12/21/2010 1

Blaise Pascal: Proving God?? A Mathematical Interpretation of Pascal's Wager

Jamie Mosley

12/21/2010 2

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 3

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 4

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 5

Three Aspects of a Mathematical Understanding

Pascal‟s Geometric Influence

Pascal‟s Probability Theory

Pascal‟s Famous Wager

12/21/2010 6

Pascal’s Geometric Influence

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.

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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 8

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 9

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 10

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 11

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.”

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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.

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

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 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 13

The Theory Applied to MathematicsA 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.

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.”

12/21/2010 14

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.”

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.

12/21/2010 15

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 16

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 17

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 18

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.

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.

12/21/2010 19

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.

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.

12/21/2010 20

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 21

Pascal’s Famous Wager

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.”

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 22

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 23

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)

“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)

12/21/2010 24

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.

“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)

12/21/2010 25

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 26

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 27

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.

Case f1 f2 f3

1 F F F d2

2 F F T d2

3 F T F --

4 F T T --

5 T F F d2

6 T F T d2

7 T T F d2

8 T T T d1

d1 is true only in case 8

12/21/2010 28

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)

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.”

12/21/2010 29

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 30

Concluding Remarks

Pascal‟s Wager is mathematically

oriented

A Response to the Critic of Pascal

What does this mean for us?