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!
