If beauty is a guide to truth, what happens if our idea of beauty changes?
David: You are not one of those who think that mathematics is woven into the fabric of the universe?
MG: So, the question of mathematics being the language of God, so to speak, or, sort of, the blueprint of reality, right? There is no question that there are patterns in nature. They are repetitive and they can be described through mathematics in beautiful ways.
You have not just symmetric patterns, or almost symmetric patterns – I like to make that distinction because nothing in nature is perfect. You have periodicity in the orbit of planets, and things like that. Clearly there is order, but I think to just say that this sort of symmetry in nature is the hidden code, so to speak – that all you have to look for is that symmetry – is missing half of the story.
David: What’s the other half?
MG: The other half is the role of asymmetry in nature. There is a lot of imbalance in nature, and it’s really through the complementary roles of symmetry and asymmetry that nature creates. A lot of stuff happens because of this imbalance between the two.
Ard: Give an example.
MG: I have lots of examples. One good example is in life. It turns out that proteins, which are made of chains of amino acids, like big molecules, and these amino acids they are basically molecules and they have a carbon in the centre, and then they have four sticks coming out of it. And it turns out that they come in two ways. They can be what we call left-handed and right-handed, jby the way in space they look like. And it turns out that if you go and you synthesise an amino acid in the laboratory – it was Pasteur that discovered this – you get 50/50: fifty percent with the left-handed shape and 50 percent with the right-handed shape. When you look at the amino acids in all living creatures, from bacteria to a sequoia, they all come only in the left-handed shape.
Ard: So, why…?
MG: The right-handed shape just is not there. So there is a fundamental asymmetry between the two, and without that asymmetry, the lock-and-key mechanism that proteins need to, kind of, fold and split and create, be part of reproduction, etc., wouldn’t work.
MG: And we do not know why this is true, okay? We just know it’s true. It’s there. It’s fundamental for life. So that’s an asymmetry which is very important, for example.
Ard: How about matter and antimatter?
MG: Exactly, so there you go. You know, that’s the good physics example: the fact that Dirac’s equation predicts that there should be as much matter as antimatter in the universe. And antimatter is nothing so esoteric that goes up instead of down, or anything like that. It just means a particle that has an opposite electric charge but the same mass. So, for example, the electron, which is negatively charged, has an antimatter particle called the positron, which is positively charged.
In principle, they should come in equal amounts, but when you look out, you find out that there is no antimatter out there – very, very, very little. And that’s good because if there were as much matter as antimatter in the universe, we wouldn’t be here.
David: We’d have all gone, pfft!?
MG: Exactly, because matter and antimatter, when they come together, they disintegrate into a puff of gamma rays – very high-energy radiation. So if you find your anti-person walking around, don’t shake hands. And so that’s the story, and we do not know. I spent a long time trying to understand what sort of causal processes may have happened early in the history of the universe that would have biased one form over the other, and there are all sorts of ideas – none of them is very compelling right now.
Ard: Okay. That’s a big mystery.
MG: So you need both. And I think it’s this yin and yang kind of thing, you know? You can’t just look at this reverential perfection, symmetry, as, kind of, the language of God, where nature is showing you that you really need both to make sense of things.
Ard: But there is some beauty to this combination between symmetry and broken symmetry.
MG: I think so. I have been proposing that there is what I call the aesthetic of the imperfect.
MG: Physics is a little old-fashioned, in a way, in thinking that it’s really the perfection that counts. It’s truth, right? The arts and music, they moved away from that in the early 20th century, and I think we’re still stuck in it.
Of course, symmetry’s fundamental: you cannot be a serious scientist, and physicist in particular, without having deep respect and veneration for symmetry. But symmetry is often an approximation to the real thing.
There’s this joke about the physicist that looks at a cow, and he says, ‘Consider a spherical cow as the first order approximation to what a cow is.’ And it works quite well for many things, right? If you want to collide cows at high speed, it’s a good approximation. But it’s not a good approximation if you want to milk the cows and things like that.
Ard: So do you think that when we understand biology better, that this aesthetic of symmetry won’t be the right way of thinking about it?
MG: Yes. I think life is a great example of the importance of asymmetry. You know, I have another example, which is Marilyn Monroe. So, Marilyn Monroe had a beautiful little mole. Imagine if she had two equidistant moles, how ugly she would look. So symmetry is not always beautiful. There is this breaking of symmetry, and I think we should embrace a combination of both.
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: I liked the way you said, ‘the crystalline part of mathematics’. So mathematics has a certain austere beauty. I was just curious whether you think that narrative has a different kind of beauty sometimes, or maybe richer forms of beauty than just mathematics. A lot of mathematicians talk about beauty in mathematics, and you've spoken about this as well.
BO: What do you think mathematicians mean when they talk about beauty in mathematics?
Ard: Well, I think I recognise beauty in mathematics, and the interesting thing is we all, all of us mathematically orientated people, tend to agree on the beauty, and we tend to think this is more beautiful than that. But when you say to me, define it exactly, it's hard to put. I can recognise it without necessarily always being able to define it.
BO: Okay, I have a question for you then. Can the beauty in mathematics, in an equation, can it be there even when the mathematics is wrong?
Ard: Sometimes it can be. So sometimes we speak about a very beautiful theory which is ruined by experiment: an ugly fact. But in general, this is definitely true... there are very beautiful theories that are wrong, but, in general, if I have two competing theories for the same bit of nature I need to explain, the more beautiful theory is more likely to be the true one.
BO: But have there been cases where the ugly theory has turned out to be the truth?
Ard: Unfortunately it has.
BO: Because you have quite a few of that in literature. You do. Things are ugly. The first people who read Moby Dick didn't see its beauty. The first set of people who saw Demoiselles d'Avignon didn't see its beauty. We still don't see the beauty in a lot of cubism. A lot of modern music we still think of as being too fractured for us. So sometimes something can seem ugly.
Ard: But there are mathematical things that seemed ugly or seemed trivial at the time but ended up being very profound later, so part of that is just our inability to perceive.
BO: It's the same with literature, absolutely the same with literature.
Ard: There are mathematicians whose work was forgotten while they were alive, and then after they died we realise there was something unbelievably profound.
BO: I think it's also the nature of perception that we're only capable of seeing that which we've seen before. We're only capable of perceiving that which we've perceived before, and when a slightly new order of perception comes along, we can't see where it diverges. We can't see what it is. And because it doesn't conform to what we've seen before, we think it's either trivial, unimportant, insignificant or not beautiful at all.
You think of the great works that people did not understand at first. You think of something like Waiting for Godot, or you think of many Shakespeare plays that people didn't understand, didn't think beautiful at first, but which we've come to learn to perceive its correct inner beauty.
Ard: Is there a logic to narrative or to poetry that has to be conformed to or...?
BO: The best poetry and the best novels, the best narrative, absolutely. You can almost create a divine logic. What you're struck by, more than anything else, is the shocking clarity of thinking. You read Dante and from one line to another there are no absurd leaps. Everything has to come; it has to add up. Let’s just say there's a great rigor to the best parts.
You can always tell when a writer is faking it because the images don't add up. The narrative doesn't add up. It makes like sudden leaps that are not prepared for meticulously.
Ard: As if it's less beautiful somehow? Is that what you're saying? It's less beautiful, less aesthetic? Or is that the wrong way...?
David: Or is it less truthful?
BO: It always feels less truthful, and it is. Sometimes something can be wrong in its inner logic and still have beauty, and I think that's because beauty partially transcends logic. I think it's implied in it. I think beauty is implied in logic. I think that which we find to be beautiful is something which is logical to our aesthetic sense.
Ard: For a physicist like myself, beauty plays a large role in understanding the world. Paul Dirac famously says his fundamental belief was that the world is described in terms of beautiful equations.
Ard: For a biologist, is it the same kind of beauty? Or is that beauty metaphor story only really true in physics?
DN: I think it’s universal, but it gets to be interestingly different in different domains of science, it seems to me.
DN: I can understand why a physicist says the equations are beautiful. And if you look at the equations of relativity theory, my goodness are they beautiful! So that kind of beauty I can fully understand.
What seems to me to be the problem is that people then go on to think that that’s the only kind of beauty there can be in science and in hypothesising.
When I found what I thought was a good explanation for the rhythm of the heart, in terms of the interactions between the various protein molecules and the membranes and cell system, I had to say that I had something beautiful there. Of course, it was, as it were, with equations, but actually they were pretty miserable equations: they were differential equations requiring all sorts of initial and boundary conditions. This is not the beauty of the equations of relativity theory! But if you look at it beyond the equations themselves, there is a beauty there that can happen.
Ard: And so do you think that kind of richer beauty that you’re looking at in this biological system, is that a guide to the truth about the system?
DN: I think when you find that you’ve got that kind of beauty, it is a guide in a similar way, though not operating in quite the same way as happens with seeing a beautiful equation. But it still seems to me to be not a bad guide because you get excited by it, because you really do think you got to, at least, part of the truth.
David: Because it’s beautiful?
DN: Because it’s beautiful in the sense of saying, ‘My goodness, the logic of that is so nice. It has to be like that.’ Now, you might be wrong, of course, and this has to be done with the humility of admitting that one might always be wrong. But it’s hard to avoid that feeling that this is logical. It works because of seeing the logic of how it works. And that, for a scientist, is what makes one appreciate the beauty of it. But it doesn’t have to be the beauty of a very simple equation like E=mc2.
David: But it’s still very odd, though, don’t you think? Why should the human mind find truth to be beautiful? Why should these two things overlap? Don’t you find that strange? I mean, why should that be the case? Why should things that we find beautiful also turn out to be true of the universe?
DN: Yes, it’s like asking the question, ‘How is it that mathematics can be applied to the universe?’
David: Well, it makes it worse.
DN: It makes it worse, precisely.
David: The fact that it does is strange to start with, and then that it’s beautiful as well adds another level of strangeness.
DN: Yes, indeed, and my reaction to that is humility. I’m puzzled