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#31 Swinburne on the Kalam Cosmological Argument

November 19, 2007

Dr. Craig,

In The Existence of God, by Richard Swinburne, he critiques your Kalam Cosmological Argument, because he disagrees with your position on the possibility of an infinite number of things in time. Its on pg. 138 of the Second Edition in the footnotes. What is your response to his critique? I think I agree with your position, but I would like to see how you’d argue it.


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Dr. craig’s response


It’s always a source of fascination to me to see how an admittedly brilliant person who is unpersuaded of the soundness of the kalam cosmological argument proposes to turn back the force of the argument. If the proffered refutations strike us as lame, we may come away from the discussion re-confirmed in our belief that the argument really does have something going for it. A case is point is Swinburne’s brief handling of the argument in the second edition of his book The Existence of God (pp. 138-9).

The kalam cosmological argument may be stated as a simple syllogism:

1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. Therefore, the universe has a cause.

With respect to (1) Swinburne says, “But it seems to me. . . this, like ‘the universe began to exist’, can be given only an inductive justification.” Great! I’m more than happy to accept the truth of (1) on purely inductive grounds. While the kalam argument itself is a deductive argument, that does not imply that its premisses are not to be supported by inductive evidence. On the contrary, I myself have made extensive appeal to the inductive evidence supplied by science as justification for both premisses of the kalam argument.

So what about (2)? The first supporting philosophical argument I’ve defended is based on the impossibility of the existence of an actually infinite number of things:

1. An actually infinite number of things cannot exist.
2. A beginningless series of events in time entails an actually infinite number of things.
3. Therefore, a beginningless series of events in time cannot exist.

In response to this argument, Swinburne expresses two misgivings. First, with respect to (1) he says, “But I suggest that we can allow what seems to me the obvious logical possibility of there being an infinite number of things (e.g., stars), without adopting Cantor’s mathematics, or this kind of way of applying it.” This response is bewildering. In the first place, the argument does not try to prove the logical impossibility of the existence of an actually infinite number of things, but its metaphysical impossibility. I have emphasized that the argument does not in any way deny that Cantor’s set-theoretical universe is, given its axioms and conventions, logically consistent, in that no contradiction has be shown to follow from its axioms.

But Swinburne’s assertion is, not that axiomatized Cantorian set theory is logically consistent, but that we can affirm the logical possibility of an infinite number of things without adopting Cantorian set theory. One can only wonder what Swinburne is talking about. Axiomatized Cantorian set theory is the standard form of set theory in mathematics today. Would Swinburne have us go back to pre-Cantorian mathematics with respect to the infinite? If so, how would that serve to vindicate the existence of an actually infinite number of things, since prior to Cantor only potential infinites were recognized? Or does he have in mind some sort of Intuitionist mathematics, which recognizes only infinites that are constructible? Again, it’s not clear how this would serve to avert the absurdity attending the existence of an actually infinite number of things. Without some further explanation, we’re just left in the dark as to what Swinburne would substitute for Cantor. In any case we can be confident that any non-Cantorian alternative to standard set theory would be virtually universally rejected by mathematicians.

But then Swinburne adds that we needn’t in any case be committed “this kind of way of applying” Cantor’s mathematics. Again, it’s unclear what he means. Hilbert’s Hotel, the illustration I use, is the brainchild of David Hilbert, one of the twentieth century’s greatest mathematicians and an ardent defender of Cantor’s set theory (though a sceptic about its actualizability in the real world). Hilbert knew well how to illustrate infinite set theory in the real world. So, we wonder, where did he go wrong? How would Swinburne apply the theory so as to the avoid the sorts of absurdities that attend the existence of an actually infinite number of real things? We could use Swinburne’s own example of an infinite number of stars to generate such absurdities—just assign a natural number to each star and then mentally execute similar moves as with the guests in Hilbert’s Hotel.

Swinburne’s second misgiving about this argument is that it assumes “that events that are all now past are in some sense actual. But in that case all the members of the infinite series of periods of unequal length, of 1/2 hour, 1/4 hour, 1/8 hour, etc., which have already occurred during the past hour, are also now actual, which . . . is not possible.” The point of this refutation is to show that on the kalam proponent’s own principles he finds himself committed to the existence of an actually infinite number of things, namely, temporal intervals of unequal duration.

The appeal to the actuality of the past is really a red herring, for Swinburne could have made the same point about spatial intervals. But, as I’ve argued elsewhere, the problem with this argument is that it just assumes our mathematical modeling of time and space as sets of points is descriptive of reality, which has never been proven. Moreover, the model assumes that an interval, whether spatial or temporal, is composed of points rather than points’ being constructed out of an interval. One may take the view instead that an interval is not composed of points but exists logically prior to any points that one might care to specify on it. That is to say, you don’t begin with an actually infinite number of points and build a line out of it; rather you begin with the line and start to make divisions of it. Thus, the series of divisions envisioned by Swinburne is potentially infinite only, in that the process of dividing can go on forever. Infinity in this case is merely a limit concept; there never is an actually infinite number of intervals.

There is a second kalam argument for the beginning of the universe based on the impossibility of forming an actually infinite collection of things by adding one member after another:

1. The series of events in time is a collection formed by adding one member after another.
2. A collection formed by adding one member after another cannot be actually infinite.
3. Therefore, the series of events in time cannot be actually ­ infinite.

Noting that Immanuel Kant expounded a similar argument, Swinburne responds, “But Kant’s claim that an infinite series cannot have a last member holds only of infinite series with a first member—which a beginningless series would not have.”

This response is far too facile. (In fact, I’ve been tempted for some time now to write an article “Was Kant a Dummkopf?” based on the sort of reasoning some of his critics have attributed to the titan of Königsberg.) To express premiss (2) as the contention that an infinite series cannot have a last member is misleading because it mutes the role which temporal process plays in the argument. It’s easy to think of an infinite series with a last member, for example, the series of negative numbers: . . . , -3, -2, -1. But Kant’s point is that it seems impossible to get to that last member by proceeding one member at a time. Just as one cannot count to infinity, so one cannot count down from infinity. To say that the infinite past could have been formed by successive addition is like saying that someone has just succeeded in writing down all the negative numbers one at a time, ending at –1, which seems impossible. Kant’s argument is not refuted simply by pointing out that the only infinite series which have a last member are also beginningless.

So much for the philosophical arguments; what about the scientific confirmation of the beginning of the universe? On this score Swinburne agrees: “My assessment of the present state of science is that this is what it does tend to show.” Of course, conclusions supported by scientific evidence are always provisional, but in Swinburne’s view that evidence does support the conclusion that the universe came into existence at some time in the finite past.

Swinburne thus accepts both premisses of the kalam cosmological argument on inductive grounds. This means that it enjoys the same sort of support as Swinburne’s own favored version of the cosmological argument. Given the modest force which he ascribes to his own version of the argument, the kalam cosmological argument therefore deserves to stand shoulder to shoulder with Swinburne’s own argument in his cumulative case for theism.

- William Lane Craig