The Semantics of Nothing

0.   Introduction

The word ‘nothing’ has interesting semantic features. It is a ‘negative existential’, in the sense that it refers to a non-existing thing. This is perplexing, because if ‘nothing’ is a simple referring term, then the semantic role that it plays in contributing to the meaning of a sentence it features in is to point to its referent. As it has no referent, how can it play this role successfully? There are two general strategies for dealing with this puzzle; one is to treat the idea of nothing as a sort of thing, and the other is to treat it as a case of failure to refer at all.

1.   Creation from nothing

The term ‘nothing’ is deployed as part of one of the supports for the Kalam cosmological argument. The first premise of that argument is: ‘whatever begins to exist has a cause’. One of the lines of support for this premise is the familiar dictum ‘nihilo ex nihilo fit’, or ‘nothing comes from nothing’. When pressed on why this is true, a typical line of defense is that ‘nothing has no causal powers’. I say that this sentence is ambiguous, due to the word ‘nothing’. On one account the sentence treats ‘nothing’ as a referring term; something like ‘the complete lack of any object’. On the other account, the term expresses a failure to refer to any thing. The first reading (which I shall call the ‘referential sense’) is the intended sense, but it strikes me as ad hoc (and I will explain this more below). The second sense (which I shall call the ‘denotative sense’) expresses a different proposition altogether – one that fails to support the premise in any way.

2.   A Toy Example

The ambiguity can be brought to the surface if we consider the two semantic accounts of the word in more detail. Before we look at the sentence ‘nothing has no causal powers’, I want to first play with a less controversial example, to get the distinction clear. So my toy sentence is:

1) ‘Nothing will stop me getting to work on time’

First, let’s look at the referential sense of ‘nothing’, as it applies to this sentence. On this account, ‘nothing’ is just another referring term, like ‘John’, or ‘Paris’, or ‘my favourite type of ice cream’, etc. The referent of ‘nothing’ is ‘the complete absence of any things’, or something along those lines. It’s like an empty void with no contents whatsoever.

The sentence is essentially of the form ‘x will stop me getting to work on time’, where ‘x’ is an empty variable waiting to be filled by any constant (or referring term), like ‘John’ or ‘my favourite type of ice cream’, or ‘nothing’ etc. Let ‘Wx’ be a predicate for ‘x will stop be getting to work on time’. If ‘a’ is a constant that refers to my friend Adam, then the proposition ‘Wa’ means that Adam will stop me getting to work on time. I will not get to work on time, because I will be stopped by Adam from doing so. Something that Adam will do, such as physically restraining me, or hiding my keys, or just distracting me with an interesting philosophical discussion, etc, will prevent me from getting to work on time. That’s what Wa is saying.

Let ‘n’ be a constant that refers to ‘the complete absence of anything’. We could put the logical form of 1) as follows:

Ref)   Wn

Ref says that I wont get to work on time because ‘nothing’ is going to stop me. This mirrors the logical form of the sentence above where Adam prevented me from getting to work on time. But this seems wrong, as 1) doesn’t seem to say that I won’t get to work on time because of nothing (i.e. the complete absence of any thing) getting in my way. We don’t seem to be expressing the idea that ‘nothingness’ is going to hide my keys, or engage me in a philosophical discussion, etc. We are not expressing that I will not get to work on time. Rather, we are expressing something close to the opposite of that; the sentiment expressed by ‘nothing is going to stop me getting to work on time’ is that I will be on time to work, come what may. So the referential way of reading the term ‘nothing’ is not appropriate here.

Let’s look at the second account, the denotative account. On this reading 1 gets analysed out as the following (note that we still have the predicate Wx, but use a quantifier and a bound variable and so don’t need the constant ‘n’):

Den)    ~(∃x)(Wx)

On this reading, we are saying that it is not the case that there is a thing such that it will stop me getting to work on time. We could re-write Den as follows:

Den’) (∀x)~(Wx)

Den’ says that for every x, it is not the case that x will stop me getting to work on time. This captures very well the sentiment that come what may we will not let anything prevent us from getting to work on time. We would say that the denotative proposition is true in this situation, and that seems right.

Thus the two analyses are very different. They render propositions with a different logical forms and different truth-values in this case. In the referential case, we are referring to an entity, and saying of that thing that it will succeed in preventing me from getting to work on time. So the logical form of the proposition, when analysed referentially, is wrong. In addition to this semantic or logical issue, we also have a metaphysical or ontological worry. We may feel that the entity referred to in Ref is of dubious ontological status. Nothing doesn’t exist; it isn’t a thing as such. Successful reference seems to have as a presupposition that the referent exists in some sense or other. If that is right, then when we successfully refer to ‘nothing’ then there is something which is the referent for the term ‘nothing’. But if there is some referent, then ‘nothing’ doesn’t mean the complete absence of any thing. It may be that the combination of this model of reference with the insistence of ‘nothing’ meaning the complete absence of any thing is incoherent. So we can feel dissatisfied with Ref here for both ontological and logical reasons.

We may want to avoid this problem by postulating that ‘nothing’ refers to an entity, yet what it refers to is not an existing thing. Nothing is, even though it doesn’t exist. It is a something, just not an existing something. I find this way of talking almost unintelligible. It seems to me as a bedrock metaphysical principle that there are no non-existing things. There is not two types of existence; rather there is only one type of existence. If ‘nothing’ is, then it exists. The terms ‘is’ and ‘exists’ are synonymous. In this regard, I find Russell (On Denoting) and Quine (On What There Is) to be instructive.

Den, on the other hand, does not refer to any thing of dubious ontological status. When recast in the form of Den’ it clearly and explicitly quantifies over all the things that there are and says of those things that none of them are going to stop me getting to work. So it has going for it that it captures the intention behind the sentence, in that it captures that I will not be stopped. Den doesn’t require postulating two types of existence. We don’t have to say that ‘nothing’ is yet does not exist. We do not directly refer to ‘nothing’, we just refer to what there is (and say that it is none of those things).

The difference between Ref and Den could be put like this: the former is a successful reference to something that does not exist, the latter is a failure to refer to anything which does exist.

3.   The Main Case

Let’s apply this to our example of ‘nothing has no causal powers’. Let’s rewrite having no causal powers as being ‘causally inert’, and represent that as a predicate, ‘Ix’. On the referential reading, the sentence has the form:

Ref2)   In

This says ‘nothing is causally inert’. As we have seen, the model of reference used here treats nothing as a referent of the term n, which means it is the thing referred to by n. The proposition is true only if the referent of n, i.e. nothingness, is actually causally inert. And nothingness, as conceived as an empty void with no contents whatsoever, is plausibly causally inert. So the claim seems to capture well the intention behind the apologist’s assertion here. The reason that the universe couldn’t have ‘popped into being from nothing’ is that ‘nothingness’ has no abilities to make things pop into existence. It cannot do anything; it is causally inert.

The denotative reading would be as follows:

Den2) ~(∃x)(Ix)

This says that it is not the case that there is a thing such that it is causally inert. Recast in universal terms, it says:

Den2’) (∀x)~(Ix)

This says that everything is such that it is not causally inert; everything has causal powers. On this reading, we are effectively saying that abstract objects, and similar proposed causally inert entities, do not exist; there are no abstract objects, etc. This is because abstract objects are causally inert, and Den2 says that there is no causally inert thing.

One would suppose, looking at this that in the case of nothing having no causal powers, we should take the referential reading, as this makes sense of the apologist’s claims about how the universe had to have a cause. It is clearly not their intention to assert that causally inert objects don’t exist; they mean to assert that the complete absence of anything cannot itself cause something.

In the toy example, when we distinguish the referential and denotative sense of ‘nothing’, it is clear that the referential sense is incorrect. It entails something which is clearly not intended by the speaker, that I will not get to work on time, when we meant to express that come what may I will get to work on time. In the apologetical example, the analysis seems to go the other way; the denotative sense seems to entail a proposition which clearly isn’t what the apologist intends. So, while the toy example is denotative, the apologetical example is referential.

I have two worries with this conclusion:

a) If we take the referential reading of ‘nothing’ in the phrase ‘nothing has no causal powers’, then we are referring to an entity that is of questionable ontological status. It is the referent of the term n, yet it is the complete absence of any thing. So it is a thing that does not exist. We might want to follow Russell in On Denoting, and Quine in On What There Is and disallow such talk of non-existing things. Indeed, we may consider such talk of nothing as a dubious case of reification. Nothing is not a thing of any type whatsoever.

b) This is my main worry. It seems to me that most cases of the word ‘nothing’ are denotative, and almost none are referential.

Here are a few examples:

  • ‘There is nothing to split the two candidates with only days before the election.’
  • ‘There is nothing I like better than ice cream’
  • ‘Nothing pisses me off more than ice cream’
  • ‘You mean nothing to me’
  • ‘There is nothing in the fridge’

The first four cases are clearly denotative (just plug in the different readings of ‘nothing’ and see for yourself in each case). Possibly in the last example, we may want to use the referential sense, but the denotative sense seems at least as plausible. Are we expressing that there is an absence of any thing in the fridge, or that there is not any existing thing in the fridge? Neither seems preferable.

My question is: can there be an example of a sentence that uses the word ‘nothing’, and isn’t the clearly apologetical ‘nothing has no causal powers’ etc, or some other esoteric metaphysical example, for which the referential reading is clearly the correct one (and not the denotative one)?

Are there ever cases where the referential sense is the correct one, apart from the use in things like supporting the Kalam? If the answer to this question is ‘no’, then the use by the apologist is ad hoc in the support for the Kalam case. This is an open question (feel free to suggest candidate sentences in the comments section). If there is a plausible looking case, then the charge of ad hoc-ness can be deflated.

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Creation ex nihilo

0. Introduction

I have recently come across a blog written by Richard Bushey, which has lots of typical apologetical arguments summarised by the author. As such, it is an interesting place to look around to find typical bad arguments to straighten out.

Here I want to look at one in particular, not because there is anything original about it, but really because there is nothing original about it. The post is an example of the sort of regurgitation of arguments made by people like William Lane Craig that one often encounters on the internet. Here is the post, entitled ‘Can a universe emerge from absolutely nothing?‘. In it, Bushey explores the idea of the creation of the universe ex nihilo (or ‘from nothing’), and rehearses some of the common arguments for why this isn’t possible.

  1. Setting

The setting for the topic discussed in the post is ultimately the cosmological argument (probably specifically the kalam cosmological argument popularised by William Lane Craig, on which I have written before). The idea is that one of the arguments put forward to prove the existence of God is that the existence of the universe requires a causal explanation, which could only be God, as a necessarily existing being. The response to this that Bushey is addressing here is to basically call into question whether the universe requires causal explanation. As he explains:

Many people seem to take it for granted that things do not just appear with absolutely no cause. But it would be quite convenient for the atheist if it were the case that this were a possibility. Atheism would then be able to deflect one of the seminal arguments for the existence of God. We need to be able to provide some justification for thinking that universes cannot emerge from absolutely nothing.

Bushey offers five distinct points, and I want to look at three of them (I have nothing to say of any note about quantum vacuums, and am happy to grant that God doesn’t need a cause to exist, at least for now). The three points I will address here are labelled by Bushey as:

a) ‘Nothing’ has no causal powers.

b) What if universes could come from nothing?

c) A good inductive conclusion.

      3. ‘Nothing’ has no causal powers

As the title of this section suggests, Bushey is arguing here that the reason the universe has to be caused by something, such as God, is that nothing is itself not able to cause anything. As an intuition pump to get you in the mood to agree with him, Bushey offers the following examples:

If your co-worker was taking a day off, the boss would naturally ask, “Who is going to cover your shift?” If the coworker said, “Nobody,” the boss would be concerned. ‘Nobody’ has no causal powers. They cannot perform the function of the job because ‘nobody’ designates the absence of somebody. Similarly, if I said that “There is nothing to eat,” my stomach would be empty. If I said that there was nothing that could stop the invasion of a particular army, I would be expressing that the military force would go unchallenged. 

Now we have the idea of what it means to say that ‘nothing’ lacks causal powers. ‘Nothingness’ cannot play the role of a co-worker, satisfy an empty stomach, or impede an oncoming army. Nothingness can’t do anything. Given that primer, here comes the beef:

So when atheists tell us that a universe could emerge from absolutely nothing, or attempt to provide accounts of how nothing could have produced the universe, they are expressing an incoherent thought. If ‘nothing’ designates the absence of anything at all, then it follows that there are no causal powers. If there are no causal powers, then it lacks the capacity to produce universes.

Given that nothingness cannot fill-in for an absent waiter’s shift in a cafe, it seems perfectly reasonable to extend this to think that it cannot manufacture universes either.

So, what is wrong with this? Well, we might already be suspicious of the first example. The boss might be concerned with the fact that nothingness has no causal powers, but I would suggest that it is more likely that he is really concerned about the lack of something to fill in which has the relevant causal powers. And these are not two ways of saying the same thing. It is not like the co-worker said ‘Don’t worry boss – nothingness will fill in for me’, to which the boss replied ‘Oh no, not bloody nothingness again! It’s complete lack of causal powers always ends up causing me grief when it comes time to tidy up at the end of the evening!’ By saying that nothing (or nobody) is going to fill in for you at work, you are saying that there is no thing about which it is true that that thing is going to fill in for you at work; you are not saying that there is this thing called ‘nothing’, about which it is true to say that it is going to fill in for you at work. We must keep these two subtly different understandings entirely distinct when we think about this, or else we are led down a garden path of confusion by Bushey here.

Consider Russell’s treatment of negative existentials in On DenotingI might want to express the fact that I don’t have a sister by saying ‘my sister does not exist’. On face value, we might think that the best way to think about the semantic value of such a phrase is as a referent about which it is true that she doesn’t exist; as if I refer to a non-existent entity. However, says Russell, far better would be to think about it like this: we are simply saying that for all the things that do exist, none of them are my sister. The propositional function ‘x is my sister’ is false for all existing things.

Let’s apply this to the boss example. Is the boss worried that a) there is a non-existent entity, who has no causal powers, filling in for a shift, or is he worried that b) for all the things that there are with the relevant causal powers, it is false that any of them is filling in for the shift? I see no reason at all to suppose that the best way of reading that situation would be by stipulating a), and every reason to suppose that it would be b). Unless Bushey has some additional argument as to why this reading is not acceptable, we at least seem to have an unproblematic rendering of this example here.

Let’s apply this to the universe example. If an opponent of the cosmological argument (who may or may not be an atheist) suggested that maybe nothing caused the universe to exist, which of the following would be be better to render this as:

a) Before the universe existed, there was nothingness, and that caused the universe to come into being.

b) For all the things that there have ever been (in any sense whatsoever), none of them caused the universe to exist.

Again, I see no reason to think that a) would be the intended meaning of such a suggestion, and every reason to think that it would be b). When someone says that ‘nothing caused the universe to exist’, they just mean the propositional function ‘x caused the universe to exist’ is false for all values of x, not that there is a value of x, called ‘nothing’ about which it is true.

Even saying that ‘nothing lacks causal powers’ is already wrong. ‘Nothing’ isn’t a thing. It is shorthand for ‘it is not the case that there is a thing’, i.e. the negation of an existential quantifier: ¬∃. So, taken literally, the phrase ‘nothing lacks causal powers’, would be rendered as follows (where ‘Cx’ is ‘x has causal powers’):

¬∃x (¬Cx)

Using nothing but the definition of the universal quantifier, we can prove the following equivalence in classical logic:

(¬∃x (¬Cx))  ↔  (∀x (Cx))

This just shows that the phrase ‘nothing lacks causal powers’ logically just means the same as ‘everything has causal powers’. Reifying ‘nothing’ to the status of an abstract object, with no causal powers, is just to misuse language; a crime which is unforgivable when there is a logically straightforward, and existentially unproblematic, analysis available.

4. What if universes could come from nothing?

Bushey has another go at providing some reason for thinking that the universe could not have come from nothing. This time he picks up on another well rehearsed argument from William Lane Craig. The idea this time is that if someone wants to hold that the universe might have come into being out of nothing, then why think that only universes could come into being out of nothingness? Here is how Bushey puts it:

Suppose for a moment that it were true that things could appear without any cause at all. If that were the case, then our rational expectations for the universe would seem to be unjustified. It would become inexplicable why anything, and everything did not emerge without a cause at all. This point was charmingly made by Dr. William Lane Craig in his debate with Dr. Peter Slezek. He pointed out that nobody is concerned that as they are sitting in this debate, a horse may have appeared uncaused out of nothing in their living room and is currently defecating on the carpet as we speak.

The idea seems to be that if we grant special exemption to universes being able to come from nothing, we would be rationally compelled to extend this to cover everything. We should expect random things popping into existence all the time, yet we don’t. We implication is that we don’t have this expectation because we know that things require causes to come into being, and cannot come into being in the absence of causes.

So, should we give a special pass to universes? Isn’t that special pleading if we do so? I say it isn’t, and that again there is a subtle but powerful misunderstanding about nothingness which is driving this line of argument.

Take the idea of a horse just appearing in front of you and defeacting on the floor. We know this isn’t going to happen (setting quantum probabilities to one side). But why do we know this? I say that the reason for this isn’t because we know that things cannot come from nothing. That idea isn’t even relevant. If you are at home in your front room wondering if a horse is about to suddenly appear, that isn’t an example of nothingness! What you know is that the relevant causal properties of what exists around you isn’t sufficient to produce a horse. You know that a horse cannot be produced by this particular type of something.

Let’s turn to the idea of the universe. Given the understanding gained from section 3 above, we do not have to think of ‘nothingness’ as preceding and causing the existence of the universe. We could just say that there is no thing (in any sense) that preceded and caused the universe. The beginning of the universe is the beginning of everything. So, the context which was not conducive to a horse popping up in front of you in the previous example has no counterpart here. There is no ’empty space’ into which the universe pops. There is no ‘nothingness’ waiting to be filled with a universe.

Could an infinite empty void of nothingness suddenly give rise to a universe? I don’t know. Could the universe simply be all that there is? I don’t see why not. Pointing out that horses don’t suddenly appear in front of us randomly is completely irrelevant.

5. A good inductive conclusion.

This last point is quite similar to the previous one, and has a similar root of misunderstanding with it. Here is Bushey again:

Common experience indicates that things have an explanation. They do not just appear, uncaused, out of absolutely nothing. The entire project of science is predicated upon this premise. Science is the search for causes within the natural world. If we were to establish the premise that things appear without a cause, then the project of science would be wholly undermined. Scientists who searched for causes of natural phenomenon would be engaging in a fruitless endeavor. It may just be that their specimen emerged without a cause. Why does a fish have a particular gill? Perhaps it appeared, uncaused, out of nothing.

It is quite easy to spot the error here. Take the fourth sentence in that quote: “Science is the search for causes within the natural world”. I don’t think this is the best definition for science one could find, but it is particularly bad that it is the one Bushey uses in this context. If science is the search for causes within the natural world, then there is no reason to think that it applies to things beyond the natural world. Just because things in the universe behave a certain way, doesn’t mean that the universe itself has to display those behaviours. Say everything in the sea floats, would it follow that the sea floats? If there is no causal explanation for the universe, which simply is all that there is, it would not follow that things that actually exist could not be described by science, or that we would have no reason to think that every particular fact in the universe had a causal explanation.

6. Conclusion.

There is no reason provided in Bushey’s post to think that the universe has to have a cause. One should resist the temptation to reify nothingness into an amorphus blob lacking in certain properties. Don’t slide from a failure of reference to an existent thing, to a successful reference to a non-existent thing. The universe didn’t pop into existence from a pre-existent state of nothingness. It just has a finite past.

At least, maybe it does. I don’t know whether the universe was created or not. Maybe a loving personal god made it in order to teach me about morality. Maybe it popped into existence from a pre-existing state of nothingness. Maybe it is just all there is. My point is that you don’t get to prove the first of these by undermining the second, given that there is a coherent third. That would be a fallacy of false dichotomy.

Craig’s List – Omniscience and actually existing infinities

Introduction

William Lane Craig has famously argued for the ‘Kalam cosmological argument’ (in many places, but for example in Craig & Sinclair [2009]). Here is the argument:

  1. Everything that begins to exist has a cause.
  2. The universe began to exist.
  3. Therefore, the universe had a cause (Craig & Sinclair [2009], p 102).

The argument is clearly valid, as it is a version of modus ponens. Thus, in order to deny the conclusion, one must argue that the first or second premise is not justified.

Most people have argued against premise one, disputing whether all things which begin to exist have causes for their existence, or the fact that a fallacy of composition may be at play with the generalization from all things in the universe to the universe as a whole. I will not be pursuing this line of argument here, but will instead look at premise two.

Premise two seems to be supported by physics, specifically cosmogony, which some say indicates that the spacetime we exist within came into existence at the big bang. People who know more about this than I do tell me that this is actually a misconception of this theory, and that it is not really a theory about the origin of spacetime at all. However, we can avoid delving any further into the details of the physics, because Craig does not rest his argument on the interpretation of the big bang theory. There is a logical argument Craig spends time going into, according to which the universe must have had a beginning – that it is impossible for the universe to have always existed. Here is that argument:

2.1. An actual infinite cannot exist.

2.2. An infinite temporal regress of events is an actual infinite

2.3. Therefore, an infinite temporal regress of events cannot exist. (ibid, p 103)

It is on this supporting argument that I wish to focus. Specifically, it is the first premise of this argument that I will be spending time going into here. If we can knock this premise out, then it undermines the entire supporting argument, and with it the credibility of the main argument. If we can deny 2.1, we can avoid having to assent to 3.

Hilbert’s Hotel

In order to motivate 2.1 (that an actual infinite cannot exist), Craig uses the example of ‘Hilbert’s Hotel’. In this imagined hotel there is an infinite number of rooms. Infinity has a distinctive property, according to which a proper subset of it can be equal in cardinality to the whole, there are various counter-intuitive consequences, which Craig uses to motivate the idea that this could not actually exist. For example, if the hotel is full but a prospective guest arrives asking for a room, the hotel manager can simply ask each occupant to move into the next room, thereby making room number one free. Because there is an infinite number of rooms, there will be room for every occupant, thus making a newly free space for the new guest to stay in, even though the hotel was full. Even if infinite new guests turn up, the hotel manager can make room by getting everyone in the hotel to move into the room with the room number that is twice the number of their current room (so room number two gets room number four, room number four gets room number eight, etc.). This frees up an infinite number of rooms, even though the hotel was full. Craig comments:

“Can anyone believe that such a hotel could exist in reality? Hilbert’s hotel is absurd. But if an actual infinite were metaphysically possible, then such a hotel would be metaphysically possible. It follows that the real existence of an actual infinite is not metaphysically possible” (Craig & Sinclair [2009], p. 109-110).

If this is correct, then because a universe with no first moment would constitute an actually existing infinity, it follows that the universe had a first moment. Thus, the idea is that it is no objection to simply say that maybe the universe always existed. It couldn’t have always existed, says Craig.

However, it is not clear to me that his objection really applies to the universe, and I will spell this out in more detail now.

Pinning down the absurdity

One might wonder what specifically it is about Hilbert’s hotel that Craig finds absurd. It seems that the sheer scale of the hotel, the fact that it has infinite rooms, is not itself absurd to Craig. If it was, then the example would simply have been:

‘Imagine that there is a hotel with infinite rooms in – that’s absurd!’

Given that the example was more complex than this, it seems that just saying that the hotel is infinite is not enough for Craig to bring out the absurdity. Nor does simply adding that the hotel actually exists constitute the absurdity, otherwise the example would have been:

‘Imagine that there is a hotel with infinite rooms in, and that it actually exists – that’s absurd!’

Surely, when picturing Hilbert’s hotel, one pictures it as actually existing. Adding that it actually exists is somewhat empty as a property, and surely not enough on its own to make the difference between not absurd and absurd. So what is it that pushes us over this threshold?

It seems to me, given the examples used to illustrate the absurdity of Hilbert’s hotel, that Craig’s idea is as follows. The factor that gets us across the line is what we might call the behavior of the hotel. With an infinite hotel, given certain conditions obtaining, contradictions can be manifested, and contradictions are absurd. So it took the new guest to arrive, and for everyone to shuffle up one room, for an absurdity to become manifested; namely, the hotel is full, but also has a room available for a new guest. If the guest does not arrive, or arrives but is turned away by the manager, then where is the absurdity? How do we generate a contradiction without interacting with the hotel? It seems like the only way we could imply an absurdity in that case would be simply pointing out that the hotel has infinite rooms. But if this was on its own enough to constitute absurdity, why bother with the example of the guest arriving? Is it just for rhetorical effect? It seems to me that the answer is that without the guest arriving and the creation of the new free room, Craig thinks that nothing absurd is present.

If this right, then we could employ a distinction between active and passive infinities. An active infinity is one that manifests absurd behavior (like being full but also making room for a new guest), whereas a passive infinity is one that does not (like a Hilbert’s hotel which never admits new guests). Now, it should be noted that a passive infinite retains the potential to manifest absurdity; it is passive just so long as it doesn’t actually do so.

This makes the distinction between ‘actually existing’ and ‘not actually existing’ slightly wide of where the beef is here. It seems we could have an actually existing Hilbert’s hotel, which remains passive, and for all Craig has said, this would not be absurd. The absurdity only kicks in when an actually existing infinity becomes active.

The infinite universe is passive

The problem with Craig spelling out the nature of the absurdity associated with actually existing infinities like this, is that it doesn’t apply to the eternally existing universe. There are models where we could make his objection apply, but the most natural way of cashing it out avoids his problem, as I will explain.

Imagine a number line that contains all integers running from minus infinity, through 0 all the way up to positive infinity. Now think of 0 marking out this very moment now. This is a bit like the most natural way of thinking about the eternal universe; each moment has infinitely many earlier moments and infinitely many later moments. If this is how Craig is characterizing the eternally existing universe, then it is a passive infinity. There is no corresponding example to making a free room, or withdrawing a book. One cannot add a moment to time, nor take one away. It is a ‘closed’ infinity. In fact, it is arguably metaphysically impossible to add a time or take one away. Thus, Craig may be correct that active infinities are metaphysically impossible, but because the eternal universe is not one of these, then he has no objection to the eternal universe.

As I said, there are ways of cashing out the eternally existing nature of the universe according to which Craig’s point holds. For example, consider the ‘growing block’ theory of time. According to this theory, the past is a fixed set of facts, which is growing as time moves forwards. We continually add new truths to the stock of settled past truths. If this were the model, then we would have an infinite list of past truths, but we would be able to add to it. In a sense, this would resemble Hilbert’s Hotel and thus make the universe an active infinity.

It should be noted that even on this growing block theory, there is room to doubt whether this really counts as an absurdity. With the hotel example, we can derive a sort of contradiction, in the sense that the hotel was full, but had room for a new guest. If being full means that there is no room, then this is a contradiction. But it is not clear what is the contradictory sentence we are supposed to be able to make out of the growing block theory here. Sure, there are infinite past moments, and then a new one gets added to the pile as time moves forward. The only contradiction I can see here is that the cardinality of the past moments is the same, even after a new one is added to the block. If so, then we have our candidate.

It is a weak candidate, as it seems to me that we ought to simply accept that this is what an infinite block would be like. However, let’s assume that Craig has scored his point here, and that the growing block theory is absurd for that reason. No such account can be leveled at the eternal universe outlined above. It has an infinite number of moments, but there is no possibility of adding new moments or taking them away, so it is passive. It seems like we can block Craig’s argument by simply explaining clearly what an eternal universe looks like, and that while it is infinite in extent, it manifests no absurdity.

In fact, this will form one horn on a dilemma I wish to place Craig in. As we shall see, if there is a problem with the growing block theory, then it also affects Craig’s version of God. The dilemma will be that either the universe is infinite in temporal extension, or God doesn’t exist.

The Infinite God Objection

Craig’s God is omniscient. This means that ‘God knows only and all truths’. Watch him commit to this position here:

It is uncontroversial that there are mathematical truths, like that it is true that 2 + 2 = 4. God knows all these truths as well (Craig explicitly makes this point at 6:20 in the video above). To make the point as simple as possible, God knows the solution to every equation of the form x + y = z, where the variables are natural numbers. As there is an infinite number of such solutions (with a cardinality equal to the smallest infinity, ℵ0), it follows that God’s knowledge is correspondingly at least as infinite as the cardinality of the natural numbers (and obviously greater if he also knows all real number solutions as well).

Let’s consider Craig’s God’s knowledge of these arithmetic solutions as a list of truths, which we could call ‘Craig’s List’. It would be an infinitely long list. So Craig’s God’s knowledge is infinite.

But, according to the Hilbert’s Hotel argument from above, the infinite cannot actually exist. Therefore, an omniscient God cannot actually exist. Craig’s God is omniscient. Therefore, by his own argument, Craig’s God cannot exist.

Call this the ‘Infinite God Objection’.

God’s knowledge is of induction schemas

It could be objected here that God does not need to know every arithmetic truth, such as 2 + 2 = 4, because as long as he knows the base case and all relevant induction schema, he would know enough to deduce the answer to any similar equation. If this were the case, then it would drastically limit the amount of propositions God would need to know, from infinite to a mere handful.

My response to this is that if this were all that were required to know all mathematical truths, then I know all mathematical truths. After all, I know the base case (that 0 is a number) and the relevant induction schema. God and I both have the same resources at hand, and if this is all it takes to know all mathematical truths, then we both know all mathematical truths. This is an awkward consequence, to say the least.

But this consequence is not just awkward. It is intuitively true that there are lots of arithmetical equations that I do not know the answer to, even though I could work them out given my knowledge of the induction schema. It seems more natural to say that I do not know the answers to these questions, but I know how to work out the answers. This makes the response in the God case inadequate though. To concede that God does not know the answer to any mathematical question, but knows how to work out the answer, is just to concede that there are things he does not know. The fact that he could work it out it not a defeater to the claim that he does not know it.

On the other hand, perhaps the similarity is only apparent, and that due to my limited nature, as compared to God’s unlimited all-powerful nature, there is a meaningful difference between the two cases. Perhaps it is the case that I slowly lumber through, applying the schema to the case at hand to derive the answer, and with the possibility that I could always go wrong on the way. In contrast, God applies it at lightening speed, without the possibility of getting it wrong on the way. In this case, there is no arithmetic question you could ask God to which the answer would be ‘I don’t know, but I will work it out for you’; as soon as you have asked the question he has already worked it out. Therefore it is never true that there is something he does not know.

But I could just stipulate an equation, without asking God directly. Even though, were he to think about it he would get the answer immediately, given that he is not currently applying the schema to the case, it is not true that he knows it. So there is something he doesn’t know. So he is not omniscient.

And if we avoid this by saying that he is constantly applying the schema to all cases, then we are right back to the original case, where he knows an infinite number of truths.

Thus this escape route will not help.

God’s knowledge is non-propositional

Craig could say that God’s knowledge is non-propositional, as in the Thomist conception. On this idea, God does not know lots of individual propositions, but rather has one unified knowledge of himself, which is perfectly simple.

To begin with, this contradicts his statements in the video above, where Craig explicitly states that God knows all propositions. Perhaps we can let this slide, as it is him talking somewhat informally.

In a paper entitled ‘A Swift and Simple Refutation of the “Kalam” Cosmological  Argument?‘ (1999), Craig considers a very similar objection, namely that if mathematical truths are just divine ideas, then God’s mind has infinitely many ideas. In defense of the divine conceptualist, Craig offers the following reply:

“[T]he conceptualist may avail himself of the theological tradition that in God there are not, in fact, a plurality of divine ideas; rather God’s knowledge is simple and is merely represented by us finite knowers as broken up into knowledge of discrete propositions and a plurality of divine ideas.” (Craig, (1999), p 61 – 62).

This theological tradition goes back to Thomas Aquinas, and as an explanation of this, Craig cites William Alston’s paper ‘Does God have beliefs?’ (1986). In that paper, Alston says the following:

“[C]onsider the position that God’s knowledge is not propositional. St Thomas Aquinas provides a paradigmatic exposition of this view. According to Aquinas, God is pure act and absolutely simple. Hence there is no real distinction in God between his knowledge and its object. Thus what God knows is simply His knowledge, which itself is not really distinct from Himself. This is not incompatible with God’s knowing everything. Since the divine essence contains the likenesses of all things, God, in knowing Himself perfectly, thereby knows everything. Now since God is absolutely simple, His knowledge cannot involve any diversity. Of course what God knows in creation is diverse, but this diversity is not paralleled in the intrinsic being of His knowledge of it. Therefore ‘God does not understand by composing and dividing’. His knowledge does not involve the complexity involved in propositional structure any more than it involves any other kind of complexity” (Alston, (1986), p. 288).

Thus, if the divine conceptualist can avail himself of this Thomistic tradition of God having non-propositional knowledge, then Craig himself could make the same move to avoid the charge that God knows an infinitely long list of arithmetical truths.

There is a problem of going the Thomist route here, as Aquinas himself is very explicit about whether God knows infinite things:

“Since God knows not only things actual but also things possible to Himself or to created things, as shown above, and as these must be infinite, it must be held that He knows infinite things” (Aquinas, Summae Theologica, Q14, A12).

Alston is perhaps trying to spell out a Thomist inspired view, rather than a Aquinas’ actual views. Even if Aquinas insisted that God knows an infinity of things, perhaps a non-propositional knowledge model can be adopted whereby God knows all mathematical truths without knowing an infinite list of truths. Indeed, Alston turns to F. H. Bradley’s idealism to spell out this possible model. Aston says that on Bradley’s view, the ‘base of our cognition is a condition of pure immediacy’, in which there is no distinction between different objects of knowledge. It is like taking in a painting as a whole, without focusing on any one particular bit of the painting. We can ‘shatter this primeval unity and build up ever more complex systems of propositional knowledge’, which would be like focusing on a particular brush stroke rather than the scene as a whole. This second mode of understanding is more discursively useful, but lacks the ‘felt oneness’ of the primeval apprehension. In contrast to these modes is the nature of the ‘Absolute’ itself – the world beyond our comprehension, which ‘includes all the richness and articulation of the discursive stage in a unity that is as tight and satisfying as the initial stage’. God’s knowledge, says Alston, could be modelled like this.

Wes Morriston, in his paper ‘Craig on the actual infinite’ (2002) considers this move by Craig, and concludes that Alston’s idea is of no help here:

“On Alston’s proposal, then, God’s knowledge is certainly not chopped up into a plurality of propositional states. On the other hand, it is said to have ‘all the richness and articulation’ of discursive thought. Even if this ‘richness and articulation’ does not consist in a multiplicity of propositional beliefs, it must surely involve some sort of distinction and variation and multiplicity within the divine intellect. However ‘tight and satisfying’ the unity of God’s knowledge, it must be thought of as a unity within a multiplicity – a one in a many” (Morriston, (2002), p. 159).

Ultimately, Alston’s idea is just that a God’s knowledge is a sort of synthesis of multiplicity and unity, and Morriston’s reply is that this does not eliminate the multiplicity. So it is not really any help to Craig.

Thus it seems that the non-propositional nature of God’s knowledge is not really a way of getting out of the claim that God is infinite.

Craig’s God is a passive infinity

Given that we now have the distinction between the active and passive infinity at hand, it could be that Craig’s reply would just be that God’s knowledge of arithmetic truths is a ‘closed totality’ of knowledge, and as such is passive. Just as no new moments can be added to the timeline, no new arithmetic truths can be added or subtracted from the totality of mathematical truths. As such it is infinite, but can never manifest absurdities as a result. As such, God can be infinite in this regard and not get chewed up in the teeth of Craig’s argument.

This would be a satisfactory response by Craig, but for one thing. Craig’s God has a very distinctive relationship to time, because Craig has a very particular theory of time. This makes Craig’s God particularly vulnerable to the actively infinite God objection.

Craig’s God and Time

Craig has a fairly nuanced view about God’s relationship to time. Roughly, God existed in an atemporal manner before he created the universe, but then entered into time and became temporal.

“God exists changelessly and timelessly prior to creation and in time after creation” (Craig [1978], p 503).

Craig also believes that the correct theory of time is the ‘A-theory’, according to which the fundamental temporal relations are tensed (like ‘it is now raining’, or ‘it will be sunny’, etc), rather than tenseless (like ‘raining at t1 is earlier than sunny at t2’, etc). For Craig, there is a fact about what is happening now which is metaphysically basic, and continually changing as time rolls forwards. God, being a temporal entity in time, has knowledge of this now, of ‘where he is’ on the timeline so to speak, and consequently what is presently happening:

“As an omniscient being, God cannot be ignorant of tensed facts. He must know not only the tenseless facts about the universe, but He must also know tensed facts about the world. Otherwise, God would be literally ignorant of what is going on now in the universe. He wouldn’t have any idea of what is now happening in the universe because that is a tensed fact. He would be like a movie director who has a knowledge of a movie film lying in the canister; he knows what picture is on every frame of the film lying in the can, but he has no idea of which frame is now being projected on the screen in the theater downtown. Similarly, God would be ignorant of what is now happening in the universe. That is surely incompatible with a robust doctrine of divine omniscience. Therefore I am persuaded that if God is omniscient, He must know tensed facts” (taken from http://www.reasonablefaith.org/god-time-and-eternity, which is a transcript of a paper given in Cambridge in July 23rd 2002)

This makes Craig’s God an ‘temporal epistemic agent’, that is one who is continually updating his knowledge set with new facts about reality as time passes; namely what is presently true. He doesn’t just know that at t1 it is raining – he knows that it is now raining.

Craig’s God is an active actually existing infinity

According to Craig then, God comes to know new things as time moves forwards. But he already knows an infinite number of truths, all the mathematical truths etc, and then he adds to his knowledge as time passes. However, the cardinality of his knowledge, how many truths he knows, stays the same – it is still infinite. So he knows more things, but also the same number of things. This is a manifestation of absurdity, just like Craig complained about with Hilbert’s Hotel, and at least as convincing as the growing block problem. Thus, by his own arguments, Craig’s God cannot exist.

Dilemma

It could be that Craig objects to the distinction between active and passive infinities. Perhaps it was made for rhetorical force only. If so, then his objection should be characterized as:

‘Imagine a hotel with infinite rooms, that’s absurd, therefore it couldn’t actually exist’.

If so, then I find it very implausible. In order to accept it, we would need to have something to justify it, and all Craig offers is that one can derive ‘absurd’ consequences from it, by which he means something contradictory. I agree that if we can derive contradictions from something, then it is to be rejected. However, we have seen that the only way we can get anything absurd from Craig’s examples is if we interact with the infinity, by getting the manager to free up a room for us, etc. Craig has never offered an example of any absurd consequences from thinking of actually existing infinities that are passive. Thus, if he wants to take this option, he still has all his work ahead of him for motivating the first premise of his supporting argument. Until he has provided this motivation, we are free to refrain from assenting to it, and consequently refrain from assenting to the conclusion of the Kalam argument.

But then if Craig accepts the active/passive distinction, then he has a pair of serious problems. Given the eternal universe model, it is infinite but passive. So not absurd. So it can exist. In addition, Craig’s A-theoretic nature of God means that God manifests absurd behavior. Therefore, he cannot exist.

The conclusion, then, is that either Craig has a lot of work to do explaining why actually existing infinities cannot exist, or he has in fact argued himself into a corner where an eternal universe could exist and God cannot. It seems there are big problems for Craig’s God.