Pascal vs. Paulos, Round 1: Pascal’s Wager

Previous Entry: Pascal vs. Paulos, Setting the Stage

Like all good mathematicians of the 17th century, Blaise Pascal’s life was characterized by three primary activities: gambling, drinking, and womanizing. After coming to Christ in a dramatic way, Pascal set out to persuade his contemporaries (who were caught up in a similar lifestyle) to follow him in Christian belief. So Pascal’s wager argument is set in the context of being presented to an audience that is familiar with gambling theory.

It should be noted that Pascal’s coming to faith was not based upon this argument. In fact, Pascal did not believe any argument was sufficient for faith. In the Pensees (which is a collection of his thoughts), when speaking of the relationship between reason and faith, he concluded by stating:

That is why those to whom God has given religious faith by moving their hearts
are very fortunate, and feel quite legitimately convinced, but to those who do not have it we can only give such faith through reasoning, until God gives it by moving their heart, without which faith is only human and useless for salvation. (Thought 282)

Now, you may be asking, “how does gambling theory apply to religious belief?” An excellent question, thank you for asking. Pascal offers two keen insights into the nature of life. First, he notes that life itself is a risk. Everyday you have to make decisions have before you have all the evidence on a given topic. This is particularly true in regards to religious belief. A decision for God or against God must be made during this life, prior to having indisputable evidence for His existence.

The second insight that Pascal offers is that we always develop strategies when making decisions that serve to maximize our potential for gain, and minimize our potential for loss. So then, based on these two insights, applying an argument based around gambling theory seems like a sound approach to the question of religious belief.

Pascal’s wager argument is based on calculating expected values. Suppose you find yourself at a local fair or carnival and there is booth that says “Dice Game.” To play this game you must pay \$3 (a relative bargain for local fair prices). Once you have paid you get one roll of a standard six-sided die. If you roll any number between 1 and 5 then you lose (sorry), but if you roll a 6 then you win \$15 (a profit of \$12)!

Expect Value (EV) of “Dice Game” = (Event of Winning \$12) x (Probability of Rolling 6) – (\$3 Cost of Playing) x (Probability of Losing)

EV(Dice Game) = (\$12) x (1/6) – (\$3) x (5/6) = -1/2 (or -\$0.50)

This means that if you played this game over and over for an extended period of time, you would average losing \$0.50 per game. Though this example is simplified, this is essentially why casinos are profitable businesses.

So then, in general the formula for an expected value can be given as follows: the expected value of an event E equals the probability of winning multiplied by the payout amount, minus the cost. In other words:

EV (E) = Pr(Winning) x Payout – Cost

Pascal then poses the question, “supposing the evidence for God’s existence is inconclusive, then is it rational to remain in unbelief?” What if you aren’t sure if God exists or if He doesn’t? In that case, suppose the probability of God’s existence is 50/50 (any numbers result in the same outcome, even if it is 99/1, but we’ll discuss this further in another post).

Pascal then supposes to paint belief in God in the worst possible light, to show that his argument still demonstrates that one should believe in God over atheism. So then, suppose the cost of believing in God is significant, such as the hardships you would endure in life and supposed freedoms you would have to forgo for your faith. Even if the cost is significant, ultimately if God exists, then the cost is only finite and temporary; whereas the payout for betting on God if He exists is infinite and eternal.

For atheism, the cost might be zero (in reality it probably isn’t, but again, just to paint the argument most favorably toward atheism, we’ll assume this is the case). The payout for betting on atheism could be significant if God doesn’t exist, in terms of the freedoms you could indulge in during life. But even if it is significant, if God doesn’t exist then this life is all you have, so the payout no matter how large is always finite. So then returning to the formula for expected value we have:

EV (Theism) = 0.5 x Infinite Payout – Significant but Finite Cost

EV (Atheism) = 0.5 x Significant but Finite Payout – 0 Cost

The result then is that the expected value of theism is infinite. Any number times infinity is infinity, and subtracting any finite number from that still yields infinity (the concept of infinity will be taken up in future posts).

The expected value of atheism could be significant, but is ultimately a finite number. You get this same result no matter the probabilities you insert into the equation. Even if the probability against God is 99/1, the result still holds. The expected value of theism is always greater.

Therefore you should bet on God. What exactly that means is a topic for another day.

Pascal vs. Paulos, Setting the Stage

I recently picked up the book Irreligion: A Mathematician Explains Why the Arguments for God just don’t add up, by John Allen Paulos. Here is basic summary of the work from inside the front cover:

Are there any logical reasons to believe in God? The mathematician and bestselling author John Allen Paulos thinks not. In Irreligion he presents the case for his own irreligious worldview, organizing his book into a series of chapters that refute the twelve arguments most often put forward for believing in God’s existence.

Now, I’ve always heard that you should never judge a book by its cover. I’m not sure that maxim takes into consideration the inside flaps of the cover, because those always seem to sum up the book pretty well. In fact, it seems like you would be an idiot not to judge a book by its cover. Perhaps the saying predates cover flaps. But anyway, for the sake of argument let us assume that the saying still holds true. So in order to give Paulos his due, I thought I would actually read through the book. Here is what I found: this book represents a perfect example of how our value system impacts the way in which we do mathematics (be those Christian values or not).

I thought it would be instructive to examine some of the arguments for God that Paulos interacts with. At some point in the future I might try to post a comprehensive review of the book as a whole, but for now I simply want to focus on some of the better arguments, starting with possibly the best argument (although the ranking of these arguments is in itself another argument): Pascal’s Wager.

We’ll look at this in three rounds:

• Round 1: An explanation of Pascal’s Wager
• Round 2: A summary of Paulos’ critique of Pascal’s Wager
• Round 3: Critique of the critique (A response to Paulos)

Should be fun.