Interview by Tor Nørretranders
How to be a Mathematician

Copenhagen, 1996, and my second TV interview. Tor Nørretranders is a Danish science journalist and intellectual who is extremely well-known in Denmark. Only one of his many books, the Danish best-seller Maerk Verden, on information theory and consciousness, is available in English, as The User Illusion. This interview was part of Tor's fabulous Mindship institute, an entire summer of intense dialogue between artists and scientists, held in spectacular abandoned waterfront Navy buildings, including a boiler factory (kettle smithy) and a submarine assembly hall.

N: What's the daily life like for a mathematician?

C: What's the daily life like for a mathematician! Oh my God, I don't know how to answer that one! Well, it's marvelous when one has a good idea. Because then you can throw your whole personality at it.---And I can just speak personally.---I can throw my whole personality at it, and then I'm completely wrapped up in it, and since I live alone, that means that I can just exist just for the idea, until I develop it. And when I'm in this state, I feel I see things much more clearly, I'm much more alive. I feel sort of like one feels when one hikes up a mountain. You know, you don't have a clear view---maybe you're going through cloud layers---and then you get to the top, and there you are with a marvelous view, maybe you've gone above the clouds, there's bright sunshine, and it's a wonderful experience! Then of course it's a bit of a let-down when you come out of that! So the problem is the holes, when you don't have an idea that you can run with.

But if I look back at my life, every few years I have a major idea, fortunately. So it's kind of tough in the middle, but, you know, if you look at many scientists, their fertile period is relatively short. Einstein is unusual, he has a fertile period from 1905 to 1915, from special relativity to general relativity, basically. But if you look at Schrödinger, it's much shorter than that. I had the idea for my theory, the first idea, when I was 15. And I just had what I think is a major idea just three years ago, that I've been working on since then to develop... I didn't think that it would last this long! And I guess I'm lucky that I have a problem which is interesting enough that I can give it my whole life. Otherwise, what do you do with the rest of your life?!

I would say that for any significant problem the minimum quantum of effort is sort of a life-time. I've been obsessed basically with one train of thought since I was a child. And I think that's sort of the minimum effort you have to make to understand a little bit better a really fundamental problem. And in fact many people would be willing to do that if they thought there was some chance that it would work! But the most likely thing of course is that you're obsessed with a problem and you give it your whole life and you don't advance even one little step. So I consider myself lucky that this sacrifice paid off. Otherwise I should just be considered to be a lunatic who threw his life away.

N: And this idea you got when, when you were... You were starting thinking about this as a child and then as a teenager at the age of 15 you got an idea that you've been working on basically since.

C: Right. Well, I was interested in, originally it was quantum mechanics and general relativity, I think, as a child, that fascinated me, because they seemed so deep and so exotic, sort of like the magic stories I'd liked as a really small child. My view is that science is crazier than magic. The real world is crazier than science fiction or fantasy. But to read the physics you have to understand some mathematics, and so I tried to learn some math. And there seemed to be something rather mysterious, that sort of in a way was analogous to Einstein's theory and to quantum mechanics, in math. And that was Gödel's incompleteness theorem. And it fascinated me and it was very mysterious. So I was interested in that fairly young.

But the idea I had at 15 was a definition of lack of structure or lack of pattern. And that's basically what I've spent my whole life doing since then. And it took me a while, a few years, to realize that it was very relevant to the question of the limits of mathematics, which is Gödel's incompleteness theorem. Originally I wasn't thinking of that, consciously, but maybe it's not surprising that it turned out to be exactly the idea that I needed.

N: And this interest in Gödel's incompleteness theorem came at what age?

C: Well, I don't know, 11, 12, 13, 14, ...

N: If you can sort of wheel back the tape to that age, how did you see Gödel's incompleteness theorem at that time?

C: Well, it looked very mysterious. It's a mathematician using mathematical methods to say that mathematics has certain limits. You're using mathematical reasoning to criticize mathematics. That's already a bit strange. And the other thing about it was that I could read the proof, Gödel's proof, sort of step by step, and every step sort of seemed okay, but the whole thing escaped me, it seemed very paradoxical. It's sort of as if you're sort of going crazy! You know, it has this funny paradoxical aspect!

So somehow I didn't feel I understood it at all, not doing it the conventional way, the way that Gödel did it. And I think the only way to understand a mathematical result is to prove it yourself, to find your own proof. When you struggle for it, then you understand it! Reading somebody else's proof in a book doesn't give you understanding. I would say the same thing about computer programs. To understand an algorithm you have to program it yourself. I think there's no substitute. So as I struggled to find my own proofs of Gödel's result---going through Turing's technique, basically, starting from him...

N: Alan Turing, the English mathematician?

C: That's right. Starting from there, I was able to find a series of proofs that had my own personality, and then I started to understand Gödel's incompleteness theorem.

N: And what happened to you when you were starting to understand it?

C: Well, the first step, the very first proof I found on my own of Gödel's incompleteness theorem, was the summer between my high school and my first year in college...

N: You were like 16, 17, 18?

C: I think, maybe, I don't know, 16 or 17. And then the paper that I wrote, the first major paper I wrote on my definition of randomness, I did it, it was the summer between my first and second year at college. That was one year later, so I was 18 then. And it got published a year later, when I was 19. And it was very, very intense! I guess I was a bright kid.

Of course there was a price I paid for that. One of the prices I paid for that was---and in Copenhagen it's very easy to think about this!---was that I didn't chase girls very much. I mean, I was crazy about them, but from a respectful distance! What I was doing all the time was carrying math books and gobbling them up, piles of books at my home! But I made up for it later!

I read a rather romantic book with biographies of mathematicians, it's by Eric Temple Bell, [See also his Mathematics, Queen & Servant of Science, a more serious book, which I adored as a child.] it's called Men of Mathematics, and it has become popular now to say it's a very bad book, a very inaccurate book, and it doesn't have women in it, and all kinds of criticisms. But for me as a child it was a very inspiring book, a very romantic story. And it told about Galois and Abel, and these were very young mathematicians who died young but did some marvelous work before they died. So as a joke, I said to myself, if I don't have a great idea by 18, I'll never get it! I didn't think this seriously! But the funny thing was, that I sort of did!

N: And you did certainly discover something very, very significant, but you were...

C: To me! Not to the average person. To the average person, the limits of mathematics, they don't understand what the problem is.

N: So what's the problem?

C: Well, you see, if I go up to someone and say, hey, mathematical reasoning has certain limits, there are simple mathematical questions that mathematical reasoning will forever be powerless to solve, a lot of people will say, first of all, I don't care about mathematics, and second of all, well, I mean, there are problems everywhere, there are limitations everywhere, you know, I don't have enough money to pay the bills, why should mathematics be any different? Why did you ever expect that mathematics had no limits?

And so, I guess that my friend Walter Meyerstein...

N: ...the Spanish philosopher...

C: ...he puts this in a historical context. He points out that there's a whole school of philosophy going back to Plato, and maybe even to Pythagoras, saying that the whole point of philosophy is to reason, that a rational man is someone who does things not because of belief or because of coercion, but because it's the reasonable thing to do. And this gives reason a fundamental significance. It makes things that can be demonstrated by reason much more solid than things established by superstition or social convention. So perhaps the fact that even in pure mathematics reason has very large limits, perhaps that should suggest that we shouldn't be surprised that reason has much bigger limits applied to human affairs.

N: So this very fundamental idea of Western civilization that reason is what can be trusted, and the reason---excuse me!---that we cannot solve problems, is that we haven't yet learned how to apply reasoning and rational arguments to that problem, that sort of basic idea of Western culture, meets somehow its limits with the proof of Gödel.

C: I think so. It's a wonderful fantasy, especially when you have several human beings, because reason should be absolute, it shouldn't depend on the person, and then reason would give a way that one could agree on the proper course of action in human affairs, or on the ethical course of action, for example. So if reason were sufficiently powerful, then it might give us a way to avoid human conflicts, not just disagreements about mathematical facts, but perhaps disagreements about how we should behave with each other!

So it's a beautiful fantasy, but my suspicion would be that Gödel's work and Turing's work and my own work, should make one very cautious about this. Of course, the other extreme would be to say that reason is powerless and it's just going to be superstition or force! And that's not a very nice idea either, so I don't know where the truth lies. But I think that it's a very interesting question to worry about...

Einstein has a very interesting remark in his intellectual autobiography, I think he calls it his epitaph. And that remark is, I think it goes something like this: even the positive integers, 1, 2, 3, 4, 5, ... are clearly a free invention of the human mind, invented because they help us to organize our sense impressions. So if that's true, there is no necessity... the positive integers are not a necessary tool of thought. If they are a free creation, we're free to make modifications, if it helps us to organize our mathematical experiences. And I think that we should feel more free to do that.

My work does suggest that mathematical questions which escape our power are common, they are not unusual. The question is, are these interesting mathematical questions or not, are they natural or not?

There's also a remark, by the way, of Gödel's which I think also goes in the same direction that I'm talking about. Now Gödel has a completely different view than Einstein. Einstein is an empiricist, he's a scientist, he believes in the physical world, right, that mathematics is all invented. Gödel believes that mathematics exists, that mathematical reality is just as real as physical reality. And he believes we observe, we discover mathematical truths, we don't invent them. We don't invent mathematics, we just discover it, we just observe it. And that's a very different philosophical position from Einstein. But the funny thing is that it leads Gödel to the same conclusion, to the same point that Einstein said. Because if mathematical reality is just as real, it's different, but it's just as real as physical reality, if 1, 2, 3, 4, 5, ... are just as real as an electron or an electromagnetic wave, then why can't we sort of use the scientific method, and if we find a new mathematical principle that helps us to organize our mathematical experience, maybe we should just add it to mathematics as a new axiom, the same way that physicists would!

Here's an interesting fact. I've gotten old enough that I'm not even sure that I believe in mathematics at all any more! I mean, not just because, you know, maybe I prefer to have a family and a more normal life. But also because I don't really believe in real numbers anymore and I don't even think I believe in positive integers anymore.

N: 1, 2, 3, ... ?

C: That's right, I'm not sure I believe in them anymore because they go on forever, so where are these large positive integers? There are an infinity of them, so how do they fit in the physical world, if I take a very empirical point of view. So let me give an example. There's this business called commutativity of addition of integers---that just means that 3 plus 5 is equal to 5 plus 3---and this should be true always, even if the individual numbers are so big that you can't even write them down in the physical universe no matter how small you write. Now do these numbers exist if they are so big you can never write them down, even if you only use one atom per digit? Now real numbers are even worse. A real number is a measurement with infinite precision, you know, 3.1415926... where you go on forever. Now no physicist has ever measured a number with more than I think it's 20 digits, maybe even less. And those experiments are miracles, I think, when they get that accurate!

Now physicists know that there are some questions they shouldn't ask, because they get meaningless numbers out, they get infinities out, so they don't ask those questions. But it seems to me that that's really telling you that these physical theories have problems, and you shouldn't push them too far in a certain direction. And now physicists say, you just shouldn't have asked that question, but if they find a better theory, they may realize that nature was trying to give us a hint. The fact that this theory gave an obviously ridiculous answer was really a hint that something was wrong, maybe that real numbers should be removed from physical theory, that it's all discrete. But we're not smart enough to imagine a better theory ourselves, we need a hint from experiment!

N: In a way, your work and Gödel's work has shown us that any description will have a limit, any logical, any sort of formal system describing something will have a limit.

C: For describing just 1, 2, 3, 4, 5, ... and addition and multiplication! Any attempt to capture all the truth about 1, 2, 3, 4, 5, ... and addition and multiplication will not get it all!

N: So there is no, sort of, full and perfect theory?

C: No final truth! Yeah, I don't think mathematicians have a direct pipeline to God's thoughts! They used to think they did. I think that mathematics is also tentative, and of course the problem is where do you get the axioms from, that's one of the serious problems. Because, you see, that problem was a problem about reasoning, and it was realized to be a problem, I'm sure, by the ancient Greeks. You can't prove the axioms, you've got to start from somewhere, you've got to take something as the starting point, and that problem has always been there. I guess what Euclid said is, he said something like axioms are self-evident truths.

But following Einstein, I don't believe that there are any self-evident truths! I believe that we're fooling ourselves! You've got to deal with uncertainty, you've got to deal with all those things in one's daily life, and it's true that some people escape to mathematics because they think that in mathematics they have this beautiful absolute certainty. It's not messy, it's not like being married or making love or all those messy things in real life, which are great!, but the numbers are clean, pristine and beautiful and you can understand them---I think that's an illusion, perhaps! Life is messy! It's wonderful, but we have to deal with uncertainties...

N: At this meeting in Copenhagen, you were confronted with artists, there are many musicians here, and so on. You find that fruitful, to discuss with such people?

C: I find it fascinating! These are all remarkable people here: extremely bright, they are extremely articulate, and they live in a completely different world than I do! But it's all connected; I think that you selected some good subjects.

I was fascinated to see the following connection the other day, which is that the biologist William Hamilton said that maybe one reason that many animals find a symmetric mate beautiful is because---this is going to be a caricature!---because if that animal were sick, its plumage might not be so regular and perfect, if it had parasites, for example. That would mean that animals would tend to find a mate attractive if it was very symmetrical and regular, for example. As my friend Walter Meyerstein has pointed out, that was the Greek notion of beauty: beauty was symmetry was mathematics! Plato connects all of these things. So then that notion of beauty is sort of like the notion of simplicity, the absence of complexity, a simple structure. And then that would connect with my ideas!

So for one wonderful moment while William Hamilton was talking, I said, wow, that's why you put us all together in this meeting: here I see a connection of ideas from biology to philosophy to mathematics. It was a wonderful moment!

N: It would even seem that when you as a youngster were chasing symmetry and simplicity in mathematics instead of girls, William Hamilton had the explanation for that!

C: It's possible! You mean, it was a good substitute for a while! The girls fascinated me, I was just a little scared of them at that time!

I've always felt that there was something sensual about a beautiful mathematical idea. I mean, I wouldn't want to make love to it, but if it's written beautifully on a page, if it's a beautiful equation or a beautiful idea, you know, I almost feel that reading such a proof and understanding it, is almost like caressing it! I wouldn't want to go too far with that analogy, but I am very sexual, and some of that rubs off in the way that I feel about mathematics, I guess!