Why Prove it Again?

Alternative Proofs in Mathematical Practice

• John W. Dawson, Jr.
Book

1. Front Matter
Pages i-xi
2. John W. Dawson Jr.
Pages 1-6
3. John W. Dawson Jr.
Pages 7-11
4. John W. Dawson Jr.
Pages 13-18
5. John W. Dawson Jr.
Pages 19-23
6. John W. Dawson Jr.
Pages 25-39
7. John W. Dawson Jr.
Pages 41-49
8. John W. Dawson Jr.
Pages 51-57
9. John W. Dawson Jr.
Pages 59-91
10. John W. Dawson Jr.
Pages 93-110
11. John W. Dawson Jr.
Pages 111-147
12. John W. Dawson Jr., Steven H. Weintraub
Pages 149-170
13. John W. Dawson Jr.
Pages 171-186
14. John W. Dawson Jr.
Pages 187-200
15. John W. Dawson Jr.
Pages E1-E2
16. Back Matter
Pages 201-204

Introduction

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

Authors and affiliations

• John W. Dawson, Jr.
• 1
1. 1.Penn State YorkYorkUSA