My guest today. Steven Strogatz is a professor of mathematics at Cornell University and has done some of the most influential math research of the last few decades. But Steven Strogatz isn’t what you’d expect from a mathematician.
STROGATZ: It was insane to have a paper that had, as its examples, the power grid of the Western United States; the neural network of the one organism whose neural network had been completely mapped at that point; and the graph of Hollywood actors who have been in movies with each other, popularly known at the time as “the six-degrees of Kevin Bacon.”
Welcome to People I (Mostly) Admire, with Steve Levitt.
Steven Strogatz studies real world problems and he has a gift for making math fun and accessible, even to people who have math phobia. Through his bestselling books, his New York Times column, and now his podcast, The Joy of Why, he’s probably done more to make people appreciate the power and beauty of mathematics than anyone else in modern times.
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LEVITT: I want to start by saying that I thought of you this morning. I’ve been reading Infinite Powers, which is your book about calculus, and I was thinking about it absentmindedly as I got in the car. I’ve lived in Chicago for a long time, which is perfectly flat, but right now I’m living in Germany and I live on the top of a big hill. Coming out of my garage, my driveway goes downhill. being the absentminded professor that I am, I didn’t bother to survey the situation and there was ice covering the driveway. And so as I began to drive down it I realized the brakes had no effect at all. There’s zero friction. I wasn’t moving very quickly, but it is a long downhill driveway. Actually had time to think about you and Galileo. And I knew enough about how acceleration works that I knew by the time I got to the bottom I was going to be going pretty fast. And indeed, I slammed into the embankment at the bottom of the driveway fast enough that the entire front of the car shattered into pieces and fell off. So that is proof that calculus is relevant in everyday life.
STROGATZ: I shouldn’t laugh. What a horror story. Yeah. Calculus can be dramatic in its effect — I don’t want to say impact. Woah, geez.
LEVITT: So you’ve managed to do something that I would not have thought possible. You’ve built a career as one of the world’s leading mathematicians by asking questions that are actually interesting to regular people. I’ve known 15 or 20 academic mathematicians, and every one of them works on a problem that’s more abstract than the one before them — N-dimensional manifolds, homological algebra. When they start explaining to me what they’re doing, I’m lost within a sentence. I actually thought it was against the rules of modern mathematics that research be comprehensible.
STROGATZ: It’s not against the rules if you do the part of math that we call applied math. There is pure math, which you could think of as math that’s inward looking, that thinks about its own structure, and it’s a wonderful, beautiful part of intellectual inquiry. But there’s also the outward-looking part of math that’s applied math where we take inspiration from nature, from the real world of engineered objects. You can look at anything mathematically. You do some of the same in your work. I do take a lot of inspiration from everyday life — little things that catch my attention. There’s that movie The Sixth Sense, where the big secret is, “I see dead people.” For me, it’s I see math. I look at anything and I see there’s a math problem there.
LEVITT: Within economics, there’s a real hierarchy. The more abstract you are, by and large, the more respected you are. Is that same hierarchy present in math?
STROGATZ: Yeah, I think it may transcend our two disciplines and just cut across the whole academy. It tends to be that the applied practitioners, I don’t know why, they’re often seen as the runts or something. But I like both parts of math. We really do need each other. We do provide great problems for each other. Also, it’s kind of a modern thing. If you look back at the greats historically — Archimedes or Isaac Newton or Leonhard Euler, Gauss — there was no distinction in their head between pure and applied. They were just interested in math, and what it could describe in the world. But it is true, there is a hierarchy. Most departments, if you were to look nationwide or worldwide — dominated by pure math. And applied mathematicians, sometimes they’re in engineering colleges or sometimes they’re in departments with funny names like operations research.
LEVITT: When I started graduate school, I was very attuned to the hierarchy. I had a basic rule, which is: I’d always been good at school and I was going to grad school and I should try to be one of the best. And so, I looked around and I could see that the macroeconomists, who are very theoretical, they were the king of the hill at M.I.T. And so I tried to be a macroeconomist. And I was terrible at it. I wasn’t good enough to do it. And then I said, “The theory guys, the people who do pure theory, they’re No. 2.” And I took a shot at that and I was bad at that too. And I wound my way down till I got to the bottom; to doing the most applied work possible. And I was just lucky I was bad at all those other things because, ’cause clearly the applied research was what suited my temperament and my interests. Did you follow the same path?
STROGATZ: I wouldn’t put it quite like that. When I was in high school — there’s a fairly standard curriculum that at least my generation went through of some algebra, geometry, then more algebra, then pre-calculus; depending on when you stopped. And then at the top of the high school hill was calculus. That’s what math looked like to me as a teenager at that early stage. But then when I went to college, I started to hear about subjects that I really didn’t know a thing about: Linear algebra. What was that? So I got dunked in the cold water first semester of freshman year with a teacher who really shouldn’t have been teaching freshmen. You know, this was really not someone comfortable being around people. No question he was an exceptionally great research mathematician, but had no social skills; didn’t give us any idea what linear algebra was about. He just went straight into the subject with definitions, theorems, proofs, more definitions. And oh man, it was dreadful. And I started to understand why people have math phobia because I used to feel myself getting really nervous before tests. I could barely do the homework. When you ask about, did I end up where I was because I bombed out of the other options? Yes and no. I did really badly in that first course. And I think that course was intended to weed out — to use that awful expression that people sometimes use — I came to feel like a weed. I got very discouraged and thought maybe I don’t have the right stuff to be a math major. I tried again with sort of the wizkid second semester course because I was placed in the wizkid first semester course, but I was getting crunched in there too. And my advisor fortunately said to me, you don’t have to keep taking this stuff. Maybe you would like the math that the engineers take. So I took that and I could do that all day long. That was easy; and I loved it. But I also liked some of the pure math, depending on the teacher. What was missing from that first course was intuition. It wasn’t visual enough. I couldn’t picture what was happening. So just bumbled my way through. But I didn’t realize it until by the time I graduated, that what I had loved all along was applied math. Anything to do the real world scientifically was interesting to me, but especially the math that was lurking underneath reality that I really loved.
LEVITT: So you say “bumbled through,” but you did graduate summa cum laude. So you couldn’t have done too much bumbling.
STROGATZ: I did actually bumble through, in that all my lowest grades in college were in math. I never actually got a B in anything except math. And I got several, including B minuses, and I was lucky to even get some of those. I just had the good fortune to work on a senior thesis about D.N.A. I had asked what I thought was the closest thing to an applied mathematician in the math department to supervise my senior thesis. I ended up having a very good experience. I actually made real inroads on a genuine scientific problem using math. So that’s why they graduated me summa cum laude, but I do remember my advisor said to me, “Why’d you have to do so badly in so many courses? You really made it hard for us to give you summa, but your thesis was so good.”
LEVITT: So let’s talk about that first problem you ever worked on, which relates to how D.N.A. is arranged within our cells. Can you start by just describing the basic puzzle? What makes this an interesting problem?
STROGATZ: Let me try to phrase it in human terms, because it’s hard to think about D.N.A.; it’s so small. There’s a lot of it that’s the mystery. There’s a lot of D.N.A. to make the instructions for any organism and it somehow has to fit in a tiny space, which in our case is the nucleus inside each of our cells.
LEVITT: So what’s a lot, like in terms of inches or centimeters?
STROGATZ: If you scale everything up by a factor of about 10,000, use the analogy of hair — the thinnest hair on your head; that’s a very fine thread. Think of that as the D.N.A. molecule. Now imagine that it is about 10 miles long, and now you’re going to pack it inside a tennis ball. That’s the feat that D.N.A. achieves when it fits inside the nucleus of each of your cells.
LEVITT: It’s one of the most mind-boggling, awe-inspiring things that I’ve ever encountered. We have 10-trillion cells in our body, and every single one of those cells manages to fit a 10-mile thread into the size of a tennis ball.
STROGATZ: Exactly. If you took all the D.N.A. in your body, because you mentioned the 10-trillion cells, and if you took out the D.N.A. from each cell and then you lay the D.N.A. end to end. It actually would make 50 round trips to the sun.
LEVITT: Totally crazy. The other point that comes to mind is that I hear you talking about D.N.A. and cells, and my knee-jerk reaction is to think that’s not what mathematicians work on.
STROGATZ: Some of them do. But you’re right, a lot of them don’t. It sounds like you’ve been hanging around with a certain kind of math person. Chicago does have some very pure mathematicians. Yeah, there are people of this type, but it’s true that we don’t really tend to get the headlines. There’s no Fields Medal in applied math. We’re not at the top of the hierarchy. I’ve had my colleagues say to me, “Where’s the math in what you’re doing?” They don’t even recognize it as math. It’s that alien to them.
LEVITT: So you laid out this problem. It’s an unbelievable amount of D.N.A. and it has to go inside the cells. And, obviously, it can’t just be stuffed in there because it’s being used constantly to make life possible. So how did nature solve the problem?
STROGATZ: It’s good that you point that out. You’ve got to organize the D.N.A. in some fashion so that enzymes can get to it and read it, and carry out all the functions of life. And then there’s also the issue that cells don’t just stay put. They divide. And then even more amazing, remember that D.N.A. is a double helix. So what that means is you don’t just have one thread, there’s two threads, and they’re winding around each other in this helical fashion, like the two banisters on a spiral staircase. So it’s very easy for those two threads to get tangled with each other. And that’s bad. If when the cell divides, the daughter D.N.A. has to go to the new cell and the parent D.N.A. has to stay in the old cell. How are you going to pull these threads apart? The way we would do it, if we were storing a lot of thread, we use a spool. We wrap the thread around some kind of shape that looks like a cylinder. And nature actually has something like that inside of our cells that are called nucleosomes. They’re shapes that D.N.A. winds around very tightly.
LEVITT: Are they shaped like spools?
STROGATZ: It is actually a lot like a spool. So that’s the first level of compaction how to solve this packaging problem. By the way, I want to make clear, it’s not like I’m the one that discovered the mystery of the packaging problem. Watson and Crick understood that, everybody for the past 70, 80 years has known it’s a miracle that D.N.A. can fit in inside of cells. And it’s always been a big mystery how exactly. But anyway, the first thing is you have the double helix. You wind it around these nucleosome protein spools. And then the next question was, it’s not enough to just have one spool. It was discovered in the mid-70s, like a few years before I was an undergraduate, that D.N.A. winds around each spool only one-and-three-quarters times. One-and-three-quarters is a weird number. If I said two-times around, you could picture that, like two full wraps. D.N.A. doesn’t quite go two full turns around a spool, it goes one-and-three-quarters, which means it goes around once and then when it’s trying to go around the second time, it comes out at a right angle, 90 degrees to the direction it entered. It doesn’t make the full two turns.
And so you have many of these spools — there’s a lot of them. But okay, what did I do? I had started out, just my math advisor gave me some questions about D.N.A. Some of my friends said, “Hey, we have an expert on D.N.A. packing over in the biochemistry department. You should go talk to Abraham Worcel.” I went to talk to him and gave him my first idea for something to work on, which was so ridiculous that he literally laughed me out of the office. It was the proposition that D.N.A. is not really a double helix. I had mathematical reasons for saying that, but he said in his Argentinian accent, “If you write your thesis about that, it will be a laughable matter.” He kicked me out of there. But I went back with another idea and he said, “Oh, you again. You know, this is also not a good idea, but if you really want a good problem, here’s the problem.” And so he started to describe this certain question to me, but I said to him, “I could do that problem this afternoon.” And he said, “No, you cannot. Did you hear me? This is the unsolved problem that everyone’s breaking their heads on.” That was the phrase he used. I said, “I know, but I know how to do it. I’ll show you tomorrow.”
So here was the question: Abe Worcel was trying to figure out how do you go from the D.N.A. double helix to the next level up, which the biochemist called “beads on a string.” The beads are the nucleosomes, the string is D.N.A. And they could see those under an electron microscope. It looks like a necklace with beads on string, but then somehow the beads move in relation to each other and get tightly packed to make the next level, that’s a thing called the chromatin fiber. How do you go up from nucleosomes to the next level? And so Worcel had a model of the chromatin fiber, a mathematical model. And he said, “I think this is how nature does it, but there’s something about it that I can’t calculate, which is a mathematical number called the linking number. Can you calculate the linking number for my structure?” And that’s what I said to him, “I can do it this afternoon.” And he said, “I don’t believe it.” But I knew how to do it because I had read an article by Francis Crick, of Watson and Crick fame, about this chromatin fiber problem, where Crick pointed out, if you go to a store and you buy a piece of ribbon and then you make a model out of ribbon of the D.N.A. winding pattern.
One thing I should say, don’t picture ribbon with free ends. The ends of the ribbon have to be joined together to make a loop like you would do with your belt. So anyway, Crick’s point was if you make this ribbon into the shape that you want, and then lay the whole thing down on the floor and flatten it out, you could just count how many twists there are in the ribbon; that would give the answer to the question he was asking me: what’s the linking number of his structure? So if you picture your belt and you put a few twists in your belt before you close the buckle, those twists are locked in there. And no matter how you deform your belt, the number of twists is fixed. If you try to untwist part of it, it’ll make more twists somewhere else. In math we would call that a topological invariant; the number of twists that cannot be changed, it’s invariant.
Anyway, the point of the story is I went out, I bought the ribbon, I made the structure and it ended up that Worcel’s model agreed with what was known about the linking number of D.N.A. as it was measured. So when I told him this, he started jumping around the room. Literally hopping up and down and he said, “This is going to be big. You don’t understand, this is a big problem. This is going to be big.” And so it was big. It got published in the biggest biology journal, The Proceedings of the National Academy of Sciences. That’s why they gave me summa cum laude. I solved this big problem. But I didn’t do it with my brain. I just did it by being a person who makes connections. I had read Crick’s article; I understood the implications.
LEVITT: Did you literally do it in the afternoon like you promised him?
STROGATZ: Yeah. I did it that day.
LEVITT: Why did he have any confidence that you had actually computed the answer properly You’re the same guy who had just argued that D.N.A. was not a double helix.
STROGATZ: Well, he didn’t believe me. I had been a crackpot in his eyes until that moment. He said, “You got to show me how you did that.” So I said, “fine.” And I went back to my room, and I got the ribbon and I came back and I showed it to him and I said, “Remember Crick explained this in his 1976 article?” He said, “Yeah.” He read that article. I mean, he and Crick were friends. So I said, “Let’s do what Crick says to do, but we’ll do it for your structure.” We did it together. He took the ribbon out of my hands. He did it himself. But then he said, “We can’t publish this. You can’t explain to people, ‘make a ribbon model of the structure.’ You got to give me a mathematical proof.” So once you know what the theorem is, it’s much easier to find a proof than if you don’t even know what you’re trying to prove. So I did have to work on that. And then I did come up with a proof and I showed him the proof and he said, “Okay, that’s it.” And after that, whenever I would show up at the lab, he would come running over to me and his grad students looked at me like, who’s that undergraduate? What’s going on?
We’ll be back with more of my conversation with Steven Strogatz after this short break.
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LEVITT: Many of my guests on this podcast took a winding path with many ups and downs before they finally found their way to what was right for them. But it doesn’t sound like that’s your story at all. You were doing this breakthrough research in college, and you got your Ph.D. by the age of 26, and you had a dozen publications by then. You got an assistant professorship at M.I.T. How early in your life did you know that math would be your calling?
STROGATZ: I would say the calling was to be a teacher, honestly. That’s what I think of myself as more than a mathematician. I really see myself as being in the business of explanation and savoring clarity of thought. I happen to like teaching math, but it’s really teaching and explaining that’s the thing that wakes me up in the morning and makes me want to go to work every day. But I had one teacher in high school who said something once that was absolutely a pivotal experience. He mentioned that there was a certain problem that he had never seen any student solve. I was in my sophomore year of high school at that point, and I’d never heard a teacher say something like that. Until that point, I had liked math. I just thought it was fun to compete with my friends and that kind of thing. But it wasn’t special to me. But when this teacher said, “I’d never seen anyone solve this certain problem about a triangle.” And then he said, “Actually, I don’t know how to solve it.” He had gone to M.I.T.; he was really a very fancy teacher in terms of his credentials. Mr. Johnson was his name and I just thought, “Wow, Mr. Johnson can’t do this triangle problem. What’s so hard about it?” It didn’t sound like anything different than any other geometry problem.
So I tried working on it and I couldn’t get it, which was already interesting because I could always do any geometry problem within an hour or two if I worked on it. But this one I couldn’t do. And I would think about it day after day and then it became weeks and then months. And I ended up spending about a half a year before I finally got a proof that I thought was correct and I showed it to him and he wrote a little note to the headmaster: “Steven solved this hard geometry problem I’ve been posing for years. He has real talent.” And it meant a lot to me. But also what meant a lot to me was this process of struggling because so many times I thought I had solved it and then it didn’t work. And I started to like it. I started to like that feeling of the fight. Of course, after I did it, I wanted the next problem. I started making up my own questions I didn’t know it but I was doing research, because that is the research impulse. You ask yourself something that you’re curious about and then you get stuck and then you work on it because you’re stubborn.
LEVITT: So let’s talk about another piece of academic research you did, which was trying to understand how thousands of fireflies all come to flash in synchrony.
STROGATZ: Hmm. I’ve always been interested in this puzzle of how does order emerge out of chaos? To me that’s a very nice visual example of it. There are some species of firefly, especially in southeast Asia, places like Thailand, Borneo, Malaysia, where the male fireflies will gather in the mangrove trees along riverbanks every night of the year, and mile after mile of mangrove trees will all light up in unison. And it’s unbelievably spectacular. It looks like Christmas trees.
LEVITT: So you’ve seen it yourself?
STROGATZ: I’ve seen it in movies. I have never gone to that part of the world to see it.
LEVITT: I’m surprised you haven’t. I would’ve pegged you as a person who would’ve done that.
STROGATZ: Well, we don’t know each other that well. You and I have only met each other once. I don’t really want to go. I would get sweaty. I would get bug bites. I’m actually a very theoretical applied mathematician. This is a dirty secret and I’m ashamed to admit it but I don’t really care about nature per se. I like nature as a source of inspiration for beautiful math problems. I don’t really care what’s actually happening. I’m a putz about reality. I have a very gauzy, idealized view of reality, and that’s what’s in my head.
LEVITT: So it’s miles of fireflies, thousands and thousands.
STROGATZ: Oh, yeah. But of course, the question is, look, these are not the fanciest organisms on earth. These are just little beetles, fireflies that flash. How do they do it? And it’s been a question for hundreds of years. Everyone who lived in that part of the world has probably wondered forever. But when Sir Francis Drake took his crew there in the 1500s, they wrote in their ships logs about the glow worms. Some people thought there was a master firefly, like a conductor, and everybody is following the leader. But that doesn’t seem very biological because if a bird eats that one firefly, what, it’s not going to work the next night? Some people thought it was about the weather, but it doesn’t seem to be because it happens no matter what the conditions are.
LEVITT: They always flash at the same speed?
STROGATZ: Well, the species I’m talking about, which is called Pteroptyx malaccae, there’s a very characteristic tempo — might be, like once a second or maybe three times in two seconds or something like that. But it’s pretty fast. By the way, at the beginning of the night, they’re not synchronized.
LEVITT: Okay, they’re not synchronized.
STROGATZ: No, they’re out in the daylight. They can’t see each other. But then as the sun goes down and it gets darker and darker, they start to notice each other, and you get little duos that are in sync and then trios and it spreads. And then pretty soon it’s a whole tree of thousands in the tree. And then pretty soon the trees get synchronized. It builds up over the night. By the end of the night it’s amazing. And the biologists thought that what’s happening is when a firefly flashes, if another firefly sees it, it adjusts its own rhythm. It can’t help it. So fireflies aren’t just emitting flashes, they’re also seeing flashes, and they adjust the timer of their flash organ to be a little bit faster or a little bit slower depending on where in their cycle it was when they received the flash. It’s like musicians in an orchestra can keep time together even without a conductor, by everyone adjusting based on what they’re hearing. We know if you flash a little pen flashlight at a firefly, it will change the timing of its next flash in response to when you flashed at it. And the thought had always been that somehow out of this cacophony of flashing, if everyone’s making adjustments according to mathematical rules, that sync will emerge inevitably. But the biologist didn’t have the training to prove that. They could measure individual fireflies, but they didn’t know how to show a group would inevitably cascade into sync. And so that was something that my friend Rennie Mirollo and I did for a very simplified model of what the fireflies were doing.
LEVITT: And what branch of math would you even call that?
STROGATZ: That would be called dynamical systems. Or nonlinear dynamics, or popularly known as chaos theory. It just means anything that changes its behavior in time according to certain deterministic rules. So deterministic meaning, it’s obeying some rule. It’s not happening at random. It’s got rules that determine the future given the present. So we try to study the implications of rules, by working out their consequences over long time scales into the future.
LEVITT: It seems like this firefly problem is a narrow one, but just judging by the fact that your papers on the topic have thousands of citations. The implications are obviously much broader, right?
STROGATZ: I think so. That’s what really attracted me. Again, I’m not so much interested in fireflies per se. I’m interested in the question of how does a group of individuals who may be dissimilar come to consensus? How do you get self-organization in a group? Think of the cells in your heart that are responsible for triggering the rest of your heart to beat and pump blood. Those are called the pacemaker cells, and there’s about 10,000 of them. They have to essentially solve a problem that’s very much like what the fireflies have to solve. They have to all fire in unison or the heart won’t work properly. So we see this question of collective synchronization all over the place, but you don’t even have to be alive. Non-living things like electrons, in a material called a superconductor, they all start to oscillate in unison in a certain sense. And it’s that property that enables them to transmit electricity with absolutely zero resistance. That’s why they’re called superconductors. So, yeah, as an applied mathematician, this is where I want to be. I like to be flying overhead, looking down at the landscape, seeing the connections between superconductors and societies and fireflies and pacemaker cells. I’m looking for the mathematical unity in all of those, at the cost of maybe getting the science a little bit wrong, hopefully not too wrong, in each case. That’s what abstraction is about; I want to abstract the essence of these phenomena and solve the math problem that I think is the essential mystery.
LEVITT: There’s one publication in particular of yours that has proven just to be enormously impactful. It’s the paper you wrote with your student, Duncan Watts. It was published in the journal Nature in 1998. And it’s one of the hundred most-cited research papers of all time. It’s a little three-page paper that’s called, “Collective Dynamics of Small World Networks.” And I’ve read it; and I understood it. It doesn’t seem like there’s that much to it. When you wrote it, did you think you were writing one of the papers that would become one of the hundred most cited research articles of all time?
STROGATZ: The honest answer is I had a feeling something like that could happen. You’re not supposed to say, “Yes,” to that question. You’re supposed to say, “Oh my God, no. Of course, it never occurred to me.” I didn’t know it would be a hit. I knew it had the potential to be a hit.
LEVITT: Can you explain what that paper’s about and why it’s so influential?
STROGATZ: At one level, it’s about the concept of the small-world phenomenon, which is something that we all know about. When we get on an airplane, we start talking to somebody and we pretty soon figure out we know someone in common or we can figure out a chain of acquaintances that will tie us together and then somebody slaps their forehead and says, “Oh my God. It’s a small world, isn’t it?” So that’s the small-world phenomenon. We all find it surprising because we know that there’s 8-billion people on earth. And we each know maybe a few hundred or a thousand people? I mean, we don’t know a million people. It’s just hard to figure out how can there just be such a short chain that we’re all six handshakes apart, even with a peasant in Mongolia? I always like mysteries as a source of my math problems. And I had seen the movie with Will Smith and Donald Sutherland and Stockard Channing — Six Degrees of Separation. I remember thinking it’s a cool movie. But my student Dunc Watts. He was supposed to be working on crickets, chirping in unison. He actually was — that was his thesis problem.
We have crickets in Ithaca where I teach at Cornell that are the champions of synchronization. We have snowy tree crickets that live out in the orchards of Cornell, and you can capture them and measure them. And the reason that I wanted Duncan to do that is because there had been a lot of theories about synchronization, but essentially no experimental tests. Just a lot of math with no data. So I thought we could collect our own data right here with these crickets. So while he was out there collecting them, he started to think which cricket listens to which? Who’s hearing who? Can they all hear each other or do they only hear the ones right next to them? There was some kind of network structure that was unknown that he thought might be important to solving the problem of how do real crickets actually do their chorusing in unison. And so he asked me, “What do people know about synchronization on networks?” And I said, “Well, not very much. That’s a hard problem. Because there’s a lot of different ways you can connect dots to make a network.”
And also, I’m not a graph theorist, which is the branch of math that does network theory. I knew that social scientists had thought about it in social-network theory, but I, again, I’m not a sociologist either. But then Duncan very quickly said, “You realize if we could solve this problem about how the crickets are connected and how that affects their synchronization, that would be much bigger than crickets because what if they’re connected in a way that’s like six degrees of separation? Then that would have implications for everything about society.” and I said, “Let’s work on it in secret.” Because if we could figure this out, this will have implications for neuroscience. Because my friends in neurobiology had always said to me, “Every brain cell is just six synapses away from every other brain cell in the human brain.” This idea was out there. And then I had a student who was working on the power grid for her project in my chaos course. She showed me a map of the whole Western United States and Canada power grid of which power plants are connected to which, through high voltage transmission lines. And I started to realize there’s all these networks out there of the brain, of the power grid, of the crickets. But the people in graph theory never talk about data. And the people in sociology, they think the small-world problem is solved. They already think they solved it back in the ‘60s with Stanley Milgram.
LEVITT: So you said you wanted to work on it in secret. Is that because you didn’t want anyone to scoop you or because you were embarrassed you were working on it?
STROGATZ: Because we were embarrassed. We were maximally incompetent. We did not know the right math. We did not know the right sociology. At that time, I didn’t even know the word small-world problem. We had no business working on this except that, Duncan, he seemed like the kind of person who would venture into totally unknown territory and not be afraid. I told him, “Don’t read the literature, don’t read about this, because then we’ll end up doing it the way other people have done it. Let’s just think about it ourselves and let’s see if we come up with something good.”
LEVITT: So basically, if I understand what you did, you took this basic graph that people had used and it’s called a regular graph. And in those settings, people pretty much only know their neighbors, and that leads to a very high degree of clustering. And what you and Duncan did essentially is just free that up a tiny bit. So every once in a while, somebody would get to know someone far, far away. And that little tiny change, so maybe one in a thousand people knowing somebody far away turned out almost miraculously to completely change the way the network performed. Is that capturing the gist of what you found?
STROGATZ: That is definitely one aspect of what we found. The reason that the paper made a splash was it was the first one to do a comparative study of networks. If it had just been a result about graph theory of the type that we’re trying to discuss, about regular networks morphing into more random networks, that would’ve been a dud. It was insane to have a paper that had, as its examples, the power grid of the Western United States; the neural network of the one organism whose neural network had been completely mapped at that point, which was a tiny worm; and the graph of Hollywood actors who have been in movies with each other, popularly known at the time as “the six-degrees of Kevin Bacon.” Because people were playing the Kevin Bacon game — which actors have been in movies with Kevin Bacon or they’re two degrees from Kevin Bacon, because they were in a movie with someone who was in a movie with Kevin Bacon. We got lucky I think, in that this Kevin Bacon game was in the zeitgeist. It was very popular at the time. Keep in mind that Facebook did not exist yet. Google did not even really exist yet. So we were thinking about networks in a data-driven way before other people were thinking about them. And there were very few networks that had been mapped at that point. As far as what our discovery was, it came about by asking the question, “What does it take to create the small-world phenomenon?” Like, what kinds of networks would have the property that everyone is only a few handshakes from everyone else? And so the reason we thought about regular graphs, meaning something like a checkerboard, where you could just picture the white and black squares on a checkerboard and maybe you’re only connected to the four squares next to you.
LEVITT: And that is not a small world, right?
STROGATZ: That’s not a small world, right. Not at all. Because if you have a — I think about just a small checkerboard of eight by eight squares and I start at one corner and now I want to make a path going through neighbors to get to someone on the far end of the checkerboard. It’s going to take me something like eight squares to get there.
LEVITT: But now take a checkerboard that’s as big as 8 billion people. It’d take you forever to get there. We know that’s not the way human society is organized. It’s not set up like a checkerboard.
STROGATZ: It’s not set up like that, exactly. But the point was that all the theory that was out there from the physics and largely math side was assuming this very regular structure, which could definitely not explain where small worlds were coming from. So the other possibility, which had also been thought about, was, like, what if I tell 10 people and each of my friends tells 10 people? So you can do that thing. That’s pretty clear that will lead to a small world because suppose I have a hundred friends and each of them has a hundred friends, then you might think two degrees of separation from me will be a hundred times a hundred people. That’d be already 10,000 people. But there’s something really wrong about that, because each of those hundred friends of my hundred friends, a lot of them are the same people that we already counted. Because the way social networks work is that friends of friends are often friends themselves. Like you and I met through our friend Joe Bowler. It’s not really correct to multiply by a hundred each time because there’s social structure. The calculation we did do with the multiplying by a hundred every time. That would be correct if people were connected totally at random. But that’s not right either.
So that was the thing that we did was we asked “What kinds of networks live in the middle ground between totally random and totally regular?” And what we found is, as soon as you moved a little bit away from totally regular, if you introduced any tiny amount of randomness, you instantly got to a small world. Where you got something that was still very structured and clustered with friends of friends, often being friends themselves, the way that real social networks are, and also the way that real power grids are, and the way neural networks are. This property of clustering is very common in real life. Yet, these networks also had very small numbers of degrees of separation. That had been the mystery. How do you get clustering and small degrees of separation simultaneously? And we showed that an easy mechanism is to just have people have a very few far-flung connections. That is enough to suddenly make the world small. Not just the world of people, but the world of neurons in the brain or the world of power plants in the power grid.
LEVITT: And it’s surprising because it’s such a non-linear change, right? This tiny permutation to dramatic results, which is remarkable, but just pops right out of your model.
STROGATZ: It does, it pops right out of the model. Now that model was not meant to be realistic. We wanted to see the simplest, explanatory model for the purpose of pedagogy, for anyone reading the paper so that they would instantly see what the idea was. And we had been led to think about this idea of morphing from one end to the other by an idea in physics that people call a “phase transition.” Where you imagine, if you have water and you start cooling it down, it’s still going to be water. And it just stays water until you hit a magic temperature called the freezing point, and then it qualitatively changes and becomes solid. The thing that we call ice, where the molecules can’t move around anymore. So we were wondering, as you go from a regular world to a random world, would there be a phase transition somewhere in the middle where you would suddenly have a small world? And to our astonishment, there was no phase transition in the middle. The phase transition was all the way jammed up at one side of that transition. As soon as you put the tiniest amount of randomness into the regular world, it instantly became small. Then the question became, how could you avoid a small world? If any tiny amount of randomness — which would be inevitable — how do you avoid the randomness to keep the world big was really the question. Like any little bit of randomness, bang, you’re in a small world. And so we argued in that paper that’s why epidemics were so dangerous in the modern connected world. One person hopping on a plane — this was before people were really thinking about that much. So it’s turned out to be spooky that we even have a sentence in that paper about, like, the alarming and less obvious point is how few of these shortcuts, basically the people hopping on a plane, are needed to make the world small.
You’re listening to People I (Mostly) Admire with Steve Levitt and his conversation with Steve Strogatz. After this short break, they’ll return to talk about Math 101.
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If you’re a regular listener to this podcast, you know that I’m not at all happy about what we teach kids in high school math classes. I’m curious to hear Steven Strogatz’s view on this issue. In my experience, professional mathematicians are the staunchest defenders of our current math curriculum. But Steven Strogatz is not the typical mathematician, so I hope, and I suspect, that he’d like to see things done differently.
LEVITT: You tell a story in one of your books called The Joy of X, about a time when your second grader asked you for help on a multiplication problem. And in the end she berated you because you were using the old school method, and apparently you didn’t know the lattice method that your child had been taught. Now that’s an experience that almost every parent has had. But I would’ve suspected that you would be immune from it. Is that really a true story?
STROGATZ: Of course it’s true. I mean she absolutely looked at me with such contempt. “What? You don’t know how to do lattice multiplication? You’re a math professor?” And I have to say, I still can’t quite remember how to do lattice multiplication. But when I did learn it, it was nifty.
LEVITT: What do you think it is about the teaching of math that by the end of school almost everybody self identifies as being bad at math?
STROGATZ: That’s a tragedy, isn’t it? Actually, what you really hear is more complicated. People say, “I liked math until,” and then there’s a different stopping point for each person. “I liked math until we got to fractions.” You agree with that?
LEVITT: Oh, totally. I think my own math experience as a student is perfect in the sense that by any objective standard, I was an excellent high school math student. I had a real talent for memorizing tricks and rules for solving problems, and recognizing which problem required which trick. And I got A’s; I did great on the S.A.T. And I don’t think I understood or even tried to understand why the tricks worked. And I certainly never even considered the possibility that math, other than arithmetic and maybe a really simple algebra, that I’d ever want to use it for anything other than exams. So I would not have self-identified as being bad at math, but I think, still, I was a complete failure. My interaction with the system was not at all what we want out of students.
STROGATZ: Well, I have heretical thoughts like that myself. I think there is something seriously wrong. Is it that we’re so busy teaching to the test because we need to — there are practical considerations. Kids need to get into college if they’re college bound. Teachers have to teach them a certain set of things. And there’s rules that the school district says you have to cover this year. If I didn’t have any restrictions, if I or some other teacher could do whatever we wanted to help students learn math, would we do a better job?
LEVITT: Look, you’re someone who loves math and you thought so hard about how to communicate mathematical ideas. How would you teach high school math if you could redesign it from the ground up and you weren’t bound by this constraint that kids had to be really good at solving problems that they would literally never, ever solve again the rest of their life, the day after they walked out of that math class.
STROGATZ: I have thought about this a little, and I want to outline a utopian dream. It seems utopian to me. It might be a nightmare for someone else. I picture something like three pathways through high school, and one of them would look a lot like what we currently have, where people get trained to be future professionals in the STEM — science, technology, engineering, math fields. That doesn’t work for most students I think we could agree. Another path would be, to learn what you described as the math that people actually use in their everyday lives or in their jobs. That would probably involve learning a lot more computer skills about things related to data that you can put in spreadsheets, or more sophisticated programs. Learning how to graph data, how to interpret graphs, how to extract meaning from data. So data science is the buzzword that people use for this nowadays. Every person, every citizen will benefit from that, whether it helps them in their job or just to read the newspaper or understand the problems of our times with climate change and the economy, you name it. You can’t go wrong by being more data savvy. So the third pathway is the one that nobody talks about and where I feel like a kind of a crazy person for bringing it up. Do you know the course called Music 101 or Psych 101? Which is a course designed to help people love music, learn the music around the world?
LEVITT: So music appreciation course or an art appreciation course.
STROGATZ: Yeah. Music appreciation, art appreciation, psych appreciation. Everybody knows, hey, I go to this museum. I can understand a little bit of what I’m seeing when I see a painting by Fragonard. You don’t need to know who Fragonard is. But I like it that I once took a course in baroque and rococo painting, so I get a little shiver when I see a Fragonard like, hey, I appreciate what that dude was trying to do. It just makes my life a little richer. Okay? I want that course for math. I want a Math 101 course in high school for a person who certainly doesn’t care about the calculus that Catherine Johnson needed to put the astronauts on the moon. This is just someone who I would like to understand, by the time they’re done with high school, that math is a beautiful thing the way that music and art are. That math has changed the world. I would want them to hear stories about the different cultures in India and China and the Mayan civilization, as well as Europe. So there would be math history; there would be the math in nature. I want them to be able to see the Fibonacci numbers on pine cones, and I get a little thrill from understanding that nature uses math. And understand what Catherine Johnson did in the same shallow way that I can understand what Mozart did. I can’t play music like Mozart. I can’t compose, but I know enough about music that I get a thrill from seeing Amadeus. Like, people don’t have any idea what math is because we’re not teaching Math 101. So I want that. And then some of my students are going to be so fired up they’re going to want to take these other courses. Just like there are some musicians who are going to want to really learn to play the cello, not just listen to Yo-Yo Ma. Right now we’re offering a course that is pre-professional math training. And that’s so stupid. That’s so narrow. That’s really our biggest mistake. So in writing the book, The Joy of X, I was asked by the op-ed page editor of the New York Times: “Can’t you write for my readers, the people who read the opinion page — just an educated, smart person who doesn’t know anything about math — what’s it all about? What are you guys talking about when you say math is elegant?” And I wish in school we did the same thing.
LEVITT: My only quibble with that is that I think it would be a terrible loss not to force the STEM and the data people to do the math appreciation course first. As soon as I hear you say that idea — it’s not an idea I’ve ever heard anyone else say before — I’m in love with it. It seems like exactly the right idea. And it seems more important than any particular set of tools you need to learn. Let me take my own example. I got to a a Ph.D. program at M.I.T. in economics, and I didn’t know integral calculus. I’d never taken integral calculus. I was so hopelessly behind on math that one would’ve thought I couldn’t have survived. But it turns out, when you know what math you need to learn, and you have a reason to learn it, well you can learn math really quickly in that setting. What I was missing, honestly, was math appreciation. I came to M.I.T. with the view that math was worthless. And it was only when I saw applied problems where I could really use it. It was only when I decided that I wanted to be able to not lose so much money at the horse racing track, betting on horses, that I realized that data science was the answer to that, that I learned how to do data science. So I love what you’re talking about. Look, the two books I’ve read of yours that are for a popular audience, they’ve done more to change my view of math than anything I’ve ever experienced. It’s the first time since I was a young child that I felt a kind of wonder, almost giddiness about the power of math. And that primes me to think that math appreciation is possible. And I might not have thought that Math 101 could be a good course, but more or less, I think we just make people read your books. And that goes a long ways towards it.
STROGATZ: Well, okay. A population I think would especially benefit are the math teachers. Because they’ve been brought up in our system too. A lot of them don’t know these stories or the history or the applications, and it’s just super motivating for everybody to realize the broader context of math. They could do their job more easily too. The pushback — people could say, “Where’s the math in math appreciation?” It is possible to do it. It is possible to give genuine mathematical experiences to people where they’re working on puzzles or games or actual math problems. And I’ve done it. I do it here at Cornell where we teach a course that is called Math Explorations, and it’s for the students who have to take math before they graduate, but they don’t want to. They thought they were done with it in high school, but some college is insisting that they take some math, and a lot of these kids already did well even on the advanced placement test in calculus. It’s not that they can’t do math — jump through the hoops. They’re great hoop jumpers. They just don’t know what math is. They don’t know what genuine math is. And so this course gives them a taste of that. And I think that could be the basis for this Math 101 course. It would be nice if everyone had to do some course like that and if the country would agree, I’d be happy to sketch out a nationwide curriculum. But, that’s not how we do things in the United States.
LEVITT: There are a few things harder to change than high school math. But look, I think it’s worth a fight.
STROGATZ: I’m glad that you’re drinking the Kool-Aid here with me.
LEVITT: So I will be happy to help you, but you’re the general and I’m the private. You tell me what to do and let’s figure out how to make this a reality.
STROGATZ: Okay, well, I think you sound serious and I have some time over the next few years to do this. This could be more important than any math paper or anything else I could do with my career. So if you really are a believer, maybe we should do this. We have a major problem in the country and I do think this would help.
LEVITT: All right. So we have a pledge, we have a blood pledge to go out and do something about this.
STROGATZ: Did not see this coming.
It’s rare that I hear a new idea that I like as much as Steven Strogatz’s suggestion for a math appreciation course in high school. It’s so obvious once he says it. I wonder how it never occurred to me before. Which makes me think: Why stop there? Why in school are my kids memorizing the faces of mitosis in the valence structure in atoms? All that accomplishes is turning them off to science. Popular science books and T.V. shows and podcasts — they’re about big ideas; about wonder. People consume them voluntarily, even choose careers based on these books. Shouldn’t all introductory science courses have that as their goal? Right now, much of high school is devoted to memorizing things the kids will never, ever use again. If we got rid of that nonsense, we could both inspire kids with big ideas and free up the space and time to teach kids the subjects that adults actually struggle with. Things like: how to make good life decisions, the basics of understanding data, mental health, conflict resolution, financial literacy. Why don’t we teach those things? It’s such a mystery to me. So what do you think of my vision of high school? Inspirational? A recipe for disaster? Either way, we are always eager to hear your thoughts. You can reach us via email at firstname.lastname@example.org. That’s P-I-M-A@freakonomics.com. Now, to our listener question.
LEVEY: Hi, Steve. So you’re a huge Taylor Swift fan. Did you try to get tickets for her new tour?
LEVITT: I didn’t. I’ll be in Germany and I don’t think her European tour dates have been released yet.
LEVEY: So for people who don’t know, tickets for Taylor Swift’s tour went on sale a few months ago through Ticketmaster. The presale was an absolute disaster. There was such high demand that the site crashed several times. The public sale the next day was canceled because the presale went so awry. Ticketmaster merged with Live Nation in 2010. And now Live Nation Entertainment controls about 70 percent of the market for ticketing and live events venues. So people are saying the company is a monopoly and the government should do something about it. Steve, what do you think?
LEVITT: Well, there are all sorts of formal rules that economists use to try to decide antitrust policy, but I take a much simpler heuristic view and it has only two conditions. The first one is that a company has a really large market share and with 70 percent of the overall market, Live Nation Entertainment certainly qualifies on that condition. And the second condition is that everybody hates them. And that’s certainly true about Ticketmaster and Live Nation. I don’t know if you ever buy tickets, Morgan. I do sometimes, and I’m certainly not price sensitive, but every time I get to the end of a purchase and I see that my $50 tickets have somehow gotten multiplied and doubled because of fees for Ticketmaster, I shake my head in disbelief that that’s the market equilibrium.
LEVEY: Okay, so since Live Nation Entertainment is a monopoly and people hate it, what do you think should be done about it?
LEVITT: Before we talk about a solution, we actually have to figure out what problem we’re trying to solve. But first we just have to recognize that it’s a tricky problem because not everybody has the same objective. Economists think about efficiency, and that means you allocate the seats to the people who are willing to pay the most for the seats, say with an auction mechanism. And that would lead to the greatest revenue for the venues and for the artists. And it would put the people in the seats who are willing to pay for it. But that’s not what the Taylor Swift fan wants. They want the seats to be cheap and to go to the most loyal fan. So there’s this real trade off between efficiency and fairness, and there isn’t going to be one solution that’s going to solve everybody’s problem. But then once you’ve made that choice, I think we could do things way better than we’re doing with Ticketmaster. Each concert, the venue would set out a set of rules of how many tickets people could buy. And there would be the possibility that any number of firms, Ticketmaster and any competitor, would be able to compete on even footing providing a user interface that people could choose between, to try and figure out how they wanted to buy their tickets for that concert. And really, I’m not describing anything radical, it’s exactly how we buy airline tickets. It’s exactly how we do hotel rooms. If you want to get a hotel room, you go to Booking.com, you can go directly to the hotel, you can go to Travelocity. There’s lots of ways to buy the same hotel room and people figure out how they want to do it. And those companies are free to charge different fees, but in the end, the equilibrium is that they charge very low fees because if you charge a high fee, no one’s going to use you. And so it’s really strange to me that the concert market has ended up so different than the hotels market or the car rental market, or the airline ticket market. And I don’t see any good reason for it.
LEVEY: Thank you to those of you who wrote in about the Taylor Swift Ticketmaster debacle. If you’re still curious about the live event ticket market, Freakonomics Radio has an episode called “Why is the Live-Event Ticket Market So Screwed Up?” If you have a comment or question for us, our email is email@example.com. That’s P-I-M-A@freakonomics.com. It’s an acronym for our show. Steve and I read every email that’s sent and we look forward to reading yours.
LEVITT: And you know what the real lesson here is Morgan?
LEVITT: Don’t mess with Taylor Swift.
In two weeks. We’ll be back with a brand-new episode with Suzanne Simard, professor of forest ecology. I find her discoveries about trees and the way they communicate absolutely mind blowing.
People I (Mostly) Admire is part of the Freakonomics Radio Network, which also includes Freakonomics Radio, No Stupid Questions, and Freakonomics M.D. All our shows are produced by Stitcher and Renbud Radio. This episode was produced by Morgan Levey and mixed by Jasmin Klinger. Lyric Bowditch is our production associate. Our executive team is Neal Carruth, Gabriel Roth, and Stephen Dubner. Our theme music was composed by Luis Guerra. To listen ad-free, subscribe to Stitcher Premium. We can be reached at firstname.lastname@example.org. That’s P-I-M-A@freakonomics.com. Thanks for listening.
STROGATZ: And so I thought, “Let’s take this adventure, but if this doesn’t work, you’re going to do crickets. Because I know we could solve that problem.”
- Steven Strogatz, professor of applied mathematics at Cornell University; author; and host of The Joy of Why podcast.
- “Collective Dynamics of ‘Small-World’ Networks,” by Duncan J. Watts and Steven H. Strogatz (Nature, 1998).
- “Synchronization of Pulse-Coupled Biological Oscillators,” by Renato E. Mirollo and Steven H. Strogatz (SIAM Journal on Applied Mathematics, 1990).
- “Structure of Chromatin and the Linking Number of D.N.A.,” by Abraham Worcel, Steven Strogatz, and Donald Riley (PNAS, 1981).
- “Linking Numbers and Nucleosomes,” by Francis Crick (PNAS, 1976).
- The Joy of Why, podcast by Steven Strogatz.
- “Why Is the Live-Event Ticket Market So Screwed Up?” by Freakonomics Radio (2017).
- Six Degrees of Separation, film (1993).