I recently came across an article on Stack Exchange that asks readers what proofs they think other mathematicians should know and why. There are 23 answers on the post (it is a closed question now unfortunately) with many different proof suggestions. This reminded me of Paul Erdös’ common reference to The Book.
Although an atheist, he spoke of “The
Book”, an imaginary book in which God had written down
the best and most elegant proofs for mathematical
theorems. Lecturing in 1985 he said, “You don’t have to
believe in God, but you should believe in The
Book.” He himself doubted the existence of God, whom he
called the “Supreme Fascist” (SF). He accused the SF of
hiding his socks and Hungarian passports, and of keeping the
most elegant mathematical proofs to himself. When he saw a
particularly beautiful mathematical proof he would
exclaim, “This one’s from The Book!”. This
later inspired a book entitled Proofs from THE
BOOK. [1]
The book entitled Proofs from THE BOOK by Martin Aigner and Günter M. Ziegler is a draft of the book mentioned by Erdös. The book is sectioned into five different categories: Number Theory, Geometry, Analysis, Combinatorics, and Graph Theory. Each proof (or set of proofs when multiple proofs are given for the same statement) is its own chapter with 40 chapters total. The authors give references at the end of each chapter, which is a neat and useful addition to the book. What do you say? Do you believe in The Book, as Erdös insists you must? What are some of your favorite proofs?