A Beautiful Math John Nash, Game Theory, and the Modern Quest for a Code of Nature (2006) / Chapter Skim
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7 Quetelet’s Statistics and Maxwell’s Molecules--Statistics and society, statistics and physics
7 Quetelet’s Statistics and Maxwell’s Molecules--Statistics and society, statistics and physics
Pages 126-143
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From page 126... ...
-- 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.
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From page 127... ...
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.
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From page 128... ...
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.
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From page 129... ...
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.
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From page 130... ...
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.
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From page 131... ...
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)
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From page 132... ...
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)
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From page 133... ...
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.
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From page 134... ...
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.
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From page 135... ...
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.
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From page 136... ...
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)
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From page 137... ...
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.)
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From page 138... ...
"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.
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From page 139... ...
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.
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From page 140... ...
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.
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From page 141... ...
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.
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From page 142... ...
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.
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From page 143... ...
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.
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