Saturday, March 8, 2014

William Lane Craig and the Argument from Successive Addition

I’ve been on a bit of a roll with William Lane Craig-related blog posts recently. I thought I might continue the theme by addressing another one of his arguments today. So far I’ve just been looking at various aspects of his moral argument, but now I want to switch focus and look at part of his defence of the Kalam Cosmological Argument (KCA).

Those who are familiar with the KCA will know that its second premise reads as follows:

  • (KCA2) The universe began to exist.

They will also know that Craig supports this premise of the argument with four separate sub-arguments, two of which are “in principle” arguments, based on the concept of an actual infinite, and the other two of which are “in fact” arguments, based on existing scientific theories. Although it is often great fun to debate these scientific theories, they are really only a sideshow. Craig himself acknowledges that the primary warrant for premise (2) comes from the “in principle” arguments.

Those two arguments have different targets. The first one targets the general possibility of an existent actual infinite (i.e. it says that an actual infinite cannot exist). The second targets the possibility of an actual infinite being formed by successive addition. I’ve looked at the first argument before, when discussing Hedrick’s critique of the Hilbert’s Hotel Argument. In this post, I want to look at the second argument. This argument really only kicks-in if the first argument fails. If an actual infinite cannot exist at all, then it certainly cannot exist through successive addition. The consensus of critics seems to be that this is a good thing for Craig since the second argument is the weaker of the two.

Anyway, my discussion of the successive addition argument will be broken down into three parts. First, I’ll look at Craig’s argument itself and present it in somewhat formal terms. Second, I’ll outline two initial criticisms of the argument. Third, I’ll take a longer look at one of the analogies Craig uses to support the argument (the reverse-countdown analogy).

Nothing I say here is original. I’m going to be basing most of this off the work of Wes Morriston, drawing in particular on these two articles.

1. Craig’s Successive Addition Argument
A firm grasp of the concept of an “actual infinite” is crucial to understanding Craig’s argument. An actual infinite is best defined in terms of set theory. To give a succinct definition: a set can be said to contain an actually infinite number of members if the set is equivalent to a proper subset of itself, where “equivalency” is understood in terms of the ability to put members of respective sets into a one-to-one correspondence with one another.

This probably sounds terribly obscure, but it makes sense if you use an example. Take the set of all natural numbers (0, 1, 2, 3…). This set is an actual infinite because it is possible to put all the members of that set into a one-to-one correspondence with a proper subset of the natural numbers (e.g. all the even numbers). As follows:

(0, 1, 2, 3, 4, 5, 6….)
(0, 2, 4, 6, 8, 10, 12)

The point is that an actual infinite is a complete set with an actual infinite number of members. This is to be contrasted with a potential infinite, which is simply a set that is constantly growing without limit.

Craig’s claim is that if the universe never began to exist then it must contain an actually infinite number of past events. Now, he would like to say that an actual infinite number of past events cannot exist at all, but if he can’t say that he would like to make the narrower claim that an actual infinite number of events cannot be formed through successive addition. This is exactly what has to happen if the universe is infinitely extended into the past. For each past event that occurs becomes the member of a set, and each subsequent event gets added into this set, resulting in a set that must have an actual infinite number of members.

The problem according to Craig is that this could never happen. You cannot form an actual infinite by adding members to a set like this. Consider an example. Take the set of years since 1914 (I steal this from Morriston). The set currently contains 100 members. Next year it will contain 101. The year after that it will contain 102. And so on. Suppose this sequence of adding years continues forever, will the resultant set ever end up containing an actual infinite? No; no matter how long this goes on it will only ever contain a large but finite number of members. The same thing is true of the set of past events.

To summarise, Craig’s argument is the following:

  • (1) A temporal series of past events is a collection formed by successive addition.
  • (2) A collection formed by successive addition cannot be actually infinite.
  • (3) Therefore, the temporal series of past events cannot be actually infinite.(from 1 and 2)
  • (4) If the universe never began to exist, then the temporal series of past events would have to be an actual infinite.
  • (5) Therefore, the universe must have begun to exist (from 3 and 4).

And this, of course, is just the second premise of the KCA.

Is this argument any good? Let’s see.

2. Does the argument beg the question?
The key to this argument is the first step (from 1 & 2 to 3). I just added in the second step to show how you get from there back to the KCA. It is possible to critique both of the premises of this first step in the argument. For instance, premise (1) relies on an A-theory of time, which is contestable. Nevertheless, we’ll ignore objections to premise (1) here and focus instead on premise (2).

The simplest objection to premise (2) is that it begs the question against the defender of the beginningless universe. Look back to the analogy we used to support it. Starting with a first member, it does indeed seem to be true that you could never reach an actual infinite number of members, but that’s only if we assume that we have to start with a first member. In other words, the analogy is only compelling if we assume that the sequence had a beginning. But that’s exactly what is in dispute when it comes to the history of the universe.

As I say, this is the simplest objection to premise (2). It’s no surprise then to learn that Craig is aware of it and tries to evade it in various ways. I’m not sure he ever succeeds in doing so, but let’s explore a couple of his methods of evastion now. The first method of evasion forces us to re-orient our perspective on the problem. Instead of (erroneously) imagining a sequence with a first member and working forwards from there to the present, instead we are asked to imagine working backwards from the present. Doing so, we see that in order for the present event to occur, so too must the event prior to that, and then the event prior to that, and then the event prior to that, and so on ad infinitum.

The problem? Well, if every event requires a prior event in order to come into existence, and if the sequence of events extends forever in the reverse-temporal direction, it seems like the present could never have arrived. As Craig puts it:

Before the present could occur, then the event immediately prior to it would have to occur; and before that event could occur, the event immediately prior to it would have to occur; and so on ad infinitum. So one gets driven back and back into the infinite past, making it impossible for any event to occur. Thus, if the series of past events were beginningless, the present event could not have occurred, which is absurd. 
(Reasonable Faith, 122)

This is an odd argument. It is really a claim about the impossibility of a infinite causal sequence (which turns it into a Thomist cosmological argument). But even then there doesn’t seem to be serious objection to the notion of a beginningless past. It may be true that every event needs a cause, but with an infinite past every event does have a cause. Causation is perfectly well-defined at every stage in the sequence, it just happens that the whole sequence itself does not have an external cause of its existence. But if you claim that it must have an external cause, you’re getting into a different argument.

Morriston puts the point rather nicely. He argues that what Craig is doing here is confusing a claim about our inability to trace back an actual infinite sequence of events, with a claim about the impossibility of an infinite sequence of events. But that’s like claiming that there cannot be an actual infinite number of natural numbers simply because we cannot count them all. The latter does not imply the former.

3. The Reverse Countdown Analogy
The other strategy Craig uses to defend premise (2) is something I am here calling the “reverse countdown analogy”. This is a thought experiment that Craig presents in virtually all his debates and scholarly writings. I’ll leave him to explain it:

…suppose we meet a man who claims to have been counting from eternity, and now he is finishing: −5, −4, −3, −2, −1, 0. Now this is impossible. For, we may ask, why didn’t he finish counting yesterday or the day before or the year before? By then an infinity of time had already elapsed, so that he should have finished. The fact is, we could never find anyone completing such a task because at any previous point he would have already finished. 
(Philosophical and Scientific Pointers to Creation Ex Nihilo, p. 189-90)

It’s important to realise that this analogy is being used to support the same basic point as the previous argument, namely: that if the past were infinite then the present could never have arrived. Nevertheless, this analogy is harder to deal with than the previous argument.

As Morriston notes, what Craig is really doing here is considering two separate historical series: (a) the series of past times (TS); and (b) the series of past counting events (ES). The man we are asked to imagine is enumerating the members of the series of counting events (E-n…E0), but he is doing so while overlaid on the series of past times (T-n…T0). Craig is then asking us:

Craig’s Question: Why, if this man has been counting down from eternity, does he reach E0 at T0 and not at T-1 or T+100 or whatever Tn we care to imagine?

This seems to commit Craig to the following argument (this is Morriston’s formulation):

  • (6) If a beginningless count is possible, then there must some reason why the whole series of counting events is located at the series of temporal locations that terminates in the present (i.e. there must be some answer to Craig’s Question).
  • (7) No such reason/answer can be given.
  • (8) Therefore, a beginningless count ending in zero is not possible (and hence the present moment T0 can never arrive).

There are a few problems with this argument and the analogy that is used to back it up. For one thing, the original thought experiment could be said to confuse the concept of counting an infinite number of negative numbers with counting all the negative numbers up to zero. But leave that aside. A bigger problem is with premise (6), which seems to demand that reasons be given for any coincidence of this sort.

As Morriston sees it, to demand such reasons is to fall back on the much-contested principle of sufficient reason, i.e. on the belief that everything must have a reason for its existence. But this seems an extravagant demand, particularly when it comes to explaining coincidences between our measures of time and past events.

To understand this point we must ignore some features of Craig’s thought experiment. We must realise that the counting man is a distraction. What Craig is really concerned with is why the set of all past events (Morriston calls it the set of macro-events) not just the set of counting events would terminate at T0. But this concern assumes that the flow of all past events is distinguishable from time — and hence that there is some reason for the coincidence between the two sequences. That doesn’t seem right, even on an A-theory of time: the flow of events surely is the passage of time? They are the same thing. If that’s right, then Craig’s demand for some explanation of the coincidence is getting tangled up in questions about what explains our metrics of time. And there won’t be any interesting answers to questions of that sort. Consider the question: “Why does the series of times end at this time rather than at some other time?” The answer will simply be: because that’s how we have chosen to measure time. Nothing more definitive can be said.

Morriston uses an analogy to underscore this observation:

Suppose that we have a bolt of cloth, and a measuring stick, calibrated in inches, that we want to use to measure a ten inch swatch of cloth. Obviously, we can line up the end of the cloth with the end of the measuring stick, or we can line it up with the one inch marker on the measuring stick, or with the two inch marker, and so on. It’s completely arbitrary which we decide to do. As long as we can do simple subtraction, we’ll have no trouble measuring out a ten inch swatch of cloth. Now suppose someone asks, “Why is the edge of the stick lined up with the end of the cloth? Why not the one inch mark?” This is hardly a question that “cries out” for a “sufficient reason” type answer. 
(Must metaphysical time have a beginning? 2003, p. 294)

The ultimate point is this: Craig tries to put the burden of proof on the defender of a beginningless countdown to explain the coincidence between ES and TS, but there is no reason to think that such a coincidence demands a reasoned explanation, particularly if those series are the same thing.

Let me close with one final observation, this time from Keith Yandell. Recall, how Craig is trying to show that there is something absurd or contradictory inherent in the notion that the past had no beginning or that the present moment has arrived from a beginningless past. Yandell suggests that this is not the case:

[T]o say that the universe is beginningless is to say that, for any past time T, the universe existed at T, and at T-1 as well. For any such time T you mention, there is a finite distance between T and now. So the universe could have chugged along from T until now. There is hence no past time such that it is impossible for the universe to chug along from that time until now. What the idea of the universe being beginningless entails is that, for any past time T, the universe actually has chugged along from it until now. Since that it not impossible, it is not impossible that the universe is beginningless. What, exactly, in Craig’s argument shows that this line of reasoning is inconsistent [or absurd]? 
(Does God Exist? The Craig-Flew Debate, p. 106)

”Nothing” is the answer.


  1. Here is your answer to the last question from Yandell:

    "How is the claim that from any given moment in the past there is only a finite distance to the present even relevant to the issue? For the question is how the whole series can be formed, not a finite portion of it. Do Mackie and Sobel think that because every finite segment of the series can be formed by successive addition the whole infinite series can be so formed? That is as logically fallacious as saying that because every part of an elephant is light in weight, the whole elephant is light in weight, or in other words, to commit the fallacy of composition. The claim that from any given moment in the past there is only a finite distance to the present is simply irrelevant."

    – William Lane Craig
    Blackwell Companion to Natural Theology, p. 120.

  2. "It may be true that every event needs a cause, but with an infinite past every event does have a cause. Causation is perfectly well-defined at every stage in the sequence, it just happens that the whole sequence itself does not have an external cause of its existence."

    You also commit the fallacy of composition.

    "But that’s like claiming that there cannot be an actual infinite number of natural numbers simply because we cannot count them all."
    Well, Craig DOES claim that there are no actually infinite number of natural numbers. Not in a Platonic way, nor in reality. Therefore, this is not an objection. Don't confuse mathematical entities with reality.

    1. I don't think he committed the fallacy of composition in the text you quoted.