## Aquinas’ Third Way argument II – Another counterexample

0. Introduction

In the previous post, I looked at Aquinas’ third way argument, as presented by apologist Tom Peeler. He proposed a causal principle, similar to what Aquinas proposed. Aquinas said:

“that which does not exist only begins to exist by something already existing”.

Peeler said:

“existence precedes causal influence”.

But basically, they are arguing for the same principle, namely:

Causal Principle) For something to begin to exist, it must be caused to exist by some pre-existing object.

From now on, let’s just call that ‘the causal principle’. Peeler was using this principle to support the first premise of his argument, which was:

“If there was ever nothing, there would be nothing now”.

The idea is that if Peeler’s principle were true, then the first premise is true as well. In the previous post, I argued that even if we accept all this, the argument does not show that an eternal being exists. Rather, it is compatible with an infinite sequence of contingent things.

In this post, I want to make a slightly different point. Up to now, we have conceded that the causal principle entails that there are no earlier empty times. However, I want to insist that this is only true if time is discrete. If time is continuous, then the causal principle dos not entail that there are no earlier empty times. I will prove this by constructing a model where time is continuous and at which there are earlier times which are empty, and later times which are non-empty, yet there is no violation of the causal principle.

1. The causal principle

I take the antecedent of this conditional premise, i.e. “there was ever nothing”, to mean ‘there is some time at which no objects exist’, which seems like the most straightforward way of taking it. Therefore, if the causal principle is to support the premise, the causal principle must be saying that if an object begins to exist, then it must not be preceded by a time at which no objects exist.

Strictly speaking, what the principle rules out is empty times immediately preceding non-empty times. Take the following model, where we have an empty time and a non-empty time, but at which they are not immediately next to one another on the timeline. Say that t1 is empty, and t3 is non empty:

In order to use the causal principle to rule this sort of model out, we need to fill in what is the case at t2. So let’s do that. Either t2 is empty, or it is not. Let’s take the first option. If t2 is empty, then t3 is immediately preceded by an empty time, and we have a violation of Peeler’s principle. Fair enough. What about the other option. Well, if t2 is non-empty, then t3 is not a case that violates Peeler’s principle, because it is not immediately preceded by an empty time. However, if t2 has some object that exists at it, then it is a case of a non-empty time immediately preceded by an empty time, because t1 is empty. Therefore, this second route leads to a violation of Peeler’s principle as well.

The point is that if all we are told is that there is some empty time earlier than some non-empty time, without being told that the empty time immediately precedes the non-empty time, we can always follow the steps above to rule it out. We get to a violation of the causal principle by at least one iteration of the sort of reasoning in the previous paragraph.

However, this whole way of reasoning presupposes that time is discrete rather than continuous. If it is continuous, then we get a very different verdict. That is what I want to explain here. If time is continuous, we actually get an even more obvious counterexample than model 2.

2. Discrete vs continuous

Time is either discrete, or it is continuous. The difference is like that between the natural numbers (like the whole integers, 1, 2, 3 etc) and the real numbers (which include fractions and decimal points, etc). Here is the condition that is true on the continuous number line, and which is false on the discrete number line:

Continuity) For any two numbers, x and y, there is a third number, z, which is in between them.

So if we pick the numbers 1 and 2, there is a number in between them, such as 1.5. And, if we pick 1 and 1.5, then there is a number in between them, such as 1.25, etc, etc. We can always keep doing this process for the real numbers. For the natural numbers on the other hand, we cannot. On the natural numbers, there just is no number between 1 and 2.

A consequence of this is that there is no such thing as the ‘immediate successor’ of any number on the real line. If you ask ‘which number is the successor of 1 on the real number line?’, there is no answer. It isn’t 1.01, or anything like that, because there is always going to be a number between 1 and 1.01, like 1.005. That’s just because there is always going to be a number between any two numbers on the real number line. So there is no such thing as an ‘immediate successor’ on the real number line.

Exactly the same thing imports across from the numerical case to the temporal case. If time is continuous, then there is no immediately prior time, or immediately subsequent time, for any time. For any two times, there is a third time in between them.

This already means that there cannot be a violation of Peeler’s principle if time is continuous. After all, his principle requires that there is no non-empty time immediately preceded by an empty time. And that is never satisfied on a continuous model just because no time is immediately preceded by any other time, whether empty or non-empty. However, even though the principle cannot be violated, this doesn’t immediately mean that it can be satisfied. It turns out, rather surprisingly, that it can be satisfied.

2. Dedekind Cuts

In order to spell out the situation properly, I need to introduce one concept, that of a Dedekind Cut. Named after the late nineteenth century mathematician, Richard Dedekind, they were originally introduced as the way of getting us from the rational numbers (which can be expressed as fractions) to the real numbers (some of which cannot be expressed as fractions). They are defined as follows:

A partition of the real numbers into two nonempty subsets, A and B, such that all members of A are less than those of B and such that A has no greatest member. (http://mathworld.wolfram.com/DedekindCut.html)

We can also use a Dedekind cut that has the partition the other way round, of course. On this version, all members of B are greater than all those of A, and B has no least member (A has a greatest member). This is how we will use it from now on.

3. Model 5

Let’s build a model of continuous time that uses such a cut. Let’s say that there is a time, t1, which is the last empty time, so that every time earlier than t1 is also empty. The rest of the timeline is made up of times strictly later than t1, and they are all non-empty:

The precise numbers on here are just illustrative. All it is supposed to be showing is that every time up to and including t1 is empty, and that every time after t1 is non-empty. There is no first non-empty time, just because there is no time immediately after t1 at all. But there is a last empty time, which is just t1.

This model has various striking properties. Obviously, because it is a continuous model, there can be no violation of Peeler’s principle (because that requires time to be discrete). However, it is not just that it avoids violating the principle in this technical sense. It also seems to possess a property that actively satisfies Peeler’s causal principle. What I mean is that on this model, every non-empty time is preceded (if not immediately) by non-empty times. Imagine we were at t1.01 and decided to travel down the number line towards t1. As we travel, like Zeno’s tortoise, we find ourselves halfway between t1.01 and t1, i.e. at t1.005. If we keep going, we will find ourselves half way between t1.005 and t1, i.e. t1.0025, etc. We can clearly keep on going like this forever. No matter how close we get to t1 there will always be more earlier non-empty times.

So the consequences can be expressed as follows. Imagine that it is currently t1.01. Therefore, it is the case that some object exists. It is also the case that at some time in the past (such as t1) no objects existed. Whatever exists now could have been brought into existence by previously existing objects, and each of them could have been brought into existence by previously existing objects, and so on forever. So, it seems like this model satisfies Peeler’s version of the causal principle, that existence precedes causal influence, and Aquinas’ version of the principle, that “that which does not exist only begins to exist by something already existing”. Both of these are clearly satisfied in this model, because whatever exists has something existing earlier than it. However, it does so even though there are past times at which nothing exists.

4. Conclusion

The significance of this is as follows. If we assume that time is discrete, then the causal principle entails that there are no empty earlier times than some non-empty time. So if t1 is non-empty, then there is no time t0 such that t0 is empty. So if time is discrete, then the causal principle entails premise 1 of the argument (i.e. it entails that “If there were ever nothing, there would be nothing now”).

But, things are different if time is continuous. In that case, we can have it that the causal principle is true along with there being earlier empty times. The example of how this works is model 5 above. Something exists now, at t1.01, and there are times earlier than this which are non-empty. Every time at which something exists has times earlier than it during which some existing thing could have used its causal powers to bring the subsequent thing into existence. There is never any mystery about where the causal influence could come from; it always comes from some previously existing object. However, there are also empty times on this model, i.e. all moments earlier than or equal to t1. This means that the antecedent of the conditional premise is true (“if there ever was nothing”), but the consequent is false (“there would be nothing now”). So even though the causal principle looks true, the first premise is false. So if time is continuous, then the causal principle (even if granted for the sake of the argument) does not entail the first premise, and so does not support it being true.

## Aquinas’ Third Way Argument

0. Introduction

I recently listened to a podcast, where the host, David Smalley, was interviewing a christian apologist, Tom Peeler. The conversation was prefaced by Peeler making the claim that he could prove that God existed without the use of the bible.

The first argument offered by Peeler was essentially Aquinas’ ‘Third Way’ argument, but done in a way that made it particularly easy to spell out the problem with it. In fact, Peeler gave two arguments – or, rather, I have split what he said into two arguments to make it easier to explain what is going on. Once I have explained how the first argument fails, it will be obvious how the second one fails as well. The failures of Peeler’s argument also help us to see what is wrong with Aquinas’ original argument.

1. Peeler’s first argument

Peeler’s first argument went like this (at about the 23 minute mark):

1. If there were ever nothing, there would still be nothing
2. There is something
3. Therefore, there was never nothing

As Peeler pointed out, the argument is basically a version of modus tollens, and so is definitely valid. But is it sound? I will argue that even if we grant that the argument is valid and sound, it doesn’t establish what Peeler thinks it does.

Here is the sort of consideration that is motivating premise 1. In the interview, Peeler was keen to stress that his idea required merely the fact that things exist and the principle that “existence precedes causal influence”. There is an intuitive way of spelling out what this principle means. Take some everyday object, such as your phone. This object exists now, but at some point in the past it did not exist. It was created, or made. There is some story, presumably involving people working in a factory somewhere, which is the ‘causal origin’ of your phone. The important part about this story for our purposes is that the phone was created via the causal powers of objects (people and machines) that pre-existed the phone. Those pre-existing objects exerted their causal influence which brought the phone into existence; or, more mundanely, they made the phone. The idea is that for everything that comes into existence, like the phone, there must be some pre-existing objects that exert causal influence to create it. As Aquinas puts it: “that which does not exist only begins to exist by something already existing”.

One way to think about what this principle is saying is what it is ruling out. What it is ruling out is that there is a time where no objects exist at all, followed immediately by a time at which some object exists.

Imagine that at time t0, no objects exist at all. Call that an ‘empty time’. Then, at t1 some object (let’s call it ‘a‘) exists; thus, t1 is a ‘non-empty time’. This situation violates Peeler’s causal principle. This is because a has been brought into existence (it has been created), but the required causal influence has no pre-existing objects to wield it. We can picture the situation as follows:

At the empty time, t0, there is nothing (no object) which can produce the causal influence required to bring a into existence at t1. Thus, the causal influence seems utterly mysterious. This is what Peeler means by ‘nothing can come from nothing.’ So we can understand Peeler’s causal principle in terms of what it rules out – it rules out things coming into existence at times that are immediately preceded by empty times, or in other words it rules out non-empty times immediately following from empty times. Let’s grant this principle for the sake of the argument to see where it goes.

If we do accept all this, then it follows that from the existence of objects, such as your phone, that there can never have been a time at which no objects existed (i.e. that there are no empty times in the past). That’s because of the following sort of reasoning. If this time has an object, such as your phone, existing at it, then this time must not be preceded by a time at which no objects existed. So the phone existing now means that the immediately preceding time has objects existing at it. But the very same reasoning indicates that this prior time must itself be preceded by a time at which objects existed, and so on for all times.

We can put it like this: if this time is non-empty, then so is the previous one. And if that time is non-empty, then so is the previous one, etc, etc. Thus, there can never be an empty time in the past if this time is non-empty.

This seems to be the most charitable way of putting Peeler’s argument.

2. Modelling the argument

For all we have granted so far, at least three distinct options are still available. What I mean is that the argument makes certain requirements of how the world is, for it’s premises and conclusion to be true. Specifically, it requires that a non-empty time not be immediately preceded by an empty time. But there are various ways we can think about how the world is which do not violate this principle. A model is a way that the world is (idealised in the relevant way). If the model represents a way that the world could be on which the premises and conclusion of an argument are true, then we say that the model ‘satisfies‘ the argument. I can see at least three distinct models which satisfy Peeler’s argument.

2.1 Model 1

Firstly, it could be (as Peeler intended) that there is a sequence of non-necessary objects being caused by previous non-necessary objects, which goes back to an object which has existed for an infinite amount of time – an eternal (or necessary) object. Think of the times before t1 as the infinite sequence: {… t-2, t-1, t0, t1}. God, g, exists at all times (past and future), and at t0 he exerted his causal influence to make a come to exist at t1 alongside him:

On this model, there are no times in which an object comes into existence which are immediately preceded by an empty time, so this model clearly does not violate Peeler’s principle. Part of the reason for this is that there are no empty times on this model at all, just because God exists at each time. Anyway, the fact that this model doesn’t violate Peeler’s causal principle means that there is at least one way to model the world which is compatible with Peeler’s argument. The world could be like this, for all the truth of the premises and conclusion of Peeler’s argument requires.

But, this is not the only option.

2.2 Model 2

Here is another. In this model, each object exists for only one time, and is preceded by an object which itself exists for only one time, in a sequence that is infinitely long. Each fleeting object is caused to exist by the previous object, and causes the next object to exist. On this model there are no empty times, so it is not a violation of Peeler’s principle. Even though it does not violate the principle, at no point is there an object that exists at all times. All that exists are contingent objects, each of which only exists at one time.

Think of the times before t1 as the infinite sequence { … t-2, t-1, t0, t1}, and that at each time, tn, there is a corresponding object, bn:

Thus, each time has an object (i.e. there are no empty times) and each thing that begins to exist has a prior cause coming from an object. No object that begins to exist immediately follows from an empty time. Therefore, this model satisfies Peeler’s argument as well.

2.3 Model 3

There is a third possibility as well. It is essentially the same as the second option, but with a merely finite set of past times. So, on this option, there is a finitely long set of non-empty times, say there are four times: {t-2, t-1, t0, t1}. Each time has an object that exists at that time, just like in model 2. The only real difference is that the past is finite:

In this case, t-2 is the first time, and b-2 is the first object.

However, there might be a problem with this third option. After all, object b-2 exists without a prior cause. It isn’t caused to exist by anything that preceded it (because there are no preceding times to t-2 on this model). Doesn’t this make it a violation the causal principle used in the argument?

Not really. All that Peeler’s causal principle forbids is for an object to begin to exist at a time immediately following an empty time. But because there are no empty times on this model, that condition isn’t being violated. Object b-2 doesn’t follow an empty time. It isn’t preceded by a time in which nothing existed. It just isn’t preceded by anything.

Now, I imagine that there is going to be some objection to this type of model. Object b-2 exists, but it was not caused to exist. Everything which comes into existence does so because it is caused to exist. But object b-2 exists yet is not caused to exist by anything.

We may reply that object b-2 is not something which ‘came into existence’, as part of what it is for an object x to ‘come into existence’ requires there to be a time before x exists at which it does not exist. Seeing as there is no time before t-2, there is also no time at prior to t-2 at which b-2 does not exist. So b-2 simply ‘exists’ at the first time in the model, rather than ‘coming into existence’ at the first time. Remember how Aquinas put it: “that which does not exist only begins to exist by something already existing”. There is no prior time at which b-2 is “that which does not exist”. It just simply is at the first time.

No doubt, this reply will seem to be missing the importance of the objection here. It looks like a technicality that b-2 does not qualify as something which ‘comes into existence’. The important thing, Peeler might argue, is that b-2 is a contingent thing that exists with no cause for it. That is what is so objectionable about it.

If that is supposed to be ruled out, it cannot be merely on the basis of Peeler’s causal principle, but must be so on the basis of a different principle. After all, Peeler’s principle merely rules out objects existing at times that are preceded by empty times. That condition is clearly not violated in model 3. The additional condition would seem to be that for every non-necessary object (such as b-2), there must be a causal influence coming from an earlier time. This principle would rule out the first object being contingent, but it is strictly more than what Peeler stated he required for his argument to go through.

However, let us grant such an additional principle, just for the sake of the argument. If we do so, then we rule out models like model 3. However, even if we are kind enough to make this concession, this does nothing to rule out model 2. In that model, each object is caused to exist by an object that precedes it in time, and there are no empty times. Yet, there is no one being which exists at all earlier times (such as in model 1).

The existence of such an eternal being is one way to satisfy the argument, but not the only way (because model 2 also satisfies the argument as well). Thus, because model 2 (which has no eternal being in it) also satisfies the argument, this means that the argument does not establish the existence of such an eternal being.

So, even if we grant the premises of the first argument, it doesn’t establish that there is something which is an eternal necessary object. It is quite compatible with a sequence of merely contingent objects.

2. Peeler’s second argument

From the conclusion of the first argument, Peeler tried to make the jump to there being a necessary object, and seemed to make the following move:

1. There was never nothing
2. Therefore, there is something that has always been.

The fact that the extra escape routes are not blocked off by the first argument, should give you some reason to expect the inference in the second argument to be invalid. And it is. It is a simple scope-distinction, or an instance of the ‘modal fallacy’.

There being no empty times in the past only indicates that each time in the past had some object or other existing at it. It doesn’t mean that there is some object in particular that existed at each of the past times (such as God). So long as the times are non-empty, each time could be occupied by an object that exists only for that time (as in our second and third models), for all the argument has shown.

The inference in the second argument is like saying that because each room in a hotel has someone checked in to it, that means that there is some particular individual person who is checked in to all of the rooms. Obviously, the hotel can be full because each room has a unique individual guest staying in it, and doesn’t require that the same guest is checked in to every room.

When put in such stark terms, the modal fallacy is quite evident. However, it is the sort of fallacy that is routinely made in informal settings, and in the history of philosophy before the advent of formal logic. Without making such a fallacious move, there is no way to get from the conclusion of Peeler’s first argument to the conclusion of the second argument.

3. Aquinas and the Third Way

In particular, medieval logicians often struggled with scope distinctions, as their reasoning was carried out in scholastic Latin rather than in symbolic logic. That they managed to make any progress at all is testament to how brilliant many of them were. Aquinas is in this category, in my view; brilliant, but prone to making modal fallacies from time to time. I think we can see the same sort of fallacy if we look at the original argument that is motivating Peeler’s argument.

Here is how Aquinas states the Third Way argument:

“We find in nature things that are possible to be and not to be, since they are found to be generated, and to corrupt, and consequently, they are possible to be and not to be. But it is impossible for these always to exist, for that which is possible not to be at some time is not. Therefore, if everything is possible not to be, then at one time there could have been nothing in existence. Now if this were true, even now there would be nothing in existence, because that which does not exist only begins to exist by something already existing. Therefore, if at one time nothing was in existence, it would have been impossible for anything to have begun to exist; and thus even now nothing would be in existence — which is absurd.” Aquinas, Summa Theologiae, emphasis added)

This argument explicitly rests on an Aristotelian notion of possibility. The philosopher Jaakko Hintikkaa explains Aristotle’s view:

“In passage after passage, [Aristotle] explicitly equates possibility with sometime truth, and necessity with omnitemporal truth” (The Once and Future Seafight, p. 465, emphasis added)

This is quite different from the contemporary view of necessity as truth in all possible worlds. On the contemporary view, there could be a contingent thing that exists at all times in some world. Therefore, being eternal and being necessary are distinct on the modern view, but they are precisely the same thing on the Aristotelian view. We will come back to this in a moment. For the time being, just keep in mind that Aquinas, and by extension Peeler, are presupposing a very specific idea of what it means to be necessary or non-necessary.

We can see quite explicitly that Aquinas is using the Aristotelian notion of necessity when he says “…that which is possible not to be at some time is not”. This only makes sense on the Aristotelian view, and would be rejected on the modern view. But let’s just follow the argument as it is on its own terms for now.

The very next sentence is: “Therefore, if everything is possible not to be, then at one time there could have been nothing in existence.” What Aquinas is doing is imagining what would be the case if all the objects that existed were non-necessary objects. If that were the case, then no object would exist at every time, i.e. each object would not exist at some time or other. That is the antecedent condition Aquinas is exploring (i.e. that “everything is possible not to be”).

What the consequent condition is supposed to be is less clear. As he states it, it is “at one time there could have been nothing in existence”. We can read this in two ways. On the one hand he is saying that if everything were non-necessary, then there is in fact an earlier time that is empty. On the other hand, he is saying that if everything were non-necessary, there could have been an earlier time that is empty.

Let’t think about the first option first. It seems quite clear that it doesn’t follow from the assumption that everything is non-necessary that there is some time or other at which nothing exists. Model 2 is an example of a model in which each object is non-necessary, but in which there are no empty times. If Aquinas is thinking that “if everything is possible not to be, then at one time there could have been nothing in existence” means that each object being non-necessary implies that there is an empty time, then he is making a modal fallacy. This time, the fallacy is the other way round from Peeler’s example: just because each guest is such that they have not checked into every room of the hotel, that does not mean there is a room with no guest checked in to it. Think of the hotel in which each room has a unique guest in it. Exactly the same thing applies here too; just because every object is such that it fails to exist at some time, that does not mean that there is a time at which no object exists. Just think about model 2, in which each time has its own unique object.

Thus, if we read Aquinas this first way, then he is committing a modal fallacy.

So let’s try reading him the other way. On this reading he is saying that the assumption that everything is non-necessary is compatible with there being an empty time. One way of reading the compatibility claim is that there is some model on which the antecedent condition (that every object is non-necessary) and the consequent condition (that there is an empty time) are both true. And if that is the claim, then it is quite right. Here is such a model (call it model 4):

On this model, there are two objects, a and b, and they are both non-necessary (i.e. they both fail to exist at some time). Also, as it happens, there is an empty time, t2; both a and b fail to exist at t2. So on this model, the antecedent condition (all non-necessary objects) and the consequent condition (some empty times) are both satisfied.

However, while this claim is true, it is incredibly weak. The difference is between being ‘compatible with’ and ‘following from’. So for an example of the difference, it is compatible with me being a man that my name is Alex; but it doesn’t follow from me being a man that my name is Alex. If we want to think about the consequent following from the antecedent condition, we want it to be the case that every model which satisfies the antecedent condition also satisfies the consequent condition, not jus that there is some model which does. But it is clearly not the case that every model fits the bill, again because of model 2. It satisfies the condition that every object is non-necessary, but it doesn’t satisfy the condition that there are some empty times.

So what it comes down to is that the claim that there are only non-necessary objects is compatible with the claim that there are empty times, but it is equally compatible with the claim that there are no empty times. Being compatible with both means that it is simply logically independent of either. So nothing logically follows from the claim that there are only non-necessary objects about whether there are any empty times in the past or not.

So on the first way of reading Aquinas here, the claim is false (because of model 2). On the second way of reading him, the claim is true, but it is logically independent of the consequent claim. On either way of reading him, this crucial inference in the argument doesn’t work.

And with that goes the whole argument. It is supposed to establish that there is an eternal object, but even if you grant all of the assumptions, it is compatible with there not being an eternal object.

4. Conclusion

Peeler set out an argument, which was that if nothing ever existed, there would be nothing now. The truth of the premises and the conclusion is satisfied by, or compatible with, model 2, and so does not require that an eternal object (like God) exists. The second argument was that if it is always the case that something exists, then there is something which always exists. That is a simple modal fallacy. Lastly, we looked at Aquinas’ original argument, which either commits a similar modal fallacy, or simply assumes premises which do not entail the conclusion.

## Infinity, Hume and Euclid

0. Introduction

Can something be infinite, yet also exist, as it were, ‘in reality’? Many people say that the answer to this question is ‘no’. However, the arguments in favour of this go over terrain that is very difficult to navigate without getting lost. Here I want to look at one very small part of the issue, and what seems problematic about it to me. As I said, this stuff gets very deep very quickly, and although I know the literature a bit, I am not a specialist in this area. There plenty of discussions of this I should probably read. But these are my thoughts at the moment. Hopefully it will help me become clearer about it, and might be helpful to people trying to understand this area a bit better too.

I am thinking about the argument in relation to the ‘Hilbert’s hotel’ style of defence for the second premise of the kalam cosmological argument. In particular, imagine that an apologist uses the infinite library analogy as follows:

Suppose there were a library with infinite books in it. If you withdraw a book from the library, then there is one fewer book in the library, yet there is also the same number of books in the library. There cannot both be fewer books and the same number of books, because that is absurd.

That is the type of claim that I am looking at here. I am not looking in particular about the other difficulties which could be brought out from the infinite object examples. There are other things one might say to motivate this part of the kalam, but here I am looking at this way of motivating it. The claim I am interested in is that the truth of there being fewer books and the same number of books is itself an absurdity.

1. Equinumerous and fewer than.

There are two principles which need to be kept in sight. They involve connecting our intuitive ideas ‘equinumerous’ and ‘fewer than’ with mathematical counterparts.

We already have a fairly clear idea about the relationship between the two concepts when we use it in casual conversation. In particular, it seems quite clear that they are mutually exclusive:

i) If the number of A’s is equal to the number of B’s, then the number of A’s is not fewer than the number of B’s.

ii) If the number of A’s is fewer than the number of B’s, then the number of A’s is not equal to the number of B’s.

In ‘real life’ examples, we say that if the number of forks in my dinner set is the same as the number of knives, then I do not have fewer forks than knives, etc. The mathematical counterparts of these terms need to preserve this highly intuitive relationship; the result must be mutually exclusive too. We will consider two linking-principles, one which links the notion of ‘equinumerous’ to a mathematical idea, and one which links the notion of ‘fewer than’ to a different mathematical idea. They are chosen because they look like they express what the intuitive ideas are getting at, and because they preserve the mutual exclusivity relationship (at least, they do to begin with).

2. Hume and Euclid

The first of these is sometimes called Hume’s Principle (HP). It can be seen as a definition of the notion of ‘equinumerous’ (or ‘the same number of elements’, or of having the ‘same cardinality’ – all three meaning the same thing here). Being the same cardinality is linked to being able to be put in a one-to-one correspondence. Imagine that I could put every fork from my dinner-set with a unique knife, and have no knives left over. According to HP, this condition is what must hold for it to be true that I have the ‘same number’ of forks as knives. The idea with HP is that “The cardinality of A is equal to the cardinality of B” means that the elements of A can be put in a one-to-one correspondence with the elements of B.

Euclid’s maxim‘ (EM) effectively says that the whole is greater than a part (this principle is advocated by Euclid in the Elements). We can see this as a definition of the notion of ‘fewer-than’. Firstly, we need to be clear about what a ‘proper subset’ is:

A is a proper subset of B if and only if

• everything which is an element of A is also an element of B, and
• there is an element of B which is not an element of A.

So the set of knives is a proper subset of the set of ‘items of cutlery in my dinner-set’, because there are items of cutlery which are not knives (i.e. forks). There are fewer knives than there are items of cutlery. So this the idea with Euclid’s maxim is “There are ‘fewer’ A’s than B’s” means that A is a proper subset of B.

Here they are side by side:

Hume’s principle: A is equinumerous with B if and only if the elements of A can be placed in a one-to-one relation with the elements of B.

Euclid’s maxim: There are fewer A’s than B’s if and only if A is a proper subset of B.

3. The Problem

If the number of elements that can be in a set is finite, then these principles are mutually exclusive. So if A and B are equinumerous (according to HP), then neither is fewer than the other (according to EM), and if one is fewer than the other, then they are not equinumerous. This is because, if A and B are finite, then there being something in B that is not in A entails that one could not put their respective elements in a one-to-one relation. If I am missing a knife, then I cannot place each knife with a unique fork, without having a fork left over. So far so good.

The problems come in if A and B are allowed to be sets that have infinitely many elements. When we make this move, these two intuitive principles cannot both be correct. Let A be the set of all natural numbers, [0, 1, 2, 3 … n …), and B be the set of even natural numbers, [0, 2, 4, 6 … n …). The elements of A and B can be placed in a one-to-one correspondence with one another, as Cantor showed. So by Hume’s principle, they are equinumerous. Yet, it is also clear that every element of B is an element of A, while there are elements of A that are not elements of B (i.e. the odd numbers). This means that B is a proper subset of A. So by Euclid’s maxim, B has fewer elements than A.

The problem is that A and B have the same number of elements, but B has fewer elements than A. As we saw, the intuitive relationship between being the same number and being fewer is that they are mutually exclusive. So it should never happen that A and B are both equinumerous and that one is fewer than the other. Clearly, something has to give here if we are to avoid an inconsistent result.

4. What is going on?

The issue here is that we have a pre-theoretical idea of the terms ‘equinumerous’ and ‘fewer than’, and we have set-theoretical expressions which looked like they gave the meaning of the intuitive notions. However, our intuitions about what those terms mean differ from how their mathematical counterparts operate in certain circumstances.

So to avoid the problem, we have to reject one of three things:

1. The idea that equinumerous means being able to be placed in a one-to-one correspondence (i.e. HP)
2. The idea that fewer-than means being a proper subset (i.e. EM)
3. The idea that being ‘equinumerous-with’ is mutually exclusive with being ‘fewer-than’.

If we try to keep all three of these things, we run into the problems that give rise to the apologist’s charge of ‘absurdity’.

5. Rejecting 1

One way of proceeding is to reject HP. This means rejecting the claim that when we say that the number of A’s is the same as the number of B’s this means that the A’s can be placed in a one-to-one correspondence with the B’s. The main problem with this is that it is unclear what else being ‘equinumerous’ could mean. Possibly, it could mean something like if they were both counted, then the final number reached would be the same in each case. It is not clear whether this is actually any different however. Imagine that I count my knives by picking each one up and saying a cardinal number out loud (like the Count from Sesame Street), and then placing them off to one side in a line according to the number they received. So I put the first one down, then I place the second one next to that, and the third one next to the second one, etc. When I come to count my forks I could do exactly the same thing. If I arrive at the same number when I have finished counting each one, this just means that the two lines of cutlery would be lined up one-to-one. So this doesn’t even seem to be a different result to HP. And what else could ‘equinumerous’ mean?

The real action is about which one to reject out of 2 or 3.

6. The case for rejecting 2

One way to reject 2, but to keep 3, would be to modify the claim made in EM. At the moment, EM says that there are fewer A’s than B’s iff A is a proper subset of B. We could add another condition as follows:

Revised-EM) There are fewer A’s than B’s iff

• A is a proper subset of B, and
• A and B are not equinumerous

The second condition isn’t needed in the case of merely finite sets, because no finite set A can be both a proper subset of B and equinumerous with B. Thus, the original EM and the revised-EM are identical with respect to finite sets. When we move to the case where sets can be infinite, then the second condition kicks in. The set of the even natural numbers is a proper subset of the natural numbers (so the first condition is satisfied). But the set of even natural numbers is equinumerous with the set of natural numbers (in that they can be placed in a one-to-one correspondence). Because this second condition is not satisfied, this means that it is false that there are ‘fewer’ even natural numbers than natural numbers. And this means that, according to revised-EM, there is no case (finite or infinite) where A is both equinumerous-with and ‘fewer-than’ B. And thus we have resolved our problem.

According to this strategy, there is nothing wrong with equinumerous meaning being able to be put in a one-to-one relation, and there is nothing wrong with the intuitive idea that equinumerous and fewer-than are exclusive. All that is rejected is the assumption that all there is to the notion of ‘fewer-than’ is being a proper subset. In addition to this, we also need to rule out being equinumerous. Only with both in place do we have a proper mathematical equivalent of ‘fewer-than’.

7. The case for rejecting 3

On the other hand, we could proceed by rejecting 3, the mutual exclusivity of equinumerous and fewer-than. On this view, infinite sets show us clear examples of when the A’s are equinumerous with the B’s, even though the A’s are also fewer-than the B’s. One might argue that our intuitions about the relationships between these terms is based on our experience of finite things, and we mistook a property of finite things to be a logical relationship between two terms. The mutual exclusivity of equinumerous and fewer-than is not a logical truth, but is actually a contingent truth, which applies only to those cases where the sets are finite. According to this view, we should revise our notions in light of this mathematical insight.

So take some case involving infinity, such as the infinite library. The number of books left in the library after I withdraw one book is ‘fewer’ (i.e. according to the original EM) than the number of books before the withdrawal, even though there is also the same number as before the withdrawal. There is only a problem with this if you insist on the mutual exclusivity between ‘same number as’ and ‘fewer-than’. If we let go of that presupposition, and let the mathematics guide our understanding, we see that the two notions are only mutually exclusive for certain cases and not others.

This sort of revision in how we use terms guided by scientific insight is not that strange. Imagine that at some point in history we discovered androgynous frogs. Prior to that we would have said that the terms ‘male’ and ‘female’ were mutually exclusive when it came to classifying frogs; if a frog is male, it is not also female and vice versa. But after the discovery we have a choice about how to proceed. We do not, I take it, say that these are not frogs, merely because it is true that no frog is both male and female! Rather, we say that, despite what we have previously thought, ‘male’ and ‘female’ are not mutually exclusive for all frogs. We revise our understanding of ‘male’ and ‘female’, being led by the discovery.

This is what it is like in our case too, if we reject 3. We originally thought that no sets could be fewer-than and equinumerous, but this was only the case with the finite sets we had considered. Once we look at these other cases, we find out that some sets are both fewer-than and equinumerous. Once we accept this, and drop the requirement that they are always mutually exclusive, we have avoided our issue from before. Saying that the library has both the same number and fewer books is like saying that this frog is both male and female. Sometimes that is what it is like.

8. Comparison

So we have two strategies. The disagreement is over the following. Imagine A is an infinite proper subset of B, such as A being the even natural numbers and B being all the natural numbers. A is equinumerous with B. But are there ‘fewer’ A’s than B’s? The first strategy says:

No, there are not fewer A’s than B’s (because they are equinumerous)

The second strategy says:

Yes, there are fewer A’s than B’s (because one is a proper subset of the other)

It is fairly clear at this stage that if one wanted to use the Hilbert’s hotel argument as a way of bringing out an absurdity, then option 2 causes a big difficulty. This is because it denies that there is ever a case where any two sets can be equinumerous and fewer-than. In particular, the infinite case is protected from this happening by the second condition in revised-EM. In these cases, the equinumerous nature of the two sets cancels out either being fewer-than the other. The very thing the apologist wanted to point to and say ‘Look at this! It’s absurd!’ is forbidden on this view.

Indeed, the third option also causes grave issues for the apologist too. On this view we have revised our notion of ‘fewer-than’ in such a way that it is no longer mutually exclusive with ‘equinumerous’. It would be like after the discovery of androgynous frogs; if I say to you ‘This frog is male’, you could reply ‘Yes, but is it also female?’ This reply wouldn’t be ‘absurd’ at all, because these terms are no longer thought to be mutually exclusive. The same thing would apply in our case too. On this view, there being both the same number and fewer books in the library after I withdraw one is not an example of two mutually exclusive things being true at the same time. Therefore it is not absurd on this view either.

9. Conclusion

So the issue I have looked at in this post is only a very small issue in the wider context of defending the kalam. It isn’t even the only issue that is brought up in relation to the Hilbert’s hotel style of defence, or even arguably the most serious. However, it is there, and people often talk as if this issue on it’s own causes problems. People often talk about the absurdity of there being both the same number of books and fewer books after the withdrawal in this sort of setting, even if they also develop additional worries.

I think there are broadly two strategies that one can adopt in response to this line of attack. The first would be to insist that there are really no situations where there are both the same number and fewer books, and provide a precise explanation of ‘fewer-than’ according to the revised-EM above. This clearly avoids the issue. Secondly, one could embrace the presence of what seemed like two mutually exclusive terms, but explain how the mathematics shows us that the two terms are not mutually exclusive for all cases.

I find each of these approaches to be independently quite plausible, and this largely discharges the force of the attack.

## 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.