May 12, 2022
How can we have System-2-type reflection but with System-1-type speed? How can math be considered to be a "fast" way of solving problems? Is math discovered or invented? How can we use math to think better in everyday life? How can math education be improved? Do mathematicians have a snobbish preference for "pure" maths over applied maths? How can math be used to tell stories?
Marcus du Sautoy is the Simonyi Professor for the Public Understanding of Science and Professor of Mathematics at the University of Oxford. He is author of seven books including his most recent book, Thinking Better: the Art of the Shortcut. He has also published a play, I is a Strange Loop, which was performed at the Barbican in London in which he was also lead actor. He has presented numerous radio and TV series including a four-part landmark TV series for the BBC called The Story of Maths. He works extensively with a range of arts organisations bringing science alive for the public from The Royal Opera House to the Glastonbury Festival. He received an OBE for services to science in the 2010 New Year's Honours List and was made a Fellow of the Royal Society in 2016. Follow him on Twitter at @MarcusduSautoy or find out more about him at www.simonyi.ox.ac.uk.
JOSH: Hello and welcome to Clearer Thinking with Spencer Greenberg, the podcast about ideas that matter. I'm Josh Castle, the producer of the podcast, and I'm so glad you've joined us today. In this episode, Spencer speaks with Marcus du Sautoy about the mathematical view of the world, math education, and pure versus applied mathematics.
SPENCER: Marcus, welcome.
MARCUS: Great to be on with you, Spencer.
SPENCER: A topic that's very dear to my heart is: how do we become clearer thinkers? We literally named our website Clearer Thinking. I think you have a lot of really interesting ideas in this space, especially ideas that come from your perspective as a mathematician, how we use tools from mathematics to improve our thinking. I'm really excited for this conversation with you.
MARCUS: Great. Well, I'm very interested to see that you also have a mathematical background as well. So I hope we're going to have a fun conversation.
SPENCER: Awesome. I'm sure we will. Recently, I had Daniel Kahneman on the podcast. He's famously the author of Thinking Fast and Slow. He talks about these two different modes of thinking sometimes called System One and System Two, System One being this fast, automatic, and intuitive thinking, like if you're adding one plus one, the answer just jumps immediately into your mind, versus System Two which is this slower, reflective and more cognitively demanding type of thinking, like if you had to multiply 14 times 29, you would have to think it through carefully to get the right answer. I think you have interesting things to say about how we can have System Two-type reflection but still make it fast. Do you want to start there?
MARCUS: Yes. I wrote this new book which is called Thinking Better, bouncing off Kahneman's title about thinking fast and slow. I think that our fast thinking, our heuristic thinking is often very faulty. Because we base our decisions maybe on our local environment, the things that happen there, and believe that it should extend to a global narrative and very often that bails and breaks down. Kahneman's book is showing, time and again, how faulty that heuristic intuitive thinking can be. You've got to engage your analytic mind and perhaps take things slower. But I think that we've actually developed a fantastic suite of tools which enables you to think analytically, but not have to do it in a slow manner. In fact, I think mathematics, my own subject, I'm celebrating in this book, calling it The Art of the Shortcut, all those clever ways of honing in on the solution without having to do things the hard way. In a way, I'd say calculus, which may frighten a few people, but actually is an amazing tool that we came up with in the 16th/17th century to fast-track our way to finding the most efficient solution to a complex problem. That's what it's very good at. You don't have to try out all the different possibilities. The calculus just finds you the most efficient way to do something. This is an incredibly powerful shortcut in order to analyze a problem with many changing variables for example.
SPENCER: So the idea would be, if you want to find the minimum of a function, in order to do that, you might think, "Oh, I have to laboriously try the function at every single point or do some crazy calculation. It's really complicated." But instead, you could just take a derivative and that might, as long as the function test has certain properties, just point us right to the minimum right away. Is that an example?
MARCUS: That's exactly it. It's almost like a bit of magic. Because you might say, "Don't I have to just (as you say) calculate lots of examples, perhaps plot a graph, see where the thing minimizes or maximizes whatever you're trying to achieve?" But how extraordinary to be able to just take the equation describing your situation, apply a differential to it, find where it's zero, and that is your minimal energy point, for example, or your maximum profits. So I think this is an extraordinary tool that Newton and Leibniz came up with to understand a world in flux. Everything's changing all the time. Yet this thing is able to pick out that moment where things are optimal.
SPENCER: I'm trying to mind-read my audience a little bit here. I wonder if some people are gonna say, "Wait, you call that fast? You call that fast thinking, having to do a derivative?" That sounds scary and hard for a lot of people.
MARCUS: Exactly. It's a really interesting point here because I think people fall into the trap of thinking that shortcuts should be lazy thinking. Actually, sometimes these shortcuts require you to be really on the ball and to learn these skills to be able to apply them. So sometimes shortcuts take a long time to learn how to use but, once you learn that language, then they're very powerful and give anyone who knows this language incredible advantages over everyone else who is taking the long way round.
SPENCER: I see. So even if technically, it's actually quite difficult to carry out the derivative calculation by hand, at least conceptually, you can jump right to the answer. You can say, "Oh, I need to find the minimum of this function," and conceptually have the shortcut that points you right to the way to do that. Maybe in some cases we might have to outsource the actual calculation to a computer.
MARCUS: Well, actually I'm trying to avoid the computer in some sense. This book I wrote was a counter to my previous book which was about artificial intelligence and creativity. AI is becoming so powerful these days that it's eating up so many jobs but we do feel human creativity is one thing it shouldn't be able to do. But my book's arguing, "No. Even that, it seems to be able to make some inroads into." Actually, I did an interview with a journalist for The Guardian. He was so depressed at the end, he said, "Well, what's left for humanity?" I said, "Actually, you see, sometimes a computer won't mind doing the long, hard, laborious, boring way of answering a problem because it doesn't get tired, whereas we get tired." I actually argued to him at the time, and it was the spark for writing this book, "Well, actually, we are a bit lazy at heart so we try to avoid doing unnecessary, laborious, boring, hard work which the computer will be absolutely happy with. So that forces us as humans to come up with cunning ways to think about things which our hardware is able to cope with." Actually, maybe calculus was a dangerous place to start for us. Because it is something generally, you'd have to outsource to us as a mathematician to be able to do. You're running a company where you will employ a mathematician to do the calculus and don't expect everyone to be able to do calculus. But I think there are some simpler things which illustrate the power of this mathematical way of thinking of the world. I started the book and it's actually the beginning of my own journey as a mathematician. It'll be a story you know works very well, Spencer. The story of the young Gauss who was asked to add up the numbers from one to 100 in his class. The teacher thought, "Oh, this should keep the class occupied for ages," because it takes quite a long time to add up 100 numbers. But before he'd even finished asking the question, Gauss had written down this number on this slate board, slammed it down in front of the teacher. The teacher thought he was being impudent but when he looked at the number, it was correct, 5050. So how had Gauss done that so quickly without doing the hard work of adding up all those numbers? Well, he thought about this clever way of combining the numbers. Rather than doing the journey from the beginning — one, two, three, four, all the way up to 100 — he combined the beginning and the end of the journey. So he said, "Well, 1 plus 100 that's 101, 2 plus 99 that's also 101, 3 plus 98, also 101. So you've got these 50 pairs of numbers, adding up to 101. He immediately sees 5050. For me, that's the mindset I'm trying to encourage people to achieve in this book, to step back, don't do the hard work of adding up numbers to 100. That's boring and you're gonna make mistakes. Is there a way to look at problems and somehow the book is a suite of different ways to look at a problem which would shortcut all the hard work and give you a nice feeling of, "Aha! I can see how to do this quickly." The beauty of Gauss' little calculation is, even if the teacher had said, "Okay, what about the numbers from one to a million?" The same trick still works. Whilst the plodder — the one who's doing it one number at a time — will take even longer, the computer as well will take even longer. Even if you give that to a computer, at some point it's going to hit a number that is just too big to do it the slow way. I think that little story is one my teacher told me at school, and I fell in love with math at that point. I'm about 12 and I was thinking, "Wow! That is a really beautiful way of thinking." And my teacher said, "Well, that's what this subject is all about. Finding clever shortcuts to problems that other people are just going to spend ages laboriously going through."
SPENCER: Sounds like you had a really good teacher.
MARCUS: Yeah. I dedicate the book actually to all math teachers, but him in particular.
SPENCER: It reminds me of this idea, which I think is highly connected, that for even really difficult problems, there's often a way of looking at the problem where it becomes much simpler. So this is a useful reframe, I think, from how do I solve this difficult problem to how do I make this difficult problem simple?
MARCUS: In fact, one of the strategies I talk about in the book is the power of changing language. Very often, if something's couched in one language, it can be very opaque. But if you change the language, suddenly you're able to see things in a new way and that language gives you access to how to solve the problem. A cute little example, there's a fun game I sometimes play with kids where you've got the numbers from one to nine. You have to take it in turn to take one of those numbers. What you're trying to do is to get three numbers which add up to 15. It's fun because you start taking your numbers, but then you see, "Oh, the opponent's about to get their 15." So you have to take the number thereafter. It's quite hard to keep track of until you realize that actually what you're playing is Noughts and Crosses, but on a magic square where the numbers one to nine are arranged such that rows, columns, and diagonals all add up to 15.
SPENCER: Could you translate Noughts and Crosses to American language?
MARCUS: Tic tac toe, I think, putting zeros and x's in a three-by-three grid to try and get a column, row, or diagonal. So once you've put it in that more visual setting, you can forget the arithmetic. That's been done for you by the magic square. Anyone very quickly learns to play Noughts and Crosses (Tic tac toe) but changing that game from one thing to another suddenly makes the game very easy. There's also another game, Nim, for example, which if you turn it into binary numbers, gives you a strategy to play that game. But if you don't do that, it's really difficult to find a strategy which will help you to win. So I think you're right, sometimes a problem can look really complex but reframing it, changing the language, can suddenly get you an in.
SPENCER: This is a great example of the more general concept of math of isomorphisms, where basically you have a mathematical description of a problem. Then you realize that it can be converted to a seemingly very different mathematical description that essentially is the same in all the detailed characteristics that matter. One of my favorite examples, personally, is the idea of Fourier Transforms, where you can have a problem that seems really, really difficult to solve. Let's say it's some function that changes over time, you calculate the Fourier Transform, which changes it from being a function changing over time to a frequency analysis. So the bars that jump up and down on some stereo systems telling you how much of different frequencies there are, you go from time space to frequency space. Then suddenly, in frequency space, the answer is sometimes just totally obvious or trivial to solve. Maybe you'll turn a difficult (seemingly) to solve differential equation into a simple algebraic expression that you can just solve using high school math. And it's just very beautiful.
MARCUS: It's like a little tunnel into another world and you pull out the other side and things are just so much simpler on that side. I'm a group theorist as a mathematician, which is studying symmetry. For me, I think this is a wonderful transformation that happened in mathematics where there's symmetry. First, it looks very geometric, it's things with reflectional symmetry, rotational symmetry and actually is a very slippery concept. For example, one of my favorite places to go to in the world is the Alhambra in Granada which is a wonderful Moorish palace covered in symmetrical tiles. Then you're interested, "Well, these two walls look very different but do they have the same kind of underlying symmetries?" And it was actually a change of language where we took this geometric symmetrical world into an algebraic language created by one of my heroes in mathematics. He has such a romantic story, Evariste Galois, who died in a duel at the age of 20 over love and politics. But before he got shot, he'd actually developed this wonderful way of translating geometry into algebraic language to understand symmetry. That's the language I use every day as a practicing mathematician. You can discover things by manipulating the language that you would never be able to have found if you just stuck to that kind of original geometry. Then you can go back and see what the implications of the exploration in this algebraic language. Even, I suppose, Descartes' ideas of taking geometry into Cartesian numbers. The fact that I'm located here today in East London, you're in Manhattan, we have two numbers which will describe our longitude and latitude, that's changed the geometry into numbers. Descartes had this idea. Sometimes to understand geometry, it's much better to change it into the coordinates to describe those points. But the amazing thing with that dictionary which translate one thing to another is that in the 19th century, we realized, "Hey, hold on. Although the geometry side runs out after three dimensions, on the other side of these coordinates, we can add another coordinate and we can start talking about four directions, five directions." It enabled us to make cubes in four dimensions, spheres in five dimensions that we could never see if we just stuck to the geometry. So the power of changing language just gives you insights that you never would have had if you stuck to just the geometry.
SPENCER: I find it interesting that you're using the phrase 'changing language' because some might think it's more natural to say 'changing perspective.' But I actually like your usage of 'language' because I think of mathematics as a language. I think of it as the language of describing patterns. So you're just saying, instead of describing the pattern using this part of the language, you describe the same pattern using a different part of the language. So it seems very different even though you have this isomorphism.
MARCUS: I think that's right. That teacher that inspired me with that story of Gauss, one of the books he recommended to me as a kid was called The Language of Mathematics. It's funny because at the time, I was fantasizing about — not being a mathematician (I didn't actually realize that was a job.) — I actually wanted to be a spy. I thought I had to learn lots of languages like Russian and Arabic but I found them very frustrating because there were all these irregular verbs, strange spellings. When my teacher gave me this book called The Language of Mathematics, I was intrigued. I was, "Oh, I never thought of mathematics as a language." But as you say, this is a great language to describe patterns. Actually, you could say it is nature's language, it's the universe's language. Often, to understand what's going on in the physical universe, we reduce it to mathematics. But what I love about this language is that it doesn't have irregular verbs. Everything makes logical sense. Yet, it does have strange twists and turns. That's what makes it a charming subject. Everything isn't obvious. But I did realize, "I'm going to become a spy in the world of mathematics." You try and understand this language because it is very powerful, as you say, in revealing patterns because that's another one of my shortcuts actually, which is often, if you've got some data but you can spot some rhythm to that data, some pattern in there, you can start to produce maybe an equation, an algebraic formula, which will extend that pattern. Well, that gives you a very powerful tool to predict what the pattern might do next. Maybe that pattern is floods of the Nile and if you can pick up a periodic behavior in that, then maybe you know every seven years you don't plant crops because it's going to be flooded. That's why time and again, I think these mathematical innovations have really given civilization extraordinary tools to be able to control and even change their environment.
SPENCER: That's interesting. I think that touches on this idea of "How do we actually use math in the real world?" The way that I think about it is that, very often when we use math, what's happening is that someone has looked at the world, noticed some recurring pattern, then abstracted away all the details that are relevant to that pattern. Then they go on to write down to math, so they use the language of math to describe that pattern. Then once it's in the language of math, they can then manipulate the math to prove things about the pattern that were not previously known, to make predictions, or generalize it, or learn new properties it has. Then people can go take those predictions and go back to the world and check that they hold. I think of this as a repeating process happening again and again.
MARCUS: That point that you made about throwing away the things which aren't important is so part of being a mathematician. That's this idea of thinking abstractly and being able to just throw away things which aren't important. I think that, so often, we just get inundated with unnecessary bits and just picking out what's essential is really key. Then the magic is... Well, I might do that in completely different settings but find I've actually got the same abstraction. Therefore I only have to do the work once and it can apply to many different settings. One of my favorite ones, people have probably heard of Fibonacci numbers, these numbers 1, 1, 2, 3, 5, 8, 13. Maybe you started to spot the pattern that you get the next one by adding the two previous numbers together. These are abstract patterns there. But this guy Fibonacci recognized that, time and again in nature, these numbers seem to be popping up. It pops up in number of petals on a flower or the way rabbits grow from one generation to the next. So anything you can tell about those numbers will give you some insight perhaps into the way things grow in nature. However, actually, they shouldn't be named after Fibonacci because he wasn't the first to discover them. I found out that they'd actually been discovered in India several centuries earlier, not by mathematicians but by musicians and poets because they discovered that these numbers also describe the number of different rhythms you can make with long and short beats. A tabla player in Indian music was trying to show off with a lot of different rhythms that they can make and they realized that there was this algorithmic way that the shorter rhythms can be put together to make the next rhythms along. It's the same numbers, yet in Europe, you've got Fibonacci using them to describe nature and, in India, they're the numbers that are being used to describe rhythms in poetry and music. But anything you discover about those numbers will apply to both those rather different settings.
SPENCER: I go back and forth on how amazed we should be at this kind of thing where you see the same pattern popping up again and again. On the one hand, it's miraculous how few mathematical constructs we need to describe so much. Once you have the polynomials and the exponentials, you get the inverses of all these things, you don't need that many different mathematical constructs to have most of what we use to describe almost everything. It's shocking. On the other hand, you think about the Fibonacci numbers appearing everywhere. That's just like saying, well, y equals x squared appears in a lot of places, too. Just because the Fibonacci numbers are slightly more complicated (and so less intuitive), it's not really that crazy to us that y equals x squared appears in a lot of places. Maybe we shouldn't be so surprised by this. I'm not sure.
MARCUS: Yes. There's so much in what you've just said actually. First of all, I think you're absolutely right, there's something rather extraordinary that you basically just take numbers, addition and multiplication, something that's so simple, yet out of that, as you start to explore how these interact, the things that you can do with them, the whole of mathematics seems to emerge from there. Actually, the book I'm writing at the moment is all about games. It's called Around the World in 80 Games. I'm discovering that the best games are those that have incredibly simple rules yet give rise to wonderful complexity and variety. It feels like mathematics. We're just so lucky as mathematicians to have this extraordinary game with really simple rules. Yet, as you start to play the game, just beautiful complexity arises. But then the other interesting point you've made is, why do we keep on seeing mathematics every time we try to understand the universe? There always seems to be a bit of mathematics hiding behind there. We try to understand what material is made out of. It goes down to atoms, electrons, protons, neutrons. Then we get down to these quarks but actually, what is a quark? It's really just a bit of mathematics. So the unreasonable effectiveness of mathematics, this idea that, how come just everything seems to work so beautifully with mathematics? Actually, I have a theory about that, which is, I think that one of the great challenges is, where did all of this stuff come from? Was there a creator? And some people call it gods. But my kids always say, "Yeah but who created gods?" You get this infinite regress. So if you're going to explain how we got all of this stuff, you need something which is outside of time, because it shouldn't be created itself. I think mathematics is a good candidate for something which is outside of time, which is just about relationships between things. Maybe what we're seeing is a physicalized piece of mathematics around us. That's why time and again, we just got mathematics bubbling under there because actually what we are seeing around us is just physicalized mathematics.
SPENCER: It's really interesting how you talk about this because I think you take a different philosophical perspective on mathematics. Maybe you treat it as more real than I do. I treat it as very much, essentially literally, a language. In other words, I think of mathematics as something that humans invented and it's a language for describing patterns. It could have been different than it is. I think one way to see that is, first of all, the way we develop math is we develop it in bits and pieces. Someone develops planar geometry. Then eventually someone else figures out how to put that in a more general framework. You can view all these little mini pieces of math as just mini languages and then over time, we wanted to systematize it all so we start sticking more and more of them together. Then we find isomorphisms that allow us to put everything in terms of (let's say) sets, and try to reinterpret all of our math in terms of some basic axioms on sets. But even then, we don't get a single unique mathematics. If you try to put together the axioms, there are axioms that you can't really agree on. There are axioms that you have to take them or leave them but nobody has a good way to decide whether to do so, like the continuum hypothesis. From my point of view, it really is just a language we constructed and then we're just studying the universe. So I don't think of the universe as being made of math. I think of math as our pattern language. I'm curious to hear, what is your perspective on that?
MARCUS: I think I am sympathetic to your perspective here but what I would say is that I still think you'd be hard pushed to say we created the fact that...it's a human construct that the number 17 is prime and the number 15 isn't. What I feel that you're articulating is actually: the stories we tell about these, and the things that we find interesting, and the discoveries that we make, they are a very human side of doing mathematics. There'd be moments in history when new insights come like the square root of minus one. People didn't think there was a number which when you square it, equals minus one yet we imagined this number. Then it's part of our human creativity to move mathematics in that way. I'd normally say, for example: prime numbers, there are atoms, there are quarks, there are hydrogen and oxygen. I think it'd be a very hard push to say that's a human construct. I think the stories we tell about those primes are things that we discover. The ancient Greeks proved that there are infinitely many of these numbers. My first book, The Music of the Primes, is all about Gauss' discovery about how the primes thin out. Then Riemann, you mentioned Fourier analysis, Riemann discovers there's a kind of Fourier analysis you can do of primes which turns the primes into something else which we now call the zeros, the Riemann zeta function, which are basically just frequencies that help us tell how the primes are distributed. Now, I think that's very much part of our human storytelling about the primes. I think if you went out into space and you found an alien culture, I think we'd be able to start with the primes and swap them around and recognize that we'd both discovered basic building blocks. In fact, of course, Carl Sagan uses prime numbers at the beginning of Contact, as the way his alien culture makes contact. But I think we might find that what an alien culture, the things that they find interesting about primes, and what they've discovered so far might be quite perpendicular, in a way, to what we're doing. So I agree that there's a human side to it. It's like a composer, the music you compose is very dependent on your culture, your period of history, but those notes that are there are ready for anybody to compose the music they want to. Now there's another really interesting story. There's so many things in what you just said. To address the last one — because you mentioned the continuum hypothesis which is really fascinating — I think a lot of people will find it quite new (maybe) that there are different sorts of mathematics. Somewhere there are infinite sets between whole numbers and real numbers. Somewhere there are no intermediate infinite sets. And for me, that's not frightening. That's the beauty of maths. We live in a multiverse. For example, geometry, there's Euclidean geometry, spherical geometries, hyperbolic geometries. They're all different from each other but they're all internally consistent. Now, that is very different to being a scientist. Because you see, I'm quite happy to move from a world where the numbers have these different sorts of infinities or those where they don't, where the continuum hypothesis is true or false. But a scientist, once a theory doesn't fit the actual physical universe that we're in, it gets thrown out. Even if it's beautiful and consistent, if it doesn't fit our universe, it's not interesting. For me as a mathematician, it will still maintain interest, even if it's not describing the physical universe around us. So I feel lucky that there's an ability to be more creative in mathematics because of that, because you can make worlds, provided they're consistent, and you can explore those which are quite fanciful and not anything to do with reality. Whilst once a physicist realizes the theory of supersymmetry isn't right, it's not interesting. But for me, supersymmetry is really fascinating as a theory in its own right.
SPENCER: I guess the way I think about this is, we can imagine aliens came up with the idea of primes. If they did, if aliens had the idea of primes and they also came up with the idea of integers — obviously they're going to call them something else, not primes and integers — but if they came up with the same ideas, then all these different things are gonna have to follow from that. They don't get to choose. There are implications of those definitions. However, where it becomes arbitrary to me is that you don't have to think in terms of prime. Prime is a convenient way to think but you could imagine coming up with other ways of describing integers that are not primes. It might be less convenient. But maybe for some purposes, it'd be more convenient. So it's conditioned on certain definitions. You no longer have any choices. All the logical implications of those follow but the choices seem to me where human arbitrariness comes in. I would extend this even to physics. People think of atoms as being fundamental. Well, I actually think atoms themselves are still just human choices. Divvying things up into atoms seems like a choice we make because it's convenient for the human mind. Whereas if you actually look at deeper reality, it's probable our best theory says something more like fields, not even atoms. Atoms are maybe not the most basic thing.
MARCUS: That's very interesting. I suppose we're getting into the realm of [inaudible] games, word games, that we just see these things around us. We have to agree on the name and properties of that game. Again, I think that's very interesting because we try and describe the physical world around us by matching it to things that our human mind and body and our embodiment can cope with. It's interesting because I had a challenge, actually, from an AI researcher. I was making a program, it was for a touring anniversary for the BBC. So it was pre all this machine learning, amazing AI heatwave that we're in. He was saying that we really have to understand artificial intelligence by putting it in a body. If we're trying to understand our intelligence, you can't do it without embodying it. He challenged me that he thought that all of the mathematics that we had discovered would actually be a result of our embodiment and our engagement of our bodies with the world around us. I thought that was interesting because, most of the time, I think of mathematics as hugely abstract and not embodied. The point is, I love playing with mathematics in my mind and going places: that four dimensional cubes, five dimensional spheres, infinite dimensional spaces. I offered them a challenge: what about the square root of minus one? We see that anyway. But where did that come from? We've traced it back. I suppose square roots came from the idea of length. The square root of two for example, something the Babylonians tried to calculate. The ancient Greeks discovered you can't describe the square two by a fraction. There's no fraction which, when you square it, equals two. So in some ways, square root of two is almost as imaginary because it's this infinite decimal thing. I mean, if quantum physics is true, everything's quantized and there's no length which is the square root of two, in a way. But the square root came from... If you look at the ancient Babylonians, they were taking a unit square and looking at the diagonal across the square and they realized that if the sides have unit length and that length across has length square root of two. So even the square root of minus one has a journey which began with just the ancient Babylonians wanting to know what's the length across the unit square. So I think that plays to your side of the tennis court in a way that he was saying, there's probably a lot of mathematics that we will not maybe ever discover because it doesn't begin with embodiment. Even though we might analytically continue our insights, everything starts with trying to understand the physical universe around us.
SPENCER: It seems to me that mathematics is an infinite game in really the deepest sense, and that there are mathematical truths that literally a human mind could not even comprehend. That just seems obvious to me. I'm wondering if that's obvious to you.
MARCUS: I totally agree. We can always prove that because you can write down any statements of mathematics which are either true or false. There are infinitely many possibilities of how you write those down. It's saying the same thing as, technically, there are infinitely many books that it is possible to write. Obviously physical constraints of the universe, we couldn't write them all down. But if the book's allowed to be as long as it wants, then there is infinite possibility. I think mathematics is an infinite game. What's really curious is, what are we finite mathematicians doing? Well, we're making choices about which of those things are, first of all, interesting to talk about. It's a bit like one of my favorite stories, The Library of Babel by Borges, which is about a library which contains every book it's possible to write. Actually, the books all have 410 pages. There aren't infinitely many books. At first you think that library has everything, but actually it has nothing because nobody's made any choices. So again, I think I'm probably playing to your side of the court again in this argument about, "Does math exist outside of being human?" Actually the mathematics that we talk about is about the choices of those proofs and stories and theorems that move us, that take us on an interesting journey and not just boring calculations. I think that's unexpected to people. But if mathematics is infinite, you're absolutely right. There will be statements that are so complex and the proofs are so long, our hardware, our brains will never be able to navigate them. I think there is a fear that we're reaching some limits at the moment that the conjectures that we've been left with — like the Riemann hypothesis — trying to understand how primes are distributed. We think we know how they work but what if the proof is so long and complex that it's such a long game that the lifetime of a human will never be able to navigate it? There will be theorems out there which have that kind of property and so we have limitations about what we will know about this infinite game.
SPENCER: It's interesting to think about the idea that there could be an alien species that exists (or will one day exist) where proving things like the Riemann hypothesis is something you do in grade school because they are just so much smarter than us and they're capable of doing math that, to us, is literally incomprehensible. You can imagine the same thing with, potentially one day, advanced artificial intelligence that may be able to do — not just calculations, which computers can already do much faster than us, much longer than us — but do symbolic manipulation and prove theorems in a way that is just so far beyond our comprehension, that makes us look silly.
MARCUS: That was one of the threads in my AI and creativity book called The Creativity Code. It was an existential challenge to myself that, are we at a point where AI... We saw it being creative in that game of Go, Move 37 Game Two (I think) was an act of creativity to come up with a new move that won at the game that most people thought was a really bad move but it showed how to use it in a very powerful way.
SPENCER: [interrupts] Are you talking about the AlphaGo match?
MARCUS: The AlphaGo match, exactly, with Lee Sedol. I think that was the first hint of, you allow a piece of AI to start playing a game or learning and changing itself, it might find new things that we've missed. I was intrigued. How far away are we from an AI actually — as you say, not just doing boring computations, we've been doing that for decades, and using a computer to help us prove things like the four-color map problem by giving it all the boring bit of the proof to just check — but I'm interested, when will we get AI learning the proofs that we really like and being able to come up with its own offerings? There's already a move afoot to see how AI can learn from computerized proofs. But it's, in a way, not making the aesthetic choice. It's certainly proving theorems but most of the theorems I looked at were just not interesting. They weren't giving one new insight. So it feels like the AI has to learn the beauty of maths, not just the utility of maths, in order to be able to really start maybe offering things to journals for the first time. I think it's still a little bit of way. But math is a very new thing. Humans haven't been doing mathematics for that long. I think the visual world, the human brain has developed millions of years to navigate visual things, oral as well. Painting or music, I think, is a very deep-rooted human activity. But mathematics is actually quite new. I don't think it's too fanciful to think that maybe an AI can get the experience. I mean, how did you and I become mathematicians? We spent time going to lectures and learning the ropes and seeing how they worked. Then we started to do our own stuff. It won't be too long before an AI can munch up all the theorems of the past and the proofs and start to get a feel for where to go next. So I think we're looking actually in the future to maybe an exciting period where we're in collaboration with an AI offering new ways of looking at things that will help us make breakthroughs on the things we're stuck on.
SPENCER: Taking this conversation in a different direction, I wanted to ask you a little bit about the topic of thinking better and how it relates to all this. Because that's the title of your book, Thinking Better. It seems like that title is suggesting that we can use mathematics to think better in our daily lives. So I just wanted to see what practical insights we can give to the listener about, "Okay. This is all really cool but what does it have to do with me? How do I use this stuff?"
MARCUS: I think patterns are a good way. Because very often you'll find that perhaps you're reinventing the wheel every time you're doing something. You realize, actually this is the same thing happening over and over again so I think just being sensitive to picking up a pattern. For example, if you're running a restaurant and you start to see the dynamics of the waxing and waning of different tastes throughout the week. That will mean that you can get much more control on what you order in and the wastage you'll have. I think the other thing is just, one of the reasons machine learning has been so successful is that we're in a data overload kind of world. The digital data world is just so huge now, and trying to navigate that is often quite difficult for us. One of the shortcuts I talk about is, how much data do you need to look at to be able to be sure that the insights you're beginning to get about the whole data set are really true? In a way, it goes back to Kahneman's idea of heuristics. Very often you look at just the things happening around you and what can you distill from that about what's happening more generally. We develop some very good ways of looking at just a very small section of a load of data which can actually give you huge insights into what's going on. For example, there was an advert that I used to love when I was a kid which was for cat food. The commercial very confidently declared eight out of ten cats prefer Whiskas, which is the name of the cat food. We had a cat and I never remember anybody coming in asking my cat what cat food it liked. I was like, "Well, how can you be so confident in saying that eight out of ten cats prefer this particular cat food? There are 7 million cats here in the UK. How many do you need to ask to be confident about that kind of claim?" Mathematics helps you to say, actually you need barely 250 cats. That data will give you — 19 out of 20 times — that will give you a percentage which is 5% out of the true value of cats and their preferences. So I think that's really important in this data-rich world, being able to know, I can get away with a small sample set to be able to tell a lot about what's going on. So I think those things are quite useful as well.
SPENCER: As long as the cats are randomly sampled. [laughs] If you get all the cats with fancy taste.
MARCUS: That's of course, the problem with Kahneman and heuristics, very often we're just locally sampling around our local environment. It isn't giving us the global insight you need to be more global.
SPENCER: So how do you use math in everyday life? Do you have examples of ways it helps you? Obviously you're a mathematician, you use it to make money. But in ordinary life?
MARCUS: One place I do use it actually is... I'm a musician. I'm a very bad musician. But I play the trumpet and learning the cello at the moment. Actually, I do use the idea of seeing a pattern in a particular bit of music because that helps. It's almost like, rather than reading a book by looking at letter by letter, if you can actually spot a word there then obviously the book makes much more sense. We learn to read by clumping letters together into words. Actually, I found that very powerful musically. There are so many patterns in music, and if you can pick them up, it means that, once you've learned one, you learned many more. Even though they may be shifting in pitch, it's just the same pattern, but just moved around. Now see, that's why as a musician, you spend a lot of time learning your scales and your arpeggios, for example. Because those are very often the building block patterns that any composer will use. Although it's a bit boring, doing that learning scales and arpeggios, it's a fantastic shortcut for then not having to read every note individually. You can see a pattern and then apply the things you learned to be able to do that quickly.
SPENCER: I feel like a lot of the benefits I've gotten from mathematics in daily life are actually ways of thinking more so than actually using math per se. Is it something you relate to?
MARCUS: Yes, certainly. I think the idea of abstraction, for example, being able to throw away what's not important, that's often an extraordinarily important shortcut. For example, I've got a whole section on diagrams. I think being able to come up with a very good diagram to crystallize genuinely what's going on can be really important and also help to communicate your insights to other people. One of my favorites, of course, is the London Underground map here in London which I use every day. I travel around the city, because that's a topological map, not a geometric map. Topology is this bendy geometry. The map doesn't physically represent where the stations are across the City of London. They just show you how they're connected together. This was a great insight of Beck, who came up with that iconic map in the 1930s. First of all, they used a map which was a more geometric map and tried to draw a picture of where the actual stops were precisely located, their distances, but it was hugely cluttered. You just couldn't see what was going on. So, that idea of saying, you don't really care the distances between these things or whether they're oriented exactly relative to each other. Much better to just push and pull this thing around so you can understand how to get from one place to another. So that's a great example of just throwing away the unimportant stuff, which was the geometry and leaving still the topology, the way the thing is connected, which is the thing that you're interested in.
SPENCER: I think that's a nice example. An even simpler example that might illustrate the point is, imagine that you're trying to plan fields. We're gonna grow crops, a kind of ancient problem that humans have been dealing with for a long time. You want to know how many crops can I grow in this land. You can imagine how hard that would be before people knew any math. But then once they know some math, they can abstract away the field and say, "Okay. This field is basically a rectangle. All we need to know is the length of one side of the rectangle and the length to the other. Then we can multiply them to compute the area. That's how much you can grow there." Once you do that, you've abstracted away all the other information about the land. You've just gotten down to these two simple numbers, that's really cool. Then you can then take that description, that mathematical description, you could say, "How much more could I grow if I extended one of the sides of the rectangle?" Then you can do that calculation. Now that should be able to be applied back into the world to make a prediction about what would happen in the real world which of course, will work. So you get this give and take, where you look at the world, you abstract it, you turn it into math, you manipulate the math that lets you learn something new that you didn't know before about the world.
MARCUS: I think what's lovely about that particular story is it shows the power of: you didn't have to just experiment over and over again with fields to find what the dimensions are that will maximize my crop. You do little thought experiments and you start to see, "Actually, if I've got a set length of fence for example, and I want to get the maximal area, then symmetry is going to give me the way to maximize this." Because it's the square in that particular case. Or if you can even get to a circle, that will even be better. That's the shape which encloses the largest amount of area for that particular length of fence. So I think that's what's lovely about maths. It enables you to explore possibilities and find the optimal solution, and then build the thing. I often call mathematics the ultimate fortune teller because it allows you to look into the future and plan your pathway to where you want to end up without having to try all the paths in advance. That's why if you want a shortcut, it's the best way; rather than just exploring the whole terrain and trying to find it, it hones in on that shortcut for you. Engineers are always using this. They will make the thing in the mathematical world first before they build the bridge or construct the building because the mathematics is going to tell them this is a nice energy-efficient way of putting together these things which is going to make this building efficient and wonderful to live in, for example.
SPENCER: Changing topics again, I'm interested to hear your thoughts on math education. There's a thing that a lot of people complain about. I think a lot of mathematicians, especially, are grumpy about the math education. So I'm curious to hear your take on how do we do it better.
MARCUS: I think my teacher in my school really had it right because what he did was tell us stories outside of the boring curriculum. I often feel like the mass education system at the moment is almost trying to get you to play a musical instrument. But all it lets you do are those scales and arpeggios. It never actually plays you any wonderful music that you can aspire to play (maybe) in the future if you dedicate yourself to this subject. What I was very lucky to have was a teacher who would tell us stories about prime numbers or infinities and how there are different sorts of infinity, or the fact that you can make shapes in geometry in dimensions beyond our three dimensions. They were all wonderfully spacey ideas and wonderful stories. I don't think the class really understood completely what was going on. But it just inspired us to, "Well, I would love to be able to navigate a four- dimensional cube. That sounds so cool. So I will dedicate myself to learning coordinate geometry." The first time I looked, rather boring, but if that's going to give me access to this big story, I think that was very motivating for me. I think that that's what we're missing. I don't know how it is in the US. But the UK education system, for example, in English, has two streams. One is what we call English Language, which is just learning spelling, grammar, and how to write letters. Very, very boring. But then we have English literature, which is reading Shakespeare, poetry, George Orwell, Animal Farm, things like this. That's really the lovely side of studying English. My feeling is, why don't we do the same in maths? We got the math language, the stuff you have to learn in order to be able to find your way in this world, to learn to multiply, do some areas and things like that. Why don't we actually have another stream which is telling these big stories, kind of Shakespeare or the Mozart of mathematics. I suppose in a way, that's what I've tried to do with my books. The first book that I wrote, Music of the Primes, is a story of prime numbers and the extraordinary journey we've gone on — still working on — to try and understand these numbers. My feeling is that's missing. We're being a bit too timid in maths, and not brave enough to tell some of these wonderful big stories out there.
SPENCER: I think almost everyone can relate to the idea of math being incredibly boring at times, even people who love math. I found some of my high school math classes really boring, even though I truly loved math. I think of there being a few different solutions to that. You've touched on a couple of them. One of them is, you tell stories interwoven in the math, like the Gauss story that you told earlier which is a lovely story that makes the math more exciting. Imagine teaching history as just a series of facts, that's really boring. But if you can talk about the stories of the people that lived through it, that makes it more interesting. That's the first. A second is inspiring awe, the way that contemplating the galaxies can get you interested in astronomy. You can use things like, how can we understand infinity? What are the different ways to define infinity? That can be very awe-inspiring, and a way to hook people into a math interest. A third thing I would say is making it relevant. There are things that kids already care about. Can you show them that math is relevant to the things they care about? If they're a YouTuber, and they're trying to increase their YouTube views, maybe then you can use YouTube statistics as a way to teach statistics. Or if they like sports, maybe you can talk about batting averages as a way to teach about percentages. Then the fourth and final way I'll mention is trying to evoke curiosity. If you pose an interesting problem, and then you get kids engaged with, "How would I solve that?" and they're actually thinking about how to solve it before you present the answer. I think that can be a fourth route. So we've got stories, we can use awe, we can try to make things relevant to their existing interest, then we can evoke curiosity and use these four tools.
MARCUS: It's clear Biden needs to come and make you education secretary in the US because, absolutely, I think that's a wonderful suite of four different approaches. I think that's what's really important because we have different ways of learning and getting excited about things. I feel we're a bit just too one-dimensional in the way that we're teaching maths. We need to find these multiple ways. Sometimes it will be utility. For me, utility is less important. I'm a very pure mathematician. I don't need to see my things being applied. But others, they want to see prime numbers used in cryptography, for example. That then will get them interested, "If I know about primes, I can crack credit cards going across the internet." But I think your last one is really important, because I think that puzzle element is often how people get into mathematics. For me, also, I was brought up on Martin Gardner's puzzles in Scientific American. I think just that joy of wrestling with a problem not seeing how to do it, and then suddenly going, "Oh, I get it. If I look at it this way, the problem plays out." I think that 'Aha' moment, which is what I crave as a research mathematician, and it's what I'm sitting in my desk every day trying to go, "Aha! Yes, I see what's going on." I think you get out of a well-set puzzle. Actually, there's something important here, which is, mathematics is a hard subject. But actually, I'm not frightened to celebrate that. I think that it's part of its charm. If things are just too easy, they get boring. But you just want to get things balanced well, so it's not too hard that you'll just get put off. I think you want to make mathematical education to try and find that sweet spot for each student where they're pushed and they enjoy the challenge. Then they enjoy the satisfaction of overcoming that challenge. One of my favorite movies is Matt Damon's Good Will Hunting, which I think illustrates this because you've got the math department — as you say in the US — they're working on these problems and they just find them so difficult, and they put them up on the blackboard. But then Matt Damon, who plays the janitor, he comes in at night, sees this problem on the board and solves it immediately. Now they're all shocked when they come in the next day. Gradually as the film goes on, they discover who it is. Matt Damon has this challenge of whether he wants to become a mathematician. He chooses not to become a mathematician. I think it's because he finds it too easy. It's just not interesting to him. What he finds really difficult is understanding his girlfriend and that's what he goes off chasing at the end of the movie, trying to understand the mysteries of how women work. I think very often we choose to become mathematicians because we enjoy the challenge. I'm working on conjectures that I've been trying to crack for 15 years. When I finally crack it, it's going to be immense. That subject would not be interesting if everything was just rather boring and obvious.
SPENCER: There really is such a deep satisfaction thinking hard about a problem for a really long time and then having it suddenly click into place. I don't know if maybe I just get this because I'm not as good a mathematician as you but afterwards I often think, "How did I not think of that? That's so obvious in retrospect."
MARCUS: The other interesting thing you said about education, I think that storytelling, but in particular, the history of the subject is underused in education. Because I think if you explain to a student where a new idea came from, that there was a person involved in coming up with that new idea. Even the discovery that zero was a number that was invented by (or discovered) — there's another interesting philosophical discussion — by the Indians in the seventh century, that negative numbers had to be thought about and conceptualized. I think it helps a student on their own journey of new discovery to learn about: Where did these ideas come from? Why were they coming up with them? What was the challenge that they were trying to solve that meant that they came up with negative numbers? That was about transactions and people owing money, for example. I found that history is a very powerful thing to use in telling mathematical stories. I often use history as a way into a subject in the books that I tell because I think non-mathematicians find historical narrative very easy to follow. But a mathematical or scientific narrative is very unnatural for them. So to use history has been a very powerful ally, I can think, in what I've been doing. I made a program for the BBC called, The Story of Maths, which was four one-hour programs about the history of my subject. It's amazing. I learned things that I didn't know about my subject through making that program which gave me new insights into my own subject. I know a lot of teachers have been using clips here in the UK from that program to explain to kids the volume of the pyramids: you can give them just the formula, the area of the base times the height divided by three. Well, that's very dry and boring. But if you then show them the Rhind Papyrus and the fact that the Egyptians — of course, they were interested in finding a shortcut formula for the number of blocks they'll need for their pyramid — that story comes alive and you begin to understand that was the motivation, and how did they come up with that formula. You want to know the secret of the story of where that came from and how it got in the Rhind Papyrus.
SPENCER: You mentioned earlier this idea that you're a pure mathematician. Something I've seen come up quite a bit among mathematicians is that they'll say, "Well I don't care about applications of this." I've even gone to a lecture where the first thing they did was say, "Now I know this sounds like an applied math topic. But I want to assure you, this is definitely not applied," as though it's superior to not be applied. I'm wondering, have you seen this feeling that there's something better about doing pure things? If so, where do you think this stems from?
MARCUS: We are very snobby about it. We enjoy celebrating, often, thought for its own sake rather than getting bogged down in some messy differential equation which is describing passage of oil through a pipe. I tell you what, I think there was another book my teacher recommended to me when I was a kid which probably has helped — certainly, in my mind and also in others — to inspire this bit of snobbery between abstract pure maths and messy applied maths. It was called A Mathematician's Apology by G. H. Hardy, who was a number theorist interested in primes, very famously brought Ramanujan over from India, this self-taught mathematician in India that he recognized was a genius. There's wonderful biographies and books about that and also movies as well. In fact, I was involved in a play here in the UK about that story. He wrote this lovely book, and I do recommend anyone to read it. It's not very long but it's described — very perfectly, I think — what it was like to be a mathematician. When I read it, it helped me to fall in love with this dream of becoming a mathematician. In that book, he's described the mathematician very much as a creative artist. He compares a mathematician to being like a painter or poet, we're maker of patterns. In that book, he's very snobby, he says the only real mathematics is pure mathematics. All the mathematics that actually gets used out there is not beautiful at all and it's just not worth considering. Now, I think that is not true. I think we're in a much more fluid place between these two subjects. For example, his subject of prime numbers is now, as I mentioned before, the heart of internet cryptography. I think he would be quite shocked to see his very pure subject of mathematics is now helping to run e-commerce in our world. So actually, I think it's very important that we have these applications because it actually justifies our funding. I think it's very important to celebrate them because, very often, what somebody needs to make something might have...the motivation might have just been the passion for creating something for its own sake. I think we need to celebrate blue skies thinking and allowing people to just follow their intuition, follow their loves, follow the beauty of the subject, because it's interesting how long it will take. Maybe it's a decade, or in the case of primes, basically, it was discoveries of Pierre de Fermat that ultimately in the 20th century were used for cryptography. So that's quite a long lead time and maybe a government won't be happy with paying for something now which is going to take 350 years to come to fruition. But I think that it's really exciting to see how often just pure abstract thinking gives rise to something which ultimately is applicable. Take for example, Google, the Google algorithm is tapping into very esoteric-looking bit of maths to find that website that you want to find. Again, this is one of the shortcuts I talk about in the book. If you use something called the eigenvalue of a matrix which is something we use actually in quantum physics to work out energy levels. But actually, this turns out to be the key to locking on to the website that's best connected to the search question that you put into the engine. So it was Sergey Brin and Larry Page, they learned their mathematics, and then they realized, "Hey, this is exactly what we need to tune our algorithm to find these sweet spots to find the searches that you want."
SPENCER: You're talking about the PageRank algorithm where it looks at, for every website, what links are coming into that website. In other words, what websites are linking to that website and then it says, "Okay, if the websites linked to it are good websites, then that makes it good." But then if you try to define things this way, then it actually becomes a recursion. Because in order to know how good a website is, you have to know how good websites there are linking to it. But in order to know how good they are, you have to know how good the things linked to them are and so on. Then the solution is recursion ends up being a problem in linear algebra, and that's where eigenvectors and eigenvalues get involved, right?
MARCUS: Exactly. One way to try and understand the problem that you illustrate is you equally value every website, and then you start to redistribute value according to what websites they link to. But that's a dynamic system, exactly. An important website will be giving more of its value to anything that it's linked to but you don't know it's important yet. So you have to let the thing run to see how the value is distributed, but that can take time. The wonderful thing about these eigenvalues is that they can just take the system and identify immediately how to distribute the balls, the stable points that that system will eventually settle on. That's an incredible shortcut. Rather than running the whole system, seeing how the value distributes itself, this just locks in immediately and tells you exactly where all the value will end up.
SPENCER: I'm someone who has always really cared about applications. I think it's important to distinguish mathematics as a game versus mathematics as a tool. You can think about chess, for example, everyone knows that chess is not doing something literally important. It's not saving lives. Maybe it helps you develop your mind by playing it but the fundamental purpose of chess is that it's a game. It's supposed to be fun and supposed to be challenging. It's supposed to be competitive. Some people, I think, treat mathematics that way. They treat it as a game. It's fun, it's challenging, and it's incredibly infinitely challenging, as we talked about. It's competitive. People are competing to get papers in top journals. Then there's another view of math, which is that math is a tool where you're saying: my goal is to change something in the world. I want to make people's lives better or want to do something better. Then math is just the way that I try to accomplish that. I'm curious to hear your thoughts on that distinction.
MARCUS: Well, I think you've missed something which is really important here, which is actually, I think of mathematics as storytelling. Writing a novel may not be useful but it will enrich your way of seeing the world and then sometimes it is useful. My feeling is you're right. Quite often we start being fascinated in mathematics because we like the game element of it. We like that often competitive play. But I think when you get to my level, I feel like I'm not just trying to churn out true statements about numbers, playing one game after another. I'm really after the things which help to transform people. That when I tell these to my fellow mathematicians, they see suddenly a new connection between two things which they never thought were connected before. There's a real emotional engagement in this. That seems to be really important. I want to feel like I'm taking my audience or my readers almost on a journey from the Shire to Mordor and to take them somewhere really dramatically new. And that will be really important, that dynamic in the choice of mathematics that I will focus on. So if something's too obvious and boring then I won't write that down and celebrate it, or if it's just too messy and complex. It might be messy and complex and help you to understand something about the world. It might have a practical implication, and therefore, it's important to go through the messy stuff. But that never really interested me. I wanted to tell these great big stories, and my feeling is that mathematics, it really is getting you to fundamental insights about the truth of our universe. That is genuinely so exciting to feel like the discoveries I've made are the power of proof, when I've actually proved something, it's there forever. It's actually Hardy in his book A Mathematician's Apology said, immortality is probably a silly word but mathematicians may have the best chance of achieving immortality. Because those proofs that the ancient Greeks wrote down that there are infinitely many primes, they're as true today as they were 2000 years ago. That's why I feel I'm still on my side of the tennis court about: is maths just a human creation or is it something more platonic? I'm a Platonist. I'm a fully signed up member of the Platonist organization. I believe that we are actually getting access through our mathematical proofs to universal truths about the universe. That, for me, is one of the motivations for doing it.
SPENCER: I feel that you really touched on two things there. One of them you might call math as art or math as inspiration where you're trying to do something that's more akin to what an artist is trying to do. Then the second there is math as exploration of universal truth, or it's like an astronomer might want to try to figure out how the universe began even if we can't do anything with that information that's practical. We can't save lives with that information. A lot of people really, really care about that and would view that as fundamental and important. So maybe we have at least four perspectives on math: as a game, as a tool, and also as a form of truth-seeking.
MARCUS: I would agree with that. I think that's quite nice description of the lay of the land.
SPENCER: Before we finish up, any last topics you wanna talk about quick?
MARCUS: I think one of the exciting things about math is its ability almost to look at itself and say things about mathematics itself. Actually, at the end of the book, I talk about the power of mathematics, sometimes to show when a problem doesn't have a clever shortcut. That's actually one of our great unsolved problems, something called P versus NP. I think that it's interesting that we can use math sometimes to show that, for example, the traveling salesman problem says, "Is there an algorithm which can find you the fastest route round a set of cities in the shortest distance to cover them?" We think there may not be such an algorithm that you can't get away with, without just trying all of the routes and seeing which is the smallest. I think another great example of that is one of the great theorems of the 20th century, Gödel's Incompleteness Theorem, which says there are true statements about mathematics within any system which you cannot prove are true within that system. Now, that's extraordinary. Mathematics was turned on itself and was able to show it has limitations. There are things that are true that you're not going to be able to prove are true within any system for mathematics. I think that's amazing that mathematics has that power almost to talk about itself. That's why I think, one of my favorite books when I was a student — and it's still one of my favorite books — Gödel, Escher, Bach by Hofstadter, not a book actually about mathematics or Gödel or Escher or Bach or music or visuals. It's actually a book about consciousness. He actually uses Gödel's Incompleteness Theorem, the ability of math to talk about itself. He thinks that may be key to try and understand why the brain is able to think about itself. That maybe it has same level of complexity as mathematics has, that it's able to encode thoughts about itself just in the way that mathematics and an equation can actually mean just an equation about numbers, but it also might be an equation which represents a statement about mathematics itself. I think that's, for me again, one of the exciting things I learned as a mathematician, the power of mathematics to actually look in on itself and perhaps find its limitations.
SPENCER: For those that haven't really heard too much about the Incompleteness Theorems before, when you say in any sufficiently complex mathematical system, there are true statements that you can't prove are true. Can you unpack that a little bit? What does that really mean?
MARCUS: [laughs] Sorry, towards the end of your podcast, I suddenly start talking about Gödel's Incompleteness Theorem.
SPENCER: It's a great topic, but I feel like it's often thrown out and without... You've got a deeper lens on it.
MARCUS: Exactly. In fact, I spent the last year making a really cute little animation with Ted Ed's The Education about Gödel's Incompleteness Theorem. Do check that out. Mathematics is about taking axioms — this is where we all begin — the fundamental truths we think about what we believe are true about numbers. Then we use proof to see the logical consequences of that. The original hope was, I think, that any true statement about numbers — maybe you've got an equation and you think it has no solutions — that you should be able to write down a proof starting from the axioms. This is the journey from the Shire to Mordor. You start with the Shire as the axioms then you gradually go on this journey to try and reach every true statement of mathematics. What Gödel showed was... So when I say within a system of mathematics, that's the choice of the axioms that you're making and that will limit you in what might follow from those. But we thought we should be able to write down a set of axioms from which all truths of mathematics can be deduced. Gödel says that's a hopeless task, that there will always be things missing. There'll always be, for example, maybe an equation which doesn't have solutions but you can't prove within the system that it doesn't have solutions. Then the frightening thing is, of course, maybe that conjecture I've been working on for 15 years is an example of one of these problems that, yeah, it's true, but actually there isn't a proof within your system that it's true. One thought is, if it is true, why not just add it as an axiom and with Gödel's Incompleteness Theorem, everything will still remain incomplete. There's no way to complete the thing by adding another axiom. There's always still something which you're missing that you can't prove.
SPENCER: I've always wondered about the Incompleteness Theorem because it's really outside of the math that I've most studied, whether there are interesting true statements that can't be proved. There are true statements that can't be proved but are they the sort that we care about? I asked a mathematician this recently. He assured me that, yes, there are actually interesting ones that you can show that there are ones that are the sort of thing you actually want to know, like does this equation have a solution. Would you agree with that? That it's not just, these are really stupid statements, self-recursive statements of no interest?
MARCUS: The original statements that Gödel cooked up were things that mathematicians said. "Well, I'm not really worried about that." The challenge was, "Yeah, but maybe there are other things out there which we are worried about." The discovery over time was, you can cook up, within very valid system for mathematics, statements that you could very well be interested in wanting to know which are unproven within that system. We are now in a situation where we can write down statements which are really mathematical in nature, not just strange, contorted, recursive things that nobody would want to know whether they're true or not.
SPENCER: So this Incompleteness Theorem applies as soon as you get to a system of sufficient complexity. Do you want to just comment on that for a moment?
MARCUS: I think that's really interesting. Because it turns out that if you just have mathematics with numbers and addition, that isn't complex enough for it to be able to talk about itself. You really need to add in multiplication as an extra layer inside there, before it's rich enough as a language to start to encode statements about itself. One of the things Hofstadter conjectures is that maybe the brain is the same. That the brain of slightly simpler sorts of animals is like numbers with addition and actually isn't complex enough, hasn't passed a threshold to be able to encode within its language, statements which have meaning about itself, but maybe the human brain did pass that threshold. It would almost be like a moment where things are sufficiently complex that now you can do it. It's almost like an on/off switch in that case. So that's why I think he finds Gödel's Incompleteness Theorem very attractive as a model for trying to understand what is it about the moment a brain actually is able to formulate thoughts about itself and have self-awareness.
SPENCER: So maybe a slug has no notion that it is a thing that exists in the world but maybe a cat actually can think about the fact that it itself is a distinct entity and model its own future behavior. Maybe this is a turning point related to consciousness.
MARCUS: Yeah, exactly.
SPENCER: Does embedding a system into a larger system help get us out of the Incompleteness Theorem? Or does that not really resolve the issues?
MARCUS: That was the hope. Maybe we just haven't found enough axioms and you just need to put more things in. But it's a bit like the proof that there are infinitely many primes. You say there are finitely many primes, and you show that there's some missing and you add those, but you can just use the same tricks again to show you're still missing things. There are ways to exploit higher forms of logic, which — now we're gonna get quite technical — but Gödel's Incompleteness Theorem refers to first order logic. If you're allowed to quantify over sets of things, which is a second order logic, then I think that you can then make the thing complete. There is a challenge that Gödel's Incompleteness Theorem isn't actually the mathematics that we do, that actually, a lot of mathematicians are using quite second order thinking to be able to achieve talking about continuity and differentiation of functions requires. We never do that in first order arithmetic, but we're actually using already a more complex system which we're embedding the mathematics in. So some people argue that Gödel's Incompleteness Theorem is too restrictive in its conditions. That's interesting but pretty edgy stuff in mathematical logic.
SPENCER: Is this still up for debate? Or is it a settled question that there are these higher order systems that just don't have these kinds of problems?
MARCUS: I think that there are still things which we're exploring about these systems. So there are things you still don't really know. I think you mentioned, for example, the continuum hypothesis. Although we've proved that this statement is independent, there's still a lot of arguments going on about maybe the numbers that we're looking at, this axiom does apply to this, and we're trying to pin down what our numbers are, not these other sorts of numbers. So there's still quite a lot of research going on in these edgy areas of mathematical logic and set theory.
SPENCER: I think one thing we can say is you shouldn't fly an airplane whose flight depends on the continuum hypothesis.
MARCUS: [laughs] Yes. Just in case. Well, that might be a proof that it doesn't hold for our numbers.
SPENCER: Marcus, thanks so much for coming on. This was super fun.
MARCUS: Yeah. Thanks, Spencer.
JOSH: A listener asks, "How do you organize your knowledge? In other words, what tools or organization styles or formats and so on do you use?"
SPENCER: When I'm working on a single project, I will keep a Google Doc and every time I think of an idea related to it or encounter information, I'll throw it into that Google Doc. So I'll end up with this long Google Doc. Then over time, I'll clean that doc up by reorganizing it and moving stuff around inside it. But then I know everything related to that idea is in one place. When it's more like discrete pieces of information that I learn, or a fact that's interesting, or concept where I'm like, "I want to remember that," or I come up with an idea, but it's not related to one singular project, what I'll do is I'll add it to Thought Saver, which is this tool we made for helping people remember all the important things they learned and for helping them put those important things they learned into action. So I always just add a Thought Saver card. If you want to try it, you can download it right now. Just go to thoughtsaver.com, you can install our browser extension that helps you clip ideas as you're reading things on the web. You can also put your own ideas in there and we're working on a mobile app version too. I don't know when this will come out but maybe we'll even have a mobile app ready when you hear this. That is one place for all the ideas and it's searchable so I'll never lose those ideas that I put in there. Also, I'll be reminded of them regularly because Thought Saver has an automated spaced repetition component where it reminds you of what you put in there.
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