80-110: The Nature of Mathematical Reasoning
Carnegie Mellon University
Location: Doherty Hall 1115
Instructor: Rob Lewis (email)
Office: Doherty Hall 4302D (directions)
Office Hours: MWF 10:30-11:30, or preferably by appointment
This course is intended to serve as an introduction to what we could broadly call "metamathematics." Without delving too deeply into any particular mathematical theory (like calculus or topology), we'll learn some interesting facts about mathematical systems in general, and see how these ideas can be used in real-life problems. To do this, we will need to understand what exactly mathematics "is," so to speak. To this end, the course will involve both some math and some philosophy (although no background in either is required).
A central point of this class is the idea of mathematical rigor. Mathematics is, ideally, something that is perfectly precise; there is no room for ambiguity, and every question in the "language" of math has a correct answer. We'll talk about how mathematics can be used to add precision to situations that lack it, to clarify questions and make them answerable. At the same time, one can ask questions about mathematics that seem to be very open-ended. We'll examine what makes a question essentially mathematical, and what makes a question philosophical.
The course will be broadly organized in four parts. First, we'll discuss some historical background and motivation for the topic: we'll look at some mathematical and philosophical work from the ancient Greeks, contributions they made, and problems they ran into. We'll then look more generally at the notion of a mathematical "theory," informally and formally. What are axioms? Definitions? Theorems? This will lead us into a discussion of propositional and first-order logic. With this framework, we'll be able to discuss some well-known fallacies and paradoxes, and see how the machinery of logic (and mathematics in general) allows us to resolve these. If there is time, we'll finish the course by talking about set theory, a common foundation for mathematics.
If there are any (even only slightly related!) topics that you think would be interesting to discuss, please let me know! Our schedule is flexible and I'm open to suggestions.
A previous version of this course:
Lectures and Assignments
- Week 1 (1/12):
- Week 2 (1/21):
- Day 4: Conjectures and counterexamples. Read Euclid through Proposition I.12, and JtG pp. 37-48.
- Day 5: Euclidean constructions vs. conditionals. Read JtG pp. 48-53. Homework 2 (due Monday- it's short)
- Week 3 (1/26):
- Day 6: More about conditionals: converse, inverse, contrapositive. Non-Euclidean geometry. Euclidean constructions. Read JtG pp.68-80, focusing on 68-74.
- Day 7: Inductive arguments. Homework 3 (due Monday)
- Day 8: Objectivity and subjectivity, fallacies. Read JtG 155-177. (focus on the 'great theorem' pages. The rest is background.) A proof that all triangles are isosceles.
- Week 4 (2/2):
- Day 9: More on objectivity and subjectivity; infinity in Newton. Read JtG p.184-196. Homework 4 (due Monday)
- Day 10: Newton, Leibniz, Bernoullis, dealing with infinite sums. Grandi's series.
- Day 11: (Sick day, no class)
- Week 5(2/9):
- Week 6 (2/16):
- Day 15: Syntax of sentential logic (beginning OLI ch. 2). Homework 5 (due Friday)
- Day 16: Syntax of setential logic, continued. Translation practice.
- Day 17: Parse trees, recursion, truth function (finishing OLI ch. 2).
- Week 7 (2/23):
- Day 18: Truth assignments and truth tables (beginning OLI ch. 3).Homework 6 (due Friday)
- Day 19: Truth tables and truth trees.
- Day 20: Deductions, intro to the Proof Lab. Homework 7 (due Friday)
- Week 8 (3/2):
- Day 21: Proof strategies (beginning OLI chs. 4 and 5).
- Day 22: Negation rules, discussion of intuitionistic logic.
- Spring break! (Note: HW7 is due Friday, but I recommend doing it early to get it out of the way.)
- Week 9 (3/16):
- Day 23: Derived rules (OLI ch. 6). Homework 8 (Due Monday)
- Day 24: Derived rules, continued.
- Day 25: Conditional analogues, biconditionals, truth-functional completeness (OLI ch. 7)
- Week 10 (3/23):
- Day 26: Soundness and completeness of SL. Midterm exam will be open next week, Monday-Friday.
- Day 27: Extensions of SL: modal logic.
- Day 28: Extensions of SL: basics of first-order logic.
- Week 11 (3/30):
- Day 29: Sets and naive set theory. Old notes. MIDTERM EXAM IS OPEN: do the problems on OLI listed here.
- Day 30: NO CLASS, but office hours (for exam questions instead!) Find me in my office, DH 4302.
- Day 31: Set theory continued. Russell's Paradox and related.
- Week 12 (4/6):
- Week 13 (4/13):
- Day 35: Monty Hall problem and probability theory. Notes here. Homework 9 (due Monday)
- Day 36: Probability theory continued; Simpson's paradox.
- No class Friday for Carnival.
- Week 14 (4/20):
- Day 37: Turing machines and computation. See: Wikipedia (esp. the informal description, additional details, and universal machine sections), a tiny universal Turing machine
- Day 38: Turing machines, continued. Homework 10 (due Wednesday). The final quiz will be distributed next Wednesday, due Friday.
- Day 39: Formalism in the philosophy of mathematics. A reading for the remaining classes.
- Week 15 (4/27):
- Day 40: Intuitionism in the philosophy of mathematics.
- Day 41: Platonism in the philosophy of mathematics. HW 10 due, final quiz distributed. Last day of class!
- Day 42: No class: work on the quiz.
- Optional final papers due Wednesday, May 13 (or Sunday, May 10 if you are a graduating senior - no exceptions!)