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Episode 3: Can Math Be Beautiful? Transcript (Note: This Is the Result Of Episode 3: Can math be beautiful? Transcript (Note: This is the result of a hybrid between machine learning and human correction, so it may have slight imperfections.) Michael Atiyah: 00:05 We mathematicians and scientists are searching for truth. But beauty is our guiding light that leads us there. Stephen Ornes: 00:14 This is the third episode of Calculated, a podcast collection of stories about people at the intersection of math and art and culture and today we're talking about beauty. Semir Zeki: 00:24 Mathematical beauty is the most extreme form of beauty that is dependent upon culture and learning because you cannot, you cannot invite somebody to your office or to your room and tell them, is this mathematical formula beautiful or not? Unless they're mathematicians, they're not going to understand it. Stephen Ornes: 00:42 People can be beautiful. Music, beautiful. Paintings, buildings, minerals, poetry, sunrises. Those amazing bands of turbulence on Jupiter. Beautiful! But what is beauty anyway? What about math? Can math be beautiful? What does that even mean? And what about people who hate math but know beauty? What's in it for them? Paul Erdos, one of the most prolific mathematicians in all of mathtopia, was once asked to talk about how he found math beautiful. He said, it's like asking why is Beethoven's Ninth Symphony beautiful? If you don't see why, someone can't tell you. I guess that puts it in the same group as irony or obscenity. You can't define it, but you know it when you see it. Stephen Ornes: 01:26 But don't despair! Today on Calculated, we're going to try to get at the beauty of mathematics by diving deep into the brains of real mathematicians. We'll also hear from a mathematician who recently organized an exhibit of math art. Stephen Ornes: 01:40 In a 1940 essay G.H. Hardy wrote that the mathematicians' patterns like the painters or the poets must be beautiful. The idea is that the colors or the words must fit together in a harmonious way. Beauty is the first testH hardy wrote. There is no permanent place in this world for ugly mathematics. Stephen Ornes: 02:02 Last January, the math community lost Michael Atiyah, a legendary British and Lebanese mathematician. In addition to being a giant of geometry, he spoke and wrote eloquently about how math could be beautiful. This is from a talk he gave at Princeton in 2010. Michael Atiyah: 02:21 Hermann Weyl once said, famously, that in his life he searched for truth and for beauty, but when in doubt, he chose beauty. Stephen Ornes: 02:29 Hermann Weyl was a German mathematician who died in 1955. Michael Atiyah: 02:35 Now, most mathematicians thought that was, you know, terrible. Mathematicians... ultimate demand a search for truth above all else. But that's a misconception. Hermann Weyl knew what he was saying. I think what he was saying is that, that truth is something you never know whether you've got it. You're searching for it, it's your ultimate goal. What you've got may be a partial truth, halfway to the truth. And beauty is something you can immediately perceive, see, it's your personal reaction. Nobody can argue that if say I see a blue color over that you can't say it's green. So that's one thing. Secondly, I think you can say it this way around. We mathematicians and scientists are searching for truth, it's our goal, but beauty is our guiding light that leads us. Stephen Ornes: 03:14 Beauty is a guiding light. Atiyah's death was a great loss to mathematics. The connection between math and beauty is old, old, old, at least as old as the ancient Greeks. Beauty, according to classical philosophy, arises from the harmonious combination of different parts. In his Metaphysics, Aristotle argued that the chief forms of beauty are order and symmetry and definiteness. He went on to say that the mathematical sciences demonstrate these properties in a special degree. It almost sounds like he's talking about a formula or a mathematical recipe for beauty. Stephen Ornes: 03:52 In a 1914 book called Arts, the critic Clive Bell explored the idea that works of art share some particular quality. On one hand, he said, the experience of art is subjective. It's something that we feel and those feelings are different from person to person. It's emotional, but on the other hand, he argued there has to be some quality that's common to everything you might see as art. He asked, what is that quality? Semir Zeki: 04:17 What is common to all that we experience as beauty? Stephen Ornes: 04:20 This is Semir Zeki. He is a neurobiologist at University College London, whose research focuses on how the human brain obtains knowledge about the world. Over the last decade or so, he's used sophisticated scanning techniques to peer inside the human brain and see what intense emotional experiences look like from the inside. Stephen Ornes: 04:39 He's explored the neurochemistry of love, for example, to see how romantic love plays out in our heads. A study published in 2011 showed that looking at a work of art sends dopamine into a part of our brain associated with pleasure and desire and romantic love -- and also with recreational drug use and addictive behaviors. Zeki takes Clive Bell's questions about the common factors of art as a starting point, not for philosophy or art criticism, but for neuroscience. He wants to know, is there something that happens in our brain that can speak to Bell's idea about the mysterious common quality? Semir Zeki: 05:16 Whenever you open your mouth about beauty, people will tell you this is subjective. And this is something I don't buy. I don't buy it. I think there are some kinds of beauty which are not subjective. Or which are largely objective. Stephen Ornes: 05:32 A few years ago, he extended his investigation into the realm of mathematics. He recruited 16 mathematicians and a bunch of non mathematicians for an unusual experiment. Mathematicians were asked to rate 60 equations on a scale from negative five to positive five where uglier equations got lower scores and beautiful equations, got higher scores. The equations included things like a definition of Pi as the ratio of the circumference to the diameter of a circle and the Pythagorean theorem and a representation of of Fermat's Last Theorem. They also included Euler's identity, which brings together many different areas of math using Pi and e, which are transcendental numbers, and i, which is used for imaginary numbers. Two weeks later, these mathematical Guinea pigs were hooked up to a functional MRI scanner, which gives a real time report of the brain's activities. They were showing the same equations and asked to rate them again. Now here's the kicker. Zeki found that simply looking at the most beautiful- rated equations activated the same part of the brain that fires up when we look at beautiful paintings or listen to beautiful music. Semir Zeki: 06:37 The surprising thing, and I was taken aback by it, that there was activity that the experience of mathematical beauty correlated with activity in the same parts of the emotional brain as the experiences to derived from other sources, beautiful and musical or joyful and moreover, and this is quite important, the strength of the activity in the medial orbital frontal cortex was parametrically related, in other words was proportional to the declared intensity of the experience of mathematical beauty. Stephen Ornes: 07:16 Okay, so let's unpack that a little bit to really appreciate these findings. Consider the brain. Consider the math brain and consider this question. Where does math happen inside our heads? A few years ago, researchers from Stanford University gave a serious answer to that question. They identified different areas associated with math reasoning. Now get ready. We've got some 50 cent words here. The areas included the ventro temporal occipital Cortex, the posterior parietal Cortex, and the prefrontal cortex. For a study published in 2016, French researchers to brain scans of mathematicians as they studied mathematical statements. Those scientists found that advanced math subjects like topology and Algebra tickled the same part of the brain that we use for basic things like adding or multiplying. But The areas of the brain associated with mathematical reasoning were not the areas that Zeki and his colleagues identified! Semir Zeki: 08:14 I would not have been at all surprised if the increase in activity occurred in the parietal Cortex, where many areas areas are given over to mathematical tasks. Stephen Ornes: 08:27 But it wasn't! The brain became active in the region that lights up when we see paintings or listen to music. Semir Zeki: 08:36 Looking at a painting is a purely sensory source. Looking at a face is a purely sensory source. But looking at a mathematical equation is a very cognitive source. Stephen Ornes: 08:47 Zeki and his colleagues found that the mathematician brains responded to those equations in an area, in the part of the brain where we experience beauty. Nowhere near where we think mathy thoughts. Just in the beautiful place. It's tempting to think that math and art can be at opposite ends of a spectrum, but these brain scans suggest they may not be so far apart after all.
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