Now, the argument as it stands has two obvious holes. First, it assumes not only determinism, but two-way determinism. Determinism says that from any earlier state and the laws, the later states logically follow. Two-way determinism adds that from any later state and the laws, the earlier states logically follow. Fortunately for the argument, actual deterministic theories have been two-way deterministic. Second, the argument assumes that the exact correspondence between states at t_{0} and at t_{1} preserves probabilities. This need not be true. If we consider the set [0,1] (all numbers between 0 and 1, both inclusive), and the function f(x)=x^{2}, then f provides an exact correspondence between [0,1] and [0,1], but if X is uniformly distributed on [0,1], then the probability that X is in [0,1/4] is 1/4, while the probability that f(X) is in [0,1/4] is 1/2 (since for f(X) to be in [0,1/4], X need only be in [0,1/2]). But, again, in the kind of classical physics setting that underlies classical thermodynamic results like the PoincarĂ© recurrence theorem, the transformations between states preserve phase-space volume, and it is very plausible that if you preserve phase-space volume, you preserve probabilities.
Once we add two-way determinism and phase-space volume preservation, which are reasonable assumptions in a classical setting, the argument is in much better shape. (Actually, if you can still have something relevantly like phase-space volume preservation, you could drop the determinism. I don't know enough physics to know how helpful this is.) The argument is now this. Let S be the set of all possible physical states of the universe. For any real number t, the two-way deterministic physics defines a one-to-one and onto function f_{t} from S to S, such that by law the universe is in state s at time t_{0} if and only if it is in state f_{t}(s) at time t_{0}+t. Let C_{1} be the subset of S containing all states the exhibit the complexity feature C. Let C_{0} be the subset of S containing all states that would result in a state in C_{1} after the passage of t_{1}−t_{0} units of time. In other words, C_{0}={s:f_{t}(s) is in C_{1}}, where t=t_{1}−t_{0}. Then the probability of C_{0} is the same as the probability of C_{1}. Hence, if our world's present state's being in C_{1} was too unlikely for chance to be a reasonable expectation, then the Darwinian explanation in terms of the world having been in a state from C_{0} at t_{0} is no better. In particular, if a theistic design hypothesis would do better than randomness if it were a matter of generating a state in C_{1} from scratch, Darwinism hasn't done anything to weaken the inference to that theistic hypothesis since C_{0} is just as unlikely as C_{1}. Even if the evolutionary theory is correct, we still need an explanation of why the universe's state was in C_{0} at t_{0}.
This argument is on its face pretty neat. One weakness is the physics it relies on. But bracket that. The kind of measure-preservation that classical dynamics had is likely to be at least a decent approximation to our actual dynamics. But there is a more serious hole in the argument.
The hole is this. If what evolution was supposed to explain is why it is that the universe is now in a state exhibiting C, the argument would work. But that isn't what evolution is supposed to explain. Suppose C is the existence of minded beings like us. Then it seems that we are puzzled why C is exhibited at some time or other, not why
- C is exhibited now.
So perhaps the explanandum is not that C is exhibited at t_{1} but that
- C is exhibited at some time or other.
Maybe the puzzle is not about (1) or (2), but about:
- C is exhibited within 14 billion of the beginning of our universe.
One problem with this as the account of what evolution does to explain C is that currently we do not have very good mathematical estimates of how long we can expect evolutionary processes to take to produce something like C, where C has any significant amount of complexity. So perhaps we do not really know if evolution explains (3).
Another move that one can make is to say that evolution does explain (1), and it does so by giving a plausible genealogical story about C, but the evolutionary explanation does not confer a non-tiny probability on (1). If so, then the evolutionary explanation may be a fine candidate for a statistical explanation of (1), but it will not be much of a competitor to the design hypothesis if the design hypothesis confers a moderate probability to (1).
In fact, we can use the above observations to run a nice little design argument. Suppose that C is the existence of intelligent contingent beings. Then for an arbitrary time t, the hypothesis of theistic design gives at least a moderate probability of the existence of intelligent contingent beings at t, since God is at least moderately likely to fill most of time with intelligent creatures. (And Christian tradition suggests that he in fact did, creating angels first and then later human beings.) Therefore, evolutionary theory assigns incredibly tiny probability to (1)—equal to the probability of getting C from scratch at random—but the design hypothesis assigns a much higher probability to (1). We thus have very strong confirmation of theism.[note 1]
But that assumes an outdated dynamics. Whether the argument can be made to work in a more realistic physics is an open question.