“Mathematical Science,” wrote Ada Lovelace in contemplating the nature of the imagination, “is the language of the unseen relations between things.” Few have mastered that language and transmuted it into Lovelace’s “poetical science” more deftly than the trailblazing English mathematician John Horton Conway, best known for the invention of the 1960s cellular automaton Game of Life.

A fine addition to the best science books of the year, Genius at Play: The Curious Mind of John Horton Conway (public library) by Siobhan Roberts is noteworthy for many reasons, among them the non-negligible fact of being a biography of a living subject — a task generally self-defeating (Susan Sontag famously proclaimed that “no biography makes sense that isn’t written after its subject is dead”) and rarely approached with as much tenacious graciousness as Roberts’s. That the subject is a man of enormous complexity and contradiction, with Richard Feynman’s charisma and Slavoj Žižek’s contrarian edge, only adds to the feat.

Alongside her intense intellectual admiration for Conway’s genius, Roberts becomes a keen observer of the elemental human psychology that bedevils even a mind as superhuman as his. She describes her subject as an “insecure egotist” — a redundant phrase absolutely perfect in its redundancy, for it’s hard to think of an egotist who isn’t at bottom insecure, brimming with what psychoanalysts call “narcissistic vulnerability.”

This paradoxical orientation of self to world comes into play in Conway’s conflicted attitude toward the biography itself. Roberts writes:

He very much cares what other people think, and he worries that a self-portrait might come off as too egotistical. And partly because he’d have a hard time with “the fiction of humility that the conventional autobiographer must at every moment struggle to maintain,” as the occasional biographer Janet Malcolm describes the dilemma. So he’ll stick to doing what he does best. Gnawing on his left index finger with his chipped old British teeth, temporal veins bulging and brow pensively squinched beneath the day before yesterday’s hair, Conway unapologetically whiles away his hours tinkering and thinkering — which is to say he’s ruminating, or maybe he is doing some work, but he’ll insist he’s doing nothing, being lazy, playing games.

For his part, Conway is rather precise about the particular allure of this tinkering and thinkering. Reflecting on the governing rule of Subprime Fibs, one of the innumerable games he invented, he tells Roberts:

I’ll tell you what interests me about this — it’s really what interests me about mathematics. Nobody else in the whole history of the world has been stupid enough to invent this rule. That’s the first thing. But then, if they had, they would find exactly this behavior that I’m finding.

[…]

That’s a curious thing about the nature of mathematical existence. This rule hasn’t physically existed in any sense in the world before a month ago, before I invented it, but it sort of intellectually existed forever. There is this abstract world which in some strange sense has existed throughout eternity.

Imagine an uninhabited planet, full of interesting things. You land on it, and it existed for a million years, but no people have ever been there, no sentient beings. There are such places, I’m sure. Go to some remote star and there will be something. But you don’t have to go there. You can sit in this very chair and find something that has existed throughout all of eternity and be the first person to explore it.

Conway arrives at this bewitching intersection of discovery and invention by being at once a naturalist of numbers, an algebraic adventurer, and an unflinching empiricist. Roberts captures the singular spirit of his endeavor:

He turns numbers over, upside down, and inside out, observing how they behave. Why is it that when you pick a number, any number, then double it, add 6, halve it, and take away the number you started with, your answer is always 3? Above all he loves knowledge, and he seeks to know everything about the universe. Conway’s charisma lies in his desire to share his incurable lust for learning, to spread the contagion and the romance. He is dogged and undaunted in explaining the inexplicable, and even when the inexplicable remains so, he leaves his audience elevated, fortified by the failed attempt and feeling somehow in cahoots, privy to the inside dope, satisfied at having flirted with a glimmer of understanding. For his own part, he calls himself a professional nonunderstander. The pursuit is what counts…

This notion, of course, is as central to genius in the sciences as it is in the arts — something Grace Paley articulated beautifully in her advice to aspiring writers. Conway himself examines the cognitive machinery of this essential disposition of nonunderstanding:

In a fundamental way my job is thinking. You can’t see it from the outside. What does the thinking consist of? I think about how to explain whatever I am thinking about to someone. Then I explain it to someone and it doesn’t work. So I think about it some more. I tinker with it, with thinking, until I’ve simplified it. I personally can only understand things after I’ve thought about them for ages and made them very, very simple.

In a sentiment that calls to mind I, Pencil — that brilliant 1958 allegory of the division of knowledge, illustrating how everything is connected — Conway adds:

Most people just understand enough to work. For example, a mechanic doesn’t necessarily understand the physics or engineering of how a car works. I’m not putting down a car mechanic. We need practical people. I’m not sure we need theoretical people. Though I’m not going to campaign for my own abolishment.

Genius at Play is a tremendous read in its totality. Complement it with this wonderful picture-book biography of the eccentric mathematician Paul Erdős, a close collaborator of Conway’s, then revisit John Dewey on how we think.