Monday, 4 August 2008

Regarding that damn wager

After dismissing Pascal's Wager as a meek argument and not in itself a reason to believe, it was pointed out to me I hadn't applied the wager properly and thus missed the point on why it's used not only by the ignorant but by intellectuals discussing the rationality of belief. While it pained me to see Sir Robert Winston (who I have great regard for) use it in a debate, it's only now dawned on me just why it's such an important weapon in a believer's arsenal. When wielded properly, it's a formidable nut to crack. A proper refutation is in order to show just why taking a position of non-belief isn't an irrational thought.

The atheist gamble
To understand Pascal's wager properly, it's needed to address the claim of uncertainty in relation to statistical probability. It's in effect trying to work the odds of belief based on the consequences of the possible outcomes. And the possible results are thus:

x = RN - C
N is the likelihood of God existing (between 0 and 1)
C is the cost of belief
R is the reward
x is the overall cost of the wager

Belief in God:
If there is a God, then infite reward (x = ∞ - C, x = ∞ since ∞ - C = ∞)
If there isn't a God, then nothing (x = -C)
No belief in God:
If there is a God, then infinite punishment (x = -∞)
If there isn't a God, then nothing (x = 0)

Given that R is either infinity or minus infinity if God is real, it completely negates the finite cost of belief and the low change of God existing. While if there is no God, then there is no reward and the cost while being detrimental is still finite.

From this, it seems only logical to believe in God, right? We atheists have been beaten by mathematics... unless the gambit is missing something. To see what it's missing, we need to look at the assumptions.
1. It assumes that there is always uncertainty
2. It assumes the choice is only between God and nothing
3. It assumes belief is a choice
4. It assumes a finite cost of belief

The first assumption is correct, there is no way of absolutely knowing anything... Maybe, I'm not absolutely sure, but it's as correct as far as I can tell. The problem in this assumption lies in the nature of the supernatural. By definition the supernatural is beyond our knowledge, so any comment on the nature of the supernatural can be nothing but speculation. How can we say God exists, or even attribute features to God if he is beyond the realms of empirical knowledge with any certainty whatsoever? This is important to understand because it leads on to the second assumption.

Now the choice is never between God and "Not God", there are plenty of other supernatural entities out there, most of which are very different in nature to the Christian God Pascal was referring to. And what is to say that any of them are right at all? It's the supernatural, it could be of any nature at all, any number of options for attribute. An update on the formula.

x = RN/T - C
T is the total number of possible outcomes.

So if it were a choice between God and Brahman, T would equal 2. If Allah and Odin are added, then T = 4. Add The Rainbow Serpent, Zeus, Ra and The FSM, then T is 8.

And if we continue adding all the world's deities that ever entered the thoughts of humans, T grows to infinity as well. It's important to remember that infinity and 0 are conceptual numbers so doing rational calculations do not apply. What is ∞/∞? Is an infinite reward comparable with the infinite improbability of picking the right means to the reward? This alone should send out massive warning bells about using such a conceptual entity to show the logic of a belief. Is the reward for the belief itself even infinite? Certainly it promises eternal life, but even then it's not as rewarding as a belief that will allow the believer to become master of the omniverse and have more power than god does. When nothing is absolutely certain, dealing in absolutes transcends mathematics and thus the prospect of showing mathematically the rationale for belief is useless.

The cost of belief
Assumption 3 is that belief is a choice, which is arguable. Personally I don't feel we have much choice in what we believe, it's based on what we are taught, what we rationalise in our brains, and our consciousness is nothing more than justification of an autonomic process. I'm by no means certain of this position, the brain is truly a mystical unit which we know very little about the processes that go on. In any case, belief is certainly not a switch that can be thrown willingly. I like to think of the brain as INPUT -> PROCESS -> OUTPUT, much in the same way the computer works. The output in this case is the belief and the input is the rationale for that belief. The process is our brain, a black box system which we know very little about just what goes on in there.

To transpose Pascal's Wager into finite terms, think of each possible outcome as horses in a race. If there was application of Pascal's Wager, the correct horse to bet on would be the 100/1 gamble, while the 2/1on favourite is a worse choice because the payout in the event of that horse winning is less. So why would anyone bet on the 2/1on horse when there is uncertainty in the result, it could do a Bradbury but it's not a safe bet by any means. The odds are there for a reason, and while there are uncertainties in a race, we use evidence to work out those odds. There's the track record for each horse, the distances they are best at running on, the starting lane, the type of surface, the weight of the jockey, the frequency of racing, and so on. Working purely on the reward of the potential gamble is not a logical measure when all evidence points to the contrary. So when there is no evidence for God's existence, how is someone meant to believe in God purely out of the potential for reward? In that way, belief cannot be a choice.

Finally there is the cost of belief, and it's quite correct to assume that the cost is finite. But being finite is only negligible when dealing in the infinite, which as demonstrated above is a useless concept. As far as certainty goes, as certain as we can be, we are here now and whether or not there is life after physical death, beliefs affect the worldview in this reality. It alters thought processes, it changes how an individual views the world. Beliefs are filters that are placed over senses that change the internal interpretation. As there are limitations on the human brain, for some there is a want to understand the natural world as accurately as possible. Having beliefs that go beyond an empirical understanding can cloud the ability to see reality for what it is. So to an empirical rationalist, the cost of belief rises with the amount of uncertainty that is associated with it. It's still finite, but by this stage it hardly matters. Without the ability to evaluate the only reality we know, the cost of belief is too great to simply just believe.

The costs are not always negative to everyone, for some there are positives in having a belief. For some, it may give meaning, give hope, give them a reason to help out in the community, to be accepting of others. Maybe some people need that reward / punishment incentive system to be a good person. Of course not everyone needs the fear of hell or the reward of heaven to do good and be responsible members of a community, and part of the indoctrination that happens all too often in churches is pushing the lie that faith and works are conjoined twins. But that's a whole other discussion. All that needs to be recognised here is that beliefs can have positives as well as negatives, depending on the person and type of behaviour.

Empirical rationalism
Not all atheists are rationalists, and there are probably as many reasons for non-belief as there are atheists to begin with. Those conclusions may be just as rational or irrational as reasons for belief, or may simply be an extension of the default setting (we are all born non-believers, some of us never change). Whether it's an implicit or explicit rejection of the supernatural, placing the wager on that belief is only going to appeal to those who use rational arguments for not believing. There is always uncertainty, to be certain of anything to an absolute is an absurd notion. With this I agree with Pascal. But without calling to empiricism, philosophy is blind. Empiricism is a constraint on reality, a context for philosophy to operate. Our desire "to know" while being uncertain can operate quite effectively in degrees of certainty. As far as it's possible to tell, gravity exists. Yet, while we are uncertain that gravity is absolutely true, only a fool would think they can defy gravity and leap off a tall building.

When dealing with the supernatural, it's unknown by definition. It's not possible to test something empirically that is not part of the natural world, but by moving a concept beyond the natural world and out of the bounds of empiricism it becomes nothing more than speculation to give attributes to it. For an empirical rationalist, the concept of god as a supernatural entity is entirely useless, it's beyond knowing. So any attribution of parameters for that unknown entity is going to be shot down without a valid reason. The implication of Pascal's Wager is that every single human makes the bet whether they intend to or not. Yet there is not a single objective answer to the wager given it's assumptions, so as an argument for a rational reason to believe, it fails. The call to empiricism is a flicker of light in the darkness of uncertainty. While it does not ever give absolute certainty, the degree of certainty in knowledge makes empiricism a powerful tool. That's why to a empirical rationalist pure reason alone is evident of nothing. And that's why Pascal's Wager does not show that belief in God is a rational position to take, let alone the only rational position to take.


Aaron Alderman said...

And if we continue adding all the world's deities that ever entered the thoughts of humans, T grows to infinity as well.
I would be inclined to say that T could not be infinite because all human made concepts can (in theory) be countable whereas the tangible benefits of an eternal life are non countable. (Forgetting that Christian heaven sounds boring)

One could mathematically show that everyone is screwed regardless if there were three or more beliefs which offered hell for non-believers. These three negative infinities will always be greater than the one positive infinity.

Thus Pascals wager is a reductio ad absurdum and fallacious to use as an argument for any religion.

Kel said...

You're right, in theory humans are limited. Though the supernatural still would be an infinite set because there are elements surely beyond human comprehension.

"Thus Pascals wager is a reductio ad absurdum and fallacious to use as an argument for any religion."
Yep, it sure is.

Anonymous said...

Awesome work, you really spelt out your argument. However, there may be one little problem, that I can see, in that the existence of other Gods would not divide the rewards, it would show up as a cost, as Aaron points out. I would put it something like this:


For belief, n is the probability your god exists, while N is the probability some other god exists, thus n+N is equal to the probability some god exists, lets call this number P. n is equal to P divided by the number of possible gods, Z, which we take as infinite, thus n is infinitely small, while R is, of course, infinite large.

Thus for belief:


while for non-belief:


These can be simplified using our values to:

For Belief:

For Non-belief:

If this is true, then if C≤2P, then it is better to believe, otherwise non-belief is better. Thus, as long as the cost of believing in your god is less than twice the probability some God exists, it is better to believe.

However, since P is the probability that some god exists, and the proof for the existence of any god is pretty slim, P is probably pretty small. Pretty much, belief would have to be practically costless. However, one must remember that an ancillary part of Pascal’s argument was that the cost of belief was quite small, possibly, in the case of some religions, not really a cost at all, since that is the way you would live your life anyway. May as well create a God of science and be done with it, ay?