7
Quetelet’s Statistics and Maxwell’s Molecules
Statistics and society, statistics and physics

The mob has many heads but no brains.

—English proverb

The actions of men … are in reality never inconsistent, but however capricious they may appear only form part of one vast system of universal order.

—Henry Thomas Buckle

When creating the fictional science of psychohistory, more than half a century ago, Isaac Asimov didn’t bother to give the details of how the math worked. He simply said you could describe masses of people in the same way you describe masses of molecules. Trained as a chemist, Asimov knew well that the behavior of gases under different conditions could be calculated with precision, even though nobody could possibly know what any one of that gas’s atoms or molecules was doing. And so he reasoned that a sufficiently advanced science could do the same thing with people.

“Psychohistory dealt not with man, but man-masses,” Asimov wrote.1 “It was the science of mobs; mobs in their billions…. The reaction of one man could be forecast by no known mathematics; the reaction of a billion is something else again.” So while any one person could do his or her own thing, society might collectively



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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 7 Quetelet’s Statistics and Maxwell’s Molecules Statistics and society, statistics and physics The mob has many heads but no brains. —English proverb The actions of men … are in reality never inconsistent, but however capricious they may appear only form part of one vast system of universal order. —Henry Thomas Buckle When creating the fictional science of psychohistory, more than half a century ago, Isaac Asimov didn’t bother to give the details of how the math worked. He simply said you could describe masses of people in the same way you describe masses of molecules. Trained as a chemist, Asimov knew well that the behavior of gases under different conditions could be calculated with precision, even though nobody could possibly know what any one of that gas’s atoms or molecules was doing. And so he reasoned that a sufficiently advanced science could do the same thing with people. “Psychohistory dealt not with man, but man-masses,” Asimov wrote.1 “It was the science of mobs; mobs in their billions…. The reaction of one man could be forecast by no known mathematics; the reaction of a billion is something else again.” So while any one person could do his or her own thing, society might collectively

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature exhibit patterns of behavior that equations could capture. Psychohistory might not be quite as accurate as the laws governing gases, but that’s only because there are many more molecules than people. As one of Asimov’s characters explained, “The laws of history are as absolute as the laws of physics, and if the probabilities of error are greater, it is only because history does not deal with as many humans as physics does atoms, so that individual variations count for more.”2 Still, psychohistory was fiction, and using math to describe something as complex as society still strikes many people as an overly ambitious goal for real life. On the other hand, in the mid-19th century math seemed similarly useless for physicists pondering the complexities of molecular motion in gases. Gross properties of gases could be observed but not understood without a way to quantify the apparent anarchy of molecular interactions. How could anyone grasp the inner workings of a mass of molecules too numerous to count and too small to be seen? Yet the Scottish physicist James Clerk Maxwell found a way, by using statistics— mathematical descriptions of the average behavior of large groups of molecules. Calculating such averages provided amazing predictive power. Although you couldn’t say exactly what any one molecule was up to, you could predict precisely what a sufficiently large group of molecules would do in certain circumstances. Measuring the temperature of a gas, for instance, tells you something about the average speed of its molecules, and you can calculate the effect of altering the temperature on the gas’s pressure. Similar methods were developed to deal with matter in all sorts of situations. Knowing the average amount of energy possessed by molecules of various substances, for instance, allows you to predict whether a chemical reaction will proceed or not—and if so, how far. You can use the statistical approach to describe a substance’s magnetic or electric properties, or whether it will snap or stretch when under tension. In Asimov’s psychohistory, features of society corresponded to variables like the temperature and pressure of a gas or the ebb and flow of chemical reactions or the fracture of a beam in a building.

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature While Asimov’s vision remains a science fiction dream, it is now closer to reality than probably even he would have thought possible. The statistical approach inaugurated by Maxwell has today become physicists’ favorite weapon for invading the social sciences and describing human actions with math. Physicists have applied the statistical approach to analyzing the economy, voting behavior, traffic flow, the spread of disease, the transmission of opinions, and the paths people take when fleeing in panic after somebody shouts Fire! in a crowded theater. But here’s the thing. This isn’t a new idea, and physicists didn’t have it first. In fact, Maxwell, who was the first to devise the statistical description of molecules, got the idea to use statistics in physics from social scientists applying math to society! So before statistical physicists congratulate themselves for showing the way to explaining the social sciences, they should pause to reflect on the history of their field. As the science journalist Philip Ball has observed, “by seeking to uncover the rules of collective human activities, statistical physicists are aiming to return to their roots.”3 In fact, efforts to apply science and math to society have a rich history, extending back several centuries. And that history contains hints of ideas that can, in retrospect, be seen as similar to key aspects of game theory—foreshadowing an eventual convergence of all these fields in the quest for a Code of Nature. STATISTICS AND SOCIETY The idea of finding a science of society long predates Asimov. In a sense it goes back to ancient times, of course, resembling at least partially the old notion of a “natural law” of human behavior or a Code of Nature. In early modern times, the idea received renewed impetus from the success of Newtonian physics, stimulating the efforts of Adam Smith and others as described in Chapter 1. Even before Newton, though, the rise of mechanistic physical science inspired several philosophers to consider a similarly rigorous approach to society. In medieval times, the importance of the mechanical clock to

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature society conditioned scientists to think of the universe in mechanical terms. Descartes, Galileo, and other pioneers of modern science advocated a mechanical, cause-and-effect view of the cosmos that ultimately led to Newton’s definitive system of physics, published in his Principia in 1687. It was only natural that the implications of mechanism for life and society attracted the attention of other 17th-century thinkers. One was Thomas Hobbes, whose famous work Leviathan described the state of society that (Hobbes believed) maximized the well-being of all its members. Conveniently for Hobbes, a supporter of the British monarchy, his conclusion was that the people should turn over control of society to an absolute monarch. Otherwise, he argued, a dog-eat-dog mentality of unrestrained human nature would guarantee life to be “nasty, brutish, and short.” In an intriguing paper, though (published in Physica A), Philip Ball points out that Hobbes’s questionable conclusion was not as important as the methods he used to reach it. The Hobbes approach was to assess the interacting preferences of various individuals and figure out how best to achieve the best deal for everybody. The resulting theoretical framework, Ball says, “could be recast without too much effort” as a Nash equilibrium maximizing the power of each individual. As such, Hobbes’s Leviathan could be seen as an early effort to understand society mathematically, with the prescient indication that something like game theory would be a good mathematical instrument for the task. Real math entered the story a little later, as the science of statistics was invented—for the very purpose of quantifying various aspects of society. The scientist and politician Sir William Petty, a student of Hobbes, advocated the scientific study of society in a quantitative way. His friend John Graunt began compiling tables of social data, such as mortality figures, in the 1660s. Graunt and others began to keep track of births and deaths and analyzed the data much like the way that baseball fans pore over batting averages today. By a century later, in the times leading up to the French Revolution, gathering social statistics had become a widespread practice, usually undertaken in the belief that studying such social

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature numbers might reveal laws of social nature the same way astronomers had revealed the regularities of the heavens. “The idea that there were laws that stood in relation to society as Newton’s mechanics stood in relation to the motion of the planets was shared by many,” writes Ball.4 Gathering numbers was not enough, of course, to make the study of society a science in the Newtonian mold. Physics, as Newton had sculpted it, was the science of certainty, his dictatorial laws of motion determining how things happened. Statistics dealt not with such certainty, but rather displayed considerable variability. Much about human behavior seemed to depend on chance— the luck of the draw (as in games!). Dealing with people called for quantifying luck—leading to the mathematical analysis of probability. Early studies of probability theory predated Newton, starting with the mid-17th-century work of Blaise Pascal and Pierre Fermat—their idea being to figure out how to win at dice or card games. An economic use of probability theory soon arose from insurance companies, which used statistical tables to gauge the risk of people dying at certain ages or the likelihood of fires or shipwrecks destroying insured property. Probability became more useful to physics (and the rest of science) with the development of the theory of measurement errors during the 18th century, particularly in astronomy. Ironically, one of the key investigators in that statistical field was Pierre Simon, Marquis de Laplace, the French mathematician famous for his articulation of Newtonian determinism. For a being with intelligence capable of analyzing the circumstances of all the bodies in the universe, and the forces operating on them, all movements great and small could be foreseen by applying Newton’s laws, Laplace declared. “For it, nothing would be uncertain and the future, as the past, would be present to its eyes.”5 Laplace recognized full well, though, that no human intelligence possessed such grand ability. So statistical methods were needed to deal with the unavoidable uncertainties afflicting human knowledge. Laplace wrote extensively on the issue of probability

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature and uncertainties, focusing especially on the inevitable errors that occurred whenever measurements were made. Suppose, for instance, that you’re trying to measure the positions of the planets visible in the night sky. No matter how good your instruments, uncontrollable factors will prevent you from measuring positions with arbitrary precision. Each time you measure a planet’s position, the answer will be at least a little bit off from whatever the true position might be. But such random errors do not render your measurements hopelessly inaccurate. While individual errors might be random, the sum of all errors could be subjected to mathematical analysis in a way that revealed something about the planetary position’s true coordinates. If the measurements are careful enough, for example, small errors will be more common than somewhat larger errors, and huge errors would be even rarer. Laplace was one among several mathematicians who developed the math for calculating the range of such errors. Another was Carl Friedrich Gauss, the German mathematician whose name was given to the now familiar bell-shaped curve that depicts how random measurement errors are distributed around the average value (the “Gaussian distribution”).6 For repeated measurements, the most likely true value would simply be the value at the peak of the curve—the average (or mean) of all the measurements (assuming the “errors” are all due to random, uncontrollable factors, rather than some problem with the instrument itself). The math describing the curve tells you how to estimate the likelihood that the true value differs from the mean by any given amount. While Gauss got his name on the curve, Laplace’s work in this arena turned out to be more important for the human side of statistics. Like others of his era, Laplace recognized the relevance of statistics to human affairs, and applied the error curve to such issues as the ratio of male to female births. Laplace’s interest in the social side of his math led to a much broader appreciation of its potential uses—thanks to the Belgian mathematician and astronomer Adolphe Quetelet.

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature SOCIAL PHYSICS Quetelet, who was born in Ghent in 1796, made a major mathematical contribution to society that most Americans today are uncomfortably familiar with, although few people know to blame Quetelet. He invented the Quetelet index for assessing obesity, a measure better known now as the Body Mass Index, or BMI. But he had much greater vision for applying science to society than merely telling people how to know when they were overweight. As a youth, Quetelet dabbled in painting, poetry, and opera, but his special talent was math, and he earned a math doctorate in 1819 at the University of Ghent. He got a job teaching math in Brussels, where he was soon elected to the Belgian academy of sciences. During the 1820s, Quetelet expanded his interest from math into physics, and in 1823 he traveled to Paris to study astronomy, part of a plan to establish an observatory in Brussels. Quetelet later wrote some widely read popularizations of astronomy and physics for the general reader. And he often delivered public lectures on science attended by all segments of the public. Quetelet was highly regarded as a teacher and as a person by those who knew him—he was described as amiable and considerate, tactful and modest, but still a rigorous thinker who expressed his views strongly.7 During his stay in Paris, Quetelet took in more than just astronomy. He also learned probability theory from Laplace and met his colleagues Poisson and Fourier, who also had an interest in the statistics of society. Quetelet was himself strongly attracted to the social sciences, and he soon realized that Laplace’s uses of the bell curve to describe social numbers could be dramatically expanded. Quetelet began to publish papers on the statistical description of society, and in 1835 authored a detailed treatise on what he called social physics8 (or social mechanics), introducing the idea of an “average man” for analyzing social issues. He knew that there is no one average man, but by averaging various aspects of a great many men, much could be learned about society. “In giving to my work the title of Social Physics, I have had no other aim than to

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature collect, in a uniform order, the phenomena affecting man, nearly as physical science brings together the phenomena appertaining to the material world,” Quetelet commented.9 Quetelet’s key point was that the diversity of behaviors among men, seemingly too complex to comprehend, would coalesce into regular patterns when assessed for vast numbers. “In a given state of society, resting under the influence of certain causes, regular effects are produced, which oscillate, as it were, around a fixed mean point, without undergoing any sensible alterations,” he wrote.10 The same statistical laws governing measurement errors could be enlisted to disclose predictable patterns underlying the apparent chaos of historical trends and events, he believed. Understanding the “average man,” Quetelet contended, was essential for sound government based on an intelligent understanding of human nature. No single set of attributes regarded as the defining features of human nature would apply in all respects to any given individual, of course. Yet certain tendencies would show up in society more often than others, so statistical methods could be used to construct an abstract “average” representation of the typical mix of human characteristics. Quetelet illustrated his point with the analogy of an archery target. After many shots by many archers, the arrows lodged in the target form a distinct pattern, some close to the bull’s-eye, some farther away. But suppose for some reason that the outline defining the bull’s-eye was obscured. Even if no arrow had actually hit it, you could infer the bull’s-eye’s location from the pattern made by the arrows. “If they be sufficiently numerous, one may learn from them the real position of the point they surround,” Quetelet pointed out.11 Quetelet collected all the data he could find on such social variables as birth and death rates, analyzing how such rates differed by location, season, and even time of day. He cataloged and assessed evidence of the influences of moral, political, and religious factors on crime rates. He was struck by the constancy of crime reports in various sorts of categories from year to year. In any given locale, for instance, the number of murders remained

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature pretty much the same from one year to the next, and even the murderer’s methods showed a similar distribution. “The actions which society stamps as crimes,” Quetelet wrote, “are reproduced every year, in almost exactly the same numbers; examined more closely, they are found to divide themselves into almost exactly the same categories; and if their numbers were sufficiently large, we might carry farther our distinctions and subdivisions, and should always find there the same regularity.”12 Similarly, the rate of crimes for different ages displayed a constant distribution, with the 21–25 age group always topping the list. “Crime pursues its path with even more constancy than death,” Quetelet observed.13 He warned, though, of the dangers posed by interpreting such statistics without sufficiently careful thought. Another researcher, for instance, had shown that property crime in France was higher in provinces where more children were sent to schools, and concluded that education caused crime. It’s the sort of reasoning you hear today on talk radio. Quetelet correctly chastised such stupidity. Quetelet also repeatedly emphasized that the statistical approach could not be used to draw conclusions about any given individual (another obvious principle that is often forgotten by today’s media philosophers). The insurance company’s mortality tables cannot forecast the time of any one person’s death, for instance. Nor can any single case, however odd, invalidate the general conclusions drawn from a statistical regularity. Quetelet’s exposition of social statistics attracted a great deal of attention among scientists and philosophers. Many of them were aghast that he seemed to have little regard for the supposed free will that humans exercised as they pleased. Quetelet responded not by denying free will, but by observing that it had its limits, and that human choice was always influenced by conditions and circumstances, including laws and moral strictures. In making the simplest of choices, Quetelet noted, our habits, needs, relationships, and a hundred other factors buffet us from all sides. This “empire of causes” typically overwhelms free will, which is why, with

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature knowledge of all the factors affecting someone’s choice, it is usually possible to predict what it will be. In any event, the controversy over Quetelet’s views served science well, for it guaranteed that his work was to become widely known. Fortunately for physics, some of the commentaries on it reached the hands of James Clerk Maxwell. MAXWELL AND MOLECULES Maxwell was one of those once-in-a-century geniuses who perceived the physical world with sharper senses than those around him. He saw deeply into almost every corner of physics, forever alert to the hidden principles governing the complexities of physical phenomena. He mastered electricity and magnetism, light and heat, pretty much mopping up all the major areas of physics beyond those that Newton had already taken care of—gravitation and the laws of motion. And in fact, Maxwell detected an essential shortcoming in Newton’s laws of motion, too. They worked fine for macroscopic objects, like cannonballs and rocks. But what about the submicroscopic molecules from which such objects were made? Presumably, Newton’s laws would still apply. But they did you no good, because you could not possibly trace the motion of an individual molecule anyway. And if you couldn’t describe the motion of an object’s parts, how could you expect to predict the behavior of the object? For a cannonball dropped from the leaning tower of Pisa, the internal motion of the metal’s atoms made no difference to the rate at which it fell. But other forms of matter did not cooperate so willingly. Suppose you wanted to know how changes in pressure affected the temperature of the steam in a steam engine, for instance. You could not even begin to calculate the motions of the individual H2O molecules. Physicists were not helpless in the face of this question—they had devised some pretty good formulas for describing how gases behaved. Maxwell, though, wanted to know how those rules worked—why gases behaved the way they did. If he could show

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature how molecular motions produced the observed gross behavior, he would have achieved both a deeper understanding of the phenomena and have provided strong new evidence for the very existence of atoms and molecules, which was at the time—in the mid-19th century—a contentious conviction in some circles. The idea that a gas’s properties depended on the motion of its constituent molecules was not new, though. It was known as the kinetic theory of gases, originally articulated in 1738 by our old friend Daniel Bernoulli, who explained the gas laws with a crude picture of molecules modeled as billiard balls. But as the science historian Stephen Brush has noted, Bernoulli’s theory “was a century ahead of its time.”14 Bernoulli’s idea was based on the (correct) notion that heat is merely the motion of molecules, but in his day most physicists believed heat to be some sort of fluid substance (called caloric). By the 1850s, though, the kinetic theory was a ripe topic for physicists to study, as the laws of thermodynamics—constituting the correct theory of heat—were arriving on the scene. One of the major pioneers of thermodynamics was the German physicist Rudolf Clausius. In an 1857 paper, Clausius presented a comprehensive view on the nature of heat as molecular motion. He described how the pressure of a gas was related to the motion of molecules as they impinged on the walls of their container. And any given molecule was constantly battered by collisions with other molecules, so its behavior reflected the influences of such impacts (just as a person’s choices reflect the influences of the countless social pressures that Quetelet had described). In his approach, Clausius emphasized the importance of the average velocity of the molecules, and in an 1858 paper introduced the important notion of the average distance that molecules traveled between collisions (a distance labeled the “mean free path”). In 1859, Maxwell entered the molecular motion arena, exploring the interplay of gas molecules and their resulting velocities a little more deeply. In his approach, Maxwell applied the sort of statistical thinking that Quetelet had promoted. Maxwell had probably first encountered Quetelet in an article

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature by the astronomer John Herschel. (Herschel, of course, was familiar with Quetelet as a fellow astronomer.) Later, in 1857, Maxwell read a newly published book by the historian Henry Thomas Buckle. Buckle, himself clearly influenced by Quetelet, believed that science could discover the “laws of the human mind” and that human actions are part of “one vast system of universal order.”15 (I encountered one Web page where Buckle is referred to as the Hari Seldon of the 19th century.) Buckle was another of the 19th century’s most curious characters. Born near London in 1821, he was a slow learner as a child. When he was 18 his father, a maritime merchant, died, leaving the son sufficient funds to tour Europe and pursue his hobbies of history and chess. (Buckle became a formidable chess player and learned several foreign languages, becoming fluent in seven and conversant in a dozen others. He also became a prolific bibliophile, amassing a library exceeding 20,000 books.) From 1842 on, Buckle began compiling the data and evidence for a comprehensive treatise on history. Originally planned to focus on the Middle Ages, the work eventually took on broader aims and became the History of Civilization in England (by which Buckle actually meant the history of civilization, period). While presumably a work of history, Buckle’s book was really more a sociological attempt to subject the nature of human behavior to the methods of science. He criticized the “metaphysical” (or philosopher’s) approach to the issue, advocating instead the “historical” method (by which he basically meant the scientific method). “The metaphysical method … is, in its origin, always the same, and consists in each observer studying the operations of his own mind,” Buckle wrote. “This is the direct opposite of the historical method; the metaphysician studying one mind, the historian studying many minds.”16 Buckle could not resist remarking that the metaphysical method “is one by which no discovery has ever been made in any branch of knowledge.” He then emphasized the need for observing great numbers of cases so as to escape the effects of “disturbances” obscuring the underlying law. “Every thing we at present know,” Buckle asserted, “has been ascertained by studying

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature phenomena, from which all causal disturbances having been removed, the law remains as a conspicuous residue. And this can only be done by observations so numerous as to eliminate the disturbances.”17 Much of Buckle’s philosophy echoes Quetelet, including similar slams against the idea of unfettered free will. Occasionally someone makes what appears to be a free and even surprising choice, but only because you don’t know enough about the person’s circumstances, Buckle observed. “If, however, I were capable of correct reasoning, and if, at the same time, I had a complete knowledge both of his disposition and of all the events by which he was surrounded, I should be able to foresee the line of conduct which, in consequence of those events, he would adopt,” Buckle pointed out.18 Read retrospectively, Buckle’s comment sounds very much like what a game theorist would say today. Game theory is, in fact, all about understanding what choice would (or should) be made if all the relevant information influencing the outcome of the decision is known. Buckle realized that choices emerge not merely from external factors, though, but from the inner workings of the mind as well. Since sorting out the nuances of all the influences exceeds science’s powers, the nature of human behavior must be described instead by the mathematics of statistics. “All the changes of which history is full, all the vicissitudes of the human race, their progress or their decay, their happiness or their misery, must be the fruit of a double action; an action of external phenomena upon the mind, and another action of the mind upon the phenomena,” wrote Buckle. “The most comprehensive inferences respecting the actions of men are derived from this or from analogous sources: they rest on statistical evidence, and are expressed in mathematical language.”19 It’s not hard to imagine Maxwell reading these words and seeing in them a solution to the complexities confounding the description of gases. Though Maxwell found Buckle’s book “bumptious,” he recognized it as a source of original ideas, and the statistical reasoning that Buckle applied to society seemed just the thing that Maxwell needed to deal with molecular motion. “The

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature smallest portion of matter which we can subject to experiment consists of millions of molecules,” Maxwell later noted. “We cannot, therefore, ascertain the actual motion of any one of these molecules; so that we are obliged to … adopt the statistical method of dealing with large groups of molecules.”20 That statistical method, he showed, could indeed reveal “uniformities” in molecular behavior. “Those uniformities which we observe in our experiments with quantities of matter containing millions of millions of molecules are uniformities of the same kind as those explained by Laplace and wondered at by Buckle,” Maxwell declared.21 The essential feature of Maxwell’s work was showing that the properties of gases made sense not if gas molecules all flew around at a similar “average” velocity, as Clausius had surmised, but only if they moved at all sorts of speeds, most near the average, but some substantially faster or slower, and a few very fast or slow. As the molecules bounced off one another, some gained velocity; others slowed down. In subsequent collisions, a fast molecule might be either slowed down or speeded up. A few would enjoy consecutive runs of very good (or very bad) luck and end up moving extremely rapidly (or slowly), while most would get a mix of bounces and tend toward the overall average velocity of all the molecules in the box. Just as Quetelet’s average man was fictitious, and key insights into society came from analyzing the spread of features around the average, understanding gases meant figuring out the range and distribution of molecular velocities around the average. And that distribution, Maxwell calculated, matched the bell-shaped curve describing the range of measurement errors. As Maxwell refined his ideas during the 1860s, he showed that when the velocities reached the bell-shaped distribution, no further net change was likely. (The Austrian physicist Ludwig Boltzmann further elaborated on and strengthened Maxwell’s results.) Any specific molecule might speed up or slow down, but the odds were strong that other molecules would change in speed to compensate. Thus the overall range and distribution of velocities would stay the same. When a gas reached that state—in which

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature further collisions would cause no net change in its overall condition—the gas was at equilibrium. Of course, this notion of equilibrium is precisely analogous to the Nash equilibrium in game theory. And it’s an analogy that has more than merely lexical significance. In a Nash equilibrium, the sets of strategies used by the participants in a game attain a stable set of payoffs, with no incentive for any player to change strategies. And just as the Nash equilibrium is typically a mixed set of strategies, a gas seeks an equilibrium state with a mixed distribution of molecular velocities. PROBABILITY DISTRIBUTIONS Nash’s mixed strategies, and Maxwell’s mixed-up molecules, are both examples of what mathematicians call probability distributions. It’s such an important concept for game theory (and for science generally) that it’s worth a brief interlude to mercilessly pound the idea into your brain (possibly with a silver hammer). Consider Maxwell’s problem. How do the molecules in a gas share the total amount of energy that the gas possesses? One possibility is that all the molecules move at something close to the average, as Clausius suspected. Or the velocities could be distributed broadly, some molecules leisurely floating about, others zipping around at superspeed. Clearly, there are lots of possible combinations. And all of these allocations of molecular velocities are in principle possible. It’s just that some combinations of velocities are more likely than others. For a simpler example, imagine what happens when you repeatedly flip a coin 10 times and record the number of heads. It’s easy to calculate the probability distribution for pennies, because you know that the odds of heads versus tails are 50-50. (More technically, the probability of heads for any toss is 0.5, or one-half. That’s because there are two possibilities—equally likely, and the sum of all the probabilities must equal 1—1 signifying 100 percent of the cases.) In the long run, therefore, you’ll find that the average number of heads per trial is something close to 5 (if you’re

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature using a fair coin). But there are many conceivable combinations of totals that would give that average. Half the trials could turn up 10 heads, for instance, while the other half turned up zero every time. Or you could imagine getting precisely 5 heads in every 10-flip trial. What actually happens is that the number of trials with different numbers of heads is distributed all across the board, but with differing probabilities—about 25 percent of the time you’ll get 5 heads, 20 percent of the time 4 (same for 6), 12 percent of the time 3 (also for 7). You would expect to get 1 head 1 percent of the time (and no heads at all out of a 10-flip run about 0.1 percent of the time, or once in a thousand). Coin tossing, in other words, produces a probability distribution of outcomes, not merely some average outcome. Maxwell’s insight was that the same kind of probability distribution governs the possible allocations of energy among a mess of molecules. And game theory’s triumph was in showing that a probability distribution of pure strategies—a mixed strategy—is usually the way to maximize your payoff (or minimize your losses) when your opponents are playing wisely (which means they, too, are using mixed strategies). Imagine you are repeatedly playing a simple game like matching pennies, in which you guess whether your opponent’s penny shows heads or tails. Your best mixed strategy is to choose heads half the time (and tails half the time), but it’s not good enough just to average out at 50-50. Your choices need to be made randomly, so that they will reflect the proper probability distribution for equally likely alternatives. If you merely alternate the choice of heads or tails, your opponent will soon see a pattern and exploit it; your 50-50 split of the two choices does you no good. If you are choosing with true randomness, 1 percent of the time you’ll choose heads 9 times out of 10, for instance. In his book on behavioral game theory, Colin Camerer discusses studies of this principle in a real game—tennis—where a similar 50-50 choice arises: whether to serve to your opponent’s right or left side. To keep your opponent guessing, you should serve one way or the other at random.22 Amateur players tend to

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature alternate serve directions too often, and consequently do not achieve the proper probability distribution. Professionals, on the other hand, do approach the ideal distribution more nearly, suggesting that game theory does indeed capture something about optimal behavior, and that humans do have the capability of learning how to play games with game-theoretic rationality. And that, in turn, makes a point that I think is relevant to the prospects of game theory as a mathematical method of quantifying human behavior. In many situations, over time, people do learn how to play games in a way so that the results coincide with Nash equilibrium. There are lots of nuances and complications to cope with, but at least there’s hope. STATISTICS RETURNS TO SOCIETY Of course, real-life situations, the rise of civilization, and the evolution of culture and society are much more complicated than flipping coins and playing tennis. But that is also true of the inanimate world. In most realms of physics and chemistry, the phenomena in need of explanation are rarely split between two equally likely outcomes, so computing probability distributions is much more complicated than the simple 50-50 version you can use with pennies. Maxwell, and then Boltzmann, and then the American physicist J. Willard Gibbs consequently expended enormous intellectual effort in devising the more elaborate formulas that today are known as statistical mechanics, or sometimes simply statistical physics. The uses of statistical mechanics extend far beyond gases, encompassing all the various states of matter and its behavior in all possible circumstances, describing electric and magnetic interactions, chemical reactions, phase transitions (such as melting, boiling, freezing), and all other manner of exchanges of matter and energy. The success of statistical mechanics in physics has driven the belief among many physicists that it could be applied with similar success to society. Nowadays, using statistical physics to study human social interactions has become a favorite pastime of a whole

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature cadre of scientists seeking new worlds for physics to conquer. Everything from the flow of funds in the stock market to the flow of traffic on interstate highways has been the subject of statistical-physics study. So the use of statistical physics to describe society is not an entirely new endeavor. But the closing years of the 20th century saw an explosion of new research in that arena, and as the 21st century opened, that trend turned into a tidal wave. Behind it all was a surprising burst of new insight into the mathematics describing complex networks. The use of statistical physics to describe such networks has propelled an obscure branch of math called “graph theory” into the forefront of social physics research. And it has all come about because of a game, starring an actor named Kevin Bacon.