To Infinity
and Beyond: What does that really mean?
Witt Sem 100
Fall 2009
“There are more things in heaven and earth,
Horatio,
Than are dreamt of in your philosophy.” Hamlet, Act I, Scene 5
The
quote is quite apropos: Hamlet and Horatio were students at Wittenberg!
“I’m sorry but you can’t add this course” the professor said to the student. “Even though there are an infinite numbers of desks in this class room, all of the desks are filled.” Indeed as the student looked out over the classroom he saw and row upon row of desks each numbered in order, 1, 2, 3, … etc. stretching out as far as he could see, disappearing into the distance. Behind each desk sat a student. “So you see”, said the professor,” with all of the desks filled, there is no room for you”. The student turned away but suddenly a thought occurred to him. “Wait”, he said to the professor. “I think I see a way to find an empty desk”.
Can you see how
to do this? If so, you can sign up for this class.
Infinity is a difficult concept
with a long history. Zeno’s paradoxes grappled with infinite processes using
them to “prove” that motion is impossible; Aristotle wrote “The infinite has
potential existence … There will not be an actual infinite” thereby
acknowledging that while there is no largest number or no smallest fraction
(the potential infinite), the concept of grasping the infinite as a whole (the
actual infinite) was impossible. Galileo (1564 – 1642) pointed out that while
on one hand the squares like 1, 4, 9, etc. were obviously fewer in number than
the integers 1, 2, 3, … , yet on the other hand for every square there was a
corresponding integer 12 =1 , 22 = 4, 32 = 9
etc. It wasn’t until the late 19th century the Georg Cantor (1845 –
1918) grasped the “actual infinite” not only showing that the integers and
their squares were the same infinity but that there were larger infinities than
these and in fact, a (potentially) infinite number of larger infinities!
But the study of the infinite is
by no means restricted to mathematics. This course will be a study of infinity
and the infinitesimal as it appears in geometry, aesthetics (arts &
literature), science, philosophy, religion and culture as well as mathematics
and puzzles.
Textbooks for this course
1. The Infinite Book: A Short Guide to the Boundless, Timeless, and Endless by John D. Barrow, Gresham Professor of Geometry at Gresham College, UK.
2. To Infinity and Beyond: A Cultural History of the Infinite by Eli Maor
Course Syllabus
WTSM 100.29 – To Infinity and Beyond – What does that really mean?
Fall
2009
Course
Syllabus
Instructor: Brian J. Shelburne
Office: 329E Science phone: x7862 email: bshelburne@wittenberg.edu
Class Meetings: TTH 9:40 - 11:10 Rm 261 BDK Science
Office Hours: MWF 1:50 – 2:50 or anytime outside of my regularly scheduled classes or meetings
Required Texts: To Infinity and Beyond: A Cultural History of the Infinite; Eli Maor
The Infinite Book: John D. Barrow
Course Objectives: To question, to explore, to think, to learn, to understand, and then to see the wonder of it all!
Grading: The grade for the course will be based on the following criteria:
1. Class attendance, completing reading assignment before class, contributing to class discussions etc. (25%)
2. Homework assignments (50%).
3. Final Project (25%)
The letter grades will be assigned using the standard 90% - 80% - 70% - 60% breakdown for A – B – C – D. Less than 60% is an F. Pluses and minuses will be assigned using a 3% - 4% - 3% breakdown (e.g. 87% plus is B+, 83% plus is a B and 80% plus is B-)
Course
Expectations
Always come to class
If you don’t understand something, ask.
The 3-to-1 Rule: Expect to spend three hours study outside for class for every hour in class
Read each assignment three times
The first time read straight through to get the main ideas (before class)
The second time read slowly and carefully to get the details (before class)
The third time to review what you’ve read (after class)
Academic Integrity Issues: I expect you to uphold and maintain the highest standards of academic integrity as stated in Wittenberg’s Honor code. Therefore all pledged homework assignments will carry the statement
I affirm that my work upholds the highest standards of honesty and academic integrity at Wittenberg and that I have neither given nor received unauthorized assistance
________________________________________
This statement (the pledge) must be signed; work with unsigned pledges will not be graded (resulting in a zero for that assignment). Students found guilty of academic dishonesty will receive a zero for the assignment and a report of the incident will be filed with the Honor Council. A second allegation of academic dishonestly will result in an automatic Honor Board Hearing which may result in suspension or dismissal from Wittenberg.
If you have questions about what constitutes academic integrity and/or academic dishonesty it is your responsibility to ask me. It is my responsibility to make the standards for academic integrity clear. .
Class Attendance: Students are expected to attend all classes. Unexcused absences will lower your class attendance/participation grade
.
Final Note: Any student with a documented disability who needs to arrange reasonable accommodations should contact me ASAP. Early notification is highly preferable. You may speak to me after class, in my office, call me or send me e-mail. You will also need to contact Van Rutherford, Assistant Provost for Academic Services at 937-327-7924 in room 208 Recitation Hall to coordinate accommodations and receive a self-identification letter.
Infinity in General:
1. Infinity – From the MacTutor History of Mathematics archives at St Andrews University in Scotland, this is an invaluable place to start when researching the biographies of mathematicians, philosophers, and scientists who figure in our understanding of infinity. In particular check out the Biographies Index.
2. Zero: A sign found in the Boston Museum of Science. Zero is a number; infinity is a concept.
3. Calculating (approximating) pi using various analytic formulas. Check out the pi search page to find out where your birth date appears in the expansion of pi.
The Mathematics of Infinity:
1. The Monkey Shakespeare Simulator: “If you have enough moneys banging away randomly on typewriters, they will eventually type the works of William Shakespeare”. Here are Wikipedia links to articles on the Infinite Monkey Theorem and the Infinite Monkey Theorem in Popular Culture. Enjoy!
The Geometry of Infinity:
1. The Cantor Middle Third Set: A geometric object (subset) on the interval closed interval [0,1] which contains no subintervals but is uncountably infinite
2. Space Filling Curve: A Java Applet from the University of Texas that draws space filling Hilbert, Piano, and Sierpinski curves by levels. In the limit the curve of infinite length fills a finite area of space.
3. Fractals! Figures with infinite perimeters and finite volumes. Check out the home page from my 2005 Witt Sem on Fractals and Chaos.
4. Euclid’s Elements On-Line: To understand projective geometry or non-Euclidean geometry, you much first understand the “standard” Euclidean variety.
The Aesthetics of Infinity: Art and
Literature
1. The Library of Babel, a short story by the Argentine author Jorge Luis Borges, considers a library filled with books containing every possible combination of letters. The Wikipedia entry for The Library of Babel provides some interesting background to the story including some rough calculations on the size of the library.
2. The Garden of Forking Paths, is another short story by Borges which examines what happens if all futures are possible? The forking path does not refer to a bifurcation in space by a bifurcation in time. The Wikipedia entry for The Garden of Forking Paths provides background.
3. The Book of Sand (link is now broken): Also by Borges; a book with an (uncountably?) infinite number of pages. Instead try this hyper-text puzzle for the Book of Sand. Again the Wikipedia entry for the Bok of Sand provides a background.
4. Mirrored Room by artist Lucas Samaras at the Albright-Knox Gallery in Buffalo, N.Y. A room whose walls, ceiling and floor are mirrors.
5. M.C. Escher’s Official Website: On the left follow the link to the Picture Gallery. The Symmetry category contain many of Escher’s work that tile of the infinite plane.
6. The Eames Collection: A web site dedicated to the artistic/design genius of Ray and Charles Eames. Of particular interest is their numerous works involving mathematics. The Eames exhibit “Mathematica – A World of Numbers … and Beyond” (from The Eames Collection, click on Eames Exhibits – Exploratorium) has a picture of the Mobius band found in Maor’s text (page 148).
The Science of Infinity:
1. Powers of Ten: (on YouTube) The famous 1977 short file by Ray and Charles Eames which takes you on a journey of scale from Chicago across the universe and back then into the nucleus of a carbon atom. The narrator is scientist Philip Morrison. A similar-themed short film The Grand Scale of the Universe is narrated by Morgan Freeman.
The Philosophy of Infinity:
1. God and
Mathematical Infinity: A paper by Brenden Kneale.
2. Philosophy
and the infinite: A paper Edward Buckner found on his website The Logic Museum.
There are a number of links to original texts.
3. In Question 7 of the First
Part of the Summa Theologica,
Aquinas addresses the question on the Infinity of God.
Assignment
Links .pdf files
1.
Asgt 01
2.
Asgt 02
3.
Asgt 03
4.
Asgt04
5.
Asgt05
6.
Asgt06
7.
Asgt07
8.
Asgt08
9.
Asgt09
Other
Interesting Links
1.
For help in how to document papers check out Research and Documentation
On-Line by Diane Hacker. Check out the [Sciences] link on the lower right, [Finding Sources] and [Documenting
Sources]. Under Documenting Sources – CSE Style (Council of Science
Editors) follow the CSE Reference List
link and scroll down to see the reference examples.
2.
What
Does It Mean to Be Well-Educated? by Alfie Kohn.
A thoughtful article on education -
after all this is what a Witt Sem course is all
about.
3.
Code
of Academic Integrity: Academic integrity is central to the academic life
at Wittenberg; thus it is important that a student understand his or her
rights, responsibilities, and privileges.
Infinity! If you think you understand it, you probably don't