Spring 1998

T-Th 10:30 - 11:50

MMA 14

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

__Syllabus__

**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

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.

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 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:

- By handing in a floppy disk at classtime, that will be returned to you.
- By enclosing a file in email (see instructions below).
- 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.

__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)

**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

*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.

Go to Home Page for 80-110 |

Updated on January 13, 1998 by Richard Scheines.

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