This is an erudite and historical look at several famous theorems for which multiple proofs are known. Most of the book is devoted to case studies that examine and compare several proofs for each theorem, with one chapter per theorem. Usually complete proofs are given and analyzed here, although in a few cases a detailed sketch is given instead. As you would expect, the theorems with the greatest variety of proofs are also the oldest, and so the results shown are slanted heavily towards geometry, theory of equations, and number theory.
The last chapter gives briefer coverage of several more theorems, and even briefer coverage of some theorems that the author thinks deserve further investigation. The book begins with some informal remarks about two general questions: (1) The title question: mathematicians have produced and studied multiple proofs for the same theorem even since antiquity, but why? (2) How do you tell if two proofs are different enough to count as different proofs?
The book doesn’t attempt a definitive answer to either question; the author intends rather to start a discussion about them. A lot of the answer to (1) seems to be taste: some proofs are more pleasing than others, and we may seek new proofs if the existing ones are dissatisfying. The book discusses in particular the motive of making proofs more “perspicuous,” meaning that they are clearer and give more insight into why the result is true, not merely convincing us that it is true. For example, most proofs by induction are not perspicuous.
Question (2) seems not well understood. Part of the difficulty is that different proofs often prove different things: the same conclusion with a weaker hypothesis, a stronger conclusion from the same hypothesis, a generalization, or a useful special case that has a neat proof. There are extreme examples with a ridiculously large number of proofs, for which (it seems to me) it would be impossible to determine exactly how many proofs there are. Two sources cited in this book are a book that lists 52 proofs of the arithmetic mean–geometric mean inequality (p. 195) and a book that lists 196 proofs of the quadratic reciprocity theorem in number theory (p. 197). Are they really all different?
Bottom line: an interesting book to browse or study, just for the variety of techniques, even if you are not interested in the philosophical and methodological questions that inspired it. Another good book to browse is Aigner & Ziegler’s Proofs from The Book, which also presents multiple proofs of many results, even though its stated purpose is to present the “best” proof of the result.
Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.