Why Prove it Again?

John W. Dawson, Jr., is Professor Emeritus at Penn State York.

• Proofs in Mathematical Practice.- Motives for Finding Alternative Proofs.- Sums of Integers.- Quadratic Surds.- The Pythagorean Theorem.- The Fundamental Theorem of Arithmetic.- The Infinitude of the Primes.- The Fundamental Theorem of Algebra.- Desargues's Theorem.- The Prime Number Theorem.- The Irreducibility of the Cyclotomic Polynomials.- The Compactness of First-Order Languages.- Other Case Studies.

"The book motivates and introduces its topic well and successively argues for the claim that comparative studies or proofs are a worthwhile occupation. All chapters are accessible to a generally informed mathematical audience, most of them to mathematical laymen with a basic knowledge of number theory and geometry." (Merlin Carl, Mathematical Reviews, April, 2016) "This book addresses the question of why mathematicians prove certain fundamental theorems again and again. ... Each chapter is a historical account of how and why these theorems have been reproved several times throughout several centuries. The primary readers of this book will be historians or philosophers of mathematics ... ." (M. Bona, Choice, Vol. 53 (6), February, 2016) "This is an impressive book, giving proofs, sketches, or ideas of proofs of a variety of fundamental theorems of mathematics, ranging from Pythagoras's theorem, through the fundamental theorems of arithmetic and algebra, to the compactness theorem of first-order logic. ... because of the many examples given, there should be something to suit everybody's taste ... ." (Jessica Carter, Philosophia Mathematica, February, 2016)