305 – Homework IV

This homework set is due Wednesday, March 21, at the beginning of lecture.

197 – Introduction to mathematical thought. Announcement and syllabus

This Fall I will be teaching a course in the Honors College, Math 197: Introduction to mathematical thought.

The goal of the course is to present an introduction to the mathematical method, the way mathematics is reasoned, discovered, and advanced. This will be accomplished through a presentation of selected (real world) examples, and an emphasis on the key notion of mathematical proof. Particular attention is paid to aesthetic, historical, and philosophical aspects of mathematics.

Pre-requisites: Instructor’s approval.

Text: We will use several texts and articles. Particularly recommended are:

• T.W. Körner. The pleasures of counting, Cambridge University Press (1996). ISBN: 0-521-56087-X (hardback), 0-521-56823-4 (paperback).

(This will be our official textbook, but we will draw material from all three.)

 

• Sh. Stein. How the other half thinks. Adventures in mathematical reasoning, McGraw-Hill (2001). ISBN-10: 0071407987, ISBN-13: 978-0071407984.
• W. Dunham. Journey through genius. The great theorems of mathematics, John Wiley & Sons, Inc (1991). ISBN-10: 014014739X, ISBN-13: 978-0140147391.

I will provide additional handouts and references as needed. Supplementary recommendations include:

 

• I. Lakatos. Proofs and refutations. The logic of mathematical discovery, Cambridge University Press (1976). ISBN-10: 052121078X, ISBN-13: 978-0521210782.
• Ph.J. Davies and R. Hersh. The mathematical experience, Mariner Books (1999). ISBN-10: 0395929687, ISBN-13: 978-0395929681.
• S.G. Krantz. The proof is in the pudding. The changing nature of mathematical proof, Springer (2011). ISBN-10: 0387489088, ISBN-13: 978-0387489087.

Contents: We will present a series of examples illustrating how mathematics is used in the real world, and how it is conceived. We will discuss the nature of mathematical proof, and some of the philosophical issues surrounding it, as well as how it has evolved through the ages. The goal is to see how mathematicians actually reason, and how mathematical ideas are a natural part of the cultural legacy of humankind.

Specific examples may be chosen depending on the audience background and motivation. Particular examples I would like to include are the mathematics of
codes, how populations evolve through time, the mathematics of infinity, and computer generated proofs.

Grading: Grades will be determined based on homework (60%), a written project (20%), and class participation (20%). There will be no exams.

Attendance to lecture is not required but highly recommended.

I will use this website to post additional information, and encourage you to use the comments feature. If you leave a comment, please use your full name, which will simplify my life filtering spam out.

580 – Topics in Set Theory. ANNOUNCEMENT

 This Fall I will be teaching Topics in set theory. The unofficial name of the course is Combinatorial Set Theory.

We will cover diverse topics in combinatorial set theory, depending on time and the interests of the audience, with emphasis on three topics: Choice-free combinatorics, cardinal arithmetic, and partition calculus (a generalization of Ramsey theory).

Time permitting, we can also cover large cardinals, determinacy and infinite games, or cardinal invariants (the study of sizes of sets of reals), among others. I’m open to suggestions for topics, so feel free to email me or to post a comment.

Pre-requisites: Permission by instructor. The recommended background is knowledge of cardinals and ordinals. A basic course on set theory (like 502: Logic and Set Theory) would be ideal but is not required.

Grading: Based on homework.

Textbook: Combinatorial set theory, by Neil H. Williams. Elsevier Science (1977). ISBN-10: 0720407222, ISBN-13: 978-0720407228. The book seems to be out of print.

We will also use:

• Combinatorial Set Theory: Partition Relations for Cardinals, by Paul Erdös, András Hajnal, Attila Máté, and Richard Rado. Elsevier Science (1984). ISBN-10: 0444861572,  ISBN-13: 978-0444861573. Apparently, this is also out of print.

I will distribute notes on the material of these books, on additional topics, and some papers that we will follow, particularly:

• András Hajnal and Jean A. Larson. “Partition relations”, in Handbook of set theory, 129–213, Springer, 2010.
• Jean A. Larson. “Infinite combinatorics”, in Handbook of the history of science, vol. 6, 145-357, Elsevier, 2012.