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},E_{0},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 E_{0} 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.)

## 12 comments:

Alex,

Maybe you remember that Tim McGrew has a similar comment on this issue at http://prosblogion.ektopos.com/archives/2007/

04/infinite_sequen.html#comment-49821

An argument against the simply infinite being actually infinite might use similar mathematics (as mine does), and an infinite past is arguably compatible with endless sequences being potentially infinite (e.g. the infinite past contains only a finite number of changes, after some first one).

Thanks for your comments on my argument (which I've only today noticed). Denying countable additivity would stop your move to impossibility here, but then one could have 2 people counting from minus infinity, and ask about the difference in days between when they finish (as in my argument). I'd guess that would also be a problem for a relational theory of time?

I think that denying countable additivity will get out of this argument. Then p(t)=0 for all t.

I think so too; but now I don't see how PSR generates a problem here...

If one chose to count up from minus infinity, one would be choosing to have always been counting, but were time not relational the choice of which day to say "0" might just be a completely free choice. Or it might be a matter of indeterminism. And similarly if there were 2 people counting. But might not PSR explain the particular day as being due to the choice to count up from minus infinity plus the freedom inherent in that choice or the indeterminism inherent in this matter?

Even were time relational, would an explanation along those lines not be needed (with 2 people counting) to explain the particular gap between the two sayings of "0"? (Incidentally, I think I've answered your rather perceptive comment on my aforementioned argument, but I can't be sure as my reply is a bit hand-wavey.)

If time is relational, the argument indeed fails completely.

If time is non-relational, though, then free will is not sufficient for the explaining. For

whichfree will choice is the relevant one? Before each one, there is an earlier one...If there was a choice to be counting from minus infinity, then that choice (presumably made from beyond physical time).

If there was not, then it could just be a matter of inherent indeterminism behind which universe was actual. There are lots of possible universes containing someone so counting, and if there is a reason for such a universe to exist (which is a different question, although that too could be indeterministic) then that plus the inherent indeterminism.

Even under Naturalism, what is necessary is all the possibilities, plus the possible existence of something (since otherwise they are not possibilities), and in the absence fo any better reason then that very indeterminism is the reason why, no?

But as there is a God, then God's free choice to create a universe with someone so counting would be the relevant one. (If He could not make such a choice, then such a universe would just be impossible.)

I see.

A free choice will not solve the problem if one thinks that (a) good reasons are needed for each choice, and (b) no good reasons can be made for creating one world versus another world shifted timewise. (This is the Leibnizian argument for relational views of time.)

Well, in a sense that's right, of course.

But were there some good reason for creating a temporally non-relational world containing someone so counting, then there would be a good reason to make an arbitrary choice, so the making of such an arbitrary choice would be for a good reason.

It would be better to make it than to not make it; while the power to make such a choice would (arguably) not be beyond God. And since the chosen world would have been chosen by God, so there would be a good reason why it rather than any of the others existed.

I thought that I read something like that in your book on PSR, but maybe you meant a diffrent PSR; or maybe I misread it. Still, that if C causes E then, if nothing causes E then E does not happen, would be implausible, surely, were unlikely random (and so lacking particular reason) events uncaused?

...oops, scratch that last remark: you may have a deterministic take on Q.M. for all I know. But still, if you need good reasons for each choice then you need a good reason for the gap between a pair of counters to be what it is, even if time is relational. Could 0 being special be a good reason for a gap of 0? But then it would be

impossibleto have one counting -1 when the other counts -2, which seems implausible (if such counters are plausible).I deny that one can make a choice with no reason for the choice.

Whether a random non-chosen event is possible is a different question. So let's suppose it is possible. Maybe then God causes an event E (obviously beyond time, or at minus infinity) which then stochastically causes a particular count-down series. But now we have a problem. For it is not clear that stochastic causation is possible absent probabilities. But no probabilities can be assigned to E causing a particular count-down sequence.

...aside from non-countably-additive infinitesimals?

But I'm amazed that you'd limit God to being unable to choose randomly (for a good reason) whilst allowing impersonal stochastic causation!

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