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Poincaré’s Prize by George G. Szpiro

David Dillard-Wright is philosophical about mathematics.

The mere mention of mathematics leads many people to roll their eyes and drift off to sleep. The idea that a history of mathematics could be, of all things, a fun read, might sound like a contradiction in terms. Oddly enough, George G. Szpiro has accomplished exactly that in Poincaré’s Prize: The Hundred Year Quest to Solve One of Math’s Greatest Puzzles. Leibniz said that “Music is the pleasure the human mind experiences from counting without being aware that it is counting.” Szpiro’s book ultimately succeeds because he entertains so well that the counting recedes into the background here too. He can give the non-mathematician a general idea of incredibly complex topics in topology – the study of the properties of shapes as they are pulled, stretched, and squeezed – without subjecting them to equations understood only by PhD mathematicians. Szpiro also moves readers along as he introduces his cast of characters – the quirky geniuses on both sides of the Atlantic who flung themselves at one of math’s greatest problems, which concerns proving whether a shape defined in an obscure topological way is actually a sphere.

Most people probably have no idea that a subject like topology even exists: I didn’t until I read Poincaré’s Prize. But as far as I understand it, topology involves, among other things, the classification of shapes of different dimensions according to the number of holes that run through their centers. Hence, doughnuts, bagels, hula hoops and tyres, all having one hole through the center, are all topologically equivalent structures; while figure-eights and pretzels, having, respectively, two and three holes, belong to different classes. Objects can also be specified in higher dimensions. A circle exists in two dimensions, and a cylinder in three; but topology also runs into the fourth, fifth, and greater dimensions. Topological study also involves the routes that can be traced on the contours of shapes – the way a piece of string could be said to travel as it traces the outside ‘skin’ of the object. This wrapping can be expressed as numbers and equations which subsequently describe the objects under study.

If your mind is spinning, a few examples are in order. A sphere is topologically equivalent to a cube, because either object can be pulled, pushed, and stretched to morph into the other without adding extra holes. Similarly, the ‘skin’ of any three-dimensional object can be expressed as a two dimensional surface. While we weaker minds cannot fathom the higher dimensions, Szpiro’s exposition allows neophytes to at least appreciate the possibility of (someone else) thinking in higher dimensions. I always came away from reading Poincaré’s Prize feeling like my mind had been pulled and stretched in much the same way as an imaginary coffee cup is stretched into an imaginary bagel. Perhaps most impressive is the fact that Szpiro does all of this analysis without a chalkboard – although it might have helped to have a few diagrams. I had to Google shapes like ‘the Klein bottle’ and ‘the Whitehead link’.

If the math won’t keep the story going, the characters certainly do. Consider the hero of the book, Grigory ‘Grisha’ Perelman, the Saint Petersburg recluse who, working in poverty and isolation, solved math’s greatest puzzle and posted the answer on the internet for all to see. Shunning honor and publicity, he declined to accept the famous Fields Medal, the mathematics equivalent of the Nobel Prize. Then there is University of Wisconsin topologist RH Bing, whose initials were actually his first name – apparently a naming predilection common to Bing’s native Texas. Upon accepting the job at Wisconsin, he was asked what should be put on the nameplate. “Loyal to his given name, he said, ‘R only, H only, Bing.’ When he arrived at his new office, he found the nameplate next to the door with the inscription ‘Ronly Honly Bing.’”

I can’t resist one more of Szpiro’s anecdotes about RH Bing. While driving a carful of mathematicians from Madison to Chicago during a sleet storm, Bing got so involved in explaining a particular proof that the windows of the car fogged over to near-zero visibility. When Bing started reaching for the dashboard, his passengers assumed with some relief that he was clearing the condensation. To their horror, Bing started demonstrating the steps of his proof on the windshield with his index finger.

Anecdotes like these fill Szpiro’s pages and make the book a welcome read. Alfred North Whitehead said, “Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happiness.” Szpiro allows readers to glimpse over his shoulder on a journey through the ‘divine madness’ of mathematical genius, taking them on a mind-bending journey through the social and intellectual world of heroes little-known outside the field. So, the next time you need to ‘evert’ a sphere (turn it inside out without cutting it), pick up Poincaré’s Prize. The higher dimensions explored by professional mathematicians may not resemble their science fiction counterparts, but it is certainly good to know that for some, these quasi-spaces are very real and demonstrable. Who knows, by reading this book you may even want to – gasp! – take a math class.

© David B. Dillard-Wright 2009

David Dillard-Wright is Assistant Professor of Philosophy at the University of South Carolina, Aiken.

Poincaré’s Prize: The Hundred-Year Quest to Solve One of Math’s Greatest Puzzles by George G. Szpiro, New York: Plume, 2008, 309 pps. pb, £17.99, ISBN: 978-0525950240

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