An Interview with Steven Strogatz - Believer Magazine
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An Interview with Steven Strogatz

[mathematician]
by Prashanth Ramakrishna
December 15th, 2020
Photo courtesy of John Groo

The translator of mathematics is a rare breed. He requires not only a mastery of the informationally dense symbolic formulations in question, but also the ability to artfully press those formulations into the very linguistic paradigm deemed inadequate to express their meaning. Above all, though, he must be able to inspire the curiosity and wonder that motivates the rather uncomfortable cognitive contortions that accompany concepts like probability, higher dimensions, how change changes, and so on, that the primate brain did not evolve to think about.

Steven Strogatz is known amongst the mathematically initiated for his seminal contributions in dynamical systems,[1] where he studied, amongst other phenomena, the synchronous chirping of crickets, and in complex network theory,[2] where he triggered a still-continuing cascade of research with his 1998 Nature paper “Collective dynamics of small-world networks.”[3] Despite these achievements, however, he might say that his most important role has been that of a translator rather than researcher. Through his books, Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life (2003), The Calculus of Friendship: What a Teacher and Student Learned about Life while Corresponding about Math (2009), and The Joy of x: A Guided Tour of Math, from One to Infinity (2012), as well as his charmingly approachable columns in the New York Times, he has done as much as anyone to assuage the public’s lingering aversion to mathematics. His dedication to and enthusiasm for communicating mathematics is apparent in his teaching style, which emphasizes intuition, application, and real-world examples that cannot help but capture a student’s imagination. For these efforts, Dr. Strogatz has collected a list of honors too long to enumerate. Among them are the Euler Book Prize issued by the Mathematical Association of America, the AAAS Public Engagement with Science Award, the Lewis Thomas Prize for Writing about Science, and selection as both a Cambridge Rouse Ball and MIT Simons lecturer. He currently hosts The Joy of x podcast for Quanta Magazine.

I first came across the work of Dr. Strogatz while searching for supplementary material to a course on chaos theory. His volume Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering was essential to evolving my relationship with mathematics, defined at the time by theorems, proofs, calculations—all in the air-tight classroom context. Really though, the power of mathematics is in its ability to produce extremely precise descriptions of how the natural world works, then use those descriptions to make predictions about how the natural world might change in the future. Soon after, I began engaging in work involving modeling dynamic real-world phenomena—the seemingly random sleep-wake cycles of adolescent rats, blood pulse wave reflections in arteries, how organizations can use hiring as a tool to maintain ideological diversity. Indeed, this Strogatz-inspired realization that a mathematician wields mathematics like a painter wields his brush to capture some aspect of reality fundamentally altered my vision. Life is patterns. If only one could learn to discern them. My phase-change, I’m sure, is wholly unoriginal.

So, in the thick of Ithaca’s infamous collegiate winter, I traveled with much anticipation to Cornell University, where Dr. Strogatz sits as the Jacob Gould Schurman Professor of Applied Mathematics. I was intent on discussing his most recent book, Infinite Powers: How Calculus Reveals the Secrets of the Universe (2019). We talked in his office for hours—yes, about the story of calculus, but also the approximation games my father played with me as a child, the divide between humanities and sciences, what makes a good mathematical education, the limits of mathematical models, the roles of beauty, intuition, and creativity in mathematics, and so much more. Below is an edited version of that conversation. Once we finished, Professor Strogatz kindly drove me back to the slightly seedy frat house in which I was staying. Later that night, amidst a combined aroma of tobacco, marijuana, and pizza, I found myself in the company of a student who, upon my mention of Professor Strogatz, exclaimed: “The dynamics legend! The Watts-Strogatz Model—that’s him, right? Everyone kept telling me to take a class with him. Alas, the ol’ schedule didn’t allow it. There’s a regret.”

—Prashanth Ramakrishna

I: CELESTIAL ORIGINS OF CALCULUS

THE BELIEVER: To me, your book is trying to explain not just how calculus developed, but also why it works. Is that too simplistic, or does it map at all onto your motivations for writing the book? 

STEVEN STROGATZ: From my perspective, I’m trying to do three things in the book. The first two are somewhat intertwined—explaining how calculus came about and why it works, as you mentioned. The third objective I had was to show how calculus has changed human life. My goal was then to write something that included all three of those aspects: history, concepts, and applications. History is not my specialty, though. There’s a lot of folklore in math history. Just think of your teachers—some of them probably tell stories about Gauss or Riemann. I find that we like to have these conversations over the centuries. I feel like I know what Archimedes was like. I’m sure I’m kidding myself. 

BLVR: There’s an enormous tendency to mythologize with mathematicians, as though they have superpowers. The pantheon of mathematicians is filled with such diverse characters. Yet, I feel as though that diversity has a bipolar structure, split into the egomaniacal and ego-less. That duality is embodied quite well by some of the characters in your book.

SS: Yes, by Fermat and Descartes. There was also Newton and Leibniz. And, though I wouldn’t call them rivals, Galileo and Kepler were contemporaries that make a natural pairing. But I ended up leaving out an enormous amount of important history: Riemann, Gauss, Cauchy, Lagrange. I began to realize, as I was writing, how mathematically elementary I wanted the book to be. I was thinking of, say, my editor, with whom I couldn’t even assume high school algebra. If I can’t even assume that, then what’s the point of even talking about what Riemann did? I wanted to be honest about the math so that any reader could follow it from scratch. 

BLVR: Luckily, an intricate history isn’t necessary to discuss the two major pieces of calculus—differential calculus and integral calculus. Saying that now, I’m realizing that people often overlook how much explanatory value names can have in math—“differential calculus”, concerned with differentiating continuous wholes into infinitesimally small components, and “integral calculus”, concerned with integrating those many spliced parts into a unity.

SS: I see it that way too. When you put it like that, it seems very natural. Calculus is incredibly powerful because many things in the world can be thought of as continua that can be sliced finer and finer. We think of time like that, as well as space and material objects. 

BLVR: So, here’s the question: why was that insight so seismic? 

SS: If you had to boil it down, it would be because calculus is the mathematics of change. To be a little more precise, it’s the mathematics of things that change continuously. We’re not talking about things that jerk around crazily. Such things include how the amount of virus in the bloodstream of a person with HIV—“viral load” as it’s called—plummets after taking combination therapy, the three-drug cocktail. They also include how celestial objects move around in the solar system—that was one of the great original mysteries and produces an interesting question. Historically, mankind was so obsessed with the stars and the planets and the moon and the sun. But, why? Why was that the first thing people could mathematize? People knew a little bit about folk medicine and animals and agriculture, but for those you didn’t need much beyond just counting. Why was all the fancy math—to the level of calculus of trigonometry—about astronomy? I have an answer, but I wonder if you’ve ever thought about this. 

BLVR: Yes, I have a theory as well. I think there’s a two-part answer. The first part is that the movements of certain celestial objects are so obviously patterned. Even if one isn’t sure about what those patterns are, there’s a strong sense that they’re there and that one can discover them over time simply by paying close attention. The second is that those patterns can be reliably observed. And, ultimately, these are the necessary precursors for science—a sense that something is worth studying and a way to reliably study it. 

SS: Very good. Also, the timescale is just right. It’s not too fast and it’s not too slow. Things that are really fast—even dropping a rock off of a bridge—were very hard to study in those days. There weren’t any clocks good enough to measure the amount of time it would take the rock to hit the water. It was fast! Remember, also, when this work was being done. It was late at night, dark without electric lights, and the sky is very awesome in lots of parts of the world. 

I think there’s a puzzle, though, because these objects are very far away. You might have thought medicine would be the first subject because of its immediacy. Yet, it’s so complicated that the laws are still, to this day, mysterious. So, I think you’re right—regular patterns, slow enough, and observable to the naked eye. You could keep charts and records. In the end, the remotest possible subject is the first one to be understood. Calculus starts to grow out of that ancient quest to predict the motions of the planets. The planets, as they’re named, are the “wandering stars.”  

BLVR: Looking up was a natural way to isolate a phenomenon of motion. Many of the earliest advances that served as precursors to calculus, though, were a consequence of developing ways to artificially isolate phenomena of motion. I’m envisioning the Galilean experiment of the ball rolling down the grooved ramp. This was the methodological breakthrough that made examining terrestrial motion possible. 

SS: Yes, the Galilean experiment in an Archimedean spirit. Archimedes did many simplified experiments with floating objects and things of that nature. He’s like a modern Archimedes, Galileo. The other major methodological breakthrough was the deliberate ignoring of certain effects. For example, friction is important in real life. But Galileo took such great pains to make the groove that he rolled his ball down as smooth and as straight as he could because he wasn’t interested in friction. That wasn’t what he was trying to understand. He was really trying to understand gravity and how it affects the motion of objects. 

These small experiments were to determine how things in the universe change and the language for describing that change turned out to be calculus. In algebra we learn to think in terms of relations such as distance equals rate multiplied by time. That works if something moves at a constant rate, or speed as it were. But, a lot of things don’t move that way. When a planet moves around a sun, it whips around when it’s close to it and slows down when it’s far away. When Michael Jordan goes up for a dunk, he looks like he’s hovering at the top of his jump, after which he accelerates towards the ground. That movement speeds up and slows down has to be dealt with. The limitations of geometry are similar. Classical geometry can handle shapes that are made of straight lines and quantify the area under a smooth curve like a parabola with straight-line geometry. You need a new idea that can cope with smooth curves. That idea is calculus.

II: BEAUTY-ADJACENT 

BLVR: That idea of calculus took many mathematicians to develop. In that vein, I wonder if we shouldn’t give some airtime to the major characters of your book.  

SS: I wanted to tell the story with a thousand characters. But, given the constraints, I thought Archimedes was the most important mathematician of antiquity in that he solved problems that are today torturing kids on their SATs. He figured out how to quantify all these smoothly curved shapes—circles, spheres, parabolas. Calculus is about change. If a line is straight, then it doesn’t change direction. A curve, by definition, is a line whose direction is changing. So, curves are a part of this bigger story of change. Archimedes is the first to be able to measure change in the form of measuring properties of curved objects such as surface areas, volumes, and arc lengths, and he does it by taming infinity.  

BLVR: One of the ways he harnessed infinity, which foreshadowed calculus, was by cleverly taking objects apart and putting them back together. 

SS: He’s a virtuoso at chopping shapes. It’s like cubism. Braque has a kaleidoscopic image of someone walking down the stairs. Out of discontinuity, you get a sense of continuous motion. How can you build something smooth out of something jagged? The cubists knew how in the early 1900s, but Archimedes knew how in 250 BC. He decomposes something as smooth as a parabolic segment into many increasingly tiny triangular shards. It’s a fantastic image. 

It’s incredible how relevant Archimedes still is today. Shrek’s smooth belly or little trumpet-shaped ears are built out of millions of polygons. All of computer graphics is the skillful use of incredibly many tiny polygons that approximate smooth faces and bellies and ears and everything else. Facial surgeons use the same technique to predict the outcomes of reconstructive surgeries. So, using Archimedean thinking, we effectively have flight simulators for surgeons that really work. 

BLVR: And after Antiquity?

SS: We jump ahead by 2000 years to the beginning of the 1600s, when we start to quantify how things move both on the earth and in the heavens. Enter the characters of Galileo and Kepler. Both are mathematicians, but also very scientifically driven, outward-looking; they’re not doing math for its own sake. They’re also sort of imbued with a sense of awe about nature. 

Kepler is explicitly religious. Although he became a schoolteacher instead, in his youth he was planning to be a Lutheran minister. At one point he describes himself as being in a state of “sacred frenzy.” He’s a mystic about the patterns of the universe. He thinks he is discovering the sacred geometry that God used to construct the universe. Kepler is the one who figured out what laws planetary motion obeys. Well, he didn’t have the law of gravity, for instance, but he did find a lot of geometry in the motions of the planets. Copernicus, before him, had said that the planets move around the sun, but Kepler put some real mathematical teeth into that claim.

BLVR: I’m interested in your characterization of Kepler, because mathematical mysticism is another major historical through line from Pythagoras onwards.  

SS: Let’s clarify what we mean by mathematical mysticism. Here is an example of Kepler as mystic. At the time, there were five known planets besides us, those close to the Sun—Mercury, Venus, Earth, Mars, Jupiter, Saturn. Interestingly, if you draw a pentagram, then connect its corners, you create a pentagon. The other, more traditional way to think about it is that if you draw a pentagon, then internally connect its corners, then not only do you create pentagram, but you also create another pentagon nested in the center of the original.  The pentagon is a shape that gives birth to itself and can keep doing so forever. So, according to Dan Brown, and in Pythagorean thinking, the pentagon is the symbol of fertility and of femaleness. “Five” is very important in the numerology of Pythagoreanism, and Kepler is very Pythagorean. He sees that there are five planets and also that there are five special solids, the Platonic Solids—the cube or tetrahedron, whose faces are all identical regular polygons—and thinks to himself, “They must be connected!” He concluded that the universe is built on Platonic geometry. He came up with a scheme, published in his Mysterium Cosmigraphicum, of cubes inside of icosahedrons[4] inside of dodecahedrons[5] that almost explained the positions of the five planets and their average distances from the Sun—a beautiful idea that nature chose not to use. 

Is math the secret of the universe? He thought so. And, so did the Pythagoreans—“all is number”, supposedly uttered by Pythagoras after discovering that music obeys mathematical laws of harmony. This is a theme that does run through math. We don’t talk about it much anymore because we don’t like to think of ourselves as mystics. But, it’s there. People think because certain theories are incredibly beautiful—supersymmetry is an extremely beautiful generalization of quantum theory—that maybe that beauty is telling us something. It certainly worked well for Einstein. The most beautiful version of Relativity turned out to be the best one. There’s a complicated relationship between mathematical beauty and truth. Is beauty a good guide to truth? Unfortunately, not always. But sometimes. 

BLVR: In pure math, there’s an emphasis on elegance, which is more or less synonymous with simplicity. So long as beauty is associated with simplicity in this way and the yearning persists for simplicity to correlate with fundamentality, I think aesthetics will remain a guide to truth. The aesthetic sense is a big part of mathematical intuition, and perhaps one that requires a bit of un-learning. 

SS: Let’s go back to Kepler. He, like everybody before him, thought that planetary motion would involve circles, because circles were the simplest and most perfect shape. To this day, people still think of circles as symbolizing perfection and eternity. It’s not an accident that we use circles for wedding rings—“I love you forever” just like “a circle never ends.” But, when you really look at the planets, it doesn’t seem that they do move in circles. On the other hand, they almost do. Kepler tried to find, as Ptolemy and other before him also had, ways of making it work—what is classically called, “saving the appearances.” 

BLVR: The truth was beauty-adjacent. 

SS: In the end, it turned out that circles didn’t work. But, what did work were ellipses, which are very close cousins of circles. You could create a cone by growing a circle, letting it expand like a dixie cup at a water cooler. If you then cut through the dixie cup at an oblique angle, the resulting plane would be in the shape of an ellipse. Cones come from circles, and ellipses come from cutting cones. This simple shape in pure geometry turns out to be how the planets move around the Sun. That’s Kepler’s First Law. The data fit perfectly. He was right! His prejudice that God would be using geometry looked like it was being confirmed. And, it was simple. 

Another equally simple “truth” is Galileo’s discovery that if you throw something in the air, its trajectory is a parabola. “Morally speaking,”[6] basketball shots should follow a parabolic trajectory. But they don’t because there’s this annoying gunk of air resistance. So, real life. . . is beauty-adjacent. Which raises the question: do you pay attention to what is actually happening, or do you live in a more Platonic world of what should be? If you’re slavishly devoted to the truth, you won’t discover the deeper principle that is a little bit of a lie. The parabolic arc is a lie. Galileo often chose the lie; he chose to see the idealized version.

There’s a lot in common with art here. There’s a quote from Picasso that I’m rather fond of: “Art is a lie that makes us realize truth.” I think this impulse is there in a lot of the great scientists. They overlook things that are inconvenient in order to get at deeper truth. But, it’s very risky because that can also be intellectually dishonest.  

BLVR: So, 2000 years of people producing solutions to problems that hint that something like calculus should be formalized. Then, along comes Newton, who’s ready to put it all together—The Grand Synthesizer. 

SS: It’s unclear, actually. Should he be considered revolutionary, or “standing on the shoulders of giants”? Both, really. When I began learning about him, I thought he was overrated, but as I came to learn more about him, I started to think that he really does deserve to be right there alongside the best of all time—but more for how he used math to help us think about other subjects, especially physics and astronomy. He gave us the view that nature is logical, which we didn’t know until him. That’s not the normal statement. Most people would say, “He discovered the laws of motion.” He provided a cohesive explanation for all earthly and celestial motion known at that time, The System of the World, as the third volume of his Principia is called. This included things not traditionally mentioned like the tides and comets. It’s definitely not true that “he invented calculus,” as people often say. But he did put everything together that had been known before him into a systematic machine, into a collection of algorithms that is now so streamlined and finely tuned that you could teach it to average high school students. That may be the greatest testimony to what he accomplished, that he made it so mechanical that you don’t even need to know what you’re doing and you can do calculus, which is unfortunately the state of operation for many students taking it.

III: LAPLACE’S DEMON AND THE MINDSCAPE

BLVR: Calculus clearly has an insanely vast scope. But, as you mention in your book, it also has its limitations. More precisely, the determinism that calculus relies on for its predictive value doesn’t always hold. This is communicated through a thought experiment, originating with Laplace,[7] that I thought you could discuss. Why is the “Clockwork Universe”[8] overseen by Laplace’s Demon untenable? And why does that untenability spell limitations for determinism and, by extension, calculus? 

SS: There was a time, before the advent of Chaos Theory in the 1970s and 80s when it was widely thought that if a system was deterministic—meaning that it was governed by rules that don’t have any element of chance in them, no randomness—then, like a movie, if it were restarted, it would always play out the same way. Any system that was deterministic would be predictable. . . in principle at least. So, if you could measure where everything was at the start and how fast it was moving, Newton’s laws would tell you how every particle in the Universe would move every instant thereafter. Laplace’s Demon is an imaginary character of infinite intelligence that knows every position and velocity of every particle and could calculate the future of the universe, including all the emotions any of us ever felt, all of history, everything. 

As a thought experiment the picture of Laplace’s Demon presiding over this deterministic clockwork universe unfolding according to Newton’s laws was disturbing. We don’t really believe in it today for various reasons. The modern argument is, “Well, Newton’s laws have been superseded by the laws of quantum mechanics as the main laws of physics.” But even if you were living in a classically Newtonian universe, you still wouldn’t be able to make predictions arbitrarily far into the future. You can only accurately make a measurement to a certain number of digits. Traditionally, it was thought that if enough digits were measured, then one could make predictions that outlasted the age of the Universe. So, in practice, I should be able to do well just by making better and better measurements of initial conditions. But we’ve known that’s not true especially since the advent of Chaos Theory in the 1970s. We learned that even these simple mechanical systems little errors in measurements of initial conditions grow exponentially fast, leading to ridiculous predictions after a short time. We call that phenomenon “Chaos.” This is why predicting the weather beyond a handful of days in advance is very difficult. In principle, it’s never going to be possible to predict the weather more than a few weeks in advance. 

BLVR: Here’s a question about formalisms. I’ve recently been deriving a lot of entertainment from ambushing friends with this question: is there an obscure corner of the Mindscape in which the propositions “P” and “Not P” are simultaneously true. 

SS: I see—so a universe in which we don’t have an “Excluded Middle.”[9]  

BLVR: This is exactly the observation my friend made, and the one I was hoping you would make. The Mindscape is just the abstract universe of ideas. Two people with the same idea might occupy the same location in the Mindscape just as two people can occupy the same location in the physical world. So, if you can conceive it, then it exists in the Mindscape. The dirty little secret about math is that you can arbitrarily construct any system you want, and it doesn’t necessarily have to map onto reality. 

SS: Are we saying, then, that anything goes? I would have said that in math we do have certain rules. We have to agree on what the rules of logic are. We have some constraints. 

BLVR: The question of whether there’s a possible universe that such a mathematical system describes is a different question, right? 

SS: Yes, there are different questions. There’s the question of whether the imagined playgrounds of math correspond to anything out there. But there’s also the more general question—and this is what it sounds like you’re asking—of “Can we have different kinds of math? Or even different kinds of logic?” Forget about what’s real. 

BLVR: It’s funny that you use the word “real.” I suppose the deeper question underlying the one I just asked is: are mathematical laws actually real? Or are they only real because what they describe—the real world, that is—is real? Mathematical objects—ideal, perfect objects—are they real? 

SS: “Real” might be a hard word to be using here. There’s also the question of different worlds to think about. Roger Penrose, in his book The Road to Reality says that there are three worlds: the mathematical world, the world of ideas in the mind, and the “real” world of physical objects. I’m not used to the tripartite picture. Normally I would have thought that there’s the platonic realm and the material realm. But, it’s interesting to distinguish between the ideas that the mind can have and mathematical ideas, which I take it he thinks exist outside of minds, but not in the “real” world, like Pi. Pi is a real. . . I don’t know what he means. But, in a way I do, because when we mathematicians make mathematical discoveries—a certain formula, or a relationship between mathematical objects we’ve thought about—it feels like we’re discovering something. It doesn’t feel like we’re creating it the way that Chopin created Nocturne. I don’t know if he thought the Nocturnes were out there and he was discovering them. They were created. Although sometimes Michelangelo would talk about “releasing the statue from the stone.” Is the Fundamental Theorem of Calculus out there waiting to be discovered? Or the concept of Pi? Or the formula for the area of a circle? Yes, it feels to us like that’s out there. But where!? 

BLVR: Another argument for whether mathematical concepts are real—a natural manifestation of physical reality—is based on their predictive value. There are things in the universe, for example, that behave like the number e. We construct e mathematically as the value (1+1/n)n approaches as n goes to infinity. But, put bacteria in a petri dish and they will grow exponentially. At the same time, you can construct a number that is infinite and nonrepeating, just like e, that needn’t have any bearing on the world whatsoever. If I can put it this way, it seems reasonable to say that the Universe doesn’t “know” about that hypothetical number. But, somehow, it seems like the Universe “knows” about e

SS: That’s what I’m trying to argue throughout the book. A lot of people don’t like Feynman’s quote: “You have to learn calculus because calculus is the language God talks.” But he’s right in the sense that calculus is the best description we have of all aspects of the Universe, whether it’s how water moves, or air, or how heat flows, or how electricity and magnetism work, or quantum mechanics. All of those have, at their foundation, mathematical equations written in a very particular sub-dialect. Most people would say, “Math is the language of the Universe,” if they’re going to say any sentence like Feynman’s. But, it’s much more precise than that: differential equations are the language of the universe. The language of the universe is not written in algebraic topology. It’s written with the symbols of calculus, using differential and integral operators. That has been true across every domain from atomic to cosmic that we’ve looked at so far. People that know the details of quantum theory will say that at the deepest level there’s something else called a “path integral”. . . but it’s still an integral! It’s still calculus! 

BLVR: The question of “real mathematical objects” can be extended into idealized models. The tools of calculus are really only valuable insofar as they are intelligently organized into useful models. We could center the discussion on one model that you briefly worked on that I became interested in while reading your work. I want to ask you, though, why, in general, people should be suspicious of models. 

SS: Why don’t you go ahead and describe the model. It seems like you have it fresh in your mind. 

BLVR: Well, the question is: can you encourage political moderation in a society? And, the hyper-simplified model that you used in order to investigate this question divided an anonymous community into four sub-populations, one with the prevailing worldview B, two with an insurgent worldview A—one group of which is composed of zealots, meaning that none of its members will change their minds—and, finally, one of moderates whose worldview is represented by AB.  There’s a series of differential equations that you defined to describe how the sizes of these subpopulations change over time. 

SS: My coauthors and I imagined encounters between people holding different views in which one person convinces the other to “come around.” They might be successful with some probability. We don’t go into any details. It’s, as you said, a hyper-simplified picture. 

BLVR: The results were quite pessimistic. After trying to tweak the model in many different ways to encourage moderation in the long run, almost nothing worked. The one thing that worked was the introduction of an external moderating force independent of interpersonal interactions. 

SS: So, something like a top-down intervention from the authorities. Or maybe a common external event. You mentioned 9/11 off-air. Rulers have known that common enemies are the best way to unify a society since the beginning of civilization. In 1984, isn’t Oceania[10] perpetually at war? But, this whole discussion hinges on the question of whether the mathematical model is reliable for anything that matters. It’s very tricky to derive applications of calculus to human affairs because we don’t have the analogue for them to Newton’s Laws or Maxwell’s equations of electricity and magnetism. We don’t know how to predict what people will do, either individually or en masse. There are real philosophical questions about whether that’s even possible in principle. It’s game playing that we’re doing. 

On the other hand, it can be interesting to play these games because they show us the epistemological humility we should have. What do we know about human behavior or society? These models often have such surprising behavior in their own right. We found that in our little model of moderation. Making the moderates more stubborn in our model did not promote moderation; it made it easier for the zealots to take over. That is a chastening lesson for us: you think you might understand what’s going to happen, but even in this deliberately simple universe it’s hard to make the right predictions. Experience teaches us that we often aren’t right when we make claims that certain social interventions will have this or that effect. But that’s such a nihilistic and pessimistic remark. You still have to do something, don’t you? Or do you just. . . I don’t know what to say about this. 

BLVR: Well, the hope is that it’s possible to get it right in principle. But thinking about Laplace’s Demon showed us that it isn’t. So, it’s possible to some degree, such as it is with weather. Our predictive powers even about things as complicated as social progress might only be good to a certain horizon. As we move towards that horizon, our ability to make good decisions improves. But still, there is a horizon, and it may be that we’ll never be able make the best or even good decisions with regard to human affairs because they’re so chaotic. That’s depressing.

SS: There are certain simple principles of behavioral economics that seem to be good. Organ donation, for example. At the moment, you have to opt in to donate your heart if you’re killed in a car accident. By default, you’re not a heart donor. That doesn’t have to be the case, though, as Richard Thaler[11] showed us with his simple idea of “nudging.” By depending on people’s inertia—that they’re not going to opt out because it’s easier to do nothing—the majority of people become organ donors. That could be a case where you’re using a little bit of insight into human behavior to do something you believe is good. This is the research for the future. Where does math meet the social sciences? The soft sciences are the hardest of all. They’re the last ones to be mathematized. Maybe they never will be properly mathematized. It’s the so-called hard sciences where we’ve really made a lot of progress. Math seems to apply so well to them. But, in a way, that makes them easy.

[1] A dynamical system is one in which a function describes the time dependence of a point in space. Classic examples of dynamical systems include swinging pendulums, the flow of liquid through a tube, and population growth. In certain dynamical systems, seemingly random behavior can occur. This phenomenon is known as “chaos” and the branch of mathematics studying it known as chaos theory.
[2] Complex network theory is the study of graphs that represent systems in the real world such as Facebook friends and neural connections in the brain.
[3] “Collective dynamics of small-world networks”, which Strogatz coauthored with Duncan J. Watts, is the sixth most highly cited physics paper of all time. In it, Strogatz and Watts give mathematical teeth to the popular notion of “six degrees of separation.” They propose a class of “small-world networks” that has since been applied to social movements, seismic stations in Southern California, and the usability of information in large databases. The most prominent construction of small-world networks is known as the “Watts-Strogatz Model.”
[4] An icosahedron is a polyhedron with twenty faces.
[5] A dodecahedron is a polyhedron with twelve faces.
[6] Mathematicians say that a statement is “morally” right if, according to their own aesthetic intuition, they feel that the statement “should be true.”
[7] Pierre-Simon Laplace was a French polymath whose life spanned the last half of the 18th Century and the first half of the 19th. He is credited with the foundations of Bayesian probability, central to modern study of probability, and inventing the Laplace Transform, ubiquitous in modern statistical physics. Laplace is also known for his magnum opus on celestial mechanics. When Napoleon, seeking to embarrass Laplace after inviting him to present his model of the solar system, asked, “Is it true that there is no mention of God in your work?” Laplace replied, “I had no need of that hypothesis.”
[8] The term the “Clockwork Universe” was used, particularly among Enlightenment deists, to describe a conception of the Universe in which it, like a mechanical clock, works predictably, mechanically, governed by the Laws of Physics. One can think of the phrases “Newtonian Universe” and “Clockwork Universe” as synonymous. Laplace’s Demon, who has knowledge of the forces acting upon, position, and velocity of every particle in the Universe, can make predictions about the state of the Universe arbitrarily far into the future. After all, the individual state of each particle will evolve predictably according to the Laws of Physics.
[9] The Law of Excluded Middle asserts that any proposition is either true or its negation is true. It is one of the Three Laws of Thought, the other two being the Law of Non-Contradiction, which states that contradictory propositions cannot be simultaneously true, and the Law of Identity, which states that each thing is identical to itself. Here, I ask whether, in effect, a universe in which the Law of Excluded Middle does not hold can be conceived.
[10] Oceania is one of the three fictional superstates in George Orwell’s Nineteen Eighty-Four. The three parties are perpetually at war with each other in order to expend the oversupply continuously being generated by runaway capitalism.
[11] Richard Thaler is a behavioral economist at the University of Chicago Booth School of Business. He was awarded the 2017 Nobel Prize in Economic Sciences for his work incorporating psychology into the economic analysis of individual decision making. The reference here is to his book, Nudge: Improving Decisions About Health, Wealth, and Happiness, which argues that organizations can improve the decisions of their members through policies that make good decisions easier and bad ones harder. The classic example is an “opt-out” policy for organ donation.
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