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July 29 - August 13, 2018
The revolutionary idea that defines the boundary between modern times and the past is the mastery of risk: the notion that the future is more than a whim of the gods and that men and women are not passive before nature.
Without a command of probability theory and other instruments of risk management, engineers could never have designed the great bridges that span our widest rivers, homes would still be heated by fireplaces or parlor stoves, electric power utilities would not exist, polio would still be maiming children, no airplanes would fly, and space travel would be just a dream.
The modern conception of risk is rooted in the Hindu-Arabic numbering system that reached the West seven to eight hundred years ago.
Pascal turned for help to Pierre de Fermat, a lawyer who was also a brilliant mathematician. The outcome of their collaboration was intellectual dynamite. What might appear to have been a seventeenth-century version of the game of Trivial Pursuit led to the discovery of the theory of probability, the mathematical heart of the concept of risk.
Some years earlier, in the 1630s, the famed Dutch tulip bubble had burst as a result of the issuing of options whose essential features were identical to the sophisticated financial instruments in use today.
By this time Martin Luther had had his say and halos had disappeared from most paintings of the Holy Trinity and the saints.
In 1703, Gottfried von Leibniz commented to the Swiss scientist and mathematician Jacob Bernoulli that “[N]ature has established patterns originating in the return of events, but only for the most part,”1 thereby prompting Bernoulli to invent the Law of Large Numbers and methods of statistical sampling
In 1730, Abraham de Moivre suggested the structure of the normal distribution—also known as the bell curve—and discovered the concept of standard deviation.
Even more important, he propounded the idea that the satisfaction resulting from any small increase in wealth “will be inversely proportionate to the quantity of goods previously possessed.”
Dan Bernoulli on utility functions
Bernoulli’s statement stood as the dominant paradigm of rational behavior for the next 250 years and laid the groundwork for modern principles of investment management.
Whenever we make any decision based on the expectation that matters will return to “normal,” we are employing the notion of regression to the mean.
The story that I have to tell is marked all the way through by a persistent tension between those who assert that the best decisions are based on quantification and numbers, determined by the patterns of the past, and those who base their decisions on more subjective degrees of belief about the uncertain future. This is a controversy that has never been resolved.
The late Fischer Black, a pioneering theoretician of modern finance who moved from M.I.T. to Wall Street, said, “Markets look a lot less efficient from the banks of the Hudson than from the banks of the Charles.”
Check out this quote.
Our lives teem with numbers, but we sometimes forget that numbers are only tools. They have no soul; they may indeed become fetishes.
The dice and the roulette wheel, along with the stock market and the bond market, are natural laboratories for the study of risk because they lend themselves so readily to quantification; their language is the language of numbers.
hundred and sixty years later, another great English economist, John Maynard Keynes, agreed: “When the capital development of a country becomes the by-product of the activities of a casino, the job is likely to be ill-done.”2
The Earl of Sandwich invented the snack that bears his name so that he could avoid leaving the gaming table in order to eat.
Although Egypt punished compulsive gamblers by forcing them to hone stones for the pyramids, excavations show that the pharaohs were not above using loaded dice in their own games.
Those games were generally referred to as “hazard,” from al zahr, the Arabic word for dice.6
There are cardplayers and racetrack bettors who are genuine professionals, but no one makes a successful profession out of shooting craps.
Time is the dominant factor in gambling. Risk and time are opposite sides of the same coin, for if there were no tomorrow there would be no risk. Time transforms risk, and the nature of risk is shaped by the time horizon: the future is the playing field.
If we buy a stock today, we can always sell it tomorrow. But what do we do after the croupier at the roulette table cries, “No more bets!” or after a poker bet is doubled?
FInite And infinite games
To explain the beginning of everything, Greek mythology drew on a giant game of craps to explain what modern scientists call the Big Bang. Three brothers rolled dice for the universe, with Zeus winning the heavens, Poseidon the seas, and Hades, the loser, going to hell as master of the underworld.
Their failure to do so is astonishing because the Greeks had the only recorded civilization up to that point untrammeled by a dominating priesthood that claimed a monopoly on the lines of communication with the powers of mystery.
Elsewhere, Socrates anticipates Aristotle when he declares that a “mathematician who argues from probabilities in geometry is not worth an ace.”11 For another thousand years, thinking about games and playing them remained separate activities.
Up to the time of the Renaissance, people perceived the future as little more than a matter of luck or the result of random variations, and most of their decisions were driven by instinct. When the conditions of life are so closely linked to nature, not much is left to human control.
As Christianity spread across the western world, the will of a single God emerged as the orienting guide to the future, replacing the miscellany of deities people had worshiped since the beginning of time. This brought a major shift in perception: the future of life on earth remained a mystery, but it was now prescribed by a power whose intentions and standards were clear to all who took the time to learn them.
The Arabs, through their invasion of India, had become familiar with the Hindu numbering system, which enabled them to incorporate eastern intellectual advances into their own scholarship, scientific research, and experimentation. The results were momentous, first for the Arabs and then for the West.
In the hands of the Arabs, the Hindu numbers would transform mathematics and measurement in astronomy, navigation, and commerce. New methods of calculation gradually replaced the abacus, which for centuries had been the only tool for doing arithmetic everywhere from the Mayans in the western hemisphere, across Europe, to India and the Orient.
The Reformation meant more than just a change in humanity’s relationship with God. By eliminating the confessional, it warned people that henceforth they would have to walk on their own two feet and would have to take responsibility for the consequences of their decisions.
The prospect of getting rich is highly motivating, and few people get rich without taking a gamble.
The successful business executive is a forecaster first; purchasing, producing, marketing, pricing, and organizing all follow.
Without numbers, there are no odds and no probabilities; without odds and probabilities, the only way to deal with risk is to appeal to the gods and the fates. Without numbers, risk is wholly a matter of gut.
Leonardo Pisano was known for most of his life as Fibonacci, the name by which he is known today. His father’s first name was Bonacio, and Fibonacci is a contraction of son-of-Bonacio. Bonacio means “simpleton” and Fibonacci means “
His intransigence brought him not just one excommunication, but two. On the second occasion, Pope Gregory IX called for Frederick to be deposed, characterizing him as a heretic, rake, and anti-Christ. Frederick responded with a savage attack on papal territory;
Fibonacci is best known for a short passage in Liber Abaci that led to something of a mathematical miracle. The passage concerns the problem of how many rabbits will be born in the course of a year from an original pair of rabbits, assuming that every month each pair produces another pair and that rabbits begin to breed when they are two months old.
Divide any of the Fibonacci numbers by the next higher number. After 3, the answer is always 0.625. After 89, the answer is always 0.618; after higher numbers, more decimal places can be filled in.a Divide any number by its preceding number. After 2, the answer is always 1.6. After 144, the answer is always 1.618.
Fibonacci’s Liber Abaci was a spectacular first step in making measurement the key factor in the taming of risk. But society was not yet prepared to attach numbers to risk. In Fibonacci’s day, most people still thought that risk stemmed from the capriciousness of nature.
Only the Bible has appeared in more editions and printings than Euclid’s most famous book, Elements.
Actually, one westerner had suggested a numbering system at least two centuries earlier than the Hindus. About 250 AD, an Alexandrian mathematician named Diophantus wrote a treatise setting forth the advantages of a system of true numbers to replace letters substituting for numbers.
According to Herbert Warren Turnbull, a historian of mathematics, a Greek epigram about Diophantus states that “his boyhood lasted l/6th of his life; his beard grew after l/12th more; he married after l/7th more, and his son was born five years later; the son lived to half his father’s age, and the father died four years after his son.” How old was Diophantus when he died?7
x to be a negative number: −4. Without the concept of zero, which Diophantus lacked, a negative number is a logical impossibility.
As the twentieth-century English philosopher Alfred North Whitehead put it, The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought.10
It was al-Khowârizmî who was the first mathematician to establish rules for adding, subtracting, multiplying, and dividing with the new Hindu numerals. In another treatise, Hisâb al-jabr w’ almuqâbalah, or “Science of transposition and cancellation,” he specifies the process for manipulating algebraic equations. The word al-jabr thus gives us our word algebra, the science of equations.13
Omar Khayyam used the new numbering system to develop a language of calculation that went beyond the efforts of al-Khowârizmî and served as a basis for the more complicated language of algebra. In addition, Omar Khayyam used technical mathematical observations to reform the calendar and to devise a triangular rearrangement of numbers that facilitated the figuring of squares, cubes, and higher powers of mathematics;
By the year 1000, the new numbering system was being popularized by Moorish universities in Spain and elsewhere and by the Saracens in Sicily. A Sicilian coin, issued by the Normans and dated “1134 Annoy Domini,” is the first known example of the system in actual use. Still, the new numbers were not widely used until the thirteenth century.
Florence issued an edict in 1229 that forbade bankers from using the “infidel” symbols. As a result, many people who wanted to learn the new system had to disguise themselves as Moslems in order to do so.15
These people respect representations of divinity but are by no means subservient to it—a message that appears over and over again in the art of the Renaissance.
Brunelleschi’s great dome in Florence and the cathedral, with its clearly defined mass and unadorned interior, proclaims that religion has literally been brought down to earth.
The notion of double-entry bookkeeping was apparent in Fibonacci’s Liber Abaci and had shown up in a book published about 1305 by the London branch of an Italian firm.
