Mathematical Proof and the Principles of Mathematics/Introduction
Charles Dodgson, better known as Lewis Carroll, asked the rhetorical question, "Now what is it you really require in a Manual of Geometry?" His answer was not a book that lists facts about geometry but, "... a book that will exercise the learner in habits of clear definite conception, and enable him to test the logical value of a scientific argument." This reflects a long-held tradition that the study of geometry, and mathematics in general, helps the student develop skills in clear and logical thought. Such skills are extremely important in daily life and for the proper functioning of democracy.
Perhaps, if the goal is to develop clear and logical thought in general, it would be better to study texts specifically designed to do so rather than to study mathematics where what is desired is merely a useful side effect. But it is certain that the correct use of reasoning is at the core of modern mathematics, especially in the construction of proofs. This insistence on the use of logic as the basis of mathematics has not only helped steer the subject away from incorrect results, but has guided it in the discovery of new areas of knowledge whose existence no one could have predicted otherwise.
This book is intended to be a guide to the use of logical reasoning as it is currently practiced in mathematics. The standards of rigor do, in fact, change over time and what would have been acceptable a few hundred years ago would not be allowed today. But the desire for rigor in mathematics has been around for thousands of years and will likely continue for thousands more. We've included a chapter on history describing some of the changes in standards. It's meant to provide some motivation and perspective for the subject, but the rest of the book is meant to be independent of it, so you can skip ahead if you want.
There is no denying that mathematics is a difficult subject. But it can be made easier by not relying on a single source to learn from. This runs contrary to the usual practice of assigning a single book for a class with is to be used exclusively. But the fact is that different authors explain things in different ways, and some explanations may resonate better with some learners than with others. To that end, we provide a list of alternatives to be consulted when the going gets difficult.
- An Introduction to Proofs and the Mathematical Vernacular by Martin Day, (Free on-line)
- Book of Proof by Richard Hammack, (Free on-line)
- How to Read and do Proofs: An Introduction to Mathematical Thought Processes by Daniel Solow
- Mathematical Reasoning: Writing and Proof by Ted Sundstrom, (Free on-line)
- A Gentle Introduction to the Art of Mathematics by Joseph E. Fields (Free on-line)
For those who would rather watch than read, there are screencast lecture courses as well.
- Communicating in Mathematics a companion to Ted Sundstrom's book.
- Introduction to Higher Mathematics by Bill Shillito
- Mathematics for Computer Science by Albert R. Meyer and Adam Chlipala. Part of MIT OpenCourseware, stressing discrete structures and computing.
How to use this book
In a sense, no book or instructor can teach you how to do the kind of mathematics we're talking about. You must teach yourself and the only way to learn is by doing. A book can get you started on the right path and provide sign posts on the way, but no one can make the journey for you. For this reason, only a few proofs will be given here as examples, the rest will be left for the reader to fill in.
It is often said that mathematics is a language; some go as far as saying it's the language through which the universe speaks to us. This book attempts to describe the grammar and vocabulary of this language, but no language book would be be complete without some description of the culture of its speakers. So this book will also include something of the history and lore of mathematics: information which is common knowledge among mathematicians but may not be covered in a typical book on mathematics.
Logic and foundations
It's very tempting in a book like this to start digging in to mathematical logic and the foundations of mathematics. The problem is that once you pick up that shovel it's very difficult to put it down. It's not helpful for the purposes of this book either because the subject becomes more abstruse the deeper you dig. So, with the assumption that the people reading this book want to get on with actually proving theorems, we'll try to include only that foundational material which furthers our goals here.
We're trying to teach the process of mathematics here rather than serve as a complete exposition of any individual mathematical subject. So we may occasionally ask the reader to accept some results without proof if the proof is very difficult or complex. The ideal we hope to achieve is that proofs can all be filled in by the reader, starting easy and progressing to more difficult. But mathematics does not always make itself so convenient and sometimes difficult results are needed to prove easier ones, so allowances must be made from time to time.