| 0:37 | Today is September 18th, 2025, and my guest is mathematician and author David Bessis. His book is Mathematica: A Secret World of Intuition and Curiosity. David, welcome to EconTalk. David Bessis: Well, I just wanted to say I'm delighted to be here. Russ Roberts: Great. |
| 0:56 | Russ Roberts: I want to start by saying that I hope to do a good job in this interview, like I do with all interviews. But if I don't do a good job, I want to tell my listeners right now that the goal of this interview is to get you to read this book. I've read it twice. That's a big number. And, by "reading it," I mean every word. I would have read it a third time before our conversation, David, but I ran out of time. I found the book electrifying, and listeners know I don't say that often. So, just in case we don't do justice to the book in this conversation, get the book. It's sort of about math, but that's not what it's really about. There's no formal math in the book; there's nothing intimidating. It's really a book about how our minds work and how we make sense of the world, which are topics that we spend a lot of time on here on this program. So, listeners, read this book. Okay. This book starts like a lot of books do, and I love books like this. It starts off like this: 'You know, math is really not that hard. It's made out to be this complicated thing, but let me show you how it's really not that hard.' And, you get all excited because you're going to get the secret: you're going to understand what math really is. And, you read chapter two, and it's a lot of throat-clearing and, 'Soon, we're going to show you,' and they never show you. This book delivers, and I, as I said, found it extremely interesting. Let's start by talking about what mathematicians do. Well, most of us out in the non-mathematician world, we know: Mathematicians are incredibly smart; they manipulate equations using Greek letters, or even more obscure symbols that you don't even know what the names for them are, and they prove things. And, obviously, they're incredibly brilliant, way smarter than the rest of us. But, you claim, quote, "The magic power of math isn't logic, but intuition." You also claim that mathematicians are just normal people. What do you have in mind? What's your argument? David Bessis: Well, I think they're both extremely smart and normal people; and that's the interesting thing. I think they started off being normal people, and then math brought them to where they are, and that's this journey that I wanted to describe. And, I think what you said about so many books about math start the same way: 'Math is easy, and I'm going to explain why it's easy.' When I was writing the book, I probably googled all the pop math books I could find, and they were all starting with the same promise. And, every time I read the page about a new book, I thought my heart was beating. I thought, 'Okay, someone wrote the book already.' And then, I went on Amazon.com, and I could browse the first pages, and I realized it was completely different. So, it's a book that I wanted to write; I've been wanting to write it since I was maybe 25 or something. I tried to write it and failed so many, many times. And, in hindsight, I realized that many mathematicians have also tried to write this book. And, that was, for me, the moment I realized I could tell the story--it's when I understood what they tried to say, why they could not say it, and why it's possible to say it differently now. So, this sounds very abstract, but let me make it really concrete. When you read Descartes--Descartes, in the 17th century, is saying, 'Hey, I have this crazy method that made me so smart. I was a regular person.' That's really--if you read the Discourse on Method--it really starts like that. 'I'm just a normal person, I'm not smarter than anyone. I wish I was smarter, I wish I had a great memory, but I just stumbled upon a method.' And, in many of his books, he talks about the method as being the method of mathematicians. And, the same story can be found over and over again in books--not in pop math books, but in books by great mathematicians. There is this absolute genius from the 20th century called Alexander Grothendieck, who wrote maybe 2000 pages on his journey as a mathematician, and he says the same exact words: 'I'm not gifted. I wish I was gifted. But I do something special inside my head.' And, I think being a research mathematician--I've quit mathematics about 15 years ago--but I'd been a research mathematician for a while. I loved it. But, I knew when I was a mathematician that what was really interesting to me was not the mathematics: it's that kind of meta-cognition that you have to learn to become a mathematician. And this is the topic of the book. What do you do inside your head when you become better at mathematics? The issue with mathematics is it's something that manifests itself in a horrible way. It's on paper, on the blackboard; you see cryptic symbols, formulas; and this is impossible to make sense of. But, how you interact with that--how you gradually tune your intuition to build up meaning for the symbols--is the real art of mathematics. And, because these things are inside your head, it's extremely hard to talk about. And, to me, the failure of teaching mathematics--and it's something that has been going on for not just centuries, but actually millennia--is the failure to admit that we do things in our head. We play with our intuition, we play with images, and these things have traditionally not even been discussed as being part of mathematics. So, it's interesting. When you speak with really good mathematicians, they just talk about that. If you go to a conference and you have a coffee conversation--a casual conversation--with people who [inaudible 00:06:54], they will talk about their intuition in a very fuzzy way, waving hands and making sounds, and joking, and saying they're confused, and all that. But, when you write a math book, you're not supposed to tell it; you're not supposed to say that. Because, this crazy, very human part, very confused part of playing with mathematics, is not supposed to be science. |
| 7:17 | Russ Roberts: And, you confess that even you, as a professional mathematician who has made significant contributions to formal mathematical theory, can't read a math book. Now, I'm a Ph.D. [Doctor of Philosophy] economist, so I had to read a little bit of math, although I was trained slightly before the era where math really took off in economics. But I had to learn some math, obviously. And, when you get to a formal mathematical proof in economics, what happened to me when I was in graduate school--which I never told anyone because it's too embarrassing--is that I would find my eyes glazing over, my mind turning off, and just skipping down and going to the end. And thinking, 'Well, what's the point? What is this getting at?' You're not supposed to do that, of course. You read the equations and figure it out, and parse it, and see if it's correct, and that's how you understand it. And, you confess in the book that you can't read a math book. Because I would just assume that, being a professional mathematician, you would just curl up on the couch with a fire in the fireplace and start on page one of a great treatise, or even just a simple 20-page formal published article in math in your field or near your field. Or even outside your field--because you're a professional and you would just read it the way I would read a novel. But, you confess in the book that not only is that not true for you, it's not true for most mathematicians. Explain. David Bessis: Yeah. Math books are not meant to be read. They're not books in the same way as a novel is. A novel is actually telling you a story using words that you can understand. So, you open up the book on page one; you read it; it makes sense. Math books are written in a certain way that follows a certain logic that is called logical formalism. It's a kind of recipe for building mathematical objects, but the words make no sense to you when you open them, so you can't read them. And actually--so, it's interesting because this assertion that I cannot read math books still sounds like a very provocative confession, and to the general public, it's perceived as being a provocation. But, actually, many, many mathematicians reason[?] the same exact thing. It's a very common knowledge among professional mathematicians that you can't read math books and they're not supposed to be read. They're kind of devices that serve a certain function. They are here to calibrate your intuition. It's something against which you validate your intuition. It's very like the telephone book--you can't really read the telephone book. But, when you need the number of a certain person, you can look at them. Or, the instruction guide for your vacuum cleaner or for your toaster--it's not something you read except when you want to troubleshoot something. The best way to interact with a math book is to start from why you want to learn something from this book. So you have a problem with it: there's something you want to understand. So, maybe it's a definition that is on page 205. And, this definition that is on page 205, or this theorem, or this proof that you want to understand--if you take math at face value, everything leading to page 205 is supposed to be absolutely mandatory, from a logical standpoint, to understand page 205. The thing is, by the time you get there, you will be dead because you read maybe one page a week, or one page a day, or something, and you will never have the persistence to get to page 205. So, you have to open the book right in the middle, try to understand what's written; and you will not understand it, but maybe you will have a faint idea. And, maybe this faint idea will prompt another question you have that will lead you to go to page 58. And, maybe on page 58, you will be happy enough with the kind of approximate understanding you have at this point and you will stop, or maybe you will go to another page. This trick about reading math books was something I learned quite late in my life. I was about 25, I think, when I was told that by a very good mathematician who was my mentor. I had my Ph.D. advisor, and he was a previous student to my advisor. And one day, I went to him because I had tried to read an actual research math book, and he told me that, 'You should not do that. It's bad for your health. Don't do it. Nobody told you that?' So, this thing, it's crazy. It's a secret of mathematicians--they all know that. After I wrote that in my book, I found maybe 20 instances of famous mathematicians writing the same exact thing. You don't read math books; you should not read math books; they're not meant to be read. |
| 12:20 | Russ Roberts: And, it raises--one of the reasons I find this piece of your book so mind-blowing is that I think everyone assumes that causation runs in the other direction. You play with some formal things--equations, theorems--and you prove a result. And then, you ask yourself, 'I wonder why this is true?' Now, you know it's true because you've worked it out carefully. And, assuming you haven't made a calculation error in the equations, you know it's true because of the power of logical math, formal math. And, now you have to say, 'I wonder what the intuition is behind this result?' And that's what happens a lot, or people pretend it happens, in economics as well, in the application of math to economics. But, what you're suggesting in this book, which is actually subversive to all kinds of things, is that actually you start with the intuition. And, some intuitions are true and some are not, so you check them. And, the ones that are true, you say, 'Okay, I got that one right.' The ones that aren't true--and this is the extraordinary, I think, deepest part of your book--you have to then say, 'Well, why was my intuition leading me over here when it's not true?' And, you can train your intuition to get better. But, that assumes that intuition is a superpower; and we're taught from day one that intuition is just thinking with your gut. It's not just not-as-good as so-called logical formalism: it's not thinking. So, you really put on its head the way we should think about how our mind works and how we should improve our mind; how our mind has the potential to be trained. David Bessis: Yeah. So, that's the central topic. Just a mild nuance compared to the conjugate: You do have to start with your intuition, and you use logic as a device to validate or invalidate your intuition, and then to correct it. But, sometimes, actually, it can be useful to use logic to explore a very tiny neighborhood of something you already know. Personally, when I was trying to prove new theorems, I did from time to time resort to making computations by hand. Russ Roberts: Of course. David Bessis: But, I know that after two or three lines of computation, there's usually a mistake in what I have written--if I don't understand it. So, you can try from what you already know to be true and play around with the formulas, and the syntax, and the symbols, and all that, but it will not get you where you want to go. It will just be a way to explore the very tiny neighborhood, a few microns from what you already know. But, you have to go from intuition to formalism and from formalism to intuition, and it is back and forth. The goal is to align the two. And, your intuition is super malleable, and that's a very important thing. I think that's actually one of the reasons why we were not able to talk about math in a proper way until very recently, when we had no idea about how the brain operates. But, when you see a deep learning network being gradually trained and changing its way to account for new data points, this is--well, I don't know exactly how the brain operates, but it's a very good metaphor for what happens in the brain. And, that's a completely different way--and you're right, it's really turning it upside down: you use logic as a device for training your intuition. And, a common trait of mathematicians is they have a very strong reliance on their intuition. They're happy when they discover that their intuition is wrong because they have learned to overcome the fear of humiliation you're taught to develop at school. At school, you enter the room with your intuition, and the teacher is telling you that your intuition is wrong; and you reach your conclusion that intuition is bad and that you're stupid. But, the thing is, it's wrong, but it's not going to be wrong forever. You will gradually evolve your intuition if you confront it with this very special apparatus that is logical formalism. So, you don't throw it away: you use it as a device. It's a kind of treadmill for your intuition. Again, it's something that sounds provocative if you look at the canon of how mathematics is told. But, when you have very small kids, you realize this is how they're building up their intuition of numbers. I have two kids. One is six and the other one is two. And, it's very interesting to see how they develop the perception of numbers--they really start that way. They start knowing a few numbers, but then they get confused, and then they check it, counting on their fingers from three or counting three objects. It's very complicated, it's very confused. That state of mind--of being confused about something but knowing it's not going to be the end game, that at some point you will develop clarity--I think every adult knows, has clarity about numbers, integers. Everybody understands integers, but it is something very abstract. But, for most of the history of human beings as a species, it was absolutely not stable, and it's a very recent thing that we've developed numbers that go to infinity, for example. And, I've never met anyone who thinks it's hard, or too abstract, or too complex, yet they did have to build up that sense. You're not born with that sense of counting to 1000; it's not natural. It's something you learn through a device. That example of a device was Hindu-Arabic numbers. It's a very advanced technology that was developed at some point during the Middle Ages. But now, it's standard, and everybody has acquired that technology. What happened with numerals, you do the same exact thing when you want to learn very advanced algebra--concepts in algebra, for example. |
| 18:50 | Russ Roberts: Yeah. I want to talk some more about intuition. So, we had Patrick House on the program talking about, I think it's Nineteen Ways of Consciousness, or Nineteen Ways of Thinking About Consciousness. He tells a remarkable story in there. He talks about a driver, a young man who has to drive a big shot, some kind of general or officer. And, they're in Iraq, and there's a real risk of danger on the road. So, they're going from one city to another, and they're going at a very high speed because they don't want to be on the road for very long. And, they get part of the way there, and the driver--this kid--slams on the brakes and turns around. And, the officers are horrified. 'What are you doing?' He says, 'I don't have a good feeling.' You don't have a good feeling? Was that a reason? I need to get to--" whatever town it was. So, they're driving back--they keep going back. And, at one point--I might get the details a little off here, but the gist is correct--the officer says something: 'Why did you have a feeling?' He goes, 'I don't know.' So, they get back. A couple hours go by, and the officer asks the kid three hours later, 'Do you know now why you turned around?' He goes, 'Well, I'm not sure, but when I think back on it, the road was really quiet. And there are usually children playing on the side of the road. And, something spooked me. I just had a bad feeling.' He says, 'Now, if I think about it logically, I think maybe the mothers knew there was something planned, and they didn't send their kids out because they knew it was dangerous.' So, there were no kids on the road. But it wasn't a logical thinking process. It was just intuition. When I tell the story, by the way, people always then say, 'Was there a bomb?' It doesn't matter; it's not the point of the story. But, that's a great punchline though--it makes everybody feel like, 'Ah, he was a genius!' But, House's point--and I think it's your point--you say it very poetically when you talk about neurons and how many we have, more than stars in the Milky Way. It's a really beautiful thing to think about the brain. The brain is doing a bunch of stuff in the background that we don't have access to. And, House's claim is that all the data of your whole life is in there, even though you can't pull it out necessarily. And, what you're calling intuition--it doesn't mean illogical or a gut feeling. Patrick House--he's a neuroscientist--he hates that gut part because he says, 'It's not near your waist; it's up in your head.' And, he said it's all the data that you've absorbed that your brain has processed it looking for patterns, because that's what we are. We constantly look for patterns. And, it's thinking; it's just not the kind of thinking we normally call thinking. And, that intuition is not just valuable--sometimes it's wrong, it leads you astray. You get spooked and there's no reason. But, it sometimes saves your life. And, the point is, it's not irrational. It's just a different kind of thinking. David Bessis: Yeah. It's essential, and it's still very hard because you're completely correct to say that the word intuition--there's kind of a red, blinking light that says: Okay, it's going to be nonsense. It's going to be some new-age nonsense, completely confused, completely irrational or something. No, it's not. I'm talking about rationality and how it should operate. Going again back to Descartes, who I think is a central figure in that theory[?]--he's the one who brought the word 'intuition' in modern science. He used the word 'intuition' to describe the clear idea of something, and he gave a definition of truth that is based on clarity. And, I think at that time, it made no sense, and he viewed it as a very mystical thing about God creating your brain in a certain way. I think what has changed now is we can make physical sense--physiological sense--of what intuition is. Intuition is really what is produced by the interconnection of the neurons in your brain, and it's obviously much more complex, and much richer, and much deeper than anything you can articulate with language. Intuition is not about language. Language is a tool to summarize and articulate some intuition, and to validate it using a very low-bandwidth framework that's called logic, that allows you to assemble a small number of very simple truths and to consider them into a new statement that is supposed to be true. But, the semantics of any statement--the semantics you attach to any--when you say something is true, when you say the Earth is round--you have to give meaning to these words, and this meaning only lives in your intuition. It's never on paper; it's never something that you can fully characterize. So, I think we actually function that way in our daily life. An example I give in the book is that of the banana cake recipe. I think nobody thinks that a recipe for a banana cake is something difficult and abstract. But, when you read a recipe, and they say, 'Okay, go and buy some bananas,' you can imagine the bananas in your head. You're at the supermarket and you're buying some bananas. Everybody can see the bananas in their head. And, when you're in the kitchen and step one is you have to mash the banana with a fork. Okay? And, when you imagine yourself doing that with your fork, smashing the bananas in plates, everybody has peeled the banana between step one and step two. It takes no effort to do that. So, the word 'banana' is attached to thousands of different images that you can produce. And, depending on context, you will switch from one image to another one. And, there's no way you're not going to do that. If you don't do that, you cannot live on a day-to-day life. You cannot understand any basic instruction about anything. You can't have a basic conversation about anything with anyone. And, mathematics is the same, except that you have to build up the images attached to different things. So, when you have a problem with numbers, maybe depending on what the problem is about, you will view numbers as counting oranges, or maybe about measuring a length, or maybe a surface area or something, and you have to go back and forth between different images. And, the journey of becoming better at mathematics is a journey of attaching richer, deeper, and more diversified semantics to mathematical abstractions. |
| 26:10 | Russ Roberts: Just to take another example from the book to try to make clear what we're talking about: How many points on a circle does a straight line touch? The answer, of course, is one or two. It could be perfectly tangent to the circle, in which case it would be one point. But, if it pierces the circle in any way, it could touch it once when it pierces and, two, when it comes out the other side. But it can't touch it in three places. Now, how do you know that's true? And, what's wonderful about that example is everybody--not everybody, we'll make the footnote in a second--but virtually everyone can see a circle in their head, as you point out. And, it's a circle, by the way, that is a mathematical circle. Unlike the Earth, which is not round--it's not a circle; it's not a sphere, even. But, when I talk about a circle with a line, you see a perfect mathematical circle, and as you point out, you can make it big, you can make it small, you can spin it around. And then, you put a line through it--you do that effortlessly in your head, almost all of us. And, you see right away it can't be three, two at most. Maybe one in some special case. How do you know? Well, because I can see it. Okay, but how do you know? And then, you'd have to prove it formally. And, that's what, quote, "real math," or what most people think of as math, is required to do. And, that is going to take, I don't know, a page maybe, half a page. It won't be fun to read, but the image is quite clear. And, your claim is that real mathematics is visualizing; it's imagining; it's playing with that line and that circle in your head. It's a simple case that most of us can do without much training. And, obviously, as you get more advanced in math, you're looking at more complicated things. Is that a fair summary? David Bessis: Yes. This example of a circle is the one I'm using when I give a conference on the topic. It's almost like standup comedy. You have an audience of maybe a few hundred people sitting and looking at you. And you say, 'Okay, can you imagine a circle in your head?' And, everybody's nodding, 'Yeah, I can do that.' Can you make it bigger or smaller? And, everybody says, 'Yeah, I can do that.' Okay, that's strange--you see things in your head, but not in the room. You see things in your head; what's going on here? Just to acknowledge that something here is going on. You see something in your head--it's already kind of weird. Nobody told you that this thing is going on, but you know it's been going on since you were in primary school; you see circles in your head. Except--we'll talk about that--but some people cannot do that. Russ Roberts: Yeah. David Bessis: But, it is a very small number of people. Russ Roberts: We've talked about that on the program actually, that there are people who cannot visually imagine things. And, I assume most of those people are not mathematicians. David Bessis: Yeah. So, the next question is, 'Okay, what about a line, a straight line crossing a circle? Can it cross the circle at three different points?' And, people--and I encourage the listeners to do that exercise. Try to think for a while. Can a straight line intersect a circle at three points? Maybe you need a couple of seconds. And, when I do that live with an audience they all say, 'No, no, no.' And, I say, 'Are you certain?' And everybody says, 'Yeah, I'm certain. I'm really certain it cannot.' What's interesting is: what makes you certain? And then I go on and tell them: 'Did you see a kind of cartoon in your head with a straight line sweeping across a circle?' That kind of visual argument of the straight line sweeping across a circle and making what you perceive as all possible ways to intersect, and actually it is fairly correct. It is convincing. But it is a non-verbal reasoning. When you do math, your intuition allows you to get to a certain conclusion even though you're incapable of articulating. If you want to write down the proof, you need some technology. You can use maybe Cartesian coordinates; you can use whatever techniques you want to prove it [?] you'll receive it. What math does here is that you have an intuition and you believe it is correct, and it makes sense to you. So, this intuition--in the book I very often talk about visual examples because they're easy to communicate. I should emphasize that intuition doesn't have to be visual. And maybe now we can come to the example of people who have this condition called aphantasia where they cannot really visualize things. So, when the first edition of the book was published, I got, on social media, a message by a reader who was aphantasic: he could not have any images. And, I asked to interview him to get his opinion about: is it obvious to him that a straight line cannot intersect a circle at three different points. And he said, 'Yeah, it is obvious to me. I cannot explain why, but it is obvious to me.' And, I think this kind of not-even-visual intuition is also a good example of what happens in your brain. Maybe for most people and for some type of mathematical problem, the intuition will recruit your visual cortex to assist with the computation. But, maybe for other types of mathematics, for example for probabilities, maybe you get a sense that is not really visual, but you feel it in a way that makes it obvious to you even though you cannot really explain where it's coming from. And, still, it is still intuition even if it is not visual. |
| 32:03 | Russ Roberts: I want to put this in the Kahneman System 1/System 2 thinking, which you do in the book. Explain why you think we should think System 3? Explain what System 1 and System 2 are in Kahneman's work, and then why you think it needs to be augmented. David Bessis: Yeah. So, I have to go back to the canonical example from Kahneman, about the ball and the bat. So, the central theory of Kahneman and Tversky is that there are basically two kind of modules in the brain to reach conclusion. One is System 1, is your instinctive answer when you say, 'Okay, what is one plus one?' Everybody says two. You don't really compute anything: you just know it. Is an elephant bigger than a mouse? Okay, you don't think. Yeah, it is bigger than a mouse. Now, if I ask you how many days ago were you born? Okay. You know how to do it, but you don't really want to do it because you would have to take a pen and paper and write it down, and you're going to make mistakes. Maybe you need a calculator or something. But, you know how it works, but you have to make computations. It's: you just don't know it off the top of your head. It's impossible. This is System 2. And, the theory of Kahneman and Tversky is that we are lazy and evolution made us prefer System 1 when it gives us an answer because it's easy, we don't waste any energy doing things, and we have a very fast answer. Kahneman gives the example of the ball and bat. So, you have a ball and the bat, and together they cost $1.10, and the bat costs $1 more than the ball. How much is the ball? They made experiments, and basically everybody says 10 cents. That's not correct because if the ball was 10 cents and the bat costed $1 plus 10 cents, the sum of the two would be $1.20, not $1.10. So the correct answer is 5 cents: the ball is 5 cents. So, I love this example because it's something experienced it in my flesh. I really felt it. This was crazy. So, a friend of mine was studying at Princeton and doing cognitive science, and she was visiting me and she was reading the book by Kahneman. She said, 'Okay, the ball and the bat, how much is the ball?' And I did not think: I just said, '5 cents.' That's obvious.' Russ Roberts: You cheated! You're a cheater! David Bessis: She literally became white. She told me that the guy had won the Nobel Prize for proving that nobody can say 5 cents without thinking. The intuitive answer is 10 cents. If you want to give a correct answer, you have to use System 2 and it will take you a few seconds. But you just cannot say 5 cents immediately without thinking. You're not allowed to do that. It's against science. You are wrong. And then she--it took her about a minute to say, 'Okay, okay, okay, okay. It's no fair again: you're a mathematician.' And, in a way, she's correct, but she may have underestimated what it means to be a mathematician in that example. It's not that I have a super-fast System 2 that enables me to make that computation incredibly fast. No. Actually, I think becoming a mathematician makes me worse at computations because I was relying less and less on my System 2. What was I doing? What does mathematicians do? What do mathematicians do all day long? They don't use their System 2, they don't make computation all day long. They hate that, like everyone else. They just hate computations. That's actually why they become mathematicians: because they want to avoid computations. So, what they do is they kind of get stuck in a kind of meditative flow, thinking, 'Okay, my intuition told me that is 10 cents, but the computation says it's 5. Why? Why did I give the wrong answer? What's wrong with my intuition?' And then, they try to see things a bit differently. And, they're gradually doing that, they gradually retrain their intuition to self-correct. The famous book of Kahneman is Thinking, Fast and Slow. And, if I had to rewrite it, I would write it Thinking, Fast, Slow, and Super Slow. And, the Super Slow mode of thinking is that of the mathematician. I call that System 3, is the following: Whenever you catch your intuition red-handed being wrong at something, don't throw that away. Don't reject the intuition--Freud would say that you reject and suppress. Don't suppress that intuition. Explore it. Try to unpack it. Try to understand how did your intuition--what's in your intuition? How does it maturize itself? What does it evoke you? And, doing that, try to identify where it's wrong. And when you put words on that, and when you play back and forth between your intuition and formal logic--between System 1 and System 2--do back and forth until they agree. It may take you five minutes, one hour, a day, a week, a year, 10 years, 50 years--it depends. There are some things that you can resolve in 10 minutes, there are some things that will nag you for years and years; and you don't understand it until you understand it. And that mindset of never giving up on your intuition is a secret for being just good at math, but actually extremely good, because if you continue exploring your intuition and trying to locate where it's incorrect using the technology coming from formal logic, then there's basically no limits; and that takes [?]. |
| 38:30 | Russ Roberts: And of course, this has application way beyond mathematics. It has to do with how we look at the world as social scientists, which we all are. Even though we might not have a degree in economics, or sociology, or psychology, we're constantly taking the data of the world around us and trying to make sense of it. And, I find this, one of the watch-words--there's two things I think that are at the heart of this program. One is it's complicated. Often, there are things that you haven't thought of. And, the second is you have to be able to say 'I don't know' when you don't know. Because, if you can't do that, you're going to limit, you're going to ruin your chance to improve yourself. And you have a remarkable story in the book of: you're giving a seminar to some other faculty members and graduate students, and suddenly in walks Jean-Pierre Serre, who is one of the greatest mathematicians of the last hundred years, and he sits down in the second row. As an academic, a former academic at least, this impostor syndrome that all of us have to some degree--this fear that we're going to be discovered as not being as smart as people might think we are--it's a terrible, terrible disease. Tell what happened at that seminar and what you learned from it, because I think it's incredibly valuable. David Bessis: Yeah. So, having Jean-Pierre Serre walking into your seminar is both a great honor and something absolutely scary. And actually, by the way, Jean-Pierre Serre just turned 99 three days ago. When you say the last century, he was the last century by himself. He is the last century by himself. So, one thing I knew about Serre is if he takes his glasses off, that means you're dead. That means he's not listening anymore because it's boring, and he's showing you that it's boring. Mathematicians can be rude in their own way. They never pretend to be interested by something they're not interested in. But, I saw that he was listening to my talk, not because it was a good talk, but because he has a keen interest in the topics I was working on. So I was just lucky to be in his sweet spot in terms of what my similar talk was about. Russ Roberts: And, you claim in the book that you did not change your talk to reflect his presence-- Russ Roberts: Because there'd be a temptation to jazz it up and make it fancy because you want to impress him. But you left it as it was. David Bessis: Yeah. Actually, there's more to that. Whenever I had the chance of attending a seminar given by Serre, I was always impressed by the fact that it always looked, like, super easy. Everything was completely transparent. When you get back home and you try to do it by yourself, basically you just fail to do it. So, he's kind of a magician in the way he presents things. He has a very, very simple way. Every piece is fitting exactly into the next piece with no effort, but actually the architecture of the talk is absolutely fabulous. So, my seminar style when I was giving math talk was: try to make things look easy, and then maybe people will think that you're super smart. If your talk looks super easy, it's actually a signal that you might be very good at math. So, I was trying to over-play that signal, so I always try to give talks as simple as possible. And my benchmark in that very seminar talk was that there were some students at the back of room, and I was looking at them, making sure they were listening until the end. It was a seminar talk on an ongoing research that was far from over, so I did not have many theorems. And I viewed it as a very basic talk compared to my average seminar level. It was a talk with not much new information, trying to make things very easy, very simple, and everything. And, he listened to the end. It was a very long seminar. I think it was at Séminaire Chevalley[?] in Paris, and it was, like, 90 minutes. So, you're just by yourself with a piece of chalk in your hand for 90 minutes. And, you get his attention for 90 minutes. And then at the end, the talk is over, he walks to you, and he says, 'You will have to repeat that again because I did not understand a word.' And that blew my mind. He was over-playing it, of course. He had understood much more than a word, but there was something he hadn't understood. So, the meaning of what we put under the word 'understand' depends. It varies from one person to another. And, I think that for him, saying that he understands something is not just that he understands the words, the statements, and the logical flow. It is that he really understands why it should be like that. And, I think that was the meaning of his question. There are things that are true for reasons that you don't fully understand: you are able to prove that it's true, but it's kind of weird; it should not be true. You don't get it. You don't get why it really works. And, that was the meaning of the question, but I was impressed by not just the fact that he confessed that he did not understand it, but that he confessed in such a brutal way. And, my first reaction was to think, 'Okay, he's kind of showing off because he's Serre.' And when you're Jean-Pierre Serre, you have a right to walk up and say, 'I don't understand a word,' and that's fun, that's provocative, and all that. And then, I thought maybe it's the other way around. Maybe with that attitude, you can become Jean-Pierre Serre. And I tried to practice it. So, it's a very common thing when you go to a math conference. A math conference--typically about 50 people coming from many different places around the world. You get people from the United States, people from Japan, people from Europe meeting at some small place: usually it's on the mountain or by the seaside. It's very remote. Mathematicians are a bit like hermits: we like to be isolated from the rest of the world. So, you go to that place and you live together for about a week on a very specialized topic. When you're at lunch or you're in dinner, you sit with all the mathematicians. Usually they say, 'Okay, what are you working on?' The other one will say, 'Okay, I'm working on the theory of'--I don't know--'algebraic reductive,' whatever weird thing, 'co-homology of that kind of varieties.' Weird things. And usually, you don't understand a word of what they're saying to you. And, you're embarrassed. Because, when you're a professional mathematician, it's not a pleasant thing to confess that you don't understand a word in what your colleagues are talking about. So, you usually pretend to understand a bit, and then you shift the discussion to another topic and you end up not discussing math. And, that was the way I had lived it for the first years of my career. And I was already an established researcher when this happened to me. And, I realized, 'Maybe I can do the Serre trick and try to say, 'Okay, I don't understand a word. Please repeat it to me.' And, I did it on the first occasion; and that was actually fantastic. Because, the guy who was telling me his research topic was--I don't remember if it was a student or a post-doc. But, he was younger than me. So, he was kind of looking up and trying to make it look hard, and deep, and profound, and abstract, and conceptual. So-- Russ Roberts: Fancy-- David Bessis: That was what I call the tourist menu. He was showing me the tourist menu with the kind of very formal dishes that nobody really wants to eat, but they look like it's a fancy place. And, he was presenting his research mathematics the same exact way. And, when I told him, 'Please repeat. Repeat it as if I had some disability, as if my brain was damaged. Because I am jet-lagged, I'm tired, whatever, I'm stupid, I'm a slow thinker. Please be as simple as you can, I don't understand anything.' So, when I did that, I gave him the permission to serve me the true menu, the thing for the locals--how he really himself looked at his mathematics. And he was using different words and describing the situation using very simple images, examples. And it was a different thing. And, I had given him permission to be simple; and I've given myself permission to ask more and more questions. This situation of not understanding something in mathematics and being embarrassed because you're supposed to understand it, it is, I would say, the standard of mathematical conversation now. And I think everybody who has interacted with math has been in that situation--of being too embarrassed to confess that they don't understand anything. Where it just starts today: it's today that you stop, you say, 'I don't understand anything.' Just like Serre, who is doing the same thing. That's the way the greatest mathematicians work. And again, after my book was published, I saw maybe 20 different examples of great mathematicians making the same exact remark--that they ask stupid questions. It's super important for them to ask stupid questions. I mentioned the example of not reading books being a kind of secret, and asking a stupid question is a kind of secret. It looks too easy, it's not credible, but it's actually absolutely essential. Russ Roberts: Yeah, it's a really beautiful thing. |
| 49:07 | Russ Roberts: I want to talk about one of the themes of the book that's also just incredibly important to me and my own way of thinking--is the imperfection of language. And, having moved to a foreign country four years ago, I think about it all the time because translation is--every time we translate something from English into Hebrew, my Israeli friends say, 'Well, that's not a good translation.' And they assume it's Google. And I sayd, 'Well, actually we asked so-and-so to translate it.' 'Oh, that's not [?] real Hebrew.' And of course, what is that? Is it impossible to translate? If it happened once, you'd go, 'Oh, I used a bad translator.' When it happens over and over again, you realize there's a certain non-transformation between two different languages. And, it could be that there are languages that translate well from English to other languages, obviously. And, it could be that some languages don't translate well with anything--that they have their own unique syntax. I'm not a student of language. But this theme that language is inherently vague, ambiguous--it bothers us as human beings tremendously because we like to think of ourselves as rational. We only have language. What I see in my head when I see the circle, I assume it's what you see, but to verify that is extremely difficult, and that's one of the subthemes of the book. But, I want to get at a different aspect of seeing. So, so far, we've been talking about seeing as visualizing, a visual image; we were talking about some people who can't do it. But, intuition, of course, is not just the physical visual image in your head. And, I think a lot--you didn't write about this, but you could have--I think a lot about Andrew Wiles. So, Andrew Wiles proves Fermat's Last Theorem--the most famous unsolved puzzle in mathematics for centuries. He makes the front page of The New York Times. Most mathematicians never make the front page of the New York Times no matter how long they live, no matter how--I suspect Jean-Pierre Serre has never been on the front page of the New York Times. Or Grothendieck. Well, Descartes maybe. But, most mathematicians don't make it. So, he's lionized. He's a hero. He gives this proof at a conference like you're talking about. The crowd--there's this electric atmosphere because it's a secret, he hasn't told more than a couple people--and there's this incredible electricity afterwards. And, he's, like, on top of the world. And then, they get someone to check the proof, and it turns out--and there aren't many people who can check it, and those people have to suffer. They're reading a long math proof that we talked about earlier that they don't want to do. I'm sure it's not fun. Somebody--maybe it's fun. But anyway, they find a mistake. Okay: well, it happens. He'll fix it. He can't fix it. Weeks go by, months go by, and there's a documentary about this, and it's one of the most moving things I've ever seen in my life. He reflects on what happened. He said, 'I was sitting in my office.' He's probably close to potentially suicide because his life has been ruined. His family is falling apart, his wife must be going crazy. He's probably only thinking about this one thing over, and over, and over, and over again. And he says, 'I was sitting in this chair,'--they're filming him saying this--'I was sitting in this chair, and then I just saw it.' What did you see? 'Well, I saw that Kakutani's conjecture would fit with so-and-so.' He just, quote, "saw it." And of course, I don't know what he's talking about except there's something mystical, and beautiful, and human. And he runs downstairs, and he says to his wife--I think he says something like, 'I got it.' And she thinks, 'What, the groceries, the dry cleaning?' And she can't believe it. And so, that moment of 'he saw something'--whatever that means. I don't know what that means. And you talk about it, this phenomenon, in the book in many ways. I'd love for you to tell the story of Hardy and Ramanujan because it's an extraordinary version of this. But, reflect on what this is. What's going on there? As a professional mathematician, what do you think is happening there in the brain? What is that? David Bessis: The first part of your question and the second part: As a mathematician, what is happening in your brain? As a mathematician, I'm not supposed to be competent about neurology, but I do have a very strong personal history of experiencing that kind of epiphany where something crystallizes. So, I do have to elaborate my own idea of what's going on in my own brain, but I have no proof from the scientific standpoint of if this is really how it works. But, I have no other model of that. I do think that things get connected. The metaphor for that is viewing the neurons, of having the synapses being really configured in a different way. Sometimes it happens at very large levels, like two things that were not connected. Most of my ideas in mathematics came at the moment where two things that were split, living in different worlds, you realize that they kind of work together and you can connect them. It usually takes a while, and probably the more distance, the longer it takes. I think that what is special about mathematics is when you get an idea like that, you can prove it. And this is where the device of formal logic and all that comes into play--you can prove that you're correct. But in day-to-day life, we experience similar things. I think everybody has woken up one day and thinking something new or realizing something, you know: 'I'm going to do that in my job; I'm going to do things differently.' Or, 'I'm going to change this in my life.' Or, 'I understand that this person has actually this attitude towards me, and I was wrong about what this person was.' We're like that every day. What's special about math is you can prove that your intuition is correct. So, it makes you rely more and more on your intuition because you have that kind of feedback loop of being reassured that your intuition is not just crap. It's okay it was wrong at the beginning, but if you educate it, it becomes better, and better, and better, and better. In your example of Wiles--another thing I want to point at because it's something that is rarely discussed: When people talk about mathematics being the science of logic and deduction, all that, and not taking seriously what's going on at the intuitive level. If mathematics was really a logical game the way people have told for centuries, then the story you just told about Andrew Wiles would not be possible. Why is that? Because if the proof was wrong, the proof was wrong, and there would be nothing for it. But, the thing is: His first proof was wrong, yet it was directionally correct. And this notion that it's directionally correct is a notion that is impossible to define in logical terms. It's either true or false, right or wrong. There's no directionally correct in mathematics, if you take it using the legacy definition of mathematics. And, the fact that there are theorems in mathematics that are wrong all the time, but it's no big deal because they're directionally correct and we fix them after that--it's a very common pattern. And, I think just the existence of this pattern in the real life of academic mathematics is proof that the semantics of mathematical objects are not reduced to logic. |
| 57:18 | Russ Roberts: Talk about Hardy and Ramanujan. David Bessis: Yeah. So, Hardy and Ramanujan--they looked like characters from a comedy. It's impossible to get people like that in real life normally. So, Hardy was a very typically highbrow, high-class mathematician from Cambridge. And, he had a very big Chair, in Cambridge, at the beginning of the 20th century. And, he was one of the prophets of the Formalist Revolution at the beginning of the 20th century. He was a friend of Russell, and he was advocating the reformation of mathematics to go towards more logical, formal truths. And, one day, he received in the mail a letter coming from India from a guy he had never heard about--Ramanujan--saying, 'Okay, I'm just a poor'--it really looked like a Hollywood script--'I'm just a poor employee in India. I've learned mathematics by myself. But I proved a couple of theorems and I showed them to the local guys who do mathematics, and they told me that it's interesting. And, what do you think about that?' And then they were like, crazy theorems--Hardy could not prove them--but they looked true. Not just true, but deep. And, he was thinking, 'Okay, they must be correct because if they were not correct, nobody would be able to imagine such crazy theorems.' So, he did something that is kind of incredible for the time--he invited Ramanujan to Cambridge. So, this is a story that is both very intense and very short. Ramanujan just spent a couple of years in Cambridge before returning to India because he was sick, and he died very young actually. But, over the course of a few years--I don't know if the word proved should be used here, but he stated thousands and thousands of "theorems." And we should put scare quotes around 'theorem' because he was basically taking a piece of paper and saying, 'Theorem,' and then a crazy formula, and that's all. And, when Hardy, who was a very formalist-style mathematician, asked Ramanujan, 'How did you get there? How do you prove it?' Ramanujan said, 'Oh, I don't have to prove it; I just saw it in my dream, and my goddess told me that in my dream.' There's an account that I found after writing the book, and I would have used it in the book. Ramanujan is actually describing one of his dreams, saying, 'There was a curtain of blood dripping from the ceiling, and there was blood everywhere, and then some hands started to write down the formula on the screen of blood.' That was the way his ideas presented themselves to him. And, so what does it mean? Of course, he's telling the story in a certain way probably because he doesn't really have the words to express what's going on in his mind. But, he's not a mutant. He's not coming from another galaxy. He's a human being. And, there's a famous mathematician called Misha Gromov, and I quote him in my book because he said something very interesting about Ramanujan. He said--it's not the exact sentence, but it's really the spirit of the sentence, I'm just inventing the quote, but you can find the actual quote in my book--he said, 'If you want to think that there exist people with mystical powers who were kind of coming from outer space and completely alien to the normal human species, then Ramanujan is perfect for you.' You can think of him that way. This guy invented crazy theorems; they took 50 years to prove them, and these were just appearing in his dream--it looked like it's not human. But, on the other hand, it could also be an actual person. It's a much more [?] way of looking at it. And probably the way these theorems appeared in his brain--and they were proved correct after decades of work--is the sign of something universal that every human being is capable of. It's probably the same mechanisms that children use to learn their native language. And I'm really on the same page. I really think that there is something to learn from Ramanujan. Unfortunately, we don't have the details. We don't really know. He never wrote down what happened in his brain. The most important math class I took was when I was studying at the École Normale Supérieure. We had a course by a mathematician called Vigneaux. And, the goal of the course was to start with a formula from Ramanujan and try to find a way to make it intuitive. And, the course was--it was using some kind of dominoes, multicolor dominoes that you were piling, making piles, he was using certain ones. And gradually, we were learning to translate certain algebraic formulas into pictures involving these dominoes. And, at the end, after six months, we had built up a kind of visual intuition for the different components. And, this very reason to[?] one among thousands, but this very reason became obvious. And, if some brain is able to produce something, the biology of different brains is actually quite similar. It opens up an entirely new conversation about what explains the diversity of cognitive abilities between people. But, at least at the basic level, the biology of a given brain is very similar to the biology of another brain. So, if one brain is capable of doing these creations, there must be a way to kind of emulate the same approach--maybe not with the same results, but at least using your brain in the same way. |
| 1:04:14 | Russ Roberts: Yeah. So, I mean, you have a number of examples--I won't spoil the book; people should read it. But, you give a number of examples of things. Well, I love how you start it. Early on you say, 'Just because a two-year-old is babbling, you don't say: Well, I guess he'll never learn to speak, but we just won't bother teaching him language because he's not good at it.' And yet, we do that with math. We say, 'Well, he's not a math person. He's not good at math.' So, it raises the question--which is another book; you didn't write this book--but it does raise the question: I'd like to hear the first paragraph of this next book. Which is: If you were teaching--well, you're teaching a six-year-old and a two-year-old. I taught my children a ton of math and probability and economics before they got to school because I couldn't help myself. I suspect you're doing some of that yourself. But, if you think about a formal curriculum: We do it now, and there's some variety in how we teach math at the K-12 level, but it's pretty similar. There's some innovation, but it's pretty tedious. And a lot of math education is training people to work through recipes without the intuition--so, you never peel the banana. It turns a lot of people off from math; it convinces them they're stupid. If I said to you what should we do differently, have you thought about where you'd start? David Bessis: Yes, of course. And, I was forced to think about it because I have kids. Russ Roberts: Yeah. David Bessis: So, again, before I say anything, a very big disclaimer: I'm experimenting with that and I never ran any control group experiments on what works and what doesn't work. And, an issue for doing that is: I think it's very long-term. I think the mental habits you have to start to become really good at math are things you develop over years, and years, and years--is not the case. So, one thing I've been trying with my own kids is try to develop the bridge between intuitive thinking and verbal thinking. Because in my experience as a mathematician, the hardest thing was to cross that bridge back and forth between the two. And, you get used to that and you get better at it, but I remember when I was a young mathematician, I found it very hard. So, I try to develop that bridge. And, a very stupid aside that doesn't look like it has to do with mathematics is asking my kid to say his dreams. 'What did you dream about?' The idea that we should remember our dreams and be able to tell them is an idea that has been associated with therapy, with Freud, with the subconscious in kind of all definitions of subconscious. But, actually, it's a very practical exercise that I think is extremely valuable as a training exercise for your language abilities. Being able to describe objects that you saw only in your head is, I think, a very good preparation for mathematics. Why is that? Because it's a way to force you to standardize mental images. To increase the clarity. And you can actually measure that. If you do that exercise of trying to write down your dreams--maybe you don't dream--like, people say, 'I don't have imagination.' People say they don't dream. They do dream, but they forget their dreams. And, the more you practice that exercise of writing down your dreams, even if it's just a faint idea when you start, if you practice it, you get better at it. Actually, I did that at a moment in my life: I started to write down all my dreams in detail. And after a few weeks of doing that, it was so, so detailed that I had to give up because it was overwhelming. So, I think there's something about training your intuition. Same thing--when something is wrong in what they're saying, instead of saying, 'It is wrong, this is the correct way,' trying to articulate the bridge between how they see the world and what is correct. Another exercise I did, which is not supposed to be mathematical but I still think that it's training the mathematical ability, is try to point things in space. Where is the school? Where is the entrance to the apartment? In 3D, not seeing something. So, it's always play with your imagination. For young kids, I think that's the best preparation. And, I have a theory about how people become really good at math. I think something happens in their early childhood that makes them more inclined to play with their intuition, play with their imagination, try to replay things they saw, try to assemble things in their head. When you do that on a regular basis, you develop a very strong visual intuition of things. And, I think years later, it will show up as mathematical skill. |
| 1:10:00 | Russ Roberts: So, that's lovely. I love all those. And you have some examples in the book of ways to use your imagination, and you talk about fiction, which I think--all these things, I think--are underrated. But, I'm thinking about this sort of standard stepwise progression. In American high schools, it's: Algebra, Geometry, Pre-calc, to Calc. That would be the sort of standard way. You get asked this question--my wife's a math teacher; she teaches high school math. She's a fabulous math teacher and she emphasizes a lot of intuition. And, she has, of course, to deal with the question that you talk about in the book--that people always say, 'Well, what's this good for?' And the standard answer is--there's sort of a couple standard answers. One standard answer is, 'Well, you might want to be an engineer. And if you're an engineer, you have to be able to do calculus, so you better learn geometry, and algebra, and pre-calc, and trig.' And, of course, some people's answer is, 'But, I'm not going to be an engineer, I'm pretty sure. So, can I do something else?' And, we don't let them in America do something else. So, the second answer is a sort of vague, 'It's good for your brain. It just keeps your mind active.' And, your claim is very different. Your claim is it's good for your brain in a different way. It's not just that you learn a bunch of stuff you didn't know, or you use your brain to figure out things that are really hard. It's that you learn how to use your brain, and that's exhilarating, actually. So, do you have any thoughts on what high school teachers should be doing in their math classes? David Bessis: I think they should be bold. They should say, 'It makes you smarter.' People hate math because they view it as an IQ [Intelligence Quotient] test that they're failing. Russ Roberts: Yeah. David Bessis: And, it's not a test. It's a technique to get smarter. If you're failing, it's normal because you start. When you start any new sport, you suck at it. There's no way you're going to be good on Day One. If you try to surf, you will fall. You will not be able to surf on Day One. Or maybe at the end of Day One, but not at the beginning of Day One. So, you cannot teach mathematics to kids who are convinced that your mathematical ability is something that is static. It's like swimming: if we were telling kids, 'Some kids can swim, the other ones drown. You will see--it's genetic. Let's see how you fare on Day One and then you will know for the rest of your life.' If you do that, there would be a riot in the swimming pool, and nobody would be able to swim. But, this is exactly what's happening with mathematics. A combination of confusion about the nature of mathematics, and confusion about how the brain operates, and confusion about the origin of the shocking gap of abilities that are visible on a given day in a given high school makes us believe that this thing is entrenched and you're not going to be able to change it. But it's not true. You develop your mathematical abilities over time. You need to insist; you need to persist. It's not going to be an easy ride; it's going to be very difficult. But, that's the whole point of mathematics. Mathematics is a technique that, if you learn how to master the technique itself, you will develop your intelligence; you will utilize your brain in a way that you would not be able to otherwise. And, the benefits, I think, are quite sexy if you look at them. It makes you smarter; it makes you more confident; it makes you more articulate. These are very good selling points. We've not been using them enough, I think. |
| 1:13:59 | Russ Roberts: So, I want to close with rationality. I'm going to read two quotes from the book. They're a little bit long, so, listeners, bear with me. These are both quotes from the book. You've always known there's a problem with rationality. It's supposed to be the basis of our civilization. At any rate, that's what they tell you in school. We're taught to organize our ideas in a logical and structured manner. We're taught to distinguish between a reasoning that's valid and one that's not. We're taught to discount what isn't logical, rigorous, coherent. Of course, no one's stupid enough to believe this story--we just pretend we do. End of that quote. Here's the second quote. And, this quote is about Descartes, and I learned a lot about Descartes that I didn't know, from your book. I bring this up, by the way, before I finish the second quote: There's an enormous debate in our world in this moment--and it's, I think, a very tough moment for the world--about rationality. And there are, I think, fundamental misunderstandings about what it means to be rational and what it means to be scientific. And, I think your book takes what I would call a nuanced approach--that it's complicated. And it's not a popular approach, but I just want to say I appreciate it. Here's the second quote: Descartes missed an essential point. All reasoning, even the most solid, ends up coming apart the further it gets from day-to-day experience. Not for lack of rigor, but because our language itself is built on a base of sand and mud. The only exception is mathematical reasoning when it is articulated in the official language of mathematics. If this artificial language is so inhuman and so incompatible with our usual way of thinking, it's for a very simple reason--its bias is to be compatible with logical reasoning. When we want to venture far from our everyday concrete experience, logical formalism helps guide us. It's the only tool at our disposal that lets us give free rein to our impulse toward rationality without limits, without complexes or taboos. Outside of mathematics, rationality remains under constant threat from the fragility of our language and our way of perceiving the world. End quote. You could just applaud because it's your quote, but if you want to add anything, expand on it, I'd love to hear it. David Bessis: Yes. It would take--I'd be happy to come back on the podcast to just discuss rationality because I think, in itself, it's a fundamental topic. The official discourse on mathematics is wrong, and the official discourse on rationality is wrong for the same exact reason--but the consequences are quite different. Rationality is not a foundation for truth. It's a technique to consolidate human truth, human language. So, you cannot be on the side of rationality, except from a methodological standpoint. There's no absolute truth that is rational. It cannot exist. Rationality is an approach to articulate different ideas together. Let's put it like that. In a certain way, rationality is the Western tradition of meditation. It's a meditation technique of the Western tradition. It's a way to align different ideas that flow in our brain, but at the fundamental level, we are not rational machines. We are intuitive machines with perceptions of the world and very weakly correlated language that is trying to express things about the world. Rationality is a framework for making all of these representations and all these statements about the world more correct--for gathering more information and making it fit together. But, in itself, it's not a replacement for human intuition. It's a device to enhance human intuition. And, everything we say and think about the world is processed through an intuitive machine. And, talking about rationality as if it were the enemy of intuition is a fundamental misunderstanding. And, this misunderstanding is caused by our inability to accept that we're intuitive machines. Russ Roberts: You left academic life and got involved in AI [artificial intelligence]. We've done over a dozen episodes on various aspects of AI, mostly about whether it's going to destroy us or help us, what it's going to do to our humanity. What's your thinking, having been in that world? Where do you think we're headed as human beings and as intuitive thinkers? In the last few months, I've changed my thinking about it. I'm not going to talk about it now--listeners will hear it as it evolves and we continue to talk about it--but my thinking about what it's capable of and how it's going to be used has gotten more--I've gotten much more pessimistic, is how I would say. I may turn out to be wrong; it's just my thought today. But, you're much deeper into this than I am. What are your thoughts? David Bessis: Wow. It's not an easy one. Russ Roberts: Obviously, you've got five minutes. David Bessis: Yeah, I have five minutes. Okay. It's going to be a mess. Whether it's good or bad, in any case it's going to be a mess. Because, we still have to overcome very big, basic limitations of all scientific tradition, basic social organization issues, fundamental political, geopolitical issues. Everything is going to collide together, so it's not just AI in isolation, unfortunately. It's AI plus the mix of issues we're facing at the same time. I do think that a number of things that were considered to be the human privilege are going to be taken away from us. And, it's hard to view this as just good, that's for sure. On the other hand, I don't see how we'll be able to do science without AI in the near future, if not the present. So, I see it as something completely unavoidable for which we're completely unprepared. But it's going to take place. Russ Roberts: My guest today has been David Bessis. His book is Mathematica. David, thanks for being part of EconTalk. David Bessis: Thank you very much. It was a pleasure. |
What if math isn't about grinding through equations, but about training your intuition and changing how your brain works? Mathematician and author David Bessis tells EconTalk's Russ Roberts that the secret of mathematics isn't logic--it's the way we learn to see. He explains why math books aren't meant to be read like novels, how great mathematicians toggle between images and formal proofs, and why we need a third mode of thought--"System 3"--that patiently retrains our intuition and the power of imagination. Bessis and Russ Roberts swap stories about the humility of great mathematicians, how Andrew Wiles "saw" the fix to his proof of Fermat's last theorem, and Ramanujan's dream-revelations that proved true.
READER COMMENTS
SK
Oct 27 2025 at 2:05pm
Is the author of the book perhaps discounting a possible fact that just like great athletes have a genetic predisposition to excel at sports, mathematicians might be blessed with a math gene?
Bob
Oct 29 2025 at 3:33pm
Given all the things that now we know have genetic components, it’d be unsurprising if there was one. Still, just like in sports, it’s one thing to need the genetic component to be at the top of the field, and another to be able to train yourself to have some capacity to do math.
Paul Clapham
Oct 27 2025 at 6:41pm
You touched on aphantasia (the inability to visualize things) in your interview, and left it quickly, leaving the reader with the impression that aphantasics can’t do math. Well, I’m aphantasic and I’ve been a mathematician since I was a child.
When you ask me if a straight line can intersect a circle in three points, it’s obvious to me that it can’t. Sure, I can’t see a picture of the circle in my mind with a straight line moving about, but I know what a circle looks like. So I don’t need a picture of this question to answer it.
So when Grothendieck said he does something special in his head, I totally believe him. I’m just unconvinced that what he does requires seeing pictures of things. For me, numbers were always my friends and I knew their features and their relationships. No pictures required. Likewise take, say, group theory, which starts out discussing symmetries which can be visually appreciated but almost immediately go ahead with abstractions which don’t need pictures. I don’t know what special things happen in my head either but they don’t include seeing pictures of things. I do know that when I see text, I read it automatically without thinking about it, and it seems to me like most people don’t do that.
Bob
Oct 28 2025 at 9:07pm
Computer scientist here, and I agree: Aphantasia just means your “visualization” isn’t all that attached to the inputs you get from your eyes. And, if anything, it’s a superpower for many kinds of math, because the intuition trying to look at something in 2d or 3d space is actually detrimental if you are dealing with higher dimensional math.
Squinting a bit, modern AI is built on gradient descent, which a numerical method what you can do to find minimum and maximums of functions. But in a modern AI model, we aren’t finding a point in a 2d space, or a 3d space, but in a space with billions of dimensions.
Those used to regular visualization will have to throw it all away to figure out, say graph theory. Or go into something like monads, where there’s nothing to see. This leads us to a lot of silly explanations “a monad is like a burrito” Everyone tries to explain how they got to train their intuition to be closer and closer to right, but ultimately few can learn from the visualization taught by others.
Eventually math is abstract, and your eyes just harm.
John Hall
Oct 29 2025 at 11:08pm
As someone with aphantasia who is good at math but not a mathematician, I also found this among the more frustrating parts of the interview, especially since I just read the New Yorker article on aphantasia that Tyler linked to today that talks about how one of the earliest investigations into the phenomenon involved basically asking scientists and mathematicians about visual imagination and all of them replied that they didn’t have it.
David Bessis seems to argue that visual imagination is essential to being a good mathematician, but then acknowledges that people without visual imagination can do math pretty well. And yet goes on to encourage people to practice visualizing to get good at math. As he says, there might be something else going on that builds the intuition that is different from visualization.
Ideally someone would do some kind of modern survey that gives a sample of professional mathematicians the test for visual imagination (and compare to a control).
David Charlton
Oct 28 2025 at 12:22pm
You should listen to your past interview with William Duggan on Strategic Intuition, 2007. William Duggan on Strategic Intuition – Econlib He covers the role of intuition in a number of social and academic contexts.
Paul Mantyla
Oct 31 2025 at 12:56pm
This episode reminded me of a quotation by John von Neumann: “In mathematics you don’t understand things. You just get used to them.”
Dr. Duru
Nov 5 2025 at 3:26am
I was struck by the notion of expanding Kahneman’s System 1 and System 2 thinking to a System 3 for fast, slow, and super slow modes of thinking. However, I did not understand the indexing once the super slow mode is introduced. Given super slow is the process that unpacks incorrect intuitions, I would expect it to sit AFTER System 2 and System 1. It takes System 1 to conclude that System 2 made an error, correct? So I am imagining a process where System 2 passes its intuition to System 1 for examination. System 1 sends the intuition to the Super Slow process if it concludes an unpacking is needed. What am I missing?
Anyway, regardless, the core insight gave me an idea for a project where I thought the objective would be to improve an analyst’s System 1 thinking. Now I may pivot to first improve intuitive thinking.
On learning math…. I was determined to have a daughter who did not fit the stereotype that “girls hate math.” Yet, she showed VERY early that she indeed hated math. Loathed it even. Nothing I did could help her appreciate it, even when I tried to convince her that she was clearly smart enough to do well in math. To this day she hates math even though she can do very well when she is forced to do it. It astounds and disappoints me because I imagine what more she could do if she at least didn’t hate math.
My son on the other hand LOVES math. I did absolutely nothing to cultivate it. He just loved it. Loves talking about it. He loves math even more than I did at his age. And is smarter and better than I was.
I wish I could get Bessis’s high level impressions on such a situation as my experience.