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

Mathematics in Scientific Context

Prof. Wilfried Sieg

  • Review session: Mo 12/7/98, 4:00, BH 150
  • Question session: Th 12/10/98, 1:00, BH 150
  • Final exam: Fr 12/11/98, 5:30-8:30, PH A18 C

  • 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.

    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
      Th, Sep 10 6 #2 Euler's theorem (contd.)
    Geomertry: loci
    Problems 4b) and 4c) of Homework #1 are due
    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

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    © Dirk Schlimm, Last modified: August 24, 1998