Genius at Play: The Curious Mind of John Horton Conway

During a visit to the Institute in the 1970s, the mathematician John Horton Conway, then of Cambridge, spent the ten most interesting minutes of his life. Invited to deliver a talk to the undergraduate math club at Princeton, Conway made his way across town and wangled himself a private audience with the God of logic, Kurt Gödel.

Conway had recently enjoyed his self-proclaimed annus mirabilis: In a period of twelve months in and around 1969 (which he usually rounds up to 1970), he invented the cellular automaton Game of Life, he discovered the 24-dimensional symmetrical entity named the Conway group, and while playing around with trivial children’s games, he happened upon the aptly named surreal numbers. While Conway might be most popular among the masses for Life and its cult following, and while he might be most highly regarded among mathematicians for his big group, Conway himself is proudest of his surreal numbers. The surreals are a souped-up continuum of numbers including all the merely real numbers—integers, fractions, and irrationals such as π—and then going above and beyond and below and within, gathering in all the infinites and infinitesimals; the surreals are the largest possible extension of the real number line. And deferring to Scientific American columnist Martin Gardner’s reliable assessment, the surreals are “infinite classes of weird numbers never before seen by man. They provide a secure foundation on which Conway … carefully builds a vast and fantastic edifice.”

But an edifice of what? Conway wrote a paper on the surreals titled “All Numbers, Great and Small,” and he concluded by asking, “Is the whole structure of any use?” The Hungarian American mathematician Paul Halmos reckoned, “It is on the boundary between funny stuff and serious mathematics. Conway realizes it won’t be considered great, but he might still try to convince you that it is.”

Halmos was mistaken, on one point at least, and quite possibly two. Conway believes the surreals are great. There’s no “might” about it. If anything, he is keenly disappointed that the surreals haven’t yet led to something greater. And he had good reason to hope. Based on his readings of Gödel’s work, he thought the surreals might crack Cantor’s continuum hypothesis—the hypothesis proposed by Cantor speculating on the possible sizes of infinite sets, stating that there is no infinity between the countable infinity of the integers and the uncountable infinity of the real numbers. Gödel and Paul Cohen (the latter achieving a result in 1963 that Conway called the “work of an alien being”) collectively showed the hypothesis to be “probably unsolvable,” at least according to the prevailing axioms of set theory, leaving the door ajar a sliver.

To travel back in time a bit, Gödel and his wife Adele had fled Nazi Vienna and landed in Princeton in 1940. Gödel became good friends with Einstein, working on a theory of relativity that entailed a nonexpanding “rotating universe” wherein time travel was in fact a physical reality. Gödel also did his part regarding the continuum hypothesis while at the Institute. And even after having proved the impossibility of a disproof, the issue with the infinities nagged at him. In 1947, he published a paper, “What Is Cantor’s Continuum Problem?” in the American Mathematical Monthly. He tried to provide an answer, first with some reinterpretive questions. “Cantor’s continuum problem is simply the question: How many points are there on a straight line in Euclidean space?”

Conway had read this and later papers by Gödel numerous times, before discovering the surreals and after. What struck Conway during these readings was Gödel’s assertion—the Surprising Assertion, as Conway came to call it—that a solution to the continuum hypothesis might yet be possible, if only once the correct theory of infinitesimals had been found. Conway couldn’t help but wonder: With the surreals, he believed that he had found at least a correct theory of ­the infinitesimals (and he still believes so). He wouldn’t go as far as to say it was the correct theory, not before eliciting Gödel’s opinion. During his visit to Princeton in the 1970s, Conway got the chance to ask the great man himself.

Conway would never have been so daring as to simply ring Gödel and request an appointment. The meeting came about via their mutual friend Stanley Tennenbaum, a mathematician and logician who for a time lived alone in the woods in New England, but he did the rounds through Montreal, Chicago, New York, and Princeton, the last being a regular pit stop for the purpose of talking to Gödel. “Stan was a sort of pet or protégé of Gödel’s,” recounted Conway. “So he had the ins to Gödel, and he said, ‘If you like, I’ll introduce you to God’—that’s what he always called him. So anyway, Tennenbaum offered: ‘Would you like to be introduced to God?’ I said, ‘Yes, of course.’ You don’t turn an invitation like that down.”

Writing my biography of Conway (Genius At Play:The Curious Mind of John Horton Conway published last July) over several years and as many visits to the Institute from 2007–14, I found the date of Conway’s meeting with Gödel impossible to pinpoint. The year had to be less than 1978, when Gödel died, and greater than 1970, the year Conway found the surreals. Probably also less than 1976, after which Gödel was in very poor health and rarely left his home, and greater than or equal to 1972, when Conway spent the spring term at Caltech. Somewhere therein, Conway had his visit with Gödel.

But never mind what year it was. I had other unanswerable questions as well. Where did the meeting ­transpire? In Fuld Hall at teatime, or in Gödel’s office just off the mathematics library? What was Gödel like? Did he look well? I pestered Conway with these fiddly detail questions, for which he had no answers. And all my badgering made Conway, fellow of infinite jest that he is, laughingly wonder whether he’d met the great Gödel at all.

This gave me pause. Conway had already proved himself to be an unreliable narrator of his own life—and worse, an accomplished fictioneer, whenever the mood struck. I was both impressed and perplexed by his derring-do. Here I was attempting a finely drawn portrait of my subject, and against my best intentions and best efforts the biography seemed to be going a bit off the rails. Mingling with my betters at the Institute—where the world’s best scholars delve deep into the past, the history of humanity, the evolution of the universe—I was ever answering the question from people as to how one writes about a living subject. And indeed, having Conway looking over my shoulder inevitably made his vital signs a liability, mostly for him. I realized this over lunch with Heinrich von Staden, an authority on ancient science, Professor Emeritus in the School of Historical Studies. He told me about the Greek and Roman tradition of vivisection, making public spectacle of strapping a live pig to a plank and cutting him open and observing the mechanics of his beating heart. A fitting metaphor, I realized, for what this experience was like for Conway.

Doing my part to contribute to the Institute community, I presented an After Hours Conversation on my predicament, on the elusive nature of biographical truth. Most talks in this neighborhood of the intellectual firmament were more rarefied. One scholar, the fabulously monikered Aristotle Socrates, an astrophysicist, spoke on “Solar Systems Unlike Our Own.” Conway was, by comparison, high comedy, in an orbit all his own—prankish, belligerent, hijacking the process. As Exhibit A, I presented to my audience a caricature of Conway done by his friend Simon J. Fraser at the University of Toronto. The sketch was inscribed, “In homage to a diabolical mathematician,” and as such it depicted Conway with hooves for feet and a topological entity called a “horned sphere” growing from his head—and to wit, more generally speaking a horned sphere is known as a “pathological example,” an entity that is counterintuitive and ill-behaved, much like my subject himself.

Irving Lavin, an esteemed art historian and Professor Emeritus in the School of Historical Studies, took an interest in the drawing and offered his assistance in deciphering what it said about my subject’s antics. Lavin observed that Conway was in good company among artists who matched creativity with promiscuity, intellectual and/or interpersonal—Picasso, for example. Perhaps Conway’s seeming inability to distinguish fact from fiction correlated to his uncanny ability to see mathematics differently and to achieve his idiosyncratically original results. Commenting on the caricature itself, Lavin rummaged around for relevant ­references and pointed to the seventeenth-century Italian artist Gian Lorenzo Bernini as an early ancestor of artists doing exaggerated comical drawings with massive heads to malign or poke fun at their subjects. Lavin thought the caricature vividly captured Conway as rapscallion. “Very cunning!” he said. Hear, hear—cunning: showing dexterity in artfully achieving one’s ends by deceit, evasion, or trickery.

But all joking aside. Conway had met Gödel, really and truly. And luckily, I found some proof pointing in this direction archived among the Institute’s Gödel papers. In a file labeled “discussion notes” for 1974, there was a list detailing Gödel’s roving discussions with Tennenbaum, touching on everything from politics to mathematics—from Nixon, McGovern, hippies protesting the middle class, drug addicts, Vietnam, riots, and the decay of the United States, to Cohen and Dedekind, Coxeter and modern geometry, Nash and games, Chomsky and the “linguistic aspect of math ed”—I found a single word that looked like “Conway,” then an eminently legible “Game of Life.” This goes some distance, at least, as confirmation: Tennenbaum commended one friend to another, and they set up ­­a date. When pressed for details of the meeting, Conway dug around in his memory bank and supposed they talked about some generally logical things while he worked up the nerve to ask Gödel about his Surprising Assertion.

“So I had, it can’t have been much more than ten minutes with him,” recalled Conway. “Between five minutes and half an hour, because it didn’t seem to go on very long. But it might have actually just been because I wanted more. Anyway, whatever it was, I hesitantly asked him: Had he heard of the surreal numbers? And he had. And I asked him about the Surprising Assertion he’d made. I said to him that I thought I’d discovered the correct theory of infinitesimals. And he agreed. And I said, ‘Well, what about your idea that we would learn more about the continuum hypothesis?’ And he said, ‘If I said that, I was wrong. Yes, you may very well have discovered the correct theory of infinitesimals, but it’s not going to do anything for us.’ I wonder what exactly his words were. The words I remember are ‘I was wrong.’ And I do remember the feeling of disappointment. And by the way, that seemed right to me. I never understood what he meant by the Surprising Assertion, what was in his mind. I think it was probably just a passing idea that he had without any real support for it. But I’m happy to have met the great man, even if it was only for a short interval.”

Those ten minutes, give or take, count as the ten most interesting minutes of Conway’s life—even if his theory of the infinites and the infinitesimals was left bereft of greater application.

A little more than twenty years later, Conway was installed as John von Neumann Distinguished Professor in Applied and Computational Mathematics at Princeton (where he’s been ever since). The university communications office sent out a glossy press release, and the president, Bill Bowen, in announcing the hire, praised Conway into hyperspace. He was a “multifaceted phenomenon . . . one of the most eminent mathematicians of the century.”

Conway bathed in the limelight, eager to woo the masses, the students, and his fellow colleagues. “Conway is a seducer, the seducer,” said his Princeton colleague Peter Sarnak—speaking exclusively of Conway’s skills as a teacher, of course. In time, Conway became the department’s prize attraction, holding forth in the common room, usually doing nothing but piddling away his days playing more games. There he engaged Sarnak, who arrived at Princeton in 1991, in a viciously aggressive (if ostensibly playful) competition with a spinning toy called a Levitron. When Sarnak proved the superior levitator, Conway banned the Levitron from the premises. A gifted expositor, Conway taught at public lectures and private parties. And during a math department party at Sarnak’s house, Conway pulled out his best parlor trick and performed it all night in the kitchen, mostly for women. The come-on, still attempted now and then, Conway always relishes recounting: “I can make U.S. pennies land the way you want for the rest of your life!” “He was the center of the party,” recalled Sarnak, who in 2007 was cross-appointed Professor in the School of Mathematics at the Institute.

Conway is his own party, and he’s always at the center. But Sarnak also holds Conway in high regard for his profound contributions to the mathematical oeuvre, especially the surreals. “The surreal numbers will be applied,” assured Sarnak. “It’s just a question of how and when.”