1.05.2008

The Myth of Probability

René Descartes once said, "When it is not in our power to determine what is true, we ought to act according to what is most probable." Now, I'm not sure whether or not I agree with that statement, because I'm not sure whether it is in our power to determine what is most probable.


But Wait, What Exactly Is Probability?


But what does probability even mean? The classical definition of probability, according to Pierre-Simon Laplace (yes, that Laplace):
The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.
Frequency probability, on the other hand, defines an event's probability as "the limit of its relative frequency in a large number of trials" (according to Wikipedia). Note the "large number of trials." Statistical significance, people! Please! (And the law of large numbers is fascinating, but it can only be proved by testing it multiple times and seeing that it eventually works - in other words, by using the law of large numbers. It basically means that "probabilities are probably true.") Anyway, there's also Bayesian probability, which doesn't make a lot of sense to me (and I'm not the only one).


The Implications of Probability


But back to what Descartes said. My question is simple: When can probability truly be applied to real life?

We apply probability a lot more than we think. In fact, every time we say something is true, we are really saying it has a 100% probability of being true. When we say that the Earth's gravitational pull is (close to) a certain number, we mean (in frequentist terms) that the relative frequency of the limit of that number is 1. We generally assume that time will not effect the probability of certain statements or truths, including scientific propositions; in fact, one foundational truth of science is that accurate results in the past are still valid today. There is a 100% probability the laws of nature do not change. (Or is there?)

In general, I don't really mind this use of probability. It's pretty effective for what it attempts to do: to predict. But the predictive number we assign to a statement does not in the least affect its actual truth value.

Consider the classic example of probability: a deck of cards. Assuming the deck is sufficiently shuffled so that the cards are distributed randomly, the chances of drawing a certain card are 1 in 52, or approximately 1.92%. But that's not really true! That's just the best prediction we can make. The truth is, there is exactly one deck of cards arranged in exactly one order, so there is a 100% probability that a certain card will be drawn and a 0% probability any other card will be drawn. The reality is that there is no chance. As David Hume said, "Chance is only our ignorance of real causes."

This raises another question: Is there irreducible randomness in nature? Is radioactive decay really random, as Schrödinger's cat might suggest? Are there hidden variables? If there are hidden variables, probability is merely a concession either of a lack of data or of an inability to analyze the data. But if quantum mechanics is truly random, or indeterministic, what exactly happens? (I'm asking rhetorically.) How does a physicalist metaphysic (How ironic is that phrase?) explain randomness when science depends upon an orderly universe? Imagine if Newton's second law were "f = ma only at random times." But if the universe is deterministic, from whence (I love that word) do the independent relationships arise?


Probability and Belief


Descartes attempted to apply probability to entire belief systems, and so do a lot of people today. Atheists (who sometimes exalt themselves as "skeptics" or "brights"), in particular, argue that the existence of a god (whatever that means) is exceedingly improbable. (By the way, I don't like to think of things this way. I don't think of atheism as not believing in God and theism as believing in God; I think of them as beliefs in different sets of independent truths, that actually aren't all that different. More on that at another time.) But where are the numbers? How can you evaluate the probability of a god? If God's probability is either 0% or 100% (He either exists or doesn't exist), are we going to say it's at 7.3% or 50.01% or whatever? Should we only believe in God if His probability is greater than 50%?

And what factors affect the probability of God's existence (Yes, I switched from "a god" to "God." I don't care enough.)? That is itself a question that is rarely asked. I'm still not sure how to answer it.

Even assuming that physicalism can explain the entirety of our existence, does that automatically bring God's probability to 0%? Recall that probability requires more than one trial. We only have one trial of the universe, so we can't determine God's probability at all; we don't have enough trials. True probability cannot exist outside the context of a sufficient number of trials. Thus, even in this (hypothetical) atheistic utopia of a universe, God might still exist. His probability would be unknown. Which is why agnosticism makes a lot more sense.

But consider more specific questions. Are the New Testament documents reliable? Can we really assign a number to the probability that they are reliable? Should we assume that every piece of evidence has equal weight? How does this work?

The reality is that people who consider their answers to such questions "objective" are overstating the precision of their arguments. Prominent atheist Antony Flew recently converted to deism because, in his mind, the existence of God became more probable than the non-existence of God. While I'm happy a prominent atheist has renounced atheism (though he's far from Christianity), I still think this whole idea is kind of absurd. At what point did the existence of God top 50%? Which tidbit of evidence did that? Did Flew assign numerical values to the different facts on each side and weigh them up? Or did he just use what atheists rarely admit to using, a gut instinct?

We cannot rationally calculate God's probability. There is no such thing. We can examine evidence, but our final determination will be arbitrary and subjective, even if it is based on objective and truthful information. Which is why people disagree more about this than about whether or not 2 + 2 = 4. (It does.)

Does that mean we should all be agnostic? No! (I don't think so.) It does mean that reason is not the only source of knowledge in this area. And it means that our question should be rephrased. "What is the probability that God exists?" should become "Is God compatible with what we know about reality? Is physicalism compatible with what we know about reality? Are they both?" (That would be interesting.) And finally, we must ask what role pure reason plays in this all. (Hint: I don't think it's 100%.) (Get it? I referenced probability in my parenthetical comment!) (Three parenthetical comments in a row!)

God is more than a hypothesis.

3 comments:

Spaceman Spiff said...

This seems to be to be related or identical to the problem of induction. In that case I would disagree that all truth is probabilistic; when a conclusion is reached deductively and a priori (such as the law of identity, or the law of non-contradiction, and anything that results from them, or all of arithmetic) I think we're safely dealing outside induction or probability.

The problem of scientific knowledge is much trickier, obviously, and perhaps anyone claiming 100% certainty that the gravitational constant is a certain number is ignoring the dark matter problem. In any case the best any scientific test can ever prove is necessarily less than 100%, and that's a fact.

As far as the existence of God, while we cannot indeed *calculate* the probability, we are still obviously taking data and processing it, deciding how it might best be explained. I think if we believe in God, we should admit that it is as probabilistic as any other conclusion about the way the world is, but this is not a bad thing, I don't think.

Your Conscience said...

I think the conclusions you mentioned - the law of identity and the law of contradiction - cannot be rationally deducted but must be assumed for us to make any sense out of anything. We can't really conceive of a world in which "A is A" is not true.

I agree with you about the existence of God. We are taking data and processing it. But atheists are saying that God is exceedingly improbable, and that is not something they can really say. They can make the assertion that God is incompatible with the evidence, but that, in my mind, is a very different assertion.

Belief in God is probabilistic in the sense that we cannot, strictly speaking, prove or disprove God. What I would say is that our beliefs are subjectively based upon our experiences, science, reasoning, and other things. We can maximize the extent to which we want science to play a role in determining our beliefs, but there will always be a degree of subjectivity.

Spaceman Spiff said...

I think you can deduce the law of identity and the law of non-contradiction by reductio ad absurdum, which is to say, show that if they aren't true the results are absurd. Of course I guess absurd probably has to be defined as "violating the law of non-contradiction or the law of identity", so it's circular at some level, but still, they at least fit in the category of a priori. The point still stands, however, that things you can deduce from such reflections are known without induction or probability.

Anyway, I think from an external perspective, it is true that the probability of God existing is either 1 or 0, but that is also true of the value of the gravitational constant, right?

From our perspective, however, I would argue that if our knowledge of God is probabilistic, then it is possible that God is exceedingly improbable. If you agree that it's probabilistic, how can you exclude the possibility that the probability is low?

Arguments from the existence of evil, from the pervasiveness of evil are more convincing in this vein than arguments from the success of science at "explanation", and I think that's what you're talking about. Of course if science were to describe nature perfectly, that still wouldn't disprove God since we aren't talking about the god of the gaps, after all.