Donald Knuth and Surreal Numbers: I worked for six days, and rested on the seventh (40,41,42 / 97)

“Computer science experts would say that I made the biggest mistake when I took on this project.”



This is a unique event in my life. It happened in the early 70s. I met John Conway, probably one of the greatest mathematicians. I met him on the way to the University of Calgary in 71st and we had lunch together. He threw a new theory on a napkin that came to his mind, and, in my opinion, it was really amazing. This is a purely mathematical theory about a new way of determining numbers. Its essence is that they can be not only integer or fractional, but also there are infinite numbers, and the square root of infinity, and infinity of infinity, and infinity of the square roots of infinity, and all this makes sense. A year later I was on vacation in Norway and in the middle of the night the thought came to me: “Wow, this theory is so beautiful that it would be interesting to tell a story, write a book in which the characters will discover Conway’s theory.

The point is to teach people how to do research in this way, so that students can not only learn what other people have done, but also discover something new in mathematics. And all this can be represented in the form of a story where the characters figure out all these things on their own. So I thought that a really cool book could come out of this, and it could be used as a supplement. I thought that teachers in high schools could advise her to their students so that they can see how discoveries are made in mathematics.





I decided to call these numbers surreal. For you to understand, real numbers have infinitely many decimal places, and surreal ones, in turn, are even larger. This was the Conway system. Therefore, I decided that I already have a name (Surreal numbers) and the idea that the theory would be put forward by the characters of the book. I woke Jill up and said, “Jill, do you know that I’ve been working on the Art of Programming for many years and still haven’t made a step? I was thinking about writing another book. What do you say?" and I also said, "I think I can write it in a week, in fact, in just a week, as it will be a very short book."

And she said, “Okay, Don, come on. This is your year on vacation and we are in Norway, why not take the week on this project so that you can concentrate on it, then you will be satisfied, and we can live happily ever after. ”



We have planned everything. I decided to rent a hotel room for a week in the center of Oslo, not far from the place where Ibsen lived, so that I could feel his spirit and tune in the right way. And we decided that she could come and visit me a couple of times a week, and we could pretend we had an affair. We always wondered what it was like. Thus, I could be a writer for a week, and then we could fool people at the hotel so that this woman would come to me secretly. That is exactly what happened over the next few weeks. I wrote the first page of the novel in my head several times, and when I, for example, was riding the bus, I was already thinking about the next sentence.

I checked into the hotel in January and it was probably one of the best weeks of my life. A muse descended upon me, as if dictating this book to me. I lived in a cheap hotel run by strict volunteers from a Norwegian church. But this did not interfere with our romance with Jill. American students from St. Olav University, Minnesota also stayed at the hotel. Every morning we were served a Norwegian breakfast, which is widespread, it does not compare with herring in England, but in any case we were served herring, eggs and much more.

Every morning I spent an hour eating my Norwegian breakfast very slowly, which was enough to hold out until noon, and listened to what American students were talking about. It was fascinating because I had not heard spoken English for a long time. By that time, we had already lived in Norway for 6-7 months. Therefore, I could recall the dialect and some expressions, but no more. My story was written in a dialogue form, and for me it was important to make an authentic emphasis on the characters. I listened to what the students were talking about, ate, then headed to my room and started writing a book.

The book is like an opera. Do you know operas that have good music but little plot? My book had good math and only a little plot. I reproduced the mathematical part myself. I remembered that Conway sketched the theory on a napkin, but I lost it. I tried to remember what was written there and what he told me a year ago. I tried to restore this theory from memory and derive it myself, as this is what the characters in my book do. I conducted my own research and decided that if I make a mistake somewhere, I will certainly add it to the book, and my characters will make the same mistake. I wanted the book to be a genuine view of mathematical research.

Usually I finished work in the morning and at first I always had sketches from the previous day that needed to be finalized. I was thinking about the first page of the book for several months, and I was ready to start working on it. When I needed to take a break before writing the second page, I could dictate the first to you almost literally. Then I wanted to do some math, before continuing to write and I was stuck.

I have come to a standstill. I did not know what to do next. Then I went outside and walked around Oslo. It was a little snowy January, and the streets were not slippery. About an hour I just wandered around the hotel and hoped that a solution to the problem would come to me. I returned to the room, wrote a little, dined leisurely, wrote a little more, and then I could not turn off the light, because I needed to write a book. I knew from which proposals I would continue to work before I went to bed. I was all at work, as you can imagine.



I said that I would need a week to write a book. After 6 days, I finished the last chapter and rested for 7 days. It was very interesting. Day 6 was actually one of those days when my wife came to see me and we went to watch the movie “Butterflies Free” with Goldie Hawn. Then we walked around the Royal Palace in Oslo and there was a wonderful frost on these magnificent trees. I was just enchanted, looking at the snow-covered branches of trees against the dark blue sky. It was a week when I saw inspiration in everything. I understood that I should return to my room and I will finally finish the book, but I just looked at the sky and the patterns created by these icy branches, and this was one of the greatest moments in my life.

Then I returned to the hotel and added the last chapter, and on the 7th day I rested. This day I tried to write a letter to my secretary Phyllis. She was going to print the manuscript, and I had to send her a letter with instructions on how to do this. I began to write a sentence, reached the middle and could not finish it. I wrote without interruption for 6 days, and then the book was finished and my writing activity was finished. I could not figure out how to write a letter to Phyllis. In the end, the letter was written, and we sent her a handwritten manuscript.

She printed it for me and sent it in a couple of months, and I, in turn, sent it to John Conway in England, Cambridge. He politely noted that I understood one of the axioms of his theory incorrectly, that the characters in my book actually worked on a different system, although his system was actually much better. But okay, it's not so bad. We visited him in England and went to Cambridge; I stayed in his house during Easter and found out what I had to write initially.

Then I returned to Norway and took another week, but not in Oslo, but in an abandoned valley, in one of those valleys, which during the time of the Black Death lost their entire population. There was a small holiday home called Solheimen, where many people spent time in the summer and everyone there spoke Norwegian. We ate at the common table, and I did everything possible to speak Norwegian with them. But most of the time I did the book, rewriting everything with axioms in mind, which, as Conway said, I had to use from the very beginning. And everything has become much better.

I also received valuable comments from other readers. For example, there were some romantic scenes in the book and my wife advised me on how best to write them. I took all this into account, worked for another week, and the book was finished, "Surreal numbers." It was completely different from the books I wrote about computers before. It has become a kind of litmus test for people, depending on whether they are mathematicians or computer scientists.

Mathematicians would look at Surreal Numbers and say that it is quite interesting.

Computer scientists would say that I made the biggest mistake when I took on this project.

In any case, I received information from my publisher, who said that he needed 17 copies of my book. I should have sent them amendments next week. The book was translated into 8 or 9 languages, which means that there are somewhere in the world of mathematicians who appreciated this little book. It has about 90 small pages. I can never write something like that again, but it was an amazing experience.

“Have you ever wanted to write another novel?”
- No, you know, when I wrote this book, I walked along the streets of Oslo, saw birds, heard sounds and felt part of this world, because I wrote a book and it was important for me how her characters live and what they say.

This automatically made me more susceptible to the world. This is a very important part of my writing experience and I can understand why writers enjoy it. Trying to transfer your own ideas into prose tempers. Maybe I wake up one day in the middle of the night and want to repeat this experience, but I always perceived the writing of this book as what I will do for the first and last time in my life. And, as I said, many people think that I made a big mistake when I wrote it.

Translation: Diana Sheremyeva

To be continued ...

Read more

“Two heroes accidentally find a stone with ancient writings and discover new mathematical structures.”

• John Horton Conway: Life Is Like a Game
• John Horton Conway: Life Is Like a Game - The End
• Donald Knuth: I sat on the back desks and bullied jokes, and the teachers humbled themselves and didn’t beat their asses often (1,2,3,7 / 97)
• Donald Knuth: “My Advice to the Young” (93/97) and “Feeling the need to assert oneself” (9/97)
• Quantum, 1979 "Surreal numbers"

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Publishing support - Edison , a company that tests critical systems for fault tolerance , and also designs and develops software for cluster computing .