• Infinite Dreams

     William Lane Craig on the infinite – Part I


    During my path out of Christianity, I read and watched a lot of material from the Christian philosopher and theologian William Lane Craig: lectures, debates, books and journal articles.  As I began to think critically about the logical underpinnings of Christianity, I came to realise that the arguments Craig presented were not sound (even if many of his opponents in live debates were unable to adequately point this out).  I intend to explore Craig’s main arguments at length in future posts.  But one particular aspect of Craig’s thinking struck me as especially problematic, and that was his rejection of the infinite.

    Craig’s motivation for rejecting the infinite1 is that some of his central arguments attempt to establish the existence of a divine First Cause of the universe, something that “started it all off”.  Of course, if the universe didn’t actually have a beginning, then it couldn’t have a first cause.  So a beginningless universe seems like a problem for Craig.  One way a universe could be beginningless is for time to stretch infinitely into the past,2 and it is this that Craig seeks to show is impossible.

    Someone familiar with other aspects of his work might wonder why Craig would bother attempting this via philosophical arguments, as in other places he relies on the Big Bang Theory, which he takes to imply that the universe (including space and time itself) had a beginning.  But some modern cosmological theories3 interpret the Big Bang as an event leading from some state of another universe to ours (possibly in a black hole of some other universe).  So, with his First Cause arguments in mind, it seems sensible for Craig to argue that even if our universe was caused to exist (in some sense) by some other universe, which itself might have been caused by another, which was caused by yet another, then this process could not go back forever.  Craig wants to show that time itself, even if it transcends our own universe, could not be infinite in the past.

    In what follows, it will be convenient to formally refer to the proposition

    (P1)  Time could not be infinite in the past.

    This is the specific claim that Craig wishes to demonstrate.  To do this, Craig argues for the stronger proposition

    (P2)  Nothing could be infinite.

    Since (P1) is a special case of (P2), it is clear that (P2) entails (P1).  Conversely, (P2) clearly cannot be deduced from (P1), or from any single special case (or even from a handful special cases) – at least not without some extra argumentation.  But it seems that Craig’s attempts to demonstrate (P2) revolve around appeals to several special cases that he takes to be problematic.  For example, in his debate with Stephen Law, he dismissed the possibility of an infinite collections of coins, or of Jupiter and Saturn orbiting the sun since eternity past.  Elsewhere, he objects to the possibility of infinite hotels and libraries, counting backwards from infinity and other oddities.

    I do plan to address these special cases, and many others, in a subsequent post.  However, while this is an interesting path to take, it is nevertheless tangential to a critique of Craig’s main objective of establishing (P1).  As I have already said, (P2) cannot be established simply by pointing to a collection of special cases.  Such a method could only be successful if one was able to consider all special cases.  And, since (P1) itself is one of those special cases, it seems a rather futile way of proving (P2), as it is really (P1) that Craig wishes to establish in the first place!  Furthermore, the task of considering all special cases of (P2) must seem hopelessly impossible to Craig, since there are infinitely many special cases to consider.  Even if Craig objected that the special cases of (P2) form a merely potential infinite collection, I’m sure he’d agree it had turned into an actual infinite once the task had been completed.

    But with that said, I do actually think there is a reason for theistic philosophers like Craig to consider special cases of (P2).  For example, if I was suspicious of (P2), I might suggest scenarios that seem to be problematic for its proponents, and say that I could only consider accepting the proposition if such potential counterexamples could be adequately resolved.  I would not be doing this in an attempt to prove that (P2) is false.  Rather, I would be doing it to remind Craig of his burden of proof.   Since he has asserted (P2), and has not offered a general argument in its support, but instead attempted to prove it by considering various special cases, he must be prepared to refute any potential counterexample he is presented with.

    As it happens, there are many potential counterexamples to (P2), and I intend to present just one (which I don’t claim is original, as I’m aware of many similar scenarios floating around in the philosophical journals).  Consider a world in which God exists in some spatially unbounded heavenly realm.  Rather than creating a universe, God instead decides to create an infinite number of angels to keep him company.

    Now what is wrong with this possibility?  I don’t see why, if there really is a God that is as powerful as the one the Bible speaks of, he could not accomplish such a task.  Does it lead to logical absurdities?  I don’t really see how.  Perhaps we could grant that strange things might happen.  For example, if God could communicate telepathically with the angels, and if they could teleport instantaneously (neither of which seems unlikely, from reading various Biblical passages), then God could cause the angels to arrange themselves into an infinitely long line.  He could then cause them to rearrange themselves into two infinitely long lines, or three, or even infinitely many infinitely long lines.  But what is wrong with this?  On another day, God could choose to send infinitely many of the angels out of his throne room in such a way that there would still be infinitely many left with him.  Or he could have dismissed a different infinite collection of angels so that only three remained with him.  Sure, we don’t have experience of infinitely long lines of people in our everyday lives, or of varying answers to “subtraction problems”,4 but what is to say that such scenarios are impossible even for an omniscient and omnipotent God to accomplish?  There just doesn’t seem to be any glaring logical impossibilities.  A proponent of (P2) has the burden of demonstrating conclusively that such a scenario really is logically impossible before anyone else should even consider accepting (P2).

    In conclusion, I don’t know whether or not time extends infinitely in the past – in our universe, or a multiverse, or even just some logically possible world.  And I don’t think anybody else does either.  But it seems clear that Craig has not successfully demonstrated that time does not (or cannot) extend infinitely in the past.  It seems to me that neither a finite nor infinite past leads obviously to logical absurdities.  In fact, I would be more surprised if we could resolve such questions than if they remained unanswered forever (whatever “forever” might mean).



    1.  Going back to Plato and Aristotle (look up), Craig distinguishes between two kinds of infinities: “actual infinities” and “potential infinities”.  I won’t go into too much detail here, but the former is meant to correspond to some kind of real existent infinite collection of things (such as an infinite number of books), while the latter merely reflects infinite possibilities (such as a universe or heavenly realm that extends indefinitely into the future, even though all the “days” have not happened yet).  There are blurry edges and, for example, some philosophers disagree on whether a (real or hypothetical) infinite past and/or future would constitute an actual or potential infinity; for example, see Wes Morriston’s article Beginningless Past, Endless Future, and the Actual Infinite.

    2.  Another way would be for time to be an “open interval”.  In this scenario, there would be a point in the past beyond which time did not reach, but there would not be an actual moment of time at this point.  By way of analogy, let S denote the set of all real numbers x such that 0<x<1.  Although the number 0 is less than every element of S, 0 is not itself an element of S.  In fact, S does not possess a minimum element.  So, for every element of S, there is a smaller element of S.  For considerations like this relating to the logical possibility of a beginningless universe that only extends finitely in the past, see Quentin Smith’s article A Cosmological Argument for a Self-Caused Universe.

    3.  Smolin Selection Theory or Chaotic Inflation Theory or Cyclical Models, to name just a few.

    4.  These examples respectively correspond to the fact that any infinite set may be partitioned into a finite or countably infinite number of equinumerous subsets, or that any infinite set has various equinumerous subsets with complements of varying cardinalities.  While these properties are not shared by finite sets, one should not be surprised that infinite sets behave differently to their finite counterparts.


    Category: Cosmological argumentGodInfinityMathematicsWilliam Lane Craig


    Article by: Reasonably Faithless

    Mathematician and former Christian