#73 Steven Strogatz: Exploring Curiosities
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A great discovery that no one appreciates is not really a great discovery because science is a social enterprise. It's not just enough to do the work. You have to communicate it and help other people understand why it matters. Hello and welcome. I'm Shane Parrish and you're listening to The Knowledge Project, a podcast dedicated to mastering the best of what other people have already figured out.

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Today I'm talking with Steven Strogatz, professor of applied mathematics at Cornell University. I wanted to talk with Steve after reading his most recent book, Infinite Powers, which explores how calculus unlocks the secrets of the universe. Together we'll explore how math helps us better understand the world and make better decisions. And as you'll see, Steve makes learning things fun and practical. Let's get started. It's time to listen and learn. ♪

© transcript Emily Beynon

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How did you get interested in math? Was it a story that pulled you in? Like, did this—were you innately interested in it? Was it something like a teacher? Like, how did this happen? Well, I was interested in all of school, and I still am. I like all the different subjects. But so math didn't stand out for me. I'd say really pretty much for most of my childhood. It was just one among many. But then

I do remember one or two moments where it started to take special significance. One was I had two friends when I was about, well, let's see, we used to call it junior high school, so it was 7th and 8th grade. I must have been about 12 years old. Two other boys, and they were both good at math and vocabulary and geography and everything else. I used to like to compete with them. We would play chess against each other and other things, and at that age, just...

Being better than your friend meant a lot to me. I don't know. Or having them feel better than me inflicted a kind of pain. So I do remember one time getting a little higher score than this other kid on a math exam, and it just felt good. So I'm not saying this is a noble reason to be interested in math, but it was just an early memory that...

It was very objective. You know, you could say, I'm better than you. And it actually meant something, which to a 13-year-old boy, that was important. But on a more elevated level, I'd say the real turning point was in my—would have been my sophomore year of high school. So by then I was about 14 or 15. And a teacher said something that took me aback. I had never heard any teacher say something like this. He was—

This was Mr. Johnson, who had a beard and had gone to MIT. And to me, he looked like what I imagined a brilliant math professor might have looked like. He was very serious, not a smiler, but fair. And he said that there was a certain geometry problem that he had never seen any student solve. Just offhandedly mentioned it to the whole class. And I thought that was interesting because I hadn't always been able to do any geometry problem any teacher asked me.

So he said, yeah, he didn't, no one had ever solved this certain question. And then he also said that he didn't know how to solve the problem. And that was very surprising because I had never heard a teacher say that. You know, he didn't know how to do the problem. So I started thinking about it. I could tell you the problem in case you're... Yeah, please do. It was a question about a triangle that sounded like many other geometry problems. It says if the angle bisectors of a triangle are congruent,

prove the triangle is isosceles. So what that means is angle bisectors, that would be a line that cuts through the angle. Picture a triangle like, say, you know, standing there flat on the table with its point sticking up. If you divide the angle in half at the base, there are two of them down there, if they're divided in half by a line and then that line meets the opposite side of the triangle...

That's the angle bisector. And so if you have two of those crisscrossing and they're equally long, that's what congruent means. So two angle bisectors of equal length prove that the triangle is isosceles, meaning that those two angles at the base are actually the same angle, have the same number of degrees.

And so it sounded like many other questions. There are similar questions if two angle, two perpendicular bisectors are congruent or something called medians. All these questions are pretty easy to prove that you would get an isosceles triangle. But something about angle bisectors made the question very hard. And when I tried it, I couldn't do it. And that already caught my interest. And I

I worked on it a lot. I couldn't do it. To spend a day or two on it, I still couldn't do it. And then days became weeks and weeks became months. And I think I spent about maybe six months on this question, trying things, getting stuck, trying.

trying other things, getting stuck. You know, and sometimes it would come close to working out, but I could never get the argument to go all the way through. And my friends would get annoyed with me, like, come on, let's go to the movie. No, I feel like thinking about the angled bisectors. Or in French class, you know, when the... We used to have this, like, conjugation train where the teacher would give...

some verbs to conjugate, and you'd feel that it'd be going around the room, and you could feel it's going to be my turn soon. But I was still thinking about the angle bisector. So I was obsessed with it, and I didn't realize it at the time, but I was doing research.

where I was thinking about something for just the pure pleasure of trying to understand it. And ultimately, I got something that I thought was a proof. It was a boarding school. My teacher lived nearby, and I asked if I could come over to his house on a Sunday morning. I could still remember him there in his beard and his pajamas and his little kids running around in his house at breakfast. And he checked this proof line by line very slowly and

you know, with his serious, stern face, said, "Well, yes, that's a correct argument. Very good." And then he wrote a little note to the headmaster of the school, which meant a lot to me that, you know, my teacher said something like, "Steven has real talent."

So I don't know. It was definitely a pivotal moment. It's interesting what a difference a teacher can make. Yeah, of course. I mean, partly that he really set the bar very high, but also he was humble. He was strong enough to admit there was something he didn't know. And apparently, he probably mentioned that in other years, and other students hadn't either risen to the challenge or weren't interested. Why would you do homework if you didn't have to? But

Something about it caught me. And, yeah, he was a very inspirational teacher. And it's interesting, too, that he's not the Hollywood inspirational teacher. It wasn't like, you know, such a loving, sweet guy or certainly not interested in building your self-esteem. He was just all business. But there's a line in that movie about another great teacher, Stand and Deliver, about Jaime Escalante, who taught in East Los Angeles.

who says in one critical scene in the movie, the students will rise to the level that the teacher expects. And so if the teacher expects something really hard, kids don't even know it's impossible. They'll try. I try to do that myself with some of my students. And what do you see? Does that prove out? Sometimes it does, especially younger ones. You can really, you know, by the time they're seasoned, if they've been in college a few years, they start to know the game.

But freshmen don't really know what college is going to be like. And so you can do mercilessly unreasonable things to the bright freshmen in math. And they will do, they will astonish you with what they can do. Yeah, we see that all the time. You didn't have an easy path in math. Was it first year university where you sort of got the worst grade you've ever gotten and you switched to physics? Or was that, when was that? That is true. That was my first year.

Well, that was that class. So I was just describing how our freshmen, the first year students, can do astonishing things. I was...

supposedly one of those students when I went to college. So they grouped a bunch of us together in like whiz kid linear algebra. And this was for all the kids who had done extremely well in calculus in high school. And it's sort of like you had a little mark on your head. This is a future professional mathematician, or at least potential to be. And the other thing, though, is that professors know that kids who are good in math in high school, a lot of them are going to get, so to speak, weeded out. They don't really have the right stuff to be

mathematicians of the future. At least that was the thinking at the time. It's a pretty nasty pedagogical philosophy, I must say. But they would throw all of us in with one of the worst teachers in the school. I don't know why they assigned him to us. I mean, this was a really pathetically terrible person for teaching. I mean, the first day of class, he came in and sort of slithered along the wall, didn't make eye contact with the students. Very shy person.

who didn't say welcome to Princeton, welcome to the, you know, this is going to be exciting, nothing. You just started with the definition of the key ideas at the beginning of linear algebra, the definition of a field.

Blah, blah, blah. Here are the axioms it satisfies. Anyway, extremely dry. And I couldn't understand what the heck was going on. And the textbook had no pictures. And as a visual thinker, I found that very problematic. I couldn't really understand what was happening. And weeks would go by. I could feel myself falling behind and not, you know, my homework was getting worse and worse, grades and everything.

I didn't know certain basic things like, well, try reading a different book, you idiot, or go ask for help from the teaching assistant. I didn't know. I had never done that stuff before, so it didn't occur to me. I thought I should be able to understand, and I was having trouble. And so I did end up getting a B-, which was a pretty low grade, and...

I wish I had your transcripts. Well, okay. But it was a low grade. And it was discouraging because I didn't deserve a B minus. I really knew that I knew nothing. I probably should have failed, but they didn't want to do that for some reason. So it was a real, yeah, very demoralizing experience. And it made me think, wow, maybe I don't have the right stuff for this. Maybe math is different in college.

I stuck with it for one more semester and took a second course in that same whiz kid sequence, Honors, this time Honors Multivariable Calculus. And I could feel the same thing was happening again in that second semester. I was on track for another lousy performance. Well, and meanwhile, I saw other kids doing fine. I mean, they were raising their hand. They seemed to know everything. So I, you know, even in a relative sense, not just an absolute sense, I could see that I was weak compared to the people around me. And...

It was a formative experience because I think it's given me a lot of sympathy as a teacher when I see students struggling. I don't always assume, you know, that they're hopeless or they can't be rescued or they have no future. In my case, I mean, it has stuck in my craw now 40 years later. You know, when I teach linear algebra, I realize...

It's not really that hard. You can do better than my old teacher did. Show some pictures, give some intuition, talk about how it's connected to the real world, tell some history. You know, bring it alive. Jesus, this is a really exciting subject. You don't have to suck all the blood out of it. Also the idea of being made to feel like a weed. You know, you hear this idea of weeding out the people who have the right stuff from those that don't. I really don't accept that idea. People have a lot of potential, more than...

That kind of superficial analysis would lead you to think. So I was practically, I was, I could have easily been weeded out, except that I just loved the subject so much. I stuck with it, even though I was always one of the weakest math majors in my cohort. I just loved it more than most. That's a fascinating story. Like, it strikes me that math is something that we develop this mindset about for whatever reason that

we're just not mathematical or math isn't my strong. And, you know, this sort of like happens around grade four or five or six. And it sort of like follows us. Is that a teaching thing? Like, why do most of us find math so boring and dry? You're right about that, aren't you? I mean, that is if you just look around and ask your friends, the average person did not have a good experience in math class, although it tends to be a little more complicated than that. Most people will say, I did like math until...

And then they'll tell you until we got to fractions. Those were really confusing with the common denominators. Or I liked it until we got to algebra and then it was all those Xs and Ys instead of numbers and I didn't really know what was going on. Or I even liked algebra but then I lost it in geometry or I hit the wall in calculus. So different people have these different places. But it seems it sort of happens to everybody at some point or almost everybody.

And so why? There's a few explanations. You know, one is that the subject, the way it's traditionally taught, is made to seem like a tower. You know, each thing builds on the topic before it. And if you fall off at any stage, it's a long way down off the tower. You know, that is, it's a very linear architecture. Whereas the reality is that math is a web, not a tower. That you can jump in at different points in the network of ideas,

and then find your way around from there. And so it's not really true. I mean, it's complicated. In some ways, it is sort of like a tower. The idea of algebra and variables and functions, and I know you know computer science, so you will know what I'm talking about here. You know, those are fundamental ideas. If you can't think abstractly about a variable rather than a concrete number, you're going to have trouble with a lot of math because that's a fundamental idea. So if you don't get that,

Pre-calculus is going to be trouble. Algebra 2 is going to be trouble. So some things are indispensable. But there are other things which, OK, you know, maybe you missed the idea of an asymptote. You can get on. You can still keep going without that. So there's that. I mean, it's partly the structure of the subject. Partly it's often taught in a very top-down way, you know, that the teacher presents material almost like pouring water.

liquid into a student's head. It's almost like a curriculum dump too, right? It's like, we have to cover all of these things and we're going to start here and we're going to progress. And I don't care if you're interested or not. It's just, this is the standard curriculum. You've hit a bunch of important things there. So one is the feeling that there's a lot to cover. I once heard a teacher say, don't try to cover the material, try to uncover it.

Right. I mean, there's a lot to be discovered, a lot to be uncovered, to remove the fog, to remove the crust of of, you know, difficulty or misunderstanding, reveal something rather than cover it. But, OK, that's just a silly use of the word cover and uncover. But, yeah.

The notion that there's a standard curriculum sometimes stifles exploration. There is a certain amount of exploring that even a very young student can do, where if, you know, the experience of being a mathematician is so different than that of being a student, where we're frequently in the dark and we're poking around and trying to make sense of some new territory, mathematical or mental territory, and you're an explorer. You're lost. You're in the jungle.

How do you get out or how do you make progress? And that's a very valuable skill in all aspects of life when there's uncertainty and sometimes fear, but excitement, exhilaration. You know, the thrill of problem solving and inching your way forward applies to everything. And so we could be spending more time on that rather than

Here's an established body of techniques that you have to learn, and I'm going to dump it or pour it into your head. That's a very artificial picture of the mathematical enterprise. And why do we do it? I don't know. I mean, maybe there's all kinds of standardized testing. The teacher has to get through certain material to satisfy local authorities or because tradition says this is what an educated person needs to know. And there's some truth to that. So I don't know. I mean, I don't have an easy solution to this, but

Times I've taught courses with this more exploratory mode where there was no place we had to get to. We just had to do the process in an honest way. I mean, the way a real mathematician would do it. Students love that. Math is not boring then. Math is, you know, that's creative. Oh, yeah. I would love that. Yeah.

Well, we do that. I mean, I teach a course called math explorations with students who are required to take a math course before they graduate and they dread it. They don't you know, these are the people who are the real dead enders, you know, like they're putting this off until their last year of college.

But they, you know, and they have to get out. They have to go through it to graduate. Then once I teach it this way with them doing explorations together, activities, working in groups and thinking and sharing ideas. Can you give me some examples of like the stuff that you cover? Oh, yeah, sure. So here's one.

You know, we're used to a standard kind of geometry, but there are other geometries. Here's an example. Suppose we're in New York City or some other city that has a grid of streets. So if I say, how far is my location from some other location?

You could say it in terms of miles or kilometers as the crow flies, but that's not very relevant if you're driving on the grid. You might be more likely to say you have to go three blocks north and, you know, eight blocks west or something like that. So you would give units of blocks as measured on the grid. Now, that's an interesting kind of geometry. If you define the distance between two points in a plane as being...

the number of units north-south plus the number of units east-west. You can define a distance like that. It's not the Euclidean distance. It's not the standard one because you're not allowed to take diagonal paths in this geometry. You can only go on the grid. Now, if you ask someone a question like, what does a circle look like in this geometry, that's an interesting question. In other words, what's the set of all points three blocks away from a given point?

So try to picture that in your head. You could go three blocks north or you could go two blocks north and one over, either east or west. You could go one up and two over. And if you can picture what those points look like, I don't know, are you picturing it? I'm trying to, yeah. Yeah, do you want to draw? Do you have a piece of paper in front of you or anything? Or maybe you have it in your head. I have it in my head.

OK, you want to say are you or should I not make you? No, no. I'm already going to embarrass myself enough in this conversation. OK, all right. Then I'll say so. I mean, if we if. OK, well, it will look it should look like a diamond. I mean, it should look like a tilted square. Yeah. OK. So so when I when my students discovered that.

One person started screaming. She was saying, that's wrong. That can't be right. That's crazy. So I said, what's the problem? She said, a circle is round. This thing has points. I mean, this has corners. This doesn't look round. So I said, well, who says it should be round? Just because the Euclidean circle is round in traditional geometry, this is a new world. We're making new rules. This doesn't have to, it doesn't have to, who says it has to look round? It doesn't look round. So, and then you can do more. You could say, what is pi?

to calculate pi in this geometry. Now, that takes you back to fundamentals again. Well, what is pi? I mean, some people have memorized 3.14159 like that, but that's just memorizing. That's not thinking. Thinking would be what does pi actually mean? It means the ratio of the circumference of a circle to the diameter of a circle. And so with this diamond shape, you have to now think,

Can you just explain, like assume nobody knows anything, including myself. So circumference is the area around the circle and the diameter is the width of the circle? That's right. Exactly. Good. Right. So the circumference would be just that. It's the distance you would travel if you moved around the circle, the total distance following the circle around on its rim. And exactly, the diameter is the widest distance across the circle.

So, you know, again, you have to try to calculate that or see what it would be for my little example with the circle of radius three in this funny geometry. And the radius is the point from the middle of the circle to it's half the diameter, I'm guessing. That's right. Sure. That's right. The radius would be half the diameter. Right. So from the center of this. It's been a long time since I did this. Good. Okay. I'm glad it's coming back to you.

So I don't know. I mean, we don't maybe it's not so good to do it in this audio format. But if I'll tell you, the answer comes out to be four. If you calculate the circumference and you calculate the diameter, you'll end up getting that they're always in the ratio. I think I'm doing it right. It's always going to be four times.

The diameter will be the circumference, which is interesting. The pi is actually a whole number. It's four in this geometry instead of 3.14. So, I mean, okay, what's the use of this? It's not like it's so important to have the geometry for getting around cities with grids. It's just to make the point that this is a playground. This is the realm of the human imagination. You can think of alternative geometries and explore them and explore.

And the point being, it's very empowering. The students feel, hey, I can do this. I mean, they can, I don't tell them the answers. The whole course is based on me never lecturing. I just give them puzzles. And then they try to figure them out. Yeah, they do. And they do figure them out. And if they're stuck, I say, okay, let's figure out what to do. I mean, now you're stuck. What are you going to do?

And then people learn how to get unstuck, which is really valuable to have problem-solving strategies for coping with frustration. We also talk about emotional stuff, like, okay, we're stuck. How does that feel? Well, I'm frustrated. I'm mad. You know, I'm curious. Whatever. And people talk. Now, there's a lot of...

people these days criticizing the notion of safe space. You hear this, that this is like a snowflake thing. Okay, we're not, shouldn't talk politics, but I'm sure your listeners know what I mean. That you hear about in education, especially higher ed, that there are things that we're not supposed to talk about because it triggers people and this is a safe space where we're not going to trigger anybody. And that's usually presented as a very negative thing that stifles free speech, et cetera. But I want to

bring up the possibility, because I've lived it with these students, that when you make a space that is safe for mathematical confusion, meaning nobody's going to feel stupid here, we're all confused. Confusion is the normal state of affairs when you're trying something really hard and when you're exploring the unknown. So it is a safe space in the sense that you can trust us, that we're all on the same team trying to figure this out together.

And don't worry about looking stupid. I'm confused over here, too. That sounds like a great environment. It is. It is. And when you collaborate on research at the cutting edge of math where nobody knows the answer, I mean, you can't look it up. You can't ask a professor because no one knows. And it's the same thing with science or any other thing at the cutting edge of knowledge. When you're collaborating, it really helps students

to be vulnerable and to have a safe enough relationship with your collaborators that you can say, I don't get this. Could you go over that again? Or I don't see what to do. Or to suggest a stupid thing that someone else, rather than jumping down your throat, or they could tease you. Maybe they do jump down your throat a little. But it's ultimately safe to take intellectual risk. That's the point.

As you were saying that, I remembered a momentary panic. One of my kids brought home some homework last week and I was looking at it going, oh my gosh, I don't know how to do this.

And I was sort of like, what does that mean? Does it mean he's going to think less of me? Does it? I mean, both my kids are pretty good at math, but I was like, oh, and you just start thinking and all these memories start coming back. And I was like, this is actually a really hard problem. You're in grade five. Like, this is pretty difficult. I don't remember being this difficult. Yeah.

Well, so what, can you tell what you did? Tell us if you don't mind. I posted it on Twitter. I even tagged you, I think, in the end of it, because I was like, you're going to be my best friend. Oh, good. I missed it. I don't think I saw it. Did I respond? Oh, no. It was sort of, so the question I posed, so my kids are both

pretty decent math. They're way beyond me already. And it was like, you're standing on this 40 meter building, which is 20 meters away from a tree. And at a 45 degree angle, you see like the height of the tree, I think was like 15 meters or something like how, how tall is the tree?

I see. I was like, oh my God. I might have pictured that, but I was like, holy cow. Grade five, I thought I was doing like multiplication and division. What the heck is going on here? But the whole point of that was basically you have two daughters. Yes. How should parents...

engage their kids in math? Like, rather than give them the answer, or how do we get kids excited? And how do we find that in ourselves again, as we're helping our kids? Yeah, because I think the parents are critical to this in that a parent who says, well, I never used math in my life, so it doesn't matter.

That's not going to help. You know, we have to try not to pass on our own anxieties to our kids. What I try to do and what I would recommend other people do is to not be afraid to admit that you don't know something. That's a big...

strong reaction to say, I don't know. Let's figure it out. You know, either we can figure it out by thinking about it or we can look it up on the web. I mean, that's maybe a second best choice, but sometimes that's the best you can do. But the key being,

It's okay to not know everything. Now, I suppose in some models of parenting, the parent is the authority figure, and to relinquish authority is a big concession. And so these are maybe in more traditional homes, the parent will never do that. You're not supposed to be the equal or even the lesser of your kid. But in intellectual matters, I happen to like honesty. And if you don't know something, why are you trying to pretend? You're going to get found out anyway. Right.

You know, and if you don't, it's going to be ugly. There's a scene I'm trying to remember. Yeah, it's in My Left Foot. Do you remember this Daniel Day-Lewis movie? He's the, what was the name of the guy? He was a poet, Paddy, no, what was his name? I'm not remembering, but you know the movie I mean, right? He's completely paralyzed, except that he can move his left foot, the young boy. And he later goes on, it's a true story, goes on to be a magnificent artist who draws fantastic pictures.

and does paintings with the toes of his left foot. But early in the movie, we see him with his father, and he's thought at that time to be mentally challenged, the young Patty, because he has trouble talking. A question comes up. His older sister is working on her homework, and her father is there sort of reading a newspaper or whatever, and the daughter says, what's 25% of a quarter? And...

The father says, "That's a stupid question. You can't take a... 25% is a quarter. You can't take a quarter of a quarter." And then young Paddy, in a very dramatic scene, starts making noises over in the corner of the room. Nobody's been paying attention to him the whole time. And someone says, "He's trying to say something. What are you saying? Go, Paddy. What are you trying to say?" And they put a piece of chalk between his big toe on his left foot and his, you know, pointer toe.

And he starts scratching out something on this chalkboard. And you can tell as you're watching, he's trying to do 1/16 because 1/4 times 1/4 is 1/16. He has figured out 25% of a quarter is a 16th. But it's pretty hard to draw the 1 and the slash and the 16 with his foot.

And they it's sort of a sad scene because you can find it on YouTube that the father says, ah, he's an idiot, you know, just ignore him because they can't figure out what he's drawing. But he he's giving the right answer. Anyway, the point being that this father who didn't know how to do the problem could have stifled all discussion in the family, except that Patty broke through.

So I looked this up. I looked up the question my grade five came home with, and it's a telescope is set up on the roof of an office building so that the lens is 50 meters above the street. Okay. A scientist notices that when he looks out the telescope at a 45 degree angle to the horizon, he sees the top of a nearby tree that is along the same street. Okay.

Wow.

All right. I'm drawing a little picture here. 50 tall. Yeah. It's not obvious. The, you know, momentary of panic. I was like, I don't know, but we can figure this out. I'm sure. And we ended up doing a lot of YouTube and figured out, I think how to solve it. But, um,

But you could easily just be like, I don't know, and then you leave this grade five to sort of like figure this out on his own, which probably is not an optimal strategy. Did you ever come up with the answer? I think I have an answer. How did you do it? I'll tell you if mine's right. Well, I don't know. I may be misunderstanding the question, but I've drawn a picture with something that's a line standing up that's 50 unilaterals.

units tall, 50 meters tall. - Yep. - Now when you said 45 degrees viewed from the top of the building to the horizon, it wasn't clear to me if that means 45 degrees down, downward. - No, 45 up, sorry, yeah. - 45 upward from the horizon? - Yeah. - That's one heck of a tall tree. - Yep. - Okay, so let's see, all right, so 45 going up, that wasn't what I had drawn. But, so if I do that,

That 45-degree angle makes an interesting... If I go over to where the tree is and then continue up vertically from the top of the tree, let's see. No, sorry, the tree is very tall. I guess I'm hitting the trunk of the tree. It looks to me like I have to go 50 plus 10.

I mean, well, is that wrong? 50, it's, geez, I'm going to embarrass myself here, the professional mathematician. But it looks like it's a 50-foot tall building. And then because you said 45 degrees, that's going to make a half of a square, right? That's a 45-45-90 triangle, so to speak. That's a half of a square, a right isosceles triangle, in other words.

So it was 10 units over east or west to the tree. Then 10 units north would be to the top of the tree. So it looks to me like 10 plus 50. I would have thought 60. Yeah, I guess we probably didn't do it right. What we did was... Well, I may be doing it wrong right now. No, no, you're doing it wrong. We drew a triangle basically from the top of the building. And all we had to do was calculate the...

the height of the angle opposite the 45 degrees from the tangent, I think. Oh, I think so. How do you figure that? It was like a, we'd use a calculation. So it wasn't exactly. You could use the tangent, but it's, but with a 45, a right triangle that has a 45 degree angle as one of its angles,

The other angle is also 45 degrees. Right. Because 45 plus 45 plus 90 makes 180. Yep. So that's what I mean by saying it's a half of a square. It's a square sliced along its diagonal. That's a 45-45-90 triangle. Okay, yeah, yeah.

So it's like you're stacking a square on top of the whole picture. And so the fact is that when it's a square, the 10 units on the bottom translates into 10 units on the vertical side of the square. Oh, that makes so much more sense. Okay. But that's what you would have gotten from your tangent. Your tangent of 45 would give you 1. Yeah. So you would have concluded it's 10 units up on top of the 50 that were there to begin with.

So, yeah, I think 60. But OK, well, this is look, it's not usually good to do to do math over a podcast. What are what are the other things that parents can do to help their kids with math? And by help, I mean,

I encourage curiosity. And for you, it's probably fairly natural with your daughters. But for many of us, people have the same reaction I probably do. And my kids come home, I'm like, oh, no, today's homework day. What sort of YouTube videos am I going to have to look up when he goes to bed tonight so that I can answer questions in the morning? Well, one thing, I suppose a parent could directly try to help him or herself first. If the parent started to like math and felt more secure, that would be a big help. So when I wrote...

Back in 2010, I was asked by the op-ed page editor of the New York Times to write a series about math.

for that kind of reader, for just the educated sort of person who's curious about a wide range of things who would be reading The Times. And so that was the proposition. Do 15 weeks of math in The New York Times starting from preschool, you know, like the idea of numbers, up to as far as you can go to graduate school or beyond. And make it understandable. Make it fun.

And so it was a fantastic challenge. I really enjoyed that. That later grew into the book, The Joy of X. But it was always written with the parents in mind. It was meant for adults. And so there's a lot of references to things that only adults would know about literature or philosophy or sports or history or whatever. And yet I find that these columns tend to get used a lot in schools because they do cover the standard curriculum all the way through grad school.

But especially a lot of elementary school and middle school and high school. Anyway, so I think parents could try reading those columns. They're free if you have the time subscription or you can get the first 10 of them free. You could also—there's so many things on YouTube now. You mentioned YouTube, and there are a lot of good resources. There are just excellent—it's like a golden era of math communication. There's someone named 3Blue1Brown.

Well, he sets his handle on YouTube, who makes really great videos about everything. Just at a more pedestrian level, there's Khan Academy, where you can learn all kinds of things. He's very good. It's like his vision is bringing education to the world at no cost. And to a large extent, I think he sort of succeeds. He's really very good. But there are others, a person named Mathologer.

It's interesting because now the best teachers have unlimited reach. They do. Right? The internet has enabled, like before it used to be that you would get, you know, if you're lucky, you would get the best math teacher in your community. And now you can access, by and large, the best math teachers in the world. Yep. That's how it's done. And the kids are already doing it. So I had occasion recently to meet a seven-year-old boy whose mother died.

said he was very excited. She's actually a professor at my school at Cornell. And the mother said that her son, who is seven, you know, that he was very excited to find out that a mathematician whose books he reads for fun actually teaches at the same school as his mom. Could he come meet me? So I was talking to this little boy and there's actually a clip of that, my encounter with him, because I didn't know anything about him. I just thought I was going to shake his hand.

and, you know, who knows what, assign a book for him or something. But he's got all kinds of math he wants to show me, and so I thought this could be fun. So I told his mother, take a little video of us, turn your phone on, and maybe he'll want to watch this video afterward. Anyway, she records it, and we're talking for hours,

About 40 minutes where it begins with the boy whose name is Zamir. Zamir shows me, at first I don't know what is he trying to do. He's just telling me to do some games on the calculator. Then I realize he's making a thing called a magic square. So just to remind people if they've ever heard of a magic square or played with one, this is a three-by-three square of numbers. A traditional magic square, you put in the numbers 1, 2, 3, up to 9, right?

such that every row adds up to 15 and every column adds up to 15 and every diagonal adds up to 15. There's a way of putting those nine numbers, one through nine, in the, you know, arrange them in the square so that every row, every column, and every diagonal adds up to 15. That's an ancient idea. That's a magic square.

What this boy, Zamir, had figured out how to do was something similar, except that every row, column, and diagonal multiplied to the same number rather than added. If you multiplied the three numbers, you'd always get the same number. So he's explaining to me how he figured this out, and I thought, this is pretty good. This kid is seven. This is amazing. But then he says to me, after about like 10 minutes, it also works with natural logarithms.

And this is all captured on tape. You know, you can see it on YouTube, me and Zamir talking. And you'll see my eyes pop out of my head because I think, wait a second, what is this? This kid knows natural logarithms at seven. And pretty soon he says, yeah, and it works with imaginary numbers too.

And, you know, that's incredible. But so I bet you we got onto this because you were asking, what can parents do? Zamir had already. I guess the thing was that Zamir and other little kids were.

are very aware of what's out there in the world of the internet. And so he knew me from the internet. He knew other mathematicians from the internet. And he doesn't really particularly read books. He watches videos. That's where a lot of learning is happening for kids today. So I would say parents should try to learn

I mean, sure, they could help the kids with the homework, but there's a lot of good learning to be had on the Internet that seems to connect very well with this generation. So I would try to use that skillfully. It's not that hard. If it's a, so to speak, gifted child or someone with a lot of talent, that poses a different set of questions than the kid who's very frustrated and hates math or is demoralized or even...

feeling shattered about it because it can be super, you know, soul crushing. It can also just be boring. I mean, there's all kinds of different negative reactions. Some people feel ego deflation. They really feel like they're stupid. You know, other people can do math, but they just don't see any point to it. They think it's boring. And then there are yet others who find it very exciting. And, you know, so there's challenges for parents of all three types of kids, the bored child, the depressed or...

you know, distraught child who has tremendous math anxiety. And then the kid who wants to do more math but is limited by the environment. So I don't know. I would have different ideas, I guess, for each one. That makes sense. Tailoring it to the kids. Switching gears a little bit, talking about your recent book, what is calculus?

Mmm. Calculus is one of the greatest ideas of all time. I would say it ranks right up there with relativity theory from Einstein, with quantum theory of the atom, you know, with evolution from Darwin. And, I mean, it's just had an enormous impact on the history of the world. It's the mathematics of change, if you had to say it in one word. That's what calculus is about, how to quantify things that change, especially things that change in ever-changing ways.

So it's the first part of math that can cope with the dynamics of the world. And, you know, what do I mean? So the simplest kind of change is something moving, literally moving, moving, changing its location from place to place. So you could throw a ball or you could, you know, hurl a javelin. You could be thinking about the planets moving around the sun. You could be thinking about the concentration of virus in the bloodstream of a person with HIV.

You know, after they take a combination drug therapy, their viral concentration will plummet, thankfully. And so when doctors develop strategies for the life-saving treatment nowadays, the triple combination therapy that has turned HIV into a chronic illness from what used to be a near certain death sentence,

calculus played a big part in quantifying the dynamics of how the immune system interacts with the virus and what role the different drugs that have been offered would play in all that. So that's it. I mean, calculus is the math for describing a world in flux. And since everything is in flux, you could see

that it's bound to be pretty useful to have the ability to do that. Can you tell us the story of how calculus, or math, I guess, in this case, and calculus influenced our treatment options for HIV? Well, okay, yeah. So take your mind back to, I'd say, you know, like the mid-1980s, where in the West, here in the U.S. or Canada, the HIV epidemic was really starting to hit. It was this very mysterious disease called

It wasn't so clear what was causing it. But the symptoms were very predictable, that a person who got infected would at first show flu-like symptoms. They'd feel kind of sick for two weeks, but then they'd get better. You know, they might have a fever. They'd have, you could see their T cells, important components of their immune system had a measurable change.

could detect some virus in the blood, but they'd get over it, you know. And so after two weeks, it seems like the person was sort of better. And then years could go by without any particular symptoms, except for this strange low level of T cells, these crucial components in the immune system. It seemed to be that the T cells were being depleted somehow by the

by the presence of the virus, but otherwise people weren't that, that sick. But then after maybe 10 years, suddenly,

a tremendous crash would seem to happen and they would become terribly sick and that's when HIV would become AIDS, at which point all kinds of nasty opportunistic infections would set in, weird kinds of pneumonia that you wouldn't normally see, weird cancers that were very uncommon. And then, you know, at that point, the person would only have a very short time to live, maybe a year or two. So the mystery...

I mean, what was thought to hold a clue to what might be going on was this bizarre asymptomatic period of 10 years. What's going on in the body for those 10 years when the person has HIV but they don't seem very sick? Is it the case, for instance, that HIV is hibernating during those 10 years, that it's just somehow lying dormant in the body waiting to come out and become full-blown AIDS?

If you believe that's the picture, and certainly some viruses do that. You know, people who have been infected with different types of herpes viruses, let's say, will know that they can have long periods with no symptoms in between outbreaks. So we do know, and chickenpox is a similar thing where people don't get shingles until they're much older after they had chickenpox as a little kid. So you can have tremendously long dormant periods of viruses not doing anything in the body. So if you think that's what's happening,

with HIV, then that would mean if you had any drugs to treat it, you shouldn't use them at the beginning when a person is infected. You should wait until the symptoms start showing after 10 years because you don't want the person to develop resistance to the few available drugs, this being back in 1985 or so.

So that was the way HIV used to be treated, that they really wouldn't do much until full-blown AIDS, and then it turned out that available drugs didn't help. But all of this changed around 1994 when a new wonder drug called protease inhibitor became available. And the trouble even though with that is that just one drug, people would always develop resistance to any drug you gave them and HIV would come back. So where math comes into the story is that

In the mid-'90s, Dr. David Ho, an AIDS researcher and a mathematician, Alan Perelson, worked together with a team of other researchers to figure out what these protease inhibitors were actually doing with HIV. How were they working? And they showed that after taking one of these drugs, that the levels of virus in the blood would drop exponentially fast. It would really plummet.

And what was so important about that is that by making measurements on the rate of this exponential drop, Perelson and Ho were able to show that the body was producing about a billion virus particles every day. HIV was making an enormous amount of new virus, and the immune system was clearing it out and flushing it out of the body just as fast as it was being made. So it was a completely different picture that was not the case that the virus was dormant those 10 years. In fact...

It was in this all-out fury. Just a battle of attrition, basically, with your body. Yes, exactly. It was a furious battle of attrition, an all-out war that was being held to a standstill. The immune system was holding HIV at bay. Until it gets exhausted and... That's it.

The math also showed that given this furious replication rate of HIV, it was no wonder that it was able to become resistant to essentially any drug. Mutations were happening so fast when HIV would get copied. Right. Because it gets copied inaccurately. I mean, it's an RNA virus with a bad copying mechanism. That's actually to its advantage because it can generate many variants that can escape any drug you try to hit it with. But what Perelson's math showed is that if you did

Two drugs, your odds were a little better because then the virus would have to do two simultaneous mutations. The odds of that were lower than one. But three drugs would be the sweet spot where the odds were so low that HIV could mutate three simultaneous ways.

that you could basically keep it at bay for a long time. And that's now the modern regimen, the three-drug therapy. So math was key in understanding that. It certainly didn't solve the problem on its own. You needed the immunologists and the doctors too, and of course the pharmaceutical companies. But with all of them working together, calculus was a key supporting player in helping change the way we look at HIV and certainly how we treat it.

And our effectiveness at that. And it also played a role in GPS. Can you explain that one? Or it's the fundamental role in GPS, I think. Well, that's right. Sure. Anytime we use our GPS gadgets to find our way to a strange destination or sometimes even to find our way home when we're lost after going far away, GPS is a wonder of calculus. It's got so many different aspects of calculus built into it in the

the way that it acquires signals from the satellites overhead, the way that the satellites and the whole system estimates distances by doing a complicated mathematical calculation. You know, it has to look at distances to three or four different satellites overhead.

There's also really what the GPS system does is it doesn't directly measure distances. It measures time and converts those into distances. So the time is the time it takes for a signal to go from the satellite to your GPS receiver given that it's traveling at the speed of light because it's an electromagnetic wave, this signal.

It's going to move at the speed of light. And so you have to time very precisely how long it takes for the signal to get to your receiver from the time it was emitted by the satellite overhead. Now, what's really tricky about all that is that the satellites need to keep extremely accurate time. They have onboard atomic clocks in them, the most accurate timepieces we know of.

that are based on principles of quantum theory, which itself is built on calculus as its math, its infrastructure. But more than that, when the satellites are moving so fast overhead, they're actually going fast enough that Einstein's theory of relativity applies to them in a significant way. And the clocks on board those moving satellites

run at a different rate from the clocks on the ground. In other words, time changes. Time doesn't move at the same rate. It sounds unbelievable. I mean, Einstein thought of this idea a little more than 100 years ago. Time actually changes if you're moving. It can speed up or slow down. Can you explain that for me? Well, God! LAUGHTER

- It's not an easy thing to explain. I mean, we'd really have to go into a bit of relativity theory, but it is a consequence of relativity that time can change in two ways. It can actually slow down when you're moving fast,

It can also speed up when you're in a weaker gravitational field. Okay. And when these satellites, because they're farther from the center of the Earth, are in a weaker gravitational field slightly than we are on the ground. So these are minuscule effects. We're never aware of them in our ordinary life. But these GPS satellites are so accurate in their atomic clock timekeeping that they can actually, and they do all the time,

effectively confirm Einstein's relativity predictions in this sense. If we didn't build in the Einsteinian corrections to timekeeping, the whole GPS system would fall apart in about 20 minutes. It wouldn't be able to keep accurate time. So...

Okay, it's a big long walk I've just taken here to give it to you. But I mean, there's a lot more to this. But suffice it to say the GPS has a lot of calculus and advanced physics built into it. And we don't give it a thought. You know, we're just trying to get home at night.

Let's see if I understood this correctly. So I plug my phone into my car, turn on Apple CarPlay, get directions. It's triangulating between three or four different satellites and the time that the satellites sync up their atomic clocks. And then it's triangulating between how long it gets from each of those satellites to my phone. And.

And then based on that, it can triangulate where I am. Did I? Yep. You got it perfectly. Because those clocks on those satellites are very accurately maintained and the military keeps extremely careful of that.

measurements of where the satellites are. So they have to know exactly where those satellites are and also, you know, what time it is on board the satellites. And so as you say, now they're all at somewhat different distances from your car, from your receiver. And so, yes, you get to measure, as you say, triangulate, you get to measure several distances, three or four distances to all those different satellites. And knowing those four distances, knowing where those satellites are,

That places you uniquely on the Earth. Not only that, it also places your velocity. I mean, so GPS can tell you how fast you're moving as well as where you are in three dimensions.

How does it tell you how fast you're moving, just based on where you were and where you are at the next? Okay. That's it, right. So it can measure your position a nanosecond later or whatever, a small unit of time later, and all those four distances have changed slightly. So that's, of course, one of the key ideas of calculus, figuring out a rate of change, what we call a derivative. That's an example, the rate of change of your position when a short time interval elapses

That's a quintessential calculus calculation. What are the other terms that we would use around rates of change? Well, in economics, people are always using the word marginal. So if they say the marginal utility, you know, how much extra pleasure or utility do you get from something for the next dollar that you spend? Or what's your marginal return on an investment? You invest one more dollar, how much bang do you get for that buck? Right.

So, those are rates of change, the rate of change of return with respect to investment. That would be an example and that's a common sense thing. You don't want to ... At some point, the marginal returns start going down, so it's not worth it to put in that extra dollar of investment.

So we use that idea a lot there, but certainly in physics we talk about speed. That's a rate of change of position with respect to time. We have acceleration. That's the rate of change of velocity. But even something as simple as a paycheck, you know, when I say I'm making $6 an hour,

Well, let's not say that. Talk about minimum wage. Let's say we're making, okay, $15 an hour is what they're talking about now. Everyone should be able to make that. If you had a decent wage, that's still a rate of change, right? It's dollars per hour. Anything where you say per is a rate.

exchange rates, you know, how many marks per dollar or how many pounds per dollar. Those are rates. You mentioned economics. I'm curious as to why physics math is so accurate and economics math is so wildly inaccurate. Is that because the mathematical models don't apply as well? Or, I mean, to what extent does biology give us a better model for economics than physics?

Lots of great ideas in that question. Well, okay, the short answer would be that physics is a lot simpler than economics. You know, when you measure the moon, it doesn't mind. It doesn't react. You know, it's just an inanimate thing. The laws governing inanimate bodies are just a lot simpler and more easily quantified than the laws governing populations or individual people. So the task of the social sciences is extremely hard.

they get to feed back on the system that's measuring them. It's also their ethical issues. You can't do experiments as easily or sometimes you can't do them at all on people or on populations of people.

It's much harder to do controlled experiments. You try to pin down some variables and they just pop up somewhere else. So if you think about the history of science, which sciences were solved first? Or where did we make progress? The first science to really make good progress was astronomy, which I always think is kind of surprising at first, given that the moon and the stars and the planets are very far from us. They're very remote. And you might think, like, why shouldn't biology be the subject

where we make the most progress. It's very important, medicine and helping sick people and prolonging life and all that. You would have thought we devote so much attention to that, but it's not-- we did-- you know, we had witch doctor medicine for most of history, and even to this day, there's still a lot of black art in medicine. And it's because it's intrinsically complex, where astronomy, if you think about it, has a lot going for it. The motions of the planets are very repetitive and regular.

The moon is very predictable. It's also very slow. It doesn't change that much from night to night. And it's very observable. You can just look up there and see what's going on and make measurements, pretty decent ones. So it's at a time scale where it's not too fast or too slow for naked eye observation, whereas the processes in the body are so fast sometimes, and the molecules or the cells involved are so small that

that biology is just intrinsically really hard and economics and sociology and psychology are even harder still. Do you think biology is necessarily valuable then? Or is it a matter of like, we don't,

Once we understand more, we'll be able to measure and predict with the degree of accuracy that sort of like rivals physics? Or is it something that is just, it's inherently too complex and no amount of computing power is going to be able to solve it because it's a dynamic system? Well, I think it's closer to the second, though I wouldn't be as fatalistic about it as that. We have made a lot of progress in understanding all kinds of things about, say, arrhythmias in the heart, which are ultimately electrical problems of

You know, aberrant waves propagating in the heart electrically that shouldn't be that are making muscles in the heart contract at the wrong times so that blood doesn't flow effectively. You know, so there are some parts of biology where it's really almost like mechanics, the heart as a pump. But there are other parts of biology, say, you know, emotions in the brain and mental illness, things like this are still very, very problematic or problematic.

So, you know, that's in a way it's a good thing. We have a lot of work to do that will keep us busy for generations to come. But I think it's just in the nature of biology that it's much harder. As you say, it's very complex. There are lots of parts. Also keep in mind that there's enormous diversity in biology, that my genes are different than yours, even though we're both human beings. There's right. Our molecules are the same.

But we're configured differently, whereas in physics, it's not like that. Any two electrons anywhere in the universe are absolutely indistinguishable. There is no diversity of subatomic particles. I mean, they're different particles. Electrons are different from protons. But every electron is completely the same in every respect as every other electron. Same charge, same mass. They never break. They never age. They never chip.

You know, I mean, so in that way, physics is really simple. Whereas hemoglobin molecules, you know, could all be a little bit different.

There's also a lot of noise in biology, a lot of randomness that's just because of molecular jiggling. It's more chaotic. It's just inherently chaotic. Yeah, so it's really hard, and it's going to keep us occupied for a while. But still, it is ultimately chemistry and physics. I mean, there's no—I don't believe in any vital spirit or soul or anything like that. I'm sure some people do, but I don't—I mean, to me, it's all material. It's going to ultimately be understood—

in purely materialistic, meaning I'm not talking about money, obviously. I mean, I don't think there's anything there but atoms and subatomic particles configured in increasingly subtle ways. And correct me if I'm wrong, you're more interested just intellectually in sort of the orderly side of nature than the chaotic side, correct? I would say that's true. Yeah, my interest has always been in how does order emerge from chaos? So I do find chaos interesting as a starting point, but

Whereas some people revel in the chaos, I like self-organization. I like systems that spontaneously show astonishing feats of collective behavior where they somehow get their act together on their own with no commander, no outside force telling them how to behave. Because I think that's – we see that all around us. We see –

I assume life evolved spontaneously, you know, consistent with this position I'm giving that it's all materialism. I think that's one of the great mysteries. How do you get life from non-life without a creator? You know, it must somehow be in my worldview that the laws of nature and the right conditions will lead to life emerging from chemistry. But understanding the origin of life is one of our great scientific mysteries people are working on. It's not hopeless, but we don't have the answer just yet.

Where does morality come into self-organizing? Yeah, morality. Well, good thorny question there. You know, a traditional view would be, I suppose, that cultures and religious traditions and our parents give us, they help us learn right from wrong. And morality is something that's passed along through culture. But from a different point of view, it might be an outgrowth of biology. What do you mean?

Okay, yeah, well, so I'm thinking of a series of experiments that were done, if you could call them experiments, that were computer tournaments run by a political scientist named Robert Axelrod out of University of Michigan. So Axelrod is one of the great political scientists of our time, and he asked experts in many different domains, economics, psychology, game theory, math, physics, computer science,

to come together and play, actually to submit computer programs to play this famous game called The Prisoner's Dilemma. I suspect many of your listeners will know The Prisoner's Dilemma, but I'll just remind people. I mean, The Prisoner's Dilemma is a model for thinking about lots of situations that occur in real life where you and someone else have basically two decisions. Are you going to

play nicely with each other? And are we going to sort of cooperate? The scenario that is always talked about is you and your friend are both being held by the police, accused of having committed a crime. Okay, so the name comes from this 1950s era scenario of the two guys are being questioned separately by the police.

And they're, you know, each in their own room. And the police say, look, we've got lots of evidence on both of you guys. We know you committed that burglary together. If you will just confess, we'll go easy on you and we'll go hard on your partner. We'll put him away for a long time and, you know, you'll get off easy. But you just got to let's go. You have to confess. And, of course, they're saying the same thing to the other guy.

And the question is, are you going to rat out your friend, which is considered not cooperating? So the word is used in a strange way in game theory. You're cooperating with the police, but you're not cooperating with your friend by being a rat. And it's relative to your friend that we're talking about. So that's the question. It's in each player's interest. I'll think of these as players playing a game now. It's in each prisoner's interest to

to rat out his friend. But if both of them rat out each other, then the police have a very easy case and it's bad for both guys. They'll end up going away for a pretty long time. In the normal setup, the person who doesn't cooperate, if one rats out the other and one keeps quiet, the guy who kept quiet goes away for a very long time. So that's the worst outcome is to be a sucker where you keep quiet and your friend, you know, cheats on you.

So anyway, the point being that this prisoner's dilemma is this interesting, complicated scenario where it's in everyone's narrow self-interest to be mean and not cooperate. To defect. To defect is the jargon. Exactly. To defect is the jargon used in the field. If you defect on your co-player, your partner... But that assumes like a one-iteration game, right? Yeah, that's right. So in the thinking where we're only going to play once and I'll never see you again...

then it's always presented as a rational thing to just be a tough guy and defect on your partner. And both players think that way. And why it's considered a dilemma is that if we would both just cooperate with each other, it would be good for both of us. Right. But the problem is if we both have this mindset, it's always tempting, you know, to then say, well, maybe just this one time I'll take advantage because there's so much reward to be had by doing that. So...

Anyway, so back to Axelrod. He set up—I mean, it's been a real paradox, like how to play Prisoner's Dilemma. If you imagine playing the game repeatedly, if I'm going to see you again and again, maybe then there's more chance for cooperation to evolve between us because although in the short term I might benefit from screwing you, you could always screw me back. And, you know, maybe if we just sort of develop trust, this could be good for both of us over the long haul.

So why we're talking about this is you asked me earlier about morality, and I found Axelrod's experiments very illuminating in this respect, that what he found when he ran these tournaments of all these programs playing against each other, he found that the programs that tended to do well in the tournament had four properties in common. And he summarized these four properties by saying that first, they were nice, which he defined as meaning they never were the first one to defect.

If they were playing another program, they would always begin by cooperating on the first move. And then if the other program cooperated back, they would continue to cooperate because they were nice programs. They would never defect first. So they could set up these long strings of cooperation with other players and both do really well. So being nice was principle number one. That actually works well in an environment where there's a lot of players of equal strength, equal power,

against you in this vicious prisoner's dilemma game. And what's so interesting is nobody's being nice for moral reasons. These are egoists, okay? These are classic Adam Smith, self-interested, I'm doing what's good for me players.

They're not trying to be nice. They're not altruists for any moral reason. They're just trying to do what's good for them. But it turns out being nice is good for them if they're in an environment where there's a lot of other players equally strong, you know, who can inflict as much harm on them as they can on the other player. So that's principle one, be nice. Principle two that was found out, you know, and I'm not talking again here about any kind of moral philosophy. This is just what worked in the tournament. Be nice, but be forgiving.

If the other player cheats on you sometimes, well, don't just retaliate forever. You got to let bygones be bygones after a while. So it's good to be nice and forgiving, but it's also not good to be a patsy. You should be retaliatory. You have to inflict punishment. If someone has abused you for no reason, that is, they gave an unprovoked defection on you, then you have to hit back. So be nice sometimes.

be forgiving, be retaliatory, and finally, be clear. The programs which were too confusing and too subtle and too brainiac, like there were some programs that tried to make a statistical model of what the opponent was doing. Right, but then they become unpredictable, I would imagine, in some sense. That's right. They're so inscrutable, they're so unpredictable that you don't know what they're going to do, and it's almost like you're playing them anew each time you play, so you might as well defect on them.

Because you don't know, you can't figure them out. You can't build up a relationship. So be nice and clear, forgiving and retaliatory. And that's what emerges as a way to thrive in this environment. And what's so interesting about that is that this is a culture that has evolved around the world many times. This is eye for an eye and a tooth for a tooth. This is Old Testament morality. And computers discovered it on their own. This was not programmed into them. This is just what worked.

So it becomes our system because of biology and how we evolution. Basically, it's natural selection in a sense. People who employ that strategy are more likely to spread their genes. Well, that's so, yeah, okay. I mean, the argument would be if you play this kind of style, Axelrod actually did do an evolutionary version of the tournament where programs that did well

got to reproduce more copies of themselves. - Right. - So just like you described. And so then he looked over many generations, how did the population evolve, and he found that it sort of evolved toward players that played this style called tit-for-tat. - Huh. - Which is they always cooperated on the first move,

They're really simple programs. In fact, it was the shortest computer program submitted, only four lines of code. On the first move, cooperate, and on every subsequent move, do whatever the opponent did on the previous move. You just give it back to them. If they cooperate, you cooperate. And that's what we know of as tit-for-tat. Yeah, that's the tit-for-tat. Now, but the story is more complicated, like everything in life, because although Axelrod did find that tit-for-tat worked well in his tournaments...

It's not some kind of universal best way to behave. It turns out it's a little too ungenerous. It's a little bit stern. And what's really... Here's the weakness of tit-for-tat. Sometimes people make mistakes, and not because they're mean. I mean, just mistakes happen. Sometimes you are trying to be nice, and a person interprets it as an insult. Now, if that person is following a tit-for-tat morality, and so are you, then...

When this accidental defection happens, they'll say, well, now I, okay, now I got to give it back to you. So then they'll defect. But now you being a tit for tat player have to defect on them. Now you're in this vendetta that it's very hard to get out of. I have a friend who, who, his approach to this is forgive unless it's malicious. Yeah. Yeah. Good. Well, yeah. I mean, there's a, there's a computerized version of it, which is tit for two tats. Right. Yeah.

You know, or some number of tats. And so they're a generous tit for tats.

Yeah. Anyway, it's an ongoing story in game theory. This is a branch of game theory that people call evolutionary game theory. But just to not get too lost in the weeds, the point was it starts to give the contours of a story in which morality can evolve from self-interested individuals playing against each other. Now, is that really what happens in evolution? That remains to be seen. There are some examples of it in evolution, you know, like when...

Animals that hunt, say picture, I don't know, lions, the female lions out on a hunt, it can be dangerous to try to go after a big scary wildebeest. And they can kick you or hurt you, you know, when you're trying to kill them. So maybe it's better to let the other lions get a little ahead of me and I'll just kind of bring up the rear. You know, you could be a free rider.

And it's tempting. That is, all the lions are kind of playing a type of prisoner's dilemma with each other where it's tempting to cheat on the rest of the, what would it be called, a herd? No, that's not what it's called. What is it with the lions? A pride, a pride, a pride. Okay, but anyway, so there are instances in biology where something like a prisoner's dilemma is happening. But...

Yeah, anyway, so I don't know how much it really tells us about morality, but it's an interesting story. It may tell us something. Well, let's talk a little bit about decision-making in terms of mathematics is a hyper-competitive world, I would imagine. You're an advisor to students. You're part of, there's publishing papers, there's sort of like credit and attribution. How do you think about this with your grad students? How do you advise them? How do you help them make decisions? What models do they use to make decisions?

It's something that we think about a lot because really the first issue for a student is what to work on. Getting a PhD is all about discovering something new, staking a claim in the intellectual firmament, in the mathematical landscape. You have to come up with something new and interesting that is yours. You have to innovate. And where do you find a good idea? Because everyone's trying to do the same thing. And these are also smart, ambitious people too.

So one of the things that comes up a lot is that the first few ideas that you'll try, they might seem promising, but then at some point the going gets rough. It's hard to make progress. Maybe you don't have the technical tools to solve the problem, even though you're fascinated by it, or for whatever reason it's not turning out as interesting as you originally hoped.

And then there becomes a... I know sometimes on your show you like to talk about sunk cost and other fallacies. Here's a student who has spent a lot of effort on something. They've put in the cost, right? Should they keep banging their head on that same problem? Or is it now time to give up? Like, you know...

When is it time to quit? Yeah, talk to me about that. Well, it's a very hard decision because you've spent a lot of effort. You've grown attached to a certain question. If you quit on it, you may feel like, well, that was all a waste. But on the other hand, if you keep banging your head and not making progress, then there's an opportunity cost. You could have been thinking about something else, and maybe you would have made progress on that new thing.

So it's, and since you don't know how it's going to turn out, because this is an issue of genuine discovery and uncertainty, it's not clear what to do. Does contrast also come into it too? Like contrast? Yeah. Are you trying to think of, oh, there's other students working on this, but they may not be as clever as I am. So are you trying to like also ascertain what is the thing that I like doing that other people are

not doing, or I'm thinking about this in a sense of, you know, if you take somebody who is

is really intelligent and you apply yourself to maybe not the hardest problem, but the second hardest problem, you might have a better chance of ascertaining your PhD. Does that make sense? Totally makes sense. Right. So problem selection is such an art form, you know, that you, you, sometimes it's better to go, as you say, after the second hardest problem. So maybe the, the gold ring, you know, is premature. We have to build up to it. So, um,

There are people who try for the hardest thing and they never succeed because they're going for pie in the sky all the time. There's also, you sort of hinted at it, the idea of comparative advantage. What is it that I bring or my student brings that might give us an edge that other people don't have or don't?

So, there are all these different things to think about. The kinds of edge that we tend to have, we, I mean, really, I guess me and my students, is that we're very widely interested in many things. Some mathematicians are narrow and deep. Deep is our highest compliment, actually, in math. That person is deep. That work is deep. That theorem is deep. That's the standard of excellence, but not really for me.

I mean, whether it's just because I'm not capable of it, I don't know. But I kind of like shallow and broad. Maybe shallow is, you know, too self-deprecating. But broad, there's so much to be gained by thinking out of—I don't want to say that cliche—out of the box. But, you know— Out of the square. Yeah.

Okay, out of the square. Being interested in wacky things like humanities or philosophy or sociology, that has often really worked well for us. How does that help you? Well, because I see problems that other people don't realize are problems. So an example would be, okay, so yeah, let me give you a little story. So probably on paper and maybe in reality, my most successful student is a guy named Duncan Watts.

who was Australian from... He was a physicist. He was in their naval defense force. He wanted to go to grad school to study chaos theory, which was something I worked on a lot. And so he became my grad student, and the first few problems we worked on together...

It was just like this scenario I was describing earlier, that he was getting stuck. We weren't really making good progress. One was something about the way that lymph flows in the vessels of the lymph system. It makes certain oscillations. We were interested in things that oscillate and go up and down and have cycles. You know, so as an expert in cycles, I thought maybe we work on lymphatic oscillations. That could be important. But we couldn't make progress. We kind of got stuck. We didn't know enough about the lymph system.

So then we had to do what I think of as strategic quitting, that we have to decide it's time to cut our losses. This is not going anywhere. Let's do something else. So we did. We stopped. And we, you know, we quit. The thing your college or high school coach tells you, don't be a quitter. Yes, I say do be a quitter. Sometimes you have to quit when it's not panning out.

But then don't just give up. You have to come up with something else. So we kept thinking of things to do. And finally, in, well, I want to say in desperation, we started to work on a problem about crickets.

Now, crickets, you know, chirp. Crickets chirp. They make a rhythmic chirping sound at night. And in Ithaca, New York, here where I live, where Cornell is, we have a species of cricket called the snowy tree cricket that they chorus in unison. They all chirp together in these vast choruses so that they make this enormous sound at night in the fall or in the summer. Yeah.

And that's a fantastic kind of synchronization where these are not very clever creatures, but they somehow can harmonize and all chirp in unison. And biologists have wondered, how can they do that?

And it's partly interesting because the crickets are interesting, but partly because a lot of things in biology synchronize, like the cells in our heart that tell our heart when to beat. Our pacemaker cells get in sync. Brain cells get in sync when we're having epileptic seizures. So we would like to understand how to stop that. Anyway, so crickets are one instance of synchronization. And we thought they might be a tractable one to do experiments on because they live right here in Ithaca. And we could capture them and do experiments on them.

and maybe figure out how they are able to synchronize in their chirping. Anyway, so while Duncan was working on that, he made this interesting creative leap. His father had said something to him once.

This was back in the 1990s. Have you ever heard that you're only six handshakes from the president of the United States or from anyone else on earth? And this idea, of course, everybody has heard of this six degrees of separation idea. But Duncan put this together with what he was trying to think about with the crickets, that they're out there in these orchards. He wondered which cricket can hear which other cricket. Like, are they all connected together?

just maybe to the nearest ones to them? Or are they listening to the whole field? Or are they somehow connected in this network that's reminiscent of, you know, like, what if they're all just a few handshakes away from each other in terms of who hears who? So he put these two things together, this thought from sociology about social networks. And this is well before Facebook or anything else. You know, what is the basic mathematics of things that are

that you could think of as networks where every node in the network is just a few hops or just a few handshakes away from every other node. What would a network have to be like to have that property if it's really true that the planet Earth, the social network of the planet, has this property? And what would that tell us about how crickets can synchronize and how other systems networked in this way could behave?

I'm bringing this up because look at how weird this whole thing is. We're talking about crickets. We're suddenly talking about six degrees of separation, which comes from a movie and a play by that title, which originally came from a social psychology experiment. You know, nobody in the math department is thinking about that kind of crazy stuff. This is I'm interested in pop culture. I'm interested in science. And so I think about weird things.

And sometimes there are really important and relatively easy problems just waiting for a mathematician to look on them. Is that because you're applying sort of mental models from other disciplines to the one you're looking at? I guess so. Yeah. It's different mental models, knowing what questions arise in what fields, what things have long been problematic. Like, you know, the sociologists will tell you the social network problem of six degrees of separation was solved in the 1960s. They thought that problem was done.

And maybe from their point of view, it was. But from a mathematician's point of view, it was certainly not done. I mean, it seemed to me we don't have the foggiest idea mathematically which networks would have this small world property. I say small world because of, you know, the old idea you meet someone on a plane and you start talking to them and you realize you know someone who knows someone, you know, who knows their cousin or something. So this phenomenon of it being a small world, it's right under our noses.

yet it's not understood mathematically. We don't really know why is it so small. So there's a math problem there. And recognizing what is right under everybody's noses, you know, but seeing something wonderful in it, seeing something mysterious in it, that's a kind of skill. And that's the skill that I, you know, I'm very interested in trying to develop in my students and that I've been able to leverage in my own career over and over again.

I want to come back to strategic quitting just for a second. Is there, that sounds more like an art than a science.

There's a lot of things that come into it, you know. I mean, you have to decide how frustrated you feel, how much time you have left. Like for a grad student, there's often real-world questions. Do I—maybe I have a family. I need to graduate. I need to get a job so I get really paid properly. In that case, we better quit that much sooner and make sure your next problem is that much more tractable. You know, it's almost like with stocks and bonds and things. They're safe investments that don't return that much.

There's riskier investments that might return a lot but could also go belly up very easily. So you have to think about similar issues, right? Like what's your time horizon? When do I need to get out of here? How much payoff am I looking for? How much am I willing to gamble? So all those different things go into the thinking about what question should I work on next? Does anything else come to mind when you're asking that question of your students or yourself?

Yes, a big thing is personality. I mean, in Duncan Watt's case, since I was just talking about him with the cricket problem that became the small world problem, and maybe I should just say, I mean, it's gross. It's going to come off like bragging, but just to finish that thought, it's

This paper is now among the 100 most highly cited papers of all time in any scientific discipline. I mean, it's the biggest home run I'll ever hit in my life, for sure. And long after I'm dead, it's the one thing that Duncan and I will be remembered for. This was a monumental, game-changing...

bit of work that started from, you know, from just a crazy question. Yeah. I love those solutions where you're pulling something from other disciplines and you're seeing something that nobody else sees, even though it exists within the world.

Yeah. As far as like, what does it take to, you know, how do you decide about the quitting? I knew something about Duncan's personality, which was that before I had even really gotten to know him, I had observed a picture of him on his office door of him hanging by his fingertips, literally on a cliff 100 meters above the sea in Australia, a place called Point Perpendicular.

He's a rock climber and he's a little bit of a crazy person. He's also physically very, very impressive. He looks like a Green Beret, like someone that could kill you if he wanted to. And he's the kind of guy who hangs off of precipices by his fingertips and does it for fun.

So it seemed to me like with that student, we could work on something extremely risky and extremely exciting, and he would not be afraid, and he would find a way to make it work. He wasn't the technically best mathematician. He's still not. There were other kids that were more math Olympiad caliber who could solve any textbook problem you gave them. Duncan wasn't like that, but he could muscle his way to a solution. It wouldn't be elegant, but he could get it done. And he was fearless, right?

And so we worked in this area that... I mean, I had to tell him at the beginning, look, I know nothing about social networks. That's going to use a part of math called graph theory, which I'm not an expert in. If we were talking about crickets and oscillations, I know about that, but I don't know about networks. And actually, nobody really knows. So we're going to be way out of our comfort zone here. We might embarrass ourselves. And so we were very secretive about what we were doing. We didn't tell people. We worked on it for maybe a year or two. And...

You know, we had nothing to show for it for that whole time, but the hope was that it could be really big when it finally came out if we got lucky, and it turned out we did. To what extent does that raw skill matter in comparison to what other attributes do you see in students that make them successful? Well, raw skill certainly has its place.

You know, if you don't have any skill, and I'm talking like skill in the sense of technique, like if you were a composer and you couldn't really play, you know, on the piano to hear what you're trying to compose, you're going to have trouble. And in math, you have to have certain technical skills. And I'm sure in every discipline in computer science, if you can't program with some facility, there's only, you know, so much you can do.

But skill is so much what we emphasize. Technique, I guess I should call it. Technique is so much what we emphasize in school. And it's just one part of the portfolio. Courage is a big part. Judgment. You know, these are hard. They're intangible. I don't even really know how to define a lot of these. But a nose, having good taste, knowing what's going to be cool if it works, knowing what other people might find interesting.

Like my dad, my dad didn't go to college, neither did my mother, but my dad had a shoe store and he said to me, it doesn't matter what shoes I want to buy. You have to know what sells, you know, you have to know what the people want to buy. And so it's sort of true in research. I mean, it might sound a little mercenary to put it that way, but

We're sort of in the business of selling ideas. I don't literally mean for money. I mean, our currency is status and getting nice jobs and getting recognition and also the thrill of discovery and all that. But still, a great discovery that no one appreciates

is not really a great discovery because science is a social enterprise. It's not just enough to do the work. You have to communicate it and help other people understand why it matters. It reminds me of this story where a guy goes into a tackle shop and looks at the lures and says, do the fish eat these green, do they go after these green lures? And the guy working there says, I'm not selling to the fish. Ha ha ha ha ha ha ha ha ha ha ha ha.

Well, that's funny. That's good. I mean, I wonder about that story. What do you think? I mean, that's sort of not really right, is it? Because if the fish don't bite, you're not going back to that store.

Well, I think it sort of goes back to that organized morality and this sort of one-off, are we playing a multiplayer game? That's what I'm thinking. Are we playing a one-time game? And it was interesting. I was traveling with my kids this summer and we sort of went to this tourist trap and we ordered a bottle of water without even thinking. And it was eight euros.

And I was like, oh, well, let's work through this problem. And we're sitting there at the table. Like I was like, we just got, you know, we just learned something here. Like, let's pay attention to it and try to get the kids to learn a little bit. And I'm like, well, what would the difference be if this restaurant was full of local people? Do you think that they would do that? No. Okay. So if it's full of tourists, how can they do that? Well, because they probably will never be back.

And so you can get away with this, not saying it's right. There's no sort of moral judgment on my part here or wrong. But you can get away with it. Whereas if you were serving to an audience that is not a tourist place or a customer base, you can't sort of get away with that type of behavior because you would go out of business because people just wouldn't come back.

But when you're the place in town for this particular type of piece of cake that the town is known for, you know, everybody goes there and you order cake. And then, of course, you get thirsty and you, you know, the cake is cheap and the water is ridiculously expensive. No, it's a good example, a real world example of what we're talking about. There's the one shot businesses and there's long term relationship kinds of businesses.

Yeah. And they, and they will tend to evolve different strategies. It's understandable. What's really fascinating to me is like so often we, we know we're in a long-term relationship with somebody and we'll convince ourself that we'll make it up to them if we take advantage of them. Right. So if you think of four permutations of relationship, there's win, win, win, lose, lose, lose, lose, win. But in,

If we map that to biology, only one of those survives across time, right? Which is the relationship has to be win-win. But I find a lot of people, I'm going to take a little bit of advantage of you right now, not too much, but I'll make it up to you later. And then we never sort of, we always know what it feels like to be on the lose side of that relationship. And, um,

it never sort of like happens that it sort of equalizes or gets made up or so even if you think you're playing a multi iteration game, we often make these trade offs, I think in our head where it's like, I'll make this up to you in the future. And then we are inaccurate at that sort of scorekeeping, if you will. And I think that's so often how a lot of our relationships that work become after we leave work. I mean, I've noticed a trend with a lot of my

A lot of people I know, colleagues who retire, and then they go from having sort of all of these friends to not really having many friends, right? So their friends came from their position or sort of at work anyway. Their friends came from their position in part because they had all these trade-offs that they sort of realized later that they achieved success maybe in a way that was mutually exclusive from relationships of meaning. Uh-huh. Uh-huh. Hmm.

Very interesting. There were actually, you know, in a primitive way, computer programs that tried a similar thing in this Axelrod tournament after he announced to the community what the results of the first one were, that it was, at the time, good to be nice and forgiving. And then you would think many players, I mean, it was a much bigger tournament the second time, many more entrants, and you would think a lot of them would submit tit for tat now that they learned how well it did.

They didn't, though. Everybody thought they were cleverer than that. And they would submit things that were they thought basically could do pretty well against tit for tat by occasionally taking advantage of it. Just rarely, just like testing it, you know, or test the other program. And then if there was too much retaliation, they'd back off and go back to being nice. And it turned out.

that that little variant, that kind of mutation, if you will, was a really deleterious mutation because it led to exactly what you're talking about. It would lead to a breakdown of trust that had been delicately established before then, and it was very hard to get it back. Yeah, especially in human systems, right? Because you can appear to trust somebody but not actually trust them. So you can sort of trick them in a way that maybe the simulations in the computer can't do. That's right.

Which is sort of makes everything a little bit more complicated, but also really more interesting from a sense of you should go positive and go first always. And my friend's principle of sort of forgiving unless it's malicious seems like a really good life strategy. Uh-huh. Uh-huh. Yeah, it is. It sounds like this is a nice friend to have. You know, generally a sweet soul. Yeah, definitely. Listen, Steve, this has been a fascinating conversation. Yeah.

I've really enjoyed talking to you. Oh, thanks a lot, Shane. It was a pleasure to be with you. You can find show notes on this episode as well as every other episode at fs.blog slash podcast. If you find this episode valuable, share it on social media and leave a review. To support the podcast, go to fs.blog slash membership and join our learning community. You'll get hand-edited transcripts of all the podcasts and so much more. Thank you for listening. ♪