O: How did your interest in mathematics develop, because you said that you grew up in an artistic environment; your father was a film-maker.
C: Well, my father was (is!) what I think could be called a European style intellectual, really. So at home there was a lot of discussion of everything: politics, art, the human condition, fundamental questions. My parents came from Buenos Aires and their parents came from Eastern Europe... Buenos Aires is a very European city. I grew up in Manhattan and Buenos Aires.
There is a problem with creativity. I recently read an article called "Hitting the Wall" about the problems one has as one grows older, and the article made me think of the biography of Feynman by Gleick. There is a passage in that biography that really affected me. At one point in the biography, Feynman says that the greatest moment for him in his life is when he has an idea and he realizes the idea is really significant and important, and that it will reveal new things. This is a period of euphoria and extremely intense work. The interviewer then asks, "How many times in your life has this been the case?" Feynman said, "Well, maybe five times in my life!" "And how long does this period of intense euphoric activity last?" Feynman said, "Maybe two, three weeks!" So the conclusion is that the intense, creative life of Feynman, the Nobel Prize winning physicist, may be five times three weeks, maybe fifteen weeks of his life!
And then the question is, what do you do the rest of your life?! Well the answer is, you are working very hard trying to get that next idea, preparing the subconscious, trying to see where there is a new path. There is routine science, normal science and there are paradigm shifts. Routine science is the kind of thing you might do in industrial labs and it is really technology, almost. But science at its deepest level is an intensely creative activity, just like the arts.
O: The artist Christian Boltanski describes a similar situation in the life of an artist: that an artist has three, four ideas in his life when something truly new pops up.
C: Yes. When almost all of a sudden you find a new style. If you are a painter, you all of a sudden find a new way of looking at the world and your paintings change. And people don't realize how emotional this is, people think that in science you just discover things, but you don't invent. And an artist invents; Shakespeare invented his plays. People think that a scientist discovers how the world works. That may be the case, but you have to imagine a beautiful new theory before you can verify it. And most of the beautiful theories you imagine, fail. The first step is an act of imagination. There is no systematic way to ask nature to reveal something which is a quantum leap forward. Experiments don't tell you that you have to go in a particular new direction. You have to imagine that a whole new approach may be possible, that a beautiful new theory is possible, before you can get to work systematically to develop it and verify it experimentally. In science experimental work does not systematically lead to new theories. It is an act of imagination, it is a tremendously emotional thing, too. You have to throw your whole personality at the problem.
O: I had a discussion with Roger Penrose, whom I interviewed for this series of interviews. He says that the actual invention seldom happens in the laboratory or the studio, it very often happens in between, it could happen as one crosses the pedestrian stripe...
C: Right. I almost never have new ideas in my office. What I do in my office is, I type papers into my computer, for instance. But I will have a new idea while swimming or going for a drive, in the strangest places!
O: One of the main differences one often resorts to in terms of art and science is that from an art point of view, one tends to think that inventions are not cumulative, in a sense, that there is a non-linearity. It is very difficult to read art history as a sense of progress. As Man Ray once put it, comparing art to sexuality, there is no progress in art, there are just different ways to do it! Science works in a more cumulative way where actually previous inventions are becoming redundant on the way.
C: Well there is some truth in that, of course. On the other hand, outsiders often don't realize how important style or fashion are in science, just as they are in the arts. It is true that science deals with a more objective reality in some ways than the arts do. When you do a paper in mathematics you can give an objective statement to say whether there is a mistake or not, but to say whether the paper is of value or not, is really a matter of opinion. Just as much in mathematics as it is with a literary work, for example. There is no clear answer and fields that are fashionable sometimes disappear. Certain kinds of questions simply go out of fashion and fields die.
One has this beautiful, utopian notion that science marches forward step by step in understanding. The history is much more dramatic than that. Often the scientists who take an immense step forward in science, have a romantic life just like an under-appreciated artist who starves to death and is only appreciated after his death. This happens in science too. Boltzmann committed suicide, and today he is considered one of the great scientists of the nineteenth century. He was one of the inventors of statistical mechanics. Well, Boltzmann was old and in poor health, but one of the reasons he committed suicide was because the leading intellectual figure in Vienna at that time was Ernst Mach and Mach didn't believe in atoms and all of Boltzmann's work depended on the existence of atoms.
Another example is Maxwell, the other great scientist of the end of the nineteenth century. James Clerk Maxwell came up with Maxwell's equations for electromagnetism. As it turns out, I was surprised to learn that Maxwell died at about my age, which seems young to me now. And when he died his theory was not accepted. The leading physicist of his time, who is not remembered by outsiders---physicists may remember him---was Lord Kelvin, who at first was sympathetic and then decided that Maxwell was wrong. And when Maxwell died, there were only a handful of young physicists in England who thought that Maxwell was right and who fought hard to try to defend Maxwell. Actually it was the German scientist Hertz, who found experimental evidence for the propagation of electromagnetic waves, who turned the tide.
O: So they were kind of too early, almost a presentiment.
C: Well, there are many stories like this. But of course when they write a history of science, they like to take all of that out and make it sound like science is a step-by-step march forward and that everyone agrees when a new idea comes. To show how little that is the case in science, as in other fields, there is a beautiful quote from Max Planck. Planck was one of the inventors of quantum theory, a very revolutionary theory. He did this work around nineteen-hundred. It was really the first step in the direction of quantum theory.
Max Planck made the remark that a new scientific theory never triumphs by convincing its opponents. The opponents are never convinced. What happens is they die, they die of old age and they are replaced by a new generation who grew up with the new ideas and view them as natural rather than foreign! And I think that it is also the case that very often deep scientific ideas are completely impractical when they start, even though fifty or a hundred years later they may have numerous technological consequences. So, in a way, deep science is like art because artists typically are in trouble. I have several friends who are artists and they are all struggling to survive. They have another job so they can do their art, they do things like that. Well, the same is true with a lot of revolutionary science. It takes many years for a revolutionary idea to convince people and it takes many years for possible technological and economic consequences to develop.
O: To kind of catch up with it.
C: Yes. When the work is first done it is completely impractical. Nuclear physics before the Second World War was like studying Greek poetry. There were only a handful of people who studied nuclear physics. Before the Second World War there were only a handful of people doing it and it had no practical consequences.
In my life there is an interesting related situation, which is that in a way, you could say that computer technology came from---and I say this in my book The Unknowable---computer technology, in a way, was a spin off or an off-shoot of a philosophical controversy about the foundations of mathematics! In the past century there has been a tremendous amount of controversy and soul searching and inner torment and self examination by mathematicians about the foundations of mathematics: doubt about whether mathematics has a firm foundation and how to make it firm.
One of the suggestions, about a hundred years ago, was by a famous mathematician called David Hilbert. And Hilbert said that we should formalize mathematics, make an artificial language for mathematical reasoning. That project failed in a very interesting way. Because the notion of total formalization, of a completely artificial language, where it is mechanical to see what something means, is, in fact, the most tremendous technological success of this past century: the computer! These artificial languages are everywhere now. But they are not artificial languages for mathematical reasoning. They are not for doing reasoning or deduction, which is what Hilbert wanted. They are languages for calculating, for algorithms, for programming.
O: So Hilbert actually invented the computer?
C: In a way, yes. There is a clear intellectual line, a thread that you can follow.
O: And how do you see the whole cybernetic movement in relation to Hilbert? Was that influenced by him? Is there a link?
C: I think that there is a link because the key person is Turing. And Turing was trying to settle some of the questions that were very clearly asked by Hilbert. And in order to do this Turing had to come up with the notion of a general-purpose computer as a mathematical device, as a logical concept. He did this before there were any computers, in 1936. It is a fantastic piece of work; tremendously imaginative and profound. Turing was interested also in artificial intelligence, in programming computers to play chess. He was interested in morphogenesis, which is how the embryo develops and how an animal gets its shape: the design, for example of the fur or the colors on a zebra or on a bird, the emergence of pattern in biological organisms. He was interested in all of these questions and, of course, cybernetics is man and machine, biology and technology.
O: And feedback loops.....
C: Feedback loops, right. My work emerges from some of this early work, too, because there were two schools of information theory when I was young, and my work is just information theory applied to mathematics, I talk about mathematical information. There was an earlier information theory which was communications engineering which had two origins. One was Shannon at Bell Labs, the other was Norbert Wiener at MIT. Norbert Wiener in fact wrote a very popular book, an intellectual best-seller called Cybernetics. It was thanks to that book that the word was known in the United States. Unfortunately, in the United States you can't use the word "cybernetics" anymore, because a large quantity of papers which were very informal were published and created the impression that it was a field which was very superficial. So, in fact, if you do a piece of work on cybernetics, it is better not to call it cybernetics because that word has a bad reputation in the United States, at least there. Maybe in Russia they still call it cybernetics. I think they used to.
But the idea will certainly come back. I have a friend, Jacob ("Jack") Schwartz at the Courant Institute of NYU, who first was a mathematician and then became a computer scientist and who just spent the summer at Cold Spring Harbor Lab in Long Island, which is one of the great centers of research in molecular biology. I visited him there and he told me that molecular biology is really digital. You can clearly see digital information in biological organisms, and each cell is like an entire computer, it turns out. It's just amazing how complicated a cell is. The DNA is being turned on and off all of the time. It has loops, it controls itself. It's like a programming language where genes are being turned on and off all of the time. The cell is constantly doing things, rebuilding itself. Cells are constantly being removed, replaced. The body is constantly rebuilding itself. The body isn't static. Jack was wildly enthusiastic about this, and the fact that a mathematician is wildly enthusiastic about biology shows that the time is now ripe for a new cybernetics: you may use a different word, but the progress in molecular biology has gotten to a point, now, where it is clearly the most exciting thing going on at this time in science, I think. It's not physics, that's not where the excitement is now.
O: And this new cybernetics might not be called cybernetics? Will there be a new word?
C: I heard a lovely lecture by a young woman who is a professor of molecular biology at Princeton University and she was saying how genes, in fact, are split into pieces.
O: What was her name?
C: Laura Landwebber. She is interested in doing computation with DNA, among many other things. She was explaining to us at a conference that I was at in Santa Fe, New Mexico, that, in fact, some genes are split into pieces on separate chromosomes and they have holes in them and different gaps and so they have to be spliced together and the whole thing looks very much like what goes on in a digital computer in many ways.
Then at my lab I heard another lecture, by a physicist, Marcelo Magnasco, who is now doing biology at Rockefeller University in Manhattan. He was doing research on how canaries learn songs. Unfortunately he was doing it by sacrificing these poor canaries and slicing up their brains. But one thing that amazed me is that there is a gene which seems to be involved with plasticity. It's when the body is modifying itself, it seems to be a central gene. And this gene is unfortunately involved in cancer, for example, because sometimes when this mechanism fails, when something goes wrong, the body starts developing cancer rather than doing what it should. But it seems that this gene is turned on in parts of the canary's brain and this gives Magnasco a way of seeing where learning is taking place. The brain is modifying itself and he can see, for example, when it learns another canary's song, because depending on the frequency, there is a very localized response in the canary brain. And if the song is not from another canary then there is a more diffuse response because the canary doesn't recognize it. The physicist Marcelo Magnasco can actually see learning taking place in the canary brain, which is amazing, by seeing when genes are being turned on!
And he explained to us that the body constantly rebuilds itself. Genes are being turned on and off all of the time, cells are constantly being told to self-destruct. There is a name in Greek for that, apoptosis: programmed cell death. Cells are ordered to self-destruct. One of the sources of cancer is if the cell doesn't obey. When something goes wrong in a cell the body gets rid of it and replaces it; the body is constantly rebuilding. For example, the reason that exercise is good for you is because the body is constantly rebuilding itself depending on what it sees it needs. And all of this is really very exciting, I think, intellectually.
O: This also leads us to questions of interdisciplinarity. Have you been involved in interdisciplinary think tanks or meetings?
C: Interdisciplinary ideas fascinate me, personally. For several years I attended a series of conferences at a polar research station in Abisko, Sweden, organized by John Casti and Anders Karlqvist. Anders Karlqvist, the co-organizer, is in charge of polar research or used to be in charge of polar research for the Swedish Academy of Sciences. The general theme was to look for bridges, to invent a new mathematics for biology. And in a funny way, I tentatively summarized my conclusion at one of these meetings by saying that the idea that we could make biology mathematical, I think, perhaps is not working, but what is happening, strangely enough, is that maybe mathematics will become biological, not that biology will become mathematical, mathematics may go in that direction!
O: And has this been published?
C: Well, I make some comments along these lines in the last chapter of The Unknowable. They used to have an Abisko meeting every year. They in theory wanted to have one book for each meeting, but in practice, maybe only half of the meetings resulted in a book.
O: Can you tell me more about John Casti's involvement in these conferences?
C: John is very interested in these questions. This is where the excitement is really: at the boundaries between different fields. There are two kinds of science: One kind of science is when you already have an exciting field and you need to progress in that field. The most exciting thing for me is when you create a completely new field.
Now I remember what I wanted to say about the relation between the arts and science, for example, look at mathematics. Mathematicians say a proof is beautiful. They sometimes use the word "elegant" but you hear the word "beauty" very often. Physicists say a theory is beautiful or that it is too beautiful to be wrong. So what is this notion of beauty? Beauty is certainly an important word for artists too. Although some artists have told me that beauty isn't fashionable anymore.
O: It has been re-introduced in a spectacular way by an American critic, Dave Hickey, at the beginning of the 1990s.
C: To me beauty is a key thing. For example, the novelist Rebecca Goldstein and the playwright David Auburn, both of whom have written works of fiction about mathematicians, [Her novel The Mind-Body Problem and his play Proof.] both of them read the book A Mathematician's Apology by the English mathematician G.H. Hardy. And Hardy is talking about beauty. He says there is no permanent place in the world of mathematics for a proof that isn't beautiful. The first proof, usually, is ugly because pioneering work is difficult. But mathematicians are searching for beauty. They want a proof that is elegant. They want you to get a shiver up your spine. Like when you are confronted with a great work of art.
O: What would be the most beautiful proof in mathematics, the most shivering?
C: When you do the work yourself. One has that sensation when one discovers something oneself. That is the pay-off. That is what one is trying to do. For example, I had a theory and I redid it. It was ugly. I thought the theory was going in the right direction but I felt something was wrong. The ideas did not fit together harmoniously. So I redid it and I came up with a new theory where I lost some of the results, unfortunately, but the new theory is so much more beautiful...
O: Which theory is this?
C: I call it algorithmic information theory. It's my theory of information complexity. Let me give another example of beauty. When I was young, I was crazy about women and I would be completely overwhelmed if I saw a beautiful woman and I had a similar sensation when I saw a beautiful proof. Now it wasn't a sexual sensation but to me they were analogous feelings of beauty. Of course there is no sexual component to that feeling but for me it was analogous.
O: Can you describe some features of a beautiful proof?
C: Well, part of it, I think, is you think a proof is beautiful if it is illuminating. Also---I think I am taking this from Hardy---if it is surprising. It has to be surprising because it is not interesting if you already know it, there has to be a surprise, but it has to then seem inevitable. After the initial surprise it has to seem inevitable. You have to say, of course, how come I didn't see this!
The best mathematics is inevitable. The best mathematics is fundamental and seems necessary. With that kind of mathematics, it is fair to say that you discovered it, that it is not invented. Some mathematics which is more superficial, which is not as fundamental maybe, is a little bit more like literature in that, perhaps, you are inventing it. You don't get the feeling that if this mathematician hadn't done it some other mathematician would have done it, necessarily, because you couldn't avoid the ideas.
Is mathematics discovered or is it invented? When I was young, I thought things were black or white but as I grow older I understand that everything is complicated and different viewpoints are also correct. So, some days of the week I think mathematics is invented, other days of the week I think it is discovered. I mean, both viewpoints have validity and they illuminate the same subject from different angles.
O: BOTH AND instead of EITHER OR instead of NEITHER NOR.
C: Yes. There are elements of both. But I think there is definitely an aesthetic component to science. That may mean that we are imposing our aesthetics on the physical world. Maybe that is wrong. Maybe we should not have a notion of beauty, we should see what nature tells us. But the act of creation is hard. To create a new theory, you have to have strong emotional reasons to want this new theory and if you are doing it for money or for practical purposes, that is one thing, but if you're doing it for the fundamental understanding or illuminating understanding of new ideas, it is an aesthetic criterion that you are following. The same as in the arts, right? One of the things I like personally, is a theory which has a few fundamental ideas, unifying ideas. My mind works that way. I like unifying ideas. I don't like complicated technical theories.
But the physical world doesn't care what I like or don't like. Biology is very complicated. Some people say there are very few unifying ideas in biology; there are always exceptions. So it is possible that I am imposing an aesthetic criterion, but the physical world doesn't have to pay any attention. The science of the future may well get much more complicated. And, who knows whether there will be any unifying ideas? So, maybe it is a psychological need in some of us researchers to try and find these unifying ideas. But the physical world may decide that we are wrong. It may turn out that things are very complicated and messy. That is another sense in which science is more of an art than people realize.
O: And how do you see the supersymmetry of superstring theory as a unifying idea, unifying quantum mechanics and relativity?
C: Well, superstring theory is a very good example. I think that the arguments in favor of superstring theory, which is really fashionable now, are really artistic arguments.
O: Did you read Greene's book The Elegant Universe?
C: I know Brian Greene. I helped him to get a job at my laboratory when he was a high school student, because they told me he was an extremely bright boy. And I met him a few times, not so long ago.
Superstring theory has no experimental evidence in its favor. The arguments in favor of superstring theory are really of an aesthetic kind. So far there is no way to test the theory. People just say that it is so beautiful, it has to be true. They all say there is no alternative, we don't know any other theory. I'm exaggerating a little. Not everyone thinks this way. Some of the old-timers like Shelley Glashow who has a Nobel Prize in physics and used to be at Harvard, don't like superstring theory.
Let me give another example. Brian Greene has a joint appointment in the math and physics departments at Columbia University. Why? The math department thought that what he was doing was not math it was physics, and the physics department thought that what he was doing was not physics it was math. What it really is, is some strange kind of art. The argument in favor of it is that these ideas are so beautiful that all of the brightest young researchers love it! And it is tremendously difficult, you have to be very bright to work in this area. Brian Greene is very, very bright. They think that these ideas are so beautiful even though there is no experimental evidence in their favor.
O: So you would say the beauty of superstring theory is an attractor?
C: It attracts young people. By the beauty of the ideas, I think one would have to say that is the real reason. It involves some mathematics that is very difficult, and connects it with fundamental physics. So you have to be good at both subjects, really, to work in that field. But Shelley Glashow used to say, "This isn't physics, it is theology!", because there is no way of testing it. Superstring theory deals with energies which are so high that there is no conceivable way to test it in the foreseeable future, though hopefully someday they will. So what happened? The result is that Shelley Glashow left Harvard and went to Boston University where he said that there were still physicists who are interested in experiments! Everyone at Harvard is doing superstring theory and what Glashow does is now called in a dismissive way "phenomenological physics". What that means, is that people now consider that the kind of physics that he is doing is not really of interest, it is not theoretical. He is just looking at experiments and looking for patterns in the experiments but is not really grounded in theory.
O: Superstring theory took over Harvard basically?
C: It took over Harvard. Shelley Glashow had to leave, and he had been his whole career there! It is very unusual for a Nobel Prize winner to leave the institution where he got the Nobel Prize. The institution normally won't let that happen. They will keep him there at all costs. They will double his salary if necessary. If you read Brian Greene, he gives a lot of arguments in favor of superstring theory. But if you read certain parts of his book he does admit that the experimental evidence is rather tenuous at the moment. I think that people in the field admit it. Witten, the leading figure in superstring theory, says superstring theory is physics from a century in the future. We, by a quirk of fate, are getting glimpses of it, but this theory is not for now, it is a theory for one hundred years from now!
O: Witten leads me to the question about what Howard Gardner calls "Creating Minds".
C: Let me tell you my personal theory of how somebody becomes a scientific genius. Look at Wittgenstein, for example. Wittgenstein was a lunatic. He was a philosopher, not a scientist. He is considered a great philosopher, right? But there are lots of lunatics who are lunatics in uninteresting ways... Let me make my point in science. To create a new theory of science, you have to be mad. You have to have for some insane reason, this unjustified belief that all the current theories are wrong and that the physical universe is completely different. Now at the time you do this, the reason you are a genius, is because you are doing this at a time when there is almost no evidence.
Is it that by telepathy you read God's thoughts? No, what happens simply is that you have a prejudice for some reason. Maybe you believe in astrology. Maybe you have some philosophical prejudice or some religious prejudice or some psychological aspect that leads you to believe in this crazy idea. If the physical world doesn't happen to believe in this idea, then you are considered to be a fool or an eccentric. But if it happens that the physical world also believes in this idea, then you are considered a genius and everyone says, "How did he do it?!" There may have been some clues, but partly it just so happens that his madness was the right madness at this particular moment for physics to advance. But then maybe fifty years later, it is no longer the right madness.
When Einstein did his theories, he had the right psychology for relativity, but he hated quantum mechanics. And he had no interest in high energy physics, he spoke of it with contempt. He said it was like zoology, because there were so many sub-atomic particles. He wanted to understand just the electron, he just wanted fundamental knowledge. Murray Gell-Mann, who is at the Santa Fe Institute, is not at all like Einstein. Einstein was only interested in the most basic ideas. Murray Gell-Mann is interested in everything. He knows dozens of languages, he is interested in birds, he has a tremendous memory.
O: I recently saw a film by Pipilotti Rist about his archive, it is amazing.
C: Yes, he is just interested in everything. He has an encyclopedic mind. When you meet him---people told me this and then it happened to me---he tells you how your name should really be pronounced and what it really means, and from which language it comes! Many people find this very offensive but I find it very fascinating. This is the encyclopedic mind, the kind of mind which he needed to deal with high energy physics, with the particle zoo. It was a zoo and you needed a mind that could take this enormous number of facts and organize it. It helps if you have the personality that is needed at that moment to take the next major step.
O: Can you tell me about your thoughts on genius which you mentioned in relation to Wittgenstein?
C: Right, there are many degrees of this. There is an element of madness in the sciences as much as in the arts, I think. You see, you have to be crazy to think something at a time when there is almost no evidence for it and go off in a different direction from the rest of the scientific community. And the scientific community will usually fight you. Then they will erase the history of how all of your contemporaries who had the political power fought against these new theories! That will be erased to make it sound like science is progressing linearly, always going forward. Science, of course, is also full of emotion, of controversy and politics, because human beings are political animals. Science is a human activity and it is much more akin to the arts than people realize. Now, of course there is art that isn't art, and that is also true in science. There is science which is very short-term or which is done only for immediate, financial or technological gain. That science can be very valuable, very useful, the same way that bad books can make a lot of money. In the United States they will sell a million copies of a book by Stephen King. And I don't think Stephen King has any desire to write great literature.
O: There seem to be more and more science book best-sellers lately. Is this a new phenomenon?
C: It happens sometimes. It happened with Cybernetics: or Control and Communication in the Animal and the Machine by Norbert Wiener. There are science best-sellers that go back in time. I have a copy of a two-hundred year old book written by Laplace, Exposition du Système du Monde. It is an exposition of the Newtonian world-view and of astronomy.
O: It was a best-seller at the time?
C: It was a best-seller two hundred years ago. It is beautifully written, tremendously accessible but still at an enormously high intellectual level.
O: Do you want your books to be best-sellers?
C: I don't really care about best-sellers. It is not the number of books you sell that counts. I would consider my books a success if only one person reads them and that is the person who takes the next step forward from my theory as I went forward from Gödel and Turing.
O: Could you tell me about the beginnings of your work. You mentioned that you grew up in the context of an intellectual home, a film-maker played a role. Could you tell me a little about the beginnings of your entering the field of mathematics.
C: I started very, very young. I am self taught, in fact. I have a high school degree, from the Bronx High School of Science. I don't have a college degree. I only have an honorary doctorate.
O: Can you tell me about your time at high school?
C: Yeah. In 1956 when the Russians put Sputnik in orbit, the United States got terrified, so they started having special programs for gifted children to study science. I benefited from all of this. I went to a special NY high school for science called the Bronx High School of Science; there was a wonderful science and math library there. I was very fortunate that I was there at the right time, at a time when the United States was trying to make new scientists. I took university level courses in high school and they were wonderful courses because they were done by some of the best people in their fields. They were really new. They were not following the old curriculum, the old subjects. In every way they were very up-to-date and modern presentations of mathematics, physics, chemistry and biology. I didn't have to waste time with a course which was really out of date.
And I also went to a program at Columbia University for bright high school students. These were professors at Columbia University teaching, Saturday morning, bright students from high school. That was wonderful. One of the things they did which was maybe even better than the course itself was that they let me use the Columbia University libraries. I was allowed to look at the books. I was reading immense quantities of books on my own. I was an unbearable child; my mind was ablaze with mathematics and scientific ideas.
O: It was early that your interest and study of mathematics started. Was there a book or something that triggered it?
C: Well, I swallowed up many books. I looked for books that I could study on my own, books that emphasized the fundamental ideas. One of these books was A Mathematician's Apology by G.H. Hardy, that I mentioned before.
O: You read that very early?
C: I read it very, very early. It is a delightful book. The normal textbooks require that you study one by one a vast series of textbooks. I looked for books that enable you to just parachute or jump into a subject without having to do fifteen courses first.
O: I see a similar thing with your own books such as The Unknowable and the Limits of Mathematics where one can parachute in without so much prior knowledge.
C: I try to write the book that I would have wanted to read when I was a child. When I was young one of the books that influenced me was a very romantic book of biographies of mathematicians called Men of Mathematics, by Eric Temple Bell. He was a very lively writer, very witty. All of the great mathematicians are there. One of the things I noticed in this book is that one of the mathematical heroes is Galois who died in a duel because he was a subversive republican at a time when to be a republican was considered subversion. And he died maybe around twenty. Nevertheless he is famous, he invented a new field of mathematics. So, as a joke when I was a child I said to myself if I don't have a great idea by the time I am eighteen, it is finished. And I did have that idea, when I was only fifteen!
O: Can you tell me about your new book, which after The Limits of Mathematics and The Unknowable is the third book in a series published by Springer?
C: My new book is called Exploring Randomness, and it is the third book in this series. It is somewhat more technical, but there might be parts that you would enjoy anyway. I usually begin and end with a chapter that is understandable even if the middle is more technical. And just last week we worked on the cover of the book and the publisher proposed three covers and they were all wonderful covers but two of them were striking computer images. I chose a cover that was a photo of mountains seen through the mist with trees. It looks very Japanese, I think. One of the reasons I chose it was because computer images have become ever present and it is striking now to see an image of nature because we are living in such an artificial environment. I love hiking and just this weekend I climbed the highest mountain on the US east coast. It is called Mount Washington. It is 6300 feet high. It's not a climb, it's a hike. You don't need to use ropes or anything like that. I like to get away from the artificial world that human beings have created, to nature.
O: How does your Springer trilogy of books unfold?
C: Well, these are three different ways of looking at my subject. The Unknowable, which happened to be the second book, looks at the history of ideas leading up to my subject. It really talks about history, the historical context, the controversy over the foundations of mathematics in the 20th century, and my work is just one chapter in that history. A number of fields were developed to try to understand what are the foundations of mathematics, and if mathematics has foundations or not. A number of fields were invented. Mathematical logic was perfected, computability theory and the computer were invented, and my subject is algorithmic information theory. These are three fundamental new fields which were created to try to understand if mathematics has a firm foundation or not and to see how firm the foundation is.
O: So this book gives a context for your whole work?
C: It gives the context. I don't pretend that I am giving an objective history. I am giving the history the way I see it, of the ideas that led up to my work. I am giving a very biased view. And the first book is The Limits of Mathematics, which is really what I consider the more subversive or revolutionary aspect of my work, which is the message that it gives about the foundations of mathematics. The message is that mathematics is quasi-empirical, that mathematics is not the same as physics, not an empirical science, but I think it's more akin to an empirical science than mathematicians would like to admit.
Mathematicians normally think that they possess absolute truth. They read God's thoughts. They have absolute certainty and all the rest of us have doubts. Even the best physics is uncertain, it is tentative. Newtonian science was replaced by relativity theory, and then---wrong!---quantum mechanics showed that relativity theory is incorrect. But mathematicians like to think that mathematics is forever, that it is eternal. Well, there is an element of that. Certainly a mathematical proof gives more certainty than an argument in physics or than experimental evidence, but mathematics is not certain. This is the real message of Gödel's famous incompleteness theorem and of Turing's work on uncomputability.
You see, with Gödel and Turing the notion that mathematics has limitations seems very shocking and surprising. But my theory just measures mathematical information. Once you measure mathematical information you see that any mathematical theory can only have a finite amount of information. But the world of mathematics has an infinite amount of information. Therefore it is natural that any given mathematical theory is limited, the same way that as physics progresses you need new laws of physics.
Mathematicians like to think that they know all the laws. My work suggests that mathematicians also have to add new axioms, simply because there is an infinite amount of mathematical information. This is very controversial. I think mathematicians, in general, hate my ideas. Physicists love my ideas because I am saying that mathematics has some of the uncertainties and some of the characteristics of physics. Another aspect of my work is that I found randomness in the foundations of mathematics. Mathematicians either don't understand that assertion or else it is a nightmare for them...
O: It is rejected?
C: Yes, because mathematicians think there CAN'T be any randomness! A mathematical assertion is either true or false, it can't be true with probability a half. Mathematicians think they have to believe in absolute truth. Physicists, on the other hand, do believe in randomness. Randomness is one of the basic themes in the physics of the 20th century. So physicists are delighted by my work. I get invited often to physics meetings. Physicists feel much more comfortable with my ideas than mathematicians, because I took an idea from physics which is randomness and I found it in mathematical logic. But people in mathematical logic don't like that, they don't understand randomness.
O: Which leads us to the third book because the third book is about randomness?
C: My third book gives the technical heart of my theory. I wanted to have a book where I really presented the technical heart of my theory as understandably as I could. Because, you see, my first two books omit a great deal. Neither the first nor the second book gives the technical core of my theory, the fundamental mathematics. My third book, Exploring Randomness, is an attempt to present the whole thing as understandably as I can. For many years, I thought there was no way to do it. I had despaired of writing that book. But somehow it wrote itself in less than two months!
I started writing this book, mid/late August, and I finished it in late September. My brain was on fire. I couldn't sleep or eat. This was one of those great moments of inspiration like with Feynman. For seven years I had despaired of being able to explain the fundamental heart of my theory in an understandable way. I thought there was no way I could explain it. It was hopeless; too complicated, too technical. Well, this is still a difficult book but I have made it much more understandable than I had ever thought was possible.
O: It is a very different idea than people usually have of mathematical work.
C: It takes tremendous emotion to do good mathematics, it is very difficult. You have to be inspired and you have to have tremendous emotional drive to do it. You are not a machine, by any means, because the act of creation is magical. There is no rule for doing it in science, the same way there is no rule for doing it in the arts. There is no systematic way to do it. They can't teach you in school how to do it, they can just get out of your way!