Pascal's Wager

by James A. Connor

  This is where Werner Heisenberg comes into the picture. During one of the conferences at the Copenhagen Institute, he asked what would happen if we wanted to measure both the position and the momentum of a specific particle. In order for us to do this, we must be able to “see” the particle, so that we can measure it. So, to see it, we have to shine a light on it. The light has a wavelength λ. But in order to actually see clearly, the smaller the object to be observed, the shorter the wavelength λ will have to be, until at the level of electrons, we will have to use such powerful gamma rays to see it that we will change it by our seeing. In this way, Heisenberg realized (actually following the philosophy of Immanuel Kant), we will never see the electron as it truly is, but only as it is once we’ve changed it by our seeing. This gives an uncertainty to the particle’s position (Δx ≈ λ).

  This means that the change in x caused by our seeing is approximately, but uncertainly, equal to the wavelength of the light that we are using to see it with. Now, this applies if we see light as a wave. If we see it as a particle, then we can say that the light we use to see it with gives up some of its momentum to the particles when it illuminates it. How much it gives up is unknown, and unknowable, because it is immeasurable. So, in this case, the change in momentum (h) is described by the following formula:

  Δx Δp ≈ h

  We must note here that this uncertainty is not something that comes from the inadequacy of our technology or of our methods, but is so in principle. You cannot precisely measure both the position and the momentum of a subatomic particle. If you try to get a more accurate fix on one, the other will go askew, and contrariwise.

  Now things really get strange. The Schrödinger equation, which spins off of Heisenberg, demonstrates how the basic nature of things is probabilistic and not classically real. The thought experiment that he came up with took Heisenberg’s uncertainty principle and generalized it by describing certain wave functions about whose outcomes, given a large number of outcomes, the Schrödinger equation will be able to make predictions. Most people have heard about his famous thought experiment about the cat placed in a box with a vial of cyanide gas that would break or not break depending upon whether a specific sensor is hit by a subatomic particle. The chances of its being hit are fifty-fifty. For Schrödinger, as long as the box remains closed and no one looks inside, the wave function of the particle and the state of the sensor remain undecided, and the cat is both alive and dead. It is only when someone opens the box that one wave function collapses and the cat turns up one way or the other. We all cheer for the cat to make it.

  Note the term “large number of outcomes” in the last paragraph. This is the part that has a direct bearing on the letters of Pascal and Fermat. In his letter on the problem of expectations, Pascal was aware that as the number of throws of the dice increased, an effect was produced on the expectations that the players could legitimately hold in relation to the game. Later practitioners of the arts of probability have studied this effect and have used it to send boats deep into the continent of this new mathematical world, into the land of big numbers.

  One of the things they discovered is the so-called law of averages, or Bernoulli’s law. Many gamblers mistakenly think that this law means that everything will average out, that if over ten tosses of a coin you have six heads and four tails, eventually—over, say, a hundred or a thousand throws—there will be a corrective and the player will begin to get more tails to make up the difference, so that eventually the number of heads and tails will be the same. In other words, in the long run, the chances will even out. But this isn’t how it really works, though it’s a nice enough idea to have its own name, the “gambler’s fallacy,” or the “maturity of chances.” It has lost a lot of people a lot of money.

  To understand the way the “law of big numbers” works, you have to understand two things: First, every time you toss a coin or throw a die, the probability of a single outcome is always the same. The probability of getting a heads for each toss is fifty-fifty, while the probability of getting a six on the toss of a single die is one in six, while the probability of getting double sixes with two dice is one in thirty-six. This means that every toss is a fresh start, and the uncertainty of the outcome is just as great as in the first toss. This takes care of the gambler’s fallacy, because there is no guiding hand making sure that the tosses even out, and at every moment while the coin is spinning, even the powerful mathematics of probability can’t make a prediction of what will happen next.

  Let us take the example of a coin toss. The “law of averages,” or the “law of big numbers,” says that as the number of times we toss the coin grows, the number of heads we throw will grow closer, in proportion, to half the number of total throws. Therefore, as the number of tosses grows, the probability grows that the percentage of heads (or tails) will get closer to fifty. But the actual number of heads thrown will get larger, too, just as will the percentage. So, we flip a coin a hundred times and get sixty heads and forty tails, which is 60 percent to 40 percent. The gambler’s fallacy would argue that somewhere in there, ten more tails will show up to make up the difference. But you keep flipping and writing down the result. You get to a thousand flips, and the difference is now 55 percent to 45 percent, but the difference in the number of throws is no longer a mere twenty throws; it is now a hundred throws. This is the Law of Big Numbers, and odd things not only happen but become commonplace as the numbers get larger.

  Two people meet each other on the streets of New York just after 9/11, and after further discussion find that their grandparents once had a similar encounter on the streets of London after the Blitz. Then they find out that their grandparents’ grandparents had a similar encounter on the streets of Paris after the First World War. What are the possibilities of that? With a big enough human population, it would be a dawdle. And so, in a big enough universe, anything can happen. The only problem for really improbable things is whether the universe is big enough. And with a big enough universe, say some, you don’t need anything else.

  At this point in Western intellectual history, the old division between materialism and religion seems unbridgeable. Religious people believe ever more deeply in the mystery that the Greeks first experienced in the glades and forests, the running streams with intimations of divinity haunting the field. As such, they are more directly tied to the long drive upward of the human race, and find ever more creative ways to express the new cosmology in theological terms. The old ways are never forgotten; they just find new ways to get into the papers. The materialists, however, are still holding to the notion that we would be better off without all that supernatural folderol, and so they spin ever more delightful theories to make that happen.

  The cultural landscape still seems to be divided into two strategic types—the climbers and the sprawlers. The climbers live vertically, and find ever more wondrous and exotic mysteries in the everyday world, mysteries that somehow intimate God, just as the forests and glades once did. The sprawlers live horizontally. They find satisfaction in the idea of many chances and of big numbers. Anything can happen in the universe if the numbers are big enough. The vastly improbable becomes probable; the uncanny becomes ordinary. Eventually, with big enough numbers, those ten thousand monkeys could type out not only Shakespeare, but Milton, too, and half the Bible. Big numbers explain the coming of life and the evolution of intelligence. Of course, the old rhetorical project lurks behind the big numbers. Why have a God when you can have googolplexes? With big enough numbers, anything can happen.

  The ultimate sprawler’s theory is the multiple universes of John Archibald Wheeler. We can avoid the whole question of Providence, even in light of the big bang, by inventing a zillion zillion universes, separate from ours, where every conceivable quantum state can have its day. In this way, my dog who was hit by the car is alive in some other universe. The fact that such universes have no more empirical evidence for their existence than does God, and are no more falsifiable than is God’s existenc
e, doesn’t seem to matter. In fact, the whole thing might balance on the edge of Occam’s razor. But which is the simpler explanation? If you have big enough numbers, you don’t need God, and that is the heart of it. But are multiple universes any simpler an explanation than God? It seems finally to come down to choice, perhaps even to the Two Standards: people who believe in God do so because they want to; people who don’t believe don’t because they want to. Almost makes one think of efficacious grace.

  NOTES

  1. John R. Cole, Pascal: The Man and His Two Loves (New York: New York Univ. Press, 1995), 24.

  2. Cole, Pascal, 19.

  3. Blaise Pascal, Pensées and Other Writings, trans. Honor Levi, ed. Anthony Levi (Oxford: Oxford Univ. Press, 1995), 8.

  4. Let us be clear here. It is my contention, with Shakespeare, that in the Reformation/Counter-Reformation period, “all are punished.” Albrecht Wallenstein may have been ruthless, but he was no more so than Protestant generals of the same period. Too many myths about good guys and bad guys still make the circuit. For every massacre on St. Bartholomew’s Day committed by the Catholics, there was an equally horrible one committed by Oliver Cromwell and others like him in Ireland, or America, or Germany, or Bohemia. But not one of these men committed crimes to equal what was done by post-Christian butchers like Hitler, Stalin, and Mao.

  5. Pascal, Pensées, 66.

  6. Some later scholars have argued that Blaise may have had a secret copy of Euclid hidden away, which is possible. But if he had, why invent his own definitions? Was Blaise hoping that his father would catch him, as eventually he did, or was he doing this all for his own amusement, as Gilberte indicated in her Vie de M. Pascal? Who can say?

  7. Fermat’s last theorem is that the equation xn + yn = zn has no nonzero integer solutions for x, y, and z when n is not equal to 2. In a marginal note, Fermat wrote: “I have discovered a truly remarkable proof, which this margin is too small to contain.” The problem was that Fermat’s own writing was sloppy and disjointed, so much so that he could never get it printed. Fermat never mentioned this proof again, and mathematicians have been trying to rediscover his remarkable proof ever since.

  8. Nor could you have told it to the two queens, who remained unbowed. After all, Anne was a Hapsburg and deeply involved in the Spanish intrigues at court. The Hapsburgs were, after all, the über-Catholics, the ones holding the line against all those dirty Calvinist heretics, who, to the Spanish mind, were good, barring conversion, only for fuel. Marie de Médicis, on the other hand, was an Italian and wanted to convert the French court into something more Tuscan.

  In 1629, Richelieu had already conquered the Huguenots (the French Calvinists), and his power over the government seemed complete. At first, he had tried to conciliate between the Protestants and Catholics, and was prepared to be tolerant, which irritated the Spanish faction no end. Religion was the most divisive issue of the time, and extreme elements on both sides had wrecked his plans. The French Calvinists had built their own army and fortified castles. Then the Huguenots made the mistake of involving themselves in an attempt to pressure England to declare war on France. And so Richelieu declared war on the Huguenots, and led the army himself to besiege the castle of La Rochelle, where the Protestants were holed up. Meanwhile, Spain tried to solidify its power in Italy by attacking France’s allies in the north. Richelieu’s secret police caught wind of certain secret communications going on between Queen Anne and her brother, the king of Spain. Marie de Médicis, on the other hand, still stung by Richelieu’s departure from her camp, and possibly suffering from unrequited love, allowed herself to be swayed by the radical Catholics to get rid of the more evenhanded cardinal, because they said he was too tolerant of the Calvinists.

  So the two queens drafted their plans. Mortal enemies up until this time, they reconciled long enough to try to get rid of Richelieu, whom they both despised more than they did each other. Marie de Médicis, the Queen Mother, who could swear like a dockworker and was heartless to boot, had led a hard life. Her uncle, Ferdinando I de’ Medici, bought the crown of France for her for six hundred thousand crowns and married her to Henri IV. But Henri already had a mistress in place, a vicious mistress, Henriette d’Entraigues, who claimed to be the queen herself because of a foolish document Henri had signed when he was particularly smitten, and maybe a little drunk, and who did not want to share him, least of all with this foreigner from Italy. Henriette instigated open war between the king and his queen, though the queen still managed to give Henri a brace of sons. There were shouting matches and haughty exchanges until the king, happily for Marie, was stabbed to death by an assassin on the rue de la Ferronnerie. By the end of the day, Marie was the regent of France, and Henriette d’Entraigues was in a lot of trouble. One evening after Richelieu’s return from war, on what was later called the Day of Dupes, Marie met with Louis privately in her chambers. Fearing spies, she set her guard at the door, and when Richelieu got wind of the meeting, he tried to bully the guards into letting him pass but was rebuffed. So, fearing for his life, the all-powerful cardinal snuck into the Queen Mother’s apartment through the side chapel and appeared before the king and his mother. Marie, raging, told Louis that he must choose between Richelieu and her—one of them would have to go. The king agreed to fire his first minister, and Richelieu was certain that his life was forfeit, but within a week, Marie de Medicis was running for her life toward the Dutch border.

  9. Susan Griffin, The Book of the Courtesans: A Catalogue of Their Virtues (New York: Broadway, 2001).

  10. Thomas M. Kavanagh, Dice, Cards, Wheels: A Different History of French Culture (Philadelphia: Univ. of Pennsylvania Press, 2005), 14.

  11. Kavanagh, Dice, Cards, Wheels, 40.

  12. Latin text from which this translation was made can be found at http://www.fh-augs-burg.de/harsch/chronological/Lspost13/CarminaBurana/bu–caro.1html.

  13. John’s Gospel is rife with this kind of thinking. Nicodemus cannot understand what Jesus is saying, whereas the woman at the well understands after only a short conversation. The blind man sees, but the priests of the temple are blind. Though such passages may be anti-Semitic at some levels, they are primarily about a fundamental social experience—that some people hear the truth of religion, while others are deaf to it.

  14. Quatrains du déiste, stanzas 1–3. Cited in Antoine Adam, Les libertins au XVIIe siècle, 90. Quoted in David Wetsel, Pascal and Disbelief: Catechesis and Conversion in the “Pensées” (Washington, DC: Catholic Univ. of America Press, 1994), 97–98.

  15. Wetsel, Pascal and Disbelief, 99.

  16. Marvin O’Connell, Blaise Pascal: Reasons of the Heart (Grand Rapids, MI: Eerdmans, 1997), 23.

  17. William R. Shea, Designing Experiments and Games of Chance: The Unconventional Science of Blaise Pascal (Canton, MA: Science History Publications, 2003), 12.

  18. Blaise Pascal, “Letter to the Chancellor About the Adding Machine,” in Pascal: Selections, ed. Richard H. Popkin (New York: Scribner, 1989), 21.

  19. This was also true of Harrison’s clock, which he invented to solve the longitude problem. Even after the clock was invented, many sea captains sailed by astronomical tables rather than the clock because the cost was prohibitive.

  20. Shea, Designing Experiments, 12.

  21. Pierre de Bérulle, “Discours de l’état et des grandeurs de Jésus,” Œuvres complètes (Paris: Migne, 1856), 161. English translation in Anne M. Minton, “The Spirituality of Bérulle: A New Look,” Spirituality Today 36, no. 3 (Fall 1984): 210–19.

  22. In my own recounting of the history of the vacuum, I am deeply indebted to William Shea’s detailed account of all the comings and goings of research on this subject. If you want to find out more about this subject, see Shea’s Designing Experiments, 17–18.

  23. Shea, Designing Experiments, 42.

  24. Popkin, ed., Pascal: Selections, 35.

  25. Emile Cailliet and John C. Blankenagel, trans., Great Shorter Works of Pascal (Westport, CT: Greenwood P
ress, 1948), 44.

  26. Pascal’s reply to Père Noël, in Cailliet and Blankenagel, Great Shorter Works of Pascal, 49.

  27. Jacqueline Pascal, “A Memoir of Mère Marie Angélique by Soeur Jacqueline de Sainte Euphémie Pascal,” in A Rule for Children and Other Writings, ed. and trans. John J. Conley, S.J. (Chicago: Univ. of Chicago Press, 2003), 132.

  28. Jacqueline Pascal, “Memoir,” 133.

  29. Pascal to Perier, 15 November 1647, Œuvres de Pascal 2:680. Translated in Popkin, Pascal: Selections, 44.

  30. Œuvres completes de Pascal 2:682–84.

  31. Shea, Designing Experiments, 115.

  32. O’Connell, Blaise Pascal, 55.

  33. O’Connell, Blaise Pascal, 53.

  34. Jacqueline Pascal, “Memoir,” 125.

  35. At least according to Cardinal de Retz, who was no great friend of Richelieu’s. If Retz is correct, then Richelieu’s service, which so dominated the court, must have been bitter to the king, who wanted to be his own man all his life. In this account, I am relying on the memoirs of Madame de Motteville and of Cardinal de Retz for many of the details about the Fronde. Motteville was a lady-in-waiting to Queen Anne, and praises her virtues while ignoring her faults. Retz was an ambitious man ill suited to the church by his own admission. We have to take what both of them say with more than a little skepticism, because both of them were partisans of one faction or other, and both were ferociously ambitious people who used their memoirs to make political points and to excuse their lives. See Jean François Paul de Gondi, Memoirs of Jean François Paul de Gondi, Cardinal de Retz (Boston: L.C. Page and Company, 1899); and Françoise de Motteville, Chronique de la Fronde (Paris: Mercure de France, 2003).