Wednesday, October 21, 2009

Games and Surreal Numbers

At Colby College, where I did my undergrad, a visiting math professor named Leon Harkleroad gave a talk on Surreal Numbers. Being a good math major, I went to the talk, even though my interests were sharply drifting towards computer science. In high school I had been very interested in the notion of infinity, and I was quickly drawn into the material as the discussion of omega and other non-real numbers began.

In surreal-land, numbers are built from two sets of numbers with non-overlapping ranges. The sets, say L and R are then written in the form {L | R}. Since they are non-overlapping, without loss of generality, we can say that all elements of L are less than all the numbers in R. (This is the aspect that separates surreal numbers from games, wherein L and R can overlap.) Then the "value" of that surreal number is directly between the largest value in L and the lowest value in R.

Thus, you can define an infiniteish number, ω, as: {1, 2, 3, 4, ... | }. The amazing thing about surreal numbers is that you can perform basic arithmetic on ω: 2ω, ω + 1 and ω - 1 are all attainable in surreal number land. Leon gave a great talk and was very accessible to the undergraduate level, though I didn't give it too much thought for a long while.

A year after Leon's talk, however, I was lucky enough to hear John Conway give the keynote at a small-college computer science conference in Rhode Island where I presented my first paper (in poster form). He played some dots-and-boxes against a member of the audience on a small board, promising that his opponent would soon figure out how to beat him. Unfortunately, "soon" took quite a long time and his allotted time quickly began to run out; we students were quite anxious for our poster session and for the attention to be lavished on us. After the game was solved, he swiftly started drawing some basic Hackenbush figures on a new slide. Time was of the essence, but he showed how in this game, players could derive numeric values from the game positions. The figures were nothing more than line-graphs sticking out of the ground, but the intuition behind their evaluations was clear.

Then he drew another line, and used a vertical ellipsis to show that it continued infinitely skyward. Even as he was saying, "We refer to this value..." a light in my head went off: "... as ω". The connection to surreal numbers was there.

Unfortunately, time was up. John was forced to end on that note and may have missed getting to connect that to computer science. We hurried out of the room and to the poster hall.

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