Professor: Richard Scheines

Office: Baker Hall 135-E

Phone: 268-8571

Email: R.Scheines@andrew.cmu.edu

Office Hours: By appointment

Teaching Assistant: Andrew Banas

Office: Baker Hall 143

Phone: 268-8148

Email: banas@andrew.cmu.edu

• Overview
• Grading
• Homework
• Required Texts
• Topics
• Lecture Notes
• Related Reading
• Go to Home Page for 80-110

Overview

Although we spend the great bulk of our mathematical education learning how to calculate in a variety of ways, mathematicians rarely calculate anything. Instead they devote their time to clearly stating definitions, finding simple and unobjectionable axioms, making conjectures about claims that follow from these axioms, and then proving these claims or finding counterexamples to them. Although thinkers since Aristotle have devoted enormous time and energy to developing a theory of mathematical reasoning, it is only in the last century or so that a unified theory has emerged.

We will begin by motivating the subject with historical material about mathematics and the problems that provoked the development of the modern theory of mathematical reasoning. We will cover several simple examples to provide a basis for future material.

After the motivation we will move into the language of first order logic. Here we will use Tarski's World (available for Macintosh or PC) which is included with the main text for the course, The Language of First Order Logic. Having acquired some facility with the language of logic, we will learn about and do some simple proofs. For this section of the course we will use the Carnegie Mellon Proof Tutor, a program developed in the Philosophy Department over the last 5 years.

Having covered enough material in the abstract theory of logic, we will then turn to looking at the theory of infinite sets, which, along with logic, is a cornerstone upon which most of modern mathematics rests. Using the same techniques we developed earlier in the course we will prove some simple theorems about infinity that are counterintuitive and stunning.

Grading

Your grade is based on 4 components, each of which will be weighted equally:

• Homework exercises
• Test 1
• Test 2
• Final Exam

Homework will be graded on a 3 point scale: outstanding-satisfactory-untisfactory. The assignments will be given in class and posted on the class website. It is your responsiblity to obtain the assignment if you miss class. Attempting to give an excuse anywhere in the vicinity of: "I didnít know there was an assignment,"or "I missed class and my friend gave me the wrong assignment" will cause excessive irritation on the part of the instructor and TA. You are free to collaborate on homework, but not to copy answers wholesale from friends.

The tests will be all be cumulative, that is they will cover all material from the beginning of the class up to the test. Sample questions will be given out before each test, and a review sessions will be held before each test.

Homework

Homework assignments will be put on the web:

http://www.andrew.cmu.edu/~rs2l/80-110/assignments.html

Homework will be graded on a 3 point scale: outstanding-satisfactory-untisfactory. The assignments will be given in class and posted on the class website. It is your responsiblity to obtain the assignment if you miss class. Attempting to give an excuse anywhere in the vicinity of: "I didnít know there was an assignment,"or "I missed class and my friend gave me the wrong assignment" will cause excessive irritation on the part of the instructor and TA. You are free to collaborate on homework, but not to copy answers wholesale from friends. Some homework problems, or facsimilies therof, will reappear on the tests, so too much "ollaborating" will make your life easier locally but more difficult in the long run.

You can turn in homework in either of 3 ways:

1. By handing in a floppy disk at classtime, that will be returned to you.
2. By enclosing a file in email (see instructions below).
3. By turning in hard-copy (old-fashioned, but still works).

You can enclose a fully formatted file (either a Word file or a Tarski's World file) in MacMail II and other mail programs, and we sould be able to extract it intact on the other end.

Required Texts

The Language of First Order Logic. Jon Barwise & John Etchemendy, Center for the Study of Language and Information Lecture Notes, number 23. CSLI Publications, Stanford, CA.

Reading Packet: (Barrow, Peterson, and Glymour)

Tarski's World (comes with FOL)

The Carnegie Mellon Proof Tutor (CPT, available on Andrew)

Topics

1. Historical Background & Motivation

- Egypt, Bablyonia, etc.

- Pythagoras and Irrational Numbers

- Euclid: Geometry and the Axiomatic Method

- Zeno: The Paradoxes of Infinity

2. The Structure of Mathematical Theories: Axioms, Definitions, and Theorems

3. Fallacies and Rigor

- Lets Make a Deal

- Algebraic Fallacies

- Attempts to deductively prove the existence of God

4. Deductive Arguments

- The Language of First Order Logic (Tarskis World)

- Proof Construction as Mathematical Problem Solving - (CPT)

5. Set Theory: The Theory of Infinite Sets

The lecture notes will be put on the web:

http://www.andrew.cmu.edu/~rs2l/80-110/lectures.html

Related Reading

General Overviews and Historical Material:

An Introduction to the History of Mathematics (1983). H. Eves, Saunders College Publishing.

Development of Modern Mathematics (1970). J. Dubbey, Crane, Russek & Co.

Innumeracy: Mathematical Illiteracy and its Consequences, (1988). John Paulos, Hill and Wang, New York

The Magic of Numbers (1974) Eric T. Bell, Dover

The Mathematical Tourist: Snapshots of Modern Mathematics, (1988). Ivars Peterson, W. H. Freeman and Co., New York.

Mathematics in Western Culture, (1974). M. Kline, Oxford.

Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory, Probability Theory (1992). Edited by A. N. Kolmogorov and A. Yushkevich, Birkhauser Verlag, Basel.

Pi in the Sky: Counting Thinking and Being, (1992) by John Barrow, Clarendon Press.

Thinking Things Through (1992). C. Glymour, MIT Press, Cambridge, MA.

Logic and Set Theory

Abstract Set Theory (1961) A. Fraenkel, North-Holland, Amsterdam.

The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number (1884). G. Frege, Breslau, published 1950 by the Philosophical Library.

Infinity and the Mind (1995) Rudy Rucker, Princeton Science Library, Princeton University Press, Princeton N.J.

Introduction to Metamathematics (1954). S. Kleene, Wolters-Noordhoff, Groningen.

Introduction to Logic (1957). P. Suppes, D. Van Nostrand, New York.

Introduction to Set Theory (1984). K. Hrbacek and T. Jech, Marcel Dekker, New York

Logic in Elementary Mathematics (1959). R. Exner & M. Rosskopf, McGraw-Hill, New York.

Logic: Techniques of Formal Reasoning. (1984) D. Kalish, R. Montague, and G. Mar, Harcourt Brace Javanonich.

Metalogic (1973). G. Hunter, Univ. of California Press, Berkeley, CA.

Logic and Mathematical Problem Solving

How to Solve It : A New Aspect of Mathematical Method (1973). G. Polya, Princeton University Press, Princeton, N.J.

The Scientific American Book of Mathematicl Puzzles & Diversions (1959). M. Gardner, Simon and Schuster, New York.

What is the Name of this Book? (1978). R. Smullyan, Simon and Schuster, New York.

Updated on January 13, 1998 by Richard Scheines.

Send email to R.Scheines@andrew.cmu.edu