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Carnegie Mellon University, Department of Philosophy,
Spring 2001

The Nature of Mathematical Reasoning

Dirk Schlimm


Schedule

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

Week
Tuesday
Thursday
1
Lecture 1: January 16
Introduction, general remarks, administrative points.
Early history of mathematics.
  • Handout: Syllabus & Overview
  • Lecture 2: January 18
    The origins of mathematics: Egypt, Babylonia, Greece
  • Handout 1: The Language of Numbers. (Dehaene)
  • Homework: No. 1  
  • 2
    Lecture 3: January 23
    Early mathematics. Proof that the square root of 2 is irrational.
  • Handout 2: Measuring the Unmeasurable. (Osserman)  
  • Lecture 4: January 25
    Proof that there are infinitely many primes. Mathematical notation.
  • Quiz: No. 1, Homework: No. 2
  • Handout 3: From mystery to history. (Barrow)
  • Handout 4: Counting with base 2, 5, 60. (Barrow)
  • 3
    Lecture 5: January 30
    The notion of proof: deductive validity.
  • Handout 5: Terminology for proofs and arguments. (Schlimm)
  • Handout 6: Deductive arguments. (Weston)  
  • Lecture 6: February 1
    More about arguments: possibility, objectiveness, forms of arguments, inductive/deductive reasoning.
  • Quiz: No. 2, Homework: No. 3
  • Handout 7: Sceptical doubts ... (Hume)
  • Handout 8: Proofs by contradiction. (Mancosu)  
  • 4
    Lecture 7: February 6
    (Early course evaluations). Proofs by contradiction.
  • Handout 9: Letter to Russell (Frege)
  • Handout 10: Proofs (Glymour)
  • Handin Portfolios.
     
  • Lecture 8: February 8
    Definitions.
  • Quiz: No. 3, Homework: No. 4  
  • 5
    Lecture 9: February 13
    (Early course evaluation results). Axiomatic theories.
  • Handout 11: Axiom systems. Models. Consistency and Independence. (Berlinghoff)  
  • Lecture 10: February 15
    Consistency and independence.
  • Quiz: No. 4, Homework: No. 5
  • Handout 12: Foundations of geometry. (Hilbert)
  • Handout 13: Space and the Geometrization of Mathematics. (Shaw)  
  • 6
    Lecture 11: February 20
    Axiomatizations of geometry. Axiomatizations of number theory.
  • Handout 14: Natural numbers. (Landau)
  • Handout 15: The series of natural numbers (Russell).  
  • Lecture 12: February 22
    Philosophical and mathematical limitations of axiomatizations.
  • Quiz: No. 5, Homework: No. 6ab
  • Handout 16: Example of the axiomatic method in practice. (Bochenski)
  • Handout 17: Gödel's Incompleteness Theorems.  
  • 7
    Lecture 13: February 27
    Let's Make a Deal problem. Axioms of Probability.
  • Handout 18: Probability.  
  • Lecture 14: March 1
    Review. Bring your own questions! 
    8
    Midterm, March 6
    Midterm exam
  • Homework: No. 7
  • Handout 19: Descartes's Myth. (Ryle)
  • March 8
    Mid-semester break
    9
    Lecture 15: March 13
    Conditional probability. Bayes' Theorem. Deductive solution to the problem. 
    Lecture 16: March 15
    Another application of probability theory. 
  • Quiz: No. 6, Homework: No. 8
  • Handout 20: Aristotle's logic.
  • 10
    Lecture 17: March 20
    Aristotle's syllogisms. Terminology
    Lecture 18: March 22
    Propositional logic
  • Quiz: No. 7, Homework: No. 9
  •   March 27
    Spring break
    March 29
    Spring break
    11
    Lecture 19: April 3
    More propositional logic: Knights & Knaves, Natural Deduction
  • Handout 21: Natural Deduction rules.
  • Lecture 20: April 5
    Quantifiers.
  • Quiz: No. 8, Homework: No. 10
  • 12
    Lecture 21: April 10
    Non-classical logics. Mathematical Induction (I)
    Lecture 22: April 12
    Deadline: Essay topic has to be approved.
    Mathematical Induction (II)
  • Handout 22: Mathematical induction problems.
  • Quiz: No. 9, Homework: No. 11
  • 13
    Lecture 23: April 17
    Set Theory. Functions.
  • Handout: Sample solutions for Homework 10.
  • Handout 23: Important definitions: Sets & functions
  • Lecture 24: April 19
    Deadline: Essay outline has to be handed in.
    Cardinality of N, Z, Q.
  • Handout 24: The Non-Denumerability of the Continuum
  • Quiz: No. 10, Homework: No. 12
  • 14
    Lecture 25: April 24
    Cardinality of R.
    Lecture 26: April 26
    Deadline:Essay due.
    Philosophy of mathematics
  • Homework: No. 13
  • 15
    May 1
    Review
    Bring your own questions!
    May 3
    Final exam


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    © Dirk Schlimm, Last modified: Tue May 1 10:32:47 EDT 2001