John Horton Conway: The Man Who Made Life
There is a photograph taken at Cambridge sometime in the early 1970s. John Conway is on the floor of the common room, surrounded by other mathematicians, demonstrating something with Go stones on a board. He is laughing. Everyone is laughing. The Go board is not being used for Go.
This was Conway at his natural habitat: on the floor, in the middle of a crowd, doing mathematics by playing. It was not performance — it was method. Conway thought by doing, by moving pieces, by making games. And one of the games he made changed the world.
A Childhood in Liverpool
John Horton Conway was born on December 26, 1937, in Liverpool, England. He was the son of Cyril Conway, a chemistry laboratory assistant, and Agnes Boyce. He was, by his own account, a mathematical prodigy — he could recite the powers of two before he had fully learned to read, and he told his mother at the age of eleven that he wanted to be a mathematician at Cambridge.
He got there. He studied at Gonville & Caius College, Cambridge, earning his bachelor’s degree in 1959 and his doctorate in 1964, working in number theory under the supervision of Harold Davenport. He then spent his entire career at Cambridge until 1987, when he moved to Princeton — a move he would later describe as a mistake he could not undo, though he stayed for the rest of his life.
The Playful Mathematician
Conway’s colleagues describe him using words that rarely appear in mathematical biographies: charismatic, theatrical, compulsive, generous, difficult. He was, by many accounts, one of the most entertaining people in any room he entered — and he was almost always in the common room, available to anyone who wanted to think with him.
He juggled. He solved Rubik’s cubes in under two minutes. He invented games — real games, with boards and pieces and rules — as a way of thinking about mathematical structure. He carried bags of coins, toys, and physical puzzles everywhere. He was known to lecture while lying on his back on the lecture room floor.
None of this was distraction. It was how he worked. Mathematical objects, for Conway, were not abstract entities to be manipulated symbolically. They were things you could touch and move and see. His mathematics was fundamentally spatial and tactile.
What He Actually Cared About
Conway is best known for the Game of Life, but he regarded it as a minor work — an accident, really. He was more proud of:
Surreal numbers. In 1969, analyzing the structure of the Go game Go, Conway invented a new number system that encompassed all real numbers, all ordinal numbers, and vast infinities beyond them — and discovered that this system had been lurking implicitly inside combinatorial game theory all along. The surreal numbers are now a standard object of study in mathematics.
The Conway groups. He discovered three previously unknown symmetry groups — now called Co₁, Co₂, and Co₃ — in 1968. These are among the most complex finite symmetry structures known to mathematics.
Monstrous moonshine. With Simon Norton in 1979, Conway noticed a bizarre numerical coincidence between the dimensions of the Monster group (the largest sporadic finite simple group) and the coefficients of the modular j-function. This observation, which Conway named “monstrous moonshine,” seemed absurd — the two objects appeared to have nothing to do with each other. It took Richard Borcherds a decade to prove the connection, earning him the Fields Medal in 1998.
Knot theory, combinatorial game theory, the look-and-say sequence, the doomsday algorithm for computing the day of the week for any date — these are all Conway’s.
He was, in the assessment of his Princeton colleague Simon Kochen, “the most creative and most unusual mathematician of his generation.”
The Game of Life: An Accident of Attention
The Game of Life began, as many of Conway’s ideas did, as a problem someone else had posed.
John von Neumann, in the 1940s, had designed a two-dimensional cellular automaton that could self-replicate. It worked — but it was fiendishly complicated, requiring 29 states per cell and a neighborhood of 5 cells. Von Neumann wanted a proof of principle, not an elegant one. Conway wanted to know: could you do the same thing — compute universally, potentially self-replicate — with a much simpler rule?
He spent the late 1960s experimenting with rules on a Go board in the Cambridge common room. He had a set of criteria: the rules should not obviously grow forever, should not obviously die out, and should allow for behavior complex enough to be unpredictable. He tried hundreds of rules. Most were boring — they either exploded or collapsed immediately. A few were interesting.
The rules he settled on — birth on exactly 3 neighbors, survival on 2 or 3 — were found by trial and observation, not by derivation. There is no equation that produces them. They were found empirically, by a man sitting on a floor pushing stones around a board.
Martin Gardner and the Explosion
In October 1970, Conway described his game in a letter to Martin Gardner, who was writing the Mathematical Games column in Scientific American. Gardner recognized immediately what he had: a game that anyone with graph paper and a pencil could play, that was deep enough to absorb a professional mathematician, and that touched on the most important ideas in computation and philosophy.
Gardner’s column ran in October 1970. The response was unlike anything Gardner had seen in twenty years of writing the column. Readers wrote in with patterns. Readers computed behaviors by hand for dozens of generations. Professors devoted their courses to it. Bill Gosper at MIT set his team to work on it and discovered the glider gun within months, winning Conway’s $50 prize and proving that infinite growth was possible.
Conway himself was slightly overwhelmed. He had not expected this.
Princeton Years
Conway moved to Princeton in 1987, joining the Institute for Advanced Study and later the Princeton mathematics department. He was a wildly popular lecturer — known for bringing bags of props to class, for asking students to lie on the floor and arrange themselves as configurations, for turning abstract mathematics into something you could perform.
He was also, by his own admission, not easy to live with. He was married three times. He was absent in ways that his family found difficult. He had periods of depression and what he called “mathematician’s remorse” — a sense that his best work was behind him.
In 2009, he suffered a stroke. He recovered, but the recovery was slow and imperfect. He continued working, continued performing, continued being present in the department.
He died on April 11, 2020, at the age of eighty-two, from complications of COVID-19. It was three weeks after Princeton had sent its students home for the pandemic.
What He Left Behind
Conway’s mathematical legacy is measured, in part, by the depth and breadth of the problems his work opened up. The surreal numbers are now a mature theory. Monstrous moonshine led to string theory and vertex operator algebras. The Game of Life spawned the entire field of artificial life and inspired a generation of computational biologists, physicists, and complexity theorists.
But there is another legacy, harder to measure: he demonstrated that mathematics could be done joyfully. That the difference between a game and a theorem is smaller than it looks. That a mathematician could be funny, warm, available, and human — not in spite of being rigorous, but as a natural expression of genuine curiosity.
“The most important thing in mathematics,” he said once, in a recording that has circulated widely since his death, “is to play. You have to play. You have to be willing to not know what you’re doing.”
He spent his life not quite knowing what he was doing, and the results were extraordinary.