John Conway
Genius Behind The Numbers

If you've never heard of John Horton Conway, you'd be forgiven. Unlike his famous counterpart John Nash, who received star treatment in the 2001 film A Beautiful Mind, Conway remains a mystery to many.

Despite his charisma, his talent for storytelling, and the fact that he’s made some of the most significant mathematical discoveries of the last fifty years, the eccentric professor of mathematics is as yet undiscovered by the general public though he’s long been revered by the maths/nerd community worldwide.
Conway is perhaps best known for inventing the cult classic cellular automaton called “Game of Life”—a grid of cells that transform according three simple rules of Conway’s devising (try entering “Conway’s Game of Life” into Google and watch what transpires – no really try it.) Life inspired the likes of musician Brian Eno and eco-warrior Stewart Brand. More than just a cool fad, It was applied broadly in the complexity sciences, modeling behaviors of things like ants and traffic, evolution and artificial intelligence, clouds and galaxies, and beyond. Indeed, Life provides an analogy for all of mathematics, and the entire universe.

Late last year author Siobhan Roberts released the biography Genius At Play: The Curious Mind of John Horton Conway. It took the better part of seven years to finish, because of Conway’s wayward and erratic behaviour. After finally giving in, Roberts had free reign to capture John Conway in all his charismatic glory. Legendary rumours of him dating Germaine Greer and almost killing Stephen Hawking made him all the more an intriguing subject. But away from all the fables, Roberts captures a man absorbed in a reverie taken by the brilliance of numbers. Whilst most of us go about our daily lives immune to this beauty,  Conway has spent a lifetime enraptured by it.

The name John Conway is very famous within the world of mathematics, but outside of that field much less so. John you’ve been described as “the greatest living genius unknown to the general public.” Do you feel perhaps that there’s an under appreciation of the work that you’ve done in your life?

JOHN CONWAY: The general public knows nothing of mathematicians, so if people know anything of me then that’s pretty good going. It’s interesting, the great mathematical biography is Sylvia Nasar’s biography of John Nash, who was my colleague here at Princeton. And Sylvia Nasar praises Siobhan’s book very much. In fact, as I go around Princeton here, people talk to me about the book. They always tell me, “I’ve read your book”, and I say, “No, it’s not my book.” None of them know how to pronounce Siobhan’s name. But they do say that she’s a wonderful writer, and she really is.

SIOBHAN ROBERTS: My aim was write a book that anyone and everyone can read, and get a glimmering of the ideas, and I think it’s a story that anyone can enjoy, even though it’s about mathematics. Being a writer focusing on math and having no mathematical background myself, I’m still surprised and dismayed by the level of math-phobia out there.

JOHN CONWAY: True mathaphobes won’t pick the book up. So that’s the end of that.

It’s been said “Mathematics is an effective tool to interpret reality. God gave us the integers, the rest came from man fiddling with them.” What do you make of that statement?

JOHN CONWAY: It’s a famous quote. I’ve heard it many times. Leopold Kronecker who made the statement was really talking about Cantor’s set theory. Cantor was a great German mathematician who discovered the infinite numbers. His work was very contentious initially, and Kronecker was really against it. So one should understand the quote in terms of that.
I have a way of talking about this kind of thing using cats, and this in a way does take up from the business about Cantor and infinite numbers and so on. Even with the finite numbers like ‘2’ and ‘3’ (and they’re my favourites really), the question is, What are they? Many people think they’re concepts. But if you think about what the word concept means, ‘2’ and ‘3’ aren’t concepts because they were around a long time before there was anyone to conceive of them. That sentence was true before there were any people to utter it. So it’s not entirely clear what ‘2’ is. It’s abstract.

If you take a cat and push it in some direction, let’s say you push it north, you’ll discover that it wants to move south. It pushes you back. It’s got a certain obstinacy. It doesn’t do what you want it to do. Neither does ‘2’ do what you want it to do. There’s no use trying to make 2 + 2 = 5. Numbers have a certain obstinacy, too. I don’t know what numbers are. They’re abstract. They’re not concepts. They really exist and the same is true of higher things.

I like your description of yourself as a philosophical mathematician. With that perhaps in mind, how do you reconcile such an experimental, abstract existence with the practicalities of day to day life?

JOHN CONWAY: I’m no good at the practicalities of day-to-day life. I don’t drive a car.

SIOBHAN ROBERTS: Right, and that might be the first in a long list of concrete examples that your wife Diana could enumerate for us.

JOHN CONWAY: I need a lot of minding. Siobhan minds me in a sense. So does my wife Diana. So do various other people.

SIOBHAN ROBERTS: I think at some point in John’s life he took the decision that he wasn’t going to be bothered with all these endless practical details. People have certainly speculated that this was his strategy, and it might explain his absentmindedly negligent ways.

JOHN CONWAY: Well you know there’s a well-known concept, the absentminded professor. I’m one of those. I walked around for about six weeks after discovering the surreal numbers in a sort of permanent daydream, in danger of being run over. I was just thinking all the time, “Nobody else in the world has ever seen this, nobody ever, and you’ve seen it John.” I thought about stout Cortez walking across the Americas. He saw the Atlantic Ocean, then he saw the Pacific Ocean, and the Americas in between. Nobody else had ever seen this greatness—well, American Indian tribes had doubtless seen it before. But I think Cortez must have felt something of the same way that I did, except what I’d seen was all in my head. It was an amazing thing, to see something that was too big to fit in this universe, because there are many more surreal numbers than there are particles in the universe. How can that possibly be?  There was another person in this respect, Georg Cantor, who discovered the transfinite numbers in the decade from 1870 to 1880, and he must have had similar feelings. Then he went mad. [Laughing] Maybe I’m going mad.

Cantor died in a lunatic asylum. In the eighties I went to see it, just before the Berlin wall came down. We walked around the asylum and like every building in East Germany it was falling to pieces. We looked through dust covered windows, and then we drove back to West Berlin and had some kaffee und kuchen. I was in a West Berlin train once that did stop at an East Berlin station by accident [when trains weren’t supposed to]. Something went wrong with the train and the doors automatically opened. Armed guards ran up and we all froze. After a time the doors closed and the train carried on. It’s very strange how this physical world, and politics and all of that craziness is going on at the same time as the permanence of mathematical things. So strange.

"I’ve never thought I was competing with anyone. I do my own thing. I don’t look at what other people do. The idea of competing with them is absurd, because there’s no competition with me. How can there be?"

John Conway on his peers

Taking you back to the late sixties at Cambridge, there was a twelve month period where you made three major discoveries: the Conway group in the field of mathematical symmetry, the very famous cellular automaton Game Of Life, and the surreal numbers, of which you’re proudest. How do you feel looking back on that period? Is it difficult to comprehend what you achieved?

JOHN CONWAY: Before that period, when I was in my late twenties, I had what I think of still as my black period. I had showed promise at an earlier stage in my life, and then I didn’t live up to that promise, and I wondered why not. Then suddenly the damn broke. It was quite amazing. When I discovered my big group, I really thought that another mathematician John Thompson (the best group theorist in the world) should have discovered it. I tried to persuade Thompson because I thought it was his job more than mine. And every now and then I’d ask him, “Have you thought about it?” “No,” he would say, he hadn’t. Eventually I said, “You’re not going to think about it are you?” And he admitted as much. I said, “Well what do I have to do to persuade you to think about it?” And he said, “Well you find, just how big, how symmetrical this thing is, and then I’ll listen to you.”

Then I found this wonderful, supremely interesting, fantastically symmetric thing. That changed my life, utterly. I was invited to speak about it all over the world. I flew to New York, gave a twenty minute talk, and flew back home again. My life, from being in this black period, suddenly changed to being one of an international jet-setting mathematician. Then I discovered various other things. Everything I touched suddenly turned to gold.

Outside of the academic environment, what can mathematics do for the real world? What kind of implications does your work and the work of your contemporaries have in terms of, for example the economic crises we face, or the many other insurmountable issues?

JOHN CONWAY: Forgive me for saying so, but that’s not my problem. There are many other people much better able to cope with those issues than I am. With mathematics, for me the problem is the thing, not its application. It sounds sort of selfish, but that’s what I’m good at, and historically mathematicians have always thought things before the applications come along. I am good at what I’m good at, and I’m good at teaching what I’m good at. So that’s my answer, in a way a rejection of your question.

What I loved about the book was how genuinely playful and riveting it was. Something I found very interesting was this sense that the narrative was leading up to something bigger. You were building almost to a point of enlightenment. 

SIOBHAN ROBERTS: I think that’s what attracted me to the prospect of writing about John’s life in the first place… The challenge of this book was to piece all the tales together, fit it together like a puzzle that somehow came together to form a full picture. […] It’s simply about the pursuit of knowledge and following one’s curiosity, and satisfaction and reward in doing that. John bills himself at once as a “know-it-all”—he wants to know absolutely everything, that is his goal in life—and as a “professional non-understander,” since on most subjects he starts from a place of not knowing. But the nature of his pursuit is very meandering, following one tangent after another in his playfully circuitous way, and the book follows John along on this journey. In life we rarely take the time to just explore, without knowing where we’ll get by day’s end. So I think if there is a takeaway, it’s that there’s value in doing more of that.

There’s an interesting line in the book, “mathematics is wedged uneasily between art and science and within the discipline there are artists and there are scientists. Conway is an artist, Thompson is a scientist.” Could you elaborate on that Siobhan?

SIOBHAN ROBERTS: In mathematics, there’s the more rational side and objective side of things, and the obstinate as John calls it. And that’s what you expect in mathematics. But the there’s also the creative side, the more imaginative and subjective, and that isn’t so well known. Mathematics straddles these two worlds. There’s the famous C.P Snow quote about the two cultures of art and science and that they exist in isolation. He was lamenting this and making a case for their integration. Writing the book, I encountered both artists and scientists. Having said that, certainly there’s overlap—there’s a creatively-minded artist in John Thompson, and there’s a rationality-minded scientist in John Conway.

JOHN CONWAY: I don’t like the word rationality. And there’s one word you said, wedged uneasily, why uneasily?

SIOBHAN ROBERTS: I don’t know. Maybe just wedged? Wedged unexpectedly?

JOHN CONWAY: Yes it’s true that mathematics is wedged I think between art and the sciences. I’m there for the aesthetic reasons, but then I think most mathematicians are. I make my mathematical name by studying this Leech lattice, a way of packing spheres in 24-dimensional space. I’ve never seen anything in 24-dimensional space. The closest I’ve come is I’ve seen a little bit of 4-dimensional space. We don’t live in anything higher than three dimensions but there are beautiful things going on, and when there are beautiful things going on I’d like to be in the front seat watching.

Siobhan, from the book I felt that there was a strong sense of competitiveness and ego in the maths world. In the research that you did, and conversations that you had with people, did you get a sense of a lot of people trying to outdo each other in the field?

SIOBHAN ROBERTS: The competitiveness does exist under the surface and there are occasional example of it coming to the fore more prominently. There were characters I encountered, Stephen Wolfram for one, for whom I think there was a bit of an unacknowledged competitive tension with John.

JOHN CONWAY: I think he acknowledges it actually. I can’t quite remember what he says in the book, but he thought I was competing with him. I’ve never thought I was competing with anyone. I do my own thing. I don’t look at what other people do. The idea of competing with them is absurd, because there’s no competition with me. How can there be? That’s a very egotistical statement, but let it stand.

"I found this wonderful, supremely interesting, fantastically symmetric thing. That changed my life, utterly. I was invited to speak about it all over the world."

John Conway on discovering The Conway Group

If we could turn to the future for just a moment, I don’t know if you might reject this question as well, but there was a moment in the book, in the chapter on the Game of Life, where Siobhan you talk about artificial intelligence. I’m interested to know John if you have an opinion of where maths and science and technology is leading us and the implications science might have for us in the future?

JOHN CONWAY: It’s very difficult to know in which particular ways things will change. A long time ago I was very interested in computers, before computers even got off the ground. And I’ve been astonished at just how the endeavours to make computers faster turns out to have been such a good idea. My attitude many years ago would be, I don’t really see the point. I’m very much a live-and-let-live person. The fact that I may not be interested in something doesn’t mean that someone else shouldn’t be interested, or that I disapprove at all. But I have my own interests.

Finally, I’m fascinated by something you’ve been exploring with your college Simon Kochen called the Free Will Theorem. Could you tell us a bit about that work?

JOHN CONWAY: Philosophers have been arguing about free will for ages and ages, two thousand years or more. And they’ve never come to any conclusion as to whether we actually have free will or not. I don’t expect much will change there, but with the Free Will Theorem we did answer another question about free will. We proved, subject to various known things, that if we humans do have free will, then in a certain sense so do elementary particles. As I say, it’s not the question people were asking about free will, but it’s the question we can answer, and have answered. I better stop there because it’s rather tough to explain. It takes a long time and it’s also very easy to misunderstand.

SIOBHAN ROBERTS: The thing I find intriguing about the Free Will Theorem is that you and Simon are still grappling with it. It’s this ineluctable thing. The theorem has a precise technical formulation, but you’re still trying to grapple with the details.

JOHN CONWAY: It’s funny that you used this word ineluctable, a very strange word really. It means you can’t struggle out of it. When Simon and I discovered this thing we weren’t trying to understand the nature of free will. We were just trying to understand how the world works. That perhaps seems a rather bigger problem. Anyway, we were standing near the blackboards in Simon’s office and suddenly the scales seemed to have fallen from our eyes in some way. We didn’t quite realise what we’d proved, but we realised we’d proved something. And then I said, “We’ve proved that if we have free will then so do particles.” And then Simon thought for a time and said, “I suppose you could put it like that.” That wasn’t what we’d hoped to prove, but it’s something that I think is a very deep and significant thing. But ineluctability is the right word. We can’t grab it, we can’t wrestle it into the shape we want, it’s just there. There’s something about free will that’s a nebulous concept. It’s surprising that you can prove anything about it on a mathematical level. But we did.

Genius At Play: The Curious Mind of John Horton Conway | Siobhan Roberts

1. WITH COMPUTER: Kelvin Brodie, The Sun News Syndication
2. IN OFFICE: Dith Pran, New York Times, Redux Pictures
3. SHAPES 1: John Horton Conway
4. OFFICE: John Horton Conway Circa 1969

A Comprehensive Dictionary of the English Language, but Ernest G. Klein. It’s a wonderful etymological dictionary. I lost it somewhere in my apartment, and then it turned up the other day.

It’s a big symmetry group, which I hope to understand before I die

He’s the mathematician I’d most like to talk to.