Why Numbers are Music to Our Ears (Update)

Mathematician Sarah Hart has twice been a guest on this show, first in 2021 and then again in 2023. We’ve taken the very best of those two conversations and turned them into today’s bonus episode.

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My guest today, Sarah Hart, has taken on a mission that many might think would be impossible: To make math fun and interesting to everyday people.

HART: We like patterns, we like structures, we like symmetry, and those things come out in whatever forms of creative expression we invent, whether that’s music or art or literature.

Welcome to People I (Mostly) Admire, with Steve Levitt.

Sarah Hart holds two academic positions. First, she’s a professor of mathematics at Birkbeck College at the University of London. A pretty standard academic appointment. But her second appointment is more unusual. She’s the professor of geometry at Gresham College, a position established in the 16th century that to this day upholds its original mission to provide free lectures to the public. I can’t wait to talk with her today because I’ve been on a crusade to make math education more engaging, and she’s someone who’s actually figured out how.

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LEVITT: So I’d love to talk about your efforts to popularize mathematics, which is how I became aware of you. And you’ve given this series of wildly popular public lectures, and the ones that first caught my attention were on the relationship between math and music. And as little as I know about math, I know far less about music. My mother forced me to take piano lessons when I was young and I had neither talent nor interest, but I do think if my piano teacher would have explained music the way you do, I think I might’ve loved it. 

HART: Oh, thank you. Well, maybe you weren’t ready at that age. —

LEVITT: So you have a little keyboard with you. And one thing you do in one of your lectures is you just play a note, say a C note, and then you play a note an octave higher and then you play the two different C notes at the same time. Could you just do that now? 

HART: Yeah. Let’s try this out. Okay, so that’s the C, and then as you say, you can play another one an octave higher. (keyboard sound)

LEVITT: Okay. Those sound really different to me. So, play them together. 

HART: (Keyboard sound) That’s them together. So here’s a C. (Keyboard sound) Here’s a high C. (Keyboard sound) And here they are together (Keyboard sound).

LEVITT: So, the crazy thing is, it really sounds like you’re playing one note. Honestly, my entire life, I don’t think I’ve ever had anyone do that simple thing you just did, which is say, “How do you know two C notes are the same? Well, I’ll play you them separately and together.” It’s a great example of how, when you show people  something instead of telling them, it’s so much more powerful. 

HART: Exactly. What we do when we’re learning something or experiencing something is we need to actually see it happening or to hear it — we need something to grab onto, to hold on to. And in mathematics, that’s super, super important, because I could say to you, “Well, here are the laws of music and write down some equations,” and that is not going to speak to you in the same way that me saying, “Okay, those two things sound the same.” And you can think about situations where a group of people are maybe singing something together and you will be just naturally doing that exact phenomenon. The men will be singing an octave lower than the women, usually, and you don’t even think about it. If you’re singing happy birthday at a child’s birthday party, you’ll just instinctively do that. And you’ll all feel that you’re singing the same tune, and to that extent you are, but you’re not because you’re not all singing the exact precise note. You are singing notes octaves apart and these simple relationships that hold between the frequencies of those notes and what makes them sound, to our ear, the same thing when we play them together.

LEVITT: So, why do all C notes sound the same to us? 

HART: It’s because when we hear a note at all — if a note has a pitch at all, it’s because what’s happening is — the air is vibrating and it’s vibrating regularly, periodically, and that vibration will have a certain frequency. So when we think of the pitch of a note. (Keyboard sound) This note here, it will be caused by vibrations of air and they vibrate in our ears and that frequency terms the pitch. Now, the notes that are an octave apart, they have this very beautiful relationship. That (Keyboard sound) low note there — the higher one is exactly twice the frequency of the lower one. So it’s this lovely, simple mathematical ratio of the frequencies. Now, that alone isn’t enough to explain why they sound the same, but that’s the first step along the way. And this was noticed even back as far as Pythagoras, who did some experiments with stringed instruments, and he realized that there was this relationship between the lengths of strings and the sounds that they made. Now we know that’s related to the frequency. There are other nice notes that don’t quite sound the same as each other, but they sound nice together, so, here’s another one — (Keyboard sound) that’s my C. If you multiply the frequency, not by two, but by one and a half — three over two — you get (Keyboard sound) this note. So (Keyboard sound, chord) those sound quite pleasing together. (keyboard sound) And there are other ones. If you multiply by four over three — so again, very simple relationships — you get this. (Keyboard sound) “Here Comes the Bride,” or something like that. (Hart plays the first few chords of “Here Comes the Bride” on her keyboard) So they’re very pleasing relationships between certain pairs of notes — these things called intervals — and they all have these nice whole number relationships between them, or very simple fractions.  

LEVITT: So, does the pleasing come because of these whole numbers — they tend to have their peaks and their valleys more or less at the same time?

HART: What — eventually — it took hundreds of years to realize this, but what’s really going on with these frequencies is that when you take a string and vibrate it, you pluck at the string and it vibrates at some frequency, but there are other vibrations that can fit into that string. So, if you imagine a string, it can have a wave moving along it — or you can fit two whole waves or three whole waves or four whole waves in that length. And, so, what you get are these nice whole-number multiples, they’re called overtones. And the exact composition of this will depend on the musical instrument you’re playing. Violin sounds different from a flute, sounds different from a piano, but you will get some kind of combination of these whole number multiples of a particular frequency. And so, when you double the frequency, you’re getting that frequency that’s twice as high, you’ve already got a little bit of that as an overtone in the lower note. So your mind is somehow prepared for that. It’s already a little bit there. So when you add in that higher note, you’re giving your brain something it’s already prepared for and that’s why they sound so pleasing together.

LEVITT: So, here’s something that puzzles me. You’ve already explained that a high C vibrates twice as fast as a C one octave lower. In between those notes on a piano though, there are 12 keys and those keys, if I’m using the right language — are each a semitone apart. Is there a reason why we divide an octave, where “oct” means eight into 12 pieces? I mean, eight pieces would make sense or even 16. How do we end up at 12? 

HART: So, this is a really fascinating question. And the reason, again, it goes back, probably, to the Greeks and we don’t — this is like the Western classical tradition — it’s not true in all musical traditions, but the reason we divide into 12 is because originally the octaves were seen as nice things — double the frequency. (Keyboard sound) There’s your nice octave. (Two keyboard sounds) And then these intervals, which I showed you before, where you multiply the frequency by three over two, which we now call a perfect fifth — so (Keyboard sounds) that thing — it’s called a perfect fifth because if I just play you like the — this is the white notes I’m playing on the piano. 1, 2, 3, 4, 5, right? (Keyboard sounds as she speaks) So that’s why it’s called a fifth, but that’s begging the question for later. Now, Pythagoras discovered that if you start at a very low, for example, C, (Keyboard sound) and you go up by octaves (Keyboard sound) and you keep going (Keyboard sound) if you go up by seven octaves, you’ll have doubled the frequency seven times, and if you’ll just do that in your head, you’ll find that you’ve multiplied the frequency by 128. Okay. Remember the number 128. Now, if instead you start on this low note and instead of going up by octaves and doubling, you go up by these fifths, which are a nice pleasing interval. (Keyboard sound) You start here, (Keyboard sound) you do a fifth, (Keyboard sound) you do another fifth, (Keyboard sound) you do another fifth (Keyboard sound) and you keep going and keep going. What happens is after you’ve done this 12 times, you get to this seven octaves higher note that we did when we went up in octaves. So, in other words, 12 fifths on a piano equals seven octaves. Now, that means this 12 number, if you just go up and down octaves, you can fit all of those 12 different fifths that you did — they give you the 12 different notes in an octave. Now, that is nice, but there’s this puzzle at the heart of a piano, because if you actually think, what is that interval of a fifth? You’ve multiplied by three over two — one and a half. If you multiply by three over two, and you do that 12 times you’ll get like three to the power of 12 divided by two to the power of 12. That is not 128. It’s 129.7 and a bit. Twelve fifths are not equal to seven octaves. So what on earth is going on?

LEVITT: Yeah, what is going on?

HART: Well, the thing that’s going on is that actually when we’re, you know, playing music, we don’t tend to require a seven octave range in 12 fifths. So this approximation that is used — and that’s what it is — it was good enough. And you can never, actually — you could never get a whole number of fifths into a whole number of octaves because fifths involve the number three, this three over two thing, and octaves just involve doublings, and however many times you multiply threes together, you’re never going to come out with an even number. It’s just never — it’s not ever going to be possible. So, because of that, there’s actually no number that you can divide an octave up into such that everything works out nicely in terms of fifths and octaves. So because you can’t do it, then you say, “What’s a good approximation?” And 12 is a good approximation. 

LEVITT: So I’m surprised the way you’re describing it. I would have thought that the musical — I don’t know the right word — canon. That the ideas behind octaves would have far preceded the math and the math would have come later to make sense of it. But it sounds like you’re saying the development of music was highly integrated into mathematical thinking from the very beginning. Is that right? 

HART: Absolutely. These very early experiments about the lengths of different strings making these lovely ratios, it was all bound up in this idea of how everything has a wonderful universal harmony, and these numbers one, two, three, four, and their ratios. This is how the universe was created, believed the Pythagoreans. It was all mixed up into one idea, and even — there was this belief that actually mathematics — well, and I believe this — underlies everything. So, if you’re thinking of music, that’s kind of number turned into sounds, and astronomy is number as it goes through time and space. and arithmetic is number just studied for itself. And geometry is number as it relates to space. So you’ve got all of these different understandings of number and pattern and music is just a part of that. 

LEVITT: So I might be wrong, but I’m guessing that the idea that mathematics applied to astronomy probably held us back hundreds of years and actually figuring out what was going on because things were not so perfect, as we really hoped they would be when the earth — oops, it wasn’t really the center of the universe.  

HART: Although, it has to be said that theology got in the way as well, a little bit, because there were writers amongst the Greeks who had suggested that the sun was at the center and the earth was going around it. So it wasn’t just this new idea from Copernicus and Galileo. It had been suggested previously, but then others had intervened and said, “No, no, the earth must be in the middle because — religion.” So there are fixed ideas. You get people like Kepler, I feel so sorry for Kepler, because he was so, like, adamant that there had to be some mathematical reason why the orbits of the planets were the distances they are from the sun. And he drew this amazing picture, fitting the five platonic solids — nesting them inside each other. And at the time there was six planets known, and so you put a platonic solid in between each one and that’s why the ratio of their orbits — and of course, now we know, no it doesn’t. 

LEVITT: It’s totally wrong.

HART: Sorry, and there’s more planets than that anyway. So, yeah, a lot of good minds perhaps spent time trying to fit things into the theory that they had — wanting things to be circles when actually they’re ellipses. Because we’re humans and humans get ideas that they want to be right and mathematicians are humans as well. It can happen to the best of us.  

LEVITT: How much of what sounds good to us is just familiarity versus the dictates of math? So, my wife loves hardcore metal music — crazy music from bands with names like “Five Finger Death Punch.” And when I first met her, I couldn’t even listen to it, but now I kind of like it much to my surprise. Just having heard it a lot.  

HART: Yeah. So, I would say the octave is probably the only thing that is universal in that sense. That sound that when you play them together, (Keyboard sounds) those sound similar to everybody. But then once you get beyond that — this division of the octave, which really comes from the, perhaps, obsession with or liking for these perfect fifth intervals. Actually, there are other ways to think about things and if you favor different intervals or gaps, then you might get different answers to these questions. So, yes, we can say that mathematically speaking, we understand because of the mathematics of frequency and the laws governing frequency, that there are some things that perhaps to our ears might sound more resonant, but that doesn’t completely answer the question, because if we talk about these small ratios — we like a ratio of four over three or something — but why don’t we like seven over five, which isn’t that much bigger, but it sounds horrible and there are also intervals that a few hundred years ago would have been absolutely not allowed. So, the fourths and the fifth that we mentioned, these were approved of for 2,000 years in Western music. But when Monteverdi comes along — who’s written, you know, some amazing music in the Baroque period — he was including things like intervals of a third, which is this one. (Keyboard sound.) Those were referred to as “licentious modulations” by the church and like — just going to drive the kids crazy and put them off their worship. So, at different periods of time different things have been in favor. You know, even something like — you’d think half an octave would be a lovely sound. 

LEVITT: Yeah, you would think that.

HART: So, what do I mean by half an octave, really? It’s halfway up, musically, so the frequency — it’s got to be not halfway along, but root two of the way along to get that geometric mean. That thing is called a tritone — half an octave. And you don’t — (Keyboard sound) you don’t hear that very much until maybe the last century, I suppose. But, of course, you hear it all the time nowadays, if you have ever watched T.V., because (Keyboard sound, intro to Simpsons’ theme) it’s the first two notes of The Simpsons theme tune. But until that came along you didn’t hear much of that. 

LEVITT: The 20th century classical — what we call classical composers of the 20th century — they really did rely heavily on mathematical foundations it seems. But, is it not true that in the end, the marketplace didn’t really buy those ideas? My impression is that people listen much more to Bach and Beethoven and Wagner than most of the composers of the 20th century, who were doing such fascinating things trying to bring math into music. Do you think that maybe math wasn’t enough — and something more was needed? 

HART: Well, response one is: are these the licentious modulations of the 20th century? So there are composers who have tried, as you say, consciously to incorporate mathematical ideas, and I think there are various levels of success. And one potential pitfall is something that can happen in mathematics as well. You box yourself in. So, you can prove amazing theorems but no one cares because you’ve got too much structure and not enough room for creativity and playfulness. So, if you are really over-specifying every single thing in your music and saying, “Right, I will have an algorithm for what the next note is. I will have an algorithm for what the length of the notes are. I’ll have one for whether they’re loud or quiet.” You can take it to an extreme that no one wants to listen to it, including the composer. So, I think what happened in the 20th century to some extent is that ideas got pushed to their limits, and then — it’s strange because it’s almost as if it sounds random at some point. If you put so much structure in that you have absolutely no control of what’s going on, then it’s almost indistinguishable from just random sounds. And then, you got to really think, ‘What is the purpose of this? What am I doing this for?’ Whereas, the more successful music that involves structure is where there’s enough freedom still to find the constraint invigorating rather than repressing. So you’ve got people like, Hindemith whose “Ludus Tonalis,” which is like tonal games — he was playing with these ideas of having different keys, and he had some kind of cool structures in there. This is a whole kind of series of pieces all linked up together, and working through all the different starting notes, and the last one and the first one, they were the rotation of each other mathematically, which is cool and difficult to do, but that did not stop him from writing a beautiful piece of music. If you listen to these they are full of structure and they’re fantastic, but they aren’t off-puttingly sterile, which is always the risk.  

LEVITT: Have you ever tried composing music? 

HART: Well, yes, but only in the sense that — only in the sense that teenagers write poems. Yes, to be truthful, but no to be, like, sensible.

More of my conversation with Sarah Hart after this short break.

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Sarah wrote a book called Once Upon a Prime, which dives into the magical overlap between literature and mathematics.

HART: I have had a blast doing this book because when people say what? What do you mean there are connections? And then it’s a great joy to be able to show people what they are. Mathematics is really our way of understanding structure and pattern. And if you think about it from that angle, then you start to see, okay, literature has, for example, poetry. I can see that there’s structure in poetry. You can see that there are rhythms and patterns and so perhaps it’s believable mathematics is involved there and it is, poetry is full of mathematics. But there’s also structure in all sorts of literary works, in novels. There are people that use structures in their work consciously, but it’s also there, even without thinking about it.

LEVITT: Let’s start with maybe the simplest form of structure that authors impose on their work. And I guess that might be haiku?

HART: So Japanese poetic forms don’t have what we might use in English in Western poetry, which might be rhyme schemes and things like that. In a form like a haiku, it’s all about counting the sounds. We sometimes say syllables. That’s not quite exactly what it is, but sounds might be a better term. And in a haiku you have 17 sounds and it’s split into five, and then seven, and then five. And if you’re a mathematician, hearing those numbers, five, seven, 17, it jumps out at you that they’re prime numbers, right? So why 17? Why not 16? Or why not some other number? Why is it split up into these three — another prime number — different parts? And thinking about it in terms of those choices that have evolved in the poetic form, you start to say, alright, so that five and seven, the first two parts of a haiku, that makes 12 sounds together. Actually, if you split it evenly six and six, that’s less of an interesting thing. It doesn’t have this dynamic structure of leading onto — five leads onto seven. It makes it just a little bit interesting, but also five and seven being prime, you can’t break them up any further. You can’t divide them by anything. So there it makes for a cleaner break and it more emphasizes that little separation. So these are the kind of things you can notice and think about even in that very simple constraint of 17 sounds.

LEVITT: You talked about the mathematics of vibration and how some combination of sounds naturally sound good together. But it does seem like a little bit more of a stretch when you apply it to literature. You think that’s fair?

HART: I think it’s perhaps less on the surface, that’s one thing. So when you get into looking at how various kinds of poetry or literature are put together, it ceases to feel like a stretch. I’ll give you an example of a book. There’s a book called The Luminaries by Eleanor Catton that won the Booker Prize in 2013. And that book has a mathematical structure underneath it, which is that every chapter is half the length of the one before. So that kind of halving thing, mathematically we call it a geometric progression, that isn’t necessarily easily spotted. You see perhaps that something’s going on when you’re reading the book, but it’s not easily spotted necessarily. But it gives a really wonderful feeling of impetus to that book and when you see what’s going on, and she leaves a few clues in the text as well, which is cool — when you see what’s going on, I think it adds to your appreciation of that novel. Perhaps in music, because you have your beats and you have your frequencies and things like that, you can see the mathematics coming in there, perhaps more obviously. But it’s definitely there in literature.

LEVITT: So you talked about the natural structure of primes in haiku. How do you explain the sonnet? I don’t see anything natural about a sonnet?

HART: Well, yeah. So this is interesting. So a sonnet, you’ve got a prescribed number of lines and you have a prescribed rhythm and a prescribed rhyme scheme, which is which lines rhyme with other lines. So there are 14 lines in a sonnet. 

LEVITT: That’s not prime.

HART: That’s not prime. Well, it’s seven couplets and two and seven are prime. But with a sonnet, my feeling is it probably developed organically from smaller collections of lines. So a quatrain, four lines, seems to be quite a natural thing. In my book, I asked my little girl to write a poem and she wrote me a quatrain, four lines, just without any further instructions sort of thing. So a sonnet, at least in English, a sonnet is usually three quatrains, which is like three lots of four lines, and then a rhyming couplet to wrap it up and let people know you are done. Like a lot of scenes in Shakespeare’s plays end with a rhyming couplet. I suspect it developed from that fairly organically and then became crystallized into a specific form. But when I talk about sonnets in the book, actually the reason I talk about them is because of this French poet Raymond Queneau, who wrote a little book called Cent Mille Milliards de Poem, which is like a hundred thousand billion poems. And he did that essentially by writing a three-dimensional poem, which is very cool. He wrote 10 sonnets, and then these 10 sonnets all matched up with each other in the sense that all the first lines rhyme with each other, all the second lines rhyme with each other. So he’s got this sort of three-dimensional thing he’s building up. And if you do that, kind of like a child’s flick book, you can pick any one of the first lines, any one of the second lines, any one of the third lines, and make yourself your own sonnet from these 10 basic ingredients. It’s an interesting discussion as to what extent all these sonnets exist, but encapsulated in that book of 10 starter sonnets, you can make 10 times, 10 times, 10 times 10 — 14, 10s multiplied together, which is a hundred-trillion potential sonnets. So it’s a really fun way to look at the amazingness of a combinatorial explosion of possibility there.

LEVITT: Now, one of the most basic assumptions of economics is that constraints are bad, at least in classical economics. People are always better off, or at least no worse off, when you relax constraints. So more money is better than less money and more hours in the day would be better than fewer hours in the day. Do you think literature is an exception to that economic logic that constraints really make it better?

HART: So I think there’s some sweet spot, right? If you constrain everything, then you’re trapped, and it’s a really rigid box; there’s no room for you to be creative. With absolutely no constraints at all, you’re out in the wilderness. But with a few simple constraints, like you might have in a poetic form, that doesn’t stop you being creative. It spurs you to creativity. And so there’s this great quote from the Irish poet Paul Muldoon, where he said that poetic “form is a straight jacket in the sense that straight jackets were a straight jacket for Houdini.” They give you something to push off from. They give you an impetus to be creative within the structure that you’ve got.

LEVITT: I’m not sure how familiar you are with six-word stories, which are an extreme version of this constraint. Stories that are told in their entirety in six words, and the most famous one, which is attributed, probably falsely,  to Ernest Hemingway, is this one — I think it goes: “For sale: baby shoes, never worn.”

HART: Yeah. Oh yeah.

LEVITT: Which, which, I find to be an extremely haunting and powerful and memorable story because it says so much with so little. “For sale: baby shoes, never worn.”

HART: Yeah, exactly. And the brevity of it is part of the greatness of that tiny story. Chekhov, his short stories are some of the greatest pieces of literature there are. What you can tell in a few pages — and with that restriction, I’m not going to give myself a whole novel. I’m going to tell this beautiful jewel, this crystalline perfection in a few pages. That makes it better. These short stories would not be better as novels. They are perfect as short stories.

LEVITT: My Freakonomics coauthor, Stephen Dubner, he started to get obsessed with these six-word stories and he was invited to be part of a book where they asked famous authors to write their own memoir in six words. And I still remember his. It was, “On the seventh word he rested.” Which I find interesting because it reflects Stephen Dubner’s identity as a writer and a creator. And also, his unusual relationship with the Old Testament because he’s someone who converted to Judaism. But there’s also something, I don’t know, sacrilegious to it because he’s saying, “On the seventh word, he rested,” which is somehow making this direct comparison between himself and God. So it’s — I don’t know. It just makes me think, which I guess is the best you can hope for in six words.

HART: Well, exactly. And I really like that because as you are hearing the six words, and you don’t get it till the very final one, there is no seventh word and here’s why. And that tiny little story’s told you why we’ve stopped at six as well. It’s fantastic that; I love it.

LEVITT: Exactly. So the people who took this constraint idea to their extreme, there was a French group of writers. What were they called again?

HART: The OuLiPo, it’s shortening of the first two letters of Ouvroir de Littérature Potentielle. So Workshop for Potential Literature. And the hundred-trillion poems were by Raymond Queneau, he was one of the OuLiPians. And they were all about exploring ways of making potential new kinds of literature with different constraints, really, mostly mathematical in nature. But the most famous one probably is Georges Perec, who wrote a whole novel called La Disparition, which doesn’t contain a single letter E. And this is a really good example where the first question probably anyone would ask is, why do that? And that’s a really important thing to address because if we are just making up constraints and rules for no reason and playing intellectual games with them—

LEVITT: But that’s what he was doing, right?

HART: Well, but it’s more than that. The novel without the letter E — that wasn’t the first book to be written emitting a single letter. So there was an earlier one which didn’t use the letter E in the 1920s, but no one’s heard of it. And I think the reason no one’s heard of it is because it doesn’t do anything cool with that restriction. Yes, it cleverly avoids using the letter E. It’s a lot of work. It’s very hard to do, but there’s nothing in the story that’s relevant to that. Whereas, Perec’s book, called La Disparition, The Disappearance, and in English it’s called A Void, which are both clever titles. The book itself is about something that’s missing; something that’s disappeared. And then there are clues in the text. The characters know something’s not right with the world. They’re looking for something. There’s an encyclopedia with 26 volumes, but volume five is missing. There’s a hospital ward where there’s no patient in bed number five, because the fifth letter of the alphabet is E. And there are further layers towards this. So they work out eventually, the characters, what’s going on. But if you think about Georges Perec, himself, lots of letter E’s in his own name. He lost both his parents during the Second World War. In French, you can’t say family, mother, father — you can’t say those words without the letter E. Perec cannot say his own name without the letter E. So this novel, which is about absence, disappearance, it has echoes within his own life around loss and things not being there. So that for me is what makes the use of a constraint interesting and worth doing. If you actually do something with it, it’s not just a random choice, and that’s exactly what we do in mathematics. We don’t just randomly think of rules. We say, okay, these things seem to be what’s happening in say, geometry. So you can set up some basic rules like we do in geometry at school. This is what a line is. This is what a point is. This is what a circle is. You need a starting point in mathematics. You don’t choose the starting points randomly, otherwise, it too would be just a sterile game. And mathematics is not that. Mathematics can help us understand so much because we don’t do things at random. We choose what are our constraints going to be? And then we play in a beautiful, wonderful playground of mathematics.

LEVITT: So that is interesting what Perec did by leaving out the E. But then he wrote another book where he only used E’s. Did he have a good story for that or was he just having fun?

HART: Well, he said it was all the E’s that he hadn’t used in La Disparition were sort of lonely and sitting around waiting to be used. So yeah, he wrote a book only with E’s and I think that was just, you know, having a bit of fun. And why not?

LEVITT: You know, I’ve always been fascinated by books where the reader gets to make choices that influence the story’s outcome. Books where it says, “What should Sarah do? Should she call the police or investigate the crime scene on her own? If it’s called the police, turn to page 61. If it’s investigate the crime on her own, turn to page 273.” There must be some really interesting math underlying this kind of story telling.

HART: It links to an area of mathematics called graph theory, which is about networks and links and connections. So if you imagine, one of the ways we encounter graph theory without knowing it every day is if we do a search online. There are gigantic networks of web pages that are interconnected to each other. And when a site like Google gives you the results of your search, it’s got that huge network that it’s looking through, looking for connections, finding the ones that are most likely to correspond to what you want. In a story with many different branches that you can follow and different parts that you can go down, you can think of all of those paths being linked together. And you can imagine a gigantic network that you could study. And one of the things that mathematics can do for us is to help us understand networks and how to find our way through them. But also, the gigantic number of paths through and number of potential outcomes would really make books like that or any other kinds of structures like that just impossibly big, unless we’re super careful to limit choices a bit, but without the reader realizing that they are being gently steered in one direction or that not all of their choices have gigantic consequences. And this is where some of the interesting stuff happens, right? There’s always at this point of a balance between some structure that you need to stop the story getting out of control, and ability for people to make free choices and involve randomness. And those little tiny tweaks, something that’s a bit different but only a little bit different from completely random, can totally change the structure you end up with. And you see this in mathematics all over the place, in modeling, any kind of situation with big networks and societies. If fully random, that’s one extreme. Just take away a tiny bit of the randomness and you can then really make progress and understand what’s going on.

LEVITT: In the book you talk about a movie called Bandersnatch, which is based on the science fiction show Black Mirror. It’s a choose-your-own-adventure film. So at different points in the movie, viewers are asked to make a choice, and that influences what happens on screen. So the making of that movie must have been a fascinating application of graph theory.

HART: Yeah. Exactly. So there, if you think about how many different scenes are we going to have to film for the audience to make — let’s say you’re going to make 10 choices, right? So you watch the first scene, you make a choice, and that gives you two possible next scenes and you make another choice. So now your two bifurcates again, and there are four possible scenes. And make another choice, it’ll be eight and 16 and 32. And we all know how doubling goes. Nine choices would give you 1,024 and then you get 2,000 and so on. With just a few choices would give the poor old actors who are going on strike by this point, like a thousand scenes to film unless — so behind the scenes there’s got to be a mathematical jiggery, pokery going on to prevent the actors having to film a thousand scenes. So these kind of big networks, these graphs that you can draw, actually, they have to connect with each other. You have to loop back and things have to connect up in a more structured way because otherwise numbers become unwieldy so quickly because of this doubling. Powers of two are a very, well, powerful thing.

LEVITT: And do you know how many scenes they actually filmed in Bandersnatch and how many ways there are through it?

HART: They didn’t film a thousand scenes. The total length of it might be about three hours, but you don’t see all of it. You see bits of it. But that’s a really tight, small amount. The total amount of minutes of television that there is filmed is very small, really, compared to the amount of choice that you have to go through it. 

LEVITT: Because if they just did two to the ninth, say —

HART: If they did two to the 10, so two multiplied by itself, 10 times, they would have 1,024, scenes. Right? Which is too many.  

LEVITT: They’d have to film a hundred times as much movie as you actually see. But if you’re saying that the total filming in Bandersnatch was something like three hours, then they only film maybe twice as much. So they managed to cut the extra material 50 fold to make it work.

HART: When I was writing the book, I was lucky enough to talk to Sir Ian Livingston, who has written dozens of these choose-your-own-adventure type books, which some of us, if we’re of a certain vintage, may have had when we were kids. When I talked to him, I asked okay, how much do you not see? If you’re reading the book one time and you’re following a particular way through, how much of the book will you see? How much of it will you never see because your choices have eliminated that part? And he said, actually, they design it — and they draw these graphs themselves by hand. They design it so you encounter about a third of the book on any trip through it, which I was surprised it was so much because that implies just very clever use of the structures. So that they put in pinch points, they call them, which is places where the plots all converge and then you have to go there and then you proceed to the next bit. It’s amazingly clever what they do, and it’s that mix of creativity, structure, the interplay between them. You don’t want it too structured. You don’t want it completely random. And it just — that sweet spot is where the magic happens.

LEVITT: Now going back to Bandersnatch, of course, the title Bandersnatch is a reference to one of Lewis Carroll’s stories. And with Lewis Carroll being of course the most famous mathematician-author of all time. He must have used math in his stories like Alice in Wonderland. Although I couldn’t tell you how, even having read them as an adult.

HART: It’s the whole outlook of the books, the whole tone of voice of the thinking. Because as a mathematician, he was very interested in logic, in the rules of logic and following the rules and working out what you could deduce given a particular collection of statements or something like this. That’s what he does in the Alice books, except that the initial setup is, you know, completely insane, right? But then you go with it. So if a small child comes up to you and says, “You are an ocelot and I’m a space hedgehog,” then you are, and you just go along with it. You follow the rules of the game, the internal logic. In Alice In Wonderland, the internal rules of the game say that you can grow and shrink by eating bits of cake and drinking potions and things. If you can grow and shrink, then it would be possible to swim in a lake of your own tears. And that’s exactly what happens to Alice. And so it’s like, if this is possible, then that. The whole thing is a gigantic reductio ad absurdum, right? It’s proof by contradiction idea that we have in mathematics. You follow a line of logic, see where it takes you. Maybe at some point you’ll prove that one equals zero. Okay, that meant your original assumption was wrong. This whole of Alice in Wonderland is he’s doing, he’s playing. Just like what mathematicians do, seeing if you can break the structure. But he also mentions a few numbers in there, and his favorite number is 42. Which is also the favorite number of another book that came much later, The Hitchhiker’s Guide to the Galaxy. But Lewis Carroll seems to have loved the number 42. He drops it in here and there, he hides it in some of the patterns in Alice in Wonderland and in Through the Looking Glass. And then games. Alice in Wonderland is based on cards. There’s the Queen of Hearts and those things. Through the Looking Glass, the sequel, is based on chess. And he actually said, you could play that book as a genuine game of chess. Now, I haven’t tried to do this. I think it’s perhaps — might be a bit tricky to work out what move is next. But Alice is traveling from one end of the chess board to the other, and at the end, she might get to become a queen herself. 

You’re listening to People I (Mostly) Admire with Steve Levitt. After this short break, Sarah shares what she’s up to now.

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Sarah Hart’s book Once Upon a Prime is so joyful. Unlike most nonfiction books it doesn’t have a goal of helping you live longer or write a better resume. It’s just pure play and a joy to read.

HART: I suppose what I would like people to come away from the book with a chance to do is to have a look again at the books they love, the writing they love, and maybe it will give them a whole ‘nother layer of enjoyment to that reading because, yeah, what is life for if it’s not for producing beautiful things, for enjoying beautiful things, and talking about and sharing and loving those things. For me, both mathematics and literature are just wonderful oases of happiness. I’m just so excited to share all these lovely things that I’ve spotted and noticed in literature, and I hope it will help people as well to see in their favorite writing, those things. 

LEVITT: So your time as the Gresham Professor, it’s soon coming to an end. Is that sad for you? 

HART: Yeah. I’ve got one more year. I’ve so loved this role, Gresham Professor of Geometry. It’s such fun thing to be because this has been going since 1597. And someone writing about it in the paper a week or so back, he said, “It’s a position so old that the first incumbent invented long division.” What a great lineage to be part of, right?

LEVITT: It must still be an enormous amount of work. You’re obligated to present many public lectures. Is it all consuming?

HART: It’s quite time consuming, yes, for sure. But I love doing it so much. I love finding out like the cute little angle, the nice hook that gets you in. Showing people things that they haven’t seen or a new angle on things that they may think they know something about and they perhaps do, but here’s a totally different way to look at it. And we talk about such a range of different things. It’s exciting for me to prepare. And I have found my true passion is really communicating — talking about, writing about mathematics. It’s something I so love doing and I want to spend the rest of my life doing it.

LEVITT: It’s interesting ’cause in my own way, I had that same realization. I loved writing academic economic papers until I had the chance to be more public facing and to think about the issues differently. And after that, all of the allure of writing academic papers left me. Have you reached that same point where the thought of writing a math paper for an audience of seven total readers is no longer very appealing?

HART: I’m getting there. I wrote an academic paper actually about the mathematics in Moby Dick that came out in 2021 and that was downloaded like 3,000 times or something within the first few months. And my next downloaded mathematics paper, which was in some kind of esoteric abstract algebra — super interesting to the 12 people who downloaded it, you know? What’s our responsibility in our short time on earth? We do the thing that only we can do. We do the thing that gives us joy and makes us happy. And that maybe is the thing that is unique mixture of our passions, our abilities. And the mathematics papers that I’ve written, I’ve very much enjoyed doing. I have loved proving the theorems that I’ve proved. But if you think about what can I bring to the world and try and make the world a teeny, tiny bit better in some way, the way I can do that is perhaps sharing my joy in mathematics with a wider audience.

Sarah Hart has definitely made my world a teeny, tiny bit better by sharing her joy in mathematics. Since we were replaying her episode, I reached out to Sarah to see what she’d recently been working on.

HART: When I was last on the show, my book, Once Upon a Time, about the wondrous connections between mathematics and literature had just been published. Since then, I’ve been honored to be invited to speak on this topic to many audiences in the U.K., U.S. and further afield. I’m next in the U.S. in March 2025, where I’ll be giving the Einstein Lecture of the American Mathematical Society. It’s a tragedy that so many people come away from school thinking that mathematics is just about dull calculations. It’s a bit of a mission for me to try and change these perceptions and share how gloriously creative and beautiful mathematics really is. As part of that, I’ve recently been speaking and running workshops with school teachers and professional teaching organizations, looking at how we can work across subjects to explore with youngsters the fascinating links between mathematics and other creative disciplines, whether that’s literature, the visual arts or music. And in fact, this last is the stepping stone for my next big project. Going back to what Steve and I chatted about the first time I was on the show in 2021. Mathematics is woven into every aspect of music from the beautiful number patterns underlying harmony to the geometrical symmetries of melody and the ancient algorithm that predicts the rhythms most favored by musicians across the world. I’m excitedly in the midst of a deep dive into all this and more, because right now I’m planning my next book, which will be all about the beautiful mathematics behind music. 

Next week I’ll be talking with my old friend Jonathan Levin. He’s a pathbreaking economist who’s now the president of Stanford University.

LEVIN:  Sometimes in a leadership role of an organization you can be a little bit like a Tyrannosaurus Rex. You have very small hands in terms of getting things done, but when you sort of turn in one direction, your tail can wipe out the time of a lot of people.

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People I (Mostly) Admire is part of the Freakonomics Radio Network, which also includes Freakonomics Radio and The Economics of Everyday Things. All our shows are produced by Stitcher and Renbud Radio. This episode was produced by Morgan Levey, and mixed by Jasmin Klinger and Greg Rippin.  Our theme music was composed by Luis Guerra. We can be reached at pima@freakonomics.com, that’s P-I-M-A@freakonomics.com. Thanks for listening.

HART: So I’m like, “Okay, why is a centaur not an insect?” And I thought, this will stump her all the way to school. But unfortunately she came straight back with, “Well, do they have an exoskeleton?” And I have to admit that they didn’t.