Philip Ording - a 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

Philip Ording, 99 Variations on a Proof, Princeton UP, 2019.


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An exploration of mathematical style through 99 different proofs of the same theorem.This book offers a 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. With a rare blend of humor and scholarly aplomb, 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. 99 Variations on a Proof is a mathematical take on Queneau’s Exercises in Style, a collection of 99 retellings of the same story, and it draws unexpected connections to everything from mysticism and technology to architecture and sign language. Through diagrams, found material, and other imagery, Ording illustrates the flexibility and creative potential of mathematics despite its reputation for precision and rigor.
Readers 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. Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape.


Book draws unexpected connections to everything from mysticism and technology to architecture and sign language.


This book offers a 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.
With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice of mathematics, according to a review on the Princeton University Press website.
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.
99 Variations on a Proof is a mathematical take on Queneau’s Exercises in Style, a collection of 99 retellings of the same story, and it draws unexpected connections to everything from mysticism and technology to architecture and sign language.
http://www.arabnews.com/node/1440851/lifestyle

“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

“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

“A charming and indeed stylish feat of metamathematical storytelling, rich in history and wit.”—Siobhan Roberts

“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

“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

“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

“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




99 Variations on a Proof is, unsurprisingly, inspired by Raymond Queneau's classic variation-work, Exercises in Style.
       As Ording explains in his Preface:

As soon as I learned about the Oulipo and Queneau's book, I wanted to see what effect constrained writing strategies would have on a mathematical narrative -- a proof.

       So he took a cubic equation -- itself based on, as he explains in his Postscript, an: "algebro-geometric reading of the story that forms the basis of Raymond Queneau's Exercises in Style" -- and offers 99 variations on proofs of it (or a hundred, if you count the first, 'Omitted', which simply does without).
       The cubic equation in question and claim is:

If x³−6x²+11x−6=2x−2, then x=1 or x=4.

       In his Postscript Ording acknowledges (and finds it appropriate) that it's a: "degenerate cubic equation, with its awkward, non-standard form" (as it can just as easily -- and arguably more obviously -- be expressed as: x³−6x²+9x−4=0), but expressing it this way does more easily bring some of the variations into play: if the first example, 'One-Line', simply offers: "By subtraction, x³−6x²+9x−4=0, which factors as (x−1)²(x−4)=0", the third -- 'Illustrated' -- graphs the two sides of the equation as he originally expresses it, cubic y=x³−6x²+11x−6 and the line y=2x−2, the proof then neatly visualized in the two points where they intersect. Indeed, one of the impressive things among these variations is how many do not rely strictly on numbers, but rather demonstrate the same thing visually (or otherwise, too ...).
       For each variation, Ording presents the proof -- generally on one page, though several extend across two or more -- and then provides an explanation and some commentary on the reverse side -- the one clever exception being 'Back of the Envelope' where, appropriately enough, only the variation-number (example 63) and name is presented recto, while the proof is scribbled, along with the commentary on it, verso.
       The variety is staggering, and extends far beyond the simply numeric. Ording offers, among much else: the 'Wordless' (diagrams and numbers), the unlikely 'Auditory' -- a musical composition scored for two violins (each 'playing' one side of the equation, the proof found at the points where they play the same note) --, a nice, full-color 'Chromatic' display (the spectra representing the two sides of the equation), a slightly tongue in cheek 'Hand Waving' proof (that is, unsurprisingly, not exactly among the most convincing), or a 'Psychedelic' illustration.
       Several of the proofs are variations on each other, including some that play directly off one another: the wordlessly illustrated 'Mystical'-proof is explained in his commentary, but the next proof also addresses it: as 'Refereed', it is an example of how material submitted to a professional journal is evaluated (though, as Ording admits, in this particular case it probably would never have made it past the editor). Or, for example, the lengthy proof from 'Antiquity' is then repeated, but with 'Marginalia' -- the common annotations that prove quite helpful. Some of the proofs, however, are and remain, essentially indecipherable -- even when elaborated on in a following one: 'Ancient' presents the proof in cuneiform, and even the succeeding proof, 'Interpreted' (in Hindu-Arabic numerals), probably doesn't get most readers much further. Still, that's still nothing compared to 'Paranoid', which simply: "contains all the letters used by another exercise in this book (which one ?) arranged in alphabetical order" -- yes, one of the proofs restated simply in alphabetical order (i.e. it begins with a long string of "a"s ...). (Other variations are similarly pared down to the symbol-essence but at least easier to make sense of -- 'Prefix' and 'Postfix', for example, though as Ording notes re. the latter: "The downside is pretty obvious -- can you find the typo ? I've corrected the ones I last found, but there always seem to be more").
       Essentially all the variations are also variations of expression, which highlights the many different ways in which a problem can be seen (as well as solved). These range from simple translation -- Ording presents his proof both in French and in German, or, for example, in sign language ('Another Interpreted') -- or, amusingly, puffed up as 'Jargon', but also more elaborately: there's the proof as 'Blog'-post, as tweet ('Social Media'), newspaper report ('Newsprint'), and even seven-page 'Screenplay'-scene (Ording noting in his commentary that: "The screenplay format here is based on guidelines published by the Academy of Motion Picture Arts and Sciences"). Several suggest the back and forth of setting out a proof, Ording cleverly offering both a traditional 'Dialogue' (between a master and a disciple), and, for when there's no one to bounce ideas off, the fall-back of 'Interior Monologue'.
       Ording also suggests approaches from the useful and common 'Open Collaborative' -- a back and forth with numerous participants, based on the Polymath Project -- to the more implausible (but still intriguing) 'Patented', in the form of a patent-overview of a process.
       Many of the proofs are far from straightforward, in a variety of ways, but it's interesting to note that among the less useful ones are 'purely mathematical' ones such 'Statistical' which, as Ording points out in his commentary: "achieves a very weak result -- it begins with an unproven assumption [...] and it ends with only an estimate of a solution". Others are surprisingly neat and precise -- 'Origami' astonishingly illustrates that all it takes is eight folds of paper to satisfy the proof. There are quite a few visual, geometric representations -- including a picture of 3-D model -- and Ording uses pictures and drawings in a variety of ways, as this is also a thoroughly illustrated volume. These go so far as the traditional tools of maths-teaching -- one proof is 'Blackboard' (yes, a photograph of a chalk-written proof on a blackboard), another includes a drawing of a graphing calculator (and instructions)
       Like Queneau's work, 99 Variations on a Proof shows there are an astonishing number of ways of telling and seeing, and it's a particularly useful exercise in suggesting how many different ways mathematical questions can be seen and addressed. Obviously, much of this is playful fantasizing -- in most instances, many of these proofs wouldn't be first (or tenth) choice in addressing the problem at hand -- but even the more far-fetched ones Ording presents can help shine a light on maths and how it is done. It is a good introduction into the workings of mathematics -- the field; or fields, actually, since it covers so many. Ording also writes with a keen historical awareness, as his examples cover a variety of times, and the approaches from these.
       Obviously, this is also somewhat of an 'insider' work, and mathematicians will likely get more out of it than those who aren't as familiar and comfortable with the field, but 99 Variations on a Proof is worth engaging with even for those who might stumble over some of the maths. Implicit throughout is also the reminder that awareness of alternatives (from perspectives to approaches) can open additional worlds, a lesson not only for professionals in this particular field but rather across the board.
       Well done, and accessibly presented, 99 Variations on a Proof may seem an odd little exercise, but proves to be one that's quite valuable -- and entertaining, as Ording shows a good sense of humor in it, too. - M.A.Orthofer
http://www.complete-review.com/reviews/maths/ording.htm