BEAUTY AND TRUTH IN MATHEMATICS

David: Why did you get interested in beauty? Because it’s not the normal thing that neuroscientists get interested in.

SZ: Well, it’s part of a more general question, which is I’m interested in the visual brain: how the visual brain functions, how we see. So the next step, really, is to ask, how does a visual input arouse an emotional state? And one of these is beauty. In a sense, what is the point of learning all about the visual brain and not being able to say what happens in your brain when you experience something which is visually beautiful? So that was the inevitable next step.

David: And what were you expecting when you started? Were you expecting that beauty would be a separate thing?

SZ: No, I think in this instance we just did not have any hypothesis. We just thought that it would be interesting to see what happens when you experience something which is visually beautiful: not only beautiful portraits, beautiful landscapes, but also beautiful abstract art. And then you go and say, well, philosophers have spoken of beauty in the abstract, so I must look at musical beauty as well. And then you reach the ultimate question, which is mathematical beauty, because mathematical beauty is one of the reasons why mathematicians speak of mathematical beauty in poetic terms. One mathematician described one of the beautiful equations as the equivalent, the mathematical equivalent, of the soliloquy in Hamlet. So you see how impressed they are. And indeed, when we did the experiments on mathematics, some of the subjects were in tears.

David: Really?

SZ: Yes, yes.

David: So, wait a minute, tell me what you did in this experiment.

SZ: Well, all we did was to give them sixty equations to classify as to how they experienced them as beautiful. We gave them a scale from one to nine: one was very beautiful and nine was very ugly. And each one classified them according to their own subjective experience, and then they came into the scanner and looked at these same equations and reclassified them. So, we now knew that they had a category of equations which to them were beautiful, ones that were ugly, and ones that were indifferent. And the experience of mathematical beauty correlates with activity in the same part of the brain as experiencing musical beauty or visual beauty.

David: Can you show us?

SZ: This is part of the emotional brain. So this is a brain looked at in mid-section.

SZ: This is the front of the brain. This is the back of the brain. So you’ve bisected the brain, and you’re looking inside there, and this shows you the area of activity in the medial orbitofrontal cortex, which correlates with the experience of mathematical beauty.

David: So when your mathematicians said, ‘That’s beautiful’, that’s the bit that lit up?

SZ: Yes, yes, exactly. Not when they said it, when they experienced it was beautiful. Then if you look at the regions of the brain which are active with musical and visual beauty, now you’ve got the same area. You see this area in yellow is common to both. There’s a huge area of overlap, but it’s the same area of the brain that’s active when you experience mathematical beauty.

David: So what does that tell you, Semir?

SZ: Well it tells you a number of things. First of all, as a neurobiologist, let me just rephrase this question, ‘What does it tell me?’ You have to understand what am I looking for. I’m not looking to explain beauty or to tell you what art is, nothing like that. I am really trying to only find out, because I’m a scientist, what are the areas of the brain which are engaged when you experience beauty.

So the first thing that it tells me is that the experience of beauty, regardless of source, correlates with activity in a given part of the brain, number one. Number two, it is part of the emotional brain. Number three, that the activity there, I have not shown you this, but the intensity of activity is related to the intensity of the experience. In other words, if you find something extremely beautiful, then the intensity of the activity is much higher than if you’re indifferent to it or something.

But it also raises questions which inevitability make us trespass into other fields which do not properly belong to us, which is, what is the use of beauty? What does it indicate? And why is there a common area in the brain for the experience of beauty from such diverse sources?

And this question especially imposes itself in respect to mathematical beauty. What is it about mathematical beauty that people experience? Now, this part of the brain is also part of the reward centre of the brain and pleasure. But then, you see, beauty is never divorced from reward and pleasure: beauty is a rewarding experience, it’s a pleasurable experience, so the two are mixed. And if one was to read the philosophies of aesthetics, they are always talking of the three – of pleasure, reward and beauty – almost synonymously.

So the question I would ask is, what is it about mathematics that people find so beautiful? It’s a very difficult question, and the answer I would give is that they find something in the logical deductive system of mathematics that makes sense. It’s entirely based on the logical deductive system of the brain, and where did this logical deductive system develop? It developed in the universe.

So it’s an interesting question to consider: if you take a very sophisticated mathematical equation, for example, quantum mechanics, and you have an equation which does not make sense to the logic of the brain – the brain’s deductive logical system – will that ever be considered as beautiful? And if not, will it ever be considered as true?

What’s his name? Eh…

Ard: Dirac?

SZ: Paul Dirac, thank you. Dirac did say that the guide to the credibility and the truth of a mathematical equation lies, above all, in its beauty before its simplicity. If it is beautiful, then chances are higher that it will be true. But ‘beautiful’ implies that there is something in it that satisfies the brain. In the case of mathematics, I would say you’re satisfying the logical deductive system of the brain.

Ard: I have experience myself of studying physics, learning about the Dirac equation, and just being blown away by its beauty. And part of the reason was because Dirac took this theory of quantum mechanics, of small things, and special relativity of fast things, put them together for the electron, and out popped the positron. You know, that was anti-matter that was predicted by taking small things and fast things and putting them together. So he took two unrelated things, put them together and a third, completely unexpected, unanticipated, unimaginable thing happened, which was you predicted a new particle, which was anti-matter.

SZ: Now, there is another story of Hermann Weyl. Hermann Weyl and his attempt to reconcile the theory of relativity with James Clerk Maxwell’s electro-magnetism led to mathematical formulations which were rejected at the beginning. He accepted them only because they were beautiful. Einstein objected to them, and it was only after they were published – ten years after they were published or so – and the event of quantum mechanics, that people began to see that these were true. So the guide to the veracity of the equation was its beauty.

Ard: So that’s a really surprising thing.

SZ: Yes, indeed, extremely surprising.

David: It’s more than surprising, it’s just mysterious. Why should it be that way?

SZ: Well, I’m still surprised by it. You know, I’ve got the authority of people like Paul Dirac and Hermann Weyl and Michael Atiyah and others who speak about these things. It is shocking in a way that you find something… You said you were ‘blown over by it’. That’s a dramatic turn of phrase, but apparently it is true.

Now, in the same way, I think there are people who are extremely moved, and indeed are blown over, by the first sight of the Pietà of Michelangelo. It’s a very deep, emotional experience which is very difficult to recapture outside this frame of actually seeing it. So these are the sort of things which lead you to ask the question, what is the use of beauty? Darwin saw it as only a question of sexual selection, which of course it is, but that’s not the only thing. It’s doing a lot more.

David: But isn’t there something strange that the bit of the brain that says that bit of mathematics is beautiful is the same bit that says that this sculpture is beautiful. Why should that be?

SZ: I don’t think that bit of the brain says anything about this bit of mathematics is beautiful. I think all that happens is that when the Michelangelo Pietà, or the mathematical equation, satisfies something in the brain, then you experience beauty, and then you have activity that correlates with the experience of beauty. I don’t think there is an area which… It’s a question of satisfaction and pleasure and reward.