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Mathematics: The mercurial mathematician

Michael Harris relishes a biography of the playful, complicated group theorist John Horton Conway.

Genius At Play: The Curious Mind of John Horton Conway

Bloomsbury: 2015. 9781620405932 | ISBN: 978-1-6204-0593-2

If you want to read about what John Conway has done and why his peers shower him with superlatives — “most creative”, “best combinatorialist”, “one of the most eminent mathematicians of the century” — there are a number of popularizations. The marks he has left on mathematics are diverse and profound, but some of their depth can be grasped given curiosity and patience.

John Conway, seen in his office at Princeton University in New Jersey, has contributed to group theory, geometry, surreal numbers and combinatorial game theory. Credit: Dith Pran/The New York Times Redux/Eyevine

You should, however, read Siobhan Roberts's Genius at Play if you want to know what it feels like to be with Conway, and glimpse what it must feel like to be him. Roberts breathes more life into the stories of a living mathematician than I thought possible. “He's high-maintenance, he's generous. He's emotional, he's impassive. He's a sweetheart, he's an asshole,” she writes. In Conway, Roberts has found a personality neither tragic nor austere, like so many biographized mathematicians. He is loquacious, joyous and most of all, playful: as he said more than 30 years ago, “if you or your readers saw what I actually did, they'd be disgusted. They'd say, 'Good money is being paid out to support these people'.”

What does he do? My work tends to the abstract, so I know Conway mainly as a central player in the successful classification of finite simple groups, the elementary structures of symmetry. The ATLAS of Finite Groups (Clarendon, 1985) was a 12-year collective enterprise that aimed to record all the groups' “interesting properties”; run under Conway's guidance, it involved colleagues including Robert Curtis and Simon Norton. Conway is also famous for his 'Monstrous Moonshine Conjecture' with Norton, a bridging of two disparate fields — finite-group and complex-function theory — that was proved by Conway's student Richard Borcherds in 1992 (although not to Conway's satisfaction).

Conway made contributions to geometry, including work on sphere-packing, polytopes and knot theory; for surreal numbers, the largest possible extension of the real number line, which he constructed in the form of a game; and (with Simon Kochen) for the 2006 Free Will Theorem, which purports to prove that if humans have free will, then so do elementary particles. There is also Conway the combinatorial game theorist, often introduced — too often for his taste — as “best known for his invention of the 'Game of Life'”. This landmark in the history of cellular automata (and in Martin Gardner's Scientific American 'Mathematical Games') column is notoriously addictive.

Conway's most memorable contributions have the appeal of a good puzzle, even when not directly inspired by games. Roberts's “kaleidoscope of inquiry” is a marvel for its virtuoso juggling of narrative speeds, reminiscences, implausible digressions and long passages of precise, comprehensible mathematics. She packs it all into a tidy chronology framed by the story of a road movie starring Conway; she plays his amanuensis, occasional driver and “back channel” through which the world communicates with this most mercurial and untidy of mathematicians.

“Thou shalt stop worrying and feeling guilty; thou shalt do whatever thou pleasest.”

“I'm confused at some times,” Conway says. “In fact ... it's a permanent state.” He was speaking of mathematics, but his casual attitude to the mundane details of his personal history poses a challenge, even for a biographer as accomplished as Roberts. Conway encapsulates his philosophy of life (and work) as a “Vow”: “Thou shalt stop worrying and feeling guilty; thou shalt do whatever thou pleasest.”

There are glimpses of the abyss. As Conway attempts to explain the ATLAS to Roberts, he exclaims, “I know all the theorems. But there's still something that to me is unknown, unknowable ... It makes me sad that I'll probably never understand it.” Roberts shows us his private abysses: three marriages and three divorces, with hints of numerous affairs; two heart attacks, two strokes and a suicide attempt.

But Conway's playfulness surfaces and resurfaces. He notes that surreal numbers “is the thing I'm proudest of ... Because it pokes fun at people who do things in complicated ways.” And in research guidance to his students, he writes: “No no no no no! You're being far too reasonable.”

To see this motley of Conways squeezed into one outlandish personality is to want to join the chorus of his admirers. Roberts has masterfully untangled Conway's complexities. His ways of being in the world, in Roberts's telling, amount to a class of adjectives yet to be invented, to join his mathematical innovations.

In search of the best ways to talk about numbers, groups, shapes and games, Roberts has rediscovered the power of talking about the people who dedicate their lives to their study; and what an enjoyable discovery that is.

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Correspondence to Michael Harris.

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Harris, M. Mathematics: The mercurial mathematician. Nature 523, 406–407 (2015). https://doi.org/10.1038/523406a

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