Drunkard's Walk - Leonard Mlodinow [39]
IN 1662, a few days after Pascal died, a servant noticed a curious bulge in one of Pascal’s jackets. The servant pulled open the lining to find hidden within it folded sheets of parchment and paper. Pascal had apparently carried them with him every day for the last eight years of his life. Scribbled on the sheets, in his handwriting, was a series of isolated words and phrases dated November 23, 1654. The writings were an emotional account of the trance, in which he described how God had come to him and in the space of two hours delivered him from his corrupt ways.
Following that revelation, Pascal had dropped most of his friends, calling them “horrible attachments.”12 He sold his carriage, his horses, his furniture, his library—everything except his Bible. He gave his money to the poor, leaving himself with so little that he often had to beg or borrow to obtain food. He wore an iron belt with points on the inside so that he was in constant discomfort and pushed the belt’s spikes into his flesh whenever he found himself in danger of feeling happy. He denounced his studies of mathematics and science. Of his childhood fascination with geometry, he wrote, “I can scarcely remember that there is such a thing as geometry. I recognize geometry to be so useless…it is quite possible I shall never think of it again.”13
Yet Pascal remained productive. In the years that followed the trance, he recorded his thoughts about God, religion, and life. Those thoughts were later published in a book titled Pensées, a work that is still in print today. And although Pascal had denounced mathematics, amid his vision of the futility of the worldly life is a mathematical exposition in which he trained his weapon of mathematical probability squarely on a question of theology and created a contribution just as important as his earlier work on the problem of points.
The mathematics in Pensées is contained in two manuscript pages covered on both sides by writing going in every direction and full of erasures and corrections. In those pages, Pascal detailed an analysis of the pros and cons of one’s duty to God as if he were calculating mathematically the wisdom of a wager. His great innovation was his method of balancing those pros and cons, a concept that is today called mathematical expectation.
Pascal’s argument went like this: Suppose you concede that you don’t know whether or not God exists and therefore assign a 50 percent chance to either proposition. How should you weigh these odds when deciding whether to lead a pious life? If you act piously and God exists, Pascal argued, your gain—eternal happiness—is infinite. If, on the other hand, God does not exist, your loss, or negative return, is small—the sacrifices of piety. To weigh these possible gains and losses, Pascal proposed, you multiply the probability of each possible outcome by its payoff and add them all up, forming a kind of average or expected payoff. In other words, the mathematical expectation of your return on piety is one-half infinity (your gain if God exists) minus one-half a small number (your loss if he does not exist). Pascal knew enough about infinity to know that the answer to this calculation is infinite, and thus the expected return on piety is infinitely positive. Every reasonable person, Pascal concluded, should therefore follow the laws of God. Today this argument is known as Pascal’s wager.
Expectation is an important concept not just in gambling but in all decision making. In fact, Pascal’s wager is often considered the founding of the mathematical discipline