siobhan_roberts's picture
Director’s Visitor, Institute for Advanced Study, Princeton; Author, Genius at Play and King of Infinite Space
Surreal Numbers

Merriam-Webster’s 2016 word of the year is surreal: “It’s a relatively new word in English, and derives from surrealism, the artistic movement of the early 1900s that attempted to depict the unconscious mind in dreamlike ways as ‘above’ or ‘beyond’ reality. Surreal itself dates to the 1930s, and was first defined in a Merriam-Webster dictionary in 1967. Surreal is often looked up spontaneously in moments of both tragedy and surprise…”

One of the lesser-known applications of the word belongs to the Princeton mathematician John Horton Conway who discovered surreal numbers circa 1969. To this day, he wishes more people knew about the surreals, in hopes that the right person might put them to greater use.

Conway happened upon surreal numbers—an elegant generalization and vast expansion of the real numbers—while analysing games, primarily the game Go, a popular pastime at math departments. The numbers fell out of the games, so to speak, as a means of classifying the moves made by each player and determining who seemed to be winning and by how much. As Conway later described it, the surreals are “best thought of as the most natural collection of numbers that includes both the usual real numbers (which I shall suppose you know) and the infinite ordinal numbers discovered by Georg Cantor.” Originally, Conway called his new number scheme simply capital “N” Numbers, since he felt that they were so natural, and such a natural replacement for all previously known numbers.

For instance, there are some familiar and finite surreal numbers, such as two, minus two, one half, minus one half, etcetera. Cantor’s transfinite “omega” is a surreal number, too, as is the square root of omega, omega squared, omega squared plus one, and so on. The surreals go above and beyond and below and within the reals, slicing off ever-larger infinites and ever-smaller infinitesimals.

They are not called surreal for nothing.

Over the years, the scheme won distinguished converts. Most notably, in 1973 the Stanford computer scientist Donald Knuth spent a week sequestered in an Oslo hotel room in order to write a novella that introduced the concept to the wider world—Surreal Numbers, a love story, in the form of a dialogue between Alice and Bill (now in its twenty-first printing; it was Knuth, in fact, who gave these numbers their name, which Conway adopted, publishing his own expository account, On Numbers and Games, in 1976). Knuth views the surreal numbers as simpler than the reals. He considers the scenario roughly analogous to Euclidean and non-Euclidean geometry; he wonders what the repercussions would be if the surreals had come into existence first.

The Princeton mathematician and physicist Martin Kruskal spent about thirty years investigating the promising utility of surreal numbers. Specifically, he thought that surreals might help in quantum field theory, such as when asymptotic functions veer off the graph. As Kruskal once said: “The usual numbers are very familiar, but at root they have a very complicated structure. Surreals are in every logical, mathematical and aesthetic sense better.”

In 1996, Jacob Lurie won the top prize Westinghouse Science Talent Search, for his project on the computability of surreal numbers. The New York Times reported the news and ran a Q&A with Lurie. “Q: How long have you been working on this? A: It’s not clear when I started or when I finished, but at least for now, I’m finished. All the questions that have yet to be answered are too hard.”

And that’s pretty much where things stand today. Conway still gets interrogated about the surreals on a fairly regular basis—most recently by some post-docs at a holiday party. He repeated for them what he’s said for ages now: Of all his work, he is proudest of the surreals, but they are also a great disappointment since they remain so isolated from other areas of mathematics and science. Per Merriam-Webster, Conway’s response is a combination of awe and dismay. The seemingly infinite potential for the surreals continues to beckon, but for now remains just beyond our grasp.