Why Prove it Again?
Alternative Proofs in Mathematical Practice
Authors: Dawson, Jr., John W.
Free Preview Contains comparative studies of alternative proofs of various wellknown theorems
 Stresses the informal notion of what constitutes a proof, as opposed to the formal notion of proof in mathematical logic
 Will appeal to a broad range of readers, including historians and philosophers of mathematics, students, and practicing mathematicians
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 About this book

This monograph considers several wellknown mathematical theorems and asks the question, “Why prove it again?” while examining alternative proofs. It explores the different rationales mathematicians may have for pursuing and presenting new proofs of previously established results, as well as how they judge whether two proofs of a given result are different. While a number of books have examined alternative proofs of individual theorems, this is the first that presents comparative case studies of other methods for a variety of different theorems.
The author begins by laying out the criteria for distinguishing among proofs and enumerates reasons why new proofs have, for so long, played a prominent role in mathematical practice. He then outlines various purposes that alternative proofs may serve. Each chapter that follows provides a detailed case study of alternative proofs for particular theorems, including the Pythagorean Theorem, the Fundamental Theorem of Arithmetic, Desargues’ Theorem, the Prime Number Theorem, and the proof of the irreducibility of cyclotomic polynomials.
Why Prove It Again? will appeal to a broad range of readers, including historians and philosophers of mathematics, students, and practicing mathematicians. Additionally, teachers will find it to be a useful source of alternative methods of presenting material to their students.
 About the authors

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

“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 firstorder logic. … because of the many examples given, there should be something to suit everybody’s taste … .” (Jessica Carter, Philosophia Mathematica, February, 2016)
 Table of contents (14 chapters)


Proofs in Mathematical Practice
Pages 16

Motives for Finding Alternative Proofs
Pages 711

Sums of Integers
Pages 1318

Quadratic Surds
Pages 1923

The Pythagorean Theorem
Pages 2539

Table of contents (14 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Why Prove it Again?
 Book Subtitle
 Alternative Proofs in Mathematical Practice
 Authors

 John W. Dawson, Jr.
 Copyright
 2015
 Publisher
 Birkhäuser Basel
 Copyright Holder
 Springer International Publishing Switzerland
 eBook ISBN
 9783319173689
 DOI
 10.1007/9783319173689
 Hardcover ISBN
 9783319173672
 Softcover ISBN
 9783319349671
 Edition Number
 1
 Number of Pages
 XI, 204
 Number of Illustrations
 54 b/w illustrations
 Topics