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Interview with John Horton Conway

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Interview with John Horton Conway Dierk Schleicher his is an edited version of an interview with John Horton Conway conducted in July 2011 at the first International Math- ematical Summer School for Students at Jacobs University, Bremen, Germany, Tand slightly extended afterwards. The interviewer, Dierk Schleicher, professor of mathematics at Jacobs University, served on the organizing com- mittee and the scientific committee for the summer school. The second summer school took place in August 2012 at the École Normale Supérieure de Lyon, France, and the next one is planned for July 2013, again at Jacobs University. Further information about the summer school is available at http://www.math.jacobs-university.de/ summerschool. John Horton Conway in August 2012 lecturing John H. Conway is one of the preeminent the- on FRACTRAN at Jacobs University Bremen. orists in the study of finite groups and one of the world’s foremost knot theorists. He has written or co-written more than ten books and more than one- received the Pólya Prize of the London Mathemati- hundred thirty journal articles on a wide variety cal Society and the Frederic Esser Nemmers Prize of mathematical subjects. He has done important in Mathematics of Northwestern University. work in number theory, game theory, coding the- Schleicher: John Conway, welcome to the Interna- ory, tiling, and the creation of new number systems, tional Mathematical Summer School for Students including the “surreal numbers”. He is also widely here at Jacobs University in Bremen. Why did you known as the inventor of the “Game of Life”, a com- accept the invitation to participate? puter simulation of simple cellular “life” governed Conway: I like teaching, and I like talking to young by simple rules that give rise to complex behavior. people, and I expected to enjoy myself. I used to Born in 1937, Conway received his Ph.D. in 1967 say about myself that if it sits down, I teach it; if it from Cambridge University under the direction of stands up, I will continue to teach it; but if it runs Harold Davenport. Conway was on the faculty at away, I maybe won’t be able to catch up. But now Cambridge until moving in 1986 to Princeton Uni- it only needs to walk away and I won’t be able to versity, where he is the von Neumann Professor. He catch up, because I had a stroke a short time ago. is a fellow of the Royal Society of London and has Schleicher: Has that ever happened to you that Dierk Schleicher is professor of mathematics at Jacobs Uni- people have run away from you in your teaching? versity in Bremen, Germany. His email address is dierk@ Conway: Yes, after five or six hours being taught jacobs-university.de. something, people edge away. You know, British We would like to thank Joris Dolderer and Zymantas Darbe- students aren’t so disciplined as German ones. nas for their contributions to this interview and Alexandru Schleicher: At this summer school, there are stu- Mihai for some of the pictures. dents from twenty-five different countries, with dif- DOI: http://dx.doi.org/10.1090/noti983 ferent levels of politeness, and they have not edged May 2013 Notices of the AMS 567 away, quite the opposite. You are a professor in Princeton, and people come to Princeton to study with people like you… Conway: Yes, to some extent. Schleicher: …and now you come here to the stu- dents. Of course, the students we have here are even younger than most undergraduates in Princeton. Is it different for you to teach the summer school students? Conway: I never treat anybody any differently, roughly speaking. I teach graduate courses in Princeton, and I teach undergraduate courses. I come to places like this and other similar events in the States. I never really change my style of teaching. I don’t change the topics I teach very Students are clearly not edging away. much. If a student is a younger person, I don’t go into so many details, but it is the same stuff for me. I am a very elementary mathematician, in a sense. Schleicher: Let me ask the question anyway. What Schleicher: Then you are a very deep elementary do you consider your greatest idea, your greatest mathematician. achievement? Conway: I will accept your compliment, if that is Conway: I don’t know. I am proud of lots of things, what it was meant to be, but… and I don’t think there is one thing I consider Schleicher: Yes, it was. my greatest achievement. My colleagues would Conway: …How might I say this? It’s harder to take probably say the work on group theory. I don’t easy, baby-type arguments and to find something consider that to be my greatest achievement. I new in them than it is to find something new think it’s pretty good, but that’s about it. I am in arguments at the forefront of mathematical glad they value it, so it means that in their eyes I development. All the easy things at first sight am not regarded as totally frivolous. In my eyes, I am totally frivolous. But I have two particular appear to have been said already, but you can find things I can mention. One is fairly recent, the that they haven’t been said. Free Will Theorem, and the other, rather less Schleicher: That puzzles me a bit, because you are recent, is the surreal numbers. I value them, or one of the people at the forefront of mathematics— estimate them, in different ways: with the surreal perhaps at the forefront of unexpected mathemat- numbers I discovered an enormous new world ics. of numbers. Vastly many numbers, inconceivable. Conway: I think the last clause I can accept. But it Nobody else has discovered more numbers than is my own mathematics. I discovered the surreal I have. In a sense, it beats so-called conservative numbers, which are absolutely astonishing. The mathematicians at their own game, because it definitions are absolutely trivial. Nobody had produces a simpler theory of real numbers than thought about them before, nobody was trying. In the traditional theory, which has been on the books mathematics, there are worthwhile subjects. Very for nearly two hundred years now. So I am very famous and deep mathematicians have investigated proud of that and astonished I was so lucky to find them, and it’s very hard to find anything new. Most it. of my colleagues at Princeton make one topic their own and become world experts on it. I don’t do Free Will Theorem that. I am just interested in a lot of things. I don’t The other more recent work is the Free Will get very deeply into any of them. I get moderately Theorem, which I found jointly with a colleague deep. of mine, Simon Kochen; I would certainly have Schleicher: I take this more as an indication of your never found it by myself. It says something at a modesty. provable level about the notion of free will, which Conway: Here is another thing I say about myself: philosophers have been arguing about for ages, I am too modest. If I weren’t so modest, I’d be two thousand years at least. It wasn’t something perfect. I am working on the modesty. everybody wanted to know, but it’s proved at Schleicher: Good luck with that work! Speaking of the mathematical level of certainty—very nearly, your modesty, what do you think… anyway. Personally, I am proud of it because I have Conway: I don’t think I am modest at all, but carry never thought about anything like that before. I’ve on. read philosophy books, but I’ve never imagined 568 Notices of the AMS Volume 60, Number 5 Schleicher: Could you make a simple statement about what exactly, or intuitively, the Free Will Theorem says? Conway: Yes. [Throws a piece of paper.] I just decided to throw that piece of paper on the floor. I don’t believe that that was determined at the start of the big bang, 14 billion years ago. I think it’s ludicrous to imagine that the entire development of the universe, including, say, this interview, was predetermined. For the Free Will Theorem, I assume that some of my actions are not given by predetermined functions of the past history of the universe. A rather big assumption to make, but most of us clearly make it. Now, what Simon and I proved is, if that is indeed true, then the same is Mathematical toys and games, such as the true for elementary particles: some of their actions Rubik’s Cube, are always an exciting topic of are not predetermined by the entire past history mutual interest for Conway and his audience. of the universe. That is a rather remarkable thing. Newton’s theory was deterministic. In the 1920s, making any progress. In general you don’t make Einstein had difficulties believing that quantum me- progress on philosophical problems. Instead of chanics was not deterministic. That was regarded tackling the problem as everybody else did, I was as a defect of quantum mechanics. Certainly when thinking about something else, thinking about I tried to learn quantum mechanics and didn’t physics, and, whoosh—something about free will succeed, I thought it was a defect. It’s not a defect. If could be said. the theory could predict what one of those particles Schleicher: So could it be that this or other discov- could do, then that theory would be wrong, because, eries of yours were shaped by coincidence or by according to the Free Will Theorem—supposing we luck? do have free will—a particle doesn’t make up its Conway: Partly luck.
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