Did you like math in school? Join two physicists and one mathematician talking about how math has an impact on every single part of our life from your favorite numbers or equations to cooking an egg or writing a book. This is a perfect fireside chat to enjoy during your end of the year break accompanied by Hannah Fry, Simon Singh & Kevlin Henney.
What’s your favorite equation?
Kevlin Henney: Let me just start by saying thank you very much. Your talks were absolutely wonderful. Let's pretend this is TV. What's your favorite equation?
Simon Singh: I should say I'm not a mathematician. So my degree was in physics.
Hannah Fry: So you're opting out of the first question?
Simon Singh: No.
Kevlin Henney: I do have a degree in physics, and there are equations in physics. I'm very sure of that.
Simon Singh: We have a mathematician on stage and sometimes, I'm mathematician singular. And then somebody who writes about maths from time to time. And I think Euler's identity is the one, isn't it?
Hannah Fry: Quite riveting.
Simon Singh: It was polled in the Institute of Physics actually for the favorite equation amongst physicists. And that came first. And I think E equals mc squared was second, and actually no, maybe Maxwell's equations were second.
Hannah Fry: The thing is I think Euler's equation is a bit like the pumpkin spice latte of the maths world. It's like the one that everyone says. The one like everyone loves to love and then I love to hate. I think that's not what I'm gonna go for instead.
Simon Singh: What are you going to go for?
Hannah Fry: I'm gonna go for Navier-Stokes equations. So these are the equations that describe how fluids move. That's fluid but it includes gases. Because in a mathematical world there's very little difference, obviously, between fluids and gases, just a question of viscosity. But the thing about these equations is you can write them in two lines on an A4 sheet of paper and they will tell you everything from how a tiny little drop of dew on a leaf will move, to the formation of a galaxy in the sky. Nothing I've ever seen is as articulate as those equations at being able to describe the universe. I just think they're extraordinarily beautiful.
Kevlin Henney: That's a couple of really interesting ones. First of all, Euler's identity is more likely to end up as a piece of graffiti than anything else. In fact, I can actually tell you where.
Hannah Fry: Is that something?
Kevlin Henney: Yes. I live in Bristol and I can tell you there's a about 10 minutes walk from my house, bath buildings, it's graffiti there. I photographed it last year. And people go like, "Wow, you live in a really good area, Kevlin." So that's very respectable. You're quite right about the Navier-Stokes, But if I remember correctly, Clay Mathematical Institute, isn't that one of their seven Millennium Problems?
Hannah Fry: It is, yeah. First of all, if I need to get a spray can and start tagging Navier-Stokes across London.
Kevlin Henney: Yeah, you need to upvote Navier-Stokes.
Hannah Fry: Yes, it is. So this is around the turn of the millennium where the Clay Math Institute's decided to come up with a list of the seven greatest unsolved problems in mathematics. And one of them, as you rightly say, is the Navier-Stokes equations. Because we have these equations, you can write them down, and you can solve them in special cases. So throw away a dimension or two and suddenly they become much easier. And in pretty much every situation, you can use a computer to crunch through and work out what numbers work inside the equations. But nobody has ever proved that you can always solve it. If you can prove that there is the existence and smoothness of a solution to these equations in every domain, in every regime, then you win a million quid. It's quite hard though, I should say. It's quite hard.
Kevlin Henney: Yeah, let's just put that into perspective. Let's see. They were set in 2004?
Hannah Fry: Yeah.
Kevlin Henney: That's seven challenges. One has been solved by Grigori Perelman, who famously refused the reward money. And you actually feature him in your book and he's the ultimate eccentric in this case and refused the Fields Medal as well.
Hannah Fry: He solved the Poincaré conjecture and he wasn't attached to a university. He was a bit of a rogue mathematician. I'm sure Simon probably knows this story better than I do. But this proof emerged on the scene and everybody in the world was like, "Where did this come from?" And then over time agreed that he had indeed solved this really important problem in mathematics. But the award ceremony, I can't remember whether it was for the Fields Medal or the million-pound prize, was in Barcelona. And he reasoned that to collect the award, to collect all the money, he would have to spend a day traveling from where he lived in Moscow to Barcelona, a day collecting the prize and then a day returning home. And that was three days when he wouldn't be doing mathematics so it wasn't worth it.
Kevlin Henney: I mean, how hardcore is that? But there's also another thing there, which I find kind of interesting. I came across Navier-Stokes once upon a physics degree. And in physics you can calculate. You just knock off those extra dimensions. You take those special cases and leave the rest for the mathematicians. In physics, we can compute with it. And then I went to computing and, honestly, we just throw stuff at it. And it sits on the wall until somebody complains about it, and you say, "Oh, we just tweak the UX." And that's it. Sorry, everybody who's a programmer out there, I've just revealed your secrets.
So there's a really interesting point, because Simon in your presentation earlier. It's not so much a burden of proof but the fact that a proof is everlasting in that sense. There is a timeless sense. When somebody said, "This theorem, it's done, that's it, attach a name to it, we're good." There's no recourse to negate here.
Simon Singh: It’s the beautiful architecture of mathematics where people build their ideas. You do have this situation where people have a conjecture, and they're pretty confident that it's true. So you build on top of it. And you just hope and pray that that conjecture will one day be locked into place with its own proof. But if it's not, then this whole other area of math just tumbles down.
Simon Singh: Yeah. So we didn't get on to the proof. But Sir Wiles was plugging away at this proof from time to time. I should say by the time he became an undergraduate, he became a Ph.D. student, he became a lecturer, and then a professor. He worked on other things, you know. He had to publish papers, he had to do some kind of serious mainstream work. You couldn't build a career out of trying to prove something that was considered impossible.
But then there was this link to the Taniyama-Shimura conjecture that was developed by Yutaka Taniyama and Goro Shimura in Japan, two young maverick kids really who believed that they could build a bridge between two very different areas of mathematics. The way I explain it, it's as if I have electricity here and magnetism here, and you build this theory of dynamics to electromagnetism to bridge these two subjects and show that they're really reflections of each other. And they tried to do the same in mathematics, but it was a conjecture, they couldn't prove it.
If Fermat's Last Theorem was false then there is an equation where x to the n plus y to the n is z. There are numbers that fit that equation. There's an X, a Y, a Z, and there's an N, and it all fits beautifully.
Now, if there is a counter-example, let's take those four numbers and let's plug them into the Taniyama-Shimura conjecture. Let's make an elliptic curve, which is what Taniyama-Shimura is all about. If you make this elliptic curve, it falsifies Taniyama-Shimura and Taniyama-Shimura then falls apart. Conversely, if you can prove that Taniyama-Shimura is correct if it's largely true, then these numbers can't exist in Fermat's Last Theorem, and if they can't exist in Fermat's Last Theorem, then Fermat's Last Theorem is true.
Suddenly there was this link between a 350-year-old problem that people had sort of given up hope on and something relatively modern and fundamentally important to the heart of mathematics. That's when Wiles returned to it because he thought, "Well, look, if I dedicate myself to try to prove Taniyama-Shimura, I'm doing mainstream mathematics. I'm doing something that's worthwhile. And even though I can't prove it, I might develop interesting ideas, interesting insights, interesting tools." And he didn't tell anybody what he was doing because proving Fermat's Last Theorem was crazy, trying to prove Taniyama-Shimura was pretty much impossible as well. People had given up on that.
He really kind of locked himself away. He didn't want people to think he was crazy. He didn't want people to badger him with questions all the time, "How's it going? How's it going?" He didn't want other people to maybe join the chase. But the clever thing he did from an academic point of view was that just prior to embarking on this attempted proof of Taniyama-Shimura, he had a stack of papers he hadn't quite published. And they were waiting to be sent out to different journals. Every year or two he'd send a different paper out to journals. So everybody thought that's what he's working on. But in fact, that was old work he'd done a while ago, now he was working full steam on Taniyama-Shimura in order to get to Fermat's Last Theorem.
Kevlin Henney: So that's career advice, just in case anybody is thinking about doing that. I'm going to just reframe it into a software thing I told a client one day. They'd spent two, three years improving the performance of their application by 25% and then I stumbled across a 200% improvement and I said, "Do not release this all at once."
The Taniyama-Shimura’s approach to problems and failures
Hannah Fry: Simon, can you tell the story about Taniyama-Shimura's approach to problems and failure? Because this is, honestly, the most beautiful thing.
Simon Singh: Yeah. So Taniyama, sadly, took his own life when he was fairly young. I think he was in his late 20s. He was a very young child. But nobody understands why this happened. He was in love, his career in mathematics was going very well. And then he took his own life. His best friend, Goro Shimura, was completely baffled by this. I interviewed him many years later. And he was telling me about Taniyama as a friend, but also Taniyama as a mathematician. He said, "The great thing about Taniyama was that when he was doing mathematics, he would make mistakes. But the mistakes would somehow illuminate new areas of maths and new insights that will lead to new breakthroughs in their work." And he said he tried to copy Taniyama in making mistakes like this. And he said, "It's really hard to make good mistakes." It's a real skill.
Kevlin Henney: I think there's a beautiful balance between the two of them. One is perhaps more methodical in his approach, and this is counter-balanced by somebody who really is very, very different. And you've got to learn to make good mistakes. That turns out to be quite an art. But there's also this fascinating idea of an immense amount of creativity. I think this is one of the things that are often underestimated in maths, and perhaps the route that people take through education is the amount of creativity that's involved in reaching certain conclusions. Often there's a twist, there's a surprise or something that is potentially either humorous or the light bulbs go on.
Obviously, these are eco-friendly lightbulbs these days. But they go on and suddenly, there's this thing. But some of it comes about through this idea of play or very, very lateral thinking, or looking at something very unusual. And that’s the thing I always find fascinating about mathematicians, there's this really hard problem over here, here's another one, and they say “But I'll do this one. This looks more interesting”. Now, you know, that's the sort of the classic distraction technique and yet it turns out to be the basis of a whole discipline.
Hannah Fry: I think that's right. Although I also think that once you're in that hard problem, then you say, "Okay, let's throw away absolutely everything and imagine what happens in the case where n equals 1," right? Simplify it so that it's ridiculous. For instance, during my Ph.D., which was on the Navier-Stokes equation. So it was all about how a droplet moved along a plane. And after four years, I'll tell you how far we got, I managed to get a two-dimensional model that didn't include gravity or surface tension.
Kevlin Henney: To be continued.
Hannah Fry: Exactly. Exactly.
Kevlin Henney: I think there's this approach of abstraction, "Let's eliminate that. Let's eliminate that." You were talking about the spherical bastard thing, and in physics, there's a very common thing of, "Well, let's just treat this as a sphere." It’s actually a real statement from a paper. Let's just assume the cows are spherical and therefore there are all kinds of conclusions that you can reach there. Spheres are beloved by mathematicians and it turns out physicists as well.
I think they have this kind of obsession with things. And they also have a mild obsession with donuts. And it turns out that one is not the same as the other. Except if you buy donuts from certain stores, they are sort of equivalent to the other kind of donuts. So there's this is kind of really interesting thing about this abstraction, this whole idea about you've got a whole branch of mathematics devoted to the idea that this is, in fact, a sphere. [shows a bottle] No, actually, it's a plane. Oh, yeah, I can make it a sphere.
Hannah Fry: You can make it.
Kevlin Henney: Yeah, you have to make a sphere. That's good.
Hannah Fry: Or a plate.
Kevlin Henney: Yes, a plate. In fact, I can turn it into working one of these. [shows a notebook] There are lots of possibilities.
Simon Singh: But I think humans are donuts, aren’t we?
Hannah Fry: It depends. I spoke about this with a biologist, in fact, Adam Rutherford, my co-conspirator as you've described him. And almost all biological life is based on the idea of a tube. But if you get really technical about humans, then you also have a hole in your eyes, your tear duct. So I think that we worked it out that we're seven-hole donuts.
Kevlin Henney: Okay.
Hannah Fry: Does anyone have any idea what we're talking about? It seems like we've just jumped into a different dimension.
Kevlin Henney: Should we rewind here? Never ask a topologist to help you lay out the house. Okay? You leave that to the geometrist. Geometrists know where everything is and they can say various things whereas topologists abstract this idea. It's known as a rubber sheet.
Simon Singh: Topologists are the kind of people that say, "If I wear my pants inside out, they contain the whole universe." That kind of thing.
Kevlin Henney: So we're learning some things here. So we got this whole idea that if you can reshape without cutting. Without cutting, that's the whole thing.
Hannah Fry: Or punching any holes in it.
Kevlin Henney: Or punching any holes. Then I can show that one form is equivalent to another. As long as I can reshape and stretch, everything's really good. And we can flatten this one down without doing anything into a plate but not until I've finished. [shows his bottle again] And it turns out that there are a number of things that are donuts, and we mean the disappointingly not jam-filled ones. A Taurus, that is a cup. Therefore, never ask a topologist for a cup of tea, somebody's going to get wet and hot. And it also turns out that that is topologically equivalent to an "Asteroids" screen. Have you ever played "Asteroids"? If not, what have you done with your life? We've had the whole retro gaming phase at least two or three times since the '80s. And so there's this idea that these form a group. So as long as you can stretch it. So that's one hole. But you're saying humans, we're seven-hole donuts?
Hannah Fry: Yeah. I spent quite a lot of time thinking about this at one point. And now I can't remember any of it. So certainly, tear ducts. There's like one on each side. And ears don't work. If you imagine you've got a bit of plasticine and then you poke a hole in it, that's like an ear. And then you can remold it, so it's out. I think that maybe because your nose connects up to the same tube, maybe that's one. And then I don't know, maybe other people can work that out. I mean, this is the very limit of what I know about biology.
Kevlin Henney: I think we're learning here some important things about how math's really done.
Outside your comfort zone
Simon Singh: So you're a proper mathematician and you sometimes write about maths. And I'm not a proper mathematician. I'm not a proper anything. So I'm always outside my comfort zone writing about cosmology or cryptography and so on. I really enjoy that process of learning about something that's completely new, whereas half of your work is within your comfort zone, I feel, and half of it is out. How does that play? And which do you prefer?
Hannah Fry: Although I think that the key thing is teaming up with somebody like Adam, who is at the opposite end of the spectrum to me because then I've got maths and physics pretty much covered. I mean, I'm not an expert, I'm not a physicist, but there's very little in physics that I can't read and understand. So maths, physics, and computer science, I'm down that end. He's got biology and genetics. And we meet in the middle where neither of us knows anything about chemistry.
Kevlin Henney: I was gonna say I'm sensitive as something being missed out here. And it's just like, "Oh, it's poor old chemistry." So apologies to the chemists out there. So you're an improper mathematician. I'm going to use it because you just referred to Hannah as a proper mathematician. And I like to make this kind of complete set.
Simon Singh: I think there's a difference between people like Sir Martin Rees, who is an astrophysicist, and who writes brilliant books about astrophysics. Or Steve Jones, who writes brilliant books. And yourself and Adam, who also write your own books about those specialized topics. It's fun for me just to go out and explore and find different things and almost do a degree because it's been three years just immersed in a topic. But I've also co-authored a book once with Professor Edzard who was a professor of complementary medicine. And it wasn't on my list of books I'd written. I don't know why I'd omitted it. But he was very much the academic. And his position was, that he'd done all this research on alternative therapies and what doesn't work and what does work. So he was very much the font of all knowledge. But there was this nice relationship where I could challenge him and say, "Well, look, I don't understand that. Or why is that true? Or check this out? Or what does that study mean?" And he could interpret all the papers and I could challenge him and try to then interpret them. I enjoyed being a co-author. I thought it went well.
Writing a book in collaboration with academia
Hannah Fry: Did you clash, though, on what was interesting? I don't think I've ever told you this, Simon, but I read that book when I was like 15. I can't imagine the thousands of people who told you this. But Fermat's Last Theorem really was the book for me, that made me decide I want to be a mathematician. And the thing about it is that you've got a real knack for finding the stories that are incredibly intriguing without getting bogged down in the detail. And I wonder when you worked with an academic who cares totally about the detail, did you clash on what to include?
Simon Singh: First of all, thank you very much. Always love hearing that. And you must know that as well. You know, he said, "Look you're the author, you do the writing." It's a very different book from my other books. In my other books, I would say, "Cosmology, I can pick the best stories in cosmology and put them together somehow and some of the more boring stuff or irrelevant stuff." It's not an authoritative history of cosmology, it's my own version of the stuff I find most interesting.
And the same with cryptography. There were some areas of cryptography that I didn't write about because I just didn't think they were that interesting. But I could pick my own pathway through. And Fermat is terrific because all the stories are fantastic and they all fit together so beautifully. And Fermat touched on so many different areas of maths that you can talk about whatever you want.
But with alternative medicine, the challenge was, this is a book that will tell you what works and what doesn't work. So how do you take that topic, which could be quite dry, it could just be a list of here's therapy A, here's condition B, what's the evidence? So the challenge was to spice that up somehow. And there are stories that spice that up. Sometimes they're tragic stories. Sometimes they're stories from history. So that was the challenge. But we didn't really clash. I think the clash was my naivety as a physicist where if you accumulate enough data, you'll know whether it's true or not. Working with a doctor who says, "Well, you know what, it's a bit fuzzier in medicine because every human is different and every dose is different and every condition is different, and so on." It took me a long time to understand that there weren't going to be clear-cut conclusions every time.
Kevlin Henney: So that's an interesting point about collaboration and pulling in different directions. Hannah, you were highlighting your work with Adam and disciplinary-wise, you're in different spaces. It's easy to hear the voices when we listen to the radio show. But reading through the book, you can kind of sense that every now and then there's like a Hannah bit, and then there's an Adam bit.
Hannah Fry: Oh, do you think you can tell? Sometimes we actually write them. Yeah.
Kevlin Henney: The bits you say, but there are actually a couple of bits where I can tell.
Hannah Fry: Oh, really?
Kevlin Henney: Yes.
Hannah Fry: Interesting. And where did you come across them?
Kevlin Henney: I co-authored a couple of books, and it is kind of interesting, every now and then I can go back and say, "I know exactly who wrote that. This chapter is actually mine, the other guy just changed the punctuation and I changed it back." Or, in fact, I did work with an academic and he put lots of references in. I'm industry, so I took half the references out. He would query references. I had to say at one point, "Actually, that was a conversation with my wife I'm referring to. That was at the kitchen table, but I don't think that qualifies."
Even when you smooth it out, there are some differences. But there are bits where also there's an interplay. We know about the music taste differences, that comes through quite a lot. And some of the film differences as well. Those are made explicit. But there's also some very nice counterbalance there as well. And you're stuck together writing a series. So clearly, something works there. But how do you find that that affects your thinking or your outlook or your approach? Are you saying, "Right, I'm going to come to the table with this, and Adams is going to bring that, or he’ll add to it or turn it around, or offer something I wasn't aware of?” Would you simply agree because this is actually the whole story?
Hannah Fry: That is a really good question. I think that Adam and I are not just opposite in terms of where we are on the spectrum of science. I think we are opposites in almost absolutely everything. I was been thinking about your Taniyama-Shimura story earlier. I think I am just very highly strung, very meticulous. Always do my homework. Really, I mean, I prepare, prepare, prepare. And Adam waltzes in at the last moment and is just amazing. And it's very annoying to me. And that's what he's like.
The real genesis of the book was actually not only because both of us know different things but we're very, very good friends, and we often just send each other messages when we come across stories that we think will make the other one laugh or think the other one will find interesting. Not for broadcast, not for publication, nothing, but just we had this whole bank of stories, and Adam was like, "Well, let's just put them in a book." But then I was like, "No, this book needs to be ambitious. I want to talk about free will. I want to talk about infinity. I want to talk about four-dimensional spheres." So it was me pulling it in that direction and then him adding in all of that joy.
Kevlin Henney: Yeah. Yeah. You did get all of those topics covered. It did not read like a series of text messages. It's “I want to write a book about everything.”
Hannah Fry: It wouldn't have taken three years when we would have just publish that.
Kevlin Henney: We'll just copy and paste this all in. I mean, that's, honestly, how software development's done. We use different systems.
Hannah Fry: Slack.
Kevlin Henney: If it's on Slack, then it's real. If I pushed it to GIT. So this whole idea was a shaping process. There's a kind of a balance there in that sense. As you said it reminds me of the collaboration of the Taniyama-Shimura. Is there this idea of, "Okay, let's do something and create a shape there?"
Simon Singh: The thing about Andrew Wiles's story is it is, he is one mathematician working in solitary confinement for seven years.
Kevlin Henney: He's the outlier, isn't he?
Simon Singh: People get the impression that there's a richer context for mathematics in a lot of it. So I remember going to the Newton Institute in Cambridge, it must be 25 years old now, and there was a building that was dedicated to bringing mathematicians from around the world to brainstorm subjects in six-month sections, and then six months later you have a different topic and a different group of mathematicians coming in. It was specially designed without corridors. Because they didn't want mathematicians to hide away anywhere. Everything was designed so that mathematicians will be continually bumping into each other and collaborating and chatting and so on. The majority of mathematics is about the way the majority of society works, and the way the majority of breakthroughs take place.
Hannah Fry: They've got chalkboards in the ladies loos at the Newton Institute.
Kevlin Henney: Okay, that's proper.
Simon Singh: And the gents' loos.
Kevlin Henney: There is this interesting idea that you've got this kind of spectrum, this wave in terms of degrees of collaboration, and Andrew Wiles is an outlier in the modern mathematics sense.
Simon Singh: Do you know about Polymath?
Hannah Fry: Go on. What do you mean?
Simon Singh: I think Polymath is a blog by Tim Gowers.
Hannah Fry: Oh, yes, I do know what you mean.
Simon Singh: Tim Gowers is one of the Fields medalists. So just for people who don't know, there's no Nobel Prize in maths, but there is the Fields Medal that gets given every four years. I think you have to be under 40. And Tim Gowers has one of these Fields Medals. He’s one of the great British mathematicians. He created this blog where he would post a problem and people would add their contribution in the comment section. They've solved a few problems and they publish the papers. And everybody who's added a comment gets to be a co-author on the paper, I think. So that's in the very other extreme end of global collaboration.
Kevlin Henney: Otherwise, we see that a lot in experimental science. CERN is the classic. There's just like thousands authors type thing.
Simon Singh: And even again, Andrew Wiles, we should finish this story. He worked seven years on this problem and eventually felt he had complete proof. He then went back to Cambridge, the town where he grew up, and gave a lecture, I think a series of three lectures, gradually revealing this proof. Never ever saying where this was going to lead to. Although I think by the third lecture, people were beginning to sense where this was going to go. It was going to be a sufficient proof of Taniyama-Shimura to lead to Fermat's Last Theorem. The lecture theater was packed at the Newton Institute.
He revealed the final line and said, "I think I'll stop there." And a huge round of applause broke out and cheering. He was in newspapers around the world the next day, you know, "Mathematician solves 350-year-old riddle." It was a math story of the century. And then it had to be published, it had to be refereed. This was a 200-page proof in seven chapters. It was sent off to seven pairs of referees. And in one chapter on one page in one paragraph, there was a problem, and it couldn't easily be resolved. There were lots of problems and they could generally be fixed quite easily, but not this one problem. Andrew Wiles tried to fix it quickly before the news got out because this was the story of the century and it was turning into the embarrassment of the century.
There's this lovely line by the Dutch mathematician Piet Hein, I think where he says, "A problem worthy of attack, proves its worth by fighting back." And that's what Fermat was doing, he was fighting back one last time. Eventually, it became clear there was a problem because no paper had been published, and every other mathematician was then trying to fill the gap because if they could fill the gap, they would have proven Fermat's Last Theorem. Andrew Wiles invited one of his former Ph.D. students, Richard Taylor, to come and join him in Princeton, and they collaborated on this. And again, having people work on ideas together, bouncing ideas off each other, you know, it leads to richer results.
Kevlin Henney: Actually, that resulted in two papers. One that was joined between the two of them, and then Wiles's final proof.
Simon Singh: Exactly. So, I wanted to correct the idea that even Wiles's story, the most extreme story of solo genius effort, did have a collaboration in there as well.
Science and humor
Kevlin Henney: Even as an outlier story other people were involved. And I think that's quite an interesting one. There's another element here. I'm gonna pick on your last book, which was "The Simpsons and Their Mathematical Secrets." The revelation that actually the Simpsons scriptwriters are most math nerds and they're collaborating in a way that is very similar.
Hannah Fry: I wouldn't say most of them are mathematicians, I would say that if you look at the different populations of writers, the biggest population are mathematicians. And these are people who have Ph.D.’s in maths. Jeff Westbrook was a professor at Yale. J. Stuart Burns was doing a Ph.D. and abandoned it in order to join the Simpsons' writing team. Ken Keeler has a Ph.D. in applied math from Harvard. Lots and lots of these are serious heavyweight mathematicians. And I got to meet them, to spend a week with them, going to some of their script meetings and readings of the scripts, and some of the editing suites and got to really spend a lot of time talking to them.
And I asked them all "Why are there so many mathematicians here?" They gave several reasons, and one of them was, "Writing comedies is intellectually really difficult, and having a background in mathematics means that you know how to tackle difficult problems and you have the intellectual stamina to get through a comedy script." And an intuition as well. Again, you mentioned the idea of creativity in math. You have to have that intuition of knowing where the answer might be, and then the stamina to pursue that. And similarly, in comedy, you need to know where the funny is, where might that joke be? But also, they all said, "It's a collaborative thing. We worked on math problems together when we were at university when we were at college, and we work on comedy scripts together. You know, somebody comes up with an idea, and you add to it, and you build to it." A lot of them felt it was the same as doing maths together.
Kevlin Henney: A lot of what you produce individually but also in collaboration. I've seen videos with Matt Parker, and certainly, everything you've done with Adam Rutherford, there's a great deal of humor. I don't think you go out doing standup?
Hannah Fry: No.
Kevlin Henney: But there is an immense amount of humor. It's incredibly engaging. I think a lot of people, are drawn to that. It's just like, "Hey, guess what? Science, math, it's all fun. This is enjoyable. You're allowed to do this. You're allowed to not be po-faced and all the rest of it." Now, is that something that you notice in your work style, in other mathematicians? Is that something that is general? Or are we seeing a landscape, which is much drier than the one that we've been offered?
Hannah Fry: I don't know really. I think it's quite hard not to find some of this stuff quite funny. Like, for example, Adam and I were thinking very deeply about the scientific method and what it means to accept failure and to recognize that you're wrong. And I was like, "Ah, I remember there's a brilliant story about this that we can put in the book." And it was Christopher Columbus, who was famously not a nice man, but sort of built his whole legacy on the world being round, right? He was also, by the way, absolutely terrible at sums. Really bad at it.
So on one of his voyages over to the Americas, he got his maths wrong and miscalculated where the North Star was in the sky. And he was jumping around all over the place. He just couldn't handle an equation. And so he looked at his charts and he was like, "Well, this doesn't make any sense." So he decided to come to the only possible conclusion that he thought was rational at the time, which was that there was no way the world could be round. There was no way his sums could be wrong, there was therefore no way the world could be round. So he wrote a letter to the king of Spain and he says, "I have reached the conclusion, my entire life's work, everything I built everything on, it was all wrong, and the world is not round after all, it is shaped like a pear. And at the end of the pear where the stock would be, is something akin to a woman's nipple." And my favorite bit, "Sailing up the nipple means you're sailing closer to heaven." Now, here's the thing about that, how can you not find that funny? I think it's impossible to be po-faced about that.
Kevlin Henney: So this is centuries before Freud and yet... And this is one of those things where sometimes there's science or the activities of misapplications of science and math in the world. We learn more about the person who's making the mistake than we do about the mistake. I mean, come on, Columbus, literally pear-shaped, and also that nipple thing going on there. Are there some issues?
Hannah Fry: You got flat earthers, now you got nipple earthers.
Kevlin Henney: I believe that's underrepresented in the modern kind of online conspiracy theory domain.
Hannah Fry: I know. To answer your question in seriousness, though. Hardy, for example, the book that you described earlier, “A Mathematician's Apology” is a very eloquent, beautiful description of what it is like to be a mathematician. And I think that there are tons and tons of people who do that and do it incredibly well. But for me, the main reason why I like doing this, in part, is because I think it's really important. I think, actually, our entire society benefits from better education in science and mathematics. But mostly, it's because I just find stories like that nipple story, they're so ridiculous. And it's like having a secret that you want to share at the pub. That is genuinely how I feel about it. And it's like, "I just want to go and tell people because it's so crazy." I think if that comes across as irreverent or, I don't know, tongue-in-cheek then I think it's just because that's the way that I see them.
Kevlin Henney: I think that's great. This is one of the human connection things. Here's this huge mountain of intellectual stuff, and isn't that funny?
Hannah Fry: Yeah, of course.
Kevlin Henney: And that's a bit of a laugh. “Oh my goodness, when I had this idea”! There's a sort of joy there.
Hannah Fry: Scientists and mathematicians aren't these grand figures. They do stupid stuff, they make stupid mistakes, they're just totally normal people who happen to really like math and science. And I think pretending otherwise, actually, isn't that helpful.
What’s your favorite number?
Kevlin Henney: Yeah, that's very true. And somehow I need to segway from that into the next question. Let's start nice and easy. This is gonna warm you up. What is your favorite number?
Simon Singh: I'll just check it doesn't relate to my PIN codes or anything.
Kevlin Henney: Ah, yes, and the name of your first dog.
Hannah Fry: My maiden name.
Simon Singh: It used to be 26. I'm not a mathematician and I used to think that math was something you learn about in order to do physics. It's quite useful, sometimes it's interesting, but physics is the real thing. Okay, and I still love physics. But then I made a documentary about Fermat's last theorem, and I started writing a book about it, and I remember learning that 26 is the only positive whole number that's sandwiched between a square number and a cube number, between 25 and 27. It's kind of remarkable because in the infinity of numbers, that's the only time that happens, and it's kinda remarkable because we can prove it. And the proof is not so difficult. It was one of the first proofs I went through that wasn't trivial but wasn't so hard that I couldn't follow it. And so, that was a really interesting moment for me. It was a moment in which I realized that this is gonna be an exciting book for me to write because I'm gonna learn some cool stuff along the way.
Kevlin Henney: That's a good case made for 26.
Hannah Fry: Okay. I'm gonna tell you what my number is, but I can't remember the actual number, so bear with me.
Kevlin Henney: Now, I do remember you've spoken at computing conferences and you do code. So, that is a classic coding response.
Hannah Fry: I just need to look it up on Google. I know what the answer is, I just need to look it up on Google.
I'll tell you what it is. The thing is that there are pure mathematicians and number theorists who really love numbers. I fully respect all of that, but it's just not my bag. I don't have a relationship with numbers in the same way that other people do. I much prefer human stories, and for that reason, the number that I like the most is Ramanujan's number. So, you see why I can't remember the actual numbers.
Kevlin Henney: 1729.
Hannah Fry: One thousand seven hundred and twenty-nine. Basically, there was this mathematician called Ramanujan. He was the character on which the film "Good Will Hunting" was based. He was born in India, in a very small village, self-taught, unbelievable in terms of his intellect. I think it's fair to say there have been only been a handful of people who have ever existed who were in his league. He wrote letters to Hardy while Hardy was at Cambridge. There's one story about Hardy seeing these letters and thinking Ramanujan made up all of his own notation. He didn't know the exact way to write sine and cos and all of that, so he made up his own notation. Ramanujan wrote these letters to Hardy saying, "I've come up with this mathematical thing," and Hardy looked at it and thought, "Well, this is an absolute load of gibberish," and put it in a drawer and forgot about it. And then a few months later, Hardy was in a tutorial or a lecture at Cambridge, and somebody was writing something on the blackboard, and he was like, "Hang on a second. I've seen this idea before." And went back through and realized that actually, Ramanujan was doing this work on his own. I mean, it's incredible.
Ramanujan, very tragically, was brought over to the UK, worked in Cambridge, and he just didn't suit the weather at all and got very sick and very tragically died very young. All of the viruses that were in the UK, are not the same as those he'd grown up with. While he was laying in bed, in hospital, I think it was Hardy that went to go and see him. Hardy, who was a very British gentleman, didn't really do that well with emotion, sat by his bedside, and didn't really know what to say, this desperate tragedy on his mind that was about to pass. And so, he tried to make conversation, "The taxicab that I got here had a very uninteresting number. It was 1729." And Ramanujan, who was literally on his deathbed said, "What do you mean it's not an interesting number? It's the first number that cannot be written as the sum of two cubes," or something.
Kevlin Henney: That can be.
Hannah Fry: That can be. The smallest number that cannot be written as the sum of two cubes.
Simon Singh: In two different ways.
Hannah Fry: Right, I told you I don't have that relationship with numbers. The human story I've got. I think, it underlies just how extraordinary he was, and just how much Hardy, one of the greatest mathematicians of all time, revered him. And that story is the thing that he told after Ramanujan’s death.
Kevlin Henney: And those are known as taxicab numbers as a result. What's interesting is you're talking about the relationship with numbers, and it was John Littlewood that said that every one of the positive integers is one of Ramanujan's personal friends. He'd be able to tell you everything about it. It's a deep intuition that is incredibly rare. That was a good number.
Hannah Fry: I should probably remember it for next time somebody asks me a question like that.
Kevlin Henney: I'll just throw in my favorite number here, 37 because it's color-coded. I was talking to Hannah about this earlier. I have synesthesia, 37 is really good because it's got a strong yellow with a blue. We can have that conversation later. And 1729 actually also has a very specific color. So, that's why I remember it. It's not because I'm a math whiz.
Simon Singh: It has a specific color for you or for synesthesia in general?
Kevlin Henney: I think probably only for me. My youngest son has color grapheme synesthesia. We do not agree on colors.
Simon Singh: Right.
Kevlin Henney: He's just like, "Dad, that's wrong." We don't agree on the colors of seasons and words, and the color of his name, stuff like that. Anyway, that's a way to talk about the fact that you can have a very personal experience that's unshareable. But speaking of sharing, on the topic of randomness, did Hannah's bucket of numbers only contain twos? I'm not gonna spill the beans on this.
Hannah Fry: I cannot believe that you guys don't believe in probability. You need to update your priors, that's all I'm saying.
Kevlin Henney: Speaking priors, within five minutes' walk of here is the cemetery where Bayes, as in Bayes' Theorem is buried. This was shown to me by somebody at a conference a couple of years back. We were walking through the cemetery and she was saying, "Yeah, yeah. It's somewhere around here." We couldn't quite establish. She said eventually, "It's this one, probably." That seemed to make the day.
What’s your favorite way to cook eggs?
Kevlin Henney: Let's go for something a little bit lighter here. What's your favorite way to cook eggs? This is a human-interest story, we've just been talking human interest. You don't have to be a mathematician.
Simon Singh: It's a microwave.
Hannah Fry: Microwave? Simon Singh, I respected you.
Kevlin Henney: All that past tense is so stinging. You drove me into maths and now you're exploding eggs in microwaves? That's a good physics answer, though. That's good.
Hannah Fry: Obviously poached is the right answer.
Kevlin Henney: Okay, poached. Okay. Well, that's not a full answer. Is it runny or is it boiled?
Hannah Fry: I mean, anything that's not in a microwave, to be honest.
Is maths invented or is it part of nature?
Kevlin Henney: The not microwaved egg, this is non-trivial. Is maths invented or is it part of nature? We've never heard that one before. And I don't know if we've had enough drink for that, but short answers.
Hannah Fry: I'll give it a go. I mean, I did a three-part series on this.
Kevlin Henney: You did, yes. That's how I know the answer is not short.
Hannah Fry: If you want to properly dig into it, you can go look at that. But I think actually most mathematicians agree that there are certain aspects of mathematics, which is constructed and created. Like, what a plus sign looks like, or the particular type of notation that you use for something. But the thing is that once you go off and you start to explore the relationship between things like numbers, take 26 for example, or even deeper more topological ideas, there is no doubt in your mind when you're doing this that someone has been there before you. And I don't mean someone in a sort of mystical sense. There is no doubt in your mind that when you're exploring these ideas that you are voyaging on a landscape that existed before humans got there. And there is no mathematician that you'll speak to who will say anything else. And so, I think that it's really a combination but mostly discovered.
Kevlin Henney: The way you're describing it is exploring and revealing. You're revealing what is there but we've not accessed it yet.
Hannah Fry: I think it's also certainly true that if you had some alien species who were using different notation, who were using different ways to represent relationships between things, it might be that some of the stuff that we find really hard, like the Riemann hypothesis or whatever, actually they'd find that really easy. And things that we find really easy, like, quadratics, would be incredibly difficult for them. I think that you're driven to discover a certain landscape by the notation that you've chosen. I don't think anybody who's experienced exploring high-level mathematics would say anything other than that you are really discovering.
Kevlin Henney: So, I think that Richard Feynman actually had a really nice observation on notation. He said, "do not laugh at notations; invent them, they are powerful”. It changes the way you look at a problem, and therefore the journey that you might take.
Simon Singh: We did just have an answer from a very clever lady who's made a three-part series on this. You don't want to ask a guy who cooks his eggs in a microwave, it's not gonna end well.
What's your favorite mathematical object?
Kevlin Henney: Actually, you're right, Simon. I mean, obviously, you know, this is a character defense or character assassination actually. The eggs defense, yeah. And if you want more details, I believe there's a three-part series that covers this. Now, I don't know if that's one for the category theorists, but what's your favorite mathematical object?
Simon Singh: Gosh, I'm not sure.
Hannah Fry: I don't really have one. Honestly, people must rank stuff way more than I do. They keep, like, a running list.
Simon Singh: There's something called the Collatz Conjecture.
Hannah Fry: Yeah.
Simon Singh: I don't know if people have come across the Collatz Conjecture, but there's this idea that if you take any number, let's say they take five, it's odd, so I'm gonna triple it and add one, three times five, 15, add one, 16. It's, even so, I half it and I get to eight, half it I get to four, half it I get to two, half it I get to one. Okay? Let's pick another number, like 10. Oh, well that goes to five straightaway. Okay, the idea is any number you pick, you follow this path: if it's odd, triple it and add one, if it's even, half it. If you keep doing that, you always seem to get back to one. It's sometimes called the Hailstone Conjecture because I think hailstones get lofted up into the air by thermals and then they grow and fall down, and they get lofted up again. And that's what sometimes happens with these numbers. They loop up and then they fall down. They always come back down to one, but nobody's ever proven it. I've got two sons of 11 and 6, and it's one of the earliest ideas that you can teach to kids. I say, "Look, here's a question. You can do the math, you can do multiplication by three. You can half things, and you can test it out, but can you ever prove it?" I have very fond memories of that because it was a nice way of introducing my older son to math and him understanding that there are some really interesting questions out there. And even with simple, basic operations, you can explore and nobody has gotten near proving the Collatz Conjecture.
Hannah Fry: There's a really nice video on the Collatz Conjecture, actually, by Veritasium that came out about a month ago. It's lovely.
Kevlin Henney: Okay. But you're going with the whole fact that, people clearly have ordered lists.
Hannah Fry: I probably should, I mean, there are loads of good ones.
Is communicating science harder than doing it?
Kevlin Henney: I'm gonna tell you my technique. My wife always used to say, "What's your top five this? What's your top five that?" This used to be the thing. I used to hedge horribly if she asked me for top five, I'd give her top 10. If she asked for the top three I'd give her the top five. Somebody recently asked me, "Kevlin, you were talking about films when you did that workshop. What's your top three science fiction films?" I gave them ten. Because what you do is you give people the discussion. You say, "I'd also consider this." You never actually give them the answer. Again, career advice. I was asked once for my five favorite books. The editor sent back, "I've seen what you've done there, you named 30 books." That's the alternative approach, don't settle on one. Okay, this is a career-related question. Is communicating science harder than doing it?
Hannah Fry: Oh, wow. That's such a good question.
Kevlin Henney: Why?
Hannah Fry: I think it depends on who you are.
Simon Singh: I think there's a big push, that scientists, in the broader sense of the word, should go out and talk about their work, and I don't think they need to. The scientists are there to do science, and that's what they should get on with. They should do it the best they can and they should love it and enjoy it. But if you happen to be one of those people that love science and can talk, by all means, go out and spend your time doing that. In a way, I'd almost rather have fewer people doing it, but doing it better. And I think things have changed in the last 10 or 15, 20 years, and we have a group of people who do an outstanding job in different mediums. It just depends on what people find easy. Adam finds it easy to get up on a stage and just improvise. Other scientists would find that incredibly difficult. It's horses for courses, and I think the lucky thing is we've got a lot of people that can do it well, so there's no obligation on other scientists to do it, they can do what they love.
Hannah Fry: I think I agree with that and I'd add that the one thing that has changed in the last, maybe, decade in trying to get scientists to talk to the public more, is as Simon says, some people are not particularly good at it. There was an old attitude that I think was really problematic, and I think it's best encapsulated by the professor who taught me as an undergrad, and he had a phrase, which was, "The world fears mathematicians. Let's keep it that way." Which is, I think, extraordinarily closed-minded and gatekeeper-y. And I think that when you have that attitude, not only you are refusing to share this gigantically important body of knowledge, the limit of human knowledge, I also think that actually it really does end up meaning that you don't attract other people into your subject who don't see that there's a place for them in there, and I think that stepping outside is really important.
Simon Singh: So, you were on the "Lauren Laverne Show" this morning.
Hannah Fry: Yeah, talking about your book actually.
Simon Singh: Oh, really? I missed that bit. But that's amazing. On a radio show for the general public, 10:00 in the morning, and you're a regular?
Hannah Fry: Every week.
Simon Singh: Every week. That's fantastic, that's amazing. And the fact that Life Scientific is on there pretty much every week and the fact they're in "Numberphile", and "Numberphile" gets half a billion downloads. It's amazing. You know, when I grew up you used to have "Tomorrow's World" every night in-between "Terry and June" and "Top of the Pops".
Kevlin Henney: Oh my god.
Simon Singh: And you used to get QED, and you used to get "Horizon" on every week. And then, what was the one on ITV with Miriam Stoppard and Magnus Pyke? And now there's less science on mainstream, prime-time TV, but it's everywhere else. And yeah, Lauren Laverne is mainstream, prime-time radio, so it's fantastic that we've got people out there doing it.
Kevlin Henney: Okay, so...
Simon Singh: We often say we want more science in the news. The more science that's covered in the press the better. And Ben Goldacre said that we don't necessarily want more science, because the more science gets in there, it tends to be a bit spurious, a bit sensationalist. There are some odd studies that's got some curious result that's gonna make a headline. What you want is sometimes a bit less science in the news, but you want more credible science in the news. But that's a different thing from documentaries and features and radio programs. That's a news issue, I suppose.
Reengagement with the beauty of math as an adult
Kevlin Henney: That somehow segways to the final question, the last question. Somehow we can make that theorem, like, there are loads of questions. Thank you very much, these are excellent questions, and if my sense of time distortion continued, we could deal with all of them. Sadly, we are going to re-dock with the real world, so I'm gonna leave you with one last question, which somehow seems related. What advice do you have for adults who want to re-discover the beauty of maths? A sort of reengagement.
Hannah Fry: I would start with "Numberphile". I really just think that that is the most extraordinary resource of everything from very light, chirpy, silly videos through to some very deep ideas. I mean, there's a reason why it's got half a billion downloads, right? And why both of us have been on it. I think it's a phenomenal resource.
Simon Singh: I think there are two things I'd add. When I talk to sixth formers, I say to them, "Just make sure you watch "Numerphile", terrific videos," but also Vihart, who doesn't make as many, it's not as prolific, but makes beautiful videos. And then Martin Gardner. I grew up reading Martin Gardner and his stuff is still incredibly impressive and wonderful to read. So, I'd really encourage teenagers and all the people that want to rediscover their love of maths to go back to Martin Gardner.
Kevlin Henney: Excellent. I just want to thank both of you, and a big hand for our guests, please.
About the speakers
Dr Hannah Fry is an Associate Professor in the Mathematics of Cities at the Centre for Advanced Spatial Analysis at UCL where she studies patterns in human behaviour. Her research applies to a wide range of social problems and questions, from shopping and transport to urban crime, riots, and terrorism.
Her critically acclaimed BBC documentaries include Horizon: Diagnosis on Demand? The Computer Will See You Now, Britain’s Greatest Invention, City in the Sky (BBC Two), Magic Numbers: Hannah Fry’s Mysterious World of Maths, The Joy of Winning, The Joy of Data, Contagion! The BBC Four Pandemic, and Calculating Ada (BBC Four). She also co-presents The Curious Cases of Rutherford and Fry (BBC Radio 4), and The Maths of Life with Lauren Laverne (BBC Radio 6).
Hannah is the author of Hello World , The Indisputable Existence of Santa Claus: The Mathematics of Christmas, and The Mathematics of Love: Patterns, Proofs, and the Search for the Ultimate Equation.
After completing a PhD in particle physics at Cambridge, Simon Singh joined the BBC science department and worked on "Tomorrow's World" and "Horizon" — his documentary about Fermat's Last Theorem won a BAFTA in 1996.
In 1997 he authored "Fermat's Last Theorem," the first book about mathematics to become a No.1 bestseller in the UK. Since then he has published "The Code Book", "Big Bang" and "Trick or Treatment? Alternative Medicine on Trial." He has written for several national newspapers and magazines. He was sued for libel by the British Chiropractic Association in 2008, and was vindicated in 2010. The resulting Libel Reform Campaign led to the Defamation Act 2013.
His latest book is "The Simpsons and Their Mathematical Secrets".
Kevlin is an independent consultant, trainer, reviewer and writer best known for being the author of two volumes in the Pattern-Oriented Software Architecture series and editor of 97 Things Every Programmer Should Know and 97 Things Every Java Programmer Should Know.
Kevlin's development interests are in programming, people and practice. He has been a columnist for various magazines and web sites, a contributor to open source software and a member of more committees than is probably healthy (it has been said that "a committee is a cul-de-sac down which ideas are lured and then quietly strangled"). Kevlin also boasts over 500K views on the GOTO Conferences YouTube channel.
He has written on the subject of computer programming and development practice for many magazines and sites, including Better Software, The Register, C/C++ Users Journal, Application Development Advisor, JavaSpektrum, C++ Report, Java Report, EXE and Overload.