# Modern Mathematicians: John Horton Conway

I have read many books about famous mathematicians, such as Pythagoras or Gauss, who lived many years ago, however I know little about modern mathematicians. So, I had the idea to start a series where I talk about those at the forefront of today’s research. To start, I choose John Horton Conway.

John Horton Conway had been described as possibly the most charismatic mathematician of the modern day. Having studied a wide array of topics throughout his career, it is hard to condense his achievements into a small blurb.

However, perhaps the three most significant are his contribution to group theory, his creation of surreal numbers and his invention of the Game of Life.

Contributions to Group Theory

In 1965, John Leech found a dense packing of spheres in 24 dimensions, which is now known as the Leech lattice. Conway was able to show that the symmetry group G of the Leech lattice was an undiscovered finite simple group of order 8,315,553,613,086,720,000, when factored by a central subgroup of order 2. To understand the magnitude of this discovery just remember that Conway was working in 24 dimensions, meaning that he wasn’t just working with simple (x,y) coordinates, but with 24-dimensional coordinates (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)!

Creation of Surreal Numbers

In 1969, Conway discovered the surreal numbers, showing how even the number systems are a continuously evolving topic. The real charm of this discovery is that Conway was actually not trying to develop number systems, but rather was analysing the game of Go. He noticed that at the end of a game appeared like the sum of a lot of smaller games. By spotting the similarity of this behaviour in games in the behaviour of numbers, surreal numbers were born.

The Game of Life

Outside the world of mathematics, Conway is best known for his invention of the Game of Life – a cellular automation. The Game of Life is a zero-player game, which means that the evolution of this ‘game’ is determined by the starting conditions and requires no further input. Hence, the ‘player’ plays the game by inputting the initial conditions and then seeing how it evolves. More ‘advanced players’ may observe patterns by setting specific initial conditions and properties.

However, surprisingly Conway almost regrets creating the game expressing:

“Well, because I am pretty egotistical. When I see a new mathematical book for a general audience, I turn to the index, I look for a certain name in the back, and if I see this name, it shines out at me somehow. And it says, page 157, pages 293-298, or whatever. So I eagerly turn to those pages, hoping to see some mention of my discoveries. I only ever see the Game of Life. I am not ashamed of it; it was a good game. It said things that needed to be said. But I’ve discovered so many more things, and that was, from a certain point of view, rather trite—to me anyway. It is a bit upsetting to be known for this thing that I consider in a way rather trivial. There are lots of other things to be discovered about surreal numbers. And the Free Will Theorem is recent, and therefore I am still flushed with enthusiasm about it.

(From an Interview of John H.Conway by Dierk Schleicher published by the AMS.)

What now?

More recently, Conway founded the Free Will Theorem jointly with Simon Kochen. This theorem states, “if there exist experimenters with (some) free will, then elementary particles also have (some) free will.” Put simply, if some experimenters are able to behave in a way that is not completely predetermined (independent of the past) then the behaviour of elementary particles is also not a function of their prior history.

Guardian Article

Numberphile Video

The New York Times Article

Let me know who your favourite modern mathematician is below! M x