Genius at Play: The Curious Mind of John Horton Conway

I would guess that most mathematicians have a John Conway story that they could tell. Many of us have had the experience of going to a talk that Conway is scheduled to give and having him let the audience vote on the topic of the talk just as it begins. Others have been to a talk where Conway is in the audience and bickers with the speaker about some detail of a definition or example they are presenting. Some people will bring up Conway’s groundbreaking work on finite group theory, and in particular his work as the primary author of The Atlas of Finite Groups. Much to Conway’s chagrin, there is probably a larger number who immediately start talking about The Game Of Life, which is probably Conway’s most famous invention and yet one that he is tired of talking about. For still others, their memories of Conway are dominated by long nights in hotel bars, playing Dots and Boxes with him or having him calculate what day of the week July 4, 1776 was on. In any event, Conway is one of the closest things that mathematics has to a rock star, full of energy and eccentricities as well as an amount of sheer talent that make people around him incredibly jealous.

Even as I am writing those sentences, it seems to me that Conway’s personality and talents almost defy description, and yet Siobhan Roberts has attempted to capture them in Genius At Play: The Curious Mind of John Horton Conway. Very early in the book, she acknowledges the difficulty of the task she has set for herself in trying to contain Conway’s story, so she gives in and lets Conway’s own voice interrupt the flow of the book with the equivalent of a DVD commentary track. When Roberts gave a talk about Conway at the recent MOVES Conference at the Museum of Mathematics in New York City, those of us in the audience got to see the dynamic come to life as Conway several times chimed in by yelling “I never said that!” or “That’s not quite how it happened!”

Roberts first encountered Conway when she was doing research for her biography of Donald Coxeter, and found him to be, in her words, “the rare man inclined to forthright and global disclosure.” She approached Conway about writing his biography, and he stubbornly refused. But after he had a stroke in 2006 he had a change of heart and they started working together on the book. In the ensuing decade Roberts has spent many hours traveling with Conway, trying to go through the papers (and other detritus) in his office, and interviewing people close to Conway. She has managed to capture his personality on paper as well as one could imagine doing.

In addition to being indisputably a mathematical genius, Conway has had his share of issues in his personal life, with several marriages and infidelities and heart attacks and even a suicide attempt. In her book, Roberts grapples with the difficulties of writing a biography of someone who is still alive and whom she has gotten to know quite well. As she writes in her introduction:

For the most part with the biography he was cooperative, ingratiating, ever willing to talk — except when secondary sources produced an irresistibly salacious anecdote, or worse, telling discrepencies, puzzling difficulties in deciphering fact from fiction, true from false in the towers of memory. At these moments … he’d shoot me his death stare and say “Oh, hell. You’re not going to put that in the book. Are you?!?”

Roberts doesn’t ignore the more troublesome parts of Conway’s life, but neither does she dwell on them or fill the book with gossipy details. Reading the book, it is clear why Conway is someone around whom many people are compelled to spend time and whom many people find abrasive and difficult to be around — and that the two sets have a large intersection.

Another thing that Roberts does not choose to ignore is the mathematics itself. Just as one could write a biography of a musician without including MP3 samples of their work, one can certainly imagine a biography of Conway that focusses on his personal and professional lives without describing his work explicitly, but this is not that biography. While she is not writing a mathematics textbook and anyone who truly wants to learn the subjects will find better places to look, Roberts does not shy away from technical descriptions of surreal numbers and sphere packing and the theory of finite groups, not to mention describing many of the games and puzzles that Conway is probably more well-known for at this point. (Yes, this includes a long description of the Game of Life, but don’t tell Conway I mentioned that one.) In fact, the first appendix of the book gives a reprint of Conway’s wonderful proof of Morley’s Trisector Theorem. Roberts gives enough of these mathematical details so that the reader, whether they are a mathematician or not, can get a sense of what Conway is fascinated by and why it motivates him, and I expect that many readers of Roberts’s book will move on to read some of Conway’s many books, such as The Sensual (Quadratic) Form or The Book of Numbers.

Many biographers make an attempt at turning their subject’s life into a single cohesive narrative, where one event cleanly flows into another and there are overarching themes that drive the full story. Certainly one can appreciate the desire to do so (especially if there is hope of selling the movie rights to the book), and it occasionally works, but much more often it doesn’t. Most lives are not a single narrative but are instead a series of interconnected anecdotes, which often contradict each other in their lessons and themes. Genius At Play embraces this idea, and while Roberts loosely structures the book around a series of lectures that Conway is giving about his work on The Free Will Theorem, the narrative tends to jump around through time and focus on smaller stories about Conway’s experiences.

And what great stories they are! Whether it is working with Elwyn Berlekamp and Richard Guy on the Winning Ways books, playing games with high school students at math camps and graduate students at Princeton, or trying to understand the representations of the Monster group, Roberts (and Conway) always have another great story to tell. This might not make for the most cohesive biography one could imagine, but it makes for a book that is a lot of fun to read and that provides some insight into one of the great minds of our time.

Darren Glass is an Associate Professor of Mathematics at Gettysburg College and can be reached at dglass@gettysburg.edu. His first exposure to John Conway’s mathematics was when a high school teacher showed him The Game of Life, but he sure isn’t going to tell this to Conway. In the spirit of full disclosure, that high school teacher went on to write a book manuscript with Conway that will hopefully see the light of day eventually. Steve Sigur is listed in the glossary of the book under review, right next to Brooke Shields.