If a small universe is evidence for theism, is a vast universe evidence for atheism? I want to consider Craig’s reply to this, but before I do that I should introduce some basic concepts about the symmetry of evidence; that is, the possibility for evidence for entails the possibility of evidence against.
On likelihoodism observation O is evidence for hypothesis H over ¬H iff P(O|H) > P(O|¬H). Since P(O|H) + P(¬O|H) = 1 and P(O|¬H) + P(¬O|¬H) = 1, we can insert it into the prior formula to get an interesting result:
- 1 – P(¬O|H) > 1 – P(¬O|¬H)
- P(¬O|¬H) > P(¬O|H)
So, in English, O is evidence for H over ¬H iff ¬O is evidence for ¬H over H. The means that you can have evidence for a hypothesis iff you can have evidence against a hypothesis.
Craig applies this principle to an example:
David Manley was making the point that on the cozy, pre-Copernican cosmology—what C. S. Lewis called “the discarded image” of the cosmos—theism seemed vastly more probable than atheism. Like a Fabergé egg, the little universe centered on the Earth, with the spheres of the planets and fixed stars revolving about it, cried out for an explanation in terms of a Cosmic Designer. But if you agree that theism is more likely than atheism on such a view, then, Manley argued, you must also agree that a vast cosmos, such as we observe, counts against God’s existence.
Here we see Manley arguing that if a certain observation (small universe) supports theism, then ¬observation (vast universe) supports atheism. Craig agrees to this, but he softens the blow by saying:
… the degree to which the vastness of the universe increases the probability of atheism is marginal! It scarcely changes the odds at all. So while the smallness of the universe would greatly increase the probability of theism, the vastness of the universe only negligibly increases the probability of atheism.
Let’s see how he derives this conclusion. He starts with the odds form of Bayes’ theorem, which says that the ratio of posteriors = ratio of priors x ratio of likelihoods.
Here Craig picks his numbers:
Suppose we say that P(Small Universe|Theism) = .01 and P(Small Universe | Atheism) = .0001. That reflects our conviction that given a small, pre-Copernican universe, God’s existence is much more probable than atheism. This assumes that the prior or intrinsic probability of theism or atheism is exactly the same; otherwise Manley’s argument collapses. So we’ll just assume for the sake of argument that P(Theism) = 0.5.
Craig is pointing out that if the prior probability of atheism is something small like .01 (making theism’s prior .99), then evidence from a vast universe will get drowned out anyway for our posterior probability of atheism. For the sake of argument, we’ll assume the priors for atheism and theism are both 0.5.
Plugging in Craig’s numbers it turns out the posterior ratios are
So the end result is that given a small universe, theism is 100 times more probable than atheism; and given a vast universe, atheism is only slightly more probable. Craig is a happy man.
The reason this happens is because of the suspect initial numbers Craig plugs in. Why does Craig estimate P(Small Universe | Theism) = .01 and P(Small Universe | Atheism) = .0001? Those numbers look really low given the initial intuition about pre-Copernican cosmology, the Fabergé egg. That Pre-Copernican intuition should be reflected by P(Small Universe|Theism) > P(Vast Universe|Theism), which means there’s a greater than 50% chance of a small universe given Theism.
Craig hints at why he chose such low numbers:
Now I’ve read enough of the philosophical and scientific literature on fine-tuning to know that the vastness of the cosmos is not really surprising on theism. For example, John Barrow and Frank Tipler in their important book The Anthropic Cosmological Principle (Oxford University Press, 1985) emphasize that the size and age of the universe are just what we should expect to observe. For the carbon that makes up our bodies was synthesized in the interior of stars and then distributed throughout the universe via supernovae. It takes aeons for galaxies of stars to form and even more time for the carbon requisite for life to be spread abroad to become the foundation of biological life. No other element could substitute for carbon in this role. So the universe must be as old as it is for life to exist and, hence, as big as it is, since the universe is in a state of cosmic expansion since its inception in the Big Bang 13.7 billion years ago. So the size (my italics) and age of the universe are just what one ought to expect given the fine-tuning of the initial conditions of the universe (my italics), which, many have argued, is best explained through design.
It seems he’s using the fine-tuned constants, the initial conditions, and the laws (though unstated) as background knowledge. So, given that, that’s why a vast universe has such high probability on theism, according to Craig. But if we’re using that as background knowledge we need to use it for Atheism too, in which case the background knowledge does all the work and P(Vast Universe|Theism+k) = P(Vast Universe|Atheism+k) ≈ 1. The only reason the two probabilities would be different was if God performed miracles to affect the size of the universe to overrule what would naturally happen. If the background knowledge makes the two probabilities equal then we can’t compare the hypotheses and we need to choose different background knowledge.
The intuition Manley was getting at was that, in the pre-Copernican era, “like a Fabergé egg, the little universe centered on the Earth, with the spheres of the planets and fixed stars revolving about it, cried out for an explanation in terms of a Cosmic Designer.” Let’s find more reasonable numbers for our imagined pre-Copernican: P(Small Universe|Theism) = .8 and P(Small Universe|Atheism) = .1. Our pre-Copernican expects a small universe given theism and a vast universe given atheism.
A small universe would be significant evidence for Theism and a vast universe would be significant evidence for Atheism.
When evaluating the evidence, there are two questions: 1) the qualitative question of which hypothesis the evidence points to, and 2) the quantitative question of how strong the evidence is. The numbers I plugged in were subjective probabilities and will differ depending on your own personal God and atheist universe theory. How strong the evidence is will depend on your subjective theories. I think the numbers Craig plugged in completely misconstrued the dialectic as Manley presented it, since on Craig’s numbers the pre-Copernican theist expects a vast universe! No wonder a vast universe is only negligible evidence for atheism.
The main point of this post is not so much about finding the exact numbers to plug in, but that the possibility of evidence for, entails the possibility of evidence against, a point which Craig now seems to agree with.
For some practice, let’s see how this works for other examples.
- If successful prayer experiments are evidence for God, then unsuccessful prayer experiments are evidence against God. If unsuccessful prayer experiments aren’t evidence against God, then successful prayer experiments aren’t evidence for God. (I’ll leave out the contrapositive hereafter.)
- If divine hiddenness isn’t evidence against God, then divine appearance isn’t evidence for God. (Surely an odd result. There must be something wrong with the antecedent.)
- If suffering is evidence against God, then non-suffering (happiness) is evidence for God. (Notice that atheists who find the problem of evil persuasive has to admit that happiness is evidence for God.)
- If fine-tuned constants is evidence for God, then coarse-tuned (wide-ranging) constants is evidence against God.
- If finding intermediate fossils is evidence for common ancestry, then not finding intermediate fossils is evidence against common ancestry.
The above examples only speak on the qualitative (or binary) nature of evidence. The quantitative aspect will depend on your priors and particular theory. If you’re wondering about the quantitative aspect for any of these examples, plug in your own numbers to the odds form of Bayes’ like I did in the vast universe example. Elliott Sober explains in his paper (p. 16) that the reason why not finding an intermediate fossil is negligible compared to the support gained from finding an intermediate fossil is because of the low probability of finding a fossil. (He goes through the math in the paper.) Evolutionists need not worry.