Good morning everybody. Let me start by saying I'm delighted to be back in Japan. I love Japanese architecture and design. I think this auditorium is magnificent. I've been admiring it the whole time I've been here and I was really in love with the cafeteria where we just had coffee and I thought it was so beautiful. So I'm very happy to be back here in Japan. We're talking about bridging the gap and I think there are a number of gaps that we have to bridge.
One gap is the gap between east and west. I think that both east and west have something to contribute to world culture. I think we should not merge them, but keep their individual identity. Now the other gap we're thinking about in this conference is between art and science and for me there's not much of a gap between art and science either because I love art. An artist I like is Magritte, an old artist. The kind of art that is done nowadays might be difficult for artists from a previous era to recognize as art. I'm a mathematician, and mathematics and art now are based on ideas. The talk you just heard started by talking about a museum where there is no art. The art is in transit - it has moved elsewhere. And all you have left are signs or something saying `object temporarily removed'. I think that's a wonderful idea. So artists are constantly reacting against the concept of art that existed before. A lot of current art criticizes art or stands outside of art looking at it - museum installations that make fun of or mock museum installations. The same kind of paradoxical thing happens in mathematics.
The kind of mathematics I'm doing many mathematicians react to violently saying it's not mathematics - you're attacking mathematics, you're against mathematics, you're using mathematical methods to criticize mathematics. One of my recent books is called The Unknowable and an immediate reaction from someone when he heard the title of this book was `well, how can you write a whole book about something that's unknowable?' Just saying `unknowable', that's it, it's finished when you've said that.
So there is a paradoxical element in current postmodern (or is it post-postmodern?) art - it's getting more and more postmodern all the time. And that's also true in the kind of mathematics I'm trying to do where I'm using mathematical methods to criticize the ability of reasoning and of mathematics. Now of course you're pulling out the rug from under you as you do this, because the tool I'm using is mathematics, but I'm using it to try to understand better what mathematics can or cannot achieve. But when I start talking about the limits of mathematics, since I'm a mathematician doing mathematics, there is a bit of a paradox in that whole enterprise. And also psychologically it is a bit delicate because if mathematics has limits and I begin to question the ability of mathematics but I've been using mathematics to do all this, so where does this leave me? It leaves me in some kind of a strange psychological space, a little bit like when you see a museum installation which has no objects on display because they were all removed. It's a similar kind of feeling.
So let me tell you a little bit about the kind of work that I've been doing and other people have been doing, in which mathematics tries to see how far it can get, and tries to see if there are things which are unknowable using mathematical methods, if there are things beyond the power of reasoning, beyond the power of mathematics. To see if we can reason about things which we can't reason about, to see how far reasoning can take us. The first work in this area, which in a way is a postmodernist piece of mathematics done in 1931 by Kurt Gödel, was Gödel's incompleteness theorem. And how did Gödel show that there were limits to mathematical reasoning using mathematical reasoning? The idea is very beautiful. There is an old paradox, the paradox of the liar: This statement is false. What I am now saying is false. And then the question is well, is it true or is it false? And if it's true and I'm saying it's false then it must be false. On the other hand if it's false and it says it's false, that means the opposite must be the case, and therefore it must be true. So it can neither be true nor false to say this statement is false. So that's a paradox, the old paradox of the liar, or the paradox of Epimenides which goes back to the ancient Greeks.
Now Gödel made a modern version of it; he made a very subtle but fundamentally important change. He constructed a mathematical statement that does not say `I'm false', `I'm lying', instead it says `I'm unprovable'. And this is very, very important, this change. So it's a mathematical statement that says about itself `I'm unprovable', `I cannot be demonstrated'.
Now let's think about this and see where it leads us. There are two possibilities. One is that you can demonstrate this mathematical statement that says `I cannot be demonstrated'. The other possibility is that you cannot demonstrate this mathematical statement that says it cannot be demonstrated. So let's see what happens in each case. Well what happens if we can demonstrate this mathematical statement that says it cannot be demonstrated? Well that's terrible, we're demonstrating something that's false. So let's hope that doesn't happen. The only other possibility is that we cannot demonstrate this mathematical statement. But it says `I can't be demonstrated', so if it cannot be demonstrated what it says is true. And mathematics has a hole, there's a gap, there's a limit to what mathematics can achieve.
So this is this wonderful piece of postmodern mathematics from 1931, and a friend of mine, Vladimir Tasic, in a book that's coming out called Mathematics and the Roots of Postmodern Thought (Oxford University Press in the United States is publishing that any month now) says that in some funny way you can argue, it's not the truth but you can argue that all postmodern philosophy and art in a way emerges from Gödel's incompleteness theorem. That's of course a vast exaggeration, but there is a tiny bit of truth in it. I would say that Magritte or maybe the Dada movement and Gödel's incompleteness theorem are the beginning of postmodernism in a way. I think that's a more accurate statement.
So what have I added to all of this? I have my own version, my own approach to incompleteness which is very different from Gödel's. It starts from a different paradox. It's not the paradox of the liar - I don't start with `what I'm now saying is false', I start with a different paradox. So let me tell you this paradox which leads in a completely different direction from Gödel. My approach is based on the following paradox. Let's think about the `whole numbers' - positive integers. These are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, just these numbers. They are very familiar, friendly numbers for counting. Let's somehow divide these whole numbers into two groups - the ones that are interesting and the ones that are uninteresting. Somehow you distinguish between the ones that are interesting, that appeal to you, that stand out, and the positive integers that just merge into the herd, that don't really stand out from the rest in any particular way. Interesting versus uninteresting numbers. Then you get into problems.
Think of the first uninteresting number. You just keep on going until you get to the first uninteresting number. But wait a second, this just happens to be precisely the first uninteresting number. That's a rather interesting fact about it, isn't it? It means this number does actually have a unique characteristic that makes it stand out from the majority of numbers. So it would look as if there cannot be any uninteresting numbers - they all have to be interesting. So this is the paradox that my work is based on, instead of the paradox of the liar, and the way I develop it further into a result on the limits of reasoning, is I start asking, well, can I prove that a number is interesting or uninteresting? I have a definition, a more precise mathematical definition of whether a number is interesting or not. And then the question is, what if you try to prove that individual numbers are uninteresting? And the answer is, you can't. And the reason you can never prove it, even though I have a mathematical definition of what an uninteresting number is, is that it turns out the most interesting fact about this definition of the uninteresting is that you can never be sure that a number is uninteresting. You can never prove that an individual number is uninteresting. It turns out the majority of numbers are uninteresting according to my definition. But what if you try to prove that an individual number is uninteresting? Well you get a paradox. Because it basically works like this: the first number that you can prove is uninteresting is, by that very fact interesting. So it seems you can never prove that individual numbers are uninteresting, even though most numbers are uninteresting according to my definition - I have a precise mathematical definition of uninteresting numbers.
So you may find this interesting or uninteresting, that's another matter! [audience laughter] And I've gone a little further in this analysis and what I'm really thinking about is not whether numbers are uninteresting or interesting, I'm talking about whether numbers are random or not. A random number is one that doesn't have structure or pattern, and a number is not random if it has some structure or pattern. So really, instead of interesting or uninteresting, I'm really talking about numbers being random or non-random. And the interesting thing about my theory of randomness is I was able to discover areas of mathematics where things are completely random according to my definition of randomness. I've been able to discover regions of the mathematical world where mathematical truth is completely random, where it has no structure.
So let me just try to state more forcefully, this black hole that I've discovered in mathematical truth. I'm not saying all of mathematics is random, it's not, mathematics is beautiful. I'm a mathematician, I'm not against mathematics. But I have discovered an area or constructed an area where mathematical truth is completely random or structureless and escapes the power of reasoning, and will forever escape the power of reasoning.
Let me tell you more precisely what I've discovered. The normal notion of mathematical truth, and this goes back to Leibniz, the great philosopher and mathematician, is that if something is true it's true for a reason. Mathematicians particularly believe in reason. And in mathematics the reason that something is true is called a proof, and the job of the mathematician is to find proofs, it's to find why something is true, the reason. So this is also the rational notion that anything that's true has to be true for a reason. This is the notion of rationality, that the world can be understood through reasoning and you can trace this back to the ancient Greeks. So what I've discovered are mathematical facts that are true for no reason, they're true by accident. They are random mathematical facts. And since these mathematical facts are true for no reason we're never going to be able to prove them. Another way to say this is that I've discovered regions of mathematics where God plays dice with mathematical truth. In a way my inspiration actually came from physics. If you look at quantum physics (and Anton Zeilinger will be talking more about quantum physics), there are a number of revolutionary things about the physical world that quantum physics tells us. And nowadays people concentrate on entanglement, which Anton will talk about, but that has nothing to do with what I'm doing. I'm dealing with one of the old things that quantum mechanics told us which is that the world contains randomness - that the world is fundamentally unpredictable, that the laws of nature are statistical or probabilistic.
As a child I loved physics. I had originally wanted to be a physicist. I went into mathematics but I brought the idea of randomness with me into mathematics. And I found randomness in a very unusual place - I found it in the foundations of mathematics. And the result of this is that mathematicians don't like this; for them randomness is an alien idea coming from physics, but physicists tend to be sympathetic. They tend to like the fact that I found the disease of randomness not only infects physics, it also infects mathematics.
So to end I would like to tell you a few more personal paradoxes about the kind of work I'm doing. I'm talking about the limits of reasoning, and Gödel talked about the limits of mathematical reasoning, the limits of the axiomatic method, the limits of formal logic. And you may think these are negative, pessimistic results, you may say `what kind of a mathematician attacks mathematics like this?' Well I don't view these results as negative and pessimistic, not at all. I should also mention Turing. I haven't discussed his work, but my work is based more on Alan Turing's work than on Kurt Gödel's work. He and I, in revealing limitations of mathematics, have actually created new mathematics. In a way, our very results contradict what we're doing. Gödel came up with new mathematical ideas, Turing came up with new mathematical ideas, I've come up with new mathematical ideas - randomness. In Turing's case it's the notion of computability. In Gödel's case it's the notion of incompleteness. And these are fundamental new mathematical ideas, so in fact we talk about the limits of mathematical reasoning, but that's only limits of mathematics viewed statically. But mathematics is not static. As Israel Rosenfield said, mathematics evolves, mathematics changes. And the `negative', `pessimistic' results by Gödel, Turing and myself actually can be taken very positively and optimistically, as showing how mathematics gets around incompleteness by creating new concepts. Mathematics is not stationary, mathematics is evolving and changing.
Another thing I would like to say deals with the connection between mathematics and art. The normal view is that artists work on inspiration, they work on intuition. And that mathematicians are not at all like that, that mathematicians are rational, and they're cold, unemotional human beings with passionless souls. Well that's not at all the case, let me point out. Kurt Gödel, who's viewed as a mathematical saint by many - he met his wife in a Viennese nightclub. She worked in the nightclub. She was a dancer. So Gödel was not indifferent to the charms and attractions of a beautiful, sexy woman, in spite of being a mathematical saint as he's viewed nowadays! [audience laughter] Gödel used to spend a lot of his time in nightclubs when he was young in Vienna. This was normal for the son of a well-to-do Viennese family. It was natural for him to spend a lot of time in nightclubs. What was not natural was that he also did some mathematics - that was unusual!
In my own work I have to say that I work completely on the basis of intuition. It's totally irrational. In creating a new field of mathematics you have to work completely on instinct. You're looking for new concepts. You are working with unconscious emotions and it's a magic, mysterious process. Once you come up with an idea, a new idea on which to base a new field of mathematics, then there is, I agree, a rational element in mathematics which is that you have to verify that the idea works. But the act of creation in mathematics is just as magical and mysterious as the act of artistic creation. I would also say that mathematics and art are much more similar than people realize, in that I would say that mathematics is an art. I would say that good mathematical ideas have to be beautiful. A mathematical theory may start off ugly when it is first created, but as G. H. Hardy once said, `there is no permanent place in the world for ugly mathematics'. Mathematical ideas have to be beautiful or they don't survive. So in a funny way, in fact looking back at my own life, I'm crazy about women, but I haven't had that much to do with women actually. I've found for me, mathematics was almost like being with a beautiful woman. I found mathematical ideas to be very sensual and I think there's an awful lot of sex also in the arts. I would like to say that some of us mathematicians also find sensuality in mathematics, in the beauty of mathematical ideas.
Mathematics is more like an art than outsiders might realize. Of course there are many kinds of mathematician like there are many kinds of artist, but certainly in my case, I value mathematics as an art. And I think maybe the few words I've said may convince you that at least in my case, but I think also in the case of many other mathematicians, it's more like artistic creation. Mathematical creation is more like artistic creation than is commonly thought. Thank you.