Within Reason - #108 Marcus du Sautoy - Do Numbers Exist?

Episode Date: June 16, 2025

Marcus du Sautoy is a British mathematician, Simonyi Professor for the Public Understanding of Science at the University of Oxford, Fellow of New College, Oxford and author of popular mathematics and popular science books. Buy Blueprints: How Mathematics Shapes Creativity here. Learn more about your ad choices. Visit megaphone.fm/adchoices

Marcus de Sotoi, welcome to the show. Thank you for having me on. Was Shakespeare something of a mathematician? Well, I think most people would say, surely that's a contradiction in terms, as he was a word smith. What on earth has mathematics got to do with Shakespeare? And actually, I did think the same for many years, but then there was an anniversary a few years ago,

and somebody asked me to do a talk on Shakespeare and Mathematics and mathematics. And I thought, well, I don't know anything about this. But so I was very lucky. One of the wonderful things about Oxford is that you mingle with a lot of people outside your discipline. And so I was sat next to a Will Poole, who's a Shakespeare expert at my college in Oxford, New College. And I said, I've got this, you know, for this task to do this talk about Shakespeare and maths. And he said, oh, no, actually Shakespeare was obsessed with numbers and used numbers a lot in the way he wrote. So I was very intrigued, and he went on to explain some really curious uses of number

to create certain effects in his work. So everyone probably all knows that Shakespeare comes in iambic pentameter, which is, you know, 10 kind of sounds which go da-da-da-da-da-da-da-da-da-da. So that's a nice number. 10 being the number of fingers on our hand, will we count in? But when he wants to say something important, he changes that. So what's the most famous line in Shakespeare? To be or not to be, that is the question.

So suddenly you get this 11, a prime number, a number which doesn't kind of fit into any other, kind of, you know, certainly wakes you up out of 10. And so suddenly you're listening because things have changed. And so this kind of use of an indivisible. number, a prime number, 11, Shakespeare uses to kind of just rattle you out of this kind of soporific, iambic pentameter that you've got used to in Hamlet. And then there's another use again, whenever magic is afoot in Shakespeare, like Mitzamonite Stream, for example, when Puck is administering the sleeping, the love potion into the eyes of the lovers,

it goes down from 10 to 7. So seven is an indication of magical work. And again, beginning of Macbeth, you find seven syllables being used by the witches. So it seems like this was a very conscious decision. Shakespeare knew what he was doing. And there was another poem that's a sort of allegorical poem that he wrote, which uses a huge number of prime numbers throughout the whole.

poem. And again, it just can't be randomly done. It's far too deliberate. So I think he was, of course, and I think that's, the connection is poetry has a lot of structure in it. And you can use that structure for very interesting effects. And I guess one of the thesis of my new book, Blueprints, is that mathematics is the study of structure. So wherever you see structure, you're going to find mathematics creeping in. And certainly Shakespeare is very structural in nature. Yeah, the book, which is out now by the time this episode goes out, I'll link it in the description. I've been reading it a bit and it's fascinating. I've always loved the sort of strange magic of

mathematics and it's quite difficult to explain exactly what it is about maths that's just so weird and wonderful. I think your book draws it out really well and you begin by talking about prime numbers as we've just been discussing and Shakespeare's seeming knowledge of various mathematical ideas, even to the point of, I didn't know this until I read your book, in his 12th sonnet, he mentions a clock, which has 12 numbers on the face, and in his 60th sonnet, he talks about the hours, or the minutes, I think, in the minutes and now. And indeed, there's one that I'm missing. It would be, oh, the 50 second sonnet, which talks about the year.

And when you first said, I think the first example you give, sort of, well, you know, sonnet 60, mentions the minutes in an hour. I thought, okay, that's probably a coincidence. But they started to add up a little bit, and it's getting a little bit suspicious, I think. Yes. I mean, I think you can take it too far. I mean, I've seen some commentators trying to squeeze way too much numerology out of the sonnets. But I think, you know, that's, those are easy games to play with, and I'm sure it's one SheaSphere would have enjoyed. And Chase Spir wasn't the only poet at the time playing these games. So you find a lot of, kind of, I suppose, you know, even if you move to kind of composers at the time, I mean, Bach as well

was using his music to embed kind of secret messages, his name, for example, or his name sort of Kabbalah turned into numbers, which is 14, so 14 becomes suddenly a significant signature number, a bit like, you know, the number of football player plays in, that's the number that Bach plays in. So I think, you know, it was a period, actually. when people are enjoying this sort of crossover between the sciences and the arts, which, of course, at that time, there wasn't such a division as we see now. And I suppose this book is sort of fighting against a rather sort of modern take on the fact that you either choose a humanities or an arts route or a science route.

And, you know, I still think we're suffering through our education system from kind of C.P.'s nose to cultures. And this book is trying to illustrate that, you know, really there's so much more crossover between these disciplines bleeding into each other. I'm interested in what extent you think it is that someone like Shakespeare or indeed the other artists and musicians that you mentioned in the first chapter kind of know what they're doing when they're using mathematical ideas. So, for example, the idea that a prime number is quite sort of irregular and jagged and feels a bit sort of out of place with the, the rest of the number sequence. And as you've pointed out, when magic is mentioned by Shakespeare,

you get these seven syllables showing up, like when the witches are talking and they say, when shall we three meet again in thunder, lightning, or in rain. It's seven syllables. And this is the kind of thing that goes notice. Again, in your book, you write about King James I, who Macbeth was first performed in front of, who writes in a treatise on poetry.

I have the quote here. He says, always take heed that the number of your feet, that is syllables, in every line, be even, not odd as four, six, eight or ten, not three, five, seven, or nine, because there was something kind of uncomfortable about it. And so there is this like mathematical undertone, but there's also this sense in which mathematics just speaks to a truth

that can be independently discovered through the way that things feel in the world, you know? Exactly. And I think throughout the book, there are examples of very conscious decisions clearly by artists who understand the structure and are using them from some effect. But time and again, you find that artists are just enjoying investigating structure and in some sense independently discover ideas of mathematics because of their structural power in their artistic realm.

So, I mean, one very curious one is people probably heard of the Fibonacci numbers, which go one, one, two, three, five, eight, 13, and you get the next one by adding the two previous ones together, named after an Italian mathematician of the 12th, 13th century Fibonacci, who saw these numbers occurring all over the natural world. However, he was not the first to discover these, and if you look back, and this is one of the stories I tell in the book, these numbers were first. discovered by poets and musicians in India who were interested in exploring how many different rhythms you can make out of long and short beats. And so you can already hear that in the iambic pantameter. It's a short, long, short, long. But the Indian poets and especially the tabula players and the musicians were wanted to explore lots of different sorts of combinations. And they discovered that the Fibonacci numbers were actually the numbers that told you how many rhythms if you wanted, if say there were four beats and a long beat was two of these beats and short beat, one beat, then it turned out there were five different rhythms you can make, which was the Fibonacci number. And if you went then to five beats long, then the number of rhythms you can make out of long and short beats is eight.

So here were the Indian musicians and poets discovering these really important numbers. in the natural world through just wanting to understand rhythm. So, and they, you know, they probably should be called the Hamashandala numbers, not the Vibonacci numbers. I mean, it's, it was a, it was Vibonacci was several centuries after they were discovered by the musicians. So, so I love this kind of idea that often, you know, you'll find, going back to the prime numbers, the first example I talk about in the book,

is an amazing use by Olivier Messian of prime numbers in an amazing, quartet he wrote when he was a prisoner of war during the Seguer War in Stellagate. It's called the quartet for the end of time. And he wrote this quartet for four musicians in the camp. There was a rickety old upright piano. He played the piano. There was a violinist, a cellist, and a clarinetist. And they performed this piece in the prisoner of war camp.

And one of the things that Messian was obsessed with was nature and theology as well. but the piece starts with bird themes by the clarinet and the violin. But it's in the piano part where you find this extraordinary use of prime numbers to create a rather magical effect, which is that the rhythm sequence is a 17-note rhythm sequence, which just repeats itself over and over again. But the chord sequence is a 29-note chord sequence. So once the rhythm is finished, and it starts again on the next 17 notes,

the harmony is still working its way through. And when the harmony finishes, the rhythm is in the middle of its playing out. And so the use of these 17 and 29 keep these two things out of sync such you never hear the whole thing coming back together like we're here at the beginning. I mean, it does eventually, but the piece is finished by then. Now, I was intrigued. Did Messia actually know about prime numbers, or did he sort of discover these structures through sort of playing with sound? I talked to George Benjamin, who worked, his composer living today, who worked a lot with Messian. And he said, well, a lot of these structures, Messian just was drawn to for their musical power and then understood the mathematics that was hiding behind it, you know, after having discovered them for their musical power.

And there's an amazing example, actually, later on that I talk about in the book, which is a piano piece that Messian wrote, using this idea of 12-tone rows. So you use the 12 tones of the chromatic scale, and Schoenberg introduced this idea, again, very mathematical. You permute these around, you make a choice of the order of these 12 notes, and that becomes a theme, which you then use symmetry to vary this theme. And there's a piano piece where Messian takes two 12-tone rows, which he then uses to sort of entangle to create the piece,

It's called Ilda four too. But if you consider these from a mathematical perspective, in the world of symmetry, which is actually my own area of research, these two 12-tone rows viewed from the mathematics of symmetry actually create one of the most extraordinary symmetrical objects in a 12-dimensional space, which is one of what we call the sporadic simple groups. We have a kind of periodic table of symmetry, which includes lots of things with patterns in them, and then 26 exceptional symmetries which don't fit into any pattern at all. And these two 12-tone rows actually combined to create one of these 26 exceptional symmetries

that was discovered before Messian discovered them by a French mathematician Mathieu. But again, how amazing. There's definitely no way that Messier knew that he was creating something with such mathematical significance. Yet he understood the beauty of the way. these two 12-tone rows interact, create something extraordinary from a musical perspective. So there's this real sort of tension between those that are using their mathematics and structure very knowingly, and those who are sort of rediscovering it just through the aesthetics

of the art form that they're creating in. We'll get back to the show in just a moment, but first, do you struggle to focus? I know I do sometimes, and especially given my line of work, it can be incredibly frustrating when my brain just isn't able to get on with whatever it's trying to get on with. people like me are exactly who today's sponsor, Brain FM, was created to help. Brain FM is an app for professionals seeking productivity boosts. They create science-backed music, which helps you to relax easier, sleep deeper, and focus better. Opening the app, I've got four options.

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Science Foundation to improve people's focus. So discover these benefits for yourself, just like my video editor did recently, this is a real text, free for 30 days by going to brain.fm forward slash within reason. That's brain.fm. forward slash within reason for 30 days free. With that said, back to the show. Yeah, it's amazing how mathematics can just underlie things which are independently discovered in that way. And music is a great example of that. I mean, when Pythagoras started to realize that music is probably reducible to maths essentially, sort of explodes his brain. And I think there's a really interesting. interesting analogy there also with the psychology of mathematics. It's kind of a mystery to me

why a major chord sounds happy and a minor chord sounds sad. And people hypothesize that it's got something to do with the dissonant frequencies and for some reason that has an effect on our brain. But to me, I don't know if you share this intuition. I have a similar psychology of numbers in the sense that if you ask somebody if even or odd numbers are happy, or sad, I would bet that most people would say even numbers are happy and odd numbers are sad. Maybe odd numbers are jagged, odd numbers are uncomfortable, a bit like King James I thought when it came to poetry. And I wondered if you had an idea of why that might be the case.

I mean, they're just numbers after all. Yes, absolutely. But I think, you know, that idea of even and odd is also interesting in ancient times. They were even numbers regarded as female numbers and odd numbers as male. So, you know, this idea of giving these abstract structures, this kind of personalities, happy or sad or jagged, angry, uncomfortable. And I think it relates exactly to that, I think, the relationship of these structures to the natural world around us.

So, you know, we are comfortable and happy in an environment which is safe. And so, you know, when we encounter things which are even, they can be shared between two or they are good for walking, for example. That's why a rhythm of four, eights and 16s, of course, took on because it's a really lovely for us with two legs. It really fits in well with the body. So I think a lot of these things can be traced back in a kind of evolutionary way to what was helping you, what was hindering you, in sort of navigating your environment around you, and that probably led to the discomfort or the contentment with particular types of number.

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See The Roses, only in theaters, August 29th. Get tickets now. But, yeah, I think, you know, it is, I mean, your question about the emotional quality of music, I think is a really interesting one, because of course, there is a universality of music across all cultures in one sense because of Pythagoras. that we, you know, there are very few cultures which do not divide their scale up into 12 notes. I mean, why didn't we go decimal? Why didn't we go, you know, when the French came in and made everything out of tens? They didn't make music decimal because it just doesn't work. You can't divide the octave into 10 notes and it to sound at all natural in any way. There are other cultures which divided, you know, more finely. But again, the reason is because, I mean, the reason we divide into 12 is if you take a perfect fifth, which is, I mean, an octave is a one to

two relationship of frequencies. And you almost sounds like the same note, which is why we give those two notes the same name, you know, a C and a higher C, because they almost sound the same to our ear, yet one is clearly higher. But then you get the next ratio, which is a two to three ratio. And this is the one that Pythagoras realized there's something beautiful about that a combination of notes, which we call the perfect fifth, and he discovered, you know, experimenting with a string instrument, that it was a three to two relationship in the frequency or the wavelength that we were responding to.

And so, if you repeat this kind of, this combination of notes, a kind of harmonic pair, you take the top one to be the bottom one of the next harmonic pair, and you keep on doing this, you actually come full circle to an octave, seven octaves after you've done 12 of these. So there's a real mathematical reason why everything comes full circle after you've divided the scale into 12 notes. And if you keep on, it's not quite exact, and that's what's kind of intriguing. You know, the tension between physics and mathematics, the physics isn't always as precise as mathematics would like it to be. In some ways that music has this little error. It's called the Pythagorean comma, because it doesn't

quite match up. So some cultures, like in Indonesia, you see instruments which are divided much more finely into like 50, I think 53 or 51 notes, because if you keep on going full cycle, you get a bit closer to matching things up. But, you know, it means that Indian culture and Western music actually is based on that same 12 notes. However, they then make different pathways, different music, using those. And that's why, you know, you can recognize different cultural styles. And my question is, how universal is the emotional response to music? Is it actually much more historically based? I mean, I think there is quite a lot of evidence for a major and minor scale being pretty universal. But, you know, actually, in Indian music, the scales,

they don't use these kind of minor scales. They'd use very different scales, which are combination of kind of major and minor. So, I mean, I think there's a real question about the universality of the emotional response to music, which I would question, and I think is probably much more culturally based than Pythagoras maybe would have liked. But the other interesting thing is, is this emotional response to music? Because this is one of the, you know, I talk a lot about mathematics and music. I mean, I play a lot of music is very much part of my life. But I think, as you say, there's a very strong connection between these two worlds, partly because music is quite abstract in its nature. I mean, it's quite hard to take music into words, and that's

why some artists prefer writing music rather than poetry, because to express their emotional world, it's much better through music. But if you think about what is music, it is basically a lot of frequencies of particular numbers in combination. It's a hugely mathematical structure that you're making, which you're in, you know, you're navigating using your ears. But why are you having certain responses of ecstasy and emotional release as these kind of notes of particular frequencies are following each other? So my feeling is, you know, if you think about our emotional world encoded in our brain, it probably has a structural shape to it.

I mean, I think one of the most interesting, you know, I have this wonderful job, the professor for the public understanding of science, which allows me to go and roam outside of the world of just mathematics. And I made this program about consciousness for Horizon, really fascinating program going into all the labs around the world. But one of the most interesting was Giulio Tannone's lab, which is looking at the difference between the awake brain and the sleeping brain. And, you know, that's a very, if you want to investigate something, investigate what it's not is quite a good way. So, but he is a very mathematical. take on consciousness and the behavior of the network during sleep and being awake.

But we talked actually about qualia and what is actually, how does the brain encode particular qualia? And, you know, we talked about it's probably, you know, a very dynamic, high dimensional mathematical shape, which is actually the way that ideas are being encoded in the brain. And, you know, I had a theory that's something like synesthesia, which actually Messia experience, this idea that when you hear particular sounds, you experience colour, in his case. And in a way, your statement about numbers, even numbers feeling happy and odd numbers feeling sad is a kind of synesthesia. It's in a feeling that you get. And so maybe synesthesia is about the way that you're, you're not a way

that the brain is encoding, for example, sound and color is far too similar. So the brain sort of gets triggered both with sound and color at the same time. But then you think about emotions. Well, emotions probably are, you know, some very interesting structural formation, which gets triggered in the brain. And my feeling is like music has almost managed to find a code for that emotional world. So it's a low-dimensional version of the emotional state, but it's enough to trigger. So when you hear particular frequencies played in combination, that it triggers an emotional response like ecstasy or sadness or, you know, that what is happening is the actual

structural encoding of the music is very similar to the structural encoding of the emotion in the brain. And those are kind of mathematical shapes in a way. Yeah, I think I'm interested in this idea that Pythagoras discovers that musical intervals are relations of numbers and ratios. And what you mean there is that since sound is just a wave, people, listeners might be familiar with the idea of a musical interval, which just means that there are two notes and there's a certain sort of space between them. If you lay them out on a piano, if you play a note. C, and then you play a note G, which is five notes up the major scale, and you play them together, you call that interval a perfect fifth, because they're five notes apart, right? And what those are

are two vibrating waves, and the ratio of the frequency of the waves will be different depending on how big or small the interval is. And some of them sound nice, and some of them sound bad. Now, obviously, we can play a middle C and look at the frequency on a computer or something, and it would be pretty easy to see the ratios involved. How does someone like Pythagoras in the ancient world realize that the difference between a C and a G is a one-third ratio? How does that happen? Well, actually, it's using a string instrument.

So, I mean, I have a guitar behind me, and if you took that guitar and you played an open string, and then you halved the length of the string and played that note, you will get a note which sounds an octave higher. So, you know, in some ways, what's happening is you've got a, you know, a large sign wave, a wave sort of bobbing up and down on the open string, and then you cut it in half, and then you've half the wavelength. So he can actually hear, he can play with that one string and play with different lengths of strings

relative to the open string and start to explore, well, which ones sound nice and which ones don't. So, you know, that perfect fifth, what he will have done is then take that half length and then take a third of the length. And now he's got two notes, which, you know, the wavelength is in a three to two relationship and here he will have heard that very harmonic pair of notes the perfect fit. I mean, the kind of story goes

that he heard this when he was going past a blacksmith and people were banging the anvils and several of them sounded rubbish but when one anvil was being banged you got these two notes which he said, well, beautiful harmony and that's what inspired him to go and explore

what was it about the anvil and the relative weights of the hammers that cause this beautiful harmony that we all recognize us as to, you know, universally. And that's what's interesting because mathematics, of course, is a universal language. It's one that, you know, I spend a lot of time traveling the world, ended up in Russia as a postdoc in a place where nobody spoke any English and I didn't speak any Russian, but we were able to use our mathematics to communicate together. And so because, you know, the music does seem to have this inherent mathematical

quality to it that Pythagra has discovered. You know, that's why, again, across the world, everyone recognized those two notes in a two to three relationship as being the beginning of music as being the basis of what is, you know, harmony as opposed to dissonance. This universality of mathematics is interesting because, of course, two plus two is true everywhere, and at all times, at least we think. However, there do seem to be cultural and historical differences in how we interpret the data.

So, for example, we use a base 10 number system. Other cultures might use base 12, which means you sort of, we count up to 10 and then we start again at 11, 12, 13. But you might go up to 12 or you could go up to 4, you could go up however high you want until you start repeating them. Things like that, as well as, I mean, for me, the most interesting is, probably the introduction of the two extremes. One is zero, this number zero, which doesn't exist in the natural world exactly. You can't point to zero number of things, but it's invented as a concept, and the opposite end of that is infinity, which has, when treated like a number, has some weird qualities.

Like, you're not allowed to, you're not allowed to really do any kind of multiplication with it because it leads to absurd results. Same thing with zero. There are just these rules. You can't divide by it. Yeah. And so when it comes to things like that, would you consider these to be like universal ideas, universal truths about mathematics or more culturally relative? So I think this is a really interesting question because I certainly think that, you know, mathematics is actually a very human activity and the stories we go on to tell and how we tell them about these kind of universal ideas, I think do reflect periods of history, places across the world.

So I do think you can pick up a kind of French style of mathematics, which you, you know, loves, you know, like in philosophy, loves the highly abstract structural, and whilst an Anglo-Saxon kind of mathematics enjoys the kind of exceptional gritty things which don't quite fit into patterns. So, and as you say, there are certainly choices being made about different bases that we work in. The base 10 is the one we do. But, you know, why do we have 60 minutes in the hour? It's because the Babylonians worked. base 60, and frankly, 60 is a much better base because it's highly divisible number means you can divide 60 in many different ways, and that's why it works so well on a clock. You know, you can have half an hour, quarter an hour, five minutes pass. You've got, okay, he's got 12 lots of those. Ten is a bit of a rubbish number, frankly, because it's only divisible by five and two. So these choices do have effects then on the kind of sort of mathematics submerges. I mean, for example, those that didn't have a place number system like the Romans had to keep on inventing new letters as the numbers got larger and larger.

And that's why their mathematics actually was kind of limited in some sense because they didn't have the idea of a placeholder, which includes in there, is the idea of zero. But then you get, you know, so you see the Babylonians have a concept of zero as a placeholder. They need to say there are no units or no lots of 60s. but they never saw this in its own right. So, as you say, that's a big, you know, historically, culturally interesting moment when suddenly a mathematician goes, oh, hold on, I know these numbers were meant for counting things, but it'd be really amazing if there was a number for nothing. And, you know, many people in the Europe certainly said, well, that doesn't make any sense.

You know, if there's nothing there, you don't need a number to count it. But that kind of move to think of a number in its own right for representing, nothing. And of course, where does it come from? It comes from India, where the concept of the void is an extremely important part of Indian philosophy. And of course, the void, Sunyata, Sunya, was never just nothing. It was the possibility of something. And so, you know, we have sort of places, if you think about it, like black holes are where, you know, things are meant to be crushed, so infinite density and zero space. and maybe the black big bang emerged from that.

But you're right, there is this. And in some ways, quantum physics sort of denies the existence of the void that there isn't. That's actually an interesting example of the way the Indian philosophy of Sunya, that emptiness always is the possibility of something. So, you know, a vacuum is never a vacuum for very long because quantum physics starts to fill it with, you know, quantum uncertainty, will give you this kind of particle and antiparticle appearing in the void. So it's kind of interesting that concept of the void. And then, of course, infinity as well is infinity for so long was just a concept which many

regarded as something untouchable, something that we couldn't, that perhaps didn't really exist in this universe, there was the possibility of keeping on counting, but that was just you know, unlimited number rather than the concept of infinity itself. And then you get this mind-blowing moment at the end of the 19th century where a cantor says, actually, no, the concept of infinity is incredibly rich. There isn't just one infinity where things just stop being finite. There are many different sorts of infinity, and you can come up with a mathematics which compares them, so you can start to say one is bigger than the other. Turns out there are infinitely, many infinities, and this has an, you know, an extraordinary effect on mathematics. It has an

extraordinary effect on philosophy, I think, as well. And, uh, and the question then about, you know, do these abstract infinities of the mind physically exist, um, in places, um, is, is, you know, a really interesting one to chew on. But, but my feeling is, you know, that's a moment when, um, although these kind of, you know, I'm a platonist at heart. I mean, I'll come out and expose myself for, you know, I do believe in a kind of structural realm of mathematics and then, but the human is very much involved in bringing these into our culture. So that moment at the end of the 19th century suddenly helps us to see the world in color whilst before we'd just seen it in black and white as finite and infinite. And then suddenly we find these infinities are an incredibly rich and

complex thing. I mean, and I mean, you're a philosopher, Alex. I don't know whether you've what your thoughts are on Badiot. I've got completely obsessed with Alan Badiou at the moment, a French philosopher who really sees the mathematics of infinity that George Cantor came up with as incredibly important kind of philosophical idea for understanding kind of ideas of the events that those moments where something genuinely new appears is a bit like moments when certain sort of infinities appear in mathematics. So this incredibly rich concept, but it needed a moment in history for this to emerge, just as zero did in the 7th century with the Indians. So I think you're right. There's a kind of universality, and I suppose that's my platonic side, yet how we tell

our stories about these things does really reflect our cultural and historical moment. So, you know, for example, imaginary numbers, square root of minus. 1, it sounds like, well, there isn't a number which when you square it is minus 1, because we were all taught that when a number times itself, plus times a plus and minus times a minus is plus. But, you know, the 14th, 15th century, they're starting, mathematicians are starting to encounter a need for a number whose square root, you know, whose square is minus 1. And for years, people just don't know, well, that doesn't exist. You can't go there. And then when does the moment happen that suddenly people experiment with adding something quite scary like this new number. It's essentially around the French Revolution. So this is a moment when people are really trying new things. They're trying new weight systems. They're trying new days of the week. There's an openness to just trying something new. And it's at that moment that sort of the square of minus one begins to get traction and becomes part of the mathematical world. And, you know,

today, it's essentially an understanding quantum physics, early radar, you couldn't have landed planes with radar without the use of these numbers to speed up calculations. So, you know, so I love that side of mathematics, that it has a kind of universality to it, yet there is, you know, cultural and historical relevance to when mathematics appears and becomes part of the human world of mathematics. I find that label imaginary numbers quite a funny one because in a sense I think that all numbers are imaginary

you've just told us that you're a you're a Platonist which might give some of the game away here but I want to ask a question which is as with so many in philosophy something which sounds quite naive but I think is incredibly difficult to answer what is a number yes

That's, that's, uh, yeah, very, very good question. Exactly. Um, so I think, you know, in a way that was one of the things that, uh, was really challenging at the end of the 19th century that we, we had to kind of, we had an intuitive sense that, uh, yeah, number is the abstraction of wanting to distinguish when something is, uh, has the same cardinality that we can match two things. You know, why do I say my, the number of hair, fingers on? my right hand is the same as the number of fingers on my left hand, because I can match them up.

And so I will, but if I took a different, you know, if I took my thumb down, then I find I can't match them up. And so I want a different name for those. So we start to see difference in, there's something common to, you know, I can have five fingers and five cups on my table, and I can match them up. And I realize, okay, there's something common to those, and I want to name that. And so number becomes the abstraction of the matching of those two things. And then we start to see different number because we find we can't match. And so that's why we'll need different names for different quantities. So, you know, all of these things are physicalizations that I would say is something which is sort of pre-physicalization, which is the abstract idea of number.

So we discover number through physicalization. but that kind of abstract idea of number for me, I feel, exists and has quality. So, for example, a prime number, there's no, that's what's interesting. We talked about different cultural ways of writing number, but every culture will identify 7, 17, 11 as prime numbers, things which can't be kind of divided or, you know, if you have 17 stones, there's no way to arrange them in a rectangle

with, you know, same number of stones on each row. each column. So that's a physicalization of prime, and that's independent of your number system, and that's the quality of the number, which for me, I believe, exists. That's not something that's created by the human mind. There's big arguments about this. It's a lovely... Yeah, of course. Yeah, I mean, there's a lovely book, conversations between Jean-Jean, a neuroscientist, and Alan Con, a French mathematician, and they actually want to use the mathematics to try to understand how the brain works, but they never get beyond the fact that Alan Con believes that mathematics exists outside of the human realm. And Jean-Jot says, no, it's just a human creation.

And they just can't get beyond this because they're so non-aligned on that. But the interesting thing is, you see, that then when there was this kind of challenge of infinities appeared, which then caused this kind of crisis in mathematics, which is interesting from a philosophical point of view, because these kind of infinities and large sets started to produce paradoxes, and we realized, you know, Bertrand Russell's paradox of the set of all sets that don't contain themselves as members produce a paradox, and we're not meant to have paradoxes in mathematics. And so we needed to go right back to the beginning and sort of re-found mathematics, almost like

a new euclid. And that involved sort of defining number again. And what's so fascinating is that this go back to the idea that we, talked about of zero being the beginning of everything, not of nothing, it's the possibility of something. So when you go up as an undergraduate to learn mathematics and maybe you did some set theory in your philosophy course, you learned to build the numbers in the following way. Nothing is the empty set. But if you take the set containing the empty set, suddenly you've got something because it's containing nothing. So if you take the set of nothing, that's

is something, and that's what we call the number one, the set which contains the empty set. And then you've got two things. You've got the empty set, and then the set which contains the empty set. It's like you suddenly contained this nothingness in a kind of, in a flask almost. And then you take a larger flask which contains the nothing and the flask which contains the nothing. And this new flask contains two things. And so that is the number two. And so I love this idea. And it's actually the beginning of a play that I just read. called the axiom of choice, which is, starts with the idea, well, it's kind of a pun. So the actor declares the empty set, which of course in theatre could be the set with no staging on it.

And then suddenly the four actors that I've got in my play start to, out of the empty set, start to build number. And they physicalize this idea of containing nothing and then containing the thing which contains nothing and nothing. And gradually they build up number and you see number sort of what's so beautiful appearing sort of out of the empty set, out of nothing, which is, you know, deeply sort of resonant with a lot of kind of Indian and Buddhist philosophy that's, you know, that's where and actually in some sense physics as well, because, you know, there's a, there's a feeling that maybe everything cancels itself out, you know, these quantum fluctuations that we were talking. about in the void, that, you know, the big bang, actually there was just maybe just pure energy, nothing, but then, you know, the particles that we see around us, maybe there's a matching antiparticle which would sort of bring everything back down to nothing and emptiness, but somehow quantum physics doesn't allow that. So you start to get something emerging from nothing. So this idea of the set containing nothing being one, are you putting that forward as a way of

thinking about numbers, or do you mean sort of like that's what one is? That's a definition of what the number one might be. Yes, I think in, so in mathematical set theory, that is the definition now of what one is. And then if you want to say that something has, we talk about cardinality, the size of something. So then if you've got a mathematical structure and you want to say there are six things in this, you will, I mean, formally what you will do is match that up with the number six, which is built out of taking the set of nothing, the empty set, and then the set, and you build up this thing which has six elements in it, and you will then use a bijection, a mathematical sort of connection between that thing which is six and the other structure,

and then you will say that has size six. So yes, that's how we do it now. The reason why I asked what a number is, although it isn't an interesting question in its own right, I had a follow-up question, which was whether or not zero is a number. Of course, it's difficult to imagine what it would be if it weren't a number. But for example, if you say that the set containing nothing is one, I feel like, well, there's not much you can take away from that to point to as your definition of zero in this way of defining numbers. I'm not sure, but do you think zero is a number?

Yes, and in fact, you know, in mathematical set theory, that's where you start. So one is made out of the number zero because you take, so really everything begins with the number zero. So in mathematical set theory, that would be how everything kicks off. But I think, you know, from a sort of more human stance, your question is quite interesting because if not, you know, what is number? Number is used for counting things and therefore it seems like zero isn't, or measuring things as well. You see, that's how we start to get fractions or more interesting numbers like the

square root of two, which can't be written as a fraction. It's an irrational number. But it is a length that we would recognize, you know, the Babylonians understood that the length across a square whose size are units in length will have to be a number whose square is two. And it was one of the Pythagorean's who understood, well, actually, I mean, I think Pythagoras hoped out of his kind of understanding about music being just whole number ratios that, you know, he talked about the whole universe being made out of whole number ratios, that everything was built out of fractions. And then this discovery that the square root of two, this distance across a square, a very, you know, simple sort of idea, a measurement of a, across the diagonal

of a square, cannot be represented by a fraction. There is no fraction which when you square it gives you two was a real challenge, but there appeared a new sort of number. So, you know, it's interesting, the Pythagorean's had started to introduce fractions and then irrational numbers. And so this begins the journey, and it's quite late, that zero is introduced and then negative numbers. I mean, of course, where do negative numbers emerge first in places like China, which had a very sophisticated banking system, which wanted to, understand the idea of debt, and then suddenly these negative numbers, again, you know, that's not only nothing there, there's an absence of, you know, it's almost you throw stuff in

and it's still nothing there. So these negative numbers were quite an abstract idea. But so, so the concept of numbers sort of just broadens as not just something for counting and not just something for measurement. But, you know, again, you get the square of minus one. You have different sort of numbers like the Quaternians, which are kind of four-dimensional sort of numbers, which a guy, mathematician in Ireland Hamilton, discovered and then carved them very famously onto a bridge. So the idea of number actually sort of starts and then grows because of the way numbers can build other numbers. So in set theory, if one is defined as the set containing nothing, what is zero defined as?

The set of nothing. The empty set. So zero is the empty set, which is nothingness. And one is the set containing the empty set? Yeah, so it's the set which contains nothingness, exactly. Gotcha. So it's not the set that contains nothing in the sense of having nothing in the set, but rather that contains...

No, exactly. It actually contains something. It contains this concept of nothingness. So it's got something in it, which is what's so interesting. So therefore has one thing in it, which is, it's got, yeah, so you start to get into kind of language, which seems to contradict itself. But, yeah. You know, when in your book, you write about how mathematics in pre-Elizabethan England was sometimes considered somewhat satanic. And you talk about how John D, a great mathematician from that era, was arrested for calculating, I'd put in scare quotes, in his early career.

And I have to say that in the context of thinking about what numbers are and whether they really exist in the real world, when people start talking about negative numbers existing in this sort of spooky underworld and imaginary numbers which kind of pop up just to serve a sort of mystical purpose but aren't really there, I do begin to see why people might have thought mathematics was a little bit satanic and scary. Well, it also gave you incredible power. And of course, that's what magic gives you, is the power to do things that. other mere mortals can't. And I think that was one of the reasons that, you know, Europeans felt very threatened by the introduction of these, like the idea of zero. And it was

Fibonacci who brought these ideas from the Arab world and the Arabs had learned them from India. And they were incredibly powerful in doing calculations. In the past, without a zero and without a place number system, people were using the abacus. And the power of being able to use the abacus. That's something you had to learn. So it was really, you know, it was a skill of those who held power to be able to use an abacus. And if you were trading with somebody, they could, you know, pull the wool over your eyes by just flicking these beads around. And you had no record of what had happened. So there was a real power dynamic of those who could use the abacus and those who couldn't. But once the place number system was introduced and you had this very simple way of accounting

for numbers using this Hindu-Arabic numerals, that was a real threat. It created a new sort of power to be able to do things. And that was why, for some time, for example, in Florence, the idea of negative numbers and zero gets banned and regarded as a thing of the devil, because they felt it was giving people an incredible power. I mean, the other place you see it is logarithms. So John Napier, who is the first to come up with the idea of logarithms. I mean, he was a bit of a weirdo. He kind of, I think, enjoyed creating this kind of persona of the kind of magician. So he would go around with a little spider in a cage on his shoulder and things like that. But he came up with this idea of logarithms, which again are

an incredibly powerful tool for speeding up calculations. So there was this feeling like this mathematics was almost magical in character because it enabled people to do things that's nobody else could do if they didn't know the mathematics. There's another sense in which mathematics might feel magical in a kind of different sense, which is that for somebody who doesn't believe that numbers really exist, that they're essentially just descriptive items, it's not much of a surprise that mathematics describes the natural world. I mean, it makes sense.

It evolves out of our observations. But for people who think that mathematics is like pre-physical, as you might have put it, that maths is this thing that just exists, and it just perfectly describes everything about the way the natural world works with absolute certainty. There was a paper written, I think, quite famously, the unreasonable effectiveness of mathematics in the natural world or something like this. Yes, yes, exactly. I wanted if you think that that is a bit of a spooky mystery, because for me, intuitively, I kind of get why it's an important question, but maybe it's because with the way that I view what numbers and maths are and is, that it doesn't seem too difficult

to describe why that might be the case to me, but what do you think about that? Yes, I think you're absolutely right, because how did mathematics emerge for, you know, in human culture, it was because we were trying to understand the world around us. So it seemed like, yeah, we were trying to measure things. We were building things. So, you know, the early mathematics is all about building a civilization. then, you know, the idea of proof matches in with the idea of democracy in Greece

and argument and logos. So, you know, we are, as we learn mathematics, we understand, yeah, it seems to be something we invented in order to be able to navigate the world around us. So I think most people who are not mathematicians say, well, of course, mathematics is describing it. That's what it was there for. It's a tool to understand nature.

But I really flip that around because, you know, after a while, as you get, you spend more and more time in this world, you get this kind of weird feeling like, hold on, this is just, there are too many weird mathematical things going on, which are not things that we're creating. They seem to be embedded in the natural world. So, for example, the fundamental particles make up matter. you know, in the 50s and 60s last century, we started discovering all of these different particles and it seemed to be a kind of like physics started to turn into biology, not kind of beautiful, just an electron proton and neutron, but then suddenly we were able to make sense of this because there was a kind of strange symmetrical object hiding behind this, which bound all of these things together to make sense and also to predict new parts.

particles which are kind of missing from the symmetrical object. And then physicists go, this is just too spooky. The whole thing is put together, not in some random way, but it has this incredible mathematical structure behind it, which is helping us to discover new particles. And in a way, that paper, the unreasonable effectiveness of mathematics is a recognition of how, in some ways, you know, this idea of a theory of everything emerging, the mathematics just seems to be embedded in the way the universe works. Now, my take on this is that, you know, mathematics, as I said, it's the study of structure. That's if I was going to define it. And so I don't think these structures need a moment of creation. They are, you know, there isn't a moment where suddenly a symmetrical, there's a moment when a symmetrical object is first discovered, like a dodecahedron or, for example, I had a dodecahedron behind me, built.

So there's a moment when we discover that for the first time, but the idea that 12 pentagons can be put together into a symmetrical shape does not need a moment of creation for that structure to be possible. And so my feeling is that mathematics is atemporal. It's outside of time. It is just the structures which are kind of abstractly, possible. And then what we're seeing, this is an idea that a few people have sort of suggested is one that I like is that what we are seeing around us is a physicalization of part of

this mathematical world. And so the reason that we're seeing all of this mathematics is that the mathematics, and don't ask me how that gets physicalized, because I don't understand that bit. But the mathematics is there to start with. And then we sort of get a physicalization of that, and this is what we see around us. And so, There's no wonder that we're seeing so much mathematics everywhere we look because this is a physicalized piece of mathematics. But the weird thing is that often it's a kind of rather rough and approximate version of what mathematically should be possible. So, for example, one of the structures I talk about as being very important to artists is the idea of the circle. And that is, you know, one of the most fundamental, it primes of the building blocks of numbers, circles of the building blocks of geometry.

Yet, so a circle you can abstractly define as, you know, it's the shape which every point on the circle is the same distance from the center. Yet, if you try and find a physical circle in our universe, you will fail because as you zoom in on this thing, I mean, quantum physics posits the everything is quantized. So there will be a point at which you will reach the pixelization of the natural world. And that pixelization means that you can't have a perfect circle. So I think that's kind of interesting that, you know, we don't seem to be able to realize physically genuinely a lot of these abstract ideas which can exist in the mathematical mind. Yeah, for this reason, I think David Hume, famously an empiricist, believing all knowledge, comes from our observations of the world, was not confident that if you had two parallel lines

extending on as far as you can imagine, of course, two parallel lines will never intersect with each other, but David Hume said, oh, we don't know, we can't be sure if they'll never intersect, because we'd never had the experience of genuinely parallel lines, which to me is probably pointing to a flaw in empiricism rather than a flaw in any idea of the mathematics of parallel lines. But it's interesting to think that these mathematical ideas like a circle are essentially abstractions from imperfect examples, which it's weird that the definition you describe of a circle, or indeed probably of a square or anything like it, would never materialize in the natural world. And that points to certainly the existence of mathematics as something

which is separate from the natural world, even if you're someone who thinks it just comes about as a way to describe the natural world. The actual language that we're trading in here is something else. And I think that this idea that I'm getting at that, well, you know, like, of course it just was built to describe the natural world. So of course it applies to the natural world. It's a little bit like if you built a shovel because you needed to dig something up for a particular task.

And you built this shovel and you were like, okay, I've managed to do my task. And then for some reason, it turned out that this shovel that you'd built just happened to be able to basically do anything else you could possibly want it to. And you're like, how have I, like, I feel like I haven't invented this tool now. I feel like I've discovered this like magical key to the universe. I guess that's kind of like what it's like to discover mathematics and begin to realize that same thing can describe anything. No, I think you're right. So you could take the position that I'm taking, which I think mathematics exists for everything, and then you're a physicalization of mathematics. Or you could take your position, which is, you know, we have the

physical world around us, we start creating this thing, mathematics, but then it takes on a life of its own. So we start creating things which aren't physically realizable. We have, you know, different models of the universe, the geometries that we've created. One of them will match up with the shape of our universe, but we know there are many more, which aren't physically realized and maybe physically realized in alternative universes. So, yeah, you're starting to make something which is clearly separate or goes beyond just the physical universe around us. So, yeah, which I think is, you know, exciting. I suppose it's, you know, you mentioned the shovel.

What it reminded me is the creation of music, actually, because, you know, some people believe that music emerges out of, you know, birdsong, for example. Birdsong was being used for a particular purpose. but then, you know, it's actually something extra that was not necessary but which emerged out of that, which then sort of becomes culture, you know. So it seems to, you know, where do these things emerging, sort of almost that something is made for a particular purpose, but then has kind of extraneous things which you never intended, but which then emerge out of it.

So then, what extent do you think, to what extent do you think mathematics explains anything? That is, I ask this question of physicists a lot when it comes to their trade, and I think it's maybe a harder question to answer. But do you think that mathematics explains the world, or do you think it describes the world, if you know what I mean? Yes, I think, you know, it certainly is good at describing, but I think you, you know, explanation you want to one. understand why, not just how something is happening. You want to understand why something is happening. And I do. I do think that it is a language which gives you a level of the quality of understanding why certain things must necessarily follow, sort of almost in advance. I mean, that's what the power of the scientific method is being able to apply a theory, a model, and then to make

predictions which you can then test. And that That things goes beyond just describing the universe. You're actually explaining why and how things work. And I suppose that's, you know, I get my kind of buzz out of, my spiritual buzz, I guess, from accessing these underlying patterns, model structures, which seem to explain the complexity of the world around it. And what's beautiful is how often something which looks hugely complex can be reduced to something which is much more, much simpler in nature. And I think, you know, that's been the buzz that scientists have lived on for many centuries in a way is just the extraordinary way that, you know, that chaos theory you can have a very simple formula which produces the complex.

rich world around us and you perturb that formula a little bit and it can do something completely different. Yet there is a beautiful, simple formula behind there. But there's a real challenge to that at the moment. I remember having breakfast with a biologist and he really challenged me and said, you know what, you mathematicians and physicists talk too much about beauty. And very often this beauty that we talk about is about understanding some simple structure behind the complexity of the world around. I think you've reached your limit. I think you are now going to find that physics is turning into much more about biology. That's about kind of random things emerging. There will be no explanation about the relative masses of the fundamental particles. You're trying to

look for a formula which will predict why they must be like they are. But actually, you're probably you're going to find that physics will become much more about, you know, it just happened to be like this. There's no reason why it had to be like. And that's where the mathematics will start to fail. It will just, mathematics will start to say, well, look, here are the, you know, huge number of possible scenarios. Our universe just happened to fall into one. And it may just be sort of a kind of randomness that meant we had one rather than the other, which we incredibly unsatisfying because that will be, suddenly you'll lose your explanation of why something is like it is. It just is how it is, and mathematics will just be describing it. I mean, you get it in

string theory because, you know, string theory posits that there are, I don't know, some ridiculously large number of models for the string universe. I mean, 10 to the 150 or something. And we're just one of those. And there's probably no reason, and priori, why we're one as opposed to one of the other huge number and that will, you know, then that's where suddenly the why and the explanatory nature will possibly disappear. I mean, it's been incredibly powerful mathematics in explaining why things the way they are. But that was an interesting challenge by the biologist who said, you know, maybe you've reached your limit and you're going to have to recognize that from now on, it's butterflies.

Yeah. Well, this idea of math describing things, we've been talking about quite abstractly, but your book, Blueprints, which is about specifically how maths informs creativity. Of course, you have a number of books and all kinds of different applications of mathematics, but most recently creativity, which I think does have a lot of overlap with philosophy as well, and is certainly of interest. We talked

briefly about prime numbers, and we talked a little bit about the Fibonacci sequence and circles got to mention. Those are three of the blueprints, the chapters of your book, but you cover all kinds of things, the so-called golden ratio, the platonic solids, randomness is there and at the end, and how these all relate, these concepts in mathematics, how they are used, sometimes knowingly, sometimes not, to inform our creative endeavours, our music, our plays, our writing, our poetry. And yeah, I think it's unique. and it's a really good access point for people to begin to understand what is so magical and interesting about mathematics

without being intimidated by having to study set theory or something like that, you know? So it's a wonderful book and I'll link it down in the description. And I thank you for your time today. I think that any one of those chapters could be a podcast of its own right and so perhaps we can do that again in the future. That would be a pleasure. Yeah, I totally agree with you.

So thanks for having me on and talking about the book and mathematics. more abstractly. You got it. All right. See you later.