The Life and Faith of Blaise Pascal. George W. Benthien

Transcription

1 The Life ad Faith of Blaise Pascal George W. Bethie September 14, 2018

2 Blaise Pascal ( ) Itroductio Who was Blaise Pascal? Was he a mathematicia, a ivetor, a scietist, a egieer, a literary giat, a philosopher, a theologia? The aswer to all these is yes. I his short lifetime (39 years) he made importat cotributios i all these areas. I mathematics his biggest cotributio was probably his collaboratio with Pierre de Fermat to form the begiigs of probability theory. Most studets of mathematics are familiar with a arragemet of umbers called Pascal s triagle. Pascal did t actually ivet this triagle, but he made extesive use of it i the combiatorial problems associated with probability ad was the first to publish a scietific paper dedicated solely to this triagle ad its properties. I additio, he also derived a theorem that is a importat part of projective geometry ad made importat cotributios to fidig the area ad cetroid of segmets bouded by a famous curve called a cycloid. His biggest ivetio was probably the first mechaical calculator called the Pascalie. This ivetio also ivolved a great deal of egieerig skill i order to actually brig it ito productio. He is also credited with the ivetio of the syrige, the hydraulic lift, ad the desig of the first mass-trasit system i Paris. As a scietist he performed importat experimets that helped to establish the existece of a vacuum. Although the existece of a vacuum is readily accepted today, it was hotly debated at the time. His experimets also established that atmospheric pressure decreases with altitude. He also established a law ivolvig fluid pressure that is fittigly called Pascal s Law ad is foudatioal to the field of hydraulics. Pascal was a dedicated Christia ad a member of a sect of Catholicism called Jaseism. This sect was a bitter rival of the Jesuits. Pascal composed a series of 18 letters that is collectively called Les Lettres Proviciales (The Provicial Letters) that defeded Jaseism ad made fu of the Jesuits. This collectio of letters is cosidered a masterpiece of Frech literature. Voltaire, who was a vocal oppoet of Christiaity, kept a copy by his bed ad referred to it ofte. At his death Pascal was i the process of writig a defese of Christiaity that was ot completed. The collectio of his otes have bee published uder the title Pesées (Thoughts). Pesées is well 1

3 regarded as both a work i philosophy ad i Christia apologetics. I m sure may of you have see quotatios from Pesées such as All of humaity s problems stem from ma s iability to sit quietly i a room aloe. The heart has its reasos of which reaso kows othig If we submit everythig to reaso our religio will be left with othig mysterious or superatural. If we offed the priciples of reaso our religio will be absurd ad ridiculous... There are two equally dagerous extremes: to exclude reaso, to admit othig but reaso. As you ca see Blaise Pascal was a ma of may talets. I the sectios that follow we will preset a brief biography of his life ad look at his may accomplishmets ad his faith i more detail. I preparig this paper I looked at a great umber of refereces, may of which are available olie. Those that I foud most helpful are listed i the Referece sectio at the ed. 2

4 Brief Biography Blaise Pascal was bor o 19 Jue, 1623 i Clermot Frace. Clermot is located i cetral Frace ad is surrouded by a chai of volcaoes. A picture of Clermot is show i Figure 1. Figure 1: Clermot: Pascal s Birthplace Frace at this time was a major power i Europe ad was a Catholic stroghold. Frace was i the process of rebuildig after several decades of war betwee Frech Catholics ad Protestats. Pascal s Father was Étiee ad his mother was Atoiette. He had two sisters Gilberte who was older ad Jacquelie who was youger. Blaise became very sick whe he was about two years old ad early died. He suffered from ill health throughout his life. His mother died whe he was about three years old ad he was raised by his father. His father was a tax commissioer i Clermot. I additio to beig a civil servat, Étiee was fluet i lati ad greek ad was a accomplished mathematicia. I 1631 Étiee sold his govermet positio i Clermot (a commo practice i that day) ad moved his family to Paris. He ivested i govermet bods to support his family. Étiee had defiite ideas o how to educate his childre so they were home-schooled by him. He wated them to be well grouded i laguages, history, ad philosophy before they took o more advaced subjects such as mathematics. He told Blaise that he could t study mathematics util he was 15 years old. However, Blaise was itrigued by geometry ad secretly developed some of Euclid s theorems usig his ow termiology. Whe his father foud out, he realized that his so was very gifted i this area ad gave him a copy of Euclid s Elemets to study. Étiee was also a member of a small discussio group who were iterested i the ewly emergig field of atural philosophy (sciece). This group met i the quarters of a catholic priest Mari Mersee, himself a accomplished mathematicia. Members of this group icluded, Pierre de Fermat, Gilles Roberval, ad Girard Desargues. They were also i correspodece with Reé Descartes, Christiaa Huyges, ad Thomas Hobbes. The group discussed curret topics i atural philosophy ad mathematics icludig their ow research. Blaise accompaied his father to the meetigs whe he was about 13 ad bega takig part i the discussios. The group could tell that this was a very gifted child. 3

5 I 1638 Cardial Richelieu, the First Miister, etered Frace ito the thirty years war. This was a power struggle betwee various catholic ad protestat states. I order to help raise moey for this war he defaulted o the govermet bods. This was a fiacial disaster for Pascal s father Étiee. He joied with others i a protest rally. Cardial Richelieu was eraged ad jailed most of the protesters. Étiee escaped, but had to go ito hidig. Jacquelie, Blaise s youger sister, was also a child prodigy i poetry ad drama. She appeared i a performace before Cardial Richelieu that moved him deeply. After the performace Jacquelie approached the Cardial ad begged him to forgive her father, ad to allow him to retur. The Cardial ot oly forgave him, but offered him a high admiistrative positio i Roue. The family moved there with him. Pascal s father faced a very difficult situatio i Roue. Cardial Richelieu had imposed very high taxes to pay for the war ad the people were agry. There were frequet riots. Étiee was a very capable ad hoest admiistrator ad evetually gaied the peoples trust. However, he was heavily burdeed by the volumious calculatios ivolved i keepig up with the ever chagig tax rates. Blaise would help his father with the calculatios. Blaise correspoded with the discussio group i Paris ad was kept up-to-date o curret evets. Mersee set Blaise a copy of a book by Desargues o coic sectios. Blaise was itrigued by the way Desargues used projectios i this study. I 1642 Blaise set Mersee a copy of his first mathematical paper etitled ıessai pour les coiques. I this paper Blaise derived a theorem o hexagos ow kow as Pascal s Theorem. This theorem states that if a hexago (a six-sided polygo) is iscribed i a coic sectio (circle, ellipse,... ) ad the opposite sides are exteded util they itersect, the the three poits of itersectio will lie o a lie. This theorem is pictured i Figure 2. Figure 2: Pascal s Hexago Theorem Père Mersee was overjoyed to see the first major result from this child prodigy. He passed the paper o to his may cotacts throughout Europe. Pascal s theorem became a importat result i the field of mathematics kow as projective geometry. Blaise was also able to help his father with the may tax calculatios by ivetig the first mechaical calculator kow as the Pascalie. There is a picture of oe i Figure 3. 4

6 Figure 3: The Pascalie The desig ad operatio of the Pascalie will be discussed i the sectio o Pascal s Cotributios to Sciece ad Egieerig. Not oly was the desig of this device impressive, but it ivolved a great deal of egieerig to brig it ito productio. I 1642 Evagelista Torricelli performed a experimet that brought ito questio a log held belief i the scietific commuity. He took a glass tube that was closed at oe ed ad filled it with mercury. He placed the tube iverted i a dish filled with mercury (see Figure 4). Figure 4: Torricelli Vacuum Experimet The mercury level dropped leavig a colum of mercury i the tube ad a apparetly empty space at the top of the tube. Torricelli claimed that the top space was actually a vacuum ad that the pressure of the air o the mercury i the dish was balacig the weight of the mercury colum. This caused quite a cotroversy sice most scietists at this time were followers of Aristotle ad believed that a vacuum was impossible. Aristotle had claimed that ature abhors a vacuum. Blaise s father alog with a fried duplicated the experimet i At first Blaise doubted that the empty space was i fact a vacuum, but o readig Torricelli s explaatio he became coviced. Before log he would coduct some crucial experimets validatig Torricelli s claim. I the witer of 1646 Étiee slipped o the ice ad broke his hip. Two local boesetters stayed for a period of time with the Pascals ad cared for Étiee. These youg me were members of 5

7 the Jaseist sect withi the Catholic church. The Jaseists followed closely the teachigs of Augustie ad thus had may similarities with the reformed Calviists. Through may coversatios with the two youg me the Pascal family coverted to Jaseism. Blaise. i particular was very excited about his ew-foud faith. However, there was oe of their teachigs that Blaise had difficulty i acceptig. They ecouraged their members to abado all earthly pursuits icludig sciece. Blaise felt that this would be a waste of the gifts God had give him. You ca read more about Pascal s faith i the sectio The Faith of Blaise Pascal. I , while his father was recoverig from his ijury, Blaise performed a umber of public demostratios of Torricelli s experimet with various liquids. I 1657 he retured to Paris accompaied by his sister Jacquelie. She cared for Blaise durig his frequet illesses. I Paris Blaise resumed his vacuum experimets. His latter experimets cocetrated o Torricelli s explaatio that it was air pressure applied to the mercury i the dish that was balacig the mercury colum. His most famous experimet was the oe carried out o the moutai Puy de Dome that illustrated that air pressure decreases with altitude ad thus that the mercury colum drops i height. You ca read more about these experimets i the sectio Pascal s Cotributios to Sciece ad Egieerig. Durig this time Blaise met twice with Reé Descartes to discuss their differig views o the possibility of a vacuum. I the sprig of 1649, Due to civil urest i Paris, Blaise alog with his sister Jacquelie ad father Étiee retured to Clermot ad stayed with the family of Gilberte, Blaise s other sister. They retured to Paris i November 1650 as the riots had died dow. Blaise s father Étiee died o September 24, For may years Jacquelie had wated to become a u at Port-Royal Abbey, a stroghold of Jaseism. However, Étiee had refused to let her go. After her father s death Jacquelie etered the Abbey. Blaise was ow, for the first time, all aloe. The ext few years were a difficult time for him. Blaise became frieds with a aristocrat the duc de Roaez. They shared a commo iterest i the itellectual topics of the day ad both had a desire for a authetic Christia spirituality. The duke itroduced Blaise to a ew circle of frieds. Oe of these frieds was the chevalier de Méré who posed two gamblig problems to Blaise. These problems resulted i a series of letters betwee Blaise ad Pierre de Fermat that formed the begiig of probability theory. You ca read more about these problems i the sectio Cotributios to Mathematics. O the 23rd of November 1654 Blaise had a profoud spiritual experiece that chaged his life. Blaise had always approached God through his mid usig reaso. But o this ight he experieced the presece of God i a way that chaged his heart. He ever told ayoe about this, but he wrote a accout of this experiece o a piece of paper ad sewed it ito the liig of his coat. The paper was discovered after his death whe they were goig through his clothes. This ight is ofte referred to as his Night of Fire because of the refereces to fire i his accout. There are more details i the sectio The Faith of Blaise Pascal. There were several stories that were later told about his Night of Fire. Oe story claimed that he had early died i a carriage accidet which caused him to reassess his life. Aother story claimed that he had bee deeply moved by a sermo. No oe really kows what happeed that ight except for what he had writte Although he ever told ayoe about this spiritual experiece, his frieds could see a differece i his life. His axiety ad cofusio were replaced by a calmess ad a peace. He later told Jacquelie that he was ow 6

8 ready to completely embrace Jaseism. However, he still cotiued to work i mathematics ad sciece ad remaied frieds with the duke ad his etourage. His cotiuatio of these worldly pursuits caused may of the Jaseists to questio his commitmet. The Jaseists were cotiually uder attack i the Catholic church, primarily by the Jesuits. These attacks were ow icreasig i itesity. A committee appoited by the Pope ad the faculty of the Sorboe codemed five of the core Jaseist beliefs as heretical. Atoie Arauld, the curret leader of the Jaseists, tried to defed their positio, but was havig little success. I desperatio they tured to Pascal who peed the Provicial Letters. There were 18 letters i this collectio writte over a two year period Pascal did t use theological argumets, but istead used comedy ad sarcasm to mock the Jesuits ad to egeder sympathy for the Jaseists. The Provicial Letters were very popular ad widely read. It is said that Voltaire, a oppoet of Christiaity, carried a copy with him ad kept oe at his bedside to serve as a remider of what good writig looks like. You ca read more about these letters i the sectio Cotributios to Literature. Pascal s health cotiued to worse i As a distractio he worked o the famous mathematical curve the cycloid. He was able, amog other thigs, to fid the area ad ceter-of-gravity of this curve. Pacal s cotributios ca be foud i the sectio Cotributios to Mathematics. Pascal also aouced his plas to publish a defese of Christiaity. He did t believe that the commo ma could follow the usual theological argumets. Istead he wated to preset argumets i the laguage of the people. He was ever able to fiish this work, but his otes were collected ad published after his death as the Pesés. There is more o the Pesées i the sectio Cotributios to Literature. Jacquelie Pascal died i October of 1661 shortly after beig forced to sig a documet drafted by the kig codemig Jaseist beliefs. I 1662 Blaise istituted the first omibus service i Paris. You ca read more about this achievemet i the sectio Cotributios to Sciece ad Egieerig. Later that year he became seriously ill ad died o August 19, He was buried iside the church Sait-Étiee-du-Mot ad the family placed a small plaque o the wall ear his tomb. A picture of the church where he is buried is show i Figure 5 below. Figure 5: Pascal Burial Site 7

9 Cotributios to Sciece ad Egieerig Pascal made a umber of cotributios to sciece ad egieerig. I this sectio we will describe some of the more importas oes. Fluids ad Hydraulics Pascal performed a umber of experimets dealig with fluids ad hydraulics. He determied that the hydrostatic pressure i a fluid depeded o the height of the fluid above it ad ot o the weight of the fluid. To illustrate this he put a very log small diameter tube i a barrel ad filled the barrel with water. From a high positio he poured water ito the arrow tube. Whe the water i the tube reached a sufficiet height the barrel burst from the pressure. Pascal also formulated Pascal s Law that states Figure 6: Pascal s Barrel Experimet I a fluid at rest i a closed cotaier, a pressure chage i oe part is trasmitted without loss to every portio of the fluid ad to the walls of the cotaier. This law is the basis of the hydraulic lift pictured i Figure 7. Figure 7: Hydraulic Lift Here a smaller weight produces a pressure chage of F 1 =A 1 i the fluid. This produces a force of.f 1 =A 1 /A 2 o the secod pisto. Thus the force is multiplied by A 2 =A 1. 8

10 He also developed a simple syrige to illustrate his law. A pluger i a cylidrical tube was attached to a sphere that had multiple holes. Both the tube ad the sphere were filled with water. This syrige is pictured i Figure 8 below Figure 8: Pascal s Syrige Applyig pressure to the pluger cause water to squirt out of the holes i the attached sphere equally i all directios. This is a illustratio of Pascal s law. Computig I order to help his father with his tax calculatios Pascal desiged ad developed a mechaical calculator called a Pascalie. Pascal ot oly desiged the device, but also supervised its productio. There were about 50 of these devices produced, but oly about 8 10 have survived to this day. A picture of a Pascalie is show i Figure 9. Figure 9: The Pascalie The device was ever a fiacial success. It was cheaper to hire workers to do the computatios maually. A lik to a You Tube video describig the operatio of a Pascalie ca be viewed at Vacuum Experimets Although it seems strage to us today, most scietists at the time of Pascal agreed with Aristotle that Nature abhors a vacuum. I 1644 Evagelista Torricelli performed a experimet that challeged this belief. He took a glass tube that was closed at oe ed ad filled it with mercury. Coverig the ope ed he immersed it i a dish of mercury. Whe the cover was removed the mercury fell util there was a colum about 30 iches high remaiig i the tube. Figure 10 shows this experimet. The questio was what is the empty space above the mercury colum? Torrecelli believed it was a vacuum. His explaatio was that the air exerted a pressure o the mercury i the dish that was 9

11 Figure 10: Torricelli Vacuum Experimet balaced by the weight of the colum of mercury. His explaatio was ot geerally accepted. May scietists thought there was some ivisible substace i this apparetly empty space. Possibly some type of vapor had leaked i. Pascal heard about the experimet ad duplicated it i Pascal duplicated the experimet with water ad with wie. Sice these substaces weighed much less tha mercury he had to costruct a log glass tube about 45 feet high. The followers of Aristotle thought that wie was more vaporous tha water ad therefore the vapor i the void above the colum should push the wie dow farther tha the water. However, wie is lighter tha water ad actually left a smaller void tha water. Pascal performed aother iterestig experimet. The apparatus is show i Figure 11 below. The apparatus was filled with mercury ad iverted as i the other experimets. A hole ear the top was iitially covered. The mercury i the lower straight sectio formed a colum as i the other experimets. However, there was some mercury left i the curved trap. Whe the hole was ucovered the mercury colum wet dow, but the mercury i the trap formed a colum i the upper tube. The mercury i the lower tube wet dow sice there is ow air pressure o both side of the mercury colum so there is othig to balace the weight of the mercury colum. The mercury i the upper colum wet up sice there is ow othig to balace the air pressure util the mercury colum is formed. 10

12 Figure 11: Aother Vacuum Experimet Pascal s most famous vacuum experimet ivolved takig a Torricelli barometer up to the top of a 3000 ft. moutai to see how the height of the mercury colum varied with altitude. Pascal reasoed that the air pressure was due to the weight of the air above us. Therefore, it was reasoable to assume that the air pressure would be less at a high altitude. O September 19, 1648 Pascal had his brother-i-law ad several trusted frieds take measuremets at the bottom of the moutai Puy de Dome ear Clermot ad the take the device to the top recordig measuremets as they wet. Figure 12: Puy de Dome Experimet They foud that the height of the mercury colum was 711 mm at the bottom of the moutai ad 627 mm at the top, a differece of over 3 iches. This was added proof that it was air pressure actig o the surface of the mercury reservoir that was balacig the weight of the mercury colum. 11

13 Thus, Torricelli s barometer could be used to measure air pressure. A umerical value for the air pressure could be obtaied by dividig the weight due to the colum of mercury by the crosssectioal area of the tube. Pascal the multiplied this pressure by the surface area of the earth to obtai a estimate of the total weight of the atmosphere. He was withi 30% of the geerally accepted value today. Pascal Iitiates Omibus System I 1662 Pascal had the idea of creatig a omibus system i Paris. He secured fiacig from some of his frieds ad the kig grated him moopoly status. The system started with seve horse-draw vehicles ruig alog regular routes. Each coach was capable of carryig six to eight passegers. A artist s reditio is show i Figure 13 below. Figure 13: A Pascal Omibus Carriage The omibus system was a big success at first, but soo the ovelty wore off. At this time oly the obility ad getry were allowed to ride the coaches. Soldiers ad peasats were ot allowed. As the aristocrats were ot depedet o the system, the busiess was ot sustaiable. By 1675 the omibus system was out of busiess. This proved to be a idea that was ahead of its time. 12

14 Pascal ad Mathematics I this sectio we will look at several areas i mathematics where Pascal made a importat cotributio Geometry Whe Pascal was 16 years old he wrote a paper etitled Essay pour les coiques. I it he preseted Pascal s Theorem that states that the itersectios of opposite sides of a hexago iscribed i a coic sectio are coliear. This is a fudametal theorem i what is ow called projective geometry. Figure 14: Pascal s Hexago Theorem Probability Theory It is geerally accepted that probability theory had its origi i a series of letters betwee Blaise Pascal ad Pierre de Fermat. The subject of these letters was several gamblig questios posed to Pascal by Atoie Gombaud, chevalier de Méré. The chevalier de Méré had bee wagerig that he could roll a six with a sigle die i four rolls. He had wo for a while, but was ow losig. He switched to a differet game where he bet that he could roll a double six with two dice i 24 rolls. He was losig more tha he was wiig. He asked Pascal if he was just ulucky or was he makig a stupid bet. Pascal showed him that he had a slight advatage i the first game, but was at a slight disadvatage i the secod game. I the first game the probability of ot rollig a six o a sigle roll was 5=6. Therefore, the probability of ot rollig a six i four rolls was.5=6/ 4 D 0:4823. It follows that the probability of rollig at least oe six i four rolls is 1.5=6/ 4 D 0:5177 a slight advatage. Similarly, i the secod game the probability of rollig a double six i 24 rolls is 1.35=36/ 24 D 0:49144 a slight disadvatage. The ext questio posed to Pascal was more challegig ad was the subject of several letters betwee Pascal ad Fermat. This questio ivolved the fair distributio of the pot i a ufiished game. The game ivolved settig a umber of poits for a wi. The players rolled the dice ad the highest roll received a poit. This was repeated util oe of the players had the agreed upo 13

15 umber of poits. However, what should be doe if the game was discotiued before oe of the players wo? How should the pot be divided? A discussio of the various solutios offered by Pascal ad Fermat is cotaied i Appedix B: Aalysis of Ufiished Game Problem. I solvig the combiatorial problems that arise i probability calculatios Pascal made great use of the triagular array of umbers ow kow as Pascal s triagle. A discussio of this triagle ad its properties is cotaied i Appedix A: Pascal s Triagle. The Cycloid A cycloid (i Frech la roulette) is the curve geerated by a poit o a circle as the circle rolls alog a lie (see Figure 15). Figure 15: The Cycloid Curve This curve had bee studied by may of the great mids of the time icludig Galileo, Descartes, ad Roberval. Pascal foud a way to fid the area ad ceter of gravity of ay segmet uder the curve. He also showed how to calculate the volume ad surface area of the solid obtaied by rotatig the cycloid about the lie it was rollig o. This work was a predecessor of the more geeral problem i calculus of fidig the area uder a curve by itegratio. 14

16 Cotributios to literature Pascal was ot oly a outstadig thiker, but he was also a outstadig writer. He wated his ideas to be uderstood by a ordiary perso. I this sectio we describe his two most famous compositios. Lettres Proviciales The Provicial Letters (Les Lettres Proviciales) is a collectio of 18 letters that were writte by Pascal i This collectio is cosidered a classic of Frech Literature. It is said that Voltaire, certaily ot a fried of Christiaity, kept a copy by his bed as a example of what good writig looks like. The situatio that led to the writig of these letters was the codematio of Atoie Arauld, the leader of the Jaseist sect withi Catholicism, by both the Sorboe ad a special committee set up by the Pope. It was held that five of the Jaseist beliefs advocated by Arauld were heretical. The Jaseists followed the teachigs of Augustie that God aloe determied who would be saved ad that God s grace offered to his elect could ot be refused, but would accomplish its desired effect without ay help from the idividual. The two most powerful groups i the Catholic church at this time were the Jesuits ad the Domiicas. The Jesuits believed that God s grace was offered to all, but each idividual had the freedom to accept or reject it. This grace was called God s Sufficiet Grace. The Domiica s accepted the idea of God s sufficiet grace offered to all, but that it required God s Efficacious Grace, offered oly to the elect, to take advatage of it. Pascal used a comedic style ad sarcasm to preset his argumets. His argumets were directed most strogly agaist the Jesuits who were the most outspoke oppoets of Jaseism. The first four letters are primarily a defece of the views of Aruald. Letters 5 10 are directed toward the moral laxity of the Jesuits ad their justificatio of it. The remaiig letters are a direct attack agaist the Jesuits where Pascal abados the comedic style of the earlier letters ad writes with great passio. The first 10 letters are preseted as a coversatio betwee a ordiary residet of Paris ad a fried i the provices cocerig the evets ivolvig the Sorboe ad the religious leaders that are i the ews. The later letters are addressed directly to the Jesuit Fathers. Below is a excerpt from the secod letter where the speaker is addressig a Domiica Father. But, to retur to the poit, father; this grace give to all me is sufficiet, is it ot? Yes, said he. Ad yet it has o effect without efficacious grace? Noe whatever, he replied. Ad all me have the sufficiet, cotiued I, ad all have ot the efficacious? Exactly, said he. That is, retured I, all have eough of grace, ad all have ot.... this grace suffices, though it does ot suffice that is, it is sufficiet i ame, ad isufficiet i effect! I good sooth, father, this is particularly subtle doctrie! Have you forgotte, sice you retired to the cloister, the meaig attached, i the world you have quitted, to the word sufficiet? 15

17 The Provicial Letters were very popular with the geeral public, eve after they were evetually baed by the Pope. This was due i part to Frech atioalism ad the close ties of the Jesuits with the Pope who did t reside i Frace. Pesées At his death Pascal was i the process of writig a defese of Christiaity that was ot completed. However, may of the otes he had prepared have bee recovered. We do ot kow the order he plaed to preset the material ad we do t kow if all of the otes would have made ito the fial publicatio. The otes have bee collected together i a volume called Pesées (thoughts). The material i the Pesées has bee used both by philosophers ad by Christia apologists. Probably the most famous selectio from the Pesées is kow as Pascal s wager. Below is a sample of Pascal s argumet. Let us the examie this poit, ad let us say: Either God is or he is ot. But to which view shall we be iclied? Reaso caot decide this questio. Ifiite chaos separates us. At the far ed of this ifiite distace a coi is beig spu which will come dow heads or tails. How will you wager? Reaso caot make you choose either, reaso caot prove either wrog.... Let us weigh up the gai ad the loss ivolved i callig heads that God exists. Let us assess the two cases: if you wi you wi everythig, if you lose you lose othig. Do ot hesitate the; wager that he does exist. Pascal s argumet was ot desiged for those seriously seekig to kow God, but was desiged for the secular populatio who avoided thikig about God ad other ultimate topics through diversios such as gamblig. It s goal was simply to help them see that, eve from a secular poit of view, belief i God was ot ureasoable. It was ot a method for evagelism, but could be cosidered as pre-evagelism. Pascal wated to get them thikig about God s existece. 16

18 The Faith of Blaise Pascal Blaise Pascal was a committed Christia who took his faith seriously. As metioed earlier he beloged to the Jaseist sect of the Catholic Church ad his beliefs were greatly iflueced by his associatio with this group. Let s look at some of the Jaseist beliefs. As far as salvatio was cocered the Jaseists followed a very literal iterpretatio of Augustie ad thus had a lot i commo with the Calviists amog the protestat reformers. For example, they believed i the followig 1. all share i Adam s origial si 2. ma is totally depraved ad is icapable of seekig or desirig God o his ow 3. salvatio was oly possible for the elect chose by God 4. Christ s grace offered to the elect could ot be resisted 5. Christ s atoig death oly applied to the elect Here is a quote by Pascal o the above beliefs... me are saved or damed accordig as to whether it has pleased God to choose them as recipiets of this [efficacious] grace from out of the corrupt mass of me, i which He could with justice abado them all. The Jaseists differed from Calviists i that they believed that the salvatio of the elect was ot secure ad could be lost. They also accepted most of the other beliefs of the Catholic Church ot related to salvatio. Beliefs such as apostolic successio, saithood, veeratio of Mary, ad the ature of the sacramet of commuio. Ulike the Calviists, the Jaseists taught a form of piety that ivolved withdrawal from the world ad its diversios. This was the oe belief of the Jaseists that Pascal had great difficulty with. He could t see why it was ecessary to abado mathematics ad sciece whe his abilities i these fields were a gift from God. The fact that he remaied active i these fields caused may Jaseists to questio his commitmet. Throughout most of his life Blaise approached religio the same way he approached everythig else through reaso. O the ight of November 23, 1654 this all chaged. Blaise experieced a mystical ecouter with God kow as his ight of fire. He did t tell ayoe what happeed that ight, but he wrote dow a accout o a piece of paper ad sewed it ito the liig of his jacket. The paper was foud after his death as they were goig through his clothes. The details of what happeed leadig up to this ecouter are ot kow, but here is what he wrote. 17

19 The year of grace 1654, Moday, 23 November, feast of St. Clemet, pope ad martyr, ad others i the martyrology. Vigil of St. Chrysogous, martyr, ad others. From about half past te at ight util about half past midight, FIRE. GOD of Abraham, GOD of Isaac, GOD of Jacob ot of the philosophers ad of the leared. Certitude. Certitude. Feelig. Joy. Peace. GOD of Jesus Christ. My God ad your God. Your GOD will be my God. Forgetfuless of the world ad of everythig, except GOD. He is oly foud by the ways taught i the Gospel. Gradeur of the huma soul. Righteous Father, the world has ot kow you, but I have kow you. Joy, joy, joy, tears of joy. I have departed from him: They have forsake me, the fout of livig water. My God, will you leave me? Let me ot be separated from him forever. This is eteral life, that they kow you, the oe true God, ad the oe that you set, Jesus Christ. Jesus Christ. Jesus Christ. I left him; I fled him, reouced, crucified. Let me ever be separated from him. He is oly kept securely by the ways taught i the Gospel: Reuciatio, total ad sweet. Complete submissio to Jesus Christ ad to my director. Eterally i joy for a day s exercise o the earth. May I ot forget your words. Ame. O this ight God touched Blaise s heart ad he was ot the same afterwards. His frieds ad family could see that he had chaged, but they did t kow the cause of this chage. Blaise committed himself all the more to the Jaseist cause, but he ever completely gave up his o- Christia frieds ad his ivolvemet i mathematics ad sciece. Oe of the evideces of his chaged life was a icreased compassio for the poor. Eve though his physical health was decliig, he ofte got out of bed ad walked amog the poorest of the poor i Paris ad gave them what moey he had. His sister Gilberte told the followig story: 18

20 Oe day whe he was comig home from Mass he ecoutered a fiftee year-old girl beggig for moey. She told him that her father had died ad her mother was ear death. Blaise took her to the rectory ad gave the priest moey to take care of her. He begged the priest to fid a safe place for the girl to live. I a few days he set a woma servat to check o the girl ad to provide more moey. Blaise ever told the priest or the girl his ame, so the girl ever kew who it was that had bee so kid to her. Towards the ed of his life Blaise was i the process of writig a defese of Christiaity. This was ever fiished, but his otes were collected ad published as the Pesées. His two mai purposes i these writigs was to show the wretchedess of ma without God ad the happiess of the life with God. There is more about the Pesées i the sectio Cotributios to Literature. The Pesées are ofte quoted by Christias of all deomiatios as well as by philosophers. Throughout his life Blaise Pascal loved mathematics ad sciece, but he grew to love God eve more. 19

21 Refereces 1. Kreeft, Peter, Christiaity for Moder Pagas: Pascal s Pesées edited, outlied, ad explaied, Igatius Press (1993) 2. Coer, James A., Pascal s Wager: The ma who played dice with God, Harper Oe (2009) 3. Hammod, Nicholas, The Cambridge Compaio to Pascal, Cambridge Uiversity Press (2003) 4. Pascal, Blaise, The Provicial Letters of Blaise Pascal, HardPress (2018) 5. Devli, Keith, The Ufiished Game, Basic Books (2008) 6. Bishop, Morris, Pascal: The Life of Geius, Greewood Press (1968) 7. Wikipedia, 20

22 Appedix A: Pascal s Triagle Most studets of mathematics are familiar with a triagular array of umbers called Pascal s triagle. The form usually foud i textbooks is show i Figure 16. Figure 16: Pascal s Triagle Pascal wrote a paper dealig with this triagle ad its properties etitled Traité Du Triagle Arithmétique. I this paper I will use a differet arragemet of this triagle as show i Figure 17. Figure 17: Pascal s Triagle (alterate forrm) This form leds itself better to row ad colum otatio. We will umber the rows ad colums startig with 0. Let.; r/ deote the elemet i the -the row ad the r-the colum. The umbers iterior to the triagle satisfy the relatio.; r/ D. 1; r/ C. 1; r 1/; (1) i.e, each iterior elemet is obtaied by addig the elemet oe row above it i the same colum to the elemet that is oe row above ad oe colum to the left. I additio,.; 0/ D.; / D 1 for all ad.; r/ D 0 for r >. We will show that the elemets of this triagle are equal to the biomial coefficiets. The biomial coefficiets are the coefficiets r i the expasio 21

23 .1 C x/ D X rd0! r x r : (2) Clearly.1 C x/ D factors ƒ.1 C x/.1 C x/.1 C x/ The term x r is obtaied by takig x from r of the factors ad 1 from the remaiig r factors. Therefore, the coefficiet r must be the umber of differet ways of choosig r x s from the factors. Sice there is othig special about the x s, r must be equal to the umber of combiatios of objects take r at a time. I order to show that.; r/ D r we must show that the biomial coefficiets satisfy!!! D 1 C 1 r r r 1 (3) with 0 D D 1 for all ad r D 0 for r >. This is the same relatio that the elemets.; r/ satisfy [see Equatio 1]. To derive this recursio relatio recall that r was the umber of combiatios of objects take r at a time. If we wat to fid the umber of groups of r objects take from a group of objects, we ca break the process ito two parts. We divide the objects ito a group A of 1 objects ad a group B with oe object. If we fid all the groups of r objects take from group A, these groups of r objects are part of the oes cosidered i r. There are 1 r of these groups. What we are missig are those cotaiig the object i group B. Each of those ivolves r 1 objects from group A. Clearly there are 1 r 1 of these. This argumet establishes the above recursio relatio. I additio, it ca be see from the biomial expasio that 0 D D 1 for all. Moreover, r D 0 for r > sice there are o powers of x larger tha. Sice.; r/ ad r satisfy the same geeratig equatios, we must have.; r/ D r. Pascal used the biomial coefficiets ad the associated Pascal triagle to solve may of the combiatorial problems ivolved i probability calculatios. The biomial coefficiets have a umber of other properties. For example, if we let x D 1 i Equatio 2, the we get X rd0! D 2 ; (4) r i.e, the sum of the elemets i the -the row of Pascal s triagle is 2. There is also the followig 22

24 direct expressio for calculatig the biomial coefficiets! D r. 1/. r C 1/ : (5) rš To see this we will start with ordered arragemets (permutatios) ad the proceed to uordered combiatios. To fid all the ordered arragemets of elemets, we have choices for the first etry, 1 choices for the secod, dow to 1 choice for the -the etry. Thus, there are. 1/ 1 D Š ordered arragemets of elemets. Similarly, there are. 1/. r C1/ ordered arragemets of legth r take from elemets. However, i the biomial coefficiets we do t wat ordered arragemets, but uordered sets. For each uordered collectio of r elemets take from elemets there are rš arragemets of these r elemets. Thus, the umber of uordered sets of r elemets take from elemets ca be obtaied by dividig the umber of ordered arragemets of r elemets take from elemets by rš. This completes the proof of Equatio 5. From Equatio 5 we ca easily derive several other recursio relatios, e.g.,.r C 1/! D. r C 1 r/ r! (6) ad r r!! D 1 : (7) r 1 Pascal derived these relatios alog with a umber of others i his paper. 23

25 Appedix B: Aalysis of Ufiished Game Problem The ufiished game problem (sometimes called the problem of poits) grew out of a gamblig questio posed to Blaise Pascal. It ivolved a game betwee two players havig the followig rules 1. Both players put the same amout ito a pot that will go to the wier. 2. The players agree o the umber of poits that will wi the game. 3. To wi a poit each player roles the dice ad the oe with the higher total wis the poit. I case of a tie the players roll agai. 4. This cotiues util oe of the players reaches the agreed upo umber of poits. This player the wis the pot. The followig questio was preseted to Blaise Pascal: Suppose the game was iterrupted before either player reached the poit goal. How should the pot be fairly divided amog the two players? Pascal discussed this problem with aother mathematicia Pierre de Fermat i a series of letters ad i the process formed the begiigs of probability theory. The two came up with differet approaches to the questio that led to the same aswer. They both agreed that the past history was ot importat. All that mattered was the umber of poits each eeded to wi. Suppose that the first player eeded m poits to wi ad the secod player eeded poits to wi. Both Pascal ad Fermat agreed that oe of the players would be guarateed to wi i C m 1 additioal rouds. To see this we must fid the path to a wi ivolvig the most rouds. The logest path must evetually lead to the situatio where both players eed just 1 poit to wi. For oe player must evetually reach the poit where he eeds 1 poit to wi. If the other player is ot there yet, the there ca be more rouds util that player eeds 1 poit to wi. It requires m-1 wis from oe player ad -1 wis from the other player or m C 2 rouds to reach the place where both players eed 1 poit to wi. It follows that oe of the players will always wi i m C 1 rouds. Fermat s Approach Fermat s approach was to eumerate the various possibilities for C m 1 additioal rouds ad the determie the umber that represeted a wi for each player. Cosider the case m D 2 ad D 3. The oe of them will surely wi i 2 C 3 1 D 4 rouds. If F represets a roud wi for the first player ad S represets a roud wi for the secod player, the the 16 possibilities for the four rouds are FFFF FFFS FFSF FSFF SFFF FFSS FSFS SFFS FSSF SFSF SSFF FSSS SFSS SSFS SSSF SSSS 24

26 The first player wis the pot i 11 of the 16 cases ad the secod player wis the pot i 5 of the 16 cases. Thus, it seems fair to give the first player 11=16 of the pot ad the secod player 5=16 of the pot. Of course the eumeratio method of Fermat could be very tedious if m ad are large. Some objected to extedig the game to C m 1 additioal rouds whe oe of the players may wi before that. Eve though that is true, the full extesio is coveiet i order to guaratee that each of the outcomes has the same probability. I additio it was argued that the extesio would ot chage the probability of either player wiig. This ca be see for the case we just cosidered i the table below. This table lists the possible ways the first player could wi alog with the probability of each outcome (the probability of wiig or losig a roud is 1 ). 2 Results Probability FF 1=4 FSF 1=8 SFF 1=8 FSSF 1=16 SFSF 1=16 SSFF 1=16 The sum of these probabilities is 11=16 which is what we obtaied before. Later we will see a differet approach by Pascal usig recursio i which the logic behid the coclusio is clearer. Pascal s Refiemet of Fermat s Approach Pascal came up with a alterate way to look at the eumeratio method of Fermat usig biomial coefficiets ad a triagle of umbers usually called Pascal s triagle (see appedix A). The umber of rouds that cotai two F s is give by 4 2. To see this we ca describe each pair of F s by describig their positio i a set of four letters. Let f1; 3g represets a F i the first ad i the third positio. We do ot distiguish betwee f1; 3g ad f3; 1g. Clearly, the umber of all such distict positios is the umber of combiatios 4 of the umbers 1, 2, 3, ad 4 take two at a time without respect to order or 2. Similarly, 4 3 is the umber of rouds cotaiig three F s. It follows that the umber of rouds cotaiig at least two F s is give by 4 2 C 4 3 C 4 4 D 6 C 4 C 1 D 11. This is the umber we obtaied previously by eumeratio ad coutig. Pascal s Recursio Method Pascal came up with aother method usig recursio. Let e.m; / represet the share due the first player whe he has m poits to a victory ad the other player has poits to a victory. Cosider oe more hypothetical roud. After that roud the share of the first player is either e.m 1; / or e.m; 1/. Sice both outcomes are equally likely, it seems fair to set e.m; / D 1 e.m 1; / C 1 e.m; 1/: 2 2 Cosider the case where m D 2 ad D 3. We the have 25

27 e.2; 3/ D 1 e.1; 3/ C 1 e.2; 2/ 2 2 e.1; 3/ D 1 e.0; 3/ C 1 e.1; 2/ 2 2 e.1; 2/ D 1 e.0; 2/ C 1 e.1; 1/ 2 2 We start with the last equatio ad work our way back up to the first equatio. Sice e.0; 2/ D 1 ad e.1; 1/ D 1=2, we have Sice e.0; 3/ D 1, it follows that e.1; 2/ D 1=2 C 1=4 D 3=4: Sice e.2; 2/ D 1=2, it ow follows that e.1; 3/ D 1=2 C 3=8 D 7=8: e.2; 3/ D 7=16 C 1=4 D 11=16: This is the same aswer that Fermat obtaied. The quatity e used by Pascal is what we would ow call expected value. This was a ew ad importat cocept itroduced by Pascal. 26