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B+ : fun, and very nicely done See our review for fuller assessment.
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99 Variations on a Proof is, unsurprisingly, inspired by Raymond Queneau's classic variationwork, Exercises in Style.
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: "algebrogeometric 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: In his Postscript Ording acknowledges (and finds it appropriate) that it's a: "degenerate cubic equation, with its awkward, nonstandard 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, 'OneLine', 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 variationnumber (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, fullcolor '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 HinduArabic 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 symbolessence 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 sevenpage '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 fallback 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 patentoverview 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 3D 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 mathsteaching  one proof is 'Blackboard' (yes, a photograph of a chalkwritten 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 farfetched 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  Return to top of the page  99 Variations on a Proof:
 Return to top of the page  Philip Ording teaches at Sarah Lawrence.  Return to top of the page 
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