An Infinite Descent into Pure Mathematics was born out of lecture notes I wrote for teaching Concepts of Mathematics, an introductory pure mathematics class at Carnegie Mellon University, in summer 2015.

I wanted to provide my students with a free resource that helped them learn the skills required to do mathematics, such as solving a mathematical problem, defining new mathematical concepts, writing a proof, typesetting mathematics in LaTeX… the list goes on! I was unable to find a resource that emphasised these aspects whilst also covering enough mathematical topics, so I decided to write my own.

The first part of the book covers core material, including basic mathematical reasoning, symbolic logic, sets, functions, induction and relations. These are the concepts that all mathematicians should feel comfortable with. The second part of the book covers basic material in a number of mathematical areas—topics include number theory, combinatorics, real analysis, infinite sets, equivalence relations, probability theory, order theory and structural induction. Throughout, strategies are provided that help readers to see how mathematical concepts and results can be used in a proof.

Communication is the emphasis of Appendix A, which aims to help readers build their proof-writing skills. This appendix covers what kinds of features make for an effective (or ineffective) proof, and provides readers with ideas for what words to use in a proof and how to structure a proof clearly and precisely (and readably).

Another aspect of doing mathematics covered in the book is using LaTeX to typeset a mathematical document, a skill required of almost all professional mathematicians but rarely formally taught to them. LaTeX commands are provided for all new notation introduced in the book at the time that the notation appears. The LaTeX source for the book is available on Gitlab.

Much material is presented in the form of exercises that a reader should work through, in order to promote mathematical enquiry. The decision not to include solutions to exercises is a conscious one, which is discussed in this post, but hints for selected exercises are included in the appendix.

This textbook is intended to be useful for both students and instructors. With this in mind, I have tried to make it elaborate enough to be useful for learning, but concise enough to be useful as a reference. Furthermore, many of the exercises are suitable for use by instructors as questions on problem sheets and examinations and as prompts for in-class activities.

A reader intending to self-study this textbook should seek additional help from others, such as peers, tutors or even online communities such as Mathematics Stack Exchange.

Despite being used primarily in the United States, the book is written in British English—I do not think I will ever truly understand American English, so I won’t try to write in it.