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Welcome to Intelligence Squared, where great minds meet. I'm producer Mia Sorrenti. Today's episode is part two of our event recording with the Oxford professor and award-winning mathematician Marcus de Sotoy. Marcus was in conversation with Dr. Shini Samara live at Conway Hall to discuss all things maths and creativity and the surprising relationship between the two.
If you missed part one, do just jump back an episode to get up to speed. Now let's continue the conversation live at Conway Hall with Dr. Shini Samara. The number three is masculine and the number two is considered to be feminine. And when you add them, it's five golden ratio. Yes. Well, this is very interesting because, you know, that
here we see, you know, cultures already understanding that mathematics has a kind of personalities involved in it. And that, you know, Ramanujan, the famous Indian mathematician, there was a play that I helped out on with Compiste in a film. He always used to say, I know every number as my own personal friends, that every number has its own different quality to it. And what
And what's interesting is that many cultures associated odd numbers with male and even numbers with female. And the Chinese went even further and said, you know, the macho numbers are the primes because somehow you can't arrange them in any way into a kind of grid. You know, 15 is three by five, but 17, there's no way. So, but...
you said two, three, and five because that's something that Mozart picked up in the Magic Flute. So the Magic Flute has...
his last opera and one I talk about in the book and one I did one of the I mean I've done a lot of events with artists one of the things I'm most proud of is an event that I did at the Royal Opera House where we took over the Limbrey Theatre and we did a kind of a sort of kind of mathematical performance with five singers of the Magic Flute sort of bits of it exploring how much mathematics Mozart uses in that opera and
Again, you just think, well, Mozart surely wasn't interested in maths, but he was obsessed with maths. He had maths books in his library. He would always sign off his letters with numbers, and he used numbers, actually, you know, square numbers for him were always an indication of love. So if you remember the Marriage of Figaro, it starts off with a sequence of numbers,
It's the dimensions of the bed that Figaro is making for Susanna for their wedding day. And the measurements, you know, I think it's 510. It sounds fairly normal. And then it goes off on some weird sequence of numbers. You think, well, they didn't buy their bed from Ikea. That is like a bespoke bed. And it's like, why did...
Mozart choose these numbers. And if you add the numbers up at the beginning of the marriage of Figaro, they add up to 144, which is 12 squared. So Mozart is just saying, okay, I'm having a bit of a fun here. I'm using these numbers to indicate the love between, the perfect love between Susanna and Figaro. And so in the Magic Flute, his last opera, this is just after, seven years after he's become a mason.
The Masonic Lodge is full of mathematical ideas and the importance of two, three, two in particular, this idea of parity, and three. But Mozart is writing at a time when there's a real drive to introduce women into the Masonic Lodge. And so there's a lot of kind of use of numbers. And in the Magic Flute, things come in twos or threes, the three boys, the
Things are very often in pairs or fives. You never get quartets. They're always quintets. And so this is Mozart saying, we need to combine two and three to make five because we need men and women in the Masonic Lodge, something which obviously didn't happen. It's really interesting because the prime number examples you give at the beginning of the book in terms of the music, and I was listening to it, and it's kind of discordant,
But then when you're talking about geometry, I was listening to those examples of music and they're harmonious. And you say at some point in the book that
as humans, we look for patterns and we want that kind of harmony, yet there are pieces of music that are discordant and also attractive. Yes, I mean, I start with this, the whole book with this story of Messiaen using prime numbers in the quartet for the end of time, which is a piece that he wrote when he was a prisoner of war during the second war in the Stelag 8a. And he wrote a quartet for four musicians, himself on piano, clarinetist, violinist and a cellist.
I mean, it's beautiful music. I really think, I mean, but it's challenging music. Well, it was out of sync, wasn't it? Yes, so Messiaen was obsessed with bird themes, so when the piece starts, there's kind of the clarinet and the violin exchange bird themes, but in the piano part, there's this amazing
amazing structure happening where the rhythm sequence is 17 notes, kind of syncopated rhythm that just repeats itself over and over again. Prime number. Exactly, well spotted. 17 notes, but the harmonic sequence, the number of chords is 29.
Another prime number. Now the effect of this is that the piano basically is incredibly patterned. You just hear this syncopated rhythm over and over again, and you hear the same chords, 29 chords over and over again. But because he's chosen these prime numbers, the thing never gets, it's always different.
when it's starting its 29 chords it's at a different place in the rhythm sequence and so it's a completely different rendition of those 29 chords rhythmically and as the rhythm finishes the chords are in a different place and you have to hear 29 times 17 chords before the thing comes back to the beginning again by which time the piece is over i mean it doesn't go on that long so i think this
But when you're listening to it, you hear that. No, but you know something's repeating it. And I think that's... Music remains interesting when you can't kind of unravel it. But you know there's something. And your brain is fighting to find the patterns there. And then you talk about dance music and how it's always 4-4. Yeah, incredibly boring. Because we dance on two legs. But then you get the musicians who want to disrupt this and...
and kind of make you unsettled. So, you know, my favorite is Radiohead. I've always been a big Radiohead fan. And when I started looking at their music, I understood why, because they use, instead of using lots of fours and eights and sixteens, they're disrupting things continually with fives, sevens. It feels like the thing is just,
you know, added a beat or it's kind of glitched because it's held back. And it was very interesting because I fell in love with Messiaen in my youth orchestra when a conductor came one weekend and brought Tarangalila and we played, I played the trumpet and we played through this piece. It was amazing.
I think it was when my trumpet, it was at its best. It's been downhill ever since and it needed to be 'cause I fell in love with Messiaen then and that's when I started to explore the amazing amount of mathematics that Messiaen has. But I found out that Johnny Greenwood, who's one of Radiohead's band and is responsible for a lot of the music, I understood that he also fell in love in his youth orchestra with Messiaen when they played it at a different time.
And then I understood, oh, of course, that's why Radiohead does all of this disruptive stuff. It's actually amazing. I've got two... Okay, let me follow through on that thought. It's amazing how many artists love science, love maths, like Zaha Hadid, Dali, Anish Kapoor. They, you know, Anish Kapoor wanted to be an engineer. And it's amazing how they're...
deep appreciation of mathematics influenced their work and allowed them to be successful. But the question I really want to ask you, because I also want to ask you about God, I have a minute to ask this question, and please do get your questions formulated in your mind, because we've got 30 minutes where I will hand over to you for Q&A. I wanted to ask you about mysticism and God. I'm not going to have time.
You talked about glitches with this out of sync music example. So my last question to you is, is there anything about mathematics that you find ugly? Yes, I do. And sometimes that ugliness is quite fun.
So, you know, that's why I said at the beginning, it isn't just about beauty and aesthetic. Sometimes surprise and being taken somewhere you were not expecting to go, which maybe the beauty was leading you there, can sometimes be really interesting. So I think...
You know, I had a conjecture that I was working on for about 15 years and it was beautiful. It was about the symmetry of certain equations and it was surely right because it was so beautiful. And then I had this, one of my PhD students came in and he looked terrified. I was like, what's wrong? And he said, no, no, no, no. And he started showing me some of his stuff and then towards the end of our session, he said, I,
I'm afraid I have to show you this. And he showed me an example where my conjecture broke down and he was so terrified I was going to be angry because the beauty had gone. But in a way it created a new texture to the whole thing because it then became, you know, why some examples have this beauty to them and others, this thing breaks down and it produced a kind of complexity that was interesting in its own right. So I think that it's not just about beauty that there can be, and that's of course a
true for artists as well. I mean, the breaking of things is often as interesting as making something that first sight looks perfect. That's what Bach does in the Goldberg variations. It seems to be beautifully constructed, these 30 variations on this aria, except when you get to the 30th variation, all the symmetry that's built up to that point comes crashing down. The last movement is a quadlibet, a musical joke, has absolutely nothing to do with everything that's gone on before.
but reveals how much structure there was there that you get this, whoa, where did that come from? So the beauty is in the imperfection. Yeah. And at some point you talk about maths being perfection. So it's this kind of, yes, we've come full circle. Yes. I mean, within perfection, you can have imperfection, which is equally perfect. That's very Godillion. Yeah.
So on that note, I'm going to pass over to you. We have roving mics. So questions, please.
Hello, thank you. It's been very fascinating and I'm excited to read your book. I've come to this as a poet and more specifically, I write bespoke poems on the spot for people. So I take prompts and they can be, I suppose, somewhat random. Now, although I have my own sort of system, I suppose, of themes I draw on and sort of how I go about my work, I was particularly interested when you were talking about music and sort of jazz poetry
maybe think of some of the bebop jazz musicians like Charlie Parker when you talk about birds. So I suppose my question is, to what extent is improvisation possible? I.e., as you were saying with Jackson Pollock, are there structures sort of behind the decisions we make and are these
are we cognizant of these? Yeah, to what extent is improvisation a possibility? Oh absolutely, but I think the best improvisation again comes back to like that Stravinsky quote that, and you'll see that within, again I think most people think jazz musicians are just kind of emoting, but they're not. I mean I try to learn jazz trumpets, I'm a real classical trumpet
and I was amazed at the level of musicology of a jazz musician is way higher than a classical musician because a classical musician has the notes, you know, the composer has done all the work very often and that's an interpretation then of those notes. But the jazz musician, as you say, they're composing on the fly, but they will very often enjoy the restriction of a chord sequence and then they will,
each time try and find an interesting new way through that chord sequence so that I recognize totally as a kind of mathematical trait as well because what I love doing is kind of improvisation within mathematics I will set up a structure but then I'll be interested what what are
the different ways that I can kind of piece this together and move through this? What are the different structures that emerge from these restrictions? And I think poetry as well, you know, why not just write freeform? I mean, freeform can be a sort of poetry, of course, but
what is often quite nice is you will find that maybe some little internal structure that you're using will push you in a new direction that you were not expecting. And that is the joy of these structures that you will find yourself coming out the other side in a place that you wouldn't have without the structure and the surprise of that structure where that takes you.
Hi, I come here also as a poet, although with a background in machine learning and artificial intelligence and so forth. I was particularly interested in what you said about the Messiaen. I was aware of the 17 and the 29. I don't know if you're aware, but there's a similar structure, not using prime numbers, in Philip Larkin's poetry.
His poem is called The Building, which is about hospital and so on. And it's got a rhyme sequence, a very complicated rhyme scheme of eight lines. And you wouldn't notice it really because the first line rhymes with the seventh or something like that.
It's a cycle of eight. What's complicated is that the stanzas are seven lines each. The rhyme scheme starts at a different point at each structure. You have, after
nine stanzas of seven lines. We've got 63 lines, okay, and one incompleted cycle. And he's put one line on its end, separated right at the end to make 64. Now, I've never seen anything else like that. I was wondering if you're aware of that sort of structure. I think what it reminds me of is... So I really thank you for that because that's...
book two, the beginning of book two. Shakespeare, who also, I mean, I illustrate in the book, you know, everyone thinks Shakespeare, wonderful wordsmith, but he also loved using number in a very interesting and almost like code or as a disruptive way. So everyone knows that Shakespeare is iambic pentameter. It's 10 syllables, five groups of two. But when he wants to say something important,
he will change that. So, you know, what's the most famous line in Shakespeare? "To be or not to be, that is the question." It's 11. So there's suddenly this disruptive thing and you're, you know, you're lulling into a false sense of, you know, the rhythm in Hamlet and then suddenly Shakespeare says, "No, you've got to listen to this." And that kind of disruptive beat, a bit like the way Radiohead will jump you out as well.
So there's a prime number, but where also prime numbers come up in Shakespeare is when there's magic afoot. So whenever there's magic in Shakespeare, it goes down to seven syllables. Again, a number which doesn't fit. I mean, the king at the time said that all poetry should be written in twos, fours, eights, and sixteens. You know, that's a dancing rhythm and not use odd number syllables, but Shakespeare broke that. And so, you know, you find it in Midsummer Night's Dream when Pucks
administering the love potion, it's all in sevens. The witches, when shall we three meet again? Sevens. But what your Larkin example reminds me of is a poem, an allegorical poem that Shakespeare wrote, which is absolutely dripping with the same sort of structural framework of primes and things, which is The Turtle and the Phoenix, I think it's called.
which is, there's a huge amount of primes being involved in that, which I discovered playing for the England writers football team. I was in defence and there was a Shakespeare expert who said, oh, what do you write? Prime numbers. Oh, and he told me all of this incredible, I've got this theory about. So I think we lost because we were too interested in defence.
Not defending and talking about what? So I would have a look at that. It's a very, very strange anyway about the birds. Yes, I think it's called the Phoenix and the Turtle. Questions, yes. Hi, thank you so much. That was really thought-provoking. I was wondering, have you ever come across a piece of art where you couldn't find any mathematical structure? And what did that look like? And is it even possible?
No, no. I mean, it's a very interesting question because, you know, my feeling is that there are, there's always some structure in any piece of artwork. If there isn't, it's sort of why on earth would your brain start to engage with it? So, and because mathematics I define as a, as a study of structure, it's whether the structure is something that's kind of has been of interest to the mathematician in some sort of
kind of meaningful way. So I think, you know, sometimes where I, you know, I, I,
I started learning the cello a few years ago and I formed a string quartet because I love quartet music. And I actually find, you know, there's quite a lot of local structure in things like Beethoven or Schubert, but actually I find it quite hard, you know, with the composers of the 20th century, they're dripping with mathematics. The composers of the Baroque also dripping with mathematics.
But the Romantic period is one where I find it harder to understand things which truly resonate with kind of structures that I'm really fascinated in. So that's very interesting because I think one of the challenges is can the artist show the mathematician something that they haven't seen before?
And that's what's lovely in this book. There are quite a few examples. One of them is the Fibonacci sequence, which was named after Fibonacci, who was a mathematician of the 12th, 13th century, Italian, who saw these numbers coming up all over the natural world, number of petals on a flower, number of cells in a
pineapple, but they were discovered centuries earlier by Indian poets and musicians because these numbers actually helped them to count the different rhythms that were possible. Whilst, you know, poetry in the West got rather stuck in particular kind of rhythmic patterns, but the poetry of Sanskrit poetry and Indian music loved disrupting and just finding incredibly varied different rhythms. And so it pushed them to think differently
if you have long and short beats like an iambic pentameter, what other ways could you combine, you know, not just five groups of two going da-da, da-da, da-da, what other things could you do? And it turns out the number of different rhythms you could make using those long and short beats, which the Sanskrit poets loved exploring, is in fact the 10th Fibonacci number. And so they'd already discovered these numbers from exploring it mathematically.
Now the one that I really was fascinated in because it's affected my own research is that I did a piece of analysis of a cello piece that I love called "Nomus Alpha" by Xenakis, who's a Greek composer of the 20th century, was also an architect and worked with Le Corbusier. And I decided I wanted to try and, because this piece uses the symmetries of a cube for the variations that go throughout the piece.
And it's a very difficult piece to listen to. Most people, including myself, when I first encountered it, like, I don't know where I am in this piece. So I wanted to create an animation which explored how the piece was put together, which I've now been using with several cellists. And we do a sort of performance where I'm kind of a VJ changing the video in response to the cellist. But the cellists have said, I've understood this piece in a way that I never have before.
thanks to your signposting of what's going on in the piece. It also helps people to listen to it for the first time.
But the structure that Xenarchist uses is actually a kind of Fibonacci applied to symmetry, which I have never seen before. He basically takes symmetries, then combines the two previous symmetries to get the next one in the sequence in these variations. And then he finds that after, with these symmetries of the cube, after 18 repetitions, it comes back to the beginning again.
I've never seen this before. And it started my own research now in trying to understand what other symmetrical objects, what's the longest cycle you can have with pairs in this Fibonacci thing? Are there rules to how these numbers vary as you change from one object to another? And it's deeply mysterious. I've never seen it before. And it was Xenarchus who came up with this. Well, that's kind of interesting idea to do that, to make some music. But it's also set me off on a new research path.
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You were talking about when a piece of music or a piece of art touches you here, it moves you emotionally. Is that a mathematical thing that's happening?
Yeah, this is so interesting because why do we have an emotional reaction to a piece of music when we listen to it for the first time? That's extraordinary. Now, I do have emotional reactions to mathematics, so the jeopardy and drama and the resolution, especially if I'm making it for the first time. I mean, if you think about...
If you've seen the documentary that Horizon made about the solving of Fermat's last theorem, my colleague Andrew Wiles in Oxford, he breaks down in tears as he talks about that discovery. And he says, please don't use that. And of course, the director, who is Simon Singh, who wrote his book, said, no, that is the beginning of the program, mate. Because he was so unexpected to have an emotional response. Now, what is happening in music? Because, you know, music in
bottom line, it's just a set of frequencies and time sequences of those frequencies. So why an earth, which is hugely mathematical in structure, it's very abstract. That's why I think, you know, music and maths are the two art forms which so clearly have a connection because they're both kind of internally consistent worlds that don't need to be translated out of there into words. As soon as you put words to music, it just
you know, you realize why the composer didn't put words because it belittles it, it doesn't work, you know, the poet will say no but poetry is where I can express my ideas much better through the words not through music but what
What I think is happening is that composers through the centuries have understood the encoding of emotions in the brain. If you think about our emotional world, again, you could reduce it to the behavior of neurons and synapses in a particular dynamic sequence. It's like a piece of choreography happening in the brain as which...
You then, you know, when you're feeling sad or when you're feeling happy, there is something in the brain that's happening. And it will be the same thing happening again when you're sad, probably.
And what I think that is happening in music is that we found a kind of low dimensional version of the huge complexity of that encoding of our emotional world in these dots on the stave, in these levels of frequencies. So I think what's happening, and sometimes it's interesting, is this cultural, is the minor scale something that only the West thinks is sad? But that's still interesting because, you know,
I think those emotional signatures could also be very culturally developed. So that isn't inconsistent with different cultures having different emotional responses to music. But I think music is a form of encoding of our emotional world. So when you hear a piece of music, it triggers the response in the brain and you are just taken on this kind of journey
And the interesting thing is when is it universal? When is it cultural? When is it so specific to a small group who have grown up with the composer? So I think that's sort of maybe what's happening. Lady over there, back. Thank you for your amazing questions, by the way. Yeah, fantastic. No pressure now.
You've described maths as guided by a search for beauty, sometimes in reaction to it and sometimes in coherence with it. But I was wondering, what would maths look like if we didn't have that capacity for beauty, or that sense of beauty? Would maths look different? Would it exist at all? How might it change? I think that's a very interesting question on many levels. Partly how we define what beauty is is kind of interesting because...
I think there's a real argument for it being related to that sort of Keats quote about the connection between beauty and truth. It seems to be, we respond to things if they seem to help us to understand the world around us. And so,
we will often call something which gives us a new way of looking at something beautiful because it's got a truth to it. So it's interesting, you know, are we saying something's beautiful because we're having a response and it's helping us and perhaps bringing us more joy because we're understanding how to navigate our way through. So there's that kind of tension, I think, in, you know, is beauty almost...
is the definition of beauty almost caused by it being these kind of structures that we respond to and think are interesting. But I think the other point is, you know, what is the mathematics that we have pulled out as humans that we find exciting and interesting and beautiful or, or, or interesting
taking us on an interesting new journey with these surprises and moments of jeopardy. I think that that is very related to our cultural upbringing and our way
the way humans are in the natural world around us. So I think that the sort of mathematics that we have discovered is only a small portion of the mathematics that could exist. And so that's where almost the creativity comes in because I will be making choices of the mathematics that I wanna share because I think it's gonna make you look at the world in a new way.
And so, I mean, one of the authors I talk a lot about in this book, who's one of my favorites, is Borges, because Borges got fascinated in the mathematics and science of the early 20th century, and ideas of different sorts of infinity, high-dimensional spaces, and he didn't really have the technical language to explore them, so he used the language of storytelling. There's one story that I love, partly because it also became the
the catalyst for a play I wrote. The story is the Library of Babel. I can see that you know this story. The Library of Babel is a library made out of hexagons put together like a beehive. And this seems to be all that exists is the library. It's a kind of universe. The librarian tries to understand the shape of this universe. And what's amazing is that the shape turns out when you read the story and view it mathematically, it's actually the surface of a four-dimensional bagel that
Borges has made in this story out of this library. But the library is also interesting because it contains a copy of every single book that it's possible to write. And the librarian at first sight thinks, well, this is an amazing library. There's one copy of every single book that's possible to write. They all have the same dimensions. But after a while, he realizes, no, hold on. This library, although it contains everything, it contains nothing.
because nobody's made any choices of the books which are important. You know, my library in Oxford, the Bodleian, doesn't have every book possible to write. It has the ones that we decided were important. Now, I think that many people believe mathematics is like trying to construct a kind of mathematical library of Babel, that what I am doing as a mathematician is trying to prove all the true statements of numbers and geometry.
But that's not true at all. Most of them are boring. Most of them don't resonate. Most of them, although they're as long as the proof of Fermat's last theorem, take me nowhere
spiritually, emotionally. And so actually mathematics that we see around us and we're exposed to is very much a creative act of those bits of mathematics that take you on amazing journeys. And very often they relate to helping us to understand our place in the natural world. And I
And I would say if there's a blueprint to this book, it is the triangle because you've got mathematics on one side, the creative arts on the other. And I think why we're seeing the same structures appearing in both is that both of us are trying to understand our place in the natural world, which is the third corner of this kind of blueprints. And that's why the structures that artists find interesting, maybe there's fractals, maybe the Fibonacci sequence or something.
they will have some place, even the prime numbers, there are cicadas that use primes for their evolutionary survival, just in the same way as Massey used them in the quartet for the end of time. So my feeling is the mathematics that emerges and the things that we find beautiful relate very often back to trying to understand our place in the universe.
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