By: Aaron Boyden
To: Nicole Chardenet
Re: Atheists & Fridge El

 > I'm not familiar with Pascal's Wager.  What was it?

On the off chance you wanted a ludicrously complete answer, I can provide the
following.  The criticism of the wager bogs down in mathematics for a while,
but it ends with some non-mathematical stuff, so don't stop reading when the
math starts flying.

Pascal's wager says that if you believe and you're right, you get heaven,
whereas if you believe and you're wrong you get a minor inconvenience.  On the
other hand, if you don't believe and you're right you get a minor reward, and
if you don't believe and you're wrong you get hell.  Thus, Pascal suggests that
the only sane course is to bet on God.

Let's make the box of expected probabilities for Pascal's wager:

               God exists   |   God does not exist
               ----------   |   ------------------
 Believe           A        |           B
 -------------------------------------------------
 Don't believe     C        |           D

Now, what numbers should we put in for A, B, C, and D? Let's grant Pascal that
the reward should God exist and should we believe is infinite.  Still, what is
the probability of the Catholic God (the God Pascal wanted you to apply this
wager to), existing?  There are infinitely many possible Gods.  Thus, if the
possibility of God existing is greater than 0 (it may not be), it is likely not
greater than 1/infinity.  This is important because in a calculation of
expected utility, you must multiply the probability of an outcome by the value
of that outcome. Infinity times 1/infinity, or times 0, is undefined. Already
we have a problem, because regardless of what B is, adding it to an undefined
number will produce another undefined number, so we don't actually have a clue
how good the expected results of believing are.

Still, let's grant for the moment Pascal's claim that A is infinite.  Then we
have to look at B.  Suppose there's some God other than the Catholic God who
eternally punishes all and only Catholics?  Surely that's also a non-zero
probability, and that's one of the possibilities that must be added into B.  If
we grant Pascal's questionable math, it adds negative infinity to the value of
the outcome B. Even if there's some way to smuggle a positive infinity into B,
adding a positive and a negative infinity doesn't let you cancel, it just
produces an undefined number.  Thus, again, the expected utility of believing,
being as it is A+B, turns out to be undefined, so we have no clue whether
believing is a good idea or not.

Things are pretty bad for Pascal by this point.  Two problems have arisen which
can each independently be seen as decisively refuting Pascal's theory.  Yet,
that isn't even the end.  Parallel arguments can establish independently of C
and of D that each of them is undefined.  Now, Pascal's argument says that
because (A+B)>(C+D), we should believe. In fact, we have conclusive reason to
believe that it's not true that (A+B)>(C+D), because if any of A, B, C, or D
are undefined, then the inequality is neither true nor false, and we have
arguments for each of A, B, C, and D that they are undefined.

Having conclusively determined that Pascal's wager fails, it's probably not all
that worthwhile to consider what it would have advised had it succeeded, but
I'll mention that briefly anyway.  People cannot simply command their own
belief, they can only command their verbal assent.  Surely an omnipotent God
could see through such a ruse; in order to "win" on Pascal's wager, you must
find some way to genuinely believe, but no means of achieving this is apparent.

Still, imagine that one could command one's own believe. Even then, we must
consider some of the other alleged characteristics of the Catholic God.  Would
He really embrace someone who decided to believe in Him for such cynical
reasons?  This hardly seems compatible with the Christian renunciation of
self-interest.  Thus, even if Pascal's wager were true, it would be of no
benefit to be convinced by it.

It has been said of the argument against reason in the beginning of part IV
book I of Hume's Treatise that it is one of the worst arguments ever
constructed by a man of genius.  With competition like this, I'm inclined to
question that commentator's judgment.  Surely Pascal's argument is quantum
levels of badness beyond any mistake Hume, or even Kant, ever made.