Mind Games

The inventive powers of the imagination infuse the work of a famed mathematician.

Interviewed for Scientific American about Genius at Play: The Curious Mind of John Horton Conway, the biography of his life authored by Canadian journalist Siobhan Roberts, the renowned mathematician reminisces about his encounter with the fabled French mathematician Nicolas Bourbaki. This would have been a meeting of the minds. Bourbaki had been behind the “New Math” movement that briefly swept the American school system in the wake of the Sputnik crisis. Meant to bring the American mathematical curriculum on par with the Soviet one, the movement introduced school children to Boolean mathematics, set theory and other high-end mathematical concepts.

Bourbaki’s abstract approach to mathematics must have appalled Conway who, for all his brilliance, often made little mistakes of great consequences, such as inverting the plus and minus signs. He once poured his cup over the coffee spill he had just made because he was thinking that more, not less, coffee was needed on the floor. Unsurprisingly, Conway refuses to get bogged down in mathematical proofs. Against those who, like Bourbaki, emphasized the elegance of set theoretic derivations, Conway offers brain teasers and games as the surest way to interest bright minds in ­mathematics.

Games have made Conway famous—so much so that he is one of the few living mathematicians to have gained popular fame. If you had a computer in the 1970s or early ’80s, you probably remember his “Game of Life,” a so-called cellular automaton that simply consists of a grid of square cells that are all attributed one of two initial states, alive or dead. The grid is then left to “evolve” according to simple basic rules such as “death by overpopulation,” stating that any live cell surrounded by more than three live neighbours should die, or a “reproduction” rule that brings to life any dead cell surrounded by more than three live cells. It takes but a few iterations to realize that complex, stable patterns can emerge from a random distribution of live cells. For Conway, the Game of Life was interesting because it proved to be a universal Turing Machine—that is, a system that can compute anything computable. To the biologists, economists, physicists and philosophers who appropriated it, the game proved a fascinating example showing that complexity and organization can spontaneously evolve from simple deterministic systems.

The Game of Life has somewhat eclipsed Conway’s other achievements in group theory, knot theory, number theory and combinatorial theory. And it is not for him to fail to integrate play in the rest of his work. After spending countless hours watching a British champion play Go, Conway realized that the end of any game of Go was nothing but a set of smaller games. It turned out that these end games are similar to the end games found in other games, such as Hackenbush, where two players successively take away the coloured lines composing a childish drawing. These sets of games, Conway soon understood, are nothing but numbers. From his study of Hackenbush, Conway recuperated not only both the real numbers, represented by the points on a line, as well as all of Cantor’s infinite numbers, but all the numbers, including a new kind of numbers: those existing between the two closest points on a line and the ones existing between infinite numbers. In a child’s game, Conway had found the so-called surreal numbers.

So a meeting between the set theoretician Bourbaki and the eclectic Conway would have been explosive if it had ever taken place. But the fact is that Nicolas Bourbaki never existed. It was the pseudonymous name chosen by a group of primarily French mathematicians who came together to rigorously redefine mathematics in terms of set theory. For that matter, the John Horton Conway presented in the book’s pages does not really exist either, which makes Genius at Play one of the oddest, most enticing biographies I have ever read. The biography of someone who is just a figment of the imagination—but what an imagination—that of John Horton Conway himself.

For there is a man named John Horton Conway. Born in Liverpool on December 26, 1937, the young John could recite the powers of two at age four and, by the time he was 15, could readily calculate the day of the week corresponding to any date in history. Even as a child, Conway seems to have had but one ambition: to become a mathematician. But his childhood, marked by bullying, was difficult. When he finally boarded the train for Cambridge University where he had been accepted on a scholarship, Conway decided to recreate himself as a confident, rule-ignoring genius with unkempt hair who walked the streets of the university town with a helmet that enabled him to see the world as a four-dimensional object and who constructed computers out of flush toilet parts.

In her lively and engaging prose, Siobhan Roberts, who has already won the Euler Prize for her biography of the geometrician Donald Coxeter, takes us from Conway’s student days (when he was recognized as a brilliant mathematician despite quite ordinary exam results) to his time teaching at Cambridge (where he avoided his messy office by playing games with students in the common room), to his move to Princeton University (where he attempted to find a connection between the indeterminable behaviour of quantum systems and human free will). If Roberts had hoped that the years of careful research she dedicated to Genius at Play would allow her a glimpse into the thoughts of the “real” John Conway, she must have been bitterly disappointed. As Conway neither kept a diary nor preserved his correspondence, nor has archived his work, she had to rely on his confessions as her main source of information, and Conway has worked too long on his public persona to let any of us into his private life. Genius at Play reveals little about the motivations behind his philandering, his divorces, his relationships with his children, his suicide attempts or his refusal to pay attention to mundane tasks. As he readily admits to Roberts: “I’ve been trying to remember things as they were but it’s very hard. Sometimes I remember the story better than the facts.”

But what stories. A generous biographer, Roberts gives Conway the space he needs for his tales. Thanks in part to long verbatim quotes, which are set in their own font, we are able to experience Conway’s charisma and love of mathematics on a personal level. Roberts’s research and interviews with Conway’s family and colleagues then add some much needed perspective without a prying eye. Roberts obviously understood from the very beginning that Conway would never collaborate on a book that went behind his public persona and she made the conscious decision to act as an uncritical spokesperson. This was ultimately the right decision. As shocking as the thoughts, actions or feelings of the private Conway may be, they simply would not have created the same contagious enthusiasm for mathematics and games as the public Conway has been able to convey, and this excitement for mathematics and games is the reason for Conway’s loyal following. So consider yourself warned. Even if you think you do not like mathematics, Genius at Play may very well convince you to spend countless hours doing nothing but playing backgammon, solving Sudoku puzzles or, like Conway, creating your own Game of Life.