Boxing Pythagoras

Philosophy from the mind of a fighter

Review of Craig v. Malpass, Part 1

On March 24th of this year, Cameron Bertuzzi’s channel on YouTube, Capturing Christianity, streamed a discussion between William Lane Craig and Alex Malpass. Nominally, the topic of debate was “Did the universe begin to exist?” However, their actual discussion was quite a bit more focused onto two very particular subjects. In part one of this review, we’ll look at the discussion of whether actual infinites are metaphysically possible. In the forthcoming part two, we’ll discuss the manner in which actual infinites are constructed.

As I am keenly interested in these particular questions, I was very excited for this discussion. I’ve discussed my contention with Dr. Craig’s treatment of the mathematics of infinity on a few occasions (most directly, here and here) but this particular debate brings forth some issues with which I have not previously engaged.

Dr. Craig opens the discussion by offering the position that actual infinites are metaphysically impossible. His argument, as he lays it out, is as follows:

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

Dr. Craig then offers that the crucial premise in this argument is the first and he goes on to attempt to defend that proposition. However, it is worth noting that the second premise is also contentious– indeed, Dr. Malpass will note that Aristotle, for example, agreed with (1) but denied (2). In fact, there are a significant number of modern philosophers of mathematics who would vehemently deny premise (2); and there are other reasons why one might be inclined to reject (2), as well. Indeed we will revisit this later, as I think that the argument is not at all as consistent with Dr. Craig’s position as he would like to believe.

In the meantime, however, let’s continue to review the first premise. The first and most well discussed reason which Dr. Craig offers in support of this position is the example of Hilbert’s Hotel, which (as he himself notes) is a favorite point of discussion for him. Indeed, it is rare for Dr. Craig to talk about infinities without bringing up Hilbert’s Hotel.

Dr. Craig maintains that Hilbert’s Hotel illustrates that actual infinites are absurd. The question, however, is what he means by this assertion. In philosophy, there is a technical meaning for the term “absurd.” This technical use of the word is meant to be equivalent to saying, “This set of propositions implies a contradiction.” That is to say, an “absurdity” is the recognition of an inconsistency in a logical argument. A rather classic example would be the phrase “John is a married bachelor,” which is absurd because the word “bachelor” implies “not married,” contradicting the statement that John is married. If a proposition is inconsistent then that gives us more than enough reason to reject it; after all, ex falso quodlibet.

However, that is not how Dr. Craig is using the word, “absurd,” in this case. Rather, Dr. Craig is using the term colloquially as a synonym for “strange.” Seeing this precise issue, Dr. Malpass notes that it may well be strange but it is certainly not inconsistent– a position with which Dr. Craig explicitly agrees– and one is left to wonder how strangeness is supposed to imply metaphysical impossibility. At no point does Dr. Craig actually address this. In fact, he admits that his argument would be entirely unconvincing to anyone unperturbed by this strangeness. So, it would seem that Dr. Craig’s argument is:

  1. Hilbert’s Hotel exhibits strange behavior.
  2. If Hilbert’s Hotel exhibits strange behavior then all actual infinites exhibit strange behavior.
  3. Therefore, actual infinites are metaphysically impossible.

This is clearly not even a valid argument, let alone a sound one.

As I noted, when replying to Dr. Malpass on this issue, Dr. Craig acknowledges that there are no inconsistencies in the treatment of actual infinites. Still, Dr. Craig remains convinced that despite being perfectly logically consistent it is somehow demonstrable that these actual infinites cannot be instantiated in reality. He then claims, “José Benardete, in his book on infinities, says that there’s no logical contradiction involved in these monstrosities but you have only to look at them in their concrete reality to see that this is metaphysically impossible.”


As someone who has actually read José Benardete’s Infinity: An Essay in Metaphysics, I would like Dr. Craig to point out exactly where it is that he supposes Benardete makes such a statement. I have not been able to find it and, indeed, that statement stands in direct contraposition to Benardete’s very explicit purpose in writing the book. The abstract on the inside cover of the book begins with the phrase, “This book is an attack on finitism in all its forms, philosophical and mathematical.” Throughout the whole of the book, Benardete defends the philosophy of actual infinites against both mathematical finitists, such as Kronecker and Brouwer, as well as metaphysical finitists, such as Kant and Hilbert. In his epilogue, he acknowledges that his work is a “metaphysical adventure” which “very much smacks of the quixotic,” but at absolutely no point does he ever concede that actual infinites are metaphysically impossible.

Dr. Craig’s claim, here, is a flat-out lie. I normally avoid accusing a person of intentional deception when reviewing these sorts of discussions. In general, I find it more charitable to think that the position which he’s referencing is misunderstood or that a purported implication is simply mistaken. However, Dr. Craig does not leave me such an option, in this case. He completely mischaracterizes Benardete’s work in an attempt to claim that it says precisely the opposite of what it actually, clearly, and explicitly states to be its entire purpose. Either he has not actually read Benardete and is trying to pretend that he has; or else he has read Benardete and is deliberately misrepresenting him. This is not a simple mistake. This is an overt lie.

Dr. Craig continues by referring to a particular paradox in Benardete’s work which imagines a book with an infinite number of pages from that work. Now, the reference to the infinite book is actually a discussion which Benardete does have in Infinity (pp 236-237); and Dr. Craig does lay out Benardete’s position fairly accurately, if one discounts his attempts to poison the discussion through his earlier lie. Benardete does, indeed, say that opening the back cover of the book, one would see nothing. And while he doesn’t explicitly state that attempting to touch the pages would cause one’s hand to be inexplicably stopped, he makes that claim for an exactly analogous paradox involving infinitely piled slabs of rock, so the inference that it would also be the case for the infinite pages is reasonable.

Incidentally, I disagree with Benardete as regards what one would see when opening the infinite book from the back cover. Certainly, one would not see the last page of the book, since there is no last page of the book to be seen. That does not, however, imply that one would see nothing. The problem with metaphysical thought experiments of this sort is that they run afoul of our physical understanding of the world.

One needn’t even posit an infinite book in order to have a problem. What if the book only had 110 pages? Well, in that case, the last page of the book would only be \frac{1}{2^110} inches thick. So let’s open the back cover of the book and take a gander at that final page. Considering that such a page would be thinner than the Planck length, what would it mean to even say that we were looking at it? What would it mean to try to touch such a page? Certainly, Dr. Craig would not tell us that the number 110 is therefore metaphysically absurd. Neither Benardete nor I would be inclined to think that extending this out to an infinite number of pages tells us anything more about metaphysics than does an example with 110 pages.

Now, one might object that, of course, in the actual physical world such things are impossible; however, we are talking about a metaphysical thought experiment, so such physical limitations ought to simply be ignored. However, the curiosities and absurdities which these thought experiments are supposed to be illustrating are physical ones.  What would we see? What would we touch? It seems silly to say, “Yes, this is physically impossible but it should be metaphysically possible!”

The infinite book, as well as each other paradoxical thought experiment which Benardete discusses, is absolutely no different than the idea of the Grand Hotel. Surely, the situation under discussion in each scenario is strange. This does not imply that actual infinites are therefore metaphysically impossible.


After this, Dr. Malpass introduces an objection to Craig’s position which he has co-developed with Wes Morriston. Now, Dr. Malpass completely acknowledges that this is not so much a direct objection to Craig’s arguments against the infinitude of the past as it is a claim that Craig’s philosophical view of the future is inconsistent with his view of the past.

According to Malpass and Morriston’s objection, the past and future bear a symmetry in respect to enumeration. On Malpass-Morriston, if it is impossible for the past to be infinite then it is similarly impossible for the future to be infinite. While Dr. Craig completely rejects the notion of the infinitude of the past, he holds to a notion of the infinitude of the future. So, if Malpass and Morriston are correct, then Dr. Craig must be incorrect either about the infinitude of the past or about the infinitude of the future.

In responding to Dr. Malpass’s acknowledgment that this is not a rebuttal of Craig’s argument against the infinitude of the past, Dr. Craig asserts that the Malpass-Morriston objection only applies to certain classes of people, such as “those who believe in personal immortality, or those who believe in angels, or… those who believe in God.” At this point, Dr. Malpass corrects Dr. Craig, noting that the objection applies to anyone who thinks that it is possible that the future is infinite. Immortality, angels, and God are red-herrings, and the objection applies with equal force to theist and atheist, alike.

Dr. Craig reacts to this correction by then claiming that the objection is just question-begging, in that case. He says that, “it just assumes that an endless future is possible even though you have an allegedly flawless argument that it’s not.” Of course, Malpass-Morriston does no such thing. It does not say that an endless future is certainly possible. Nor does it say that an endless future is certainly impossible. Rather, it claims that if an endless past is impossible, then it is also the case that an endless future is impossible.

After this, Dr. Craig says that if Malpass-Morriston is assumed, then it commits a person to hold that “there is no possible world in which the series of events has a beginning but no end.” Again, this is absolutely not the case. Rather, Malpass-Morriston claims that if there is no possible world in which the series of events has no beginning, then there is also no possible world in which the series of events has no end. Put another way, on Malpass-Morriston, if there is some possible world in which the series of events has a beginning but no end (as Dr. Craig claims is the case), then there must be a possible world in which the series of events has no beginning.

Unfortunately, Dr. Malpass seems to miss this distinction, himself, during the discussion (and in fact, he seems to somewhat back-up Dr. Craig’s misconception in the heat of the moment), so I have no idea how Dr. Craig might respond to this correction.

Moving on from this, Dr. Craig attempts to say that on his notion of a Tensed Theory of Time there is an asymmetry inherent in time itself which breaks the Malpass-Morriston objection. Interestingly, the moderator of the conversation steps in at this point with a rather poignant objection. Cameron Bertuzzi notes that Dr. Craig is a Presentist, and as such he believes that only the present moment exists while neither past nor future moments exist. Bertuzzi asks, “Well, if the past is not real, how can you say there is an infinite number of past events?”

Dr. Craig’s response is completely inadequate.

He says, “Um, because we can count them.” Well, what does that mean? What does it mean to count things which do not exist? If it is the case that we can count non-existent past events, it would seem that we can count non-existent future events just as well. If both are equally non-existent, then the fact that we can count them certainly doesn’t stand as a symmetry breaker for Malpass-Morriston.

Dr. Craig continues that past events “have been instantiated in reality.”  While this is certainly an asymmetry between past and future on Craig’s Tensed Theory of Time, it is not clear how this is meant to be an asymmetry between our ability to enumerate the events of the past and our ability to enumerate the events of the future. He clarifies that, “I’ve made it clear that when I say that an actual infinite cannot exist I mean that it cannot be instantiated in the real world.” However, even if the past is beginningless and even if past events have been instantiated, it does not follow that an actual infinite has been instantiated. None of those past events exist, on Craig’s view. There is no extant set of past moments which is infinite even if every past event has been instantiated and was preceded by some other past event.

This returns us to to Dr. Craig’s initial argument. As you’ll recall, his syllogism was as follows:

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

However, it seems that on Dr. Craig’s own view of time, it cannot be said that a beginningless series of events in time is an actual infinite. So even if (1) holds and even if one adopts Dr. Craig’s theory of time (which I have discussed at length, 1, 2, 3, 4, 5, 6), then (2) is not true, and the whole syllogism fails. Of course, on Dr. Craig’s view, a beginningless series of events still does not exist– but then neither does a series of events which actually has a beginning!

Perhaps Dr. Craig might insist that a series of events need not necessarily exist in order to be enumerated; however, that leads us right back to the Malpass-Morriston symmetry and implies that future events need not exist in order to be enumerated. When Dr. Craig asserts that past events have been instantiated, Dr. Malpass notes that future events will be instantiated; so that does not appear to be a symmetry breaker, either.

It is at this point that Cameron Bertuzzi breaks in, once again, to let both participants know that he will need to cut that particular discussion short in order to move on to the second half of the debate.

In summary, Dr. Craig asserts that actual infinites are metaphysically impossible; but the only two pieces of support which he offers for this claim are the fact that actual infinites are strange and the bald-faced lie that José Benardete said they are metaphysically impossible. He asserts that a beginningless series of events is an actual infinite despite the fact that this is inconsistent with his own view of the nature of Time. He claims that future events cannot be totally enumerated because they do not exist while simultaneously claiming that past events can be totally enumerated despite the fact that they do not exist. All in all, Dr. Craig does a very poor job of defending his position.

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