Fermat’s reply, written in July 1654, shows that he understood that Pascal was talking about a game that is interrupted several times, stopped and then started again, and that in each interruption the gambler is allowed to remove one-sixth of the remaining stake.
Fermat missed the point; he objected that because the stake remains intact, the probability of success in each toss is one in six, and this remains constant with each throw, so that the probability of winning the next throw is still one in six. This is where Fermat misunderstood, since what Pascal was asking was not the probability of a particular gambler winning the next toss, but the probability of that same gambler winning the game if it was carried through to its completion; he was trying to calculate the future as if it had already arrived. This required calculating the partial scores of each player, at each throw. Fermat was looking at the question in the way that the chevalier de Méré was looking at it, the way most people looked at it: as a question about dividing the spoils. But Pascal had created something new, a way of calculating the chances of a final outcome. And this was the beginning of a new science of probability. With this simple shift in focus, from the concrete game at hand to a supposed set of outcomes, the concept of risk management was born. Pascal and Fermat were not the only founders of this science, of course, but they were two of the most important. The difference between these two positions on this question—Fermat’s and Pascal’s—is the difference between setting the odds for the next throw, this next instance of luck, and of exploring the geometry of luck itself, of plumbing chance. With this letter, the modern world had, almost unnoticed, slipped into the murky world of partial existence.
Over the next few months, into August, the exchange of letters invented a range of new methods for calculating the distribution of points. Fermat’s proposal was his combinatorial method, and Pascal was giddy over its possibilities, but it proved too dependent on complex calculation. It was a lot of work, and Pascal, who had invented an arithmetic machine in order to avoid tedious calculations, looked for another method, a cheat. After all, a simple game of coin flipping was one thing, but a game of hasard played with two dice is much more complicated, with a far greater number of combinations. The time it would take to write them all down would make the combinatorial method impractical. What if, Pascal asked, the two players had previously agreed to keep playing until one of the players won three points? In this case, the calculation would be based not on possible outcomes but on the rules set by the two players at the beginning. Pascal considers three cases in which each player puts thirty-two pistoles into the pot:
Two gamblers have agreed to the game, and in the first instance gambler A quickly wins two tosses, and therefore has two points, while gambler B wins only one toss, and has one point. Either A would win with the next throw, or B would catch up with the first, and they would be tied. If the first player wins, he would walk away with the entire pot, equaling 64 pistoles. But let’s say the second player wins and then the game is interrupted; then they would walk away with their original contribution, 32 pistoles. But what if for some reason they decide not to take the next throw and leave the question undecided. How would they divide the stakes then?50
Pascal argues that in this case the first player should be able to say, “I’m certain to get thirty-two pistoles, for I get them even if I lose, but as for the other thirty-two, perhaps I will get them, perhaps you will get them; the odds are even. Let us divide these thirty-two pistoles in two, and you will give me one half plus the thirty-two of which I am certain.”51 So the first guy walks away with forty-eight pistoles, leaving his friend with sixteen. This is what Pascal called the method of expectations, which is based upon a calculation of probabilities plus an understanding of the rules of the game agreed to by the players. Here, we have an analysis of chance, but one that is made from within the context of contract law.
Pascal then moves on to another case, a variant of the first. In this case:
Two gamblers agreed to a game similar to the one mentioned above. In this case, gambler A has won twice, while his opponent has lost every throw.
If they quit the game at this point, the first gambler wins everything, but if they go on to the next throw, they are in a situation similar to the first example. If the game is interrupted at that point, they will divide up the stake in the same way as they had done in the previous game, with gambler A walking away with forty-eight pistoles and gambler B walking away with sixteen. Now, gambler A can say to his friend, “If I win, I win all the money, and if I lose, I have a right to forty-eight pistoles. So give me these forty-eight pistoles that I can claim even if I lose the next throw, and let us share equally the remaining sixteen pistoles, since you have as much chance as I do of getting them.” In this case, gambler A walks away with fifty-six pistoles, and his hapless friend walks away with eight and the sure but painful knowledge that he should avoid dicing with his friend gambler A in the future.
Finally, Pascal presents a third case:
Two gamblers agree on a game similar to the ones above. In this case, after the first toss, gambler A has won one throw, and gambler B has won none.
Now, let us say that gambler A wins the next throw. He will then have two throws to his opponent’s none. Most people can see where this is going: gambler B is going to get hosed again. This situation would then be like the beginning of the second game, and the first gambler would have a right to walk away with fifty-six pistoles if the game were interrupted. If gambler B were to win this next throw, they would be even, and if the game were interrupted, they could each walk away with their original stake. But what if they decide not to make this next toss? In this case, gambler A could argue, “If we agree not to play, give me thirty-two pistoles, the amount I am sure to receive, and let us share equally the remainder of fifty-six pistoles.” This would allow gambler A to walk away with thirty-two plus twelve (one-half of twenty-four, the difference between fifty-six and thirty-two), or forty-four, pistoles.52
Over the next few letters, Pascal and Fermat worked out several other methods, largely variants of those stated here. In his Treatise on the Arithmetical Triangle, Pascal applied these questions to work that he was doing on the arithmetic triangle, which became the basis for most of the calculations done in the insurance industry.53 In this way, insurance companies calculate what policyholders would be due under various circumstances, making the insurance industry a very complicated dice game, with people’s lives in the balance. Let us never fool ourselves that the insurance game is primarily about protection, for it is at heart a gambler’s paradise, with computers.
It was Pascal’s method of expectation that changed so many things, however. Pascal had essentially rewritten the question, so that what the chevalier de Méré had originally asked, which was how he could calculate the particular odds of something happening over a certain number of throws, became what each gambler could legitimately claim at each stage of the game if the game were to be interrupted. The future had been rewritten to fit the present. In short, Pascal asked, “What can I legitimately claim now, given the rules of the game and the likelihood of future outcomes?” To answer this question, he had to make the assumption that the players all agreed that once the money was in the pot, it no longer belonged to them. Thus, Pascal added a new twist to the old question of fairness. When people play games, they want them to be fair. Pascal took this seriously and added it to the nature of the game, so the games themselves implied a binding contract between the players, informal as this may be. The game, as a contract, allowed them certain expectations under certain conditions, which could permit them to walk away with portions of the pot or all of it, depending upon what happened. The players can enter the game at any time and they can leave the game at any time, and the method of expectations gives them a way to calculate what they would be entitled to at each stage of the game. Thus, Pascal’s solution to the division problem or the problem of points allows them to make a decision based upon calculation. The same method
applies to any decision, from dicing games to experiments in physics. How we respond to uncertainty left the oracles behind and became a question of pure mathematics from that time on. From these rough beginnings, both probability theory and decision theory began to take shape.54
This was a monumental change in thinking. Pascal knew this, and in his letter to the Parisian Academy, he writes about his new discovery as a geometry of chance:
Here the uncertainty of fortune is so controlled by the fairness of reason that each of the two players is always assigned exactly what belongs to him by right. This is the more to be sought by reasoning, the lesser it can be investigated by experiment. The ambiguous outcome of fortune is rightly ascribed to chance rather than natural necessity. This is why the issue has remained uncertain to this day. But if it has proved refractory to experience, it can no longer escape rational inquiry. We have turned it into an assured form of knowledge with the help of geometry whose certainty it shares. It combines mathematical demonstration with the uncertainty of chance, and having shown that these are not contraries, it borrows its name from both and proudly calls itself the Geometry of Chance.55
[1654]
The Night of Fire
Among scientists are collectors, classifiers, and compulsive tidiers-up; many are detectives by temperament and many are explorers; some are artists and others artisans. There are poet-scientists and philosopher-scientists and even a few mystics.
—SIR PETER BRIAN MEDAWAR (1915–1987)
Did you know that secret? The awful thing is that beauty is mysterious as well as terrible. God and devil are fighting there, and the battlefield is the heart of man.
—FYODOR MIKHAYLOVICH DOSTOYEVSKY
The mystic too full of God to speak intelligibly to the world.
—ARTHUR SYMONS (1865–1945)
Mystical experiences are what they are. Everyone has them, but most people turn them aside as moments of beauty, or as strange upwellings of happiness that comes from nowhere. Commonplace mystical experiences are too lukewarm to change lives. All too often, we turn them into psychology, into explosions of neurons in the brain, or into the working out of unfinished childhood conflicts—a lump of undigested meat, a bit of underdone potato. But mystical experiences are what they are, and they cannot be so easily parsed away. A few people have mystical experiences powerful enough to change lives. People who have had a near-death experience, for example, often come back transformed, with a new set of values that sets their lives onto a radically new course. Blaise Pascal, on a Monday night, the 23rd of November, 1654, had one of these.
Pascal had recently returned to Paris from Poitou, and in spite of all the aristocratic diversion he had enjoyed in the duke’s company, his depression had only deepened. His relationship with Jacqueline was gradually healing, and yet his headlong flight into the world had not satisfied him. For the past three years, he had been living the life his father had created for him, the life of a Christian gentleman who piously lived in the world, and yet it was not enough. Of course, he kept a close contact with the duke, for the two of them had become deep friends. The duke involved him in an investment scheme, a company that he had set up to drain marshlands in Poitou, but it didn’t make much money. The only good thing that happened in Pascal’s life at that time was an invitation to lecture on his current work to the Parisian Academy, the grand institution that Père Mersenne’s little seminar had mutated into.
Something was moving inside of Pascal, a dark night of the soul leading to a conversion. He was unhappy with his life, though it was the life he had always lived. In spite of his previous rancor with the sisters of Port-Royal, he consistently noticed on his visits to see Jacqueline and his nieces, Gilberte and Florin’s daughters, who were living at Port-Royal, that his sister possessed a serenity that he could not find in his own life. He had lived a life of the mind, and the life of the mind had not filled him up. His emotional life was stifling, and what he needed was a change of heart. He was disgusted with the world and with himself.
Just before or just after the night of his great conversion, Blaise Pascal wrote a short piece entitled “On the Conversion of the Sinner.” Before the conversion, he said, there is a deep dissatisfaction with the world: “The soul can no longer serenely enjoy the things that captivated it. Constant scruples attack the soul in its pleasure, and because of this introspection it no longer finds the usual sweetness in the things to which it once abandoned itself blithely with an overflowing heart.”56
This was a perfect description of what Pascal’s own life had become. This dissatisfaction was part of the “first stirrings” that God was causing in the sinner’s heart. “But the soul finds more bitterness in the disciplines of holiness than in the futilities of the world. On the one hand, the presence of visible things seems more powerful than the hope of the things unseen; on the other hand, the permanence of things unseen moves it more than does the frivolity of visible things. And thus, the presence of the one and the constancy of the other fight for the soul’s affection; the emptiness of the one and the absence of the other awaken its disgust.”57
This apparent contradiction in the heart of the sinner produces great confusion and discord. But this is only temporary, for eventually the sinner begins to realize that visible things are already passing, even in their enjoyment. “The soul considers mortal things as already dying and even as already dead. It is terrified by this realization, this certainty of the annihilation of all that it loves as the ticking by of each moment snatches away the pleasure at hand; it is terrified when all that is dearest slips away into nothingness at every moment, and by the certainty that there will come a day when all the sweet things of this earth will be gone, and the soul will be destitute of the things it placed its trust in.”58
A knowledge too terrible to contemplate, a truth of life too fearful to hold—this is what drove Pascal, bit by bit, from the world. This was a knowledge that he had already known as a set of platitudes, something one says at funerals, but not something one takes to heart. It is interesting to speculate how these reflections had come upon Pascal as if newly minted after he had spent a greater part of the year in the company of gamblers. Perhaps it was the chevalier de Méré and his questions, so full of the uncertainty of the future, so full of the desire to hedge his bets, that started Pascal upon this road.
Once back in Paris, Blaise changed his residence once again, moving from the rue Beaubourg to the rue des Francs-Bourgeois, not far from the house that his father, Étienne, had brought the family to when they had moved to Paris from Clermont, but, more important, near the convent of Port-Royal de Paris. It was only a short walk away, a matter of ten minutes, past the queen’s basilica of the Val-de-Grâce, the very church where years before Cardinal Richelieu’s agents had caught Anne of Austria in the act of sending treasonous letters to her brother, the king of Spain. In these new quarters, Blaise could visit his sister anytime he wished, and did so regularly.
Likely, Port-Royal’s parlor was similar to the ones used by Discalced Carmelites—a poorly lighted room with hard-backed chairs gathered near a wide grille or set of bars cut into a wall. On the other side, there would be a curtain that could be opened or closed whenever the sisters were present. The old-fashioned grille work, which is most likely what existed at Port-Royal at the time, was more like a semiopaque screen, so that the sisters would appear almost ghostly on the other side, like disembodied voices speaking out of shadows. Nevertheless, Blaise could visit his sister often, so often that Jacqueline wrote to Gilberte that “it would take a volume to tell you about all the visits one by one. This was such that I sometimes thought I had no time for any other work.”59 In these long sessions, Blaise opened his heart to her, and the rancor of all those months at the time of her profession was gone. She pitied him deeply, for in the midst of all of his busy life and all of his success, he hated his own life. He was living the very existence he described in his short tract on the conversion of the sinner; he was the sinner described
on those pages, and what appeared as a theological discourse was actually a most honest confession. And yet he felt no attraction to a God who would ask him to give up the acuity of his mind. This was his stumbling block, for although he could easily give up the pleasures of the flesh—good food, good conversation, a host of entertainment and diversions—he did not wish to give up his life as a philosopher and scientist. His own body had caused him nothing but pain for twenty years. What were the pleasures of the flesh, when the flesh had been so full of pain? But how could he give up the one thing he relied upon for the truth—his own reason, his own intellect?
Sadly, had he turned to the Jesuits for spiritual guidance, they would not have required such a sacrifice, for it was their stated belief that one could find God in the midst of all things, that finding God was not a matter of abandoning the world as the Desert Fathers had done, but of choosing Christ in the midst of one’s life in the world. One could live in the world, one could live the life of a Christian gentleman, and still find God there.
But the Jesuits’ was not the path he had taken. Blaise had always been attracted to a more rigorous theology. For Jacqueline and the sisters of Port-Royal, it was not enough to merely assent to God in the mind; it was necessary to love God in the heart. But one could not love God and any other thing. Oddly, this is where Jansenism was both right and wrong, for though one of the two great commandments that Jesus held to was that we should love God with our whole heart, with our whole soul, with our whole mind, and with our whole strength—which sounds an awful lot like loving God, and only God—the other great commandment was that we should love our neighbor as we love ourselves, so that self-love was not anathema, but a component of the love of God. Perhaps, however, Christians could truly understand this connection only after the invention of psychology.
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