Surreal Numbers by Donald Knuth

Postscript

The late Hungarian mathematician Alfred Renyi composed three stimulating "Dialogues on Mathematics," which were published by Holden-Day of San Francisco in 1967. His first dialogue, set in ancient Greece about 440 B.C., features Socrates and gives a beautiful Description of the nature of mathematics. The second, which supposedly takes place in 212 B.C., contains Archimedes' equally beautiful discussion of the applications of mathematics. R6nyi's third dialogue is about mathematics and science, and we hear Galileo speaking to us from about A.D. 1600.

1 have prepared Surreal Numbers as a mathematical dialogue of the 1970's, emphasizing the nature of creative mathematical explorations. Of course, I wrote this mostly for fun, and I hope that it will transmit some pleasure to its readers, but I must also admit that I also had a serious purpose in the back of my mind. Namely, I wanted to provide some material which would help to overcome one of the most serious shortcomings in our present educational system, the lack of training for research work; there is comparatively little opportunity for students to experience how new mathematics is invented, until they reach graduate school.

I decided that creativity can't be taught using a textbook, but that an " anti-text " such as this novel might be useful. I therefore tried to write the exact opposite of Landau's Grundlagen der Mathematik; my aim was to show how mathematics can be "taken out of the classroom and into life," and to urge the reader to try his or her own hand at exploring abstract mathematical ideas.

The best way to communicate the techniques of mathematical research is probably to present a detailed case study. Conway's recent approach to numbers struck me as the perfect medium for illustrating the important aspects of mathematical explorations, because it is a rich theory that is almost self-contained, yet with close ties to both algebra and analysis, and because it is still largely unexplored.

In other words, my primary aim is not really to teach Conway's theory, it is to teach how one might go about developing such a theory. Therefore, as the two characters in this book gradually explore and build up Conway's number system, I have recorded their false starts and frustrations as well as their good ideas. I wanted to give a reasonably faithful portrayal of the important principles, techniques, joys, passions, and philosophy of mathematics, so I wrote the story exactly as I was actually doing the research myself (using no outside sources vague memory of a lunchtime conversation I had had with John Conway almost a year earlier).

I have intended this book primarily for college mathematics students at about the sophomore or junior level. Within a traditional math curriculum it can probably be used best either (a) as supplementary reading material for an "Introduction to Abstract Mathematics" course or a "Mathematical Logic" course; or (b) as the principal text in an undergraduate seminar intended to develop the students' abilities for doing independent work.

Books which are used in classrooms usually are enhanced by exercises; so at the risk of destroying the purity of this "novel" approach, I have compiled a few suggestions for supplementary problems. When used with seminars, such exercises should preferably be brought up early in each class hour, for spontaneous class discussions, instead of being assigned as homework.

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