This is an unusual biography of a highly unusual man, the prodigiously gifted mathematician and professional eccentric John Horton Conway—creative scientist, teacher, showman, and cult figure. His third ex-wife told the author, Siobhan Roberts, that he was both “the most interesting person I have ever met” and “the most selfish, childlike person I have ever met,” and that she didn’t think she would marry again because John had “set the bar rather high.”
Writing about Conway’s work is difficult because it’s all over the map, including significant contributions to number theory, the theories of groups, knots, games, coding, and, more recently, to quantum mechanics. The arc of his story is determined not, as in many scientific lives, by the urge to ever deeper exploration of a few fundamental questions, but by a vow he made, at a critical time, to “stop worrying and feeling guilty” and permit himself to think about whatever pleased him. So, for example, he has devised a clever algorithm for calculating the day of the week on which any given date falls and regularly practices in order to use it as a parlor trick.
“Mathematicians in general don’t do . . . calculational tricks,” he says. “My colleagues in Princeton think it’s rather beneath them. They don’t think anything is beneath me.”
Thus the substance of what Conway does is inseparable from his outsized personality. Roberts, a science journalist, has therefore written an entertaining, often exhilarating, book that reads like a deeply researched magazine profile. (Its website naturally contains a blurb from Conway: “I couldn’t put it down!”) The book consists largely of anecdotes—often very funny, many presumably true—and extensive quotations from a very quotable man: “I have taste but I don’t exercise it very frequently. So I’m just as likely to be doing something that’s not worth doing as something that is.”
Every superhero needs an origin story, and Conway’s comes in two acts. As a schoolboy, he says, he was a shy, insecure outsider, the “math brain.” But when he arrived at Cambridge to begin university studies he realized that, since no one there knew him, he could start from scratch. So he willed himself to become an extrovert. (On first encountering Conway, I wondered: Is he really like that? Is it all an act? The answer, it turns out, is: Yes.)
The curtain went up on Act Two in his early thirties. After a workmanlike Ph.D. he held a teaching appointment at Cambridge but spent most of his time holding court in the mathematics department common room playing backgammon and inventing, other games. He was oppressed by the thought that he had done, and was doing, no work of importance. Enticed by a problem in a then‑hot area—the search for exotic objects called “sporadic groups”—he buckled down and quickly discovered the Conway group.
“Before,” he told Roberts, “everything I touched turned to nothing. Now I was Midas, and everything I touched turned to gold.” Newfound confidence encouraged a life spent following, insouciantly, wherever curiosity might lead.
At the same time, he also became a celebrity to a wider, nonprofessional public by inventing the Game of Life—though he is now dismayed by the possibility that it will be the only thing he’s remembered for. Life was the subject of the most popular article that Martin Gardner ever wrote for his famous Scientific American column, “Mathematical Games.” It is not a game with winners and losers but rather a set of rules for a cellular automaton. Imagine a checkerboard with a limitless number of squares. Mark the board by placing checkers on some of the squares; given any such marking, the rules of Life define the next marking, so that repeated application of the rules causes the board to evolve, step by step.
The rules of Life are simple: Whether a square is occupied in the next position depends on how many of its neighboring squares are occupied in the current one. A checker is removed from a square—having, metaphorically, died of loneliness or overcrowding—if it has too few or too many neighbors, and one is placed on an empty square if a
suitable number of neighbors are available to (metaphorically) procreate there.
Conway’s interest lay in finding simple rules that could produce complex behavior. For example, there is no general procedure for predicting how a marking might evolve (e.g., whether it will eventually die out, leaving an empty board). Some markings act like beings that reproduce by populating the board with copies of themselves; some can simulate fully general programmable computers. The Life craze began before the advent of personal computers, so addicts played by filching millions of dollars’ worth of computer time from their employers.
The reader will not learn a great deal of mathematics from this book (which is an observation, not a complaint). Its real concern is what moves Conway: philosophy, aesthetics, a hunger for knowledge (also for sex). He has a Platonist view of the mathematical world. The Conway group was not something he invented but something he found, waiting for an explorer to come upon it. He treasures simplicity because simple things are beautiful and can truly be understood. Hence his interest in Life; hence his attitude toward a celebrated theorem of his former student, Richard Borcherds, which proved the truth of a remarkable guess by Conway himself, the “Monstrous Moonshine conjecture.” (The showman likes whimsical names.)
Borcherds developed powerful abstract mathematical machinery to prove that the theorem was true; but from Conway’s point of view, the machinery was too complex to explain why. Conway also revels in the thinginess of the mathematical universe, the particularity of its inhabitants. The sporadic groups are so called because they are exceptions to a classification scheme for organizing the building blocks of group theory into a sort of periodic table. When Conway found his group, it wasn’t known whether there were only a finite number of these renegades. He hoped that they would never run out, each a remarkable one-of-a-kind. Sadly, they did.
Now, for some mild complaints. Never-explained technical terms crop up in sentences that will bewilder a nonmathematician. It is interesting and true, for example, that “Penrose tiles produce only nonperiodic tilings of the plane”—but “nonperiodic” and “tiling” are left undefined. There are some errors: “[Alan] Turing . . . showed that any Turing machine could be programmed to behave like every other Turing machine.” But what he showed is that some Turing machines could be so programmed. And I feel entitled to whine about an annoying orthographic tic: Words that name numbers are routinely (and inappropriately) replaced by numerals, as in “[he] had managed to infuriate 2 departments at once.” This is a book, not a tweet.
Siobhan Roberts is clearly fond, and in some awe, of her subject, but does not try to hide his shortcomings: “He’s high-maintenance, he’s generous. He’s emotional, he’s impassive. He’s a sweetheart, he’s an asshole.” She has produced a portrait of a lion in winter. Conway, now 77, has weathered two heart attacks, bypass surgery, two strokes, and a suicide attempt. Roberts frames her portrait with an account of a series of public lectures that Conway gave in 2009 on the free will theorem, a result in quantum mechanics proven jointly with the Princeton mathematician Simon Kochen. He is feeling his age and wonders, as the date approaches: Has he lost his mojo? Could he still dazzle an audience?
The free will theorem says, roughly, that if an experimenter can choose what measurements to make on a certain kind of physical system, and if that choice is not determined by the preceding history of the universe, then the results of the experiment are similarly undetermined. A twist in the argument shows that the unpredictability of the results cannot be accounted for by postulating some mechanism that selects randomly among the possible experimental outcomes. In Conway’s provocative formulation, if an experimenter has free will, so do elementary particles. Conway, an atheist, believes that we do have free will, and that the small freedom available at the atomic level could explain why we have it. The mathematics is not controversial, but the interpretation is.
When the first lecture began, the hall was full and people were sitting in the aisles; an overflow room was overflowing. All went well, putting Conway in an expansive mood: “I can be conceited again. I’ve got my groove back. I wondered if I was really capable of giving the blockbuster talk any more. . . . So last night was tremendously reassuring. My ego needs to be fed!”
Roberts compares Conway to the Hotspur of Henry IV Part I, quoting Samuel Johnson: “Inflated with ambition” and full of “turbulent desire,” Hotspur is “able to do much and eager to do more.” And Conway, it seems, is not done yet.
David Guaspari is a writer in Ithaca, N.Y.David Guaspari is a writer in Ithaca, N.Y.
