Math 103. Spring 2003. Infinity and the imagination.
INSTRUCTOR: Dr. Avery Solomon Office: Malott 210.
e-mail: aps5 (dec-jan averysolomon@hotmail.com)
In this course, students will explore different
mathematical meanings of infinity, including: infinite extent, infinitesimal,
orders of infinity, potentially infinite processes, and paradoxes. We will
roughly follow a sequence of five
historical periods, putting our explorations in a context of ancient and modern
mathematics, science, nature, art, and philosophy. The course will take into account different approaches to
learning, and use rational investigation, imagination, discussion and writing. Part of the course will be spent in the
computer lab exploring iterations of simple arithmetic and geometric processes
which are at the heart of dynamical systems and fractals.
The goal of this course is for students to
experience and enjoy some exciting mathematics, and to reflect on what it means
to think about infinity, an important aspect of thinking mathematically. Students should come away from the course
with a greater understanding of the power of mathematics in helping us think
about infinity, an experience of the deep connections between the physical
world and the imagination, and a deeper appreciation of the mystery of the
universe and the human mind.
Class attendance and participation is required.
Writing will be an essential part of the course.
There will be a written assignment each week, to be handed in Tuesday of the
next week. Homework will constitute about 60% of the grade.
There will be computer and other explorations to be
completed together with a partner, which will constitute about 15% of the
grade.
A final presentation/project to be done with your
partner will constitute about 15% of the grade.
There will be no exams or final, but occasional
quizes may constitute about 10% of the grade.
WRITTEN ASSIGNMENTS:
Please turn in whatever your thinking is on a
question even if only to say: "I don't understand such and such" or
"I'm stuck here". Turning in
homework every week, even if incomplete, will entitle you to revisit and revise
the problems. Your papers with comments
will be returned. Respond to comments,
use them as invitations to explore, or clarify your understanding. Keep responding (in writing or during office
hours) to comments until you understand and have completed the question. Also, ask questions in class, or after
class, on your paper or in office hours.
When I ask you to "show" that something is
true, what this means is to "give an argument that a reasonable skeptic
can understand and accept". A two
column proof is not required, but reasoned explanations are. An argument can include models, pictures and
examples .
Share your ideas with others and listen to their
ideas in and out of class. If you work
something out with others, that is fine, but make sure that you understand it
in your own way, that you communicate in writing your own words, and please write
down your collaborator's name on your paper. You may use ideas you see in
class, even to respond to a question after it has been discussed in class.
Acknowledge the class' contribution to your thinking.
Text:
A Cultural History of Infinity Maor, Eli.
Recommended also:
Images of Infinity Hemmings, Ray and Tahta, Dick Tarquin. 1995
Infinity and the Mind. Rucker, Rudy. Bantam. 1983.
Organizing Outline: Five
Historical Contexts:
A. 6th-3rd century BC: Early origins of thought about infinity in traditional cultures in India, Egypt, Babylonia, China. Focus on Greek paradoxes of infinity.
B. 1st-11th century: Hindu, Muslim, Taosit, and Buddhist views of Infinity, related to 0, equations, I-ching, and tiling.
C. 17th-18th century: Perspective, Infinite series, calculus and perspective.
D. 19th-20th century: Mathematical Logic and the infinite.
E. Late 20th-21st century: Fractal Geometry and dynamical systems.
Ancient questions of space, time and causality (Two weeks: 1-2)
Cosmology: Infinite Space? Rotating. First Cause
Matter: Smallest particle?
Motion: Zeno
Causality: Evolution: beginningless? Combined dependent origination
Greek:
Euclid: # of primes is infinite
Eudoxus: method of exhaustion
Eratosthenes
Archimedes: Sand reconer, Value of pi, Area of Circle, Axiom of Archimedes
Pythagoras: incommensurability Infinity and the irrational: incommensurability, square and pentagon, Pi, infinite lattice, square circle.
Square the circle, duplicate the cube
Causality and Infinity (one week: 3)
Problem of infinite regress and first cause: Aristotle, Augustine and Aquinas
Self referential paradoxes: liar paradoxes.
Self-similar infinity: Land of Lakes
Theology: Plotinus, Plato.
India: Infinite from the infinite yields the infinite: Shankara articles unfolding One, Infinitesimal
China: Hwa-Yen Buddhism
Infinity and 0
Cipher for 0 or sunya around 1000 years ago.
Infinity and uncertainty: (1/2 week: 4)
Paradoxes of probablity: coin toss and likelihood
Geometric probability and the dart board.
Infinite sequences and Limits. (1.5 weeks: 4-5)
Sequences and series: geometric and harmonic and Fibonacci
Spirals
Potentially infinite, limit, process.
Limits and converging: geometric ½ + Ό + 1/8
Do other sequences: 2/3, 4/3, 4/5, 6/5,
Do Harmonic series: why diverge? And how slowly. Oresme proved: 1250. In appendix.
Gregory: pi/4 = 1/1-1/3+ 1/5 1/7
Wallis: pi/2 = 2/1 x 2/3 x 4/3 x 4/5 x
Infinity and Geometry in the Renaissance and Art (1 week 6)
Parallel
Perspective
Escher
Tiling.
Logic of infinity (1.5 weeks: 7-8)
Infinity and the Real Numbers
Cantor: all the numbers, orders of infinity,
Paradoxes of infinity and sets. Infinite Hotel.
Russel: definition is crucial: set with just as many members as itself. Give self-similarity example. Ordinal and Cardinal.
The continuum and ideas of continuity
Many and One
One to one relations and diagonal arguments.
Independence of the CH: correspondence and consistency,
Cantor: order of 2Aleph > Aleph (proof in Maor)
Platonic view, Kantian, positivist.
Fractals (two and 1/2 weeks: 8-10)
Self-similarity: land of lakes, tankhas, rucker,
cantor dust
Not smooth: coastline, modeling the coast,
Fractional dimension: Why is an ant small? what has dimension 1.5
Attempt to measure smaller by larger (0), or larger by smaller (infinity)
Building fractals: Koch and Frame.
Multiple copy machines: Fern and spiral, numerical descriptions of transformations.
Genetics.
Dynamical systems: exploration of iteration (two weeks: 11-12)
Numerical iteration: multiplying and adding
Discrete dynamical systems: quadratic or complex?
Student presentations: (2 weeks 13-14)