Carnegie Mellon University, Department of Philosophy,
80-101 Freshman Seminar, Fall 1998

# Mathematics in Scientific Context

Prof. Wilfried Sieg

# Schedule

Note that this schedule will be updated frequently in the course of the semester!

The HW column tells you when the homework will be handed out. Usually, they are handed out on Thursdays and are due the following Tuesday.

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 Week Date Lec # HW Topic Supplementary Reading 1 Tu, Aug 25 1 General remarks, administrative points Th, Aug 27 2 Everything follows from a contradiction Hippocrates' quadrature of the lune p.1-17 of handout on Hippocrates Functions: examples and terminology 2 Tu, Sep 1 3 Systematic connection between parts I and II of the course Hippocrates' quadrature of the lune (contd.)Erastothenes' determination of the circumference of the earth Finish handout on Hippocrates Poetry of the universe, Chapter 1 Discuss "the impossibility of squaring the circle" in group, and write a brief summary of the argument Th, Sep 3 4 #1 Presentation: the impossibility of squaring the circle Rewiew "the impossibility of squaring the circle", read Geometry civilized p.49-71 Sets and ordered pairs 3 Tu, Sep 8 5 Structure of impossibility arguments ;Euler's theorem Read Chapter~II, Encompassing the Earth, in Osserman's book;Homework #1 due Relations Th, Sep 10 6 #2 Euler's theorem (contd.) Geomertry: loci Problems 4b) and 4c) of Homework #1 are due Functions 4 Tu, Sep 15 7 Euclid's proofs I.1, I.2, I.9, I.10 Th, Sep 17 8 #1 Reflection: what have we been doing? Review: ordered pairs, relations, functions Euclid's proof of the Pythagorean Theorem 5 Tu, Sep 22 9 Form and structure of Euclid's proofs: I.15, I.32, I.41, I.46 Euclid's proof of the Pythagorean Theorem Th, Sep 25 10 #2B Structure of proofs Proof that two sets of points are equal Proof for the Pythagorean Theorem Inverse, and injective functions 6 Tu, Sep 29 11 The development of logical calculi: Natural Deduction Calculus; Hilbert's axiomatization of geometry Arguments in and on logic (handout) Th, Oct 1 12 #3 The language of predicate logic; Discussion of Euclid's common notions; Hilbert's axiom system for geometry Surjective and bijective functions 7 Tu, Oct 6 13 Hilbert's axiom system for geometry Th, Oct 8 Midterm (1h) 8 Tu, Oct 13 14 Recap: functions; Midterm Part A; Cantor's Theorem Th, Oct 15 15 #4 Midterm review: Internal organization (proofs), context of applications, functions 9 Tu, Oct 20 16 Provability of the Parallel Postulate? The Non-Denumerability of the Continuum Th, Oct 22 17 #5 Syntax and Semantics; Geometry on the shere; Calculable functions Courant/Robbins: What is mathematics? pp.214-227 (handout) 10 Tu, Oct 27 18 Modern axiomatic presentations of mathematics The sum of the angles of a triangle on a sphere,and the ratio of circumference/diameter Calculable functions (multiplication) Think about: the sum of the angles of a triangle on a sphere,and the ratio of circumference/diameter! Th, Oct 29 19 #6 Counterexamples to inferences Interpretations of geometry (hyperbolic geometry) 11 Tu, Nov 3 20 Universe of discourse; Hyperbolic geometry;Turing Machines Th, Nov 5 21 #7 Distance in hyperbolic geometry 12 Tu, Nov 10 22 Curvature; Logical Calculi Th, Nov 12 23 #8 Entscheidungsproblem; Example for OR-elimination: Proof Hilbert: The foundations of geometry 13 Tu, Nov 17 24 Fitch-representation; Semantics for sentential logic Th, Nov 19 25 Gödel numbering; Fitch-Diagrams Church: An unsolvable problem of elementary number theory, Review of Turing (1936); Post: Finite combinatory processes; Turing: On computable numbers, Computing machinery and intelligence 14 Tu, Nov 24 26 #9 Gödel's Incompleteness Theorems Gödel: On undecidable propositions of formal mathematical systems Th, Nov 26 Thanksgiving! Happy Holiday! 15 Tu, Dec 1 27 Turing's analyis of computability and calculability Turing: On computable numbers Th, Dec 3 28 Unsolvability of the Halting Problem; Learning Machines Turing: Computing Machinery and Intelligence, Sec. 3-5 and 7 Mo, Dec 7 Review Session (with Wilfried), 4:00 Th, Dec 10 Question Session (with Dirk), 1:00 Fr, Dec 11 Final (3h) 5:30-8:30