The Joy of Math With Sarah Hart

My guest today, Sarah Hart, is the Gresham Professor of Geometry. The first woman to hold that position in its 400-year history. She has a special gift for making math interesting and accessible.

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.

Her brand-new book is called Once Upon a Prime, and it dives into the magical overlap between literature and mathematics. Back in January, I interviewed mathematician Steven Strogatz, and we had an idea to teach math appreciation, a course about the elegance and beauty of what math can do. In Sarah’s return visit to the show, I’m hoping to explore how the material in her book can actually bring math to life.

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LEVITT: So Sarah, you were on the show back in 2021, and that was a conversation that really sticks with me. We talked about all sorts of things, but especially about the links between music and math. 

HART: It was a great conversation. I still think about it a lot actually, as well.

LEVITT: We ran out of time in that conversation and never got around to talking about another passion of yours, which is investigating the connections between literature and math. And I have to say, when you told me you were writing a book on that topic, I was a bit incredulous because off the top of my own head, I couldn’t come up with a page worth of connections between literature and math, much less a book.

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. And this is why it’s been so much fun. 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. We, as human beings, we can’t help ourselves. 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. 

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: It reminds me a little bit in our last conversation where 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 are 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. 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 co-author, 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 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?

We’ll be right back with more of my conversation with mathematician Sarah Hart after this short break.

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LEVITT: So you masquerade as a normal person, but there’s a point in your book where you reveal just how exceptionally weird and geeky you really are. Do you know what part of the book I’m talking about?

HART: It could be so many parts, Steve.

LEVITT: There’s one that jumped out. So there’s something called fan fiction, and that’s when regular people write stories involving popular fictional characters. But you propose something called Fano fiction.

HART: Yeah.

LEVITT: Do you want to explain what Fano fiction is?

HART: So I was playing around with ideas around constraints and when I mentioned, this link between the constraints of poetry really giving us creative impetus to write beautiful sonnets or other kind of poetic forms and the constraints of geometry enabling us — those few little axioms that we start off geometry with — enabling us to prove beautiful theorems, like Pythagoras’s theorem. And so that got me to thinking, I wonder if there’s a geometry-like constraint that I could play with and make a new form of potential fiction. It’s a bit like your six-word stories. There is a kind of geometry, it’s a particular mathematical structure called a projective geometry where you only have a very few points. So of course, in our normal world of geometry, there are infinitely many points. In this very strange structure called the Fano plane, there are seven points and there are seven lines. And every line contains exactly three points, and it’s got this beautiful structure. And you can draw a little picture of this happening. It looks like a triangle with a circle inside it. And so what I did was to create a kind of literary form, which — I’m just waiting for my Nobel Prize for literature here at this point — so points now become words and lines are sentences. So the rules were, okay, there are only seven words that I can use and I’m only allowed seven sentences, and every sentence has to contain exactly three of these words. So within that, I wrote this little story, about some nonsense thing. But it was a really fun way to challenge myself to be creative. Because if you’ve only got seven words and they’ve each got to appear in sentences. Then some of those words have to do double duty as verbs and as nouns. So I think I had “best” as one of them. “Best” can be a verb. If you best someone, it means you can come out top in a fight. Or it can be an adjective, right? The best show in the world, which is yours, of course. So that gave me a real amusing time. Trying to make a story with seven sentences each with three words, using the same seven words in different orders. That’s my new form of fiction that I have invented.

LEVITT: Now, is it just by chance the name fan and Fano are close? Or was that part of your deeper design?

HART: Well, I wanted to use the Fano plane, that’s the name for it. It’s named after a Italian mathematician. And then of course I had to think, “What do I want to call this amazing new kind of literature?” So I was just pleased to hit onto something that sounded like fan fiction.

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 storytelling.

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 are 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.  

LEVITT: I think there are other great examples in your book that have a different feel to them. They’re applying math to answer questions that kids might intrinsically gravitate towards. And one of my favorites is what math has to say about the life quality of giants. Could you explain what math has to do with that?

HART: So we think of giants — which there are many of in literature and in stories and fairy tales — as huge, powerful figures with immense strength, but actually poor old giants would have a really difficult time of it. And the reason is a little mathematical rule called the square-cube law, which basically says, if you grow something, all its dimensions, so you double the length, the width, the height — everything — then all the lengths double. Fine. The volume of that shape, length times width times height would go up, not by a factor of two, but by two times two times two. ‘Cause all of the dimensions are doubling. So that would be eight. So the volume of this thing is increasing with the cube of the scaling, but any kind of creature moving around, the pressure on its bones, that’s got to do with the cross-sectional area of its bones. ‘Cause pressure is force divided by area, you might remember. So the area of anything that’s doubling in all dimensions, that’s only going to increase with the square of the scaling factor. So if we doubling everything, it’ll be two times two, four. So what happens is your volume and therefore your mass, your weight that’s bearing down on your bones is increasing in the case of doubling by a factor of eight, but the amount you can actually take only increases by a factor of four. So already, if you doubled everything, you are doubling the pressure on your bones. So you know, a giant that’s even only twice our size would have twice as much pressure on their bones to cope with. If you make a giant that’s perhaps ten times our size, then that giant will have 10 times the pressure on his bones and actually bone breaks at about 10 times the pressure we normally put on it. So that giant would essentially not be able to move and as soon as it moved, it would probably break its legs. You know, the giants of fiction are actually, pretty much, fifty-foot weaklings, and it’s really interesting to learn that.

But you can think about going the other way and think about tiny people like the lilliputians or any kind of little fairy-like creatures. Now, those kind of creatures will be proportionately much stronger than us. They’d be able to lift many times their own body weight. They can jump to proportionately great heights. So any of these kind of stories where a little creature is, I don’t know, trapped in a jam jar or something and can’t escape. They can, they could just jump out really easily. They wouldn’t have to worry about falling because once you get to a certain small size, you can fall from any height and not be hurt. And there’s this amazing analogy by the biologist, J. B. S. Haldane, who wrote an essay about animals being big and small. And it’s quite a graphic analogy, but he says if you drop a mouse down a thousand foot mineshaft, it’ll get to the bottom and have, get a bit of a shock, but then stand up and wander away and it’ll be fine. But if you drop a man, of course he’s going to be instantly killed. And this is the gruesome bit — if you drop a horse, he said, it would splash. Because it’s mass grows much faster in proportion to the other animals. And so if you work out things like what’s its terminal velocity, the impact on that bigger creatures’ bones when it lands will totally destroy it. So there’s all these really cool little things you can deduce about how fictional creatures would find life. The giant spiders that are in Harry Potter, for instance, as a mathematician, yes, I concur that there must be magic going on because a spider the size of Aragog would not be able to exist without the presence of magic.

You’re listening to People I (Mostly) Admire with Steve Levitt and his conversation with mathematician Sarah Hart. After this short break, they’ll return to talk about how to bring math alive in the classroom.

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As I mentioned at the beginning of this episode back in January, I had mathematician Steven Strogatz on the show. And since then, we’ve been working together to try to pilot his concept for changing the way we teach math. I want to now get into more specifics about how Sarah’s ideas about math and literature could be brought into the classroom.

LEVITT: I had the mathematician, Steven Strogatz on the show, and he expressed frustration with the way we teach math to high school students in the United States. So we tend to teach them all sorts of techniques for solving very specific problems that they will never ever be asked to solve anywhere but on a math exam. And the consequence is that almost everyone gets discouraged and in the end they conclude that they’re not a math person. So Steven Strogatz, his idea is that we should move towards math appreciation courses like art appreciation — courses with the goal to show kids the wonder and the power of math applied to interesting, real-world problems with less emphasis on rote memorization. And wow, did that conversation strike a nerve. I’ve never gotten such a flood of emails from listeners, hundreds of emails that are still coming in on a daily basis. And the only negative responses are from professional mathematicians who I’d say are split about 50-50. So I’m guessing you’re, though, going to fall into the camp of liking that idea and not disapproving.

HART: I love the idea. I couldn’t love the idea anymore. We do not need everybody to come out of school being able to do like arcane stuff with trigonometry, they’re never going to need it. It’s going to put them off. I’m good at maths and I enjoyed doing mathematical calculations, but even for me there were things that were not super interesting. And we don’t even motivate like why we’re doing it. Did you ever have a lesson in school where they said, “Why are we doing trigonometry?” I don’t think I did. Show people the wonder and the joy and the power of mathematics. I was talking to some school kids a couple years ago and they hadn’t done so well in their mathematics exams, age 15. And in England there’s some compulsory thing you have to do a certain amount of mathematics and if you don’t do so well, you have to retake until you do. So I went to talk to these kids and I wanted to talk about triangles. And I thought, let’s ask, “How do we know the world is round?” And they sort of went, “What?” How do we know the world is round? We could send a probe up into space, and take a photo. How did the ancient Greeks know the world was round and how did they manage, which they did, to measure the circumference of the earth, to work out what that would be, without any of the modern equipment that we have available. And the answer is, there’s a place where at midday, at noon on the summer solstice, nothing casts a shadow. And then you can go to a place a few hundred kilometers away where you do cast a shadow at noon on the summer solstice. And by looking at the length of the shadow — it’s a really, really simple calculation you can do just with similar triangles — that gives you an estimate for this conference of the Earth. And this estimate is like incredibly accurate. It’s only 2 or 3 percent out. You can do that with just one little tiny fact about triangles and then you think, okay, maybe that’s worth knowing. It’s not advanced mathematics, but that power, the power of that simple deduction made by a man, a brain, and a stick, you know? That’s all you need. And you can work out the circumference of the earth with nothing else. That’s the power. These simple things. And once you’ve got people to realize that and find it interesting, then they want to learn more. It’s the same with music. Sing songs with little kids — you don’t sit a kid down and go, “First we’re going to spend two years learning how to write musical scores. Then if you are lucky, we’ll sing a song.” You don’t do that. Why do we do that with mathematics?

LEVITT: So thinking of what students could do in a math appreciation course. I’m thinking back to Bandersnatch and writing one of those branching stories seems to me to be a great way to introduce kids to graph theory. 

HART: Absolutely. You could write branching stories. There are other ways of using graph theory and interconnections between networks. So there’s a fun detective story that incorporates mathematics by looking at when a particular set of suspects said they were in a place and who they saw when they were there. Because then you can draw a little graph, a little map of who said they saw who. And based on a few little facts about graphs, you can then spot who is lying. So that’s a kind of fun investigation you could do. And the other thing, with younger children even ask them to think about fairy tales and what numbers are in fairy tales. Because if you think of the stories of your childhood and the Goldilocks and The Three Bears and the Three Billy Goats Gruff. And you always get three wishes. And there are always three brothers that go on a quest. And there are three sisters. There are so many threes in fairy tales. You could send people on a quest to find numbers in literature and then think about what makes them interesting. And that’s another facet of all of this is, it’s everywhere. Numbers are everywhere, and patterns are everywhere, structures are everywhere.

LEVITT: Have you ever tried to create materials like that for the classroom?

HART: So in my work at Birkbeck College, which is where I’m currently based, I created a course called “Explorations in Mathematics,” for the first-year undergraduate mathematics students. What I want to show the students is, let’s just take our minds into a situation and then play and see what we can see and find out. So we do talk about, one of the sessions is about precisely giants and growing small and what the implications are. And of course there are real world things that come out of that in terms of scale models and how do we know whether the scale model will tell you something about the real machine or device. So we do that as one session, but it’s all about giants, really. And then, another session we talk about navigation and how we know where we are in the world and how our calendar has developed. And why do we have the system of times and we talk about different kinds of geometry in the context of if you are driving, if you’re in a taxi and you’re traveling around a city like Manhattan where it’s all blocks, actually, you don’t want to measure things by the normal distance. You want to think about only being able to travel along roads and think about your distances that way. What are the implications for geometry if you do that? That’s a fascinating path on which to travel. So we introduce all these ideas and we play with them. And we explicitly begin by having absolutely no idea what’s going on. And that is the state of mind of most mathematicians most of the time. So it’s good to get comfortable with that — to have no idea, and then to play and see what you can find out. And the best examples and the best times we have as a group doing that is when, yeah, actually I don’t really know what’s going on either. And we’re all exploring together. And then someone will come up with a great idea and we’ll follow it and we’ll just see where it leads. And you can see them, you know, these students who’ve come in and perhaps they’ve been top of their class, they’ve done lots of mathematics. They can do all the questions on the test, but they maybe don’t really love mathematics. They themselves haven’t seen what it can do. And how to confidently just dive in and play. And that’s what we really want to encourage people to do.  

LEVITT: I think we do two enormous disservices to students, and I see it as people go from an undergraduate degree to a Ph.D. The first one is that we always present things that are right, so in an econ Ph.D. course, you will read 20 papers written by 20 brilliant economists who have been awarded Nobel prizes for what they’ve done. And you get the impression that you could never yourself produce anything like that. But number two, that everything is decided. So when I teach, I try to very much do the opposite. I try to teach half good papers and half papers that have terrible mistakes because I want the students to see that in the top academic journals, it’s loaded with people who thought they were getting it right, but then with the wisdom of time we’ve come to realize they were completely wrong. That’s a much more interesting way to teach, is to understand what was wrong and how we came to see that it was wrong. 

HART: Absolutely. It’s like what we tell our kids about social media. Everyone else is presenting that best photo that they took two hours preparing for, and then you are thinking you yourself are inadequate and ugly because you don’t look like that photo that you’ve seen on social media. It’s the same thing with academic research. One of the things that makes doing a bit of the history of maths really helpful is to see actually these great, amazing people that we revere correctly for being amazing mathematicians, but they made loads of mistakes. Galileo made this mistake here. You know, people who thought that various curves had particular properties and they didn’t, and people who made mistakes proving theorems that were uncovered many years later. And as well thinking about the rules. So in the area of algebra that I’ve done the most work on, it’s called group theory. There are essentially four rules that define whatever a group is. Now, these four rules are presented to you in your first abstract algebra class as, as if handed down by the gods on tablets of stone. Let’s explore it first and see why these are the rules, and then you won’t have to rote learn it and you’ll understand it and maybe you’ll really learn to love the subject.

LEVITT: Your book Once Upon a Prime is so joyful. It’s an interesting book because unlike most nonfiction books, it doesn’t have a goal or an aim. It’s not trying to make you live longer or make you write a better resume. It’s just pure play. It’s very unusual that way. 

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 there was someone writing about it in the paper a week or so back, and 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. And if we’re successful in changing the way math is taught, every student will get to experience that joy. If you like today’s conversation, check out episode 49 of People I (Mostly) Admire, that was Sarah’s first visit to the podcast. The Steven Strogatz episode we talked about at the end, that’s episode 96.

LEVITT: And now it’s time to answer a listener question. And as always, I’m joined by the show’s producer, Morgan. 

LEVEY: Hi, Steve. So after listening to our recent episode with music producer Rick Rubin, a listener named Andre had a question about the scientific method. In the Rick Rubin episode, you discuss the value of listening to the universe when approaching a problem, not going in with a plan, but instead letting the solution evolve. This approach seems to conflict with the scientific method where you first develop a hypothesis, then test and then validate whether it’s true or not. How do you reconcile the difference in these problem solving techniques? 

LEVITT: So, at least the way I approached listening to the universe, I don’t see any conflict at all between that and the scientific method. I wasn’t actually looking to the universe to give me answers to questions. I was more looking to the universe to give me good questions to ask, sometimes to give me possible hypotheses about those questions. And those are really the inputs to the scientific method.

LEVEY: So do you believe wholeheartedly in the scientific method?

LEVITT: So I’m sure it won’t surprise you, Morgan, that I do indeed believe in the power of science and I’m deeply committed to finding causal relationships between different variables. On the other hand, I do have deep reservations with how the scientific principles are currently applied in the field of economics, which is the only field that I’m qualified to comment on. The scientific method dictates that you develop a hypothesis, you propose a test, you collect data, and then you determine whether the hypothesis can be rejected. But I have never known an economist whose research progresses like that most of the time. What we do is we get a hypothesis. We test the hypothesis, and the data almost always reject our hypothesis. So you can’t really publish an economics paper where you say, I had a hypothesis, and it turned out it was wrong in the data. So instead what economists do is they go back and knowing what the answers are in the data, they concoct a hypothesis that is consistent with the data. And then the part that’s completely non-scientific is how we write it up. We pretend that our original hypothesis was the one we made up after we saw the data, and then we come up with all the predictions that hypothesis makes, which just happened to be all the predictions that are proved true by the data. And then we show low and behold, it’s amazing. Our hypothesis was perfectly consistent with the data and therefore cannot be rejected. It’s a complete fiction. The way we write up our papers, it’s completely fictitious when you compare it to how we actually do the research and it’s borderline fraudulent or maybe outright fraudulent that the profession writes up results as if we follow the scientific method when we’re doing something completely different.

LEVEY: So, Steve, then what’s the point of coming up with a hypothesis? I realized that’s a foundational element of the scientific method. Is it to frame the data collection and the experiment? Why can’t the data collection and experiment just be more open-ended?

LEVITT: This is such a great question. And indeed in my own career, that’s what I started doing. So instead of pretending like I knew what was going to happen, I would just go to the data and I would understand what the patterns were. And then, instead of picking one theory that happens to work, what I would try to do was to come up with every plausible theory. What’s the universe of theories that any reasonable person could offer to try to explain what’s going on in the data? And I would catalog for each of those hypotheses what I would expect to see in the data if that hypothesis were true. And then I would go and I would just compare what I saw in the data to what those different theories predicted.

LEVEY: So have you had a lot of success with this? And is anyone else doing this? 

LEVITT: I haven’t had that much success with it, but it does feel good to me at least. And I can say I have managed to convince almost no one else to follow this particular path. I’m roughly the only person who does choose to write up my papers like this. Now, there has been a really great development in the profession, which is called pre-registration. Now there’s a lot of social pressure within economics that if you’re doing a randomized experiment, that before you run the experiment, you publicly record what kind of data you’re going to collect and how you’re going to analyze that data. And it really does make it impossible ex-post for people to do the kind of cheating that I talked about at the beginning of this question where you pretend after you’ve seen the data, what you are going to try to analyze. So I think that has been a really positive movement towards true science in the field of economics. 

LEVEY: That’s great. Andre, thanks so much for your question. If you have a question for us, our email is pima@freakonomics.com. That’s P-I-M-A@freakonomics.com. Steve and I read every question and email that’s sent and we look forward to reading yours.

In two weeks will be back with a brand-new episode featuring Clementine Jacoby, whose mission is to transform criminal justice through the innovative use of data. Thanks for listening and we’ll see you soon.

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People I (Mostly) Admire is part of the Freakonomics Radio Network, which also includes Freakonomics Radio, No Stupid Questions, 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. Lyric Bowditch is our production associate. Our executive team is Neal Carruth, Gabriel Roth, and Stephen Dubner. Our theme music was composed by Luis Guerra. To listen ad-free, subscribe to Stitcher Premium. We can be reached at pima@freakonomics.com, that’s P-I-M-A@freakonomics.com. Thanks for listening.

LEVITT: Can I make a confession? I believe that the only two books that I have read of all the ones you mentioned are the Harry Potter books and Alice in Wonderland.

HART: That is brilliant though because you have got so much happiness ahead of you.