18 Dec Born in 1501, Gerolamo Cardano was not a child you’d have put your money on. (LEONARD MLODINOW) | Part A’
In the years leading up to 1576, an oddly attired old man could be found roving with a strange, irregular gait up and down the streets of Rome, shouting occasionally to no one in particular and being listened to by no one at all. He had once been celebrated throughout Europe, a famous astrologer, physician to nobles of the court, chair of medicine at the University of Pavia. He had created enduring inventions, including a forerunner of the combination lock and the universal joint, which is used in automobiles today. He had published 131 books on a wide range of topics in philosophy, medicine, mathematics, and science. In 1576, however, he was a man with a past but no future, living in obscurity and abject poverty. In the late summer of that year he sat at his desk and wrote his final words, an ode to his favorite son, his oldest, who had been executed sixteen years earlier, at age twenty-six. The old man died on September 20, a few days shy of his seventy-fifth birthday. He had outlived two of his three children; at his death his surviving son was employed by the Inquisition as a professional torturer. That plum job was a reward for having given evidence against his father.
Before his death, Gerolamo Cardano burned 170 unpublished manuscripts. Those sifting through his possessions found 111 that survived. One, written decades earlier and, from the looks of it, often revised, was a treatise of thirty-two short chapters. Titled The Book on Games of Chance, it was the first book ever written on the theory of randomness. People had been gambling and coping with other uncertainties for thousands of years. Can I make it across the desert before I die of thirst? Is it dangerous to remain under the cliff while the earth is shaking like this? Does that grin from the cave girl who likes to paint buffaloes on the sides of rocks mean she likes me? Yet until Cardano came along, no one had accomplished a reasoned analysis of the course that games or other uncertain processes take. Cardano’s insight into how chance works came embodied in a principle we shall call the law of the sample space. The law of the sample space represented a new idea and a new methodology and has formed the basis of the mathematical description of uncertainty in all the centuries that followed. It is a simple methodology, a laws-ofchance analog of the idea of balancing a checkbook. Yet with this simple method we gain the ability to approach many problems systematically that would otherwise prove almost hopelessly confusing. To illustrate both the use and the power of the law, we shall consider a problem that although easily stated and requiring no advanced mathematics to solve, has probably stumped more people than any other in the history of randomness.
Gerolamo Cardano was no rebel breaking forth from the intellectual milieu of sixteenth-century Europe. To Cardano a dog’s howl portended the death of a loved one, and a few ravens croaking on the roof meant a grave illness was on its way. He believed as much as anyone else in fate, in luck, and in seeing your future in the alignment of planets and stars. Still, had he played poker, he wouldn’t have been found drawing to an inside straight. For Cardano, gambling was second nature. His feeling for it was seated in his gut, not in his head, and so his understanding of the mathematical relationships among a game’s possible random outcomes transcended his belief that owing to fate, any such insight is futile. Cardano’s work also transcended the primitive state of mathematics in his day, for algebra and even arithmetic were yet in their stone age in the early sixteenth century, preceding even the invention of the equal sign.
History has much to say about Cardano, based on both his autobiography and the writings of some of his contemporaries. Some of the writings are contradictory, but one thing is certain: born in 1501, Gerolamo Cardano was not a child you’d have put your money on. His mother, Chiara, despised children, though—or perhaps because—she already had three boys.
Gerolamo’s father was a bit of an operator. A sometime pal of Leonardo da Vinci’s, he was by profession a geometer, never a profession that brought in much cash. Fazio often had trouble making the rent, so he started a consulting business, providing the highborn with advice on law and medicine.
In 1516, Gerolamo decided his best opportunity lay in the field of medicine and announced that he wanted to leave his family’s home in Milan and travel back to Pavia to study there. Fazio wanted him to study law, however, because then he would become eligible for an annual stipend of 100 crowns. After a huge family brawl, Fazio relented, but the question remained: without the stipend, how would Gerolamo support himself in Pavia? He began to save the money he earned reading horoscopes and tutoring pupils in geometry, alchemy, and astronomy. Somewhere along the way he noticed he had a talent for gambling, a talent that would bring him cash much faster than any of those other means.
For anyone interested in gambling in Cardano’s day, every city was Las Vegas. On cards, dice, backgammon, even chess, wagers were made everywhere. Cardano classified these games according to two types: those that involved some strategy, or skill, and those that were governed by pure chance. In games like chess, Cardano risked being outplayed by some sixteenth-century Bobby Fischer. But when he bet on the fall of a couple of small cubes, his chances were as good as anyone else’s. And yet in those games he did have an advantage, because he had developed a better understanding of the odds of winning in various situations than any of his opponents. And so for his entrée into the betting world, Cardano played the games of pure chance. Before long he had set aside over 1,000 crowns for his education—more than a decade’s worth of the stipend his father wanted for him. In 1520 he registered as a student in Pavia. Soon after, he began to write down his theory of gambling.
It gives one a feeling for some of the challenges Cardano faced to note that the equal sign did not yet exist, to be invented in 1557 by Robert Recorde of Oxford and Cambridge, who, inspired by geometry, remarked that no things could be more nearly alike than parallel lines and hence decided that such lines should denote equality. And the symbol ×, for multiplication, attributable to an Anglican minister, didn’t arrive on the scene until the seventeenth century.
Cardano’s Book on Games of Chance covers card games, dice, backgammon, and astragali. It is not perfect. In its pages are reflected Cardano’s character, his crazy ideas, his wild temper, the passion with which he approached every undertaking—and the turbulence of his life and times. It considers only processes—such as the toss of a die or the dealing of a playing card—in which one outcome is as likely as another. And some points Cardano gets wrong. Still, The Book on Games of Chance represents a beachhead, the first success in the human quest to understand the nature of uncertainty. And Cardano’s method of attacking questions of chance is startling both in its power and in its simplicity.
The Drunkard’s Walk