99 Variations on a Proof

From the title, you might think that the book is showing different proofs of a crucial theorem that has been reinvented many times and hence has been proved in many different ways. The Pythagoras theorem could be a good candidate with many published proofs, but still it would be hard to find 99 different ways. No, the author wants to illustrate that there are many different styles of communicating mathematics, in particular of giving a proof of some theorem. What the theorem is, is not essential here. The choice of the author is to prove some totally unimportant statement: If $x^3-6x^2+11x-6=2x-2$, then $x=1$ or $x=4$. The cubic equation has indeed a double root at 1 and a simple root at 4.

The idea of the book is inspired by Exercises de style (1947) by Raymond Queneau, a member of the experimental writers group Oulipo (Ouvroir de littérature potentielle). What "styles" could one think of? There is certainly a tradition of doing mathematics that is inherited from historical mathematical centres in a country. There is definitely a difference between some papers written in a German or a French tradition. When looking at history, then proofs by Euclid or by Newton are certainly different and they are not at all like computer (assisted) proofs. Digging into history along this line of ideas will result in many proofs of the same statement looking very differently. One would come a long way, but still, 99 variations is a lot. Another source for differentiating is the way things are represented: graphical (geometric or other), colours, prefix or postfix notation, hand waving, oral discussion, e-mails, encrypted, sign language,... and there are some fun variations. In this way Ording arrives at 99 proofs. In fact, there is also a proof numbered 0, which has the statement of the problem with a proof omitted, which is indeed a way in which a lemma is often formulated in current mathematical papers. This is nicely represented by the cover of the book which has a ten by ten square grid of black squares, except for the first one which is white. The grid fills almost the whole cover which is possible since the shape of the book is almost square too.

Most of the proofs are short and take only a few lines. On the back of the page, Ording gives some comments, for example where he got his (historical) inspiration or how to read the proof (for geometric constructions) or explaining the notation, the symbols, or the language, etc. For example the psychedelic proof is just a black-and-white fractal plot of the attraction basins of Newton's method. Such a "proof" obviously requires some explanation. These backside notes give also cross references to related proofs in the book. Exceptionally a proof takes more than one page. The comments by Ording turn the book into a fragmented survey of many historical mathematical factoids. Even if some of the proofs are abstract and incomprehensible for the layman, it can still be considered a popular science book about the peculiarities and trivia of mathematics. We meet proofs as dialogues formatted after one of the oldest Chinese texts on mathematics, but also a dialogue as it would develop at a tea party in a mathematics institute, or a proof can take the form of a screenplay featuring Cardano and Tartaglia, etc. There is a proof in the form of a parody of the collaborative discussion modelled after the Polymath projects by Yitang Zhang and Terrence Tao to work on the twin prime conjecture. Fermi was famous for his proofs scribbled "on the back of an envelope" and that one is represented too with a hand written proof on the back of the page. There are proofs as given during an exam, or during a seminar, or on a blackboard, proofs discussed in a referee report, in a patent application, in blogs, in a newspaper article, preprints on arXiv, a tweet by Cardano (if he would have been able to twitter), ... Among the more surprising ones are the ones using a music score, colour spectra, origami, a slide rule,... In fact most of them are surprising and/or amusing.

The book can be safely considered as one produced by constrained writing as an oulipo author would produce it, and indeed as Queneau did in 1947, not for mathematical proofs, but for some surrealistic short story. The influence of constrained writing is for example obvious in a verbal proof using only monosyllabic words. There are many more hidden trouvailles like the proof called "Ancient" (in Babylonian cuneiform signs) as opposed to the one called "Modern" (with high level of abstraction). These are not placed next to each other but the first is numbered 16 and the second gets the opposite number 61. It must also have been great fun inventing parodies for styles like doggerel, paranoid, mystical, or social media. But it is more than just fooling around. There is always some rationale behind the way it is presented.

I love this book. It is so much fun to read, and there are many double layers to be discovered. It is temping to invent some extra variations of your own. It is so much more than a book about mathematics. It is indeed creative writing under constraints. I will pick it up regularly and I am sure there will be more hidden gems to be unravelled that are easily missed in a first reading.

Adhemar Bultheel
Book details

This is an exercise in using different mathematical styles to prove a theorem about the solution of a particular (otherwise not important) cubic equation. It is inspired by, and written in the style of, the constrained writing embraced by the French oulipo writers group, in particular the book Exercises de style (1947) by Raymond Queneau was a main source of inspiration.

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978-0-691-15883-9 (hbk)
24.95 USD