



1  January 14
(Lecture 1)
Topic: Introduction Handout: Syllabus & Overview 
January 16 (Lecture
2) Topic: The origins of mathematics Homework: Problem set 1 
January 18 (Lecture
3) Topic: The importance of notation: Chinese number words; Roman and Babylonian number systems. 
2  January
21 Martin Luther King day (No class) 
January 23 (Lecture 4)
Topic: Early mathematics. Proof that the square root of 2 is irrational. Homework: Problem set 2 
January 25 (Lecture 5)
Topic: More proofs: Proof that there are infinitely many prime numbers. 
3  January 28 (Lecture 6)
Topic: The notion of proof: Deductive validity. Handout #1: Terminology for proofs and arguments (pdf) 
January 30 (Lecture 7)
Topic: More about arguments: Possibility, objectivity, forms of arguments. Homework: Problem set 3 Handout #2: Deductive arguments (Weston) 
February 1 (Lecture 8)
Topic: Deductive vs. inductive reasoning. 
4  February 4 (Lecture 9)
Topic: Proofs by contradiction. 
February 6 (Lecture 10)
Topic: Definitions. Homework: Problem set 4 Handin Journal. Handout #3: Proofs (Glymour) 
February 8 (Lecture 11)
Topic: Feedback on journals. Mathematical theories. 
5  February 11 (Lecture 12)
Topic: Feedback on early course evaluations. Axiomatic theories, and properties of axiom systems. Handout #4: Axiom systems. Models. Consistency and Independence. (Berlinghoff, et.al.) 
February 13 (Lecture 13)
Topic: Consistency and independence. Homework: Problem set 5 
February 15 (Lecture 14)
Topic: Axiomatization of geometry. NonEuclidean geometries. 
6  February 18 (Lecture 15)
Topic: More NonEuclidean geometries. Axioms of arithmetic. Handout #5: Foundations of geometry (Hilbert) Handout #6: Foundations of analysis (Landau) 
February 20 (Lecture 16)
Topic: Axiomatizations; the situation in mathematics at the beginning of the 20th century Handout #7: Axiomatization of sentential logic (Bochenski) Handout #8: Introduction to mathematical philosophy, Ch.1 (Russell) Handout #9: Hilbert and Gödel (pdf) 
February 22 (Lecture 17)
Topic: Limitations of formal systems: Gödel's incompleteness theorems Homework: Problem set 6, Extra problem (pdf) 
7  February 25 (Lecture 18)
Topic: Philosophical implications of Gödel's theorems. Let's Make a Deal problem. Axioms of probability theory. Handout #10: Probability (Eels). 
February 27 (Lecture 19)
Topic: Conditional probability. Bayes' Theorem. Handout #11: Conditional probabilities and Cherry pies (pdf) Homework: Problem set 7 
March 1 (Lecture 20)
Topic: Deductive solution to the Let's Make a Deal problem. Empirical confirmation. Concept map. 
8  March 4 (Review session)
Review for exam: Bring your own questions! More questions. 
March 6
Midterm exam 
March 8 Midsemester Break 
9  March 11 (Lecture 21)
Topic: Another example for the application of probability theory 
March 13 (Lecture 22)
Topic: Discussion of midterm exam Handout #12: Aristotle's logic (Glymour). Homework: Problem set 8 
March 15 (Lecture 23)
Topic: Aristotle's syllogisms 
10  March 18 (Lecture 24)
Topic: The language of logic. Syntax and semantics. 
March 20 (Lecture 25)
Topic: Propositional logic. Truth tables. Homework: Problem set 9 (A) Homework: Problem set 9 (B) Note, you need to do only (A) or (B). 
March 22 (Lecture 26)
Choose your topic: Lecture A (Dirk) Fun with truth tables Lecture B (Henrik): BH 231 A Compactness, nonstandard models 
11  March 25 (Lecture 27)
Topic: Syntactic reasoning: Natural Deduction Handout #13: Natural Deduction rules (pdf) 
March 27 (Lecture 28)
Topic: Soundness, completeness, quantifiers Homework: Problem set 10 (A) Homework: Problem set 10 (B) Note, you need to do only (A) or (B). 
March 29 (Lecture 29)
Choose your topic: Lecture A (Dirk) More on quantifiers Lecture B (Henrik): BH 231 A Quantifiers in intuitionistic logic 
April
1 Spring Break 
April
3 Spring Break 
April
5 Spring Break 

12  April 8 (Lecture 30)
Topic: Nonclassical logics 
April 10 (Lecture 31)
Topic: Mathematical induction (I): Recursive definitions Homework: Problem set 11 
April 12 (Lecture 32)
Topic: Mathematical induction (II) Handout #14: Mathematical induction problems. 
13  April 15 (Lecture 33)
Essay outline due Topic: Introduction to the theory of sets 
April 17 (Lecture 34)
Topic: Functions and cardinality Homework: Problem set 12 Handout #15: Sets and functions (pdf) 
April
19 Spring Carnival 
14  April 22 (Lecture 35)
Topic: Cardinalities of the natural numbers, integers, rational numbers 
April 24 (Lecture 36)
Topic: The cardinality of the continuum Homework: Problem set 13 Handout #16: The NonDenumerability of the Continuum 
April 26 (Lecture 37)
Topic: Paradoxes of set theory 
15  April 29 (Lecture 38)
Topic: Questions in philosophy of mathematics 
May 1 (Lecture 39)
Topic: Summary of course, concept map 
May 3 (Last day of classes)
Review for exam: Bring your own questions! More questions. Essay due 
Exam  May 6
Final Exam 8:30am11:30am, HH B131 Handin Journal 