An argument against an infinite past

This is a version of an argument by Bill Craig, with probability in place of the Principle of Sufficient Reason. I don't actually think this argument is sound, but the premises might well be plausible to a number of people. Suppose, then, for a reductio, that it is possible for a world to have an infinite past. Let H be the following hypothesis: The world has an infinite past and future (nobody who allows an infinite past will balk at an infinite future, surely), and contains Jones, who counts up from minus infinity (not inclusive) to zero (inclusive), uttering one number a day. Thus, on some day he uttered "-4848", and on the next he uttered "-4847" and so on. Then on some day he finished by uttering "0".

For any time t, let E_t be the hypothesis that Jones has finished counting at a time t* such that t - 1day < t* ≤ t, i.e., that Jones has finished within in the 24 hours preceding t. Let p(t)=P(E_t|H). Since H does not mention any specific times, by the principle of indifference, p(t) has to have the same value for every value of t. Thus, for all t, p(t)=p(0).

But now consider the following infinite sequence of events: ...,E_{-3 days},E_{-2 days},E_{-1 day},E₀,E_{1 day},E_{2 days},E_{3 days},.... Given H, it is certain that exactly one of them happens. Thus, P(... or E_{-3 days} or E_{-2 days} or E_{-1 day} or E₀ or E_{1 day} or E_{2 days}, or E_{3 days} or ...|H) = 1. Moreover, these events are mutually exclusive, so the left hand side of this equation is equal to: ...+p(-3 days)+p(-2 days)+p(-1 day)+p(0)+p(1 day)+p(2 days)+p(3 days)+.... But each of the summands here is the same, namely p(0). If p(0) is positive, then this sum is infinite, and hence not equal to 1. If p(0) is zero, then this sum is zero, and hence not equal to 1. And p(0) can't be negative since it's a probability. Hence, impossibility ensues no matter what value p(0) has. (And, no, infinitesimals won't help. That was shown by Tim McGrew--see this paper of mine.) If all of this works, then we need to reject as absurd the assumption that an infinite past is possible. And once we reject this assumption, the Kalaam argument becomes available.

There are two weak points in the argument. The first is that there is an actual difference between the hypotheses E_t for different values of t. If one accepts an A-theory of time, according to which what time it is now is an objective feature of the universe, then one has to agree there is a difference between these hypotheses--it is an objectively different thing for Jones to finish counting today than to have finished counting yesterday. Likewise, if one takes a substantival theory of time, one will see a difference. But the Leibnizian like me, who takes time to be purely relational, will not see a difference between the hypotheses: if one shifts over the history of the world by a day, one changes nothing. The second weak point is the assumption that one can apply classical probability theory to events like E_t conditioned on H, which, again, I am suspicious of. (But I accept the Principle of Sufficient Reason, and that can be used in place of the probabilistic reasoning.)