i META MATH! The Quest for Omega by Gregory Chaitin Gregory Chaitin has devoted his life to the attempt to understand what mathematics can and cannot achieve, and is a member of the digital philos- ophy/digital physics movement. Its members believe that the world is built out of digital information, out of 0 and 1 bits, and they view the universe as a giant information-processing machine, a giant digital computer. In this book on the history of ideas, Chaitin traces digital philosophy back to the nearly-forgotten 17th century genius Leibniz. He also tells us how he dis- covered the celebrated Omega number, which marks the current boundary of what mathematics can achieve. This book is an opportunity to get inside the head of a creative mathematician and see what makes him tick, and opens a window for its readers onto a glittering world of high-altitude thought that few intellectual mountain climbers can ever glimpse. arXiv:math/0404335v7 [math.HO] 13 Sep 2004 ii Meta Math! Cover William Blake: The Ancient of Days, 1794. Relief etching with watercolor, 9 1 6 7 inches. 8 × 8 British Museum, London. iii iv Meta Math! Preface Science is an open road: each question you answer raises ten new questions, and much more difficult ones! Yes, I’ll tell you some things I discovered, but the journey is endless, and mostly I’ll share with you, the reader, my doubts and preoccupations and what I think are promising and challenging new things to think about. It would be easy to spend many lifetimes working on any of a number of the questions that I’ll discuss. That is how the good questions are. You can’t answer them in five minutes, and it would be no good if you could. Science is an adventure. I don’t believe in spending years studying the work of others, years learning a complicated field before I can contribute a tiny little bit. I prefer to stride off in totally new directions, where imagina- tion is, at least initially, much more important than technique, because the techniques have yet to be developed. It takes all kinds of people to advance knowledge, the pioneers, and those who come afterwards and patiently work a farm. This book is for pioneers! Most books emphasize what the author knows. I’ll try to emphasize what I would like to know, what I’m hoping someone will discover, and just how much there is that is fundamental and that we don’t know! And yes, I’m a mathematician, but I’m really interested in everything: what is life, what’s intelligence, what is consciousness, does the universe contain randomness, are space and time continuous or discrete. To me math is just the fundamental tool of philosophy, it’s a way to work out ideas, to flesh them out, to build models, to understand! As Leibniz said, without math you cannot really understand philosophy, without philosophy you cannot v vi Meta Math! really understand mathematics, and with neither of them, you can’t really understand a thing! Or at least that’s my credo, that’s how I operate. On the other hand, as someone said a long time ago, “a mathematician who is not something of a poet will never be a good mathematician.” And “there is no permanent place in the world for ugly mathematics” (G. H. Hardy). To survive, mathematical ideas must be beautiful, they must be seductive, and they must be illuminating, they must help us to understand, they must inspire us. So I hope this little book will also convey something of this more personal aspect of mathematical creation, of mathematics as a way of celebrating the universe, as a kind of love-making! I want you to fall in love with mathematical ideas, to begin to feel seduced by them, to see how easy it is to be entranced and to want to spend years in their company, years working on mathematical projects. And it is a mistake to think that a mathematical idea can survive merely because it is useful, because it has practical applications. On the contrary, what is useful varies as a function of time, while “a thing of beauty is a joy forever” (Keats). Deep theory is what is really useful, not the ephemeral usefulness of practical applications! Part of that beauty, an essential part, is the clarity and sharpness that the mathematical way of thinking about things promotes and achieves. Yes, there are also mystic and poetic ways of relating to the world, and to create a new math theory, or to discover new mathematics, you have to feel com- fortable with vague, unformed, embryonic ideas, even as you try to sharpen them. But one of the things about math that seduced me as a child was the black/whiteness, the clarity and sharpness of the world of mathematical ideas, that is so different from the messy (but wonderful!) world of human emotions and interpersonal complications! No wonder that scientists express their understanding in mathematical terms, when they can! As has been often said, to understand something is to make it mathe- matical, and I hope that this may eventually even happen to the fields of psychology and sociology, someday. That is my bias, that the math point of view can contribute to everything, that it can help to clarify anything. Mathematics is a way of characterizing or expressing structure. And the universe seems to be built, at some fundamental level, out of mathematical structure. To speak metaphorically, it appears that God is a mathematician, and that the structure of the world—God’s thoughts!—are mathematical, that this is the cloth out of which the world is woven, the wood out of which the world is built. Preface vii When I was a child the excitement of relativity theory (Einstein!) and quantum mechanics was trailing off, and the excitement of DNA and molec- ular biology had not begun. What was the new big thing then? The Com- puter! Which some people referred to then as “Giant Electronic Brains.” I was fascinated by computers as a child. First of all, because they were a great toy, an infinitely malleable artistic medium of creation. I loved programming! But most of all, because the computer was (and still is!) a wonderful new philosophical and mathematical concept. The computer is even more revo- lutionary as an idea, than it is as a practical device that alters society—and we all know how much it has changed our lives. Why do I say this? Well, the computer changes epistemology, it changes the meaning of “to understand.” To me, you understand something only if you can program it. (You, not someone else!) Otherwise you don’t really understand it, you only think you understand it. And, as we shall see, the computer changes the way you do mathematics, it changes the kind of mathematical models of the world that you build. In a nutshell, now God seems to be a programmer, not a mathematician! The computer has provoked a paradigm shift: it suggests a digital philosophy, it suggests a new way of looking at the world, in which everything is discrete and nothing is continuous, in which everything is digital information, 0’s and 1’s. So I was naturally attracted to this revolutionary new idea. And what about the so-called real world of money and taxes and disease and death and war? What about this “best of all possible worlds, in which everything is a necessary evil!?” Well, I prefer to ignore that insignificant world and concentrate instead on the world of ideas, on the quest for under- standing. Instead of looking down into the mud, how about looking up at the stars! Why don’t you try it? Maybe you’ll like that kind of a life too! Anyway, you don’t have to read these musings from cover to cover. Just leap right in wherever you like, and on a first reading please skip anything that seems too difficult. Maybe it’ll turn out afterwards that it’s actually not so difficult. I think that basic ideas are simple. And I’m not really interested in complicated ideas, I’m only interested in fundamental ideas. If the answer is extremely complicated, I think that probably means that we’ve asked the wrong question! No man is an island, and practically every page of this book has benefited from discussions with Fran¸coise Chaitin-Chatelin during the past decade; she has wanted me to write this book for that long. Gradiva, a Portuguese publishing house, arranged a stimulating visit for me to that country in viii Meta Math! January 2004. During that trip my lecture at the University of Lisbon on Chapter V of this book was captured on digital video and is now available on the web. I am also grateful to Jorge Aguirre for inviting me to present this book as a course at his summer school at the University of R´ıo Cuarto in C´ordoba, Argentina, in February 2004; the students’ comments there were extremely helpful. And Cristian Calude has provided me with a delightful environment in which to finish this book at his Center for Discrete Math and Theoretical Computer Science at the University of Auckland. In particular, I thank Simona Dragomir for all her help. Finally, I am greatly indebted to Nabil Amer and to the Physics Depart- ment at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York (my home base between trips) for their support of my work.