Introduction

Summary

The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no place in the world for ugly mathematics.

G. H. Hardy A Mathematician's Apology

This is a book about proofs, focusing on attractive proofs we refer to as charming. While this is not a definition, we can say that a proof is an argument to convince the reader that a mathematical statement must be true. Beyond mere convincing we hope that many of the proofs in this book will also be fascinating.

Proofs: The heart of mathematics

An elegantly executed proof is a poem in all but the form in which it is written.

Morris Kline Mathematics in Western Culture

As we claimed in the Preface, proofs lie at the heart of mathematics. But beyond providing the foundation for the growth of mathematics, proofs yield new ways of reasoning, and open new vistas to the understanding of the subject. As Yuri Ivanovich Manin said, “A good proof is one that makes us wiser,” a sentiment echoed by Andrew Gleason: “Proofs really aren't there to convince you that something is true–they're there to show you why it is true.”

The noun “proof” and the verb “to prove” come from the Latin verb probare, meaning “to try, to test, to judge.”