Pod Insights EconTalk A Mind-Blowing Way of Looking at Math (with David Bessis) About this episode 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.
Transcript 13,031 words · 2 speakers
0:02 Speaker 00 Welcome to Econ Talk, Conversations for the Curious, part of the Library of Economics and Liberty. I'm your host, Russ Roberts, of Shalem College in Jerusalem and Stanford University's Hoover Institution. Go to EconTalk.org, where you can subscribe, comment on this episode, and find links to other information related to today's conversation. You'll also find our archives with every episode we've done going back to 2006. Our email address is [email protected] . We'd love to hear from you. Today is September 18, 2025, and my guest is mathematician and author David Bessis.
0:43 Speaker 00 His book is Mathematica: A Secret World of Intuition and Curiosity. David, welcome to Econ Talk.
0:51 Speaker 01 I just wanted to say I'm delighted to be here.
0:54 Speaker 00 Great. 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
1:36 Speaker 00 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 here on this program. So listeners, read this book. Okay. This book starts like a lot of books do.
1:57 Speaker 00 And I love books like this. You know, it starts out like this. You know, math is really not that hard. It's, you know, it's, it's, it's, 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 more, a lot of throat clearing and, 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 that mathematicians are incredibly smart.
2:40 Speaker 00 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?
3:08 Speaker 01 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 is, that I wanted to describe. And I think what you said about, you know, 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, every time I read the pitch about a new book, I thought my heart was beating. I thought, okay, someone wrote the book already. And then I, I went on amazon.com and I could browse the first pages and I realized it was completely different. So.
3:54 Speaker 01 It's a book that I wanted to write. I've been wanting to write since I was maybe twenty-five 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 the, for me, the moment I realized I could tell the stories when I understood what they tried to say, why they could not say it, and why it's possible to say differently now. So this sounds very abstract, but let me make it really concrete. When you read Descartes, Descartes in the seventeenth century is saying, hey, you know, 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.
4:39 Speaker 01 I'm just a normal person, I'm not a smart man anyway, 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 of the twentieth century called Alexander Grothendick, who wrote maybe two thousand 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.
5:19 Speaker 01 And having been a research mathematician, I've quit mathematics about fifteen years ago. But I've 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 itself, but the kind of metacognition 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?
5:51 Speaker 01 This the issue with mathematics is it's it's something that manifested stuff in a horrible way. It's on on paper, on 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 to to build up meaning for these symbols is the real art of mathematics. And because these things is inside your head, it's extremely hard to talk about. And to me, the the failure of teaching mathematics, and it's something that has been going on for for not just centuries, but actually millennia, is a failure to admit that we do things in our head.
6:35 Speaker 01 We play with our intuition, we we play with images. And these things are not traditionally, they've not been discussed as being part of mathematics. They've been. 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 coffee conversation, a casual conversation with people who have a field medal, they will talk about their intuition in a very fuzzy way, waving hands and making sounds and and, you know, joking and saying they're confused and all that. But when you write a math book, you're not supposed to tell that, not supposed to say that. Because this crazy, very human part, very confused part of of playing with mathematics is not supposed to be the science.
7:16 Speaker 00 And you confess, 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, every day, I'm a PhD economist, so I 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,
7:50 Speaker 00 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 sort of skipping down and going to the end and thinking, "Well, what's the point? What is this getting at?" And you're not supposed to do that, of course. You just, 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 you're 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,
8:35 Speaker 00 you know, 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, it's for you, it's not true for most mathematicians. Explain.
8:51 Speaker 01 Yeah, so it's, 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, 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. That is, 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 sounds like a very provocative confession. And to the general public,
9:36 Speaker 01 it's perceived as being a provocation. But actually, many, many mathematicians have written 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, you know, they're kind of devices that serve a certain function. They are here to calibrate your intuition to, it's something against which you validate your intuition. It's very like the telephone book, you know. You cannot really read the telephone book, but when you need the number of certain person, you can look at them. Or of the instruction guide for your vacuum cleaner or for your toaster, it's not something you read except when you want to troubleshoot something. And the best way to interact with a math book is to start from
10:19 Speaker 01 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 this 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.
11:02 Speaker 01 So you have to open the book right in the middle, try to understand what's written, and you will not understand it. But you maybe 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. And this trick about reading math book was something I learnt quite late in my life. I was about 25, I think, when I was told that by a very good mathematician that was, you know, my kind of mentor, my PhD advisor,
11:45 Speaker 01 and he was a pretty 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, "But you should not do that. It's bad for your health. Don't do it. Nobody told you that." And so, this thing is, it's crazy. It's a secret of mathematicians. They all know that. And after I wrote that in my book, I found like maybe 20 instances of famous mathematicians writing the same exact thing. You don't read that book, you should not read math book, and not made to be read.
12:21 Speaker 00 And it raises, I mean, 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've 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 happens in economics as well, the application of math to economics.
13:06 Speaker 00 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.
13:48 Speaker 00 It's not even, it's not just not as good as so-called logical formalism. It's not thinking. And so you really, you 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.
14:07 Speaker 01 Yeah, so it's, that's the central topic, just a mild nuance compared to the account you gave. 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. Uh, personally, when I, when I was trying to prove new theorems, I, I did from time to time resort to making computations by hand.
14:45 Speaker 00 Of course.
14:46 Speaker 01 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, you can try from what you already know to be true and play around with the, 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, to explore the very tiny neighborhood, a few microns from what you already know. But you have to, to, to, 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 the very important thing. You, I think that's actually one of the reasons why
15:31 Speaker 01 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, you know, gradually trained and, and changing its way to account for a new data point. Uh, this is, well, I don't know exactly how the brain operates, but it, it's a very good metaphor of what happens in the brain. And that's a completely different way, and you're right, it's, 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. But they're very happy when they discover that their intuition is wrong.
16:15 Speaker 01 Because they, they have learned to overcome the, the, the feeling of humiliation you, you, you are taught to, to, to develop at school. When you, you have, you come, when, at school, you, you come, you enter the room with your intuition, and the teacher is telling you that your intuition is wrong. And you reach a 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 to this very special apparatus that is logical formalism. So you, you don't throw it away. You use it as a device. It's kind of a treadmill for your intuition. And it's
17:01 Speaker 01 Again, it's something that sounds provocative if you if you look at the canon of mathematics is taught. 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, you know, they they start knowing a few numbers, but then they get confused. And then they they they check it, you know, counting on the 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.
17:44 Speaker 01 And there is, I think, every adult knows as clarity about numbers, integers. Everybody understands integers, but it's something very abstract. But for most of the history of a human being as a species 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, you know, counting to a thousand. It's it's not natural, it's something you learn through a device. In that example of device is, well, Hindu Arabic numbers.
18:29 Speaker 01 It's a very advanced technology that was developed over the, at some point during the Middle Ages. But now it's it's 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.
18:51 Speaker 00 Yeah, there's a, I want to talk somewhere about intuition. So we had Patrick Haus on the program.
18:58 Speaker 00 Talking about 19, I think it's 19 ways of consciousness, or 19 ways of thinking about consciousness. He tells a remarkable story in there. 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?
19:43 Speaker 00 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 they're just as correct. The officer says, "Why do 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 what now? Why you turned around?" He goes, "Well, I'm not sure, but now I think back on it, the road was really quiet, and there's usually children playing on the side of the road. And something spooked me. I just had a bad feeling."
20:26 Speaker 00 Because, he says, "Now, if I think about it logically, maybe, 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 are 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?" Doesn't matter, it's not the point of the story, right? 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. And 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.
21:09 Speaker 00 The brain's 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 means, it doesn't mean a logical or a gut feeling. Patrick House hates, he's a neuroscientist, he hates that gut part because he says, "It's not below your waist or 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, 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, leads you astray, you
21:54 Speaker 00 get spooked and there's no reason, but it sometimes saves your life. And the point is, is it's not irrational. It's just a different kind of thinking.
22:03 Speaker 01 Yeah, it's essential and it's still very hard because you're completely correct to say that the word intuition is kind of, you know, 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. And going again back to Descartes, who I think is a central figure in that story, he is the one who brought the word intuition in modern science. He used the word intuition to describe the clear idea about something.
22:46 Speaker 01 And he gave a definition of truth that is based on clarity.
22:54 Speaker 01 And I think at that time, it made no sense. And they viewed it as a very mystical thing about, you know, "God created your brain in a certain way." I think what has changed now is we can make physical sense, physiological sense of what is intuition. And 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 what language.
23:34 Speaker 01 Language is a tool to summarize and articulate an intuition and to validate it using a very low bandwidth framework that's called logic, that allows you to, you know, to assemble a small number of very simple truths and to consolidate 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.
24:18 Speaker 01 It's never something that you fully... You can fully characterize. So I think we actually, we function that way in our daily life. An example I give in the book is that of a banana cake recipe. And I think nobody thinks that a recipe for a banana cake is something difficult and abstract. But when you look at... When you read a recipe and they say, "Okay, go and buy some bananas", and you can imagine the bananas in your head. You're here at the supermarket and you're buying some bananas. Everybody can see the bananas in their head. And then you... You're in the kitchen and step one is you have to mash the banana with a fork, okay?
24:59 Speaker 01 And when you imagine yourself doing that with your fork, smashing the bananas in a plate, 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...
25:18 Speaker 01 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 cannot have a basic conversation about anything with anyone. In mathematics, it's 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 would view numbers as counting oranges, or maybe but measuring your length, or maybe a surface area or something. And you have to go back and forth between different images.
25:57 Speaker 01 And the journey of becoming better at mathematics is a journey of attaching richer, deeper, and more diversified semantics to mathematical abstractions.
26:10 Speaker 00 Just to take another example from the book to try to make clear what we're talking about.
26:17 Speaker 00 How many points on a circle does a straight line touch?
26:23 Speaker 00 And 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 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, is not a circle, is not a sphere even. But when I talk about a circle and 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.
27:08 Speaker 00 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 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 we call, most people think of as math, is required to do. And that is going to take a, 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.
27:51 Speaker 00 And obviously, as you get more advanced in math, you're looking at more complicated things. Is that a fair summary?
27:56 Speaker 01 Yes. This example of a circle is the one I'm using when I give a conference on the topic. It's almost like stand-up comedy. You have an audience of maybe a few hundred people and sitting and looking at you and 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 know, 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, you know? Nobody told you that this thing is going on, but you know it's been going on since you're in primary school. You see circles in your head. Except, we'll talk about that, but some people cannot do that.
28:41 Speaker 01 It's a very small number of people.
28:43 Speaker 00 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.
28:50 Speaker 01 Yeah. So, the next question is, okay, what about a line, straight line, crossing a circle? Can it cross a circle at three different points? And people, you know, and I encourage people in your listeners to do that exercise, try to, you know, 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?" "Yeah." And everybody says, "Yeah, I'm certain. I'm really certain it cannot." And what's interesting is, what makes you certain?
29:33 Speaker 01 And then I go on and tell them, you know, "Did you see a kind of cartoon in your head with a straight line sweeping across a circle?" And that kind of visual argument of, you know, the straight line sweeping across a circle and making...
29:50 Speaker 01 What you perceive as all possible ways to interest, actually is fairly correct, is convincing, but it's a non-verbal reasoning. When you do math, your intuition allows you to get to certain conclusions even though you're incapable of articulating. So if you want to write down a proof, you need some technology. You can use maybe Cartesian coordinates, you can use whatever techniques you want to prove it rigorously, but what matters here is that you have an intuition and you believe it's correct, it makes sense to you.
30:28 Speaker 01 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, who could not have any images, and I asked to interview him to get his opinion about, you know, is it obvious to him that a straight line cannot intersect a circle at three different points?
31:13 Speaker 01 And he said, yeah, it's obvious to me, I cannot explain why, but it's 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 of, you know, that is not really visual, but you kind of...
31:52 Speaker 01 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's still intuition, even if it's not visual.
32:02 Speaker 00 I want to put this in the Kahneman, System One, System Two thinking, which you do in the book. Explain why you think we should think System Three. Explain what System One and System Two are and Kahneman's work, and then why you think it needs to be augmented.
32:16 Speaker 01 Yeah, so I have to go back to the canonical example from Kahneman about the ball and the bat. So, when, so the central theory of Kahneman and Fersky is that there are basically two kinds of modules in the brain to reach conclusion. One is System One is your instinctive answer when you say, okay, what's 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, you just, yeah, it's bigger than a mouse.
32:53 Speaker 01 Now, if I ask you, how many days, 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 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. You just don't know it at the top of your head. It's impossible. This is System Two. And the theory of Kahneman and Fersky is that we are lazy, and we, evolution made us prefer System One when it gives us an answer, because it's easy, we don't waste any energy doing
33:37 Speaker 01 things, and we have a very fast answer. And it gives, Kahneman gives the example of the ball and bat, so you have a ball and a bat, and together they cost $1.10. And the bat costs $1 more than the ball. How much is the ball? And they made experiments, and basically, everybody says $0.10.
34:05 Speaker 01 That's not correct. Because if the ball was 10 cents and the ball, the bats, 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, I love this example because it's something experiment in my flesh. I, I really felt it, that was crazy. So I had a friend of mine was studying at Princeton in, doing cognitive science and she was visiting me. And she was reading a book by Kahn and she said, okay, the ball and the bat. Oh, she's the ball. And I did not think, I just said 5 cents, that's obvious.
34:44 Speaker 00 You cheated, you're a cheater.
34:46 Speaker 01 I am, she, she, she literally became white, like she told me that a guy had won the Nobel Prize for proving that nobody can save 5 cents. We were thinking, the intuitive answer is 10 cents. If you want to give the correct answer, you have to use system 2 and it will take you a few seconds, but you just cannot save 5 cents immediately. We're thinking, you're not allowed to do that, it's against science. You are wrong. And, and then she, it took her about a minute to say, okay, okay, okay, okay. It's, it's no fair game, 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.
35:30 Speaker 01 It's not that I use, I have a super fast system 2 that enables me to make bad computation incredibly fast. No, and very, actually, I think being, becoming a mathematician makes me worse at computations. Because I was relying less and less on my system 2.
35:48 Speaker 01 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, 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, they, they, they kind of get stuck in a kind of meditative flow, thinking, okay, my intuition told me that it's 10 cents. But the computation says it's 5, why? Why did I give, why did I give the wrong answer? What, what's wrong with my intuition? And then they try to see things a bit differently. And they gradually doing that, they gradually retrain their intuition to self-correct.
36:32 Speaker 01 It's, it's, uh, so, uh, the, the, the, the book of, you know, famous book of Kahneman is Thinking Fast and Slow. And if I, if I had to rewrite it, I would write it, uh, thinking fast, slow and super slow. And the super slow mode of thinking is that of mathematics. I call that system three is the following. Whenever you catch your intuition red-handed, being wrong at something, don't throw that away. Don't reject the intuition. Don't, you know, do some, Freud would say that you kind of reject it and suppress. Don't suppress that intuition. Explore it. Try to unpack it.
37:18 Speaker 01 Try to understand how did your intuition, uh, what's in your intuition? How does it materialize 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, you, you, you play back and forth between your intuition and formal logic, between system one and system two, 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.
37:57 Speaker 01 It depends. There are some things that you can resolve in 10 minutes. There's something 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 not just good at maths, 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 in that exercise.
38:30 Speaker 00 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 trying, taking the data of the world around us and trying to make sense of it. And I find this, one of the watchwords, there's sort of 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
39:13 Speaker 00 faculty members and graduate students, and suddenly in walks Jean Pierre Serre, who's one of the greatest mathematicians of the last hundred years. And he sits down in the second row, which is just like, you know, as an academic, a former academic at least, this imposter syndrome that all of us have to some degree, that 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. And tell what happened at that seminar and what you learned from it, because I think it's incredibly valuable.
39:50 Speaker 01 Yeah, so I think 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 19, nine, three days ago. So it's, when you say the last century, that 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 he's boring. Mathematicians can be rude, you know, in their way, in their own way. They never pretend to be interested in something they're not interested in.
40:32 Speaker 01 But I saw that he was listening to my talk, not because it was a good talk, but because he has a keen interest for the topics I was working on. So I was just lucky to be in his sweet spot in terms of what my seminar talk was about.
40:47 Speaker 00 And you claim in the book that you did not change your talk To reflect his presence, because there be a temptation to jazz it up and make it fancy because you want to impress them, but you left it as it was.
41:00 Speaker 01 Yeah, so, um, actually, there's more to that. Whenever I had the chance of attending a seminar given by Sauer, I was always impressed by it. It always looked like super easy. Everything was completely transparent, but when you try to, when you get back home and try to do it by yourself, you just fail to do it. So it's kind of a magician in the way, it presents things a very, very simple way. Everybody is kind of, every piece is fitting exactly into the next piece with no effort, but actually the architecture of the talk is absolutely timeless.
41:39 Speaker 01 So, um, 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 what you took looks super easy, it's actually a signal that you might be very good at math. So I'm just trying to overplay 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, it was a seminar talk on an ongoing research that was far from over, so I did
42:21 Speaker 01 not have many theorems, and I was really, 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, and then it was a very long seminar. I think it was, it was at Seminar Chevalier 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, "Yeah, you will have to repeat that again, because I did not understand a word."
43:02 Speaker 01 And that blew my mind.
43:05 Speaker 01 But he was overplaying it, of course. He had understood much more than words in it. But there was something he hadn't understood. And so the meaning of what we put under the word "understand" depends from, you know, it varies from one person to another. And I think for him, saying that he understands something, it's not just that he understands the words, the statements, and the logical flow, it's that he really understands why it should be like that.
43:40 Speaker 01 And I think that was the meaning of this question. There's things that, you know, things are true for reasons that you don't fully understand. You're able to prove that it's true, but it's kind of weird. It should not be true. There's, you don't get it. You don't get why it's really worse.
43:57 Speaker 01 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.
44:07 Speaker 01 And my first reaction was to think, okay, he's kind of showing up because he's Sae. And when you're Jean-Pierre Sae, 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 Sae. And I tried to practice it. So it's a very common thing when you go to a math conference. So a math conference typically about 50 people coming from many different places around the world. You get people from the US, people from Japan, people from Europe.
44:49 Speaker 01 Meeting at some small place. Usually it's, you know, by the mountain or by the seaside. It's very, very remote, you know. It's kind of, mathematicians are a bit like hermits. They 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, and you... When you're at lunch or during dinner, you sit with other mathematicians and usually they say, okay, what are you working on? They won't say, okay, I'm working on the theory of, I don't know, algebraic, reductive, whatever thing, weird thing, homology of that kind of varieties, weird things. And usually you don't understand a word what
45:32 Speaker 01 they're telling 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 Sear trick and try to say, okay, I don't understand a word, please repeat it to me.
46:09 Speaker 01 And I did it on the first occasion. And that was absolutely fantastic because the guy who was telling me his research topic was, I don't remember if it was a student or a postdoc, 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 Fancy. That was, that was what I called the tourist menu. He was, you know, showing me the tourist menu with, with a kind of very formal dishes that nobody really wants
46:52 Speaker 01 to eat, but they look like it's a fancy place. And he was presenting his research on mathematics the same exact way. And
47:01 Speaker 01 And when I told him, please repeat, repeat it as if I had some disability, as if my brain was damaged. And because, you know, I am jet-lagged, I'm tired, whatever, you know, I'm stupid, I'm 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, you know, the thing, how really himself looked at his mathematics. And he was using different words and describing the situation using very simple images, examples, and it was, it was a different thing.
47:46 Speaker 01 And I had given him permission to be simple, and I have given myself permission to ask and more and more questions. And this situation of, you know, not understanding something in mathematics and being embarrassed because you're supposed to understand it is, I would say, the standard of mathematical conversation. And I think everybody who has interacted with mathematics has been in that situation of being too embarrassed to say, to convey 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, just know that Ser is doing the same thing. That's the way the greatest mathematicians work.
48:30 Speaker 01 And again, after my book was published, I, I, I saw like many, maybe 20 different examples of great mathematicians making the same exact remark, that they asked stupid questions. It's super important for them to ask stupid questions. It's, it's not, I, I mentioned the example of, you know, not reading boom, not reading books being kind of secrets, and asking stupid questions is a kind of secret. It looks too easy, it's not credible, but it's actually absolutely central.
49:03 Speaker 00 It's a really beautiful thing. I want to, I want to talk about one of the themes of the book that 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, you know, every time we translate something from English into Hebrew, my Israeli friends say, well, that's, that's not a good translation. And they assume it's Google. And I said, well, actually, we asked so and so to translate. Oh, it'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
49:45 Speaker 00 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. 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
50:29 Speaker 00 to get at a different aspect of seeing. So so far, we've been talking about seeing as visualizing a visual image. We're talking about some people can't do it. But intuition, of course, is not just the visual, 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 Serra has never been on the front page of the New York Times,
51:10 Speaker 00 or Grothendieck, or 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's not, hasn't told more than a couple people. And there's like this incredible electricity afterwards and
51:31 Speaker 00 And he's like on top of the world. And then they get to someone to check the proof. And it turns out, and there aren't many people 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. And I'm sure it's not fun. Somebody, maybe it's fun. But anyway, they find a mistake. Okay, well, it happens. They'll fix it. You 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. And he's probably close to potentially suicide because his life has been ruined. His family is falling apart. His wife must be going crazy.
52:14 Speaker 00 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, you know, that Cacatani's conjecture would fit with so and so. And he just quote, "saw it." And of course, that, we don't know what he's talking. 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.
52:54 Speaker 00 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. And 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?
53:23 Speaker 01 So the first part of your question and the second part, as a mathematician, what is happening in 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 brain, but I have no proof from a scientific standpoint if this is really how it works. But I have no other model of that. I do think that things get connected.
54:05 Speaker 01 If you, the metaphor for that is viewing the neurons, you know, kind of having the synapses being reconfigured in a different way. And if it's at a, sometimes it happens at a very large level. So like two things that were not connected. One of, most of my ideas in mathematics came at a moment where two things that were kind of split, living in different worlds, you realize that they kind of work together and you can connect them.
54:35 Speaker 01 It usually takes a while and probably the more distance, the longest it takes.
54:44 Speaker 01 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, you know, 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 like that every day, you know.
55:27 Speaker 01 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 a 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.
55:47 Speaker 01 In your example of Whiteface, another thing I want to point out, because it's it's something that is rarely discussed. When people talk about mathematics being the science of logic and deduction and 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've just told about Henry Wise would not be possible. Why is that? Because if the proof was wrong, the proof was wrong, and then there would be nothing from it.
56:27 Speaker 01 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
57:06 Speaker 01 life of academic mathematics is a proof that the semantics of mathematical object is not reduced to logic.
57:19 Speaker 00 Talk about Hardy and Ramanujan.
57:22 Speaker 01 Yeah, so Hardy and Ramanujan, they look like characters from a comedy. It's impossible to get people like that in real life normally. So Hardy was a very typical high-brow, A-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 and more logical, formal proofs.
58:06 Speaker 01 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's really, it looked like Hollywood scripts. "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 there were like crazy theorems, Hardy could not prove them, but they looked true. Not just true, but deep.
58:50 Speaker 01 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, invited Ramanujan to Cambridge. And 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, he, I don't know if the word "proved" should be used here,
59:29 Speaker 01 but he stated thousands and thousands of theorems, and we should put scare quotes around "theorem", because he just say, "Okay", he was basically taking a piece of paper and saying, "Okay, 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."
1:00:10 Speaker 01 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 a formula on the screen of blood. That was the way these 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.
1:00:54 Speaker 01 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, you know, it's not the exact sentence, but it's really the spirit of a 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 were kind of coming from outer space and are completely alien to the normal human species, then Ramanujan is perfect for you. You can think of him that way.
1:01:39 Speaker 01 This guy invented crazy theorem. It took like 50 years to prove them. And they 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 fertile way of looking at it. And probably the way these theorems appeared in his brain, and they were proved correct after decades of work,
1:02:10 Speaker 01 is the sign of something universal that every human being is capable. It says it's probably the same mechanisms that children use to learn their native language.
1:02:24 Speaker 01 And I'm really on the same page. I really think that there is something to learn from Romano Gian. Unfortunately, we don't have the details. We don't really know. He never wrote down what happened in his brain. But the most important math class I took was when I was studying at the Economic Math Superior. We had a course by a mathematician called Vieno. And the goal of the course was to start from the formula of Romano Gian and try to find a way to make it intuitive. And the course was, he was using some kind of dominos, multicolored dominos that you
1:03:09 Speaker 01 were piling, you're making piles, was using certain rules. And gradually, we were learning to translate certain algebraic formulas into pictures involving these dominos. And at the end, after six months, we had built up kind of visual intuition for the different components. And this very result, one among thousands, but this very reason became obvious. And if some brain is able to produce something, the biology of a different brain is actually quite similar. And it opens up an entirely new conversation about, you know, what explains the diversity
1:03:54 Speaker 01 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 this crazy thing, 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.
1:04:18 Speaker 00 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, "Well, 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, the two-year-old.
1:04:59 Speaker 00 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 through 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. And it turns a lot of people off from math. It convinces them they're stupid. If I said to you, "What should we
1:05:44 Speaker 00 do differently?" Do you have a thought about where you'd start?
1:05:49 Speaker 01 Yes, of course. And I was forced to think about it because I have kids. So, again, before I say anything, a very big disclaimer. I'm experimenting with that. And I never ran any control group experiments and 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, if not decades. So...
1:06:26 Speaker 01 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 tried to develop that bridge. And very stupid exercise that doesn't look like it has to do with mathematics is asking my kid to tell his dreams.
1:07:11 Speaker 01 What did you dream about? The idea that, you know, we should remember our dreams and be able to tell them is an idea that has been associated with certain, with therapy, with Freud, with subconscious in kind of all definitions of subconscious. But actually, it's a very practical exercise that I think is extremely valuable as 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.
1:07:54 Speaker 01 Why is that? Because it's a way to force you to stabilize 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 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 details.
1:08:36 Speaker 01 And after a few weeks of doing that, it was so detailed that I had to give 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's 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.
1:09:20 Speaker 01 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 their soul, 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.
1:10:01 Speaker 00 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 is, all these things I think are underrated. But I'm thinking about the sort of standard stepwise progression in American high schools. It's algebra, geometry, pre calc, calc. That would be the sort of standard way. And you know, you get asked this question. My wife's a math teacher, teaches high school math. And she's a fabulous math teacher and she emphasizes a lot of intuition. And she asked, 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 of standard answers.
1:10:45 Speaker 00 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 it's, 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 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 use how to use, you learn how to use your brain. And that's exhilarating, actually. So does that have, do you have any thoughts on what
1:11:30 Speaker 00 high school teachers should be doing in their math classes?
1:11:33 Speaker 01 I think they should be bold. They should say, "It makes you smarter."
1:11:40 Speaker 01 People hate math because they view it as an IQ test that they're failing.
1:11:43 Speaker 00 Yeah.
1:11:44 Speaker 01 And it's not a test. It's a technique to get smarter. If you're failing, it's normal because you start, you know. When you start any new sport, you, you circle it. There's no way you're going to be good on day one. If you try to, to, to, to, to surf, you, 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, you, you cannot teach mathematics to kids who are convinced that your mathematical ability is something that is static. It's like swimming. If you, if we were, you know, telling kids, "Some, some kids can swim, the other ones drown."
1:12:27 Speaker 01 You will see, it's genetic. "Let's see how you, how you fare on day one, and then you will know for the rest of your life." If you do that, there will be a riot in the swimming pool. And nobody will be able to swim. But this is exactly what's happening with, with mathematics. We, the 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...
1:13:08 Speaker 01 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. It's not going to be an easy ride. It's going to be very difficult. But that's the war 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.
1:13:53 Speaker 01 But these are very good selling points. We've not been using them enough, I think.
1:13:58 Speaker 00 So I'm going 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. And 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 the story. We just pretend we do." End of that quote. Here's a second quote. And this quote is from, is about Descartes.
1:14:41 Speaker 00 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's, I think, fundamental misunderstandings about what it means to be rational and what it means to be scientific. And I think your book takes...
1:15:10 Speaker 00 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.
1:15:51 Speaker 00 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 or expand on it, I'd love to hear.
1:16:22 Speaker 01 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,
1:16:59 Speaker 01 human language. So you cannot be on the side of rationality except from a methodology standpoint. There's no absolute truth that is rational. It doesn't, it cannot exist. Rationality is an approach to articulate different ideas together. Let's put it like that.
1:17:32 Speaker 01 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 around the world and a very weak, weakly correlated language that is trying to express things about the world. Rationality is a framework for making all these representations and all these statements
1:18:13 Speaker 01 about the words more correct, for gathering more information and making it 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 was the enemy of intuition is a fundamental misunderstanding. And this misunderstanding is caused by our inability to accept that we're intuitive machines.
1:18:52 Speaker 00 You left academic life and got involved in AI. 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 become, 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
1:19:36 Speaker 00 much more, I've gotten much more pessimistic, is how I would say. It may turn out to be wrong. That's just my thought today. But you're much deeper into this than I am. What are your thoughts?
1:19:48 Speaker 01 Wow.
1:19:50 Speaker 00 I mean, obviously.
1:19:51 Speaker 01 It's not an easy one. 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
1:20:07 Speaker 01 very basic limitations of our 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.
1:20:34 Speaker 01 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,
1:20:57 Speaker 01 I don't see how we will 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.
1:21:10 Speaker 00 My guest today has been David Bessis. His book is Mathematica. David, thanks for being part of Econ Talk.
1:21:18 Speaker 01 Thank you very much. It was a pleasure.
1:21:26 Speaker 00 This is Econ Talk, part of the Library of Economics and Liberty. For more Econ Talk, go to EconTalk.org, where you can also comment on today's podcast and find links and readings related to today's conversation. The sound engineer for Econ Talk is Rich Goyette. I'm your host, Russ Roberts. Thanks for listening. Talk to you on Monday.
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