Why Prove it Again?: Alternative Proofs in Mathematical Practice

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Birkhäuser, Jul 15, 2015 - Mathematics - 204 pages

This monograph considers several well-known 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.

 

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Contents

1 Proofs in Mathematical Practice
1
2 Motives for Finding Alternative Proofs
7
3 Sums of Integers
13
4 Quadratic Surds
19
5 The Pythagorean Theorem
25
6 The Fundamental Theorem of Arithmetic
41
7 The Infinitude of the Primes
51
8 The Fundamental Theorem of Algebra
59
9 Desarguess Theorem
93
10 The Prime Number Theorem
111
11 The Irreducibility of the Cyclotomic Polynomials
149
12 The Compactness of Firstorder Languages
171
13 Other Case Studies
187
Erratum
200
Index
201
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About the author (2015)

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

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