80-110: The Nature of Mathematical Reasoning
T-Th 10:30 - 11:50
Professor: Richard Scheines
Office: Baker Hall 135-E
Office Hours: By appointment
Teaching Assistant: Andrew Banas
Office: Baker Hall 143
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
Your grade is based on 4 components, each of which will be weighted equally:
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:
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:
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
Tarski's World (comes with FOL)
The Carnegie Mellon Proof Tutor (CPT, available
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
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
The lecture notes will be put on the web:
General Overviews and Historical
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.,
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,
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
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
What is the Name of this Book? (1978).
R. Smullyan, Simon and Schuster, New York.
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Updated on January 13, 1998 by Richard Scheines.
Send email to R.Scheines@andrew.cmu.edu