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 emphasised not only the technical aspects of mathematics, but also the human aspects, particularly communication and inquiry—I was unable to find a resource that emphasised these aspects and also covered enough ground, so I decided to write my own.
The first two chapters of the book cover core material, including basic mathematical reasoning, symbolic logic, sets, functions and induction. These are the concepts that all mathematicians should feel comfortable with. The rest of the book covers basic material in a number of mathematical areas. Topics include number theory, combinatorics, infinite sets, equivalence relations, ordered sets, structural induction, real analysis and probability theory.
Several features of the book focus on aspects of communication in mathematics. Early on in the book, there is guidance on structuring proofs and suggestions for writing them. Furthermore, the book provides support for using LaTeX, which is a tool used almost universally in the mathematical community for typesetting mathematical documents. In particular, there is an appendix which provides an introduction to LaTeX, and the LaTeX commands for all mathematical notation used in the textbook are provided as they are introduced. Additional features that will appear in future renditions of the textbook include descriptions of possible mathematical writing projects, and a comprehensive mathematical writing guide.
I have tried to promote inquiry-based learning in the book, meaning that much material is presented in the form of exercises that a reader should work through. 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.