Download Blaise Pascal: Proving God?? A Mathematical Interpretation of

Blaise Pascal: Proving God??
A Mathematical Interpretation of Pascal's Wager
Jamie Mosley
12/21/2010
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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.”
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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.”
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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.”
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Three Aspects of a Mathematical Understanding

Pascal‟s Geometric Influence

Pascal‟s Probability Theory

Pascal‟s Famous Wager
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Pascal’s Geometric Influence



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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.
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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.
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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.

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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.”
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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.

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.
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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.
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
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.”
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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.”
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
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.
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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.
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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.
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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.
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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.
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
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.
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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.
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
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.
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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.”
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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.
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
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.”
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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)
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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)
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“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)
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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.
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
“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)
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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)
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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.
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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.
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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
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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)
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



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.”
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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.”
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Concluding Remarks

Pascal‟s Wager is mathematically
oriented

A Response to the Critic of Pascal

What does this mean for us?
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