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 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 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.
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 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 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 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 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 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, b
ecoming 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 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
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