Review of 99 Variations on a Proof

99 Variations on a Proof
Philip Ording

Princeton University Press, Princeton, NJ, 2019
272 pp., 1 col., 25 b/w. Trade, $24.95
ISBN: 978-0-691-15883-9.

Reviewed by
Robert Maddox-Harle
June 2019

This book is “one-of-a-kind”, could be a text book but it’s not, could be a history of mathematics but it’s not, could be a philosophical treatise but it’s not – it subsumes all these possibilities into 99 investigations of one equation, – If x3-6x2+11x-6=2x-2 then x=1 or x=4!

Ording wrote this book to prove that mathematics does not only have one style:

“The received wisdom is that mathematics, the universal language of science, has one style – the mathematical style – characterised by symbolic notation, abstraction, and logical rigor This book aims to challenge that conception of mathematics.” (p. ix)

It succeeds admirably through logical rigor, a quirky sense of humour and a vast knowledge of mathematical processes.

The book is very nicely produced, starting with an introductory Preface, followed by 100 “proofs”, (you’ll have to read the book to find out why not just 99?), then a Postscript; Acknowledgements; Notes; Sources and an Index. The book does not have to be read in sequence; one may dip into any of the proofs arbitrarily depending on how inquisitive one is, the proofs have tantalising titles here are just a few examples: Slide Rule, Neologism, Psychedelic, Doggerel, Interior Monologue, World Problem; these examples will give you an idea of what you are getting into with this book.

99 Variations on a Proof is NOT for the general well educated reader. Despite its frivolity and broad scope, without a fairly high level of mathematical expertise Ording’s mathematical brilliance is unfortunately wasted. Philip Ording is a professor of mathematics at Sarah Lawrence College, New York. One of the problems with many academics is they forget what it is like to know very little about the subject that they know virtually everything about. The publishers “blurb” does not help this conundrum when they state: “Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape” (back cover). This is a misleading statement; I do not have a high level of mathematic ability and selected this book to review because of the assurance that I did not need to have one. The base cubic equation is simple enough, but some of the proofs are so complex as to be unintelligible to me. In Proof 30 for example Ording states: “The difficulty remembering Cardano’s formula isn’t just that it’s so long, there’s also a subtlety at work in the cube roots. Just as one needs to account for both positive and negative square roots in the quadratic cousin to this formula, there are additional solutions resulting from different complex cube roots” (p. 70). Oh really!

In writing this book Ording was inspired by the philosophy behind the Oulipo movement. This group experimented with literary forms constrained by various rules and restrictions. Raymond Queneau, a radical mathematician himself, was an original founder of this group. They insisted that chance and inspiration had no part in the writing of original literature. I have shown elsewhere that these particular beliefs are untenable. See my thesis, The Myth of the Freudian Unconscious and its Relationship with Surrealist Poetry (Deakin University, 2000), or more recently my essay published in Setu Journal,Creativity, Chance and the Role of the Unconscious in the Creation of Original Literature and Art” (https://www.setumag.com/2018/05/creativity-chance-Unconscious.html). Ording’s book is a mathematical take on Queneau’s Exercises in Style, a collection of 99 retellings of the same story.

“The most vivid form of mathematics for many is likely to be the exam. “Math Test” is a powerful linguistic collocation not unlike “splitting headache” or “excruciating pain” (p. 26). By analogous association, for a nonmathematician trying to understand the equations in this book similar maladies are likely to occur!

This book is a insightful addition to mathematical literature, but I believe it is only suitable for mathematicians and higher level mathematical students.