day begin? 6.
What are the time markers/units in the human life cycle? 7.
What are age-grades? 8.
How long ago was the Dreamtime? 9. Summarize the story of the two kangaroos. |
|
|
Oct. 8 (Th) |
Hand in your answers to these. 1. What is the skin name of the son of the daughter of a
Napanangka woman? 2. What is the skin name of the daughter of the son of the
daughter of a Jampijinpa man? 3. What is the skin name of the mother of the father of a
Napaljarri woman? 4. According to the Milingimbi calendar, what are the
characteristics of Barri’Mirri Mayaltha? 5. Time as event. Give an example from your experience of
some event whose date you remember and then describe how you would refer to
that same event without a date. 6. Time as limit. Give an example from your experience in
which time was regarded as a limit.
Describe how that example would have been different if you hadn’t
regarded time as a limit. 7. Describe what your average Tuesday would be like without a
clock and without any structuring that depended upon a clock. |
|
|
|
Oct. 13 (T) |
Maor: Chapters 23-29 |
Things to ponder. Vocabulary:
parallax, fixed stars, heliocentric, geocentric, Olbers paradox, Cepheid
variable, Doppler effect, cosmological principle, space-time continuum, Names: Thales, Ptolemy, Cusanus, Copernicus, Digges, Bruno, Brahe,
Kepler, Herschel, Hubble Queries: Why is the sky dark
at night? How can you prove that the
earth orbits the sun? Is the universe
infinite? What does that last question mean?
Why is the idea of an infinite universe dangerous? |
|
|
Oct. 15 (Th) |
BBC
Transcript: Parallel Universes |
1. What is matter made up of? 2. What is "the singularity"? 3. How many dimensions are there in the Universe? 4. Why is gravity so weak? 5. Why does an infinite number of universes imply an infinite number of civilizations? 6. In M-theory, what caused the big bang? |
|
|
Oct. 20 (T) |
Barrow:
Is the Universe Infinite? |
|
|
|
Oct. 22 (Th) |
none |
No class: Fall Break |
|
|
Oct. 27 (T) |
none |
Write essays answering any two of the following four
questions. Each essay should be at
least 250 words in length.
|
|
|
Oct. 29 (Th) |
|
|
|
|
Nov. 3 (T) |
none |
No class: Scheduling Day Hand in: Project proposal (bring to SH203J) |
|
|
Nov. 5 (Th) |
none |
Hand in your completed logic exercises. (handout) |
|
|
Nov. 10 (T) |
Maor: Epilogue |
Hand in your completed Total Library exercises. (handout) |
|
|
Nov. 12 (Th) |
Maor: Appendices 9 & 10 |
Contemplation: “The collection of all ordinals is called
Ω” |
|
|
Nov. 17 (T) |
none |
Hand in
your completed ordinal numbers exercises. (handout) |
|
|
Nov. 19 (Th) |
none |
Hand in your answers to these: 1. There are a. How many different chants are being chanted? Explain.
2. Explain why |
|
|
Nov. 24 (T) |
none |
No class: Thanksgiving Break |
|
|
Nov. 26 (Th) |
none |
No class: Thanksgiving Break |
|
|
Dec. 1 (T) |
none |
No class: Work on your project I’ll be in the classroom
during the regular class time if you need to consult with me about your
project. |
|
|
Dec. 3 (Th) |
none |
No class: Work on your project I’ll be in the classroom during
the regular class time if you need to consult with me about your project. |
|
|
Dec. 8 (T) |
|
Student Presentations: 1. Thornton(Klein
bottle) 2. Walworth
(Infinite Nature) 3. Eades (Hotel
Infinity Model) 4. Bulthuis
(Warlpiri painting/story) 5. Bein (M-play) |
|
|
Dec. 10 (Th) |
|
Student Presentations: 1.
Gleason(Tessellation
art) 2.
Kale & Osiecki (Infinite Food Time) 3.
Burczak (Homemade universes) 4.
Bettenhausen (Infinity: the Movie) 5.
Smith (Aboriginal
Dance) |
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Cell phones: Turn off your cell phone during class or set it to vibrate rather than ring. Violators will be assessed exponentially increasing point deductions for each offense.
Student Learning Outcomes for the Honors Program: At the end of four years, honors students will have (1) created a substantial original project resulting from interdisciplinary collaboration and independent research and (2) demonstrated critical thinking, oral communication, and self-assessment skills.
Quantitative Reasoning Goals : A student who successfully completes a Millikin QR course will demonstrate the ability to (1) use deductive reasoning in a formal, symbolic, axiomatic system, and (2) apply the theorems of the system to solve appropriate problems.
Academic Honesty Policy
All students are expected to uphold
professional standards for academic honesty and integrity in their research,
writing, and related performances. Academic honesty is the standard we expect
from all students. Read the Student Handbook for further details about offenses
involving academic integrity at: http://www.millikin.edu/handbook/.
Staley Library also hosts a web site on Preventing Plagiarism, which includes
the complete university policy. It is located at: http://www.millikin.edu/staley/services/instruction/Pages/plagiarism-faculty.aspx. Visit and carefully read the Preventing Plagiarism web site.
The
Faculty has the right and the responsibility to hold students to high ethical
standards in conduct and in works performed, as befits a scholar at the
university. Faculty members have the responsibility to
investigate all suspected breaches of academic integrity that arise in their
courses. They will make the determination as to whether the student violated
the Academic Integrity Policy. Should the faculty member determine that the
violation was intentional and egregious, he or she will decide the consequences,
taking into account the severity and circumstances surrounding the violation,
and will inform the student in writing, forwarding a copy of the letter to the
Registrar and to the Dean of Student Development.
This letter will be destroyed when the student
graduates from the University unless a second breach of integrity occurs, or
unless the first instance is of sufficient magnitude to result in failure of
the course, with an attendant XF grade recorded in the transcript. If an XF is
assigned for the course, the faculty letter of explanation becomes a permanent
part of the student’s record. If a second violation occurs subsequent to the
first breach of integrity, the Dean of Student Development will begin
disciplinary and judicial processes of the University, as outlined in the
Student Handbook.
If a student receives an XF for a course
due to academic dishonesty, this remains as a permanent grade and cannot be
removed from the transcript. However, students may repeat the course for credit
toward graduation. Some programs and majors have more explicit ethical
standards, which supersede this Policy, and violation of which may result in
dismissal from some programs or majors within the University. If you have
difficulty with any assignment in this course, please see me rather than
consider academic dishonesty.
Disability Accommodation Policy
Please address any special needs or special
accommodations with me at the beginning of the semester or as soon as you
become aware of your needs. If you are seeking classroom accommodations under
the Americans with Disabilities Act, you should submit your documentation to
the Office of Student Success at Millikin University, currently located in
Staley Library 014.