The Reason for My Life |
Interview by Guillermo Martínez The Reason for My Life Held in the historic Café Tortoni, Buenos Aires, 1998. An extremely lively interview by a talented young Argentine mathematician and writer, Guillermo Martínez. Among other things, he is the author of a remarkable novel about genius, Regarding Roderer, and has translated Tasic's book on postmodernism into Spanish. This interview was the main article in the Sunday magazine Radar of the Buenos Aires newspaper Página/12 on 7 June 1998, and it was the first time that my photo was on the cover of a magazine. The next time was when I was featured as ``The Omega Man'' [!] on the cover of New Scientist, 10 March 2001. M: What was your childhood like? I know that your father is a playwright. How did you become interested in science? C: My father is very intellectual, and we were always having deep discussions at home. I grew up half in New York, in Manhattan, and half in Buenos Aires. I came to Argentina when I was eighteen and we stayed here almost ten years. When I lived in New York, I think it was around 1957, the Russians put up the first artificial satellite, Sputnik. This terrified the Americans, and they created a series of special courses for elementary school and high school students interested in science. I went to all of them, and I also managed to get into the Columbia Science Honors program, courses for bright high school students that were given at Columbia University in New York. Also at that time the public libraries in Manhattan were very good, and Manhattan was a very stimulating place to be. M: Why did your parents come to Argentina? C: Actually, my parents were born here, they were the children of immigrants from Eastern Europe, and they decided to go to the United States after the Second World War. When we came back to Buenos Aires, in 1966, I did a bunch of things here. I joined IBM (I started working for IBM here), and I also got involved with the Faculty of Exact Sciences at the University of Buenos Aires. I gave a few courses there; it's the only time in my life that I've given ``normal'' university courses, with a final exam and everything. I enjoyed that a lot, the university environment, there were a lot of enthusiastic, bright kids there, and it's a pleasure to teach when the students are really interested. The Berry Paradox M: What were your first research interests? C: When I was very young it was the theory of relativity, quantum physics and cosmology, that's what attracted me. But to understand physics, first you have to learn some mathematics. And I got stuck there, in math, and never got back to physics! I got stuck in math because I wanted to understand what I thought was the most profound problem there, which is Gödel's theorem. I thought that it was very mysterious, and I felt that it had to be very important and just as deep as relativity theory and quantum mechanics are. When I was fifteen I had the basic idea which I've worked on ever since, so that's thirty-five years working on just one idea, which is to define the complexity of something to be the size of the smallest program for calculating it. That's how I measure complexity. M: This idea of yours comes from the Berry paradox. Is there some simple way for you to explain that? C: Consider the first natural number [0, 1, 2, ...] that can't be defined in less than a million words. Well, the paradox is that I've just defined this number perfectly and in much less than a million words! The basic idea behind all my work is to measure the minimum number of words that you need to define something. But this quantity is ambiguous and it depends on the language that you're using, so the next step in order to get a precise mathematical concept is to switch to an artificial language, and I decided to use computer programming languages for that. So my complexity measure becomes the size of the smallest program that calculates the thing that you're interested in. The Music of Randomness M: Was your original goal to obtain another proof of Gödel's theorem? C: No, I got there via a detour. My original goal was to define randomness or lack of structure using my new complexity measure, to define ``algorithmic'' randomness. The way you do that is you say that a number is random if its digits cannot be compressed into a small computer program. If there's a program for calculating the number that's smaller than the number is, then that number isn't ``random'', because its digits have some kind of pattern or structure that enables them to be compressed. On the other hand, if the most concise description of a number is to give all its digits one by one, this means that there is no structure or pattern that a clever gambler could use to make money betting on what the next digit will be. For example, the number consisting of a million 9's is a very large number with a very concise description, so that's not irreducible information. But if a number is random then the information in its digits is irreducible and cannot be compressed. This definition of randomness has the extremely paradoxical property that according to it most numbers are random, but it turns out that there is no way to give a mathematical proof that a particular number is random! These are mathematical facts that have a very high probability of being true, but it's impossible to be absolutely certain in individual cases. And that's the basic paradox on which I built my new approach for understanding the limits of mathematics. Searching for Gödel M: Did you already know all of this when you tried to meet Gödel? C: Yes, this was the new thing, the new viewpoint that I had. As you can imagine, Gödel was my hero, and I was anxious to know what he thought about my new approach, which was rather different from his way of doing things. I wanted to know how he'd react. So I called him on the phone. M: He was living in Princeton then? C: Yes, and the only person that he had any contact with was Einstein. It was 1974 and I was young, half the age I have now, and no one spoke to Gödel on my behalf. I called him on the phone, and I said, ``Look, I have this new approach, and I'd very much like to discuss it with you.'' Incredibly, he didn't hang up, he said, ``Well, send me one of your publications about this, call me again, and we'll see if I give you an appointment.'' So I sent him one of my papers, and I called him again, and he gave me an appointment! You can imagine how ecstatic I was. At that time I was visiting the Watson Research Center in New York and I found out how to get to Princeton, New Jersey by train. There I was in my office, about to leave for Princeton, when the phone rings, and a voice, a terrible voice, says that she's Gödel's secretary, and because it was snowing in Princeton and Gödel is very careful about his health, my appointment is canceled! It was spring and it shouldn't be snowing, but it was, and my appointment was canceled! I had to go back to Argentina that weekend, and I realized that I would never have another opportunity to meet Gödel. And that's what happened, because he died a few years later, and I never spoke with him again. But now I think that he was very generous with me. He even read the paper that I sent him, and he made a technical remark about it during our second telephone conversation. I wonder how I would react now if an unknown youngster were to ask me for an appointment! [Laughs] But going back to the limits of reasoning, as I told you my first loves were physics and astronomy, and I understand physicists and how they think. And a very fundamental, very controversial idea of the physics of this century is randomness. Remember that Einstein said that God doesn't play dice with the universe? Why did he say that? Because in subatomic physics, according to the Schrödinger equation, you can no longer predict exactly what's going to happen, nature is non-deterministic, you can only predict probabilities. The fundamental laws of physics become statistical, and Einstein hated that, he believed in classical, deterministic, Newtonian physics. M: He believed in hidden variables. C: Yes, he believed that there had to be hidden variables. And he thought that once they were discovered, all the randomness would disappear and you'd go back to being able to predict exactly how particles would behave. Who knows, Einstein may still be right, the universe may in fact be deterministic, but physicists now think that randomness is fundamental, it's unavoidable. I even read the discussion between Bohr and Einstein about all of this. Einstein was one of the founders of quantum physics, but he didn't believe in indeterminacy and randomness, he rejected them, which almost made Bohr cry, because Einstein was his hero. Neither of them could convince the other, but as for me, I was convinced that randomness plays a fundamental role in the world. [For an interesting reappraisal of the Bohr/Einstein debate, see Mara Beller, Quantum Dialogue, University of Chicago Press, 1999.] And at the same time that I was reading all of this physics I was also studying Gödel's results, and I began to wonder about some of the mathematical questions that have remained unanswered for centuries in spite of the best efforts of mathematicians to solve them. So I asked myself if maybe it's sometimes the case that mathematicians can't solve a problem, not because they're stupid, not because they haven't worked on it long enough, but simply because the same randomness or lack of structure or pattern that occurs in physics also occurs in pure mathematics. So in a way, everything that I've done comes from taking ideas from physics and applying them to mathematics. And physicists feel much more comfortable with my results than mathematicians do. A Bizarre Number M: That's because you showed something that's very hard for mathematicians to understand, which is that there are arithmetical facts that are true for no reason, that are true only by accident, in fact, they're random. C: Yes, I discovered a number (I call it W) with the amazing property that it is perfectly well-defined mathematically, but you can never know its digits, you can never know what the digits in the decimal expansion of this real number [I.e., a number like 3.1415926...] are. Every one of these digits has got to be from 0 to 9, but you can't know what it is, because the digits are accidental, they're random. Mathematicians believe that if something is true, then it's got to be true for a reason, and that the job of the mathematician is to find out the reason that something is true and make that into a proof. But it turns out that the digits of my number are so delicately balanced between one possibility and another, that we will never know what they are! This disgusts mathematicians, it's terrifying for them, because mathematicians believe in reason, and anything that escapes the power of reason is horrible, dangerous, it scares them. But physicists think completely differently than mathematicians do, and randomness doesn't scare them---on the contrary! Even before quantum physics, in what's called classical physics, there was a branch of physics that used statistical methods. In Boltzmann statistical mechanics, for example, there's randomness in the motions of the molecules of a gas; physicists have been successfully using statistical methods for more than a century. Asking God Questions M: In fact, this number that you've defined has another important property: its digits contain a great deal of information about all possible computer programs, and in particular about whether they ever halt or not. Some people even call it ``the number of wisdom.'' C: Yes, this number has a lot of information coded inside it, and in an extremely compressed form. If you knew its first hundred digits, you'd know a lot, you'd be able to settle a lot of open mathematical questions. Let me put it this way: if a mathematician could ask God a hundred questions---questions that have one-digit answers---then the best he could do would be to ask for the first hundred digits of this number. And that's precisely the reason that we can never know the digits of this number, because they contain too much information. There are some people who think that this number has mystical properties. The fact that it escapes the power of reason excites their imagination. But I'm not a mystic, I'm a mathematician, I'm a rational man, I'm trying to follow the tradition of rationality that comes from ancient Greece. However there is something paradoxical about all of this. I work on the limits of mathematics, but in a way my whole career is a great big reductio ad absurdum, a ``reduction to an absurdity.'' Because I'm a mathematician, and by using mathematical reasoning I can show that there are limits to mathematics. My number demonstrates these limits. So, from a philosophical point of view, this is a somewhat uncomfortable position to be in. I'm crazy about mathematics, I love math, but I can see limits to mathematics, and this sometimes makes me question what I've been doing all my life, because if mathematics is really only a game that we invent, then I've thrown away my life! So there's a personal paradox involved in studying these limits; from a psychological point of view it's a bit delicate! [Laughs] M: Anyway, only a small percentage of mathematical results are subject to randomness within today's mathematics. There randomness is the exception, not the rule. C: Yes, I agree, in normal, everyday mathematics my results can safely be be ignored. But in some fields my results do have a philosophical impact. Some mathematicians have started to do quasi-empirical, experimental mathematics; they behave more like physicists and believe results for which there is experimental evidence but no proof. This happens because it's now very easy to do massive calculations and computer experiments. And my results provide some theoretical justification for what these people are doing. M: Physics has also changed because of the computer. C: Yes, before, one wrote down an equation, for example, the Schrödinger equation for the hydrogen atom, and you sat down and solved it analytically, in closed form. But today the physical systems that one studies are very complicated, with infinitely many particles. So there are no simple equations anymore. Instead you do computer simulations to try to get an idea of how the system behaves. That's a new kind of physics, and you can only solve equations numerically, never analytically. You never get a general expression for a solution, you just look at a lot of individual cases one by one, numerically, to try to get a feel for what's going on. Supercomputers and Quantum Computers M: Can you tell me about those quantum computers that some people hope to build? How do they work? What's the idea? C: The idea is to take advantage of quantum parallelism. It turns out that inside the atom things happen rather differently than in everyday life. In the quantum world, a physical system simultaneously pursues all possible time evolutions; its behavior is a kind of sum over all possible histories. It's as if my flight to Buenos Aires arrived six hours late, and simultaneously arrived on schedule, and simultaneously crashed before arriving, and simultaneously never made it off the ground in New York! In the quantum world, the final result that you measure is a sum over all possibilities, which interfere constructively and destructively with each other. [What I'm describing is called a Feynman path integral. For more on this, see Richard P. Feynman, QED, Princeton University Press, 1985.] At first people thought that the way Nature behaves inside the atom is crazy, and Einstein hated it, but now a new generation of physicists feels very comfortable with all of this. So instead of trying to fight it, they want to take advantage of all this subatomic madness, they even want to amplify it, and to use this quantum parallelism to make computers that can simultaneously do millions of calculations! If quantum computers can really be built, one of these computers might be able to replace millions of ordinary computers! A physicist friend of mine, Rolf Landauer, thinks that this will never work as a technology, but I certainly think that it's fascinating from a conceptual point of view, to try to force Nature to be as quantum-mechanical as possible and to go to these extremes! Plus experiments in this area of physics are very cheap compared to the cost of giant particle accelerators. I don't work on all this myself, but we do have an important group working on quantum computation at my lab. I went to some of the first meetings where these new ideas developed and have been an interested observer of what's going on in this field, because I love it when a revolutionary new idea takes off, I think that's very exciting. Artificial Intelligence and the New Golem M: What's your opinion about the feasibility of creating an artificial intelligence? C: I'm glad you asked me that. I think that we have already achieved a kind of artificial intelligence, we just don't realize it. Normally people think that artificial intelligence should be like human intelligence, but not much has been achieved in that direction. Things that are easy for us, like using natural languages, recognizing faces, walking... all these things that are easy for us are very difficult for computers to do. But computers are very good at things that are hard for us, for example, symbolic computations. There's a program called Mathematica, that was created by Stephen Wolfram, that knows a lot of mathematics and how to do numerical and symbolic computations. I think of Mathematica as a kind of AI; it's a mathematical assistant that I can use when doing research. Even though it has no human intelligence, it certainly has a lot of mathematical intelligence. Then there's chess. My laboratory built the supercomputer that beat Kasparov, but it doesn't play chess the way people do, it uses brute force, it throws a massive amount of special-purpose hardware and parallel computing at the problem. It's a good example of what's called a massively parallel computer. M: Actually what I really wanted to know is what you think about Penrose's argument that Gödel's incompleteness theorem shows that an artificial intelligence is impossible, because we humans can solve problems and prove theorems that escape the power of any particular formal axiomatic mathematical theory, any particular mechanical problem-solving system. Basically, his argument is that we can use self-referential reasoning, but machines can't, which is how you normally prove Gödel's incompleteness theorem. C: Penrose's book [Roger Penrose, The Emperor's New Mind, Oxford University Press, 1989.] is very interesting, he did fine work on black holes, and he's worked with Stephen Hawking. But I have to say that I completely disagree with the thesis of his book regarding the impossibility of creating an AI. My point of view is going to seem very strange to you, because I'm a mathematician. But you have to remember that I've also done a lot of work at IBM developing new hardware and software technologies; I was a member of the team at IBM Research that designed the prototype for IBM's UNIX workstation, the IBM RS/6000. And my personal opinion is that AI is not a mathematical problem, it's an engineering problem. I don't think that Gödel's theorem applies at all. To me a human being is just a very complicated piece of engineering that's exquisitely well-suited for surviving in this world. And Gödel's theorem isn't about organisms, it's about formal axiomatic mathematical theories, which are completely different kinds of objects. In fact, it's very often the case that theoreticians can show that in theory there's no way to solve a problem, but software engineers can find a clever algorithm that usually works, or that usually gives you a good approximation in a reasonable amount of time. And I think that human intelligence is also a little bit like that, and that it's a matter of creeping up on it little by little, a step at a time, until we can usually do a good job imitating it. In fact I think that we may almost be half-way there, only we don't realize it, and that fifty years from now we'll be close to a real AI, and then people will wonder why anyone ever thought that it was difficult to create an AI. This AI won't be the result of a theorem, it'll be a mountain of work, a giant engineering project that was built piece by piece, little by little, just like what happens in Nature. As the biologists say, God is a tinkerer, he cobbles things together, he patches things up, he makes do with what he has to create new forms of life by experimenting with sloppy little changes a step at a time. There's a word for that in Spanish, isn't there? M: Un remendón? C: Yes, that's it, un remendón, someone who likes to repair things, to fix them up, not to throw them away! [Laughs] We human beings aren't artistic masterpieces of design, we're patched together, bit by bit, and retouched every time that there's an emergency and the design has to be changed! We're strange, awkward creatures, but it all sort of works! And I think that an AI is also going to be like that... M: Like the sheep Dolly. C: Yes, I think that a working AI is going to be like some kind of Frankenstein monster that's patched together bit by bit until one day we realize that the monster sort of works, that it's finally intelligent enough! So you see, here my point of view is that of an engineer, not that of a mathematician. I'm not talking at all like a theoretician! The New Renaissance M: Do you think that your work justifies a pessimistic point of view regarding science, or regarding reasoning in general? C: Some of the things that I've said may sound pessimistic, and I'm even interviewed in a book called The End of Science. The guy who wrote that book, John Horgan, thought that I would agree with his thesis that science is coming to an end, and that little that is fundamental remains to be done. But when he interviewed me I insisted that I'm very optimistic about the future of science and of mathematics. I prefer another book, one that's just been published by Oxford University Press, called The New Renaissance, by Douglas Robertson. According to Robertson, we are now going through a quantum leap in human abilities due to the PC, the Internet, and the Web, and we are going to go through a major social discontinuity, a major leap forward. According to him, what initially separated humans from animals was spoken language. Then civilization was created when we learned to read and write, which permits knowledge to be accumulated and transmitted better. Then the European Renaissance was due to the invention of the printing press, which made books available to everyone, instead of being luxuries reserved for Bishops and Kings. And now, according to Robertson, we're entering a new era, in which everyone can have at their fingertips, instantly available in their home on the screen of their PC, all of human knowledge, from anywhere on earth. The Web will become an immense library, a universal human library, and this will have to have a profound social impact. Because, according to Robertson, what's important is the total amount of information that is readily available at each of these major stages of social evolution---language, writing, printing, the Web---that's what pushes us forward. But Robertson points out that the computer isn't just changing human society, it's also having a profound conceptual impact, it's changing the way you do fundamental science. Now we can study systems that are much more complicated than we could study before. The kinds of problems that one can solve analytically, that one can write general solutions for, now look like toy problems. Now we solve problems numerically, or via simulation, not just analytically. And now there's even a new quasi-empirical school of thought regarding the foundations of mathematics, which I support. Mathematics, I think, is different from physics, but it's not as different as mathematicians would like to think. In my opinion, one shouldn't be afraid to add new principles that are justified pragmatically, via computer experiments, even if one can't prove them. I believe that not all proofs have to be absolutely water-tight, and that different proofs can carry different degrees of conviction. M: But with this new approach you lose something important, you lose the idea of elegance, of conciseness, of mathematical beauty. The idea of beauty doesn't mean anything to computers, that's something human. C: Yes, that's right, and it's precisely the beauty of mathematical reasoning that I care about. When I was young I used to think that some mathematics was as beautiful as a graceful woman. Obviously it's a different kind of beauty, but to me it wasn't that different. But mathematics is constantly evolving, and I'm afraid that problems that have simple, elegant solutions are now going to be considered toy problems. For example, the problem of classifying all finite simple groups took more than ten-thousand pages to solve! And look at the complicated proof of the four-color theorem. [Or more recently, look at all the calculations required to solve Kepler's sphere-packing problem.] But this feeling I have that math is getting more complicated and less elegant, that's just my personal opinion, it's not a majority view. Since I'm back here in the Café Tortoni, I feel that I'm a porteño [Someone from Buenos Aires.] again, and that it's okay to say what I think, even if it sounds a little bit crazy, that's what the Café Tortoni is for! [Laughs] Truths and THE Truth M: I wanted to ask you about that ten year gap between your initial theory of complexity, and your corrected, improved theory. How did it feel to have a bad theory, to have the wrong definition of complexity? Was that very frustrating? C: I think that mathematicians are actually artists. Pure mathematics is really an art form, and I'm acutely sensitive to beauty or to lack of beauty. A definition of a new concept is good if it leads to beautiful, natural theorems. The concepts in a new theory have to fit together, they have to work together well, harmoniously. When I started with my theory, I used a straight-forward definition of complexity, one that made life easier and avoided certain technical difficulties. But I thought that some of the other definitions that I had considered had some nice properties that I hated to lose just to get around those obstacles. So I took advantage of my 1974 visit to the IBM Watson Research Center in the U.S.---this was before I joined permanently, I was just visiting---to concentrate on this problem. And then I realized that yes, it was in fact possible to keep all those nice properties, all I had to do was change to my current definition of complexity, which measures the size of what I call ``self-delimiting'' binary programs. In retrospect, the correct definition seems inevitable, it looks inescapable, but in fact I had to consider many other possibilities before I could find the correct one. When you create a new mathematical theory, you have the freedom to change the rules of the game if things don't work properly. So now 99% of my theory is better, but there are a few results that I lost with the new definition that I still miss. There are a few little things that I still don't know how to repair. M: How did you feel the first time that you proved an important theorem? What does it feel like to prove a major result? Your book on The Limits of Mathematics starts with the quote, ``He thought he had THE TRUTH!'' C: Well first of all, in normal, everyday life there is no such thing as THE truth, everything is very complicated and messy, and you have to look at things from many different points of view. But we used to think that at least in mathematics we could in principle all agree on the basic assumptions, and then everything would be black or white. But Gödel's theorem, Turing's work, and my own results show that even in math you can't know the whole truth and nothing but the truth. But yes, it's true that in research there's a moment of ecstasy, of euphoria. Most of the time doing research is really very unpleasant, you're struggling and everything is ugly, nothing works, the ideas smash into each other, and you feel that you're getting nowhere, that you're wasting your life. But then all of a sudden you see the light, you discover the right way to think about the problem, and everything falls into place. It's like the time when I was going up a mile-high mountain in northern New York State, near Canada. My friends and I were in the rain inside a cloud, covered with mud, unable to see anything. And all of a sudden we made summit, the summit was just above the cloud layer, in blinding sunshine, and beneath the bright blue sky in the distance we could see the other peaks poking through a perfect white plane of clouds! That's just what it feels like when you've been struggling with your lack of understanding, and all of a sudden you discover the correct approach, and you get the exhilarating sensation that your mind is sharper and that you can see farther than you've ever done before. It's a wonderful moment, it's the payoff, it's how God rewards you for all that suffering... if you're very lucky. But there's a big price you have to pay, which is that you have to be obsessed with the problem, it has to be like an open wound, like a sharp stone that you can't get out of your shoe. At least that's the way it is for me, and Einstein said the same thing. I think that you have to be obsessed, and I wouldn't advise anyone to lead that kind of a life! Einstein had a good friend, Michele Besso, with whom he discussed a lot of the ideas of the theory of relativity. But Besso never managed to achieve anything on his own. And late in their lives, Besso's wife asked Einstein, why was it that in spite of all his talent, her husband had never managed to achieve anything in science. ``Because he's a good man!'' exclaimed Einstein. That's it, that's exactly it, you have to be a fanatic, and that plays havoc with your life, and the lives of those around you. M: Do you have anything to do with real life? For example, do you ever read the newspaper? C: Well, when I was young I used to row in the Tigre River delta, I went to the opera, to the ballet, and to the movies, and I courted the pretty porteñas. And I used to laugh at those portraits of mathematicians as dazed, self-absorbed, forgetful eccentrics. But God's revenge has been that as the years go by, I look at myself in the mirror, and that's me, that's exactly what I've become, it's not a joke after all! But the truth is, that in order to work on this kind of stuff, you really have to isolate yourself from the world. I live in the countryside, in the remote suburbs of New York City, and it takes me a fifteen-minute drive to get to the nearest café. Now that I'm back in Buenos Aires I realize how much I miss this kind of life, it's wonderful here, the streets are full of people, there are cafés everywhere. I don't live that far from New York City, it's an hour or so away, and it's a terrific city too, but I'm almost never there. I prefer to go hiking in the hills overlooking the Hudson River near where I live, or in the mountains... Anyway, that's the kind of life that I live now. M: You visited the places in Vienna where Gödel lived and worked. What was he like when he was young? C: One has this image of Gödel from his photographs as someone who's emaciated, extremely serious, and angry at the real world for intruding on his thoughts. But when he was young he used to spend all his time in Vienna nightclubs, that's where he met his wife, who was a dancer. It was normal in Vienna for sons of well-to-do families, like Gödel, to spend a lot of time in nightclubs. What wasn't normal was that he also did some mathematics! Dennis Flanagan, who used to be the editor-in-chief of Scientific American and once lived in Princeton, told me the following story about Gödel. One day Flanagan was walking down the street, and he saw Gödel, and he decided that he would go up to Gödel and introduce himself. He knew what Gödel looked like, because he had published an article on ``Gödel's proof'' with Gödel's photograph in Scientific American in 1956. But just at that moment a young female Princeton University student was passing by, and she wasn't wearing very much clothing, because it was summer, and summer in Princeton is very hot and humid. And Gödel stopped dead in his tracks to admire her and watch her go by, just when Flanagan was about to introduce himself. And that's how Dennis Flanagan lost his opportunity to shake hands with Kurt Gödel, he didn't dare to interrupt! And that just proves that Gödel wasn't a mathematical saint. After all, we're all made of flesh and blood, aren't we?
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