A Mind-Blowing Way of Looking at Math (with David Bessis)

David Bessis: Well, I just wanted to say I’m delighted to be here.

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.

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.

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.

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.

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.

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.

David Bessis: But, it is a very small number of people.

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.

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.’

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 [?].

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.

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. [More to come, 41:39]