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Blaise Pascal: Proving God?? A Mathematical Interpretation of Pascal's Wager Jamie Mosley 12/21/2010 1 Two Perspectives on Pascal Ernest Mortimer “A modern man, starting out for the office, may glance at his wrist-watch, tap the barometer, slip into the nearest tobacconist‟s shop for a purchase and receive his change from the cash machine, board an omnibus and presently settle at his desk. How remote might seem the French geometer who got mixed up with Jansenism before Versailles was built! Yet Pascal originated that barometer, invented that calculating machine, was the first man to think of an omnibus and to organize a line of public vehicles, and was perhaps the only man before the twentieth century habitually to wear a wrist-watch.” 12/21/2010 2 Two Perspectives on Pascal (cont.) E. T. Bell “We shall consider Pascal primarily as a highly gifted mathematician who let his masochistic proclivities for self-torturing and profitless speculation on the sectarian controversies of his day degrade him to what would now be called a religious neurotic.” 12/21/2010 3 A Mathematical Interpretation Rather than becoming involved in this debate, the purpose of today‟s discussion is to explore the connection between Pascal‟s mathematical bakground and the section of Pensee’s known as “Pascal‟s Wager.” 12/21/2010 4 Three Aspects of a Mathematical Understanding Pascal‟s Geometric Influence Pascal‟s Probability Theory Pascal‟s Famous Wager 12/21/2010 5 Pascal’s Geometric Influence 12/21/2010 b. 1623 in Clermont, a provincial French city Spent the majority of his life in or near Paris Educated at home by his father in a quite unique manner. 6 The Pascal Educational Model Goal: Do not overwhelm the young mind until it is mature enough to grasp the concept. Early education was instead directed toward observation. Tentative Schedule: Latin studies begin at the age of twelve and mathematical studies begin between the age of fifteen or sixteen. 12/21/2010 7 The Actual Pascal Education At the age of twelve, Pascal‟s observational instruction produced mathematical and geometrical rewards. Blaise reached the same conclusion as did Euclid in Proposition 32 of Book 1 of the Elements. As a reward for his mathematical achievement Blaise is given a copy of the Elements for study and invited to the Acadmie libre. 12/21/2010 8 The Genius Established The challenges posed to Pascal in these meetings served as a springboard for scientific achievement throughout his life At the age of 16, Blaise published a work consisting of projective geometric theorems, which focused on conic sections. 12/21/2010 9 The Effects of Geometric Study Geometric thought is present in all of his writings. Edward Craig “His outlook was deeply influenced by what he conceived to be a new way of looking at the world inspired by geometry, and most commentators would agree that his writings are impregnated with it.” 12/21/2010 10 A General Overview of Geometry The term geometry is synonymous with the term axiomatic system. Edward Wallace and Stephen West “The axiomatic method is a procedure by which we demonstrate or prove that results discovered by experimentation, observation, trial and error, or even by „intuitive insight‟ are indeed correct.” 12/21/2010 11 Four Characteristics of the Axiomatic Method Any axiomatic system must contain a set of technical terms that are deliberately chosen as undefined terms and are subject to the interpretation of the reader. The axiomatic system contains a set of statements, dealing with undefined terms and definitions, that are chosen to remain unproved. These are the axioms of the system. All other technical terms of the system are ultimately defined by the means of the undefined terms. These terms are the definitions of the system All other statements of the system must be logical consequences of the axioms. These derived statements are called the theorems of the axiomatic system. 12/21/2010 12 The Theory Applied to Mathematics A Simple Axiomatic System: Three-point Geometry Undefined terms: Point, Line, and the relation “belongs to” Axiom 1: There are exactly three distinct points in this system. Axiom 2: Two distinct points belong to exactly one line. Axiom 3: Not all points belong to the same line. Axiom 4: Any two distinct points contain at least one point that belongs to both. 12/21/2010 Theorem: Two distinct lines contain exactly one point. Proof: Case 1: Assume that two distinct lines do not contain exactly one point. Case 2: Assume that two lines share more than one point, and consider two, the simplest form of “more than one.” 13 Extending the System All theorems of the system must be consistent. Wallace and West “A set of axioms is said to be consistent if it is impossible to deduce from these axioms a theorem that contradicts any axiom or previously proved theorem.” 12/21/2010 Assuming that the geometry is consistent, the following conclusions can be made. If the original set of axioms is true, then the geometry is true. 14 The Theory Applied to Life Edward Craig “Pascal begins by conceding that definitions in geometry are nominal and not real, and that what are taken for axioms are intuitive perceptions which can neither be demonstrated or reasonably denied.” These terms would include things like number, space, movement, time, etc. Pascal calls these things “first principles” in the Pensees. 12/21/2010 15 The Theory Applied to Life (cont.) Pascal “We know the truth not only through reason, but also through our heart. It is through the latter that we know our first principles, and reason, which has nothing to do with it, tries in vain to refute them…our inability must therefore serve only to humble reason, which would like to be the judge of everything, but not to confute our certainty.” (Pensees 110) Reason or logic is insufficient to explain everything in the world, and this inclined Pascal to conclude that geometry, with its axioms, is more effective than logic. 12/21/2010 16 Pascal’s Probability Theory These theories developed from solving two problems, the “Problem of Points” and “The Gambler‟s Ruin,” in a correspondence between Pascal and Pierre de Fermat. The “Gambler‟s Ruin” problem, which begins with the dice problem, is an extension of the “Points Problem.” The dice problem will be used to gain an understanding of Pascal‟s work on probability theory. 12/21/2010 17 The Dice Problem Purpose: To determine how many throws with three dice are necessary for one to have a better than even chance of throwing three sixes in a single throw. To fully establish the principles necessary for this solution, Pascal and Fermat first had to disprove the traditional and incorrect answer to this same problem dealing with two dice. 12/21/2010 It was generally agreed that the solution for a single die was four throws. In consideration of this conclusion, the traditional answer for two dice was 24 throws. Pascal and Fermat concluded that the answer for two dice is 25. 18 The Dice Problem (cont.) Goal for two dice: the probability of rolling two sixes, p[n] is greater than 1/2. There is a (35/36) chance of not rolling a pair of sixes in a single throw. For n throws, there is a (35/36)^n chance of not rolling a pair of sixes. For n throws, the probability of rolling a pair of sixes is p[n] = 1 – (35/36)^n If n = 25, then p[n] is greater than 1\2. 12/21/2010 To this conclusion, Antione Gombauld, chevalier de Mere, responded so boldly in opposition that Pascal wrote, “This was a great scandal which made him (de Mere) proclaim loudly that the theorems were not constant and Arithmetic had belied herself.” The solution to the complete “Gambler‟s Ruin” problem is beyond the scope of this paper, but understanding its principles is important. 19 The Theory Applied Oystein Ore “Pascal never quite relinquished his interest in the newly created field…it (the wager) is at first difficult to understand…but if one recognizes that Pascal has a definite mathematical probability formula in mind, the passage becomes quite lucid.” 12/21/2010 20 Pascal’s Famous Wager Fire God of Abraham, God of Isaac, God of Jacob, not of philosophers and scholars. Certitude. Certitude. Feeling. Joy. Peace. God of Jesus Christ. “Thy God shall be my God.” Forgetfulness of the world and of everything except God. He is to be found only by the ways taught in the Gospel. Greatness of the Human Soul. “Righteous Father, the world hath not known Thee, but I have know Thee.” Joy, joy, joy, tears of joy. I have separated myself from Him. “My God, wilt Thou leave me?” Let me not be separated from Him eternally. “This is the eternal life, that they might know Thee, the only true God, and the one whom Thou hast sent, Jesus Christ.” Jesus Christ. I have separated myself from Him: I have fled from Him, denied Him, crucified Him. Let me never be separated from Him. We keep hold of Him only by the ways taught in the Gospel. Renunciation, total and sweet. Total submission to Jesus Christ and to my director. Eternally in joy for a day‟s training on earth. Amen. 12/21/2010 Were it not for November 23, 1654, Blaise Pascal would never have began the project that eventually became the Pensees. On that day, he penned following words and sewed them on the inside of his coat and carried it with him for the rest of his life. Robert Coleman “The reality which it describes changed the life of Blaise Pascal, universally acclaimed scientist, inventor, psychologist, philosopher, and Christian apologist; by any comparison one of the greatest thinkers of all time.” 21 The Geometric Basis of the Wager “Thus we know the existence and nature of the finite because we too are finite and extended in space. We know the existence of the infinite without knowing its nature, because it too has extension but unlike us no limits. But we do not know either the existence or the nature of God, because he has neither extension nor limits. “If there is a God, he is infinitely beyond our comprehension, since, being indivisible and without limits, he bears no relation to us. We are therefore incapable of knowing either what he is or whether he is.” (Pensees 418) 12/21/2010 22 The Geometric Basis of the Wager (cont.) “‟Either God is or he is not.‟ But to which view will you be inclined? Reason cannot decide this question. Infinite chaos separates us. At the far end of this distance a coin is being spun which will come down heads or tails. How will you wager? Reason cannot make you choose either, reason cannot prove either wrong. “Yes, but you must wager. There is no choice, you are already committed. Which will you choose then?” (418) 12/21/2010 “The prophecies, even the miracles and proofs of our religion, are not of such a kind that they can be said to be unreasonable to believe in them. There is thus evidence and obscurity, to enlighten some and obfuscate others. But the evidence is such as to exceed, or at least equal, the evidence to the contrary, so that it cannot be reason that decides us against following it, and can therefore only be concupiscence and wickedness of the heart. Thus, there is enough evidence to condemn and enough to convince, so that it should be apparent that those who follow it are prompted to do so by grace and not by reason, and those who evade it are prompted by concupiscence and not by reason.” (835) 23 The Geometric Basis of the Wager (cont.) The fact of God‟s existence is not the “therefore” of a proof. Instead it is the axiom of an axiomatic system that makes proof about the world possible. The Christian worldview is true iff. God exists. According to Pascal, everyone cannot see that this axiom is true because God has not revealed it to them. 12/21/2010 “If there is no obscurity man would not feel his corruption: if there were no light man could not hope for a cure. Thus it is not only right but useful for us that God be partly concealed and partly revealed, since it is equally dangerous for man to know God without knowing his own wretchedness as to know his own wretchedness without knowing God.” (446) 24 The Probability Argument of the Wager Purpose: For those, who do not believe God exists, to wager that God exists because there is really nothing to lose by wagering. Recall: Pascal has already established that a choice must be made. There are only two options, God exists or he does not, and reason cannot make the decision. “Let us assess the two cases: if you win you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist.” (418) 12/21/2010 25 Why Wager That God Exists? Expected Value = (Probability x Payoff) – Cost Pascal presented the argument as if the probabilities of God existing and God not existing are each 1/2. If God Exists: The payoff is an “infinity of infinitely happy life,” the cost is finite, and the expected value is infinite gain. If God Does Not Exist: The payoff (if any) is finite, the cost (if any) is finite, and the expected value is finite. 12/21/2010 26 Assessing the Two Options Let d1 = God exists Let d2 = God does not exist Three factors used to decipher what Pascal means by d1 and d2. f1 = God exists f2 = God is not indifferent to human behavior f3 = life after death for human beings is eternal Consider the following truth table. 12/21/2010 Case 1 2 3 4 5 6 7 8 f1 F F F F T T T T f2 F F T T F F T T f3 F T F T F T F T d2 d2 --d2 d2 d2 d1 d1 is true only in case 8 27 Assessing the Two Options (cont.) “Concentrate then not on convincing yourselves by multiplying proofs of God‟s existence but by diminishing your passions. You want to find faith and you do not know the road. You want to be cured of unbelief and you ask for the remedy: learn from those who were once bound like you and now wager all they have.” (418) 12/21/2010 w1 = live as if d1 is true w2 = live as if d2 is true The wager‟s purpose is to show that w1 is the best option. Frank Chimenti “The peculiarity of the strength of the argument is that it does not rely on finding evidence to support the truth of T1 (d1), rather it relies on the difficulty of proving beyond the shadow of any doubt that T1 is false…the provable indeterminancy of T1 cannot deflect the strength of Pascal‟s argument.” 28 Application of the Wager Thomas Morris “(Pascal‟s) recommendation is that anyone who sees the reasonableness of the wager should begin to enter into a new pattern of living and thinking, insofar as he or she finds it possible. The unbeliever should begin to attempt to conform his life to a pattern set by true believers. He should begin to think on the idea of God, he should meditate upon moving religious stories, he should attempt to pray (as far as that is possible), he should associate with people who already believe and hold religious values to be very important, he should expose himself to the religious rituals of worship. The recommendation of the wager argument is not „It is in your best interest to believe in God, so therefore go and believe.‟ Belief is not under our direct voluntary control.” 12/21/2010 29 Concluding Remarks Pascal‟s Wager is mathematically oriented A Response to the Critic of Pascal What does this mean for us? 12/21/2010 30