Welcome to the revised Lecture Notes for Math 299. This site is a work in progress. Check back often for updates.

Introduction

The development of logic and mathematics over thousands of years is one of the great achievements of human cognition.

On the one hand, its usefulness and practical importance in the modern world is hard to overstate. It provides a foundation upon which science, engineering, finance, medicine, economics, computer science, agriculture, and many other areas of human knowledge have been developed. But this fact raises an interesting question. Why?

Why is it that such a wide and disparate collection of applications from counting to cosmology all rely upon mathematics? Why are they built upon mathematics instead of something else like, say, music or poetry? Most of us recognize that mathematics is exceedingly useful, but why is it so useful?

One possible reason is that it provides us with a reliable starting point on which to build consensus. A group can accomplish much more by working collaboratively than they can by working as separate individuals. But cooperation and collaborative decision-making require a consensus about the assumptions, terminology, and facts that allow us to communicate and inform our decisions.

Mathematics and its underlying logic provide us with a tool that allows us to reason in a way that is reliable, objectively verifiable, and independent of our individual, personal, and subjective human biases. Mathematicians do not debate whether $5$ is larger than $4$, or whether $2+3=5$. The fact that mathematics can be verified objectively and reliably is what makes it a prototype for achieving consensus about mathematical facts. Other disciplines that can successfully build upon mathematics can then use it to construct a solid foundation for consensus about facts in their own subject area.

In addition to being useful, mathematics is one of the most beautiful, creative, and sublime creations of the human mind. It is inherently valuable for its own sake as a work of art, to be enjoyed and shared with others, and passed down from generation to generation for thousands of years.

For this reason, our introduction to mathematical proof must combine both the rigorous objectivity that is needed for determining and communicating mathematical facts, with the elegance and beauty that exemplifies any human art form.

The main reason mathematics and logic are so amenable to building consensus is that anyone can check a mathematical claim for themselves. There is no need to believe anyone, cite any book, or consult any oracle. We can each take personal ownership of our mathematics by proving all of it ourselves.

The goal of this course is to do exactly that. It will guide you on a personal journey to build and verify most of elementary mathematics from the ground up. It is a quest to objectively prove for yourself all of the basic elementary mathematical facts about logic, natural numbers, sequences, real numbers, set theory, functions, relations, and combinatorics.

To accomplish this, we must begin by slowly and carefully defining exactly what constitutes a mathematical proof, how to construct them ourselves, and how to express them up in a way that allows them to be accurately and elegantly shared with others.