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Summer 2000

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

Week |
Date |
Lec # |
Topic |
Reading |
Homework/ Quiz |

History & Motivation |
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1 | Mo, Jul 3 | 1 | Introduction, general remarks, administrative points | Handout: The Language of Numbers. (Dehaene) |
(No.1) |

Tu, Jul 4 | Independence Day. No class. |
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We, Jul 5 | 2 | The origins of mathematics | Handout: Measuring the unmeasurable. (Osserman) |
(No.2) | |

Th, Jul 6 | 3 | Early mathematics: Proof that the square root of 2 is irrational | Handout: From mystery to history. (Barrow) |
(Quiz 1) | |

Fr, Jul 7 | 4 | Early mathematics: Proof that there are infinitely many prime numbers. Mathematical notation. | Handout: Counting with base 2, 5, 60. (Barrow) |
(No.3) | |

Kinds of reasoning/arguments |
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2 | Mo, Jul 10 | 5 | Deductive vs. inductive reasoning | Handouts: Deductive
Arguments. (Weston)On
Induction. (Hume) |
(No.4) |

Tu, Jul 11 | 6 | Deductive validity | Handouts: Letter to Russell. (Frege)Proofs by
contradiction. (Mancosu) |
(Quiz 2), (No.5) | |

We, Jul 12 | 7 | Proofs by contradiction. | Syllabus update: New guidelines for portfolio glossary/index. |
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Th, Jul 13 | 8 | More about arguments | Handout: Proofs. (Glymour) |
(Quiz 3) | |

The structure of mathematical theories |
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Fr, Jul 14 | 9 | Definitions.(Early course evaluations). |
The Language of First-Order Logic (FOL): p. 9-13 and Section 2.8 (p. 24-28). |
(No.6) | |

3 | Mo, Jul 17 | 10 | Axiomatic systems | Axiom systems. Models. Consistency and
Independence. (Berlinghoff et al.) |
(No.7) |

Tu, Jul 18 | 11 | Axiomatizations of geometry | Handouts:Foundations of geometry. (Hilbert)Space and the Geometrization of Mathematics. (Shaw) |
(Quiz 4) | |

We, Jul 19 | 12 | Axiomatizations of number theory | Handouts: Natural
numbers. (Landau)The series of
natural numbers (Russell). |
(No.8) | |

Th, Jul 20 | 13 | Limits of axiomatizations | Handouts:Example of the axiomatic method in practice. (Bochenski)Gödel's Incompleteness Theorems. |
(Quiz 5) | |

Fr, Jul 21 | 14 | Review. |
Handout: Sample solutions to Homework 8, Problem 3. |
(No.9) | |

4 | Mo, Jul 24 | 15 | Midterm exam. | ||

Case study I: Probability theory |
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Tu, Jul 25 | 16 | Let's Make a Deal problem. Axioms of Probability. | |||

We, Jul 26 | 17 | Deductive solution to the problem. | Probability. (Eels) |
(No. 10) | |

Th, Jul 27 | 18 | Another application | (Quiz 6) | ||

The nature of mathematical proof |
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Fr, Jul 28 | 19 | Aristotle's syllogisms. Terminology | Aristotle's logic. (Glymour) |
(No. 11) | |

5 | Mo, Jul 31 | 20 | Propositional logic | FOL: Sections 4.1-4.4, 4.7 | (No. 12) |

Tu, Aug 1 | 21 | More propositional logic | Handout: Natural
deduction rules |
(Quiz 7) | |

We, Aug 2 | 22 | Quantifiers. Non-classical logics. | FOL: Sections 5.1-5.10, 5.13, and 6.1-6.5 | (No. 13) | |

Th, Aug 3 | 23 | Mathematical Induction (I) | (Quiz 8) | ||

Fr, Aug 3 | 24 | Mathematical Induction (II) | FOL: Sections 9.1, 9.3, and
8.1-8.4. Handouts: Sample solutions for
Homework 12. Claims to prove by mathematical induction. |
(No. 14) | |

Case study II: Set theory |
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6 | Mo, Aug 7 | 25 | Set Theory. Functions. | Handouts: Important definitions: Sets & Functions.The Non-Denumerability of the Continuum. (Dunham) |
(No. 15) |

Tu, Aug 8 | 26 | Cardinality of N, Z, Q | (Quiz 9) | ||

We, Aug 9 | 27 | Cardinality of R | |||

Review |
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Th, Aug 10 | 28 | Review. Last day of classes! |
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Fr, Aug 11 | 29 | Final exam. |

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© Dirk Schlimm, Last modified: August 4, 2000