Wednesday, April 09, 2008

Craig and Moreland on the Case Against Beginningless Traversals

I haven't seen this argument made explicit by Moreland or Craig, but I believe the materials for the following dilemma can be gleaned from their writings:

The Dilemma:
If a traversal of an infinite past is possible, then either the traversal requires a starting point or it doesn't. If it does, then the traversal could never get going, as one could never get a foothold in the beginningless series to begin the traversal (for there are an infinite number of moments before every day in such a past, and so to start at any point, you would have already had to have traversed an infinite number of moments to get to it). But if it doesn't, then the traversal should always be finished, for an infinite number of moments exist before every day in such a past, and thus all the time needed to complete the traversal exists before every day. Therefore, either the beginningless traversal can never get going or it's always completed. But both implications are absurd or otherwise unacceptable. Therefore, a beginningless traversal is impossible.

But I tend to think the argument is a failure. For take the first horn of the dilemma. Obviously one couldn't traverse a beginningless past by starting at some point, for there *is* no starting point in such a past by definition -- just as there is no "starting point" in the set of negative integers {..., -3. -2, -1}) . So the first point, while true, is something that nobody claims to be false.

I've just granted that a beginningless traversal, if possible, has no starting point. That brings us to the second horn of the dilemma. We've seen that Craig thinks we can rule out this possibility indirectly, via reductio: assuming the possibility of a beginningless traversal (i.e., counting, but without a starting point) leads to the contradiction that the traversal both occurs and has never occurred -- the latter being true because the traversal should always be finished. But why should the traversal always be finished? Craig's answer is that because there is an infinite number of moments before every day of a beginningless past, and thus enough time to complete the traversal before every day of such a past. But this is where the argument falls down. For this is to conflate traversing infinitely many moments with traversing all the moments that are to be traversed. But the former, while necessary for completing a particular infinite traversal, is not necessarily sufficient. For one can traverse an infinite number of members of a set, and yet not finish counting them all. So, for example, suppose I were counting down the negative integers toward zero from eternity past and am now counting -3. Then I have so far counted infinitely many negative integers (viz., {..., -5, -4, -3}, and yet I have not yet finished counting them all (I still have to count -2 and -1).[1]

In short, Craig and Moreland think that a beginningless infinite traversal is impossible, since either it could never get going or it should always be completed. But the first disjunct is true but irrelevant, while the second is (at least so far) without sufficient justification.
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Footnotes:

[1]Craig has denied that he is guilty of this charge:


“I do not think the argument makes this alleged equivocation, and this can be made clear by examining the reason why our eternal counter is supposedly able to complete a count of the negative numbers, ending at zero. In order to justify this intuitively impossible feat, the argument’s opponent appeals to the so-called Principle of Correspondence…On the basis of the principle the objector argues that since the set of past years can be put into a one-to-one correspondence with the set of negative numbers, it follows that by counting one number a year an eternal counter could complete a countdown of the negative numbers by the present year. If we were to ask why the counter would not finish next year or in a hundred years, the objector would respond that prior to the present year an infinite number of years will have elapsed, so that by the Principle of Correspondence, all the numbers should have been counted by now.


But this reasoning backfires on the objector: for on this account the counter should at any point in the past have already finished counting all the numbers, since a one-to-one correspondence exists between the years of the past and the negative numbers.”9


From this passage, we see Craig’s rationale for (3):


(R) The counter will have finished counting all of the negative integers if and only if the years of the past can be put into a one-to-one correspondence with them.


Furthermore, from the passage cited, we see that Craig thinks that the defender of an A0 past agrees with (R). But since the type of correspondence depicted in (R) can be accomplished at the present moment, it follows that the counter should be finished by now. Therefore, Craig’s opponent is committed to a view that entails the absurdity surfaced by the above reductio.

III

It isn’t clear that Craig hasn’t made his case, however. For consider the following scenario. Suppose God timelessly numbers the years to come about in a beginningless universe. Suppose further that He assigns the negative integers to the set of events prior to the birth of Christ, and then the positive integers begin at this point. Then the timeline, with its corresponding integer assignment, can be illustrated as follows:


…-3 -2 -1 Birth of Christ 1 2 3…


Suppose yet further that God assigned Ralph, an immortal creature, the task of counting down the negative integers assigned to the years BCE, and stopping at the birth of Christ. Call this task ‘(T)’. With this in mind, suppose now that Ralph has been counting down from eternity past and is now counting the day assigned (by God) the integer -3. In such a case, Ralph has counted a set of years that could be put into a one-to-one correspondence with the set of negative integers, yet he has not finished all the negative integers. This case shows that, while it is a necessary condition for counting all of the events that one is able to put them into a one-to-one correspondence with the natural numbers, this is not sufficient. For if the events that are to be counted have independently “fixed”, or, “designated” integer assignments set out for one to traverse, one must count through these such that, for each event, the number one is counting is the same as the one independently assigned to the event. In the scenario mentioned above, God assigned an integer to each year that will come to pass. In such a case, Ralph must satisfy at least two conditions if he is to accomplish (T): (i) count a set of years that can be put into a one-to-one correspondence with the natural numbers, and (ii) for each year that elapses, count the particular negative integer that God has independently assigned to it. According to Craig’s assumption (R), however, Ralph is supposed to be able to accomplish (T) by satisfying (i) alone. But we have just seen that he must accomplish (ii) as well. Therefore, being able to place the events of the past into a one-to-one correspondence with the natural numbers does not guarantee that the counter has finished the task of counting all the negative integers. In other words, (R) is false. But recall that (R) is Craig’s rationale for (3). Thus, (3) lacks positive support. But more importantly, (3) is false. This is because the scenario above is a counterexample to both (R) and (3). For (3) asserts that it is sufficient for counting down all the negative integers that one has an infinite amount of time in which to count them. But our scenario showed that one could have an infinite amount of time to count, and yet not finish counting all of the negative integers (e.g., one can count down to -3 in an infinite amount of time, and yet have more integers to count).

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