99 Variations on a Proof

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99 Variations on a Proof

by
Philip Ording

general information | review summaries | our review | links | about the author

Title:	99 Variations on a Proof
Author:	Philip Ording
Genre:	Non-fiction
Written:	2019
Length:	238 pages
Availability:	99 Variations on a Proof - US
	99 Variations on a Proof - UK
	99 Variations on a Proof - Canada

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Our Assessment:

B+ : fun, and very nicely done

See our review for fuller assessment.

Review Summaries
Source	Rating	Date	Reviewer
FAZ	.	28/6/2019	Dietmar Dath
The NY Rev. of Books	.	13/1/2022	Dan Rockmore

From the Reviews:

"Das Spektrum der Beweistechniken im Buch reicht von sehr konkreten, mechanischen Verfahren mit Balken und Gewichten (Beweis Nr. 87) bis zum ultratrockenen Algorithmus (Beweis Nr. 27). Dass Nichtmathematiker dieses Buch verstehen können, ist dabei keineswegs unwahrscheinlich; es vermeidet nämlich Fachdialekte und ist stattdessen in Idiomen geschrieben, die man fast auf der ganzen Welt versteht (Englisch, Musikalisch, Humoristisch), vor allem aber durchgängig in einer Sprache, die sogar in beliebig vielen, ohne diese Sprache aber unzugänglichen, nämlich völlig fremdartigen, dafür jedoch denkbaren Welten gilt (Mathematisch)." - Dietmar Dath, Frankfurter Allgemeine Zeitung

"The entries range from the technical to the whimsical, from real proofs to more meditative reflections on the process and activity of mathematical research. But Ording’s book is more than just a survey. Each proof is accompanied by a brief commentary, outlining the inspiration of the variation and reflecting on the culture of mathematics through the ages. (...) 99 Variations on a Proof is not really meant to be a history lesson, even if it is by necessity partly that." - Dan Rockmore, The New York Review of Books

Please note that these ratings solely represent the complete review's biased interpretation and subjective opinion of the actual reviews and do not claim to accurately reflect or represent the views of the reviewers. Similarly the illustrative quotes chosen here are merely those the complete review subjectively believes represent the tenor and judgment of the review as a whole. We acknowledge (and remind and warn you) that they may, in fact, be entirely unrepresentative of the actual reviews by any other measure.

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The complete review's Review:

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, 26 February 2019

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Links:

99 Variations on a Proof:

Princeton University Press publicity page

Reviews:

Philip Ording:

Faculty page

Other books of interest under review:

Raymond Queneau's Exercises in Style
Matt Madden's 99 Ways to Tell a Story
See Index of books dealing with Mathematics
See Index of Oulipo books

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About the Author:

Philip Ording teaches at Sarah Lawrence.

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