IN207 - Honors Seminar in Mathematics - Spring  2010

Guide: Dr. James Rauff, Professor of Mathematics 
Office: Shilling Hall 203J
Office phone: 424-6249
Office hours:  TThF 10-10:50 a.m.; MW 1-3 p.m.

Textbooks: 

 In Search of Infinity  by N. Ya. Vilenkin  ( Springer: 1995)  ISBN-10: 0-8176-3819-9  

Labyrinths  by J. L.Borges (New Directions: 2007)  ISBN: 978-0-8112-1699-9



Course description: In this seminar we undertake an exploration of the mathematics of infinity.  Our travels will take us to the higher mathematical infinities.  Along the way we will encounter parallel universes, mysticism, chaos, truth, and beauty.  Our main guides to the infinite will be mathematics and the fiction of Jorge Luis Borges. We will also be assisted by anthropology, art, linguistics, philosophy 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 each week (except the first week). Writing assignments should be typed (double-spaced), but extensive mathematical notation may be handwritten. Assignments will be posted in the Daily Planner shown below. The writing assignments may require some contemplation. I recommend that you look ahead to them and not try to do the whole assignment the night before it is due.
* 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, 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. 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. All projects will be presented to the seminar from April 27- May 6. The presentation should be about 10 minutes in length. You should discuss your ideas for your project with me by March 9 and be settled upon what you are going to do by March 23. 

Your grade will be based upon the quality of your oral and written work. Weekly written assignments with count for 65% of your grade, attendance and class participation 10%, 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.

Assignment Schedule:  Last Updated on April 19, 2010

Date

Assignment for this Day.

January 19 (T)

none

January 21 (Th)

Read pp.33-38 (stop at Infinite sets) in Vilenkin.

Vocabulary:  set, element, finite, characteristic property,  {x| P(x)},  equal, equivalent, fuzzy set.

Be ready to respond to these in class. (Note: natural numbers are 0, 1, 2, 3, 4, etc.)

1.      Give a characteristic property of a set with no elements.

2.      Are the two sets A = {x| x is a natural number less than 6} and  B = {x | x is the remainder when a natural number less than 100 is divided by 6} equal?

3.      If two characteristic properties are equivalent are the sets they describe equal?

4.      Give the characteristic property of a set of objects in your major field.

5.      Select a novel or play that you are now reading or have recently read. Open to a random page.  List all the fuzzy properties named on that page.

 

January 26 (T)

Read pp.38-47 (stop at How to compare sets in terms of size) in Vilenkin.

Hand in your responses to these (40 points):

1.      When Ion shows up at the hotel and is put into room number 1 into what room is the occupant of room 42 put?

2.      When the philatelists arrive and the cosmic zoologists are moved to new rooms into what room is the 2010th philatelist put?  Into what room does the cosmic zoologist in room 4514 move?

3.      After the zoologists all leave, into what room does the philatelist in room 51431 move?

4.      All the hotels except one were closed.

a.       Using Ion’s first solution, into what room of the only remaining hotel did the occupant of room 13 of the 5th hotel move?

b.      Using the director’s improvement to Ion’s solution, into what room of the only remaining hotel did the occupant of room 13 of the 5th hotel move?

c.       Using the president of the Academy of Mathematics of the galaxy of Swan’s solution, into what room of the only remaining hotel did the occupant of room 13 of the 5th hotel move?

5.       On the bottom of page 46,Vilenkin writes, “ Later on, when they began to serve the guests ice cream, it was discovered that each guest had two portions, although as a matter of fact, the cook had only prepared one portion per guest.  I hope that by now the reader can figure out by himself (sic) how this happened.”  How did this happen?

 

 

 

January 28 (Th)

Read pp.47-54 (stop at Algebraic numbers) in Vilenkin.

Vocabulary:  one-to-one correspondence, cardinality, equivalent, countable, densely distributed

Be ready to respond to these in class.

1.      Give an explicit one-to-one correspondence between each of the following pairs of sets.

a.       The set of even natural numbers AND the set of natural numbers that are multiples of three.

b.      The set of natural numbers AND the set of integers (both positive and negative).

c.       The set of letters in the English alphabet AND the set of cards in a standard deck of playing cards that are either hearts or diamonds.

d.      The set of natural numbers AND the set of prime numbers.

2.      Consider the set {A,B}.  A word over {A,B} is defined to be any finite sequence of  letters A and B with repetition allowed.  For example, BABA, AAAAB, and BBABBBBBAAAB are all words over {A,B}.  Describe a one-to-one correspondence between the set of natural numbers and the set of words over {A,B}.

3.      A natural lattice point in the plane is a point whose coordinates are both natural numbers.  Thus, (1,2) and (5,9) are lattice points, but (-3, 4) and (2.6, 8.8) are not.  Describe a one-to-one correspondence between the set of natural numbers and the set of natural lattice points.

 

 

Feb 2 (T)

Read pp.54-58 (stop at Uncountable sets) in Vilenkin

Hand in your responses to these (50  points):

1.       Let W be the set of natural numbers greater than 100.

a.       Prove that W is countable.

b.      Let V be the set of integers less than 100.  Prove that V is countable.

c.       Put all of the elements of W and all of the elements of V into a new set called O.  Prove that O is countable.

d.      Remove from W all the even natural numbers greater than 100.  Call the remaining set Y.  Prove that Y is countable.

2.       Prove that.

3.       An   AB-palindrome is a finite sequence of the letters A and B that reads the same left-to-right as it does right-to-left.  For example, BAAB and ABABA are AB-palindromes, but BAA is not.  Prove that the set of AB-palindromes is countable.

4.       Suppose that the set of beans in a pot of chili is a countable set. Also suppose that another pot of chili contains a trillion beans.  The two pots of chili are combined into one pot.  How many beans are in this combined chili pot?  Explain.

Feb. 4 (Th)

Read pp.58-62 (stop at The existence of transcendental numbers) in Vilenkin

Be ready to respond to these in class.

1.      A “tune” is defined to be a finite sequence of the notes C, F and/or G.  For example, “CFGCFG”  and “FFFFCCCCCGCF” are tunes.  An “omega-tune” is countable sequence of the notes C, F, and/or G.  For example, “CCCCCC…” and “CFCFCFCFCF…” and “CFCCFG…”

 

a.      Prove that the set of tunes is countable.

b.      Prove that the set of omega-tunes is uncountable.

 

2.      Make up your own example of an uncountable set.

Feb. 9 (T)

Read pp.62-70 in Vilenkin

Hand in your responses to these ( 60 points):

In #1-5, calculate  (See p.68 for meaning of symbols and .)

1.     

2.     

3.     

4.     

5.            

6.      Find an element of the ordinal  that isn’t an element of the ordinal      

7.      Prove .

8.      Decide whether each of the following sets is countable or uncountable.

a.      

b.     

c.      

d.     

e.      

f.        

9.      A countable number of boys and a countable number of girls are at an infinite dance.  They pair up (boy with girl) and begin dancing. Everyone has a partner.  Unexpectedly, a countable number of girls from a different town arrive.  The chaperones stop the music and rearrange the people so that everyone has a dance partner (again every pair is boy-girl).  Explain how this can be done.

10.   A “babble” is defined to be a countable sequence of the letters B and/or A.  Prove that the set of all babbles is uncountable.

 

Feb. 11 (Th)

Read "The Library of Babel" (p.51)  in Borges .

Be ready to respond to these in class.

 

1.      How many books are in each hexagon?

2.      How many books are in the Library?

3.      We know that the Combed Thunderclap is one of the volumes in the Library of Babel. How many books in the Library differ from the Combed Thunderclap by only one symbol?

4.      What are the 25 orthographic symbols that appear in the Library's books?

5.       Does the Library have an index of its contents?  If so, is the index in the Library?

6.       There is a book in the Library that consists only of the letter C repeated continuously without spaces or punctuation. How many times does the letter C appear in this book?

7.      How could you prove that no two books in the Library were identical?

8.      Can 410 pages of repeated MCV correspond to any language?

9.      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?

10.  Is it illogical to think that the world is infinite?

11.  How is a repeated disorder an Order?

12.  Is the Library infinite?

 

Feb. 16 (T)

Read  “Funes the Memorious” (p.59) and “The Immortal” (p.105) in Borges

 

Hand in your responses to these (60 points):

1.      How would Ireneo Funes' numeration system work in arithmetic?

2.       Ireneo remembered every leaf and every time he perceived that leaf. He logically would also remember every time that he remembered that he perceived that leaf.  Continue this line of reasoning to argue that Funes' memory was infinite. 

3.      Is Funes memory uncountable?  Explain.

4.      Why does Borges suspect that Funes was not very capable of thought?

5.      What happens to an immortal over an infinitely long span of time? 

6.      Explain Borges’ reasoning behind this passage from The Immortal (p.114): “…all our acts are just, but they are also indifferent. There are no moral or intellectual merits.”

 

Feb. 18 (Th)

Read pp.71-90 ( stop at Euclid does not rely on Euclid) in Vilenkin

Be ready to respond to these in class.

1.      How is infinity embodied in geometry and in arithmetic?

2.      What is a real number according to Weierstrass and according to Dedekind?

3.      What is a function?

4.      Vilenkin offers up six “mathematical wonders.”

a.      Marching mountain peaks

b.      Wet points

c.       Devil’s Staircase

d.      Prickly curve

e.       Koch’s Snowflake

f.       Sierpinski’s carpet

                             What makes them wonderful?

 

Feb. 23 (T)

Hand in your completed Koch Snowflake Exploration. (40 points)

Read pp.1-31 in Vilenkin

 

Be ready to respond to these in class.

1.  Use an argument like Zeno's to explain why you can't cut all the way through a banana.

2.  What are the "three eternal problems"? 

3.  Is there a direction in an infinite world?

4.  Distinguish between the actual infinite and the potentially infinite.

5.  Why were Bruno’s ideas so dangerous?

6.  Reflect on Kant’s remarks about the universe (p.14).

 

Feb. 25 (Th)

Read “Parallel Universes” handout.

 

Be ready to respond to these in class.

1.      What are the two types of religious cosmologies?

2.      How is the night sky like a time machine?

3.      What is the age of the universe?

4.      What percent of the matter in the universe is visible?

5.      What is the inflationary theory?

6.      What is the multiverse?

7.      What is the fate of our universe?

 

Mar. 2 (T)

Read “Is the Universe Infinite?” and “The Infinite Replication Paradox” handouts and the “Parallel Universes” transcript.

Be ready to respond to these in class.

1.      Describe Thomas Digges’ model of the universe.

2.      Describe Giordano Bruno’s model of the universe. What is his argument against the notion that the universe is finite?

3.      What is cold, dark matter?

4.      What is the difference between positive and negative curvature?

5.      What is the difference between geometry and topology?  Why is this difference important to the question of whether or not the universe is infinite?

6.      What is the problem of uniformity?

7.      What are some implications of the inflationary theory?

8.      What is Olbers’ Paradox and how can it be resolved?

9.      Is the universe infinite?

10.  What must occur in a universe of infinite size?

11.  How would you calculate the distance to the nearest identical copy of yourself?

12.  What are the implications of the hypothesis that there are an infinite number of different forms of living complexity?

13.  What are the implications of the hypothesis that there are merely a finite number of different forms of living complexity?

14.  What does Johnny Appleseed have to do with infinity?

15.  Is Barrow talking about countable or uncountable infinities?

16.   What is matter made up of?

17.   What is "the singularity"?

18.   How many dimensions are there in the Universe?

19.  Why is gravity so weak?

20.   In M-theory, what caused the big bang?

 

 

Mar. 4 (Th)

Read “The Garden of Forking Paths” (p.19) in Borges , “Tlon, Uqbar, Orbis Tertius” (p.3) in Borges and the “The Book of Sand” handout (I gave it to you earlier in the semester. It’s yellow.).

For discussion.

1.      Conjecture southern hemisphere Tlön verbs for the English nouns cat, sun, tree, book, and iPod.  Write English literal translations of Tlön sentences equivalent to The cat sat in the sun, It is a book about trees and The iPod fell into the river.

2.      Conjecture northern hemisphere Tlön adjectives for the English nouns cat, sun, tree, book, and iPod.

3.      What are hrönir?

4.      Give an example of an ur that would appeal to you.

5.      What is the garden of forking paths? 

6.      Write three different endings to “The Garden of Forking Paths”.

7.      How could the narrator of “The Book of Sand” determine that the small illustrations were spaced at two-thousand page intervals?

Mar. 9 (T)

Write essays answering any two of the following three questions.  Each essay should be at least 250 words in length.  The essays should show that you have considered the science and speculation in the readings. (60 points)

 

  1. Does an infinite universe imply an infinite number of civilizations? 
  2. What is Olber’s paradox? What are some possible resolutions?  Which resolution seems most likely?
  3. What are some ethical/moral/religious implications of an infinite universe?

 

Mar. 11 (Th)

Read “Cosmology and Ethnoscience” handout

For discussion:

1.      What is “cosmology”?

2.      What is the difference between modern and traditional cosmologies?

3.      What are Yin and Yang?

4.      How is the I Ching used?

5.      What is siemeigaku?

6.      What are the two sorts of meanings that numbers have?

Mar. 13-21

Spring Break

Mar. 23 (T)

 

Hand in a written description of your project (2 paragraphs)

 

Mar. 25 (Th)

Read “The Anthropology of Numbers: Time” handout

 

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.

Mar. 30 (T)

Hand in your responses to these:

1.      Identify the ten day names of the pink Tika square given to you in class on Mar.25

2.      Calculate the Maya date that is 475 days after 9.16.10.0.0  1 Ahau 3 Zip.

3.      Calculate the Maya date that is 300 days before  9.16.10.0.0  1 Ahau 3 Zip.

4.      What are some socially constructed time units in your culture?

5.      What concepts are involved in the measurement of time?

(50 points)

Apr. 1 (Th)

Read “The Local Origins of Time” handout

 

For discussion:

1.      Where is Sumba?

2.      According to the Kodi, when was time invented?

3.      What are the two major forms of the experience of time?

4.      What is the significance of “Mbyora at the banyan tree?”

5.      What are the Kodi day markers?  Compare to ours and to the Lakota.

6.      What is “chicken time?”

7.      In Kodi mythology, why does the moon have phases?

8.      Are some people privileged controllers of time?

9.      What common theme extends through the Kodi stories about the origins of temporal divisions?

Apr. 6 (T)

No class today- work on your project

Apr. 8 (Th)

Read “Mathematics in a Cultural Context” handout

 

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?

 

Read “Comparing Time and Temporality in Cultures” handout

 

Discussion:

1. What are some of the problems that all cultures deal with?

2. What is entropy?

3. What are cardinal phenomena?

4. How do humans define their place in the world order?

5. What is the primary criterion for knowledgeability?

6.  Explain sa'ah naghai bik'eh hozhoon?

Apr. 13 (T)

Scheduling Day – No class

 

Hand in your responses to these (Bring to my office or send by email):

 

Your answers to #2-5 should be given in complete sentences and paragraphs.

 

1.a. What is the skin name of the wife of Robert Jangala’s son?

   b. Calculate the Maya date that is 400 days after 9.16.10.0.1  2 Imix 4 Zip.

 

2. Pam Harris talks about time as event, limit, and distance.  Give one example of each of how people in our USAmerican culture view time as event, limit, and distance.

 

3. Harris also makes the contrast between quantity and measurement and quality and comparison. Use the data available in the articles by Pinxten and Hoskins to decide which of these two emphases is most likely to found in Navajo and Kodi tradition. Explain your answers.

 

4. What are some of the problems that all cultures deal with according to Rik Pinxten?  Explain how one of these problems is dealt with in Kodi culture.

 

5. What are the two major forms of the experience of time according to Janet Hoskins?  Give at least two examples of each in our USAmerican culture.

 

 (50 points)

Apr. 15 (Th)

Task:  Conversation on time/truth.

Start time: When we are ready.    

Finish time: When we are done.

Apr. 20 (T)

Read:  pp.117-122 (stop at A baffling axiom)  in Vilenkin,  “The Lottery in Babylon (p.30) in Borges and “Emma Zunz” (p. 132) in Borges.

Discussion:

1. What is the liar paradox?

2.  What is truth?

3.  What is the difference between an undetectable, omnipotent Company and one that doesn’t exist?

Apr. 22 (Th)

Read the “Truth” handout

Hand in your responses to the Truth Exercises (60 points)

Apr. 27 (T)

Oral Presentations:  SK, AR, ST, SP, JB&KE

Apr. 29 (Th)

Oral Presentations:  AF,  EH, AW, MW&JB

May 4 (T)

Oral Presentations:  SH, BAM, BF&JO, JL, MO

May 6 (Th)

Oral Presentations: JJ, OH, KC

 

 

 

 

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.

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.