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  • © 2015

Why Prove it Again?

Alternative Proofs in Mathematical Practice

Birkhäuser
  • Contains comparative studies of alternative proofs of various well-known 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

Buying options

eBook USD 59.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-17368-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 79.99
Price excludes VAT (USA)
Hardcover Book USD 109.99
Price excludes VAT (USA)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xi
  2. Proofs in Mathematical Practice

    • John W. Dawson Jr.
    Pages 1-6
  3. Motives for Finding Alternative Proofs

    • John W. Dawson Jr.
    Pages 7-11
  4. Sums of Integers

    • John W. Dawson Jr.
    Pages 13-18
  5. Quadratic Surds

    • John W. Dawson Jr.
    Pages 19-23
  6. The Pythagorean Theorem

    • John W. Dawson Jr.
    Pages 25-39
  7. The Fundamental Theorem of Arithmetic

    • John W. Dawson Jr.
    Pages 41-49
  8. The Infinitude of the Primes

    • John W. Dawson Jr.
    Pages 51-57
  9. The Fundamental Theorem of Algebra

    • John W. Dawson Jr.
    Pages 59-91
  10. Desargues’s Theorem

    • John W. Dawson Jr.
    Pages 93-110
  11. The Prime Number Theorem

    • John W. Dawson Jr.
    Pages 111-147
  12. The Irreducibility of the Cyclotomic Polynomials

    • John W. Dawson Jr., Steven H. Weintraub
    Pages 149-170
  13. The Compactness of First-order Languages

    • John W. Dawson Jr.
    Pages 171-186
  14. Other Case Studies

    • John W. Dawson Jr.
    Pages 187-200
  15. Erratum

    • John W. Dawson Jr.
    Pages E1-E2
  16. Back Matter

    Pages 201-204

About this book

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.

Keywords

  • Alternative Proofs
  • Desargues's Theorem
  • Distribution of Primes
  • Fundamental Theorem of Algebra
  • Pythagorean theorem
  • Quadratic surds

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 first-order logic. … because of the many examples given, there should be something to suit everybody’s taste … .” (Jessica Carter, Philosophia Mathematica, February, 2016)

Authors and Affiliations

  • Penn State York, York, USA

    John W. Dawson, Jr.

About the author

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

Bibliographic Information

  • Book Title: Why Prove it Again?

  • Book Subtitle: Alternative Proofs in Mathematical Practice

  • Authors: John W. Dawson, Jr.

  • DOI: https://doi.org/10.1007/978-3-319-17368-9

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2015

  • Hardcover ISBN: 978-3-319-17367-2

  • Softcover ISBN: 978-3-319-34967-1

  • eBook ISBN: 978-3-319-17368-9

  • Edition Number: 1

  • Number of Pages: XI, 204

  • Number of Illustrations: 54 b/w illustrations

  • Topics: History of Mathematical Sciences, Geometry, Algebra, Analysis, Topology

Buying options

eBook USD 59.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-17368-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 79.99
Price excludes VAT (USA)
Hardcover Book USD 109.99
Price excludes VAT (USA)