99 Variations on a Proof

by Philip Ording

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.

Genres: Mathematics, Science, Nonfiction, Philosophy, Art

272 pages, Hardcover

First published February 5, 2019

Reviews

[Markus · Rating 5 out of 5]

Ist Mathematik auch Literatur? Ich denke wohl! Mathematik basiert auf Sprache und verfügt über Grammatik, Syntax, Form und Stil. Ihre Ausdrücke und Berechnungen können schön sein, elegant oder witzig, manchmal verstörend und nicht selten unverständlich.

Raymond Queneau hat in seinen berühmten Stilübungen ein und dieselbe banale Geschichte in 99 verschiedenen Stilen erzählt: einfach, metaphorisch, logisch, altmodisch, distinguiert, förmlich usw... Philip Ording ließ sich von Queneau inspirieren und hat 99 Beweise für diese kubische Gleichung zusammengetragen:

Beweise: Falls x³ - 6x² + 11x - 6 = 2x - 2, dann ist x = 1 oder x = 4

Das Ergebnis ist verblüffend und zeigt, dass Mathematik keine objektive oder absolute Universalsprache ist, für die man sie oft hält, sondern eine Kulturtechnik und damit ein Konstrukt, das abhängig von Zeitgeist und Mode einem andauernden Wandel unterworfen ist.



Zufällig hat Ording seine kubische Gleichung in einem Buch aus dem Jahr 1545 gefunden, in der "Ars Magna" des berühmten Universalgelehrten Girolamo Cardano (siehe unterstrichene Passage; interessant, dass Cardano nur die 4 als vera aestimatio - die richtige Lösung bezeichnet!). Ich war wirklich überrascht und einigermaßen entsetzt, in welcher Form man sich damals mit solchen Gleichungen herumschlagen musste (p: und m: stehen für Addition bzw. Subtraktion, Tpq̃d für tertia pars numeri quadratorum, ein Drittel des Koeffizienten vor dem quadratischen Term).

Die 99 Beispiele bestehen jeweils aus dem Beweis selbst und einer zugehörigen Erklärung. Form und Stil reichen von bekannten Varianten wie arithmetisch, geometrisch, zahlentheoretisch über niedlich, dentritisch, mechanisch, experimentell oder psychedelisch bis zu sehr originellen Einfällen wie Blog, Drehbuch oder Gebärdensprache. Die Beweise dürften selbst (oder gerade) für hartgesottene Mathematiker einige Überraschungen bieten. Mein Lieblingsbeweis ist "26. Nach Gehör" und wird in Notenschrift präsentiert.



Natürlich wollte ich wissen, wie sich das anhört und hab das schnell in Töne gesetzt - hitverdächtig ist es nicht, aber immerhin ein Beweis:
26 Nach Gehör.mp3

Das Buch ist auch als Buch sehr schön gemacht, es ist quadratisch, hat sehr dicke Deckel und ist im Inneren liebevoll gestaltet. Philip Ording hat sich eine Riesenarbeit angetan, alles ist gründlich recherchiert, sehr fundiert und die Kommentare sind voll mit hochinteressanten Quellen und Referenzen, die schon für sich eine Enzyklopädie der mathematischen Form von Babylon bis heute abgäben und zu weiterführender Lektüre anregen.
Das Buch muss nicht linear gelesen werden, vielmehr animiert es dazu, es irgendwo aufzuschlagen und kreuz und quer herum zu schmökern. Man muss auch kein studierter Mathematiker sein, es reicht ein mittleres Niveau und natürlich Interesse. Trotzdem ich einzelne Beispiele nicht ganz verstehe, habe ich große Freude mit dem Buch und werde noch viel Zeit damit verbringen.

Als Appetitanreger noch ein paar meiner Favoriten, besonders der Bierdeckel hat es mir angetan:

[Randy · Rating 5 out of 5]

This might book one of the strangest math books I own. It shows 99 different proofs for a degenerate cubic polynomial - which already sounds salacious. It's point is to show that there is Style in mathematical proofs. It is also very inspiring. There is not just one way to do things, there are at least 99 ways of doing some (most) things. Some are stupid, and some are surprising and some you almost can't read. The book was inspired by an examination in literary style, and an attempt to apply the same idea to the literature of mathematical proofs. It succeeds wildly and is an inspirational book that I will flip open to a random page to remind myself that sometimes the solution to a problem doesn't have to come in the form you expect.

Surprisingly witty, and highly recommended.

[Yathish Dhavala · Rating 3 out of 5]

It's a fun read. The 99 proof variations are actually the different ways of presenting solutions to a cubic equation of no particular significance. Except that it's a degenerate cubic. Some of the proofs I liked were Wordless, Model, Origami, Clever, Geometric and Word Problem. But perhaps the best is the one named Omitted with Condescension -

There is a simply beautiful theorem which provides all solutions of the equation x^3− 6x^2+ 11x−6=2x− 2. Alas, any further explanation would deny you the satisfaction of discovering it on your own...

Having said that many of the proofs are rather dry. They are inserted ostensibly to reach the number 99 which was probably a choice made for obvious reasons. Variations like blog format, or just showing the same proof in LaTeX as a separate variation is rather dull.
Each variation of the proof is only one page long followed by a one page commentary. The commenataries are a treasure trove of trvia on how different mathematical styles were developed and the context in which they developed.

[Caroline · Rating 5 out of 5]

This is a clever project by a mathematician positing the many different ways one can present a cubic equation for solution and then ways to solve it. The author comments on each one on a separate page following the example. I am not a mathematician and I did not try to fully understand the solutions. I was fascinated by his exploration of mathematical history and different styles of mathematical writing (did you ever even conceive of styles of mathematical writing?), and also of the possible diagrammatic ways of expressing the problem and proving it. At the end he raises the issue of drug-assisted thinking and includes a psychedelic illustration. There are many fascinating international references from antiquity to the current day.

Two entries that especially peaked my curiosity were 74, illustrating the limitations of American Sign Language when it comes to mathematics, and 75, which uses a slide rule. I inherited my great uncle mathematician Edward Kasner's slide rule and have always wondered how it might be used. Now I will follow the text and perhaps find out.

[Sarra Tebib · Rating 4 out of 5]

Well if you love maths ..
If you don't, stay away
I learned a lot !

[Finnton Wentworth · Rating 5 out of 5]

Humorous but not at the expense of making a well thought point. A reminder that mathematics is something beyond formalized notation and obtuse theorems. Some of the proofs worked much better than others and discussions on why mathematics is taught and performed the way it is would have been good includes.

[Wing · Rating 4 out of 5]

Gosh, what a book. I can only understand half of it. By trade I am only a physician. I suppose you need to be a professional mathematician to fully appreciate it. It shows that there are many ways of dealing with a single mathematical problem. The author uses a cubic equation to illustrate the point. It covers areas well beyond simple algebra. Number theory, group theory, calculus, analysis, probability, geometry, etc. – you name it – are discussed. In doing so, he shows, if you think about it, how mesmerising the subject mathematics really is. I will no doubt go back to those sections that I don't understand after consulting basic texts and re-read them more intently. Four stars – despite of some of its impenetrability.

[Arav Agarwal · Rating 4 out of 5]

In a lot of ways, this book is not really what it seems. Presented as “99” proofs on the solution to a non trivial cubic, this book more echoes postmodernist literature than anything else, focusing on the stylistic presentation more than anything else. What makes this worth reading, however, is the sheer thoroughness of the author in presenting many ways to view the same simple problem, from topology to analysis to even an electrostatic solution. These solutions, coupled with a long sources list, seem to suggest this book is more of a guide to interesting mathematics than a book on proof writing, something I highly prefer.

[Alex · Rating 2 out of 5]

Возможно, эта книга не попала в моё настроение, но мне кажется, что наблюдения автора можно (и стоило) изложить в более систематичной форме. Если же одной из целей книги действительно было дать читателям представление о разных областях математики (и физики), надеюсь, что для кого-то она оказалась полезной!

[Alex · Rating 5 out of 5]

Brilliant! I can’t say I get it all, but the sense of wonder, and the creative way of teaching and discussing mathematics is really, really wonderful

[Daval Davis · Rating 3 out of 5]

Some of the proofs were very interesting. Some were really just minor stylistic differences in presentation. And some flew right over my under-educated head.

[Joe · Rating 5 out of 5]

I wish I was this creative - and good at math.

[r]

"My motivation for this project, from beginning to end, has been to try to conceptualize mathematics as a literary or aesthetic medium."

"In literary terms, we might identify substitution with 'metaphor,' as the historian and philosopher of mathematics Reviel Netz does. 'Mathematics,' he says, 'can only become truly interesting and original when it involves the operation of seeing something as something else.'"

"Like any other kind of writing, it is easier to write mathematics badly than it is to write it well."

"Proofs that borrow from a domain of knowledge perceived to be distant from the context of the theorem are deemed ‘impure’ by proof theorists. They can appear simpler than their purer counterparts, though it's not clear how precisely to evaluate observations such as this."

"Some people can't help but see mathematics in color. Physicist Richard Feynman reported, 'When I see equations, I see the letters in colors—I don't know why. As I'm talking, I see vague pictures of Bessel functions from Jahnke and Emde's book, with light-tan j's, slightly violet-bluish n's, and dark brown x's flying around. And I wonder what the hell it must look like to the students.'"

[Chandler · Rating 5 out of 5]

to be fair, this is not 99 different proofs of one theorem (and even less is it 99 variations on one proof), but "99 different approaches to proving a theorem" just isn't as snappy a title! I also want to come clean and say that i skimmed most of the highly geometric ones ... i don't follow written geometry so well, i'm afraid!

but it's a remarkable demonstration of just how varied mathematical style can be -- and a good refutation to anyone who thinks the study of polynomials has to be, pardon the pun, "monotonic"

[Hubert · Rating 5 out of 5]

Amazing! Brilliantly conceived, a cornucopia of various ways of doing mathematics, drawing upon historical and contemporary references. More than simply "99 ways to solve" a cubic equation, it's an exploration on how to think mathematically.

This has got to be one of the coolest books that has passed through my hands. I didn't get to delve into all the variations of a proof on a line by line basis, but I got a lot out of it, and a bigger and greater understanding on the multiple facets on how math works.

[dead letter office · Rating 3 out of 5]

This is a hard one to rate. It's a little niche. Some of the proofs amount to inside jokes, some of them pretty clever. If you're a mathematician you'll probably enjoy it? Especially if you also like literature? It does get across that there is real style to mathematical writing, that there are lots of ways to do and communicate mathematics, and that writing mathematics really is an art.

[Charles Eliot · Rating 5 out of 5]

99 Variations on a Proof will not be for everybody, but if you love the quixotic beauty and charm of mathematics, it's a winner. One postulate, 99 different proofs, each illustrating a facet of the mathematical world. Sheer joy for the math nerd.

[Adam · Rating 3 out of 5]

A fun curiosity, which was educational and enjoyable. Following Ordring's recommendations, I mostly skimmed the proofs but enjoyed reading his explanations, and focusing on a few that seemed interesting.

[Ben Haines]

interesting variations. fun anecdotes on math and writing. it's a good framework, would be interested in applications to other topics.

[Jon · Rating 4 out of 5]

Charming, interesting, an art book with a mathematical theme rather than a purely mathematical book.

[Pia · Rating 5 out of 5]

gosh I miss math

[Bonnie · Rating 5 out of 5]

For MathHum: 3, 6, 7, 26, 48, 52, 62.