99 Variations on a Proof

Philip Ording

Published by Princeton University Press 2019

99 Variations on a Proof

Philip Ording

https://doi.org/10.1515/9780691185422

Overview
Contents

About this book

An exploration of the art of mathematics through 99 different proofs of the same theorem.

An exploration of the art of mathematics through 99 different proofs of the same theorem

This lively and compelling book offers a unique, multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from “Medieval,” “Topological,” and “Doggerel” to “Chromatic,” “Electrostatic,” and “Psychedelic.” Along the way, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice of mathematics.

Inspired by the experiments of the Paris-based writing group known as the Oulipo—whose members included Raymond Queneau, Italo Calvino, and Marcel Duchamp—Ording explores new ways to examine the aesthetic possibilities of mathematical activity. This mathematical take on Queneau’s Exercises in Style—a collection of 99 retellings of the same story—draws unexpected connections to everything from mysticism and technology to architecture and sign language and includes diagrams, notations, and other imagery that are as amusing as they are edifying. With a rare blend of humor and scholarly aplomb, Ording illustrates the inherent creative potential and flexibility of mathematical language despite its reputation for precision and rigor.

Readers of this playful and expansive compendium of mathematical exposition and style will gain not only a bird’s-eye view of the discipline and its major branches but also new insights into its historical, philosophical, and cultural nuances. Every reader, no matter their level of expertise, will discover in these proofs surprising new aspects of the mathematical landscape.

Author information

Philip Ording is a professor of mathematics at Sarah Lawrence College. He is the coeditor of Simplicity: Ideals of Practice in Mathematics and the Arts.

Reviews

“Ording takes a central idea of mathematics—that there are many ways to write a proof and illustrate a theorem’s main point—and weaves it into a story with deep history and flavor. I love this book. It has the unique effect of making you feel like you are getting smarter the more you read.”—Raffi Grinberg, author of The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs

“According to Molière there are many ways to declare love. It could be ‘Belle Marquise, vos beaux yeux me font mourir d’amour,’ or ‘D’amour mourir me font, belle Marquise, vos beaux yeux,’ or ‘Vos yeux beaux d’amour me font, belle Marquise, mourir.’ Molière only lists five variations and skips the 115 remaining possibilities. Philip Ording proposes a poetic transposition of mathematics. Starting with an easy theorem from high school, he offers 99 variations on the same theme, 99 different proofs of the same fact, 99 love declarations to mathematics.”—Étienne Ghys, CNRS - École normale supérieure de Lyon

"Ording presents ninety-nine proofs that a specific cubic equation has two real roots. The theorem itself is fairly uninteresting, but the proofs are the stars and each of them seeks to show a different aspect of the theory, history, or culture of mathematics."---Geoffrey Dietz, MAA Reviews

“Showing us the astonishing variety of ways the same mathematical facts can be justified, Ording notes the influence of Queneau’s Exercises in Style, but perhaps Wallace Stevens’s ‘Thirteen Ways of Looking at a Blackbird’ casts its shadow, too. In each proof, thought takes flight and veers in different directions, bringing back spiral snail shells, polygonal berries, and elliptical seeds to the same stable nest: proofs, like blackbirds, are omnivores!”—Emily Grosholz, Pennsylvania State University

“To the outsider, mathematics seems like a matter of pure logic and skill, an endeavor to be evaluated along a single axis of excellence. But in this adventurous, elegant book, Ording shows to what extent mathematics is also a question of style. And mathematical style, like all style, cannot be placed on a simple gradient. Its terrain needs to be mapped, and in Ording it has found an exquisite cartographer.”—Sina Najafi, editor in chief of Cabinet magazine

“A charming and indeed stylish feat of metamathematical storytelling, rich in history and wit.”—Siobhan Roberts, author of Genius at Play: The Curious Mind of John Horton Conway

“A savant, exhilarating inquiry into the deep roots connecting mathematics to language, belief to persuasion, and truth to style. Philip Ording has one problem, 99 answers, and a world’s worth of ways to carve up reality.”—Daniel Levin Becker, author of Many Subtle Channels: In Praise of Potential Literature

“Fun, funny, and unexpectedly deep, Philip Ording’s Oulipian expedition through the far reaches of mathematical style shows there’s more than one way to skin a cubic equation.”—Jordan Ellenberg, author of How Not to Be Wrong: The Power of Mathematical Thinking

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	vi
Preface Publicly Available Download PDF	ix
0. Omitted Requires Authentication Unlicensed Licensed Download PDF	1
1. One-Line Requires Authentication Unlicensed Licensed Download PDF	3
2. Two-Column Requires Authentication Unlicensed Licensed Download PDF	5
3. Illustrated Requires Authentication Unlicensed Licensed Download PDF	7
4. Elementary Requires Authentication Unlicensed Licensed Download PDF	9
5. Puzzle Requires Authentication Unlicensed Licensed Download PDF	11
6. Axiomatic Requires Authentication Unlicensed Licensed Download PDF	13
7. Found Requires Authentication Unlicensed Licensed Download PDF	17
8. Prerequisite Requires Authentication Unlicensed Licensed Download PDF	19
9. Monosyllabic Requires Authentication Unlicensed Licensed Download PDF	21
10. Wordless Requires Authentication Unlicensed Licensed Download PDF	23
11. Exam Requires Authentication Unlicensed Licensed Download PDF	25
12. Ruler and Compass Requires Authentication Unlicensed Licensed Download PDF	27
13. Reductio ad Absurdum Requires Authentication Unlicensed Licensed Download PDF	29
14. Contrapositive Requires Authentication Unlicensed Licensed Download PDF	31
15. Matrices Requires Authentication Unlicensed Licensed Download PDF	33
16. Ancient Requires Authentication Unlicensed Licensed Download PDF	35
17. Interpreted Requires Authentication Unlicensed Licensed Download PDF	37
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20. Definitional Requires Authentication Unlicensed Licensed Download PDF	43
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22. Substitution Requires Authentication Unlicensed Licensed Download PDF	49
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24. Another Symmetry Requires Authentication Unlicensed Licensed Download PDF	53
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29. Model Requires Authentication Unlicensed Licensed Download PDF	67
30. Formulaic Requires Authentication Unlicensed Licensed Download PDF	69
31. Counterexample Requires Authentication Unlicensed Licensed Download PDF	71
32. Another Counterexample Requires Authentication Unlicensed Licensed Download PDF	73
33 Calculus Requires Authentication Unlicensed Licensed Download PDF	75
34. Medieval Requires Authentication Unlicensed Licensed Download PDF	77
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51. Topological Requires Authentication Unlicensed Licensed Download PDF	119
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53. Marginalia Requires Authentication Unlicensed Licensed Download PDF	125
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61. Modern Requires Authentication Unlicensed Licensed Download PDF	143
62. Axonometric Requires Authentication Unlicensed Licensed Download PDF	145
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67. Approximate Requires Authentication Unlicensed Licensed Download PDF	157
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69. Statistical Requires Authentication Unlicensed Licensed Download PDF	161
70. Another Medieval Requires Authentication Unlicensed Licensed Download PDF	163
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72. Translated Requires Authentication Unlicensed Licensed Download PDF	171
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77. Monte Carlo Requires Authentication Unlicensed Licensed Download PDF	185
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80. Paranoid Requires Authentication Unlicensed Licensed Download PDF	191
81. Doggerel Requires Authentication Unlicensed Licensed Download PDF	193
82. Inconsistency Requires Authentication Unlicensed Licensed Download PDF	195
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85. Exhaustion Requires Authentication Unlicensed Licensed Download PDF	201
86. Another Substitution Requires Authentication Unlicensed Licensed Download PDF	203
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88. Dialogue Requires Authentication Unlicensed Licensed Download PDF	209
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90. Retrograde Requires Authentication Unlicensed Licensed Download PDF	215
91. Mystical Requires Authentication Unlicensed Licensed Download PDF	217
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96. Electrostatic Requires Authentication Unlicensed Licensed Download PDF	227
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98. Mondegreen Requires Authentication Unlicensed Licensed Download PDF	231
99. Prescribed Requires Authentication Unlicensed Licensed Download PDF	233
Postscript Requires Authentication Unlicensed Licensed Download PDF	235
Acknowledgments Requires Authentication Unlicensed Licensed Download PDF	237
Notes Requires Authentication Unlicensed Licensed Download PDF	239
Sources Requires Authentication Unlicensed Licensed Download PDF	247
Index Requires Authentication Unlicensed Licensed Download PDF	257