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#60 Counting Down from Infinity

June 09, 2008

Dear Bill,

Me and my husband have a difficulty with a version of Kalam argument against infinite past.

You wrote in your answer to question 12 (when arguing against the thesis that the temporal series of all past events is actually infinite),

... if an actual infinite could be formed by successive addition, then various absurdities would result. ... Suppose we meet a man who claims to have been counting down from infinity and who is now finishing: . . ., -3, -2, -1, 0. We could ask, why didn't he finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished.

We are interested in what seems to us to be an implicit and crucial premise (P, for short) of the quoted argument:

(P) IF (i) the temporal series of all past events is actually infinite in its duration (as measured by equal temporal intervals), THEN (ii) there COULD be some mind/clock/counting machine/computer/angel/god which would SUCCESSIVELY pair all the past equal intervals (say, seconds) to all negative whole numbers in the corresponding order.

As we understand the argument, you assume (P), or something relevantly similar, then you argue from some other premises (including the premise "it should already have finished") to Not-(ii), and finally you derive Not-(i) from Not-(ii) and (P) by modus tollens.

Now our questions:

1. The connective "IF ..., THEN ..." in (P) suggests some kind of entailment. But entailment is a convoluted issue. Philosophers talk about different kinds of entailment (like material implication, strict implication, relevant entailment), but they seldom specify them precisely. Could you, please, explicate the entailment in (P)?

2. The word "COULD" suggests some kind of possibility. Again, scholars talk about different kinds of possibilities (like strictly logical, broadly logical, and conceptual possibility). Could you explicate the "COULD" in (P)?

3. Are there any good reasons for (P)?

For example, do they consist in a sort of imagining? If so, what is that we are imagining? Secondly, note that we can't imagine the process of counting of the given sort clearly even if we can imagine the END of such counting (like the finite sequence of "-3, -2, -1, 0, finished!"). Thus we can't argue by means of the principle "whatever is readily imaginable is possible." We admit we can imagine, e.g., a god's beginningless meter applied to a 4D model of a beginningless universe, recording how many years remain until some important event in that universe. For every second the meter would display a corresponding negative number. But then we imagine rather enumeration of a non-successive, intuitive, all-at-once apprehension by the god of each and every one-to-one correspondence between any past second the infinite temporal series with any appropriately ordered negative number. And this enumeration is different from the temporal process of SUCCESSIVE counting (pairing) referred to in (P).

Thank you very much!


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Dr. craig’s response


Your husband Vlastimil must figure I'm partial to the ladies, Pavla, since I hadn't yet selected his same question! O.K., I give in.

The argument I gave takes P for granted—granted, that is, by the proponents of the infinitude of the past. They assume that if the past series of events is infinite, then the order type of that series is the order type of the negative numbers, or in the usual symbolism, ω*. Logically, that isn't necessary. The series of past events could have the order type ω* + ω*, the order type exemplified by . . . , -3, -2, -1, . . . , -3, -2, -1. But it's obvious why proponents of the infinite past don't adopt this view: for then there are past events which lie at an infinite distance from the present. But then one could never traverse the infinite distance from, say, the first -3 to the second -3. That's why proponents of the infinite past always insist that the existence of an infinite past doesn't entail an infinitely distant starting point. The Tristram Shandy paradox about a man writing his autobiography so slowly that it takes him a year to record the events of a single day challenges the assumption that an infinite past would have the order type ω*. The only hope for proponents of the infinite past is to insist that the series of past events has the order type ω* so that every event lies at only a finite distance from the present. In that way, forming an infinite past by successive addition doesn't involve, they claim, traversing an infinite distance.

So to your questions:

1. I suppose that P is thought to be broadly logically necessary, since, as we've seen, it's not strictly logically necessary. It would be odd to think that it's contingently true, so mere material implication would not seem strong enough. The proponents of the infinite past seem to assume that in any possible world in which the past is infinite, such a pairing could be done.

2. Same seems to go for the possibility of a being's pairing the events one-to-one with the negative numbers. There's a possible world where such a thing happens.

3. I suppose that what commends P is the inconceivability of making the transition from the first ω* series to the second ω* series in a series ordered ω* + ω*. This seems contrary to the nature of temporal becoming, since the event marked by the first -1 has no immediate temporal successor, which seems mad. Given the nature of temporal becoming, it does seem that an infinite past ought to have the order type ω*. The whole discussion has an air if unreality about it, since what one is arguing is that an infinite past is in any case metaphysically impossible. But there do seem to be non-trivially true counterfactual statements with impossible antecedents, e.g., If God did not exist, the universe would not exist. Similarly, here we might say that If the past were infinite, then it would have the order type ω*.

I don't clearly understand your last paragraph. If we do imagine the sort of timeless or tenseless pairing of events and numbers that you describe, then that assumes that an infinite past would have the order type ω*. The inconceivability of animating that picture is precisely the point of the defender of the finitude of the past, given the reality of temporal becoming. What we seem to be imagining is a past in which the series of events is formed by successive addition. So we preserve in the imagined worlds the nature of temporal becoming. We then ask whether this can have been going on from infinity past, and we wind up with the problems I've discussed.

Having written all this, it suddenly occurs to me that maybe I've misunderstood the thrust of your question and have been barking up the wrong tree. Maybe your question isn't about the possibility of the order type of an infinite series of past events' being something other than ω*, but rather about the possibility, given that order type, of some being's taking note of it. If that's your worry, I guess I just don't see the problem. Why couldn't a god who has existed from eternity take note of every event as it has occurred? Indeed, maybe the events just are the counting down from infinity of such a being. If an infinite past would have the order type ω*, then I can't see why a co-eternal being couldn't enumerate them as they happen.

- William Lane Craig