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 |
|
| 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 | ||||
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© Dirk Schlimm, Last modified: August 24, 1998