On Numbers and Games — John H Conway

On Numbers and GamesThis amazing book sets out a mathematical framework for describing and constructing numbers, and then generalises this to a way of analysing certain games. You probably need a postgraduate degree in mathematics to really understand all of it. I am not quite that qualified, but I know enough to be awed by what can be done by simply starting with nothing. Literally nothing: Conway starts with just an empty set, and proceeds to show how to spin this out into integers, rational numbers, real numbers, infinite numbers, infinitesimal numbers, and (believe it or not) more.

The whole class of numbers he constructs is called the Surreal numbers, and that’s a fair evocation of the way this book stretches my mind. I can generally understand the steps in the various derivations, but it takes some time and effort to develop the intuition to really understand the later constructions and to see where they are heading. Conway obviously had this intuition: after working on the ideas for some years, he wrote this book in just a week.

You know this is a classic work because it’s often referred to by its acronym,¬†ONAG. That puts it at the same rarefied level as TAOCP, (The Art of Computer Programming) the bible of computer science, written by the superhuman Donald Knuth. In fact, it was Knuth who came up with the name Surreal numbers, in his¬†amusing novel(!) about them called Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. That book introduced the Surreal numbers to the world, but ONAG really dives into them, generalising the Surreal numbers to represent positions in a class of games. He shows how to fully analyse the game of Nim (much beloved of mathematicians and game theorists), and notes that similar ideas have been used to analyse the game of Go. Now that I would love to know about, but I think I would need another lifetime to build up enough background knowledge to understand it.

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