- P(the universe has low entropy | naturalism) is extremely tiny.
- P(the universe has low entropy | theism) is not very small.
- The universe has low entropy.
- Therefore, the low entropy of the universe strongly confirms theism over naturalism.
Low-entropy states have low probability. So, (1) is true. The universe, at the Big Bang, had a very surprisingly low entropy. It still has a low entropy, though the entropy has gone up. So, (3) is true. What about (2)? This follows from the fact that there is significant value in a world that has low entropy and given theism God is not unlikely to produce what is significantly valuable. At least locally low entropy is needed for the existence of life, and we need uniformity between our local area and the rest of the universe if we are to have scientific knowledge of the universe, and such knowledge is valuable. So (2) is true. The rest is Bayes.
When I gave him the argument, Dan Johnson made the point to me that this appears to be a species of fine-tuning argument and that a good way to explore the argument is to see how standard objections to standard fine-tuning arguments fare against this one. So let's do that.
I. "There is a multiverse, and because it's so big, it's likely that in one of its universes there is life. That kind of a universe is going to be fine-tuned, and we only observe universes like that, since only universes like that have an observer." This doesn't apply to the entropy argument, however, because globally low entropy isn't needed for the existence of an observer like me. All that's needed is locally low entropy. What we'd expect to see, on the multiverse hypothesis, is a locally low entropy universe with a big mess outside a very small area--like the size of my brain. (This is the Boltzmann brain problem>)
II. "You can't use as evidence anything that is entailed by the existence of observers." While this sort of a principle has been argued for, surely it's false. If we're choosing between two evolutionary theories, both of them fitting the data, both equally simple, but one of them making it likely that observers would evolve and the other making it unlikely, we should choose the one that makes it likely. But I can grant the principle, because my evidence--the low entropy of the universe--is not entailed by the existence of observers. All that the existence of observers implies (and even that isn't perhaps an entailment) is locally low entropy. Notice that my responses to Objections I and II show a way in which the argument differs from typical fine-tuning arguments, because while we expect constants in the laws of nature to stay, well, constant throughout a universe, not so for entropy.
III. "It's a law of nature that the value of the constants--or in this case of the universe's entropy--is exactly as it is." The law of nature suggestion is more plausible in the case of some fundamental constant like the mass of the electron than it is in the case of a continually changing non-fundamental quantity like total entropy which is a function of more fundamental microphysical properties. Nonetheless, the suggestion that the initial low entropy of the universe is a law of nature has been made in the philosophy of sceince literature. Suppose the suggestion is true. Now consider this point. There is a large number--indeed, an infinite number--of possible laws about the initial values of non-fundamental quantities, many of which are incompatible with the low initial entropy. The law that the initial entropy is low is only one among many competing incompatible laws. The probability given naturalism of initially low entropy being the law is going to be low, too. (Note that this response can also be given in the case of standard fine-tuning arguments.)
IV. "The values of the constant--or the initially low entropy--does not require an explanation." That suggestion has also been made in the philosophy of science literature in the entropy case. But the suggestion is irrelevant to the argument, since none of the premises in the argument say anything about explanation. The point is purely Bayesian.