80-110: The Nature of Mathematical Reasoning

Carnegie Mellon University

Spring 2015

Location: Doherty Hall 1115
Instructor: Rob Lewis (email)
Office: Doherty Hall 4302D (directions)
Office Hours: MWF 10:30-11:30, or preferably by appointment

Course Description

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.

Course Resources

Syllabus: here
Textbooks: A previous version of this course:

Lectures and Assignments