Pascal vs. Paulos, Round 2: Paulos Ups the Ante

Previous Entries:

Pascal vs. Paulos, Setting the Stage

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

Here is a brief summary of John Allen Paulos’ critique of Pascal’s wager in his book Irreligion: A Mathematician Explains why the Arguments for God Just Don’s Add Up. I believe there are very sound responses to each of his points mentioned below and I plan on discussing them in the next post. For now, you can read the previous entry in which I summarized Pascal’s Wager, read what Paulos has to say, and think for yourself how valid his reasoning is and how you might respond.

If you pay attention you will notice how the presuppositions Paulos brings to this matter affect the way he reasons and the way he discusses mathematics. I think this book is a clear demonstration that religious beliefs affect the discipline of mathematics, be they Christian, atheist, or something else entirely. So the real question then, is why should an atheistic or naturalistic worldview be more appropriate for mathematics than a Christian one? But I digress, a topic for another time. Back to the book…

Paulos’ Summary of the Pascal Wager Argument for God (133):

  1. We can choose to believe God exists, or we can choose not to so believe.
  2. If we reject God and act accordingly, we risk everlasting agony and torment if He does exist but enjoy fleeting earthly delights if He doesn’t.
  3. If we accept God and act accordingly, we risk little if He doesn’t exist but enjoy endless heavenly bliss if He does.
  4. It’s in our self-interest to accept God’s existence.
  5. Therefore God exists.

The first problem that Paulos finds with the argument is that:

The argument itself has little to do with Christianity and could just as readily be used by practitioners of Islam and other religions to rationalize other already existing beliefs (134).

In summarizing the mathematics behind expected values, Paulos’ first critique becomes even more apparent by his own parenthetical comments:

If we multiply whatever huge numerical payoff we put on endless heavenly bliss by even a tiny probability, we obtain a product that trumps all other factors, and gambling prudence dictates that we should believe (or at least try hard to do so) (135).

In essence, the first problem with Pascal’s Wager is that it is not faith specific and it very vague in its description of how one actually bets on God’s existence.

As an aside Paulos even brings into doubt the ability to assign a probability to God’s existence. He notes that the statement “the probability of a God” is unlike “the probability of a royal flush.” We can calculate the number of poker hands and royal flushes that are possible and determine that all hands are equally likely but, unlike a deck of cards, the universe is unique. Since we cannot calculate the number of universes there are and how many of them have a God and how equally likely they are, the statement “the probability of a God” is nonsensical.

The second problem that Paulos finds with the argument is that while it can be used to argue for a rational belief in God, by assigning such large values to the payout of God’s existence the argument can also be used to rationalize horrible actions.

Killing thousands or even millions of people might be justified in some devout believers’ eyes if in doing so they violate only mundane human laws and incur only mundane human penalties while upholding higher divine laws and earning higher divine approbation (135).

Paulos uses this notion to relate Psacal’s Wager to an argument from fear, summarized as follows (137):

  1. If God doesn’t exist, we and our loved ones are going to die.
  2. This is sad, dreadful, frightening.
  3. Therefore God exists.

This prompts a sound lampooning of religious and political leaders who lead by fear-mongering. His critique then moves into a discussion of ethics, largely focused on demonstrating that atheists and agnostics are at least if not more moral that devout religious believers.

An atheist or agnostic who acts morally simply because it is the right thing to do is, in a sense, more moral than someone is trying to avoid everlasting torment or, as is the case with martyrs, to achieve eternal bliss. He or she is making the moral choice without benefit of Pascal’s divine bribe (140).

Extrinsic rewards undercut intrinsic interest and this is a reason not to base ethics on religious teaching (141).

Tune in next time for a response to Paulos’ critique.

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.

Recommended Reading:

You can read Pensees for free here.

Tim Rogalsky, “Blaise Pascal – Mathematician, Mystic, Disciple”

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.