Phil402

PHIL 402 Philosophy of Logic and Mathematics
Spring 2007-2008

OFFICE MATTERS :

Office : Room B139 in the Social Sciences Building.
Office hours :Tuesday 14:40-15:30 & 16:40-17:30; Friday 14:40-15:30. Other meeting times with the instructor are also possible by prior appointment.
Office phone : On campus: 5339; off campus: 210-5339. You can leave a message on the answering machine.
E-mail : esayan@metu.edu.tr

COURSE DESCRIPTION :
The two “incompleteness theorems” proved by Kurt Gödel in 1931 (when he was only 25 years old) are probably the most interesting and important results in entire logic. In this course, our primary aim will be to try to understand Gödel’s proofs without recourse to any highly technical formalism of logic and mathematics. In the meantime, we shall talk about the following: the nature of philosophy and of logic, proposition, argument, deduction vs. induction, Aristotelian vs. Modern Logic, contradiction, tautology, paradox, the Liar Paradox, proof, axiom, axiomatic system, non-Euclidean geometries, Russell’s Paradox, Peano’s axiomatization of arithmetic, the question of the consistency of axiomatic systems, Hilbert’s programme, language vs. metalanguage, Cantor’s diagonal argument, transfinite numbers, continuum hypothesis, Frege and Russell’s Logicism. We shall also discuss, time permitting, the alleged relevance of Gödel’s proofs to some important philosophical issues, such as whether the human mind is a machine and whether, more generally, the materialistic view of reality is correct.

Our emphasis will be on grasping of concepts rather than on formal technicalities. Nevertheless, prior acquaintance with symbolic logic at an elementary level will prove very helpful to the student.
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COURSE REQUIREMENTS :

• There will be two midterms and one comprehensive final exam. All three exams will be composed, for the most part, of questions requiring short answers, such as true-false, fill-in-the blank, and multiple-choice questions. There may be, however, a couple of “classical-style” questions on the exams, each requiring at most half-a-page-long answer.
• Attendance to classes is required. You will be expected not only to attend but also to actively participate in class discussions by asking questions, answering questions, and showing genuine interest in the problems being discussed. Class attendance will be taken randomly throughout the semester. Students can have three unexcused absences without hurting their grades. After three absences, the student’s total score will be lowered by 3 points per unexcused absence. Class attendance may be taken at the beginning of the class period as well as at the end. When it is taken at the beginning, latecomers will be counted as absent.
• The requirements of the course will be weighted as follows:

1. Midterm : 30 pts.
2. Midterm : 30 pts.
Final : 40 pts.
TOTAL : 100 pts.
Participation in class discussion : up to +8 pts.
Every unexcused absence after 3 unexcused absences : -3 pts.

The final letter grades for the course will be assigned via an “informal curve” to be set up on the basis of the students’ total scores at the end of the semester.

• The first midterm will be given some time between March 29-31, and the second midterm will be given between May 10-12. The exact day and hour of the midterms will be decided by the vote of the students in due time. The final exam day and hour is largely out of our control—it is determined by the Registrar’s Office.
• Make-ups for the exams will normally be given in cases of certified illness or certified emergency. Be warned that the make-ups will tend to be harder than the regular exams. The make-up will be especially harder in the case of students who present less than compelling reasons to take a make-up.
• If our class hours conflict with those of another course you are taking, then you are expected to spend half of those conflicting hours in our class, rather than spending them entirely in the other class during the whole semester. At the end of the semester, you will be given a form which will be signed by the teacher of the other course as proof of the class-hour conflict, and your absences during those hours will be discounted to an appropriate extent.

SEQUENCE OF READINGS :

A. What is philosophy?: Hospers, pp.4-5; Moody, pp.1-4.
B. Basic concepts of logic.
C. Euclidean Geometry: Barker, pp.15-26.
D. Nagel and Newman, pp.1-8.
E. Cantor’s diagonal argument: “Diagonal Procedure” in Penguin Dictionary of Philosophy; “Cantor, Georg” in The Cambridge Dictionary of Philosophy; “Cantor’s Paradox” in The Oxford Companion to Philosophy.
F. Nagel and Newman, pp.8-37, 41-44, 57-59, 66-114.
G. Lucas’ argument: Pieces by Flew, Hofstadter, Benacerraf, and Steprans.

OPTIONAL BUT HIGHLY RECOMMENDED READINGS :

*. Kurt Gödel: “Gödel, Kurt (1906-78)” in Routledge Encyclopedia of Philosophy.
*. Non-Euclidean Geomety: Barker, pp.32-55.
*. The Axiomatic Method: Yıldırım, pp.235-251.
*. Gödel Sentences: Honderich, pp.350-351.
*. Meaning of Gödel’s Theorems, Examples of Undecidable Statements, Misconceptions about Gödel’s Theorems, Minds and Machines : Sections of the Wikipedia article “Gödel’s Incompleteness Theorem”
(http://en.wikipedia.org/wiki/Goedels_incompleteness_theorem).
*. The Nature of Mathematical Truth : Hempel, pp.222-237.

The longer readings in the two reading lists above are from the following texts:

Barker: Steven F. Barker, Philosophy of Mathematics (Englewood Cliffs: Prentice-Hall, 1964).

Hempel: Carl G. Hempel, “On the Nature of Mathematical Truth,” in Herbert Feigl and Wifrid Sellars (eds.), Readings in Philosophical Analysis (New York: Appleton-Century-Crofts., 1949), pp.222-237.

Nagel and Newman: Ernest Nagel and James R. Newman, Gödel’s Proof (New York U. P., 1958).

Yıldırım: Cemal Yıldırım, Logic: The Study of Deductive Reasoning (Ankara: Middle East Technical University publication, 1973).