David: For you, is mathematics something discovered or invented?
GC: You know, there is mathematics that, as Ulam puts it, fills in much-needed gaps. There are pieces of mathematics and when you find them, they seem sort of inevitable afterwards – they weren’t in advance – and when you find a thing like that, then it seems more real, then it seems that you’re discovering it. But let’s face it, from a practical point of view we’re inventing it as we go.
But it’s true that on good days, when you’ve found something that you really love, you say, ‘Oh, my God.’ You have a feeling of inevitability that you’re discovering something, that there’s this beautiful thing out there that it really doesn’t feel like any mortal could have invented it. It seems to be something from the Platonic universe of beautiful ideas or from God’s mind. Who knows?
So that’s if you’re very lucky. The whole thing seems so beautiful and so natural and so fundamental that you say, ‘How come I didn’t see it before? How come nobody saw it before?’
That’s a very wonderful feeling, I have to say. But it took ten years to get there because the mathematics was clumsy and awkward, as new mathematics always is. Then people polish it for 300 years... But discovering mathematics is messy, like making love; it’s messy.
David: But the outcome…
GC: The outcome can be…
David: Extremely pleasurable!
GC: Yeah, and it can also be a baby, another human being, who can be a great artist or something. But there are wonderful moments when, yes, you have this feeling, and then moments like that you say, ‘Well, it wasn’t me. I didn’t discover this. This idea exists out there independently of me and maybe it wanted to use me to express itself.’
But maybe it’s a way of fooling ourselves. But this is the kind of thing that helps you to do good mathematics. You have to be inspired. It’s profoundly emotional. I mean, the best mathematics is an art. It’s totally creative. You have to throw your whole personality at it. And it may be that you discover something because you’re crazy. You were the right crazy person to come up with this crazy idea, but other crazy people don’t find anything because their craziness isn’t in sync with the next discovery that had to be made.
David: I was fascinated when you said, ‘Sometimes you feel like the idea just needed you.’
GC: It needed people to express it. It wanted to incarnate, so to speak.
David: Well, I sometimes think that ideas are like a seed.
David: And that the mind is like a garden. So when people say, ‘I did this,’ I always think to myself that it would be like a little lump of dirt saying to you, ‘Look at the flower I made.’
GC: I absolutely agree with you, David. I remember Benoit Mandelbrot was in a documentary as he was dying, and you could barely hear his voice – they had to put subtitles – and he was saying, ‘I discovered a beautiful world.’
It feels like it’s out there, and you feel very lucky to have stumbled on it. But I don’t think you can take the credit. It’s like climbing mountains: you climb a mountain to get a better view, to see further, and one always feels that there are other mountain ranges that are higher. In the distance you can see still higher mountains, and the ranges never stop and it always gets higher. The further you see, the more you realise that there are.
So, for example, I’ve worked on Omega, but what is consciousness? What is the mind? How does the brain work? Can you prove that Darwinian evolution works? Where do new species come from? I mean, there’s endless questions, and each question just opens more questions.
Ard: Yesterday we talked to Marcelo Gleiser and he talked about the idea of knowledge like an island. So as you grow… an island in a sea of ignorance. So as knowledge grows, so does the size of the border that you have of the ignorance that you see. So as you get more and more knowledge, you also see more and more ignorance.
GC: That’s a very nice image. Also people don’t like talking about what they don’t know. They like talking about what they know. I’m the other way around. I prefer thinking about what I don’t know.
Certainty is bad because it’s uncreative. It means you know already – you don’t need to think any more about it. Well it’s also totally uncreative in mathematics. The idea of Hilbert was to ensure certainty He thought it was possible: he thought the possibility of doing this is what it meant to say that mathematics was black or white, that mathematical truth is more solid than any empirical truth. And it’s wonderful that mathematics refuted this.
You know, Gödel’s Incompleteness Theorem is suppressed. The mathematics community doesn’t want to take it into account, because they view it as a tremendously pessimistic, horrible fact that you can’t have a ‘theory of everything’ for mathematics, and that mathematics doesn’t give absolute truth. I think this is absolutely wonderful. The viewpoint is wrong. What Gödel’s Theorem is about… it’s not a negative theorem, it’s a positive theorem. It’s about creativity. It’s the first step in the direction of a mathematical theory of creativity – of saying that math is not a closed system, it’s an open system, just like biology. And this is totally liberating and we should all celebrate…. celebrate this fact rather than bemoaning it, beating our breast, ‘Oh my God. What happened to absolute truth in mathematics?’ Well, what happened was that absolute truth was a closed system. It was a prison: the notion of a formal theory that would give you absolute certainty.
Ard: A theory of everything.
GC: A theory of everything. Yes, a formalisation of all of mathematics in one finite set of axioms. This would have been horrifying.
Let’s say that they have this computer program which can decide if mathematical assertions are true or false.
Well, what good is it to know whether something is true or false? You want to understand what’s happening, right?
David: The why rather than the…
GC: The why, exactly. You want to be convinced emotionally that something is true. That’s why new questions are important, because what counts is not the mathematics we know – the science we know is uninteresting – it’s what we don’t know that’s interesting.
Unfortunately universities spend all their time filling your head with what’s known, but that’s totally trivial. What’s interesting is what we don’t know. That’s what all the courses should be about, so that maybe the students can come up with new ideas before they’ve been brainwashed with the current paradigms. That would be the university I would create, you know, which only would talk about what we don’t know because what we know is really very uninteresting.
David: You’ve mentioned a couple of times the importance of beauty to you in ideas. Do you find…?
GC: Everything is sexual. It’s all Eros, Eros and Thanatos. Another way to put it: it’s Shiva – destruction and creation. These are the basic, common forces. Creation is all about beauty. Sex and creation is the same thing. This is what motivates artists, and I think it’s where you get the energy to do good scientific work also.
Some of my most creative periods – I was ignoring women. I mean, I was terribly fascinated by them, but I was shy, and I put all this energy into mathematics. This was like a substitute for sex. I found mathematics absolutely sensual. I thought, at that time –teenage boy – I thought some proofs of mathematical theorems were as beautiful as a beautiful, naked woman, for example. If I had been chasing girls at that time, then there would have been no definition of randomness and no Omega number. So I made up for lost time later, but…
David: Well, I’m glad to hear it.
GC: Yeah, fortunately!
David: But was it a guide for your work? Because we’ve talked to some people who’ve said… and there’s famous stories where people have said, ‘If I find an idea beautiful, then this is what tells me that the truth is going to be that way.’ Have you found that?
GC: What I was really looking for… it’s not just beauty: I wanted to get to the bottom of things. I’m looking for the mysteries – the deeply hidden mysteries behind things. It’s sort of like looking for magic. In the Middle Ages I probably would have tried to do magic. Remember that Newton did: Newton was an alchemist. He was not a modern thinker at all, like Voltaire portrays him. He was the last of the Babylonian sorcerers, as Maynard Keynes said in that wonderful essay. Science is the same idea as magic: that there are hidden things behind everyday appearances. Everyday appearance is not the real reality. The apparent reality is not the real reality, and we want to get behind things to the real reality.
Beauty is very important. I certainly agree with beauty, but I also find these fundamental truths deeply beautiful in some way. Your notion of what is beautiful affects everything, colours everything, your whole conceptual scheme, it’s all connected.
The notion of truth and beauty cannot be separated. Now the notion of beauty changes as you go, as you create it, as you find it. But that’s what mathematics is really about at the deepest level, at pure mathematics.
Max Born has a wonderful essay, and he says, ‘Well, we make it up as we go.’ You know, in retrospect, the notion of what is beautiful is something that we create as we go based on things that have worked before. And I think it’s certainly true, because if you look at Japanese aesthetics and Indian aesthetics versus European aesthetics, they’re completely different. So he doesn’t believe in an absolute notion of truth. He believes we create a notion of truth: we create the universe.
David: What do you think of that, Greg? Where do you lie on that?
GC: Well, I think it’s more fun to take the provocative extreme instead of the conventional view, always, so you can guess that I’m going to be on the side that beauty is what counts, but that we are inventing our notion of beauty. This is part of human creativity to create notions of beauty. We create aesthetics, we create moral systems, philosophical systems, religious systems, and beauty is an absolute integral part of this. Remember that people a century and a half ago, they weren’t religious like they had been during the Middle Ages, but God still survived, at least to talk about the good, the true and the beautiful, which now are subjects that I thought you couldn’t mention, but it seems fortunately we’re able to discuss in this series.
But those were prohibited topics for a long time, because it was like being a religious fanatic if you mentioned those words.
David: Yes, it’s odd.
GC: So I’m glad to hear them mentioned again.
Ard: There’s sometimes, I think, a stereotype of scientists trying to unweave the rainbow and trying to, you know… if we once we’ve understood something, then we can reduce it to its components, and then…
GC: That destroys it. That’s reductionist.
Ard: That’s reductionist.
GC: Yeah, we’re destroying the beauty of everything. We’re destroying the sacred and making it secular and prosaic by understanding it. I don’t feel that way. I feel that it’s absolutely inspiring that puny human beings are able to have such beautiful theories.
Ard: Four forces…
GC: Yeah, there’s general relativity, there’s quantum mechanics, there’s electromagnetism and there’s Newtonian physics. And these are… are tremendous landmarks. They’re like crystals. They’re places where you can see that the universe actually… that mathematics is relevant.
Mathematics is not relevant to everything. For example, I don’t think it’s relevant to biology, and in Proving Darwin I tried to think, what is possible? What kind of mathematical theories might be possible as fundamental theories for biology? So what I see biology is all about is tremendous creativity: where new ideas come from, biological ideas. And I see that connected with mathematical creativity.
I don’t believe that there is a direct mathematical biology, like there is… mathematical physics is deeply mathematical. I don’t think that biology will ever be that way. So I propose creating a theoretical biology at one remove from real biology. It’s a toy model working with a model of biology which I call meta-biology. And that’s the idea. Instead of studying what happens when you… the evolution of making random changes in DNA software, DNA programs. What happens if you make random mutations in computer software and subject it to selection? So this is much simpler than real biology and has some mathematical ideas and is mathematically tractable. I don't know how far this idea will go, but that’s the kind of thing which I think might be possible in biology.
Understanding it is a patchwork quilt. No one idea is enough to understand biology. Population genetics is splendid, but creativity has sort of gotten lost in all of this. And I wanted to bring the focus back to that, which for me is the central issue.
Ard: There’s a very famous quote by a Dutchman, Hugo de Vries, who said, ‘Natural selection explains the survival of the fittest, but doesn’t explain the arrival of the fittest.’
David: Yes, that’s nice.
Ard: Where do new species come from? Where does biological creativity come from? And that’s in the variation. And that’s what your book is trying to do.
GC: That’s what I’m trying to do.
Ard: You’re trying to say, where does that creativity come from in the first place?
GC: I’m trying to propose a possible framework where this question can at least be asked, because in conventional population genetics, it can’t even be formulated.
David: Which leaves out the very engine of natural selection.
GC: Well, it depends if you’re interested in micro-evolution, which is sort of small variations, or if you’re interested in taking a broad perspective on the sweep of things and the major transitions in evolution, which is like when you go from single-cellular to multi-cellular, when you go from plants to animals, where the brain comes from. Those are major transitions in evolution, and those are much more mysterious.
There’s this tremendous outburst of creativity that takes place in a relatively short period where you find nature seems to be trying out every possible body plan. It’s just fabulous, and it just seems to be too fast, perhaps.
But if you think of genomes, entire genomes as software, a small change in a program can make an enormous change in what it does, in its output. So from that point of view, I think abrupt change doesn’t seem quite so mysterious.
Ard: People often think about mathematics, and even science, as just being a closed machine. And so what science is telling us is that we’re just machines, and if we’re just machines then we’re predictable.
GC: I hate… I detest that point of view.
Ard: You detest that point of view because you think it’s wrong?
GC: Well, look at Euler. Look at Ramanujan. Look at… Let’s take my favourite mathematician, Cantor: his theory of infinite sets which is really mathematical theology.
GC: So the stuff I’m working on is like a lower-level version of this. It’s a little more down to earth than Cantor’s infinities, and greater and greater infinities. But the whole question – the whole issue in both cases – is creativity, an open system.
David: You contrast one view of mathematics with another. And we’ve talked about one kind of science: the let’s just take it all apart and reduce it and claim that we’re machines. Is there something at stake, do you think, for you? For us?
GC: Well, there’s definitely something at stake. I want to save the human soul. One has to save the human race. If we think of ourselves… maybe we are machines. What do I know? But I think the idea of thinking that we’re just machines is a destructive idea. Even if we are machines, I think it’s better to think we’re not machines. I’m not saying that we shouldn’t do science to understand as much as possible how the brain works, how the body works, but if the human brain is a machine, it’s quite an amazing machine. It’s a machine on a level so different from our current machines.
David: So if we imagine that we are a machine, we need to be very careful about what kind of machine we imagine ourselves to be?
GC: This is a radically different kind of machine. And it’s a concept of a machine that will end up being totally un-machine-like. So in the end the machine and the sacred may meet.
David: In the book you said, ‘Darwin replaced God with randomness…’
GC: That’s right.
David: ‘…but randomness is lovely.’ Would you tell us that, because that…
GC: Well, that was one of the reasons that people initially rejected Darwin’s theory. One of the things they didn’t like is randomness. The idea that we’re sort of a random product – that there’s no purpose – makes everything meaningless. Now, randomness and atheism have become the new religion.
Ard: Do you think randomness is…? Do you think they’ve misinterpreted randomness?
GC: Yeah. Randomness does not mean everything is meaningless. Randomness is, sort of… You’re looking at creativity in its primordial state.
You see one of the characteristics of randomness is unpredictability. Now, something is unpredictable if you couldn’t predict it in advance: that’s creativity. So, in other words, randomness and creativity are practically different names for the same thing. Something that isn’t random is something you can predict, which means that it’s not creative. You’re sticking within your current system of concepts.
David: In my mind, some randomness is just, ‘Well, we can’t figure it out now,’ and some randomness is, ‘You will never…’ What do you mean?
GC: Yeah, I’m on the, ‘You’ll never’. It’s something that can’t be compressed. The technical definition is that a string of zeros and ones, a finite sequence of zeros and ones, is random if there’s no compact theory for it. If there’s no way to compress it into a program that’s much smaller in bits that generates it. There’s no theory… no concise theory for it.
Ard: Is that like saying, very crudely, if I put it on my computer and tried to compress it, I can’t compress it?
Ard: It’s random.
GC: Yeah, but it means no computer could compress it. It’s not just one computer. Another way to put it is that there is no concise theory: it has to be comprehended or apprehended as a thing in itself, to use Kantian terminology. There is no theory for it: the only theory is to write it out bit by bit. There is no more compressed, compact way to give it structure than just to write it out bit by bit. There is no simple theory for it. It would be experimental data for which there is no simple theory. The only thing you can say is, ‘It was zero, then it was one, then it was zero…’
David: Right, so randomness, when you talk about it, is…
GC: Lack of law, lack of structure, lack of…
David: It’s genuine randomness. It’s not something that we will figure out, we will be able to predict later. This is… You just won’t.
GC: That’s correct.
David: Well, can you tell me about that, because if I’ve understood you right, then you’re saying that this is where creativity comes from.
GC: The question of creativity… The problem of creativity is, can you have a mathematical theory of creativity? Well it can’t be a theory that will give you a mechanical procedure for being creative because then it’s not creative. So a mathematical theory of creativity has to be indirect. Creativity is by definition uncomputable. If we knew how to do it, it wouldn’t be creative.
When you have maximum creativity, it looks random because it’s totally unpredictable from what you knew before.
GC: That means that you’re really being creative. So randomness is the extreme of creativity, really. They’re…
Ard: They’re connected together.
GC: And if you can calculate… if you can calculate something, then it’s not creative because you’re working within your existing system. So there’s this paradoxical aspect. A mathematical theory of creativity is a more abstract kind of mathematics where you can prove theorems about creativity – you can describe it – maybe you can show it’s highly probable, but it won’t give you a way to mechanically produce creativity, which is the kind of thing that instrumental mathematics normally does.
Ard: You can also say what creativity is not.
GC: Yes, that’s very important too. So the more we can say what it’s not, we begin to see the complement to what it is.
Ard: It’s interesting, there’s a very famous tradition in theology which says that you can’t speak about God, you can only say about what God is not. And so there’s something interesting there with creativity. You can’t nail down creativity, but you can say what it isn’t.
GC: Well, isn’t God pure creativity?
Ard: Apophatic theology is what it’s called. You have an apophatic theory of creativity. You’re speaking about it in an indirect way, but you can never grasp it. If you could grasp it, then you wouldn’t have it. That’s what you’re saying.
GC: That’s right.
Ard: You’re saying, if you could nail it down, then it wouldn’t be creativity.
GC: It wouldn’t be creative by definition.
Ard: And in the same way a theologian would say if you could nail down God, then that’s by definition not God.
GC: That wouldn’t be God, that would be a limited being.
Ard: Yeah, exactly.
David: And it’s lovely that you’re doing it in mathematics, because most people think mathematics is doing the opposite. It’s…
GC: It trivialises things. It makes them like arithmetic – boring, uninteresting, meaningless. So this is a different kind of mathematics: the mathematics of creativity. Math as an open system, not a closed system.
I’m trying to get to the concentrated essence of the mystery: the mystery is creativity, and I think that’s deeply meaningful. I mean, in Brooksonian terms the universe wants to create us. The universe wants to create mind. The universe maybe wants to get closer to God, or maybe the universe is God and it’s trying to increase its level of perception, its level of understanding.
Albrecht von Müller, he goes to an extreme that I like, which is the idea that the whole universe is actually creative: that perhaps the ontology is not fixed in static. Perhaps it’s actually plastic, even at the fundamental level. That would be much more fun. I don't know if this universe has that property, but that might be an interesting universe that is fundamentally creative.
So I’m in favour of creativity. I’m in favour of rocking the boat. I’m in favour of new ideas. And new ideas will always be fought.
David: Do you see yourself as a religious person, Greg?
GC: No, but I’ve been sort of driven to it by the mathematics I’ve been trying to do. I don’t see myself as a religious person, but I’m very interested in mysteries and deep questions. And I also have a very… I’m very responsive to a feeling of transcendence. I love going to the tops of mountains in the snow and seeing the view, feeling closer to God, or closer to the fundamental nature of reality. It’s an illusion, perhaps, but one gets to the top of a mountain in the winter with the snow, and you have this beautiful view…
David: Is it the feeling of the sublime? You know, people talk about there’s the beautiful, in the sense of pretty, and then there’s a beauty connected with some kind of a truth.
GC: Something so beautiful that it’s sublime…
David: Do you feel the same sense of transcendence when you see certain things in mathematics? Does that give you the same sense as being on top of a mountain?
GC: Absolutely, but if you’re reading a piece of work that was already done by someone else, it’s sort of like seeing a photograph of a mountain. That’s no fun. Or take going in a helicopter to the top of a mountain, which is unworthy. But if you struggle up the mountain yourself, then you see every inch of the mountain. You see all the views. You pay for it, and then at the top you’re worthy. And in the case of mathematics, if you’ve been struggling with questions that seem mysterious and incomprehensible, and all of a sudden you discover a viewpoint that makes it clear, that’s like an illumination all of a sudden.
I remember once I was going up a mountain. We made summit, and to make summit we had just broke through the cloud level. The summit was in – I wouldn’t say blinding sunlight, but beautiful sunlight. There was this white plain of the clouds, and sticking through it were these little toy mountains, because they were just the peaks, and that was just fantastic. It had only happened to me once in that extreme from, and that was just a wonderful moment.
That’s like the moment when I had struggled ten years with a bad mathematically formulated definition of randomness. The basic idea was right, but the way I dealt with it mathematically – the techniques – were wrong. It was very clumsy. And all of a sudden everything fell into place. It was as if I had been wearing glasses that distorted everything, and all of a sudden I put on a pair of glasses and everything’s sharp. It’s fantastic.
If you reduce everything to the shopping mall and what you see on television, then what’s the point of doing anything? So you have to see the Himalayan mountains out in the distance and think maybe Shiva is there. Or that there is a transcendent god who created everything and this is really a beautiful work of art that we have to understand.
David: You [to Ard] feel that very strongly, don’t you? That there should be a transcendent set of truths, or a god, out there and that somehow that is important?
Ard: That’s important. It’s inspiring.
David: As soon as you said that, I thought, that’s Ard.
GC: People need to be inspired. You have to get the energy to get creation from somewhere. You can have sexual energy; it can be from thinking about transcendental, spiritual things – whatever it is. It can be just because you’re crazy, I don't know. William Blake created this whole world in his poetry and his wife said his feet never touched the ground. He was always somewhere else. If all that exists is what you see in front of you, it’s too damn boring. The whole thing is pointless.
So I’m not sure that there’s a transcendent reality that is sublimely beautiful if we could know it, but I think it’s better to think that and look for it, than to give up and say, ‘Oh, it’s just a big, incomprehensible mess, and who cares?’
David: Let’s go shopping instead.
GC: Yeah, let’s be consumers, right, sure. You know, what is… What are we here for? Just to go to shopping malls and buy stuff we don’t need with money we don’t have?
David: Thank you.
Ard: That was great. Thank you. It was fantastic.
GC: It’s a pleasure.