Steven Strogatz Says You Can Understand Math

Steven Strogatz: It’s a spooky thought! That here’s this thing, which is just a little piece of weight hanging off of a string, and it secretly seems to know this thing that I have just been learning about Y and X in algebra class, and it gave me this feeling that there’s a hidden world. That there’s this secret world of math underneath the world we can see. Honestly, it felt like a quasi-religious experience, to tell you the truth.


Alan: 00:00:00 Steve, this is so great that you can join me on the show, because you and I have had, over the years, so many fun conversations, and most of them about mathematics. Which word, I hope, hasn’t made people drop out of the show in the first minute.
Steve: 00:00:16 Yeah, great start.
Alan: 00:00:17 Yeah.
Because I don’t have any grasp of mathematic beyond very basic things. I taught my granddaughter the first steps of algebra on the back of a pizza menu, but she was six, and that’s the level I’m at. She could do it, but the most basic steps. And I’ve always thought that if I could understand math better, I could understand the universe better, because I think physicists, who understand it through math, see things that I can’t see and I just wish I had their vision. You think I’m wrong about that?
Steve: 00:00:55 No, you’re right about that. It would help. There’s a lot of beautiful and tantalizing and thrilling ideas about physics and other parts of science that are best seen through math, so that’s one reason why, for I think it’s about 25 years now, you and I have had these conversations. I think each little thing will help, and since I know you love science so much, this would give you a whole new angle on thinking about it.
Alan: 00:01:21 One of the things I always think about a story you’ve told in one of your books about how you became so interested in math and had a kind of, I felt a sense of awe about math when you were plotting the movement of a … what’s the thing that swings back and forth?
Steve: 00:01:41 Yeah, a pendulum.
Alan: 00:01:42 Pendulum, yeah. Couldn’t think of the word.
When you were plotting the movement of a pendulum on a chart, you suddenly realize what? What was it that came over you?
Steve: 00:01:52 Well, it’s an epiphany. It was a formative moment in my life. I was 13, I was in my first science class in high school, and our teacher … Mr. [Decursio 00:02:04] … gave each of us a little stopwatch and a pendulum that was unusual in that you could make it longer or shorter. It was sort of retractable or collapsible. Or extensible, the way an old spy glass in a pirate movie … you know, where the guy stretches it out or shrinks it back down. So it was that kind of thing.
And you could take this pendulum, make it longer or shorter, and then the instruction was, “Let it swing back and forth ten times and time it with your stopwatch. How long does it take to go ten times back and forth?” And then … it was really supposed to teach us how to use graph paper. That’s really what this was supposed to be. That on the X axis, you’re gonna put how long was the pendulum in centimeters, and then—
Alan: 00:02:52 Okay, if anybody’s listening who is not even up to X axis, that’s the line that goes horizontally, right?
Steve: 00:02:59 Right. Right, right, right.
Alan: 00:03:00 And then there’s a line that goes from the bottom up, that’s the Y axis?
Steve: 00:03:03 Yes, thank you, beautiful. Right. That’s right. Good point! Good one!
Alan: 00:03:09 I don’t want to lose anybody.
Steve: 00:03:11 No, it’s good, and it really helps actually, since we’re doing this with sound rather than pictures. Please jump in with that kind of thing, because that’s an example of me forgetting people may have been a little while since anyone did Cartesian coordinates and X and Y and all that.
Alan: 00:03:30 Quite awhile, actually. But … so you were starting to plot the movement on that graph paper?
Steve: 00:03:38 Yeah, right. So it’s a graph where, as you say, you’ve got this horizontal axis where you’re gonna say, “How long is the pendulum? Is it two centimeters long?” Or three or four or whatever. And then, “How long did it take to make ten swings?” That’s gonna be graphed as a dot, vertically, measuring the number of seconds. Was it 20 seconds to make 10 swings, or 23 seconds, or whatever?
And so as I started making a dot, and then I stretch the pendulum, and then I time it again, and now I got a new dot. And I plot these dots. After about four or five dots, I notice that the dots were all forming a pattern. They didn’t make a straight line … I thought they might, but they didn’t … but they did definitely make a clear shape, and I recognized this shape, because I had just been learning about it in my math class in algebra, and it’s a shape called a parabola.
Alan: 00:04:32 What does a parabola look like in real life?
Steve: 00:04:34 Parabola would be if you’re at a drinking fountain and you press the button to drink the water, where it makes that arc.
Alan: 00:04:41 And that’s a parabola.
Steve: 00:04:42 That would be a parabola.
Alan: 00:04:43 So why was this an epiphany for you? That you got a picture that looked like the drinking water fountain spurting from writing dots on a paper that corresponded to the swing of a pendulum. Why was that an epiphany?
Steve: 00:04:59 Because it gave me the feeling that this pendulum knows algebra.
Alan: 00:05:09 Okay.
Steve: 00:05:09 You know what I mean? It’s a spooky thought! That here’s this thing, which is just a little piece of weight hanging off of a string, and it secretly seems to know this thing that I have just been learning about Y and X in algebra class, and it’s totally inanimate. It’s not conscious, it’s not even an animal, it’s not an insect, it’s just a weight. It’s just a piece of iron or something, and it somehow is behaving in accordance with algebra, and it gave me this feeling that there’s a hidden world. That there’s this secret world of math underneath the world we can see.
Alan: 00:05:47 So now, jumping ahead to some very smart people, like [Max Tegmark 00:05:51], who says … I believe he says this and means it … that the universe is math. I always thought math could describe the universe, but are you saying, along with Max, that underneath everything is math?
Steve: 00:06:12 I think I’m probably closer to you on this one, because when I hear people say that the world is made of math, it’s hard to know what that would mean. I mean, I know he’s written a book about that. I think it’s called “Our Mathematical Universe” or something like that. And this idea has been around since ancient times. Pythagoras, around 500 BC said … or, his followers said … “all is number.” Everything is number. It’s unclear—
Alan: 00:06:41 To me, it sounds a little like falling in love the way you did with the parabola and thinking, “This is mysterious and wonderful” … which it is, by the way, that the pendulum can make the same picture as the drinking water fountain can, just by observing the way the different lengths travel … but it sounds like it’s puppy love. It sounds like Max and others maybe … I have to talk to Max about this, but it seems like they may be too much in the thrall of the wonder.
Steve: 00:07:22 Well, I mean, it’s an interesting phrase you use, “puppy love,” because I think the love deepened … if that first feeling was some kind of infatuation or … I don’t know, honestly, it felt like a quasi-religious experience, to tell you the truth. I remember having a chill of a type that I’ve felt only once or twice in my life since then. Once when I was in Florence, the first time I ever went to Florence, Italy, I stood before … as so many people have … Michelangelo’s “David”. You know, that statue of the boy and his hand is too big and he’s looking very determined at Goliath, who he’s probably scared out of his mind he has to fight.
And I mean, there was something in seeing that statue that thrilled me … thrilled is not even the word for it, I felt a shiver of some kind of transcendent beauty or perfection or something that I think was similar to the feeling I felt when I suddenly realized that math was underneath the patterns of the world.
Alan: 00:08:26 That sense of beauty is so interesting. I think civilians like me often think of mathematics as being rigid and hard and always reduce it to 2 + 2 = 4. Whereas, I’m constantly reading about how one of the attributes of a good mathematical proof is its elegance, which is a lot like saying the thrill of beauty that you got standing in front of the “David” was one of the things that makes you know a proof is a good proof in mathematics, which sounds very non-rigid.
Steve: 00:09:03 Mm-hmm (affirmative). Well, that’s right. Mathematicians don’t experience it as … I mean, hard, yes. Hard in that it’s hard for us to do. But hard, as far as cut and dried or uncreative or … no, I mean, math is supremely creative, just like any other … to us, it feels like a kind of art. Like sculpture, like music, maybe even acting. I mean, the whole person is involved in the creation of math, and in the discovery of math. Probably it’s a little different, in a way, from art, in that we do have this feeling that there’s something outside us that we discover.
Like I know you’ve written plays, you’ve written screenplays. Do you feel like you’re discovering them? Like Michelangelo himself talked about discovering the statue in the marble.
Alan: 00:09:50 Right.
Steve: 00:09:51 He had to release the statue from the marble.
Alan: 00:09:53 Cut away everything that’s not a horse, I think he said.
Steve: 00:09:56 Yeah. Yeah, yeah, like that. Do you feel you’re cutting away everything that’s not the script?
Alan: 00:10:00 Well, it is a process of discovery to me. I always like discovering something in acting or writing, rather than putting it there, forcing it into it. Finding out what the scene is about, finding out how the character feels about other people … to discover it seems to me to be more open and you’re less likely, I think, to impose a stereotype onto it. Something that you’ve seen before or something you’ve heard before. This is more innocent and raw … I always hope.
You want to arrive, not just at a sense of beauty as a mathematician, you want to arrive at a proof, which is, I presume, unassailable. Or are there proofs that other people say, “No, this proof is no damn good”?
Steve: 00:11:00 Well there certainly are. Yeah, that’s an interesting thing. So in our dreams, we think of proofs as unassailable logic, but in reality, proofs are made by people, just like anything else that’s made by people. Some proofs have stood the test of time for thousands of years. Euclid has some arguments in his book, in “The Elements”. That’s probably the first math textbook we ever have, from 300 B.C. and those proofs are just as beautiful today as they were … whatever that is, 2300 years ago.
For centuries, they were held up as unassailable. That they were perfect bits of reasoning. And yet, that isn’t true. Even in his first theorem in The Elements, where he shows how to make an equilateral triangle using just a compass … you know, that old thing with the pen in it and the pencil? You can spin around to draw an arc of a circle … and a straightedge, which would be like a ruler, except with no markings on it. You can’t measure a length, but you can draw a straight line with it. So with just a compass and a straightedge, Euclid shows how you can take a little piece of a line and build an equilateral triangle from it, and people thought this was perfect proof, except that around 1800 … or maybe it was early 1900s … a gap was discovered in the proof! It was assailable. There was a … I wouldn’t say “a mistake”, but there was a hole in the logic that needed to be patched up, that Euclid hadn’t noticed.

Alan: 00:12:30 And this was a really smart guy. I got a sense of that … remember when you gave me a geometrical problem to solve, and I spent two years … every time I’d get on an airplane, other people would be reading books or watching movies, and I’d be with a compass and a straightedge, trying to derive a proof of this … what seemed like an obvious thing. It was obvious what was taking place, but I couldn’t prove it! And I must’ve sent you 100 different proofs, or what I thought were proofs, and you kept saying, “Not yet.”
Steve: 00:13:08 Well, welcome to the world of math! You became a mathematician with that experience. It’s so interesting that you were stubborn like that, because that’s what … I’m serious, that’s the mathematical impulse. That feeling of determination that you had, that you didn’t want to give up. Why didn’t you just quit?
Alan: 00:13:26 It just, it was obvious to me that this thing was true. That … I don’t want to get into the whole complex idea, but it had to do with the corners of an equilateral triangle. And one was equal to the other in some way. And it was just obvious that it was. And I couldn’t prove that it was true in all cases, and it just seemed … it seemed too logical for me not to be able to seize the logic and use it.
I don’t know why I stuck with it. I guess I’m stubborn.
Steve: 00:14:10 Well, I think there’s an interesting little puzzle there, and actually, maybe that’s the word to use. Think of how many people like to do puzzles just for pleasure. The feeling of being stuck on a crossword puzzle or Sudoku, this is not something we do in school for homework. We do it for pleasure, just ’cause we like to think, or do logic puzzles out of books
Alan: 00:14:30 So let me ask you a kind of a fundamental question: where do you stand on the idea that some people that they’re just not math people? That there are math people, and non-math people, and they can into the world as non-math people, and they’re never going to be able to change. How do you feel about that?
Steve: 00:14:51 Well, let me give you a kind of waffle-y answer on it.
There is such a thing as talent. I do believe that some people might be more suited to becoming really entranced by math. But I do think most people give up way too early, and that … most people use this as an excuse. A very understandable one. They’ve had all kinds of experiences in school that were aversive that turned them off to math in some way, that made them feel stupid or stressed or anxious or something.
Alan: 00:15:27 I had a similar experience. I remember in high school, I went into my first class in trigonometry, and for an hour, there were these unintelligible questions being flung at the class, “What’s the sine? What’s the cosine?” I mean, I didn’t know what they were talking about, and it all seemed like some mechanical motor that had parts I was supposed to plug into the motor in the right spot, and it didn’t sound at all like what you were describing before of the fun, the beauty of math. It sounded like just a mechanical process, and I left and then never came back. I’ve regretted it all my life, ’cause I wish I understood trigonometry.
Steve: 00:16:09 Well the description that you just gave, I think many of your listeners will feel it themselves. It’s just a question of where they hit the wall. So some people will say … I mean, what I always hear, and I think all mathematicians or all math teachers will tell you this … what people say to us when they hear that we teach math is, “I always liked math until I got to,” and then it’s something different from person to person. “You know, I really liked math until I got to fractions. Those were very confusing, the thing with the common denominator.”
Or other people will say, “I really liked all of arithmetic, and I liked algebra, but there was something about geometry, I’m just not a visual person, I’m not right-brained.” So what I think is going on is that people more or less instinctively like math, just like most people like puzzles. It’s just that it can become very alienating when something goes too fast or you missed a few days, and so you missed an important definition … that it starts to seem … to use the words you used earlier … rigid or mechanical or something that feels arbitrary, and that is no longer any fun.
Alan: 00:17:17 Seems hard to believe, to me, that people don’t come in with some sense of math, of counting or being able to estimate quantities.
On the science show that I did … I can’t remember if it was birds or chipmunks or some damn animal … that could look at a dish of three pellets, next to a dish of five pellets, and could tell the difference and went for the five-pellet one. Probably they weren’t counting, I didn’t hear anybody go “woof woof woof woof woof.” But they were estimating. That seems, to me, part of what you’re talking about. Seems to be an early stage of math.
Steve: 00:18:07 Mm-hmm (affirmative). Sure. Yeah, no, the people that do psychological studies of little kids or cross-cultural studies, there’s definitely a lot of intuition about math that comes in childhood, and as you say, there are different animals that have some ability to estimate sizes and quantities and things. So yeah, I think there is something probably inborn, but of course, a lot of math has to be learned. Trigonometry would be an example. You’re not gonna just instinctively catch on to the idea of the sine function, but … I mean, one thing that does help is to connect something that seems abstract to something that’s in the person’s real life.
Like say with the sine function. I honestly didn’t know the thing I’m about to tell you ’til about a year ago.
Alan: 00:18:54 Oh come on—
Steve: 00:18:55 I’ve been thinking about the sine function—
Alan: 00:18:56 I can’t wait to hear this.
Steve: 00:18:57 Yeah, I’ve been thinking about sine waves for about 50 years now, and a friend of mine who’s … she’s actually a scientist here at my university, at Cornell, but … she does molecular biology, but on the side she likes to sew dresses. She’s a seamstress.
Alan: 00:19:17 Uh huh.
Steve: 00:19:17 You know, a person who’s really interested in sewing machines and sewing. And she showed me a certain pattern … if you want to attach a sleeve to a shirt, or a sleeve to a dress, you start with something that looks like a cylinder … that is, you take a piece of fabric and then make it into a tube, that’s gonna be the sleeve … and you don’t just attach the tube onto the body of the shirt or the dress, because that’s gonna stick out like a scarecrow from the side. You want it to drape down at an angle. And also, it has to be such that it will attach correctly to the hole in the body of the shirt or the dress.
Alan: 00:19:56 So the tube will be cut at an angle so it’s more like an oval—
Steve: 00:20:01 Right, something like that.
Alan: 00:20:01 And then the hole has to be more like an oval, too, to attach it. Is that right?
Steve: 00:20:06 Yeah, yeah, something like that. But then she says, “What if I then slit the tube open,” because when she’s working on the fabric, it’s not a tube, it’s a flat piece of fabric on the table. “What should the top of that thing look like so that when I roll it up to make a sleeve, it’ll fit right on the dress?” That is, the top of the tube, when it’s laid out flat … I don’t know if you can picture [inaudible 00:20:27]
Alan: 00:20:27 Yeah, I can. I’m picturing this scalloped shape, ’cause when you fold it over—
Steve: 00:20:30 Yeah! That’s it! That’s it! The scalloped shape.
So she drew the scalloped shape, she said, “What is this shape?”
It turns out it’s a sine wave.
Alan: 00:20:37 Oh, no kidding.
Steve: 00:20:38 It’s the sine wave of trigonometry.
Alan: 00:20:40 So how would you define ‘sine wave’ mathematically? Or scientifically? If you’re doing math and you say, “Let’s make a sine wave here,” you don’t cut out an arm for a shirt—
Steve: 00:20:56 No, that’s right.
Alan: 00:20:56 So what—
Steve: 00:20:57 Right, you don’t usually do it through sewing. You could! You could. That’s not traditional, but you could.
What we usually talk about is we imagine a point moving around a circle. So you could picture a person on a Ferris wheel. There’s someone sitting in a seat on a Ferris wheel, and they’re going up, and now they’re at the top of the Ferris wheel, and then they’re coming down, and then they’re at the bottom, and they’re going around and around. So if you imagine that person on the Ferris wheel, and the Ferris wheel is turning steadily … so it’s not speeding up, it’s not some weird jerky motion, it’s just steadily rotating around and the person’s going up and down and … and then you ask … well, it’s a bit contrived, but basically if you kept track of how high up they were at each time … so that is, they’re high, then they start coming down, then they’re low, then they’re going up high … if you kept track of their height … again, I’m asking you to visualize one of these X and Y graphs—
Alan: 00:21:52 Right, so then, it sounds like it goes up above the midline, comes down, and goes back up again, and in between, it’s got a circular top and a circular bottom.
Steve: 00:22:02 Yeah, except they’re not exactly circular. They’re curved.
Alan: 00:22:05 We’re not talking about parabolas again, are we?
Steve: 00:22:08 No, we’re not. No. It’s just the shape that … I mean, maybe I shouldn’t jump in with that. By ‘circular’, you didn’t mean literally circular.
Alan: 00:22:17 No, no, I meant—
Steve: 00:22:18 If you just meant sort of curvy.
Alan: 00:22:18 Around top and around bottom.
Steve: 00:22:20 Yeah, it’s around top and around bottom. That’s right.
Alan: 00:22:22 Yeah. That’s what I meant.
Steve: 00:22:23 So that … okay.
Alan: 00:22:26 Now, are you still teaching a course at Cornell for people who hate math?
Steve: 00:22:40 Yes.
Alan: 00:22:45 How does it work?
Steve: 00:22:47 We don’t describe it that way, by the way.
Alan: 00:22:50 Oh, I thought that’s what was in the catalog!
Steve: 00:22:52 No, it’s not.
Alan: 00:22:52 Not in the catalog. What do you call it?
Steve: 00:22:55 We try to put a little more positive spin on it.
Alan: 00:22:59 What do you call it?
Steve: 00:23:00 The course is called Mathematical Explorations, and it’s in a category of courses that we call “mathematical and quantitative reasoning,” which … honestly, what it is is that Cornell feels everybody should know a little bit of math before they graduate, and there are some kids who thought, “Oh … I thought I was done with that in high school, and I’m really not eager to do it again.” They just want to be a history major or a government major, or art, or something. They’re not interested in math at all, yet Cornell imposes this requirement on them, like the swim test. You have to be able to swim or you cannot graduate. So I’m the last person standing between them and graduating, that they have to take some math course.
So what do you do with these people? You could give them another course in algebra or something, but we think a better thing to do is show them what math really is. Actually, let’s be honest for once and not do the artificial kind of math that they suffered through in high school … I mean, for these students, it was a suffering experience. Instead, let’s give them a taste of what it really means to discover something and to make conjectures and to be frustrated and then solve it and have epiphanies.
And so this course, that’s why it’s called “Mathematical Explorations.” They’re genuine explorations where I don’t teach them how to do things. In fact, I don’t lecture at all. I give them questions that are meant to be interesting, and I have them work on the questions together … they sit at tables of four to six students … and they can talk and brainstorm and challenge each other, and then we all, as a class, talk about their ideas.
If you’ll allow me to keep going, I want to describe it a little more, because it’s a very—
Alan: 00:24:53 Yeah, I’d like to hear an example of what you challenge them with, what kind of a question.
Steve: 00:24:59 Sure. Okay.
Can I set it up first by saying that there’s a whole psychological dimension to this. I’ll give you a math example in a second, but I wanna set the stage by saying that this cohort has to be approached carefully. These are students who are very anxious and resistant, and don’t want to be there. Most of them are seniors. I mean, that’s a measure of how little they want to take this class. They’ve done four years, they’ve put it off as long as they could, and they’re taking it at the last minute so they can graduate.
So they really don’t wanna be there, they come in usually with a pretty unhappy-looking face, and so I think that that has to be addressed. Before we even talk about math, let’s talk about what this room is gonna feel like … that we’re gonna be in together … sounds like a play, doesn’t it? Like something from Sartre … We’re gonna be in this room together for the next three months—
Alan: 00:25:53 Another definition of Hell.
Steve: 00:25:55 Yeah.
So what do you do with these people? You could give them another course in algebra or something, but we think a better thing to do is show them what math really is. Actually, let’s be honest for once and not do the artificial kind of math that they suffered through in high school … I mean, for these students, it was a suffering experience. Instead, let’s give them a taste of what it really means to discover something and to make conjectures and to be frustrated and then solve it and have epiphanies.
And so this course, that’s why it’s called “Mathematical Explorations.” They’re genuine explorations where I don’t teach them how to do things. In fact, I don’t lecture at all. I give them questions that are meant to be interesting, and I have them work on the questions together … they sit at tables of four to six students … and they can talk and brainstorm and challenge each other, and then we all, as a class, talk about their ideas
Okay, so anyway, I have them write out little name plates with colored pens … you know, show your first name big … and then every day you’re gonna have your little name plate in front of you, so I’m gonna learn your name, and every other person in the room is gonna know you. We’re all gonna know each other.
And there’s a lot of political talk about safe spaces. Like the people that call those of us on the left “snowflakes”. They say that we’re so worried about making safe spaces for our students. I’m very proud of making a safe space for my student in my math class, where by “safe”, I mean … it’s perfectly fine to make mistakes. It’s perfectly fine to take a chance, to be confused, to express your confusion, to not be embarrassed. Because the truth is, to really make progress, and to do very hard math, you’ve gotta be ready to be mixed up. This stuff is hard, and if you pretend you understand something and you don’t understand it, you’re not gonna learn, and you’re not really gonna overcome all these hangups that put you in this class in the first place. So I try to create that kind of environment for them.
Alan: 00:24:53 Yeah, I’d like to hear an example of what you challenge them with, what kind of a question.
But okay, your question was, what kind of exercise would I give them, so here’s an example of one.
I asked them to think, one time, about a city … like Manhattan … that’s laid out on a grid. Where, if I want to tell you how far is it from one point to another … say on the east side … it’s commonplace not to give distances, but to say … excuse me a second.
Okay, let me try that again.
If I were giving you directions in New York City, I might say, “Go three blocks north and one block east.” That is, rather than saying distances, like how many miles or how many feet, I would say, “It’s three blocks uptown and one block east.” That’s a normal way to measure distance in the city, where things are laid out on a grid. It’s not the distance as the crow flies. You know, you could imagine a bird flying between the buildings or something. But for someone … like say you’re in a taxi cab … you’ve gotta measure distances east and west, and north and south.
And so I ask the students to figure out what a circle looks like if this is how you’re gonna measure distance. That is, draw, for me … if I put a point on a piece of graph paper … you know, graph paper picture has all these vertical lines and horizontal lines crisscrossing. That’s like the grid of the city. And where two lines meet, that’s an intersection in the middle of a road, say two roads meeting. Imagine that that’s a starting point, and now show me all the points that are three blocks away from that point. So you could imagine going three blocks east or three blocks west, or north, or south. But you could also go two blocks east and one block north.
Do you know what I’m asking here?
Alan: 00:28:56 Are all the points three blocks away going to show a circle? Describe a circle?
Steve: 00:29:01 That’s the question! Excellent! That’s the question. It sounds like a circle, right? ‘Cause everything is three blocks away from this point.
So if you draw this out … do you have a piece of paper in front of you?
Alan: 00:29:12 No, I’m doing it in my head.
Steve: 00:29:13 You’re doing it in your head? So what do you picture it looks like?
Alan: 00:29:15 That ones that are three blocks in a straight line are going to be a different distance, as the crow flies, from the ones that are two blocks in one direction and one block in another direction. It’s a different crow.
Steve: 00:29:29 Yes they will. Exactly. But what does it look like? That’s the question.
Alan: 00:29:32 Yeah.
Steve: 00:29:34 I love that you said, “It seems like a circle.”
Alan: 00:29:37 Yeah.
Steve: 00:29:37 But yet, it’s not gonna look like a circle to a crow.
Alan: 00:29:40 No.
Steve: 00:29:41 But it is a circle to a taxi cab. Because if a taxi … which cannot fly … yet … the taxis we’re talking about, that are just driving on the grid, on the street … a taxi that is street-bound, if you ask it, “What do all the points that are three blocks away from a starting point look like?” Well, I’ll tell you.
Okay, maybe you have the picture in your head, or maybe your listeners are trying it right now…
Alan: 00:30:03 I see a circle that I could draw of all of the ones that are three blocks to the east or west.
Steve: 00:30:17 Right? Or north and south, as opposed to.
Alan: 00:30:20 As long as the ones on one side of the circle are east, and on the other side of the circle are west. And north and south, the same thing. I’d have a circle of those.
Steve: 00:30:32 But are you picturing a circle to look like really a round circle, like a normal circle?
Alan: 00:30:35 Yeah, if I go three blocks to the east, and three blocks to the west, and three blocks to the north, and three to the south—
Steve: 00:30:45 Yeah, those points are all—
Alan: 00:30:46 Then it’s a circle.
Steve: 00:30:47 Those would all sit on one circle, but what about if you go two over and then one up?
Alan: 00:30:51 Then that’s going to be indented.
Steve: 00:30:54 Yeah!
Alan: 00:30:56 I’m gonna get a sine wave or some damn thing.
Steve: 00:31:00 Well you’re right that it’s indented.
So if you draw this out, you’ll see that it makes a shape that looks like a diamond.
Alan: 00:31:06 Oh really?
Steve: 00:31:08 Yeah. It actually looks like a square, except tilted, so that it looks like a diamond.
Alan: 00:31:11 I think I’m getting an epiphany here.
Steve: 00:31:14 Well, let me tell you what happened with the class! I mean, it’s great that you’re getting it. You have a positive reaction. I had a student in the class who started shouting, and she said, “This is wrong!”
Alan: 00:31:23 Why?
Steve: 00:31:24 “This is wrong! This is not right!”
Because I asked the class first, “How would you define a circle?” And they said, “It’s all the points that are three blocks away from the given point.” A circle of radius three.
So I said, “What does a circle look like in a taxi cab geometry?” … that’s what we’re call this that we’re doing, “taxi cab geometry” … “What does a circle look like?” And the kids draw it, and it looks like a diamond, and she said, “It’s no good! It’s wrong! It’s not round!”
But I said, “Well, the rules are different. This is not crow geometry, this is taxi cab geometry, so why would you expect it to look round?” And she was just very flabbergasted at this.
Alan: 00:32:01 So that’s really interesting to me. Did you learn anything from her reaction?
Steve: 00:32:05 Oh, I loved it.
Alan: 00:32:07 Did you think about her reaction and try to see it from her point of view? I’m wondering if there is something to learn.
Steve: 00:32:12 Her reaction is fantastic, because first of all, her reaction is heart-felt. She felt what she felt and let it out. Which is that this is surprising, and this is counterintuitive, and it’s upsetting. And all of that is part of the mathematical experience. It’s part of any creative experience, I think. That you have a prejudice about how things are gonna turn out, but if you use your mind, and use reasoning, sometimes you surprise yourself.
So the first stage is shock. The second stage is maybe sorta coming to accept, “Hm, my brain is telling me this, but I don’t believe it, but why don’t I believe it?” Examine your assumptions. What’s the problem?
She took us on the first step, which is to go deeper into this question to see, why are we unhappy about it? And it’s because we have been taught one kind of geometry, which is the one that you’re thinking of, traditionally called “Euclidean geometry”. It’s the geometry that Euclid did in 300 BC. And it’s what I’ve been calling the crow geometry, the distance as the crow flies. And that is the most natural geometry, but mathematicians have discovered that there are many kinds of geometry that are equally good. And sometimes, like if you’re driving in a city, the distance the crow flies is not relevant. You can’t drive through a building. This is the relevant distance.
Alan: 00:33:36 So did they discover this as soon as there were taxi cabs, or did they get—
Steve: 00:33:42 I don’t know. That’s an interesting question. I don’t know what the history of taxi cab geometry was, but somebody had to invent this idea, and then explore its implications.
I’ll just give you one last thing about it, just to finish off that story.
So I ask the students, “Okay, once you accept that this is a circle … this diamond-looking thing … what is pi?”
‘Cause you remember, there’s this idea of pi. I don’t mean a pie that you would eat, I mean the number pi, which, in geometry, we define as the ratio between the circumference of a circle … so that’s the distance around the circle … to the diameter of the circle … that’s the distance across the circle, through its center.
So I ask them, “What’s the circumference of this thing that we just did with a radius of three?” and then figure out what pi is in this geometry, and it turns out that pi equals four. Pi is not the number that we are used to in Euclidean geometry, the 3.14… In this geometry, pi is exactly equal to four. And again, the same student—
Alan: 00:34:45 She must’ve had a nervous breakdown.
Steve: 00:34:47 Yeah. She was screaming again. She said, “Pi is not four!” And the students … “Okay, so where are we going with this? I mean, who cares about this taxi cab geometry?”
Well, the big lesson here is that math is much more open and creative, and allows much more room for the imagination, than these students realized. That you can invent your own geometry that sometimes makes sense and is more appropriate than the one you were brought up on. And so math is not just black and white and two plus two equals four. Sometimes pi equals four.
Alan: 00:35:21 The creativity of math starts to get infectious, I imagine, in that kind of a class. Did the student come out of her feeling of resistance? Did she start to get it and enjoy it?
Steve: 00:35:35 Well, what was wonderful about her is that she was always ready to fight.
Let me say why I like it so much. I don’t want students who are just acquiescing to the authority of the teacher. I tried not to have any authority. I’m just asking them questions and asking them to think, and use their own judgment, use their own reasoning and their own creativity, so that they can see that they have the power.
We all have math within us, and it’s not that the teacher told you how to do it. You don’t need the teacher. You can invent a lot by yourself, and they never had this experience in their 16 years of school before. They just were following the rules, coloring by numbers. And this is a chance to paint your own painting, not just color by numbers.
So I liked her reaction. She took on this role, in the sociology of the class, where she was always the one fighting back. But I could see that she liked it, and that she thought it was kinda cool that you could have different geometries.
Alan: 00:36:39 Yeah, well, I think it’s cool, too. I love how you talk about creating your own geometry.
I wanna ask you something that I have often asked you in our conversations like this that we’ve had over the years. Can you help me understand something, anything, about something that I’ve heard about for many years, but don’t really have a clear understanding of?
For instance, the quadrilateral equation is very common in science, and I don’t know how it works. I have always heard about the [Fourier 00:37:18] transform. What a wonderful name. I have no idea what it does. And [Bayesian 00:37:24] statistics. How about Bayesian statistics?
Steve: 00:37:29 Great. Sure.
Alan: 00:37:31 Just so you know who you’re talking to, you know what level I’m at, let me tell you what I think it is and then you can work from that.
I think it’s a way of figuring out a problem that’s where something is changing all the time and you keep getting updates on the information you have about it so you can have a more accurate appraisal of it. Something like that.
For instance, I was on an island once with roads and stuff, and the only way you could get off the island was by taking a ferry. And I was visiting with a friend who was very rich, and I was curious about how rich people think. How did he get rich? Did he estimate things better than the rest of us? And we heard that the ferry had gone out of commission on one end of the island. The other end of the island had a ferry that was working, but if you took that ferry, it would take you three or four times longer to get where we both needed to go.
So I looked it up on my iPhone and found out what condition the ferry was in … the one that was broken … and I saw that it was out of commission and I took the long route.
Later I found out he kept checking on the ferry. He didn’t take one report about the ferry, he kept checking on it, and at the last minute, managed to get on the ferry and got home quicker than I did. And I thought he was using some kind of Bayesian method.
Now how close am I understanding anything about this?
Steve: 00:39:12 Yeah, I would say you’re quite close. That the idea that you have some estimate of the odds of something being favorable, or something happening, and then you update the odds as more information comes in … that’s exactly the heart of what Bayesian statistics is. So I think you got the gist of it.
I’d like to try to give you an example of just the kind of thinking that goes into it, actually, because it was instructive to me in my own teaching. It’s something that I was not trained in. I, in fact, never studied probability or statistics as a student. And so, over the years, when I’ve been asked to teach those subjects, I’m a bit terrified.
You know, there’s this old dream … a lot of people have this dream of … an anxiety dream that they’re in a course, some course in school, and they’re signed up for the course, but they didn’t know they were in the course, and they haven’t gone to any of the classes, and they’re sitting there for the final exam and they have no idea what’s on the page of that exam.
When you become a professor, you have a different dream, which is that you dream you have to teach a course that you don’t know the first thing about. And that is the way I feel when they ask me to teach about Bayesian statistics. That I don’t know anything about that.
So the first time I had to teach it, I stayed very close to the textbook. I would really just sort of inch along with the examples in the book and do everything the way the book did, hugging close to shore, ’cause I really didn’t know what I was doing. And on this one Bayesian problem that I used to assign, the students would always do it differently from the book, and I was inclined to think they were doing it wrong, except that they always kept getting the right answer.
Alan: 00:41:07 What was that?
Steve: 00:41:08 This happened year after year, and I started to realize the students had found a more intuitive, clearer way to think about Bayesian reasoning that I wanna do with you now.
Alan: 00:41:17 Oh, okay, great. Okay.
Steve: 00:41:19 Okay? This is an example of how the teacher can learn from the student. And I realize your show is about communication and trying to be clear and vivid, and I think this was a nice example of communication going in the other direction, of the students teaching the teacher that they found the right way to think about this, better than the textbook.
So here’s an example of the kind of thing that they would do. Now, the example I’m gonna give you has a lot of numbers in it, so I worry about that, but you could write them down, or I could just keep repeating them.
Alan: 00:41:51 I haven’t got a pencil and probably the listeners don’t, either, so—
Steve: 00:41:55 I’ll just keep repeating [crosstalk 00:41:55]
Alan: 00:41:55 I’ll just make sure I’m up to date with you.
Steve: 00:41:58 It’s not really important. You’ll get the gist of it anyway.
Alan: 00:42:01 One second. Let’s both clear our throat.
Steve: 00:42:06 All right.
Alan: 00:42:12 Okay, so how is this going to work?
Steve: 00:42:15 Okay, here it is. It’s a question that has to do with the real world … it’s sort of a grim topic, but it’s important. You’ll see why Bayesian reasoning is so important.
Imagine a woman who goes in for her first mammogram. You know, you’re supposed to get a mammogram, they used to say at age 40, for breast cancer screening, or nowadays they say go in at age 50. But anyway, so imagine our hypothetical woman who goes in to have a mammogram, and she’s in what a doctor would consider a low-risk group. There’s no history of breast cancer in her family, she doesn’t have any symptoms. She’s reasonably young, there’s no reason to be worried.
So here’s the first number I wanna give you. For a woman of this type … a low-risk person … her odds, according to the doctors, the probability that a woman like this would have breast cancer is 0.8%.
Alan: 00:43:16 0.8.
Steve: 00:43:18 It’s already confusing, right?
Alan: 00:43:20 Less than 1%.
Steve: 00:43:21 Yeah, less than 1%.
The thing I’m gonna give you here is an example that was used in a study of how doctors do or do not successfully use Bayesian reasoning. This is a study of practicing physicians and the psychology of how difficult these kinds of questions are.
Let me just read to you. It’ll be a bit to take, but we can go over it.
Okay, this was the question:
The probability that one of these women has breast cancer is 0.8%. That’s fact number one.
Second fact: you’re just told … and you can just accept these numbers … if a woman has breast cancer, the probability is 90% that she will have a positive mammogram.
Alan: 00:44:04 Positive meaning she’s got cancer.
Steve: 00:44:06 Yeah, well … positive meaning that the mammogram says that she does.
Alan: 00:44:10 Yeah. Okay.
Steve: 00:44:11 We don’t know that she really does, we just know the mammogram says that she does.
Alan: 00:44:14 I see.
Steve: 00:44:16 So if she does have breast cancer, the mammogram will pick it up 90% of the time. That is, it will say, 90% of the time, “you have breast cancer” when you really do.
Alan: 00:44:25 Uh huh.
Steve: 00:44:27 It’ll miss some. It’ll miss 10% of them. But 90%, when you have it, it’ll say you have it.
Now, the thing that’s making it further confusing is if a woman does not have breast cancer, the probability is known to be 7% that she will still test positive. In other words, a false positive.
Alan: 00:44:46 Is there anything in there that the person listening gets dizzy at this point?
Steve: 00:44:49 Oh, very dizzy. And so do the doctors. So do professionals. That’s the amazing thing.
So the question is … and this is the real question … imagine you’re a woman that has gone in for her first test, she’s with her physician, the physician has all those confusing numbers I just gave you, and unfortunately, her test comes back positive. The question is, how bad is this news? What is the probability she actually has breast cancer?
Alan: 00:45:14 Oh. That’s an interesting … and Bayesian reasoning can give you a better picture of it?
Steve: 00:45:21 I’m gonna give you the answer in a minute, but I’m gonna show you … I’m glad that you find it confusing, because so do the doctors. I mean, even professionals who have done this for decades cannot put all of these numbers together in a way that makes sense.
So the guy who did this study is a German psychologist named [Gerd Gigerenzer 00:45:40] and Gigerenzer describes what happened when he … he said the first doctor he tested was a department chief at a university teaching hospital who had been teaching for more than 30 years. And this guy, he says, was visibly nervous trying to figure out what he would tell the woman. He just eventually gave up. He just said, “I don’t know, I mean, ask my daughter. She’s studying medicine.”
And it wasn’t just this one guy. Gigerenzer asked 24 other doctors the same question, and some of them said the woman’s odds were 1% that she had breast cancer, other doctors said 90% chance that she had it, and there were people everywhere in between. 50%, 80%. So you could imagine a poor woman asking a second opinion and a third opinion, and some doctors would be saying, “It’s very likely you have it, I’m very sorry to tell you.” And others would say, “Don’t worry.” So what’s the right way to think about it?
Alan: 00:46:37 Quickly review the numbers again?
Steve: 00:46:39 Yeah, let’s go over them again.
So first of all, she’s supposed to be in a low-risk group—
Alan: 00:46:44 Which means that she’s got less than 1%.
Steve: 00:46:46 Less than 1%, so that’s like when you said to me, at the beginning, that Bayesian reasoning is about updating information as new information comes in. What you should think of is, before the woman goes in for the test, she knows that her odds are good. Less than 1% chance of having breast cancer, just by virtue of her age and no family history.
Okay, so her odds, before she does any test, are good. Less than 1% chance of trouble.
Then, new information comes in, the new information being she has just tested positive for cancer. So now we have to update her odds, based on two things that we know, which is that sometimes the test is wrong in one direction, and sometimes it’s wrong in the other. That is, it can give either false positives or false negatives.
Alan: 00:47:37 And what’s the difference between those?
Steve: 00:47:40 A false positive would mean she doesn’t have it, but the test says she does.
Alan: 00:47:43 Does she get as many false positives as false negatives?
Steve: 00:47:53 Well, the numbers that we were given were that … sorry, now I’ve gotta look at my own piece of paper here … no laughing matter, it’s important. This poor hypothetical woman.
Yeah, sorry. So we said that if she does have breast cancer, 90% of the time, the test will pick that up and say that she has it. But that if she does not, the test will still say she has it 7% of the time. So the information I’ve given you is that there’s a 7% chance of a false positive. That she tested positive even though she doesn’t have it.
I also gave you information about the sensitivity of the test. That it picks up cancer 90% of the time that it’s there. I didn’t give you information about the false negative. I guess I have indirectly given it to you, which is that if she does have breast cancer, 10% of the time, the test will miss it.
Alan: 00:48:53 So it sounds like the two figures that are important are the difference between the false positives and false negatives. The fact that her past history puts her at less than 1% seems irrelevant, once she takes the test—
Steve: 00:49:08 No, it’s really relevant. No, it’s very relevant.
Alan: 00:49:11 So tell me why. That’s interesting.
Steve: 00:49:14 It’s interesting that you have that intuition. That’s one of the big lessons of Bayesian thinking is that the base rate … that is, in this case, the rate that a person like her probably doesn’t have cancer … really important to know. Keep it strongly in your mind. She’s from a low-risk group. The odds are really good she doesn’t have cancer. There’s no reason to think she would. And just because the test says she does, you still shouldn’t necessarily believe it.
So here’s what my students figured out. This thing that I just gave you, totally bewildering. I mean, if you’re totally confused at this point, that’s the right reaction. And it’s because … and this is one of Gigerenzer’s big insights in his study … people, including doctors, don’t know how to think about probabilities as probabilities. I gave you all these percentages and they’re super confusing when I say it like that.
If I gave it to you as numbers, like the numbers that you learned about in elementary school, you’ll be able to do the problem easily.
Alan: 00:50:10 So go ahead.
Steve: 00:50:12 It’s numbers that we should be thinking about, not percentages.
Okay, so here’s the good way to think of it, and this is what my students would do. And this is what I didn’t realize. The book doesn’t do it this way! This is what the students hit by being smart little kids.
They just would think about a group of 1000 women. Okay, so let’s do that this way. Let me give you the same numbers that I gave you, except not as percentages but as actual numbers.
Instead of saying .8%, which is already bewildering, I’m gonna tell you that 8 out of every 1000 women have breast cancer … that are women like this hypothetical woman in the low-risk group. That’s what .8% means, 8 out of 1000.
Alan: 00:50:51 8 out of 1000 of low-risk women will have it, in spite of the fact that they’re low-risk.
Steve: 00:50:58 Right. Exactly. Perfect so far.
So 8 out of 1000. So we’re gonna imagine this hypothetical cohort of 1000. And of these 1000, 8 of them, unfortunately, do have it.
Now, of these 8, 7 will test positive on their mammogram. Why did I say 7? ‘Cause I told you earlier that the test would pick it up 90% of the time. So of these 8 that actually have it, if I do 90% of 8, that would be 7.2. So I’m just rounding to make it simpler to keep in our head, of these 8, 7 will test positive.
Alan: 00:51:34 Okay.
Steve: 00:51:34 Because the test is pretty good! It’s gonna catch it 90% of the time.
Alan: 00:51:37 Yeah.
Steve: 00:51:37 So of the 8 out of 1000, 7 will test positive.
Okay, now, what we also have to realize is that there’s 992 remaining women, because 8 of them do have breast cancer. 992 … that’s the rest of the 1000 … do not have breast cancer. ‘Cause remember, they’re low-risk. 992 out of 1000 actually don’t have it, yet 70 of them will test positive. Because we said earlier that the test will give you a false positive 7% of the time, and 7% of these 992 women is about 70 women.
In other words, what I’m saying is, 70 women will test positive even though they don’t have it, and 7 will test positive because they really do have it.
I’m sorry, is that too much to do over the … should I say it again?
Alan: 00:52:27 I’m not sure I’m still awake.
Steve: 00:52:29 I’m sorry.
Alan: 00:52:31 No, I’m trying to follow you, but when you started putting numbers onto percentages, it got hard.
Steve: 00:52:39 Okay, so I’ll say it simpler. Out of the 1000, 8 of them do have breast cancer. 7 will test positive. 70 will also test positive, even though they don’t have it. Because the test makes mistakes.
Alan: 00:52:57 Yeah.
Steve: 00:52:58 And so 7 out of the total that tested positive … which is 7 + 70 … 7 out of 77 of those that tested positive will actually have it.
So the bottom line is, only 1 out of 11 that test positive actually have cancer. The odds are 9% for our poor woman.
Alan: 00:53:19 If they come in with low risk.
Steve: 00:53:22 They’re low-risk and they test positive, the odds are still only 9% that they have it. I mean, her odds went up, but we updated the—
Alan: 00:53:22 The odds went up, but if they’re still—
Steve: 00:53:33 They’re still not that terrible.
Alan: 00:53:34 It’s not a—
Steve: 00:53:34 I mean, it’s not like she has it. She might have it.
Alan: 00:53:37 Not as horrible news as it—
Steve: 00:53:38 It’s not as horrible as it would sound.
So okay, that was difficult to do in the medium that we’re using here, but … I guess what I was hoping to convey was that … first of all, that my students taught me the right way to think about this. That percentages are confusing, whereas imaging sort of a tangible population of 1000 women, then you could just calculate things. It was much more straightforward, and that was the way they would do it. And the thing that we just did, painfully and confusingly, was an example of so-called Bayes theorem. We just did Bayesian reasoning.
Alan: 00:54:15 Who was Bayes and how did he arrive at this? Was it a practical problem he was working on?
Steve: 00:54:21 Honestly, I don’t know much about him. They always call him Reverend Thomas Bayes, so—
Alan: 00:54:26 So maybe he was figuring out the collection plate as it went down the [crosstalk 00:54:29]
Steve: 00:54:29 I don’t know what his “Reverend” had to do with anything, but yeah, he’s Reverend Thomas Bayes. I’m not sure, I think he’s some time in the 1800s and I don’t actually know why he was driven to think about all this.

So let me ask you about, what do you face in … this show is about communication a lot, and I think of teaching and the relationship between a student and the teacher as a kind of quintessential example of communication. Do you have a theory or an idea, a way of working that you go by that helps you relate to your students in a way you think is productive?
Steve: 01:16:20 Sure, yeah, lots of ideas. I think a very important one … maybe the most important of all … is … wow, I was gonna say the word “love”. Maybe it’s too strong. But that this is about friendship. That we have to care about each other. Especially I have to care about them. That if they feel my job is to judge them or to evaluate them … which is often a big part of what the teacher has to do, giving a grade … that already is a strike against the relationship.
If it’s really that we’re friends and I’m trying to help you, or I wanna share something that’s gonna be good for you or that will make your life richer or more fun or more interesting … and I really mean it sincerely, if that comes across … then I think that’s a big start, because most of the things that teachers do wrong, they wouldn’t do with a friend of theirs. You know what I mean? Like having patience or sympathy or joking around, or just any of the good things that people instinctively do with their friends. If you thought of your student as your friend, truly, I think most of us would automatically be better teachers.
Alan: 01:17:30 That’s a really interesting way to put it. We don’t often end a conversation with our friend and give them an F.
Steve: 01:17:43 No, you wouldn’t! You wouldn’t. You wouldn’t. No.
So that’s the thing, I would say that above all. Now, of course, it’s easier said than done. I mean, in reality, if I’m teaching a class … like this semester, we haven’t started yet, but I’m gonna have a class of 100 students and it’s me in front of them in a lecture. That’s already gonna be very tough. I can tell you now, I’m probably not gonna learn all their names. There’s no hope of me being their friend. And it’s not gonna work as well as it could.
Whereas when it really works, like with this class that I mentioned earlier about mathematical explorations, there I stricture it so that I do have a chance to be their friend.
Alan: 01:18:25 So how will you approach this? How will you try to overcome that roadblock?
Steve: 01:18:31 Well, just to continue the thought though, I’ll come back to your question, but …
What works with this class of 30 … the much smaller class … on the first day, I asked them to hand in their mathematical autobiography. That is, I want them to write … tell me your story, in math, about who were your teachers that inspired you, that hurt you, what happened in your elementary school math that was either good or bad? How do you feel about math? What’s the first word that comes to mind when you think of it?
So I just wanna know what it’s been like for them, and I start to get a picture of them as people in math class. Now, I guess I wouldn’t do that with my class of 100, because it’s too big! I mean, basically it’s an inhumane thing to teach 100 students at once. It’s not gonna really work. So the best I can do there is I might try to make the class active. That’s something that’s commonplace nowadays, but not in the old days, where I would ask a question and then have students talk to each other, talk to the person next to them. There’s technology you can use, like I can let them choose by tapping something on their phone … like, I could give them multiple choice questions, and after discussing with their neighbors, we could see what the class thinks the answers is … A, B, C, or D … and then after we see the results, which I can project onto a screen, we could then have them discuss again. Do they wanna explain to each other why they thought the answer was B?
I don’t know, I mean, again, you’re sort of limited what you can do in a big class.
Alan: 01:20:06 But you’re moving toward interaction, toward not just blasting your lecture at them, but hoping to make it a two-way street in some way.
Steve: 01:20:19 Yes, a two-way street, and also to have them not just being a receptacle. To the extent that they can think and struggle, which, in that kind of a format, they’ll probably be doing that with their fellow students more than they could directly with me. But still, that’s something. They can talk to each other, and then I can, of course, call on them, and then … I can diffuse a little bit of the pressure that someone would have for raising their individual hand … I could say, “Is there any group that thought they see how to think about this? And does someone volunteer to be a representative of your little group?”
And often, some kid will say, “Yeah, well, I was talking to Jimmy and we thought such and such.” And then if the class is set up in a way that people are friendly, then maybe I could say, “Well what do other groups think of what we just heard? Do you agree with that? Does that sound right, or do you see a gap in their reasoning?” And if it’s all done respectfully, and again, from a place of caring, I think then people will take chances and come up with really interesting ideas, and also, when they offer criticism, it’ll be the kind of criticism that helps instead of hurts.
Alan: 01:21:26 You know, a question I get very often … and usually from science teachers … is, “How do you help them be more curious?” Because they sometimes face a class that has blank faces, are not apparently interested in what the teacher is trying to get them interested in, and don’t seem basically curious about the whole thing.
Do you face that ever in your classes?
Steve: 01:22:04 Oh yeah.
Alan: 01:22:04 How do you handle that? How do you get people to be curious?
Steve: 01:22:08 People are curious, that’s not the problem. The problem is when you make someone answer your question instead of their question.
The trick is to have respect. I mean, think of the boring person at the bar. You’re sitting at a bar or you’re at a cocktail party, and someone’s carrying on about something that you don’t want to hear about. They’re talking about what they’re talking about, but you’re not interested in that. That’s the definition of boredom. So, you know, if I start jamming something down the student’s throat … well, okay, maybe the might accept it because there’s power. I’m the professor, they have to do what I say. But they’re not gonna be curious about that.
But if you give them a puzzle that’s truly interesting, most people are natively curious about it. So curiosity is not the problem, it’s giving them the space to express their curiosity instead of jamming your answer down their throat. Let them have their question.
Alan: 01:23:05 I love the idea that you answer their question instead of your question, but what if you’re a high school teacher and you have to cover a certain amount of your textbook or you get in trouble with the state?
Steve: 01:23:36 Yeah, yeah.
Alan: 01:23:37 You can’t just keep asking them what their questions are, you’ve gotta get them to ask the questions that are related to the textbook, don’t you?
Steve: 01:23:43 That’s true.
Alan: 01:23:44 How do you do that?
Steve: 01:23:45 That’s true, and that is difficult. I mean, I haven’t been a high school teacher, I haven’t had the constraint of having to teach to a test, so a lot of what I’m saying may be pie-in-the-sky. But my instinct is still … like if I were teaching geometry, let’s say, which is gonna be on the SAT, the Scholastic Aptitude Test that everyone has to take to get into college … I would ask students to think about … give them a geometry question, and ask them to start thinking about it.
And I wouldn’t start teaching them a method for how to solve it. I would just say, “Let’s think about this question. Here’s a puzzle. How would you do it?” And of course, they don’t know how to do it. And then we would start having a discussion. People could work by themselves for a while, then they could work in teams, and then we would discuss ideas, and things would come out. That is, the creation of the right ideas that are in the textbook will tend to come out naturally, because these ideas are natural. Students will think of them, if you give them the space, and I really do believe that when it comes from them …
I mean, it takes guidance. This takes a lot of skill. This is much harder than teaching out of a textbook or giving a canned lecture. To let the real math emerge naturally, you have to be a very good mathematician yourself. Because people will do things that surprise you, and if you say, “No, that’s not right, let’s not do it that way,” you’re gonna stifle creativity and it’s not gonna work.
There are many correct ways of doing math problems, and you have to be strong enough, mathematically, to see that, and to let people go that way, and then sort of curate it. Pull out the parts that are working, and sort of gently suppress the dead ends.
Or, actually, sometimes it’s good to let the students go into a dead end so that they realize how to get out of a dead end.
Alan: 01:25:33 Yeah.
Steve: 01:25:33 I mean, that’s a very important skill to learn.
Alan: 01:25:36 The business of tracking what they’re paying attention to, where they are, what their interest is, seems to be really important in doing what you’re saying. I remember I tried to teach one of my granddaughters the importance of the hypotenuse … ’cause I really love the right triangle and the relationship with a hypotenuse to the other two sides … so I said, “Let me tell you a story about a little girl, and her father was driving eight miles in one direction, and then they took a right turn and went seven miles in the other direction. They only had enough gas to get home … ” a certain number of miles, I forget what it was … I said, “But the little girl said, ‘I know how we can get home.'” And they were gonna take the hypotenuse. As soon as my granddaughter realized what I was talking about, she said, [inaudible 01:26:28].
I was trying to make a puzzle that she would have fun solving, where this little girl is the hero of the day. But I wasn’t paying attention to what she cared about. Even though I was trying to make it a playful story that might appeal to a kid, it didn’t appeal to that kid, and I didn’t find out first what might’ve appealed to her before I launched into this presumably instructive story.
Steve: 01:27:03 Yeah. Well it’s very difficult to think of good stories or good activities or good puzzles. That’s part of the art of teaching. To guess what might be … Like say, we talked earlier about this thing with the taxi cab geometry. Is that gonna stimulate students or not?
So one thing to keep in mind is many different things work, but not all on the same people. Like some people love stories, some want history … stories of the great mathematicians, or even the not-so-great ones. Some people like practical applications. Some like competition. So I try to do a little bit of everything so that I’m hitting—
Alan: 01:27:43 That really does sound smart, and it really sounds like you’re paying attention to what might interest them, and what you have evidence of that interests them. And if you start from that, it sounds like you have a much better chance of a happy conclusion.
Steve: 01:27:59 Well, the idea is to connect. And as I know you have talked about on your show before, to communicate, it’s a lot about relating and connecting. And so the kids in this Math So this is a universal language, and I feel like it’s a happy story that with math, we can all connect.

Alan: 00:54:44 Well, we’re reaching the end of our allotted time. Are you interesting in doing these seven questions we have? These quick questions that invite quick answers.
Steve: 00:54:57 Why not? Everyone does them, don’t they?
Alan: 00:54:58 Yeah, they’re roughly to do with communication and relating.
Steve: 00:55:04 I worry … I mean, I don’t wanna keep people past our time, but … since we can cut this little conversation out that we’re having right now, do you think we talked enough about communication? I felt like we talked too much about math and not enough about communication.
Alan: 00:55:17 So tell me what thoughts you had. What thoughts related to math, or just in general.
Steve: 00:55:24 Well so okay, [Graeme 00:55:26] had said we could talk about AlphaZero.
Set up pt 2…. AI
This is C+V and now back to my conversation with Steve Strogatz
Alan: 00:55:28 Yeah, yeah. Well let me give you a question I was thinking of with regard to that. I forgot about it.
Maybe this doesn’t lead you into what you’re thinking of, but here’s what I was wondering: do you think if we don’t even have a smattering of the ability to think the way mathematicians think, that we’re liable to be swamped by the increasing presence of artificial intelligence, and the increasing directive it has over our lives.
Or won’t that even help us?
Steve: 00:56:07 Well, I’m … “swamped”, I’m a little stuck on your word “swamped”. We’re using artificial intelligence all the time without realizing it now, and often, it’s helpful. Like when you have a spam filter in your email. If you think about, that spam filter is making a guess about which messages you wanna read and which ones you don’t, and there’s also usually a button where you can say, “This is spam,” or, “It’s not spam.” And actually, the way spam filters work is Bayesian reasoning.
Alan: 00:56:36 Oh, that’s interesting.
Steve: 00:56:38 Isn’t’ that interesting? That’s a good practical use of Bayes. Every time you hit “this is spam” or “this is not spam”, the program updates its probabilities about the tell-tale signatures of what is likely to be spam. Things that often don’t start with an address, they may be spam, but maybe not always. Maybe you have friends who never like to say, “Dear Alan.”
So you can train your spam filter, and so here’s a case where Bayesian reasoning … and indirectly, a kind of artificial intelligence … is helping you deal with your email inbox.
Alan: 00:57:13 That certainly is useful, and most artificial intelligence promises to be useful, but what I wonder about is that it’s a machine that learns by itself, without being programmed much, if at all. Where is it gonna get its values from? How is it gonna make moral judgments unless we tilt it in one direction or another?
For instance, when a car is able to decide whether or not to save the driver’s life by veering away from the little girl chasing a ball into the street, is the car going to put more weight on the life of the owner of the car than it does on the person in the street? Should it? Are we at the mercy of a machine that we’ve made and don’t understand how it arrives at its conclusions?
Steve: 00:58:16 Mm-hmm (affirmative). That’s a very important problem, of course, for the future of self-driving cars, and that’s a case where I imagine we will have to program in answers. I don’t know how the program itself would come up with reasonable moral choices, except that that’s such a difficult problem, I don’t even know how a human being would come up with the choice. What would you do?
Alan: 00:58:40 Yeah, I guess it depends on your instinctual feelings at the moment.
But it’s interesting that we’re heading more and more toward really sophisticated machine learning. You wrote an article about AlphaZero. What is AlphaZero in a nutshell?
Steve: 00:59:03 Well, it’s a chess-playing program that was created by the people at DeepMind … an artificial intelligence company that was acquired by Google … and AlphaZero is a program that taught itself how to play chess. It was given the rules of the game, it didn’t have to teach itself that. It was told what the rules were—
Alan: 00:59:27 But no strategies?
Steve: 00:59:29 It didn’t know any strategies, that’s right. So it wasn’t given any special openings that human beings have figured out to be good ways to begin the game, it didn’t learn any special endgame knowledge. It just played itself millions of times, starting from complete ignorance, other than the rules, and it did this, of course, very quickly, being a computer. It played itself millions and millions of times in a matter of hours, and it learned from its mistakes. It was based on a kind of artificial intelligence called a neural network using a method of updating its estimates of what’s important about king safety or the position or activity of the pieces and all the other things that chess players worry about. It figured out all these principles on its own, and in about four or five hours, it became the best chess-playing entity that the world has ever seen.
I say it that funny way because not only was it better than any human being has ever been, but it was better than any existing computer, which are now the best chess-playing entities.
Alan: 01:00:31 No human can beat these computers, huh?
Steve: 01:00:34 No. Not even close. ‘Cause in chess, we have rating system that tells you how strong a player is, and a beginner at chess is rated about 1400 in some arbitrary units. I’m about a 2100, which is considered an expert player. A master would be about a 2300. The world’s best human is about 2850. So 2800-something is a really tremendously strong, world-champion-caliber player. And the best computers are about 3300.
Alan: 01:01:05 Wow!
Steve: 01:01:06 You know, 500 points higher than the world champion. And AlphaZero, nobody really knows how to even give it a rating, because it completely crushed the computer world champion. It didn’t lose a single game to it. In the first match they played, it won 28 games and had 72 draws. So it annihilated the best computer-playing program, called Stockfish … which, by the way, I learned … curious, I looked this up … I thought, “That’s a kind of a weird name, Stockfish.” It turns out that’s an insult that Falstaff hurls in one of Shakespeare’s plays.
Alan: 01:01:39 Oh.
Steve: 01:01:40 Isn’t that funny?
Alan: 01:01:41 Yeah.
Steve: 01:01:42 I don’t know why the creators of it called it “Stockfish”, but anyway … I mean, “fish” is a disparaging term we use in chess for a weak player, we’ll say, “You fish.”
Alan: 01:01:51 I never heard that.
Steve: 01:01:52 Yeah, “fish” is the biggest insult you could call another chess player. So I think it’s a self-deprecating name, but anyway—
Alan: 01:02:02 You said in the article that AlphaZero displays a breed of intellect that humans have not seen before.
Steve: 01:02:12 Yeah.
Alan: 01:02:13 You mean that it’s just so smart that we … or is it a kind of smart we don’t know?
Steve: 01:02:20 Okay, well that … yeah, so let me say, what I’m saying is controversial. A lot of computer scientists have written to me telling me I don’t know what I’m talking about, and they’re right. I don’t.
But having said that, I do know what I’m talking about when it comes to chess, and I can tell you … and so do much better chess players … could tell you that the way that AlphaZero beat the best computer program … chess engine, as it’s called … it played in a style that no computer has ever played. It played what, to a human master, looks like a very risky, intuitive, creative, brilliant … the word we would like to use is “romantic”.
Alan: 01:02:59 See, now we’re getting back to those art terms that you apply to math. It’s so interesting how it keeps turning back to this aesthetic appreciation of what is considered by so many people in our culture to be something mechanical. To me, it’s the difference between dancing … which I apply to you as you approach chess, as you approach math, you dance with it … the difference between that and what they used to publish in the [Arthur Murray 01:03:36] dance lessons ads, where you put your two feet here, and you put one foot over here, then you put another foot over there, and now you’re doing the foxtrot.
You may be doing a mechanical version of the foxtrot, but you’re not dancing. Not yet, anyway. And you dance, and that’s so interesting to me.
Steve: 01:03:56 Well, when this AlphaZero plays chess, it looks to us like it’s dancing. The great chess-playing engines that have beaten a world champion, they seem to us like machines. They just calculate very far. They can see dozens of moves ahead, they can look at every possibility. They seem very mechanical to us. And that is not what AlphaZero did. AlphaZero plays in a style that looks like the greatest human players, except superhuman. It plays in a very inspired, risky-looking style where it hasn’t calculated everything to the end. It somehow makes some kind of assessment that, “My position is good and I think this is gonna work out.”
Almost like you on the island when you were talking about whether to take the ferry or not. It seems like it’s making inspired guesses. Except that it’s very very rarely wrong.
Alan: 01:04:52 And you said it seemed to show insight.
Steve: 01:04:57 Yeah.
Alan: 01:04:59 In what way can a machine show insight? How did you experience its showing insight?
Steve: 01:05:06 Because the techniques it uses look … it doesn’t look like it’s just calculating, “If I go here, the other machine will go there, and then I’ll go here, and it’ll go there.” That, to me, is just a brutal kind of calculation that doesn’t involve higher-order thinking.
You know there’s always this expression about the forest for the trees? I feel like the old computers could only see the trees. They can’t see the overall shape of the game, the strategy, the big architecture. I don’t know, what are the words you would use? Like in drama, you would speak about the arc. There’s a whole layout that is a higher-order thing than just who said what to who, “I said this, you said that.”
If you think about how to describe Macbeth … you know, Macbeth is a story about ambition and power going to your head, and going crazy from that, and … that’s not word-for-word what’s in the play, that’s a higher-order understanding of the play.
Alan: 01:06:09 I see.
Steve: 01:06:09 And I feel like, when I see the way that AlphaZero constructs its chess games, it has a grand higher vision of the game than the machine it beats, Stockfish, which is very myopic and only sees the trees.
Alan: 01:06:25 Well now I’m dizzy for a whole other reason. The world is not just passing me by, it’s passing humanity by. It’s so interesting to me that we can make a tool that’s smarter than we are.
Steve: 01:06:42 Mm-hmm (affirmative). Well, we’ve taught it how to teach itself. It’s interesting, this sort of ties back to our whole discussion of education. This is about students and teachers, except that in this case, the student is made of silicon, or software, or bits, or something.
What we did was we endowed it with the ability to teach itself. Like that old proverb about, if I teach you how to fish … what is it? I show you how to fish, or I fish for you? I’m getting it all garbled ’cause [crosstalk 01:07:12]
Alan: 01:07:12 If you give a fish or teach how to fish. And it’s so interesting. AlphaZero is left by itself to figure things out, to understand the problem, in the same what that you left your students to understand the problem. And they came up with a better way to do it in the same way AlphaZero did.
Steve: 01:07:29 Well, and this is the experience of all teachers everywhere for all time … Socrates had Plato, who had Aristotle … that the student often surpasses the teacher. And that’s one of the pleasures of being a teacher, if you have the right spirit. That you’re gonna give the student the tools to teach him- or herself, and to make new discoveries, and to go beyond the teacher, and so I feel like we did that.
I mean again, I’m sure I’m getting carried away, but I don’t think it’s very far-fetched to imagine … if it’s not AlphaZero … something coming in the future, that we’re gonna teach how to teach itself, like we’ve taught this program how to teach itself how to play chess. That’s gonna … you know, the next program will teach the next program, and this chain of teachers and students will continue, and we will be left in the dust. And that’s our inevitable future. If we’re still here. If we haven’t killed ourself with nuclear weapons or climate change.

Alan: 01:10:53 Anything that you wanted to pick up on again? Or re-say?
Steve: 01:11:00 Of what we did? Not really.
Alan: 01:11:01 Yeah. I think Graeme has … Graeme, is there anything that you wanna get into?
Graeme: 01:11:06 [inaudible 01:11:06]
Alan: 01:11:11 Yeah, I thought so, too.
Steve: 01:11:13 Yeah, if you wanna leave it there, we could.
Alan: 01:11:16 Can you get away from that?
Steve: 01:11:17 I’m happy to talk about it more. It doesn’t have to be so negative.
Alan: 01:11:24 Yeah, I had that question myself.
Graeme: 01:11:26 I’m beginning to think you might want to do [inaudible 01:11:28]
Alan: 01:11:30 Yeah, maybe.
Steve: 01:11:32 It did feel to me … honestly, it felt a little bit off.
Alan: 01:11:35 The talk?
Steve: 01:11:38 Yeah, I feel like we didn’t stick true to your show. Now maybe that’s okay with you, and maybe the show is evolving, but I feel like there’s a lot we could’ve talked about about what it is to communicate with other … what communication means between teachers and students, or between mathematicians and the public.
Alan: 01:11:57 Well let’s do a couple of minutes of that. Let me ask you about … it sounds—
Steve: 01:12:02 Are you okay [inaudible 01:12:03] Yeah, okay.
Alan: 01:12:07 Good, thank you.
Sounds a little ominous to me, that we’re headed toward that endgame with our own tools. Do you think it really will lead to that?
Steve: 01:12:19 Well … who knows, really? I don’t know. It’s pretty wild speculation. But I picture a sort of a high and a low, where there will be a time when artificial intelligence gives us a great dawning of new insight. And that will be the golden era. Like for instance, if the day comes that artificial intelligence can solve some of our most intractable scientific problems … you know, we’ve had the war on cancer for decades now … and we’re making progress, but what if the real understanding of what’s essential in cancer requires an intelligence superior to ours that is possessed by this new generation of self-teaching machines?
If they could cure cancer or other diseases that have been very difficult, if they could help us understand the mysteries of the immune system … you know, there are so many diseases that are related to inflammation or other immune system problems.
I could picture a time when medicine is completely changed, and life will be easier and healthier and longer, and better for human beings. Assuming these machines stay benevolent. That’s, of course, an assumption. There’s all kinds of science fiction scenarios here, but—
Alan: 01:13:43 The darkness keeps creeping back in here.
Steve: 01:13:45 Well, because you can imagine that their interests will not necessarily align with our interests.
Alan: 01:13:50 Yeah, once they are aware that they have interests and can act on it.
Steve: 01:13:54 Yeah, so that’s the danger. There are all those kind of Terminator scenarios that we can think about, but before we get to those, I do feel like there will be happy times, both for medicine and … for those of us who are scientists, a lot of the problems that we’ve been struggling to solve for millennia will be solved by our computer teachers. And maybe if they’re really good, they’ll not only be good at solving science problems, but they’ll become good explainers. Maybe we could work with them.
Alan: 01:14:24 I’ve wondered that. If they’re so smart … if we’re creating machines that are so smart … why can’t they spend a little of their energy making clear to us, not only what their answer is to the big problems, but how they arrived at those answers. So that we can track the reasoning and maybe say, “You know, we just took a vote in the lab here,” or in the country, “and we decided your solution is not in our interest and we don’t want to go that way, so change that a little bit.” But you can’t do that if you don’t know how it’s arriving at its decision.
Steve: 01:15:02 That’s a very exciting scenario to me, where it’s not just about answers, but about reasoning, like you said. And I could imagine that they will, at some point, be able to explain themselves. And that would be a very illuminating time in human history. If we had this superior intelligence that were our students, that are now teaching us.
Maybe again I’m sorta sticking too close to home, ’cause I’ve had this experience of my students teaching me, but I feel like this is something for humankind in the future … that our machines may, in one happy scenario, teach us for a while.
Alan: 01:15:39 I hope we make machines that are more benevolent in general than we tend to be. At times, anyway.

Alan: 00:54:44 Well, we’re reaching the end of our allotted time. Are you interesting in doing these seven questions we have? These quick questions that invite quick answers.
Steve: 00:54:57 Why not? Everyone does them, don’t they?
Alan: 00:54:58 Yeah, they’re roughly to do with communication and relating.

Alan: 01:33:15 That’s great. Thank you, Steve, I really enjoyed this talk.
Steve: 01:33:19 Thank you, Alan. My pleasure.
Graeme: 01:33:21 Well done, guys.
Steve: 01:33:28 Thank you. Yeah. I think that gives you something else to work with, Graeme. I think that that was better.
Graeme: 01:33:35 [inaudible 01:33:35]
Steve: 01:33:38 Yeah.
Alan: 01:33:49 Yes.
Graeme: 01:33:50 [inaudible 01:33:50]
Steve: 01:33:49 Yes.
Graeme: 01:33:50 [inaudible 01:33:50]
Alan: 01:33:51 No, it describes a parabola. The dots that you make—
Graeme: 01:33:55 [inaudible 01:33:55]
Steve: 01:33:57 No, Graeme’s right. That’s right.
Alan: 01:34:00 What? Oh, you mean it doesn’t—
Steve: 01:34:01 So should we clarify that?
Alan: 01:34:02 It doesn’t go down, you mean?
Graeme: 01:34:03 [inaudible 01:34:03]
Alan: 01:34:09 Well wait, it’s gotta curve. It has to curve or it’s not gonna show you a parabola, right?
Steve: 01:34:16 I think what Graeme means is there’s half a parabola, because … yeah, he’s right.
Alan: 01:34:22 I don’t get the question. Can you explain it to me again so—
Graeme: 01:34:25 [inaudible 01:34:25]
Alan: 01:34:28 Yeah.
Graeme: 01:34:29 [inaudible 01:34:29]
Alan: 01:34:31 Yeah.
Graeme: 01:34:32 [inaudible 01:34:32]
Alan: 01:34:40 Right, so you would have to decrease the lengths again.
Steve: 01:34:46 Let’s talk about it, Alan, with your grandfather clock. It wouldn’t keep time and you would lengthen it or something?
Alan: 01:34:52 Yeah. I forget exactly—
Steve: 01:34:55 Didn’t you used to … what did you do with that clock, or did you adjust some setting?
Alan: 01:34:59 Let me try and understand this this way. I had a grandfather clock that was slow, so that meant the pendulum wasn’t swinging fast enough. So I think I changed the length of the pendulum.
Steve: 01:35:24 Yeah. If it’s too slow, that would mean the pendulum is too long.
Alan: 01:35:29 So I made it shorter. And then the clock kept time, but how is that related to the parabola you drew?
Steve: 01:35:42 Well, so there’s a relationship between how long the pendulum is, and how much time it will take to tick back and forth once or ten times, or any other number of times. And so in this … it’s a kind of an abstract graph that we’re talking about, where I’m showing the length of the pendulum going … I think of a pair of numbers. There’s a number for the length of the pendulum in inches or centimeters, and then there’s the time it takes to make a complete swing. That pair of numbers, a length and a time … if I graph that pair of numbers … the length is the X and the time is the Y … they’ll fall on a curve that looks like a parabola, except it’s really only half a parabola, because it’s only the half where … you know, the length are all positive. It wouldn’t make sense to talk about a negative length. I’m only gonna show, so to speak, the right half of the parabola.
It might also be confusing to think of, in terms of a drinking fountain, ’cause this parabola will be opening upward. The parabola in the drinking fountain looks like it’s got it’s maximum on the top and it’s kind of bending downward. This will be a parabola that opens upward. It says that the—
Alan: 01:37:02 Let me ask it a different way.
Steve: 01:37:03 God this is hard to do on … without a picture!
Alan: 01:37:06 Let’s stick with the drinking fountain, the way it normally squirts.
Steve: 01:37:10 Yeah.
Alan: 01:37:11 The drinking fountain squirts from the horizon, into the air, down to the horizon again.
Steve: 01:37:16 Right.
Alan: 01:37:16 You get a picture of the low point on the graph when the pendulum is short or long?
Steve: 01:37:31 Wow. I wonder if we’re barking up the wrong tree with this? I mean, ’cause the thing that’s supposed to be important about this is that there’s underlying math in the data.
Alan: 01:37:42 Yeah.
Steve: 01:37:42 The shape of the parabola is not so much the point. The point is that there is … I mean, we could try this a different way, with formulas. I don’t know if that would work.
Alan: 01:37:51 I don’t think so, because I think what’s nice about your story is that you had an epiphany when you saw the parabola.
Steve: 01:38:01 True.
Alan: 01:38:01 And you said that the pendulum knows algebra.
Steve: 01:38:05 Right.
Alan: 01:38:06 A shorter pendulum … ?
Steve: 01:38:08 A shorter pendulum swings faster, so it’s period which is the time to go back and forth will be less.
Alan: 01:38:13 So if you wanted to make it look like the drinking fountain, you would start with a long pendulum that—
Steve: 01:38:22 There’s no way to do it is the problem. The picture for the graph I’m talking about opens upward. It looks like a bowl, roughly speaking, half of a bowl. Whereas the water fountain looks like an upside-down bowl, like the St. Louis arch. There’s no way to do it. It’s not an arch, it’s a bowl.
Alan: 01:38:45 But when you say it’s only half a parabola, you mean it starts high and goes low, but it doesn’t go back up again?
Steve: 01:38:53 I mean it starts low, and then it accelerates up going higher and higher, but it looks sort of like a hockey stick or something. I don’t know how to say it. It looks … like if you pictured something … geez.
Alan: 01:39:08 I don’t know how it gets to look like a parabola unless there’s a sense of curve.
Steve: 01:39:14 There is a curve.
Alan: 01:39:19 Why is there a curve? Why isn’t it a straight line?
Steve: 01:39:26 Well, I don’t know an easy answer to tell you why it’s not a straight line.
The time it takes isn’t proportional to the length of the pendulum. If it were, it would be a straight line on the graph. It’s because the length of the pendulum is related to the time squared. The time it takes to swing back and forth has to get squared. And actually, our teacher told us to do that. He said make a graph. Instead of graphing the time for the swing against the length of the pendulum, graph the time squared. You know, use your calculator and calculate time times time … that is, time squared … against the length. And when we did that, bingo, all the data fell on a perfect straight line. And that freaked me out even more. Like suddenly there’s a straight line appearing if I do time squared against length. And that was amazing.
Alan: 01:40:18 But you got the parabola if you didn’t square time?
Steve: 01:40:21 Right, because time against length will actually make a curve. Sort of by squaring the data, you unkink the curve. The curve becomes straight when you square one of the variables.
It’s not obvious why this would work. This is, I feel like it’s too much in the weeds. The point, basically, was supposed to be that when you graph the dots … you know, here I am, I’m 13, I’m doing all these experiments, I have no idea what’s gonna happen, and I do dot dot dot, and after I have five dots, I see they’re not a snowstorm on the page. They’re falling in a very neat pattern. And I recognize the pattern, ’cause I saw it in algebra class. And I have this enveloping sensation of fear and awe that this pendulum knows algebra! That’s all.
Alan: 01:41:11 That is a great way to say it. I think you might not even need to go into the drinking fountain.
Steve: 01:41:18 I don’t think you do need the drinking fountain!
Graeme: 01:41:20 Yeah, we should avoid the drinking fountain. [inaudible 01:41:21] Wait a minute. [inaudible 01:41:25]
Alan: 01:41:30 Yeah.
Steve: 01:41:31 So maybe I should try re-telling it.
Alan: 01:08:21 Well, I consider you my teacher, and you don’t have to worry about me surpassing you.
Let me ask you the seven quick questions.
Steve: 01:08:29 Okay.
Alan: 01:08:30 Number one: what do you wish you really understood?
Steve: 01:08:36 Well, my dog Murray.
Alan: 01:08:38 That’s great. I know Murray, I wish I understood him, too.
What do you wish other people understood about you?
Steve: 01:08:50 How in love I am with Murray.
Alan: 01:08:55 What’s the strangest question anyone’s ever asked you?
Steve: 01:09:03 Oh, gee … that’s not ringing any bells for me. Hm.
I’m gonna blank out on that. I don’t know what to tell you.
Alan: 01:09:15 Okay.
How do you stop a compulsive talker?
Steve: 01:09:23 I think I would just kinda go flat. Just not even nod. Not even murmur. Just let them talk into a vacuum.
Alan: 01:09:32 Yeah, but sometimes that’s hard. We’re programmed to nod, aren’t we?
Steve: 01:09:35 Yeah. No, don’t do that.
Alan: 01:09:37 Yeah.
Is there anyone for whom you just can’t feel empathy?
Steve: 01:09:45 Well, a person who can’t admit they’re wrong.
Alan: 01:09:49 Hm.
Steve: 01:09:51 I mean, I really think that’s important. We all make mistakes, and … be a grownup and admit when you’re wrong.
Alan: 01:09:57 How do you like to deliver bad news? In person, on the phone, or by carrier pigeon?
Steve: 01:10:04 It has to be in person. Bad news is hard enough as it is, but I wanna be able to use all the different ways of making contact … facial expressions, voice. For instance, there’s so much misunderstanding over email. So I would do it in person.
Alan: 01:10:24 Last question: what, if anything, would make you end a friendship?
Steve: 01:10:31 Cruelty. Cruelty towards me, but cruelty in general in the person’s character. I can’t stand that.
Alan: 01:10:40 This has been really fun talking to you. One of our fun conversations, of which I hope we have many more.
Steve: 01:10:48 Thank you, Alan, it’s been a pleasure.
Alan: 01:10:50 Thanks, Steve. Bye-bye.
Steve: 01:10:52 Bye.

This has been Clear + Vivid, at least I hope so.

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Steven Strogatz is an amazing mathematician and teacher. What makes him unique is that he’s inspired by the curiosities of everyday life. He finds math were you’d least expect it and he’s able to illuminate the wonders of our world through the beauty of mathematics.

Steven is the Jacob Gould Schurman Professor of Mathematics at Cornell University. And in addition to my podcast, he’s also a frequent guest on Radiolab and Science Friday.

He is the author of Nonlinear Dynamics and Chaos, Sync and The Calculus of Friendship and The Joy of x. You can find out more about Steven by visiting his web site at:

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Thanks for listening. Bye bye!

Next in our series of conversations, I talk with Steven Johnson…
All the books that I’ve written about breakthrough ideas, almost all of them follow a completely different pattern, which I called years ago … I called it the slow hunch, which is, instead of a light bulb moment or an aha moment, you get this inkling that there’s something worth exploring or some idea out there and it stays in that hunch state for months, some cases for a decade or more in some of the people that I’ve written about. It’s only over time that it actually crystallizes into something more powerful. If you set up your life looking for eureka moments, looking for epiphany’s you actually won’t succeed. What you want to do is cultivate these hints that are floating around because that’s what the truly transformative ideas are going to come.