Here's Looking at Euclid: Chapters Nine–Eleven

While we take this idea for granted in our daily lives—it is implicit, for example, whenever we read the weather forecast—the realization that math can tell us about the future was a very profound, and comparatively recent, idea in the history of human thought.

“So surely as I was inordinately addicted to the chess board and the dicing table, I know that I must rather be considered deserving of the severest censure,” he wrote.

His habit yielded a short treatise called The Book of Games of Chance, the first scientific analysis of probability.

Nevertheless, his amateur ruminations had made him one of the most respected mathematicians of the first half of the seventeenth century.

Perhaps the cause was the near-death accident in which his coach hung perilously over a bridge after the horses plunged over the parapet, or perhaps it was a moral reaction to the decadence of the dicing tables of prerevolutionary France—in any case, it revitalized his commitment to Jansenism, a strict Catholic cult, and he abandoned math for theology and philosophy.

Game A is what is called a low volatility game, while Game B is high volatility—you hit a winning combination less often, but the chances of a big win are greater. The higher the volatility, the more short-term risk there is for the slot operator.

Initially, casinos didn’t want the liability of having to pay out so much, so IGT underwrote the entire network by taking a percentage from all of its machines, and paid the jackpot itself.

The gambler’s fallacy is an incredibly strong human urge. Slot machines tap into it with particular virulence, which makes them, perhaps, the most addictive of all casino games.

Persi Diaconis and Frederick Mosteller call this the law of very large numbers: “With a large enough sample, any outrageous thing is apt to happen.”

In Luke Rhinehart’s best-selling novel The Dice Man, the eponymous hero makes life decisions based on the throwing of dice.

Still, baby-predicting companies are alive and well, as a cursory Internet search will demonstrate.

Since casinos allow players to place a bet after the ball has been dropped, all Thorp and Shannon needed to do was to figure out how to measure the speeds and process them in the few seconds before the croupier closed the betting.

The man was Manny Kimmel, a mathematically astute New York gangster and inveterate high-stakes gambler.

The man was Manny Kimmel, a mathematically astute New York gangster and inveterate high-stakes gambler.

With small edges and judicious money management, huge returns could be achieved.

Two of them, Fischer Black and Myron Scholes, created the Black-Scholes formula, indicating how to price financial derivatives—Wall Street’s most famous (and infamous) equation.

Let’s suppose that your whole net worth is your house. To insure your house is a bad bet in the expected value sense but it is probably prudent in the long-term survival sense.

The frisson of expectation was just like the feeling when you check soccer scores or financial markets.

It was, I reasoned, the same self-delusion displayed by sports and financial pundits who are equally unable to predict random events, and yet build careers out of it.

And he conjectured that the graph of bread weights would resemble a normal distribution—since the errors in making the bread, such as how much flour is used and how long the loaf is baked, are random.

Quételet’s ideas transformed the use of the word statistics, whose original meaning had little to do with numbers. The word was used to describe general facts about the state, as in the type of information required by statesmen. Quételet turned statistics into a much wider discipline, one that was less about statecraft and more about the mathematics of collective behavior.

Sets of data in those days were harder to come by than they are now, so Quételet scoured the world for them with the doggedness of a professional collector.

“The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of unreason.”

If the characteristics of a population are normally distributed, in other words are clustered around an average in the shape of a bell curve, and if the bell curve is produced through random error, then Quételet argued, the variation in human characteristics can be seen as errors from a paradigm. He called this paradigm I’homme moyen, “the average man.” Populations, he said, were made up of deviations from this paradigm.

He had thought to call his new science viticulture, from the Latin vita, life, but eventually settled on eugenics, from the Greek eu, good, and genos, birth. Even though many liberal intellectuals of the late nineteenth century and early twentieth century supported eugenics as a way to improve society, the desire to “breed” smarter humans was an idea that was soon distorted and discredited.

In retrospect, it is easy to see how ranking traits—such as intelligence or racial purity—can lead to discrimination and bigotry.

Galton wrote that the bell curve reigned with “serenity and in complete self-effacement.” Its legacy, though, has been anything but serene and self-effacing.

If I had been as indefatigable as Henri Poincaré, I would have continued the experiment for a year and had 365 (give or take days of bakery closure) weights to compare.

Poincaré once received a letter from the French physicist Gabriel Lippmann, who brilliantly summed up why the normal distribution was so widely exalted: “Everybody believes in the [bell curve]: the experimenters, because they think it can be proved by mathematics; and the mathematicians, because they believe it has been established by observation.”

“Have you ever seen a mathematician do crochet?” came the dismissive response.
The rebuff, however, made Daina even more determined to use female handicraft in the course of scientific advancement.

Despite their love of Euclid’s deductive method, mathematicians loathed his fifth postulate; not only did it go against their sense of aesthetics, but they sensed that it assumed too much to be an axiom. In fact, for 2,000 years many great minds attempted to change the status of the fifth postulate by trying to deduce it from the other postulates, so it could be reclassified as a theorem instead of remaining as a postulate, or axiom.

His mathematician father, Farkas, knew the scale of the challenge from his own failed attempts and implored his son to stop: “For God's sake, I beseech you, give it up. Fear it no less than sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life.”

While there was debate about whether the fifth postulate was an axiom or a theorem, no one had the temerity to suggest that it might actually not be true.

And when, years later, János learned that Lobachevsky had actually published a proof before him, he became haunted by the ludicrous notion that Lobachevsky was a fictional character invented by Gauss as a cunning ruse in order to deprive him of the credit for his work.

The challenge of visualizing the hyperbolic plane galvanized many mathematicians in the final decades of the nineteenth century.

They give a visceral experience of the hyperbolic plane, allowing students to touch and feel a surface that was previously understood only in an abstract way.

Henri Poincaré, for example, described Cantor’s work as “a malady, a perverse illness from which some day mathematics would be cured,” while Leopold Kronecker, Cantor’s former teacher and professor of math at Berlin University, dismissed him as a “charlatan” and a “corruptor of youth.”

In his book on Cantor, Everything and More, David Foster Wallace writes: “The Mentally Ill Mathematician seems now in some ways to be what the Knight Errant, Mortified Saint, Tortured Artist, and Mad Scientist have been for other eras: sort of our Prometheus, the one who goes to forbidden places and returns with gifts we can all use but he alone pays for.”

But this time the travelers have not come in buses; they are in fact a rabble with each wearing a T-shirt displaying a decimal expansion of a number between 0 and 1.

While these concepts were at first counterintuitive, they were assimilated quickly and we now use them on a daily basis.