Craig’s Five Point Response: A Response

1. Introduction

It has been exactly one year since I had my discussion with William Lane Craig on Capturing Christianity. Following that exchange, I wrote up my thoughts into a paper, and got it published in Mind (I posted a link to it here). This took a long time, partly because of the peer review process, and partly because the copy editing at Oxford University Press seems to be struggling over lockdown (that’s what it seemed like to me anyway). Craig was much quicker to publish his thoughts, which came out on his blog just a month after the discussion. You can read what he put here. At the time, Craig alluded to a ‘five point response’ that he had to the argument I was making. However, he didn’t get the chance to fully express those in the stream. Fortunately for us, he managed to explain them in more depth in his post.

Here I’m going to explain what each of his five points are, and respond to each of them.

2. The context

The discussion centred mainly on the argument I made with my friend Wes Morriston in our joint paper Endless and Infinite (which you can read here). The issue is whether the Hilbert’s Hotel argument, which is supposed to show that the past must have a beginning, can be used to show that the future must have an end. We gave a ‘symmetry argument’, which is just the Hilbert’s hotel argument, but ‘re-tensed’ so that it is about the future not the past.

We think the situation is perfectly symmetrical. If the past had no beginning, then the number of past events (the number of past days, if you like), will be equal to the cardinality of the natural numbers. You could assign each one a unique number, and have none left unassigned. Having this property makes it an ‘actual infinite’ for Craig; or at least, having this property makes it problematic for Craig (the ‘actual infinite’ is the problematic one for Craig). The thing is that this cardinality behaves weirdly, as it can have proper parts that are equivalent to the whole. Craig brings this out with his Hilbert’s Hotel examples, each of which involves things like there being both more guests and also the same number of guests after a new guest checks in to the hotel, etc. This is too much for Craig, who says that this shows that anything actually infinite cannot exist concretely in reality, in the same sort of way as hotels or library books. Given that there would be just as many, say, past even days as the total number of past days, the beginningless past has the problematic property that got Hilbert’s hotel into trouble. Thus the past must be finite. So his argument goes.

All we are pointing out is that this property applies to future events just as much as past ones. If the future never ends, then the number of future events (the number of future days, if you like), will be equal to the cardinality of the natural numbers. You could assign each one a unique number, and have none left unassigned. As each day passes there is the same number of days left to go as there was previously. As such the endless future is just as similar to Hilbert’s hotel as the beginningless past. There is a very obvious, at least prima facie, symmetry here. If you accept that Hilbert’s hotel is intolerably absurd, such that nothing having that property could exist, then this bars the beginningless past and the endless future equally. This is the symmetry argument in a nutshell.

But this is quite tricky if you want to hold, like Craig does, that the endless future is possible, but the beginningless past is not. At least, if you also hold that the reason to think that the beginningless past is impossible is because of these Hilbert’s hotel examples. Such a person needs to provide some kind of ‘symmetry breaker’ according to which the past and future can be treated differently with respect to Hilbert’s hotel argument.

We looked at what we took to be the various symmetry breakers that Craig has proposed in the literature, and explained why each one is insufficient. The first was his central claim, which is that the beginningless past is an actual infinite, but the endless future is merely a potential infinite. The second is that there are no future events at all; the third is that the future is pure potentialities, etc, etc. You can read it for yourself to see all the details.

I think all of Craig’s symmetry breakers obviously fail. In particular, the potential vs actual infinite is I consider to be completely dead as a response to this question. Anyway, this was the context for the discussion. Craig told me he had read the paper in advance of the discussion, so I expected these specific arguments to be addressed. Spoiler, I don’t think he read the paper very carefully. That’s the only way I can explain the following.

3. The Five Point Response: point one

Craig’s first point is about the ad hominem nature of the argument. Here is how he puts it on his blog:

The objection is either ad hominem or question-begging. As Alex recognized, the objection tends to be ad hominem, not in the abusive sense, but in the sense that it has force only against particular people, e.g., those who believe in personal immortality or, in the case of Andrew Loke’s formulation of the argument (see (4) below), against theists. If one tries to avoid this ad hominem feature by claiming that it’s clear that the series of future events can be infinite, then one seems to beg the question. For the objection does nothing to expose a flaw in the reasoning in support of a beginning of the series of past events. It allows that there is a sound argument against the infinitude of the past that also applies to the infinitude of the future. Without refuting this argument, it just assumes that an infinite future is possible, which begs the question against the argument.  So the objection is both question-begging and/or ad hominem

This response is itself split into two halves. The first half is everything up to “If one tries to avoid…”. He is right that there is something of an ad hominem about this discussion. The idea is that if someone advanced the Hilbert’s hotel argument as a way of showing that the past necessarily had a beginning, and also believes that the future at least could be (if not actually is) endless, then they have some explaining to do. They have to show how they are not committed to an inconsistent set of assumptions. And let it be clear, this description accurately describes Craig. So he has some explaining to do.

On the other hand, let’s also be clear about the consequences if we were to be right. What it would show would be that Craig has to figure out how to square his beliefs so that they are consistent. But there are various ways he could do that. For instance, he could do that by biting the bullet and holding that the future must come to an end. And this means that he could accept the symmetry argument without that in any way undermining the original Hilbert’s hotel argument, and the role that this plays in the Kalam. So in some sense then, the symmetry argument is not aimed at the Kalam directly. And it is only fair to be open about this. You could rationally buy the symmetry argument, and still think that Hilbert’s hotel establishes that the universe had a beginning (by biting the bullet about the future coming to an end).

But the point remains, even after we are super clear about this, that Craig in particular has to respond in some way. And he has done, in multiple places and in differing ways, and he doesn’t bite the bullet. Rather, he provides purported symmetry breakers to show how the future is not like the past with respect of the Hilbert’s hotel argument. So he is not giving up his belief in the endless future, nor the efficacy of the Hilbert’s hotel argument. And this requires some explaining.

But it being an ad hominem wasn’t itself an objection to the symmetry argument. It’s just a classification of the type of argument that it is, and not a marker that the argument is improper as such. So the first half of this response is really just signposting that the argument has this ad hominem aspect to it, but this isn’t really a response to the symmetry argument as such. It’s just a comment about it. So there’s not a lot more to say about it than that.

The second half of the quote makes the case that the symmetry argument (or at least the person making it) might themselves be guilty of begging the question at hand. He says that one could avoid it being an ad hominem by claiming that the future just could be endless. It’s like you could put the argument by saying ‘this is a problem for all those people who believe A, B and C’, and this would be ad hominem against those people. But you could avoid this by saying ‘this is a problem if A, B and C are true, and A, B and C actually are true’. In the second way of putting it, the emphasis isn’t on those people who believe certain things, but is just about how the contents of those beliefs are inconsistent. And you might want to put the argument like this if you have some prior commitment (maybe you think that it is just obvious, or perhaps highly intuitive) that the future could be endless. In that case, you might think that any theory that ends up meaning that this is impossible (like if you bite the bullet against the symmetry argument) is itself implausible, because it ends up deeming an obviously possible thing impossible.

I’m not so sure about this. Wes and Landon Hedrick seem to see it the second way; they think it is quite plausible that the future could be endless. I think that I’m unsure what I think about the possibility of the endless future. It is certainly epistemically possible – that is, consistent with what I take myself to know about the world. But then I’m quite open to my knowledge not extending very far towards possibilities like this. I’m at least as confident in the idea that we can never know whether the future is endless as I am in the proposition that it is metaphysically possible that it is endless. So I guess my own personal view is that I’m non-committal about whether it really is possible, or if it just seems to me like it is possible. Perhaps Wes and Landon agree with this as well. Either way, I don’t think I could reasonably be said to be begging the question here, because I’m not strongly committed to the possibility of the endless future; rather, I’m agnostic about it.

But that just makes me think that this first point just amounts to a statement about how the problem is particularly acute for people who hold Craig’s combination of beliefs, and an accusation that someone could (although I don’t think I do) beg the question by assuming that it is obvious that the endless future is metaphysically possible. And I honestly don’t see where an objection to anything is supposed to be found here.

4. Point two

In his second point, Craig gives a proposal for a symmetry breaker, saying “It is plausible that the past and future series of events are not perfectly symmetrical”. He goes on:

“On a tensed theory of time, according to which temporal becoming is an objective feature of reality, there are no events later than the present event and, hence, no future events. So a tensed theory of time entails that an actually infinite number of future events does not exist; indeed, the number of future events is 0.”

But this is exactly one of the symmetry breakers that Wes and I addressed in the paper. It’s as if he didn’t read what we put, because his response makes no mention of it. As we pointed out, the idea that there are no future events on a tensed theory of time is ambiguous, and when disambiguating Craig himself shows why this isn’t a symmetry breaker. What I mean is that we need to distinguish between how many future events there are, and how many there will be. Fair enough, a presentist wants to deny that any non-present events exist. So for them, there are no future events. However, even a presentist wants to say that there will be future events (unless she thinks that she is at the very final moment of time, which is obviously not what someone like Craig, who thinks time doesn’t have an end, would want to say).

And as we pointed out in the paper, Craig himself makes this distinction in print. Consider the following lines:

Thus, there really are no past or future events, except in the sense that there have been certain events and there will be certain others

Craig, 2001, p. 148

It’s all good to say that for a presentist there are no future events. But if you think that there will be future events (as Craig does, look up), then the question becomes: ‘how many future events will there be, if the future is endless?’ Given the distinction, it is no good to try to claim that the answer is still ‘zero’, because that means that time is coming to an end right now, and isn’t just the innocent presentist commitment to no non-present events. So this part of the response just feels a step behind the dialectic. Craig already made this point, and we already responded to it. Craig is just repeating the point and not engaging with the response. He isn’t explaining how the disambiguation doesn’t apply, or how it does apply but the answer is still zero, or anything like that. He is just not engaging.

He goes on to develop this point, and brings up what I think is a different symmetry breaker, but which Wes and I also extensively engaged with in our paper. He says the following:

The series of events later than any event in time, including the initial cosmological singularity, is always finite and always increasing toward infinity as a limit. In other words, such a series is potentially infinite. Georg Cantor called the potential infinite “a variable finite.” If the series of future events is potentially infinite, then the series of future events is finite but endless.

This is a somewhat obscure notion, but can be made totally clear. He is saying that the ‘series of events later than any event in time’ is ‘always finite and always increasing toward infinity as a limit’. But this is just wrong. Consider a mundane case. Suppose I’m going to count from one to ten. Before I start, how many future counting events will there be? The answer is ten. Now suppose I’ve already counted five numbers. How many future counting events are there now? The answer is five. I’ve counted 1, 2, 3, 4 and 5, and I still have 6, 7, 8, 9 and 10 to go. So the amount of future counting events is might be finite, but its not increasing over time. It’s clearly decreasing.

Suppose an angel starts counting now and never stops, and that time is endless. How many numbers will she count? Well, at the start of her count she has all of the natural numbers left to count; so ℵ0-many numbers left to count. But after she has counted, say, five numbers, she still has ℵ0-many numbers left to count. The amount of numbers left to count is neither increasing nor decreasing, but staying exactly the same. In a very straightforward way then, when the future is endless the ‘series of events later than any event in time’ is NOT ‘always finite and always increasing toward infinity as a limit’. Rather, it is never finite, and remains the same size forever. In short, it is not a potential infinity at all, but an actual infinity.

How come Craig sees things so differently? Well, it has to do with the weird way that Craig is using the term ‘the future’. In his paper with Sinclair, he puts it like this:

For as a result of the arrow of time, the series of events later than any arbitrarily selected past event is properly to be regarded as a limit.

Craig & Sinclair, 2009, p. 116

We are looking at the ‘events later than any arbitrarily selected past event’. Why this and not just events in the future of the present? It’s because if you look at it like this, then you get something that it much more like a potential infinite (that is, always finite but ever increasing, etc). Consider the following picture, which visualises what Craig is saying:

If we think about the length of the interval marked ‘some finite amount’, then obviously it is only finitely long. Now imagine that the point marked ‘now’ slides off towards the right as time passes. Clearly, the interval between it and the arbitrarily selected past event is going to increase but never be anything other than finite. This is obviously what Craig has in mind when he says that the future is a potential infinite.

But this is where the distinction between the simple future tense and the future perfect tense comes in. Notice that the interval in question is (always) behind the now point. It marks the bit between a past event and the present. It is as if we are at the arbitrary past point and wondering how much time will have passed as of some future point. The answer to this question is that it is always finite, but ever increasing as time passes. And I totally agree that this is a potential, and not actual, infinite. Totally. But the thing that cannot be ignored is that we are not talking about the simple number of future events any more. We are talking about how many days will be past in the future. That’s super obviously not the same thing. As Wes and I put it in the paper: “…that simply isn’t an answer to the question of ‘how many events will there be?’ It is just to answer a different question”. Again, by simply repeating this line, rather than responding to the extensive critique offered in the paper, Craig just seems to be one step behind in the dialectic.

So far, point 2 merely repeats Craig’s positions without interacting with the critiques of them. Again, it’s not really a response. It’s the thing that we originally responded to, and which still stands without a response.

Craig ends this section with the following:

Consider Alex’s premise:

1. If a beginningless series of past events is impossible, then an endless series of future events is impossible.

That commits Alex to the view that there is no possible world in which the series of events has a beginning but no end. In other words, he has to say that the view that the series of events is potentially infinite is not just false but impossible. That is a radical thesis carrying a heavy burden of proof.

This premise leaves out a very important caveat. The whole point here is that if you think Hilbert’s hotel is a good argument for the past having a beginning, then the future must have an end. I don’t buy the antecedent condition here; I don’t think Hilbert’s hotel is a good argument. All I’m saying is that if you do, then you need to explain how to avoid the seemingly obvious symmetry when applying it to the future. Remember how this has an ad hominem aspect to it.

Far from me endorsing a ‘radical thesis’, I’m highlighting how Craig, and people who share his views, need to do some explaining. One way they could do that is by biting the bullet and accepting that the future comes to an end. Another way is to give up Hilbert’s hotel as applied to the past, and say that both future and past could be infinite. What they cannot do, at least without successful symmetry breaker, is maintain an asymmetrical view, where Hilbert’s hotel shows the past is finite, but not that the future is finite. I think that advocates of the Hilbert’s hotel argument (a group I’m not in) cannot maintain an asymmetrical view, because I think all the symmetry breakers fail. Again, Craig seems one step behind the debate if he thinks that I personally believe that the ontology of time has to be symmetrical. I’m making no such claim, because I don’t buy the Hilbet’s hotel argument in the first place.

5. Point 3.

Here Craig brings up an argument discussed (though not endorsed) by my friend Landon Hedrick. Here is what Craig says:

Landon Hedrick, himself no friend of the kalām cosmological argument, has offered a version of the argument for the finitude of the past that is not susceptible to the symmetry objection. It goes as follows:

(1) There cannot be a world in which an actually infinite number of things have been actualized.

(2) If the actual world is one in which the universe is past-eternal, then there is a world in which an actually infinite number of things have been actualized.

(3) Therefore, the actual world cannot be one in which the universe is past-eternal.

This version of the argument for the finitude of the past avoids any alleged parallelism with the future.

It’s strange to me to think that this argument ‘avoids any alleged parallelism with the future’. Here is an obvious re-tensing of the argument:

(1) There cannot be a world in which an actually infinite number of things will be actualized.

(2) If the actual world is one in which the universe is future-eternal, then there is a world in which an actually infinite number of things will be actualized.

(3) Therefore, the actual world cannot be one in which the universe is future-eternal.

This argument obviously has the same logical structure as the original. The reasons for thinking that the second premise are true are exactly the same in both cases. If there are an infinite number of days, either past or future, then these days constitute a set of things that is infinite each of which either will be or was actualised.

The only real place I can imagine Craig objecting is to the first premise. Here, presumably he will say that the future things are never all actualised; rather they each become actualised but the overall number of actualised future things remains finite. But now we are just back to the simple future vs future perfect thing again. I quite agree that the number of currently future events will never have been completely actualised, but that’s a future perfect thing again. It’s just the same as thinking about the interval between a random past event and the present. What matters is not this issue, but the simple question of how many events are in the future of the moving now; not how many will be between it and some random past event.

So, not only do I think there is an obvious way that this argument can be re-tensed to make it about the future, but when it is done I think it puts us right back where we started. Again, I can only think that Craig is a step behind in the dialectic here.

6. Point 4.

Here, Craig brings up Loke’s hotel room builder argument, which was also addressed at length in my paper with Wes. Craig makes the following comments:

Alex’s response to Loke involves an illicit modal operator shift. Alex thinks that if God can bring about the existence of every future room in an endless series of events, then He can bring about the existence of all of them in the present moment. This is a mistake. It does not follow from God’s ability to bring about the present existence of any particular future room that He is able to bring about the present existence of all the future rooms. So to reason is modally logically fallacious. Thus, Loke is quite justified in denying that the possibility of an endless future implies the possibility of the existence of an actually infinite number of things, as does the possibility of a beginningless past.

Again, this seems to be clear evidence that Craig isn’t tracking the dialectic clearly. It is true that in the paper we said: “If an omnipotent God had completely unrestricted power, then he would have the ability to make a HH appear all in one go.” (p. 15). However, we went on to discuss what the consequences are for the argument when the theist insists that “God has the ability do anything metaphysically possible, but nothing metaphysically impossible” (p. 16). Given the metaphysical impossibility of Hilbert’s hotel (granted for the sake of the argument), this restriction on omnipotence has the precise consequence that “for each natural number n, it is possible that God made a hotel with n many rooms in total”, but not “it is possible that God made a hotel so big that there is a hotel room for every natural number n” (p. 16).

Basically, we made precisely the distinction that Craig drew. We are saying: assume that God could make any natural number of hotel rooms in one go, but not infinitely many. It is strange to think that you could carefully read our paper, and the response to Loke, and think that we are arguing with the assumption that because God could make any number of hotel rooms, that he could make infinitely many. I don’t know how to have been more clear about the assumptions we were working with, and how we were presuming the restricted omnipotence that blocks the idea of God making Hilbert’s hotel all in one go. Needless to say, Craig is clearly wrong in his charge here. He doesn’t begin to engage properly with the response to Loke, and so is far too premature to say that Loke is justified in making his argument.

7. Point 5.

Finally, Craig brings up “Alexander Pruss’ version of the argument for the finitude of the past”, by which he means the grim reaper paradox. Now, it might be true that there is no temporal mirror of the GRP. Perhaps. Cohen’s paper ‘Endless Future: A Persistent Thorn in the Kalam Cosmological Argument’ presents a version of this, and I developed a similar one in my post on the Dry Eternity Paradox (which I keep meaning to write up properly). So I don’t accept that it is obvious that no temporal mirror version of this argument could be made to show that the future must have an end. I think it is basically an open question, and that the various responses and counter responses have not been explored enough in the literature.

But, as Craig noted in point 1, and as I have been explicit about here, the symmetry argument wasn’t ever supposed to show that there is no successful argument for the beginning of the universe. It is not even required to defend the thesis that any such argument can be mirrored into an argument for the finitude of the future. Rather, the thought is just that the Hilbert’s hotel argument does apply to both past and future equally. Let’s suppose the GRP is asymmetrical in a way that Hilbert’s hotel is not. What would the significance of this be for the symmetry argument? It seems to me nothing. The GRP isn’t another version of the Hilbert’s hotel argument (and Pruss isn’t an advocate of the Hilbert’s hotel argument). So it would just be that an independent argument supports the second premise of the Kalam. This is basically irrelevant when it comes to assessing the merits of the symmetry argument, which is all about whether there is a symmetry breaker with respect to the Hilbert’s hotel argument to prevent it being applied to the future.

At this point, Craig’s appeal to the GRP seems like he is just appealing to another argument than the Hilbert’s hotel argument altogether. And that is fine, I’m happy to talk about the GRP instead of Hilbert’s hotel, but changing topic isn’t a response to the symmetry argument.

8. Conclusion.

I have no beef with Craig. In fact, I enjoy engaging with his published philosophy, even if I am critical of it. I wouldn’t engage with it if I thought it was low-tier material. He isn’t an idiot, nor do I think he is in bad faith (as it were). But I genuinely struggle to see how he carefully read the paper, or internalised the responses we gave to his arguments, given the contents of this five point response. The things he brings up are basically irrelevant (the ad hominem classification, or the comments about the GRP), or are just behind the discussion (as with his gesture to the asymmetry of time on the dynamic theory, or Loke’s theory). I genuinely think he skim read the paper and thought to himself “oh, this stuff again; I’ve already seen this sort of thing before” and didn’t pause to engage with it deeply. Otherwise it is hard to explain how his responses here are so superficial.

Perhaps one day he will spend some time going through it in more detail, or perhaps someone else will take up the challenge to defend him from the critiques. But his five point response really does nothing to advance the dialectic.

All the time in the world

My paper on the Kalam and successive addition argument came out in the journal Mind today. You can read it here:

The Logical Form of the Grim Reaper Paradox

[Edit: it turns out that something quite similar to this is argued for in this paper by Nicholas Shackel]

0. Introduction

The Grim Reaper Paradox (GRP) comes in various different forms. Sometimes it is about the divisibility of time, and sometimes it is about whether the past (or future) is finite. Even when we fix on which of these issues it is aimed at, there are also lots of different ways it can be cashed out. It can be reapers swinging their scythes, or placing point particles on planes, etc. Much of the discussion can be on how these details are to be understood.

Here I want to highlight what the GRP is at the most abstract level. It might be that once we think about it from this rarefied perspective, without the complications about what exactly the details are supposed to be, that we can see the paradox more clearly.

1. The Schema

Any GRP has a logical form, which I shall refer to as the schema. Let’s just have as a toy example, the following:

The past has no beginning. There is an eternal machine such that each day at midnight, it checks to see if it has printed out anything yet from its printer. If it has, then it hibernates for the rest of the day. If it has not printed anything out yet, it immediately prints out the date and then hibernates for the rest of the day.

This is enough to generate our paradox. If it had not already printed anything out, this means that yesterday it would have run the same check and printed out the date. So it can’t be that the machine finds nothing printed out today. But that applies also to yesterday too, and every previous day. So although it can’t be that no date is printed out, no date could be printed on the paper.

The way to conceptualise this abstractly is as follows. There is a rule that characterises this example (and all the others). It is a universal condition that applies at all times. That condition says that some proposition p (which might be that a reaper kills Fred, or places a point particle on a plane, or that a machine prints out a date, etc) happens at a time if and only if p does not happen at any earlier time:

For all t (p at t iff for all t’ (if t'<t, then ~p at t’))

It says that p is true at t if and only if p is not true at any earlier time.

The schema on its own is not unsatisfiable. That is to say, if there are only finitely many times, then the schema can be true. In particular, the schema is true if there are only finitely many times and p is true at the first time. At that first time, p is true at t, and on the other side of the biconditional, the nested conditional (if t'<t, then ~p at t) has a false antecedent, and as it is in the scope of a universal quantifier it is vacuously true. So both sides of the schema are true. At all other times, p is not true at t (so the left side is false), and on the other side of the biconditional we have a condition that says that ~p is true at all earlier times, which is false because (as we just went through), p is true at the first time. So in all cases, the biconditional holds.

But if there is no first time, then we run into the familiar problem. If p is true at some time t, then the right side of the biconditional says that no earlier time, t’, could have p true at t’. But then take t’. It is also the case of that time that no earlier time, t”, has p true at it either. So given the left side of the biconditional, p is true at t’. Contradiction.

If p is not true at any time, then the left side of the condition is false for some arbitrary time t. But if p is not true at any time, then its not true at any time t’ earlier than t, which makes the right side of the biconditional true, which in turn implies p is true at t. Contradiction.

2. Conclusion

Now we have a purely logical version of the argument, freed from any distractions about reapers, or point particles, or eternal machines. The GRP really just says:

1. There is no first time t
2. For all t (p at t iff for all t’ (if t'<t, then ~p at t’))

As we have just seen, you can’t have both of these together. That is the GRP.

Counting forever

0. Introduction

Here I just want to explain a simple point which comes up in the discussion of whether it is possible to ‘count to infinity’, and what that tells us about whether time must have had a beginning. Wes Morriston deserves the credit for explaining this to me properly. All I’m doing is showing two places where his point applies.

1. The targets

I have in mind two contemporary bits of philosophical literature. One is found in Andrew Loke’s work, specifically his 2014 paper, p. 74-75, and his 2017 book, p. 68. The other is found in Jacobus Erasmus’ 2018 book, p. 114. In each case, the authors are arguing that it is not possible to count to infinity because no matter how high one counts, no matter which number one counts to, there are always more numbers left to count. Here is how they express this point.

Firstly, here is Loke in his paper:

“If someone (say, George) begins with 0 at t0 and counting 1, 2, 3, 4, … at t1, t2, t3, t4, … would he count an actual infinite at any point in time? … The answer to the question is ‘No’, for no matter what number George counts to, there is still more elements of an actual infinite set to be counted: if George counts 100,000 at t100,000, he can still count one more (100,001); if he counts 100,000,000 at t100,000,000, he can still count one more (100,000,001).” (Loke, 2014, p. 74-75)

Secondly, here is Loke in his book:

“Suppose George begins to exist at t0, he has a child at t1 who is the first generation of his descendants, a grandchild at t2 who is the second generation, a great-grandchild at t3 who is the third generation, and so on. The number of generations and durations can increase with time, but there can never be an actual infinite number of them at any time, for no matter how many of these there are at any time, there can still be more: If there are 1000 generations at t1000, there can still be more (say 1001 at t1001); If there are 100,000 generations at t100,000, there can still be more (100,001 at t100,001), etc.” (Loke, 2017, p. 68)

Finally, here is Erasmus in his book:

“Consider, for example, someone trying to count through all the natural numbers one per second (i.e. 1, 2, 3, . . . ). Can the person count through the entire collection of numbers one per second? Clearly not, for no matter how many numbers the person has counted, there will always be an infinite number of numbers still to be counted (i.e. for any number n that one counts, there will always be another number n + 1 yet to be counted). Therefore, it is impossible to traverse an actually infinite sequence of congruent events and, thus, if the universe did not come into existence, the present event could not occur.” (Erasmus, 2018, p. 114)

What each is saying is that if someone starts counting now, they will never finish counting. And this is true, of course.

Think about the following mountain: it has a base camp at the bottom, but it is infinitely tall and has no highest point (for each point on the mountain, there is another one which is higher than that point). Can one start at the bottom and climb to the top of such a mountain? No, because there is no top of such a mountain.

(My dispute with Craig was not over exactly this point, but on something slightly more subtle. That was whether the following is false:

A) It is possible that George will count infinitely many numbers.

I say that A is true. All Loke and Erasmus’ sorts of considerations get you is to say that the following is false:

B) It is possible that George will have counted infinitely many numbers.

But we can leave this point here for now.)

2. My point

All I want to highlight today is that the fact that there are ‘always more numbers left to count’ only applies to certain types of infinite count. Imagine the following three scenarios:

i) George is trying to count the positive integers, in this order: (1, 2, 3, …)

ii) George is trying to count the negative integers, in this order: (…, -3, -2, -1)

iii) George is trying to count all the negative and all the positive integers, in this order: (…, -3, -2, -1, 0, 1, 2, 3, …)

We can think of the scenarios like this:

• Scenario i) is like climbing up an infinitely tall mountain that has a bottom but no top;
• Scenario ii) is like climbing up an infinitely tall mountain that has no bottom but does have a top;
• Scenario iii) is like climbing up an infinitely tall mountain with no bottom and no top.

In each case, due to the nature of infinite sets, the tasks involve counting the same amount of numbers. Simple intuition tells you that the third involves counting more numbers than the first two (and should be the sum of the first two). However, it is actually the case that each scenario involves counting sets with the same cardinality (that is, ℵ0). Put another way, each mountain is the same height as the other two.

The key point I want to make is this. It is obviously true that the Loke-Erasmus observation (that “there will always be an infinite number of numbers still to be counted”) only applies to scenarios i) and iii). It just doesn’t apply to scenario ii).

When George is starting at 0 and counting up, he always has the same amount of numbers left to count (always ℵ0-many). The same is true for when he is in scenario iii), no matter where George is along his task.

But if he is in scenario ii) he is in a very different situation. No matter where George is in his task in this scenario, he only ever has finitely many numbers left to count. He doesn’t have infinitely more mountain to climb in scenario ii). It has a top, and no matter where he is, George is only finitely far away from the top of the mountain. Clearly, he can reach the top of such a mountain if he is already some way along his climb.

And here is where the rubber meets the road. If there is a problem with scenario ii), it is not that “there will always be an infinite number of numbers still to be counted”, because that is just false. And scenario ii) is the one that Loke and Erasmus ultimately have in their sights. That’s because it is the one where George ‘completes’ an infinite task by counting an actually infinite amount of numbers.

Put simply, the argument they are making looks like this:

1. George cannot count up to infinity, because there would always be more numbers left for him to count.
2. Therefore George cannot count down from infinity.

Put like this, the fact that the argument is invalid is plain to see.

The problem, if there is one, is not about completing, or finishing, an infinite task. It might be that the problem has something to do with starting such a task. But if so, there is really no point in talking about problems that involve the impossibility of finishing, as that is a different point.

3. Conclusion

There might be other reasons to think that one cannot count down from infinity, of course. Indeed, Loke and Erasmus both have more to say about this issue. But what one often finds in discussions like this is the following sort of move: they make an observation that applies to an endless series, and try to apply that to a beginningless series. As is simple to see, sometimes the initial observation (such as that there will always be more numbers left to count) just doesn’t apply to both. And the switch from one to the other is thereby not valid.

Endless and Infinite

Philosopher Wes Morriston and I have coauthored a paper on the Kalam, and it has been accepted publication in the journal Philosophical Quarterly. Once it is actually available on their page access will probably be limited, unless you have an institutional subscription. However, for now you can download it (for free) via this link:

Endless and Infinite

Thanks,

Alex

0. Introduction

Like many people, I am participating in ‘dry January’, meaning that I am not drinking any alcohol during the month of January. I’m also thinking about the Grim Reaper paradox, and have spent much of the year thinking about the infinite future debate between Morriston and Craig. Interestingly, all of these things have come together, in a paradox I shall now dub the ‘dry eternity’ paradox.

Part of the inspiration for this comes from a paper I read recently by Yishai Cohen, called ‘The Endless Future: a Persistent Thorn in the Kalam Cosmological Argument‘(2015). In that, Cohen agrees with Morriston that if a beginningless past is an actual infinite, then so is an endless future, and thus if a beginningless past is impossible for being an actual infinite, so is an endless future. Cohen also argues that if the grim reaper argument shows that the past has to be finite, then a parallel version shows that the future must have an end as well. His argument is critiqued by Jacobus Erasmus in a paper called Cohen on the Kalam Cosmological Argument (2016). Erasmus’ rebuttal is that Cohen’s version of the grim reaper argument presupposes that it is possible for God to actualise an ungrounded causal chain, which can be plausibly denied.

1. Two versions of the Grim Reaper paradox

Here is how Erasmus sets up the grim reaper paradox:

Suppose that the temporal series of past events is actually infinite and that an actually infinite number of Grim Reapers exist. Suppose also that, at each past moment in time, a unique Reaper was assigned to issue a death warrant iff no previous Reaper had already issued a death warrant. (Cohen on the Kalam Cosmological Argument, p. 52)

This results in a contradictory state of affairs. Firstly, for all times tn, there must have been a warrant issued prior to tn. That’s because if no warrant had been issued by tn-1, then the reaper at tn-1 would issue their warrant (resulting in a warrant going out prior to tn).

But, this same reasoning also applies to tn-1 itself, giving us the contrary proposition. That’s because we can also say that the warrant won’t be issued at tn-1, because if it had not been issued by tn-2 it would have been issued by the tn-2 reaper (i.e. before tn-1).

Thus, we have both that the warrant must have been issued at some time prior to tnbut also that there is no time prior to tat which it could be issued. Contradiction.

Cohen applies this to the endless future. All we need to do is change the relevant tenses in Erasmus’ quote from above to get the following:

Suppose that the temporal series of future events is actually infinite and that an actually infinite number of Grim Reapers exist. Suppose also that, at each future moment in time, a unique Reaper is assigned to issue a death warrant iff no future Reaper will issue a death warrant.

But now we can derive a mirror image contradiction. Suppose that the reaper at t0 checks to see if any future reapers will issue warrants or not. At this point there are two options:

i) no future reaper will issue a warrant (in which case the treaper issues theirs)

ii) some future reaper will issue a warrant (in which case the treaper does nothing)

Suppose that at tno future reaper will issue their warrant, meaning that the reaper at t0 can issue theirs. If it were the case that no future reaper issues a warrant at t0, then, in particular, it is also true that at tno future reaper will issue a warrant (consider: if it it true today that I will never drink again, then it will also be true tomorrow that I will never drink again). But if it is true at tthat no future reaper will issue a warrant, this would mean that the reaper at tdoes issue their warrant! And, plainly enough, the reaper at tis in the future of the reaper at t0. So if, at t0, no future reaper will issue a warrant, then some future reaper (such as the one at t1) will issue a warrant! Contradiction.

Let’s take the other horn. Suppose at tsome future reaper will issue their warrant, meaning that the reaper at t0 can stand down. Let’s suppose it is the reaper at t1. Then we are right back to the beginning of the first horn again. For the reaper at twill only issue their warrant if none of the future ones will. But if it is true at tthat none of the future ones will, then this is also true at tas well, resulting in that reaper issuing their warrant which in turn brings about another contradiction exactly like the one from above.

Cohen discusses two objections that Koons posed to him in correspondence. The first of these is that in Koons’ version of the paradox, reapers are sensitive to what past reapers have done, but in Cohen’s version they have to be sensitive to what future reapers will do; yet it isn’t possible to have causal sensitivity to future events in the same way as to past ones. The reply Cohen makes to this is that an omniscient God could communicate the future to the reapers such that they know what the others will do, thus overcoming this causal asymmetry. Koons’ second point is that it isn’t possible for God to create beings who embody his omniscience. Even if that is true, the reapers themselves do not have to be omniscient (and can be quite ignorant of, say, how many coins I have in my pocket), just so long as God ensures that they know the behaviour of future reapers. In addition, Cohen points out that Koons’ reason for thinking that the reapers cannot embody omniscience has to do with avoiding causal loops, but it is not clear that there are any causal loops as such in this story (the behaviour of reaper n+1 does not depend on the behaviour of reaper n, etc). Thus it is far from clear that Koons has a successful reply here. One could avoid this by denying the possibility of an omniscient being that knows the future and can communicate it to reapers, of course, but a theist (in particular a Christian theist) will be unlikely to pick that option.

2. Erasmus’ objection

Erasmus’ objection comes at this from a different angle. He says that Cohen’s version of the grim reaper paradox (GRP) requires the following two principles:

K1. It is possible for God to predetermine an endless future

K2. It is possible for God to actualise an ungrounded causal chain

An ‘ungrounded causal chain’ “has a non-well founded relation (xRy, zRx, zRv, wRv, … ) because the chain lacks a first cause” (Erasmus, 2016, p 53). The behaviour of the reaper at t0 is determined by (or grounded in) the behaviour of reapers that are in its future. But the behaviour of the reapers in its future, such as the one at at t1, are themselves determined by (or grounded in) the behaviour of reapers in the future of them as well. Thus there is no ‘first cause’, or grounding, for the behaviour of the reapers. Let us suppose that this is an ungrounded causal chain, and that it is also (metaphysically?) impossible for God to actualise such a causal chain.

He then goes on to show that K2 is doing all the work in generating the paradox because it also applies to ungrounded causal chains that are purely spatial in character. Here is his spatial version of the example:

For example, suppose that time had a beginning and has an end. Accordingly, the predetermined series of future events is finite. Suppose further that space is inhabited only by an actually infinite row of successive Grim Reapers such that (1) there is a first Reaper but no last Reaper, (2) each Reaper is located at a unique spatial point, and (3) all the Reapers are facing the same direction. Now, suppose that God has predetermined that, at noon tomorrow, each Reaper will swing his scythe iff no Reaper in front of him swings his scythe. Accordingly, the same contradiction as above will result at noon tomorrow, namely, regardless of whether the first Reaper swings his scythe, it is both true and false that some Reaper in front of the first Reaper swings his scythe. The contradiction disappears, however, if (K2) is false. (Ibid)

Erasmus’ conclusion then is that denying K1 is not enough to block the contradiction, as it reappears in the spatial case. But denying K2 blocks both contradictions, and as such K1 is not the offending assumption. In effect, he is saying that Cohen’s GRP doesn’t show that the future must have an end. Rather, it just shows that God cannot actualise ungrounded causal sequences.

Now is time to present my version of the paradox that does works even if K2 is false. It does not require that God actualises any ungrounded causal sequences. All that it requires in addition to K1, is one additional assumption:

K3. God can act based on his (presently available) knowledge of future events.

Suppose God has decided to undertake an infinite version of dry January. That is, he has decided to stop drinking (say) holy water forever. However, he enjoys a drop of holy water (who doesn’t?), and wants to to have one final sip. Accordingly, he determines to obey the following rule:

Every day, God will check his comprehensive knowledge of all future events to see if he will ever drink again. If he finds that he does not ever drink again, he will celebrate with his final drink. On the other hand, if he finds that his final drink is at some day in the future, he does not reward himself in any way (specifically, he does not have a drink all day).

Again, we are caught in a dilemma:

Firstly, suppose that, at t0, God consults his comprehensive knowledge of the future, and discovers that he never again drinks after t0. He immediately downs a shot to celebrate (who wouldn’t?). But in that case, when he does his check the next day, at t1, he then will (again) discover that he will never have another drink, and immediately pour himself a drink to celebrate! So even though he rewarded himself yesterday for never having another drink, he is having another drink! Contradiction.

On the other horn, suppose that, at t0, God consults his comprehensive knowledge of the future, and discovers that he does indeed have a drink at some day after t0. Accordingly, he doesn’t celebrate by having a drink on t0. But in that case, there must be some future day at which he has a drink. Suppose it is t1. In that case, it must be that at t1 God will check to see if he will have any subsequent drinks, and find that he will not, resulting in him pouring the last drink. But now we are back at the start of the first horn, because his check at at twill also reveal a dry eternity ahead, at which point he will reward himself with another final drink! Contradiction again.

So we clearly have the exact same paradox again. This time however, it is not clear that God has actualised an ungrounded causal chain. After all, at each day God knows the future, and can merely consult his own (presently available) knowledge to see what happens in the coming days. We can imagine him writing it all down in a big book and every day he consults the book. Whatever causal story that happens each day that he consults the book, it is not clear that it is an ungrounded causal chain.

4. Replies?

One might deny that God can check his own knowledge to see what he knows about the future and act on it (K3). This would be weird. Why can’t God do that? Does he not know what he foreknows? Is he repressing it? Can he not act on what he knows? He seems to act on his foreknowledge on most versions of theism (specifically any where he has a plan, or reveals the future in prophecy, etc). Denying K3 leaves only the most austere versions of deism, it seems to me. Christianity seems hard to reconcile with its denial in any case.

Objecting to the possibility of the book doesn’t help unless it is really an objection to God’s omniscience, which a theist probably isn’t going to opt for (apart from Open Theists). Denying the possibility of an omniscient being would avoid the paradox of dry eternity though.

One could avoid the problem by denying the possibility of an endless future. The whole point of Morriston’s original reply to Craig was to say that if the past must have a beginning, then the future must have an end. This would vindicate Morriston’s challenge against Craig. It would show that either time has both a beginning and an end, or no beginning and no end, but that there is no third option.

The seemingly only other target we can find is the rule that God undertakes to obey. Perhaps it is not a proper rule, and that somehow it isn’t possible for God to undertake to obey it. Yet this seems rather strange. Consider this similar rule:

Every day, God will check his comprehensive knowledge of all future events to see if he will ever drink again. If he finds that he does not ever drink again, will celebrate with a chocolate bar. On the other hand, if he finds that his final drink is at some day in the future, he does not reward himself in any way (specifically, he does not have a chocolate bar).

Nothing paradoxical follows from this rule. Obeying this rule means that God drinks every day up to the day when he has his final drink, after which he eats chocolate bars every day. God can obviously follow that rule.

But what could stop him from undertaking to follow the rule obtained by merely swapping out the word ‘chocolate bar’ with ‘have his final drink’? Of course, it would lead to contradiction if he were to go this route, and that is a reason to think that it is (somehow) metaphysically impossible for him to swap those words around and undertake to follow the resulting rule. On the other hand, that is just to say that this is one of the things that could be denied to avoid the paradox. It doesn’t motivate thinking that it is impossible. We could ad hoc postulate anything is metaphysically impossible to avoid any paradox.

Something has to go to avoid the paradox of dry eternity.

The Grim Reaper Paradox, and the original solution, part 1

0. Introduction

Robert Koons has published a version of the ‘Grim Reaper paradox‘, also popularised by Alexander Pruss. One of the assumptions that sets up the paradox must be abandoned in order to avoid the contradiction. Koons and Pruss’ conclusion is that the assumption that an infinite past is possible should go.

It turns out that there is a potential solution to the paradox which was proposed by its originator Jose Benerdete, and more recently brought out in a paper by John Hawthorne (2000). I’m going to explain that solution here, and then (hopefully) develop it to the Koons/Pruss version in a part 2.

The Grim Reaper paradox was first put forwards by Jose Benardete in his 1964 book ‘Infinity‘ (p. 259). Although the developer of the paradox, Benardete was a passionate defender of the actual infinite, announcing in the blurb on the cover of his book that:

“This book is an attack on finitism in all its forms … A metaphysics of the actual infinite is offered as the solution to the contemporary crisis in the foundations of mathematics”

It is perhaps a consequence of the clarity with which he engaged with the topic was that his examples have since been able to be used to argue for precisely the opposite thesis; that is, for finitism. This is how the grim reaper paradox has been used in the contemporary philosophy of religion debate, primarily by Alexander Pruss and Robert Koons, who argue for a version of finitism (‘causal finitism’).

Here is how Benardete states the paradox:

“A man is shot through the heart during the last half minute by A. B shoots him through the preceding 1/4 minute, C during the 1/8 minute before that, &c. ad infinitum. Assuming that each shot kills instantly (if the man were alive), the man must already be dead before each shot. Thus he cannot be said to have died of a bullet wound” (p. 259)

Koons describes this story as follows:

“The story leads quickly to a contradiction, on the assumption that Fred [i.e. A] does not die unless one of the Reapers kills him. At least one Grim Reaper must act, since if all of the Reapers whose numbers are greater than 1 do nothing, then Reaper #1 will act. However, it is impossible for any Grim Reaper to act, since, for any n, Grim Reaper #n cannot do so unless Fred survives until its assigned deadline at 1/2n seconds after midnight. It is impossible for Fred to survive that long, since Fred’s surviving until Reaper #n’s deadline entails that no Grim Reaper with a number larger than (n+2) has acted, but, in that case, Reaper #(n+1) must have acted.” (Does the Universe Have a Cause?, p. 4)

The contradiction lies in the fact that Fred will surely die before the end of the minute, but that also there is no Grim Reaper who will kill him. What this paradox seems to show, at least according to the finitists, is that there could not be finite duration of time (such as a minute) that is actually divided into infinitely many sub-regions.

2. The original solution, part 1: Benerdete.

Benardete did not draw the same conclusions as Pruss and Koons from the grim reaper paradox, and remained committed to the possibility of actually infinitely divided durations and lengths. Shortly after stating the paradox, he makes the following comments about it:

As to the dead man, although he did not die of any single bullet wound, his death was certainly caused by the infinite fusillade of shots. Here, again, although he is already dead prior to each shot, he remains alive at any assigned instant which is prior to them all. Thus he cannot be said to have died at any moment of time whatever! (Infinity, p. 260)

These brief comments offer only a hint of how to understand the response. The outline can be made out however.

Firstly, Fred is alive at every moment prior to all the shots. Yet, for each shot he is already dead before that one is fired. It follows from this that “he cannot be said to have died at any moment of time whatever!” We can spell this out by saying that although there is no first moment in which it is true that Fred is dead, there is a last moment at which Fred is alive. And this is to be expected. The series of bullets (reapers) is an open infinite sequence; while there is a last bullet at 12:01, there is an endless sequence of bullets ever closer to 12:00, and in particular there is no first bullet. Given the continuity of time, we can think of the transition from 12:00 to later times as a Dedekind cut on the real numbers:

In the above diagram, the left line is time as it approaches 12:00, during which Fred is alive. This is a closed set (which is what the square join indicates), in that 12:00 is a member of that set, but it is the final member. So Fred is alive at every time up to and including 12:00. However, the line on the right is an open set (which is what the curved join indicates). So Fred is dead at every time strictly after 12:00. This is what Benardete means by “he cannot be said to have died at any moment of time whatever”. There is no first moment at which he is dead (rather, there is just a last moment at which he is alive).

This much of the solution seems fairly straightforward. It is addressing the time of death, and we have the following information. When is Fred alive? All the way up to and including 12. When is he dead? At all times later than 12. If time is continuous and actually infinitely divisible, this is what it is like to transition from one state to another (it is like a Dedekind cut on the real number line).

The second bit is where the difficulty lies. It involves the cause of death. Benardete has it that no bullet (reaper) is the individual that causes the death of Fred (“he did not die of any single bullet wound”). Yet, the totality of all of them does cause Fred to die (“his death was certainly caused by the infinite fusillade of shots”). But how can it be that the totality of the bullets causes him to die, when none of them individually causes him to die?

3. The original solution, part 2: John Hawthorne

This is where Hawthorne comes in. His paper Before-Effect and Zeno Causality develops Benerdete’s solution, spelling out the principles in greater detail.

Hawthorne first considers the case of a ball rolling towards an open-infinite Zeno-sequence of walls. 2 miles away there is a wall; 1 and 1/2 miles away is another wall; 1 and 1/4 miles away is another wall; 1 and 1/8 miles away there is another wall, etc. Thus, there is an infinite sequence of walls, ever closer to the point that is exactly one mile away. There is no wall which is the ‘closest’ to the one mile point (which makes it an open sequence). Suppose the walls are impenetrable and cannot be knocked over (etc). The ball is rolled towards the walls. What happens as it arrives at the one mile mark? Hawthorne’s answer is as follows:

The ball does not proceed beyond a mile and it does not hit a wall.” (p. 625)

We are puzzled by this combination partly because we have fuzzy intuitions about what happens when there is ‘contact’ between objects and open series in this sort of setting (where space is continuous and actually infinitely divided). But we can spell it out by specifying what we mean by ‘contact’ in such a way that it makes sense. Hawthorne calls ‘open-closed contact’ what happens when “A closed surface contacts an open surface insofar as there is no unoccupied space in between the two surfaces.” When the ball arrives at the one mile point, it has achieved open-closed contact, in the sense that the closed surface of the ball has no unoccupied space between it and the infinite series of walls. Here is what Hawthorne says:

Consider the fusion of walls. Call it Gordon. On reflection it is clear that the sphere contacts Gordon. Gordon has an open surface. When the ball stops proceeding at the one mile mark, there is no unoccupied space between the sphere and Gordon. Contact occurs …  So the ball is stopped by contact: The ball hits something, though the thing that it hits is not one of the walls. (p. 626)

If we find this puzzling still, Hawthorne has the diagnosis at hand. It is because we are assuming what he calls the ‘contact principle’, namely:

If y is the fusion of x’s and z contacts y, then z contacts one of the x’s.

That principle holds for the finite case. But it is false if the x’s are infinite in number. Once we are clear about this, there is no residual puzzle, nor anything further to learn about the wall case. It is clear what happens in worlds that satisfy the original description: At a mile, the ball makes contact with the fusion of walls, which is rigid and impenetrable. As a result, it does not proceed further. The ball does not, however, make contact with any wall. (Ibid)

It is weird that the ball can be stopped by making contact with the totality (or ‘fusion’) of all the walls even though it does not contact any particular wall,  but that is partly because we are taking an intuition which is applicable to the finite cases only, and trying to apply it to the infinite case. In finite cases, there will be a first wall, and contact would be defined in relation to that wall. However, this is precisely what we do not have in this case, and relying on an intuition that presupposes that there is one will lead us into trouble.

4. From walls to reapers

When it comes to the case of the infinite sequence of bullets (or reapers), Hawthorne basically applies the same analysis. Instead of the ‘contact principle’, he identifies the analogous principle and calls it the ‘change principle’:

“If x is the fusion of y’s and y’s are individually capable only of producing effect e by undergoing change, then x cannot … produce effect c without undergoing change”

This principle is also true in the finite case but false for the infinite case. Each of the bullets (reapers) are only capable of producing the effect of killing Fred by undergoing change (by being shot, or by swinging their scythes, etc). The assumption is that this applies to the totality of bullets (reapers). And it does, in finite cases. If there were only 10 bullets, then the totality would have to change, in the sense that one of the bullets would have to be fired at Fred, for it to bring about its effect. Yet, in the case of the infinite sequence of bullets (reapers), this is not the case. The totality can bring about the effect even though none of its elements (or the totality itself) changes in any way.

Going right back to Koons, we can apply this to his comments. In the quote I gave of his from above, he begins by saying:

“The story leads quickly to a contradiction, on the assumption that Fred does not die unless one of the Reapers kills him.” (Emphasis added)

What Hawthorne’s approach questions is this assumption. It is false that one of the reapers kills Fred, but that doesn’t mean that the effect is not brought by the totality of reapers. In the infinite case at hand, that is what happens. Once again, the weird behaviour of the infinite confounds our intuitions. Yet, giving up this intuition saves the situation from contradiction, and thus avoids the paradox.

5. Conclusion

Of course, some will argue that a ball cannot be stopped by a bunch of walls unless one of those walls makes contact with the ball, and the move from finite to the infinite context should make no difference to this.

Hawthorn is sensitive to the seemingly radical nature of the conclusion, admitting that it is a “big metaphysical surprise”. He goes on to finish by saying that:

“The Contact Principle, in full generality, could be given up fairly readily on
reflection. The Change Principle has a rather deeper hold on us. It seems to us scarcely thinkable that mundane causal powers—say that of killing with a machete—could combine so as to logically entail the causal power of producing some effect without the agent of the effect undergoing change. Nevertheless, surprising as this may be, the Change Principle should be rejected. The diagnosis is complete. The logic of each case is very much in order. And our puzzlement has been traced in each case to some faulty principle relating fusions to parts. Once we discard those principles, we will have no problem in accepting the required conclusions about what happens in each case” (p. 630-631)
I intend to write a part 2 to this, where I apply Hawthorne’s analysis to the version of the Grim Reaper paradox in which the target is not the actually infinitely divided duration, but the actually infinite past.

Darth Dawkins’ Failed Argument

0. Introduction

Recently Darth Dawkins (i.e. aggressive presup shouty man) has been running an argument according to which agnosticism is contradictory. You can see him make that argument in this short clip. The argument is fallacious, and I pointed that out to him recently. But just to make things crystal clear, I figured I would put it down in writing too.

1. Darth’s argument

Darth’s argument starts at 2:17 in that video. He says:

Another problem is that agnosticism is the claim that ‘I don’t know that any creator god exists or does not exist’. Now, if that statement is true, then it necessarily follows that Christianity is false. Now, if it necessarily follows that Christianity is false, then the agnostic knows that at least one major contender for the creator god does not exist, thereby contradicting the agnostic statement.

The way this is supposed to work is that the specific idea of the Christian God that Darth has in mind involves the notion of being ‘revelatory’, which is to say that it is part of the definition of God that he has revealed himself to you. On Darth’s conception, if this God existed, then you would know that he existed. It is this concept of God that he thinks agnosticism is incompatible with.

Here is  how the argument is supposed to run:

1. If Darth’s God exists, then you would know that he exists (Darth’s definition)
2. You do not know that Darth’s God exists & you do not know that he does not exist (agnosticism)
3. Therefore, you do know that Darth’s God does not exist. Contradiction!

The conclusion contradicts the second conjunct of premise 2. This is supposed to show that you cannot be agnostic about all gods, because some gods are such that if they existed you would know about them existing. Your not knowing about them existing is enough to know that they do not exist.

On the face of it, there is something fairly intuitive about this argument, and when he is aggressively shouting the premises at people it can be hard to spot where it goes wrong. But on closer inspection it is pretty clear that it is invalid, and we can bring this out very vividly.

2. The problem

So what’s wrong with the argument? Well, first of all, the argument is compressed, and there are clearly steps we haven’t made totally explicit. What exactly is the inference rule we are using to get to the conclusion? It’s not clear. So let’s make it easier. Let’s forget about the second conjunct of the second premise for a minute. Consider the following two premises:

1. If Darth’s God exists, then you would know that he exists
2. You do not know that Darth’s God exists

What follows from these premises? Well, it is basically the first two premises of modus tollens, i.e. ‘if p, then q’, and ‘not-q’. So we can apply that here and derive ‘not-p’ as follows:

1. If Darth’s God exists, then you would know that he exists
2. You do not know that Darth’s God exists
3. Therefore, Darth’s God does not exist

We can logically derive from the first two premises that Darth’s God doesn’t exist. If he did, I would know about it, but I don’t, so he doesn’t. So far, so good.

The contradiction Darth wanted to derive was using the second disjunct of agnosticism; ‘you do not know that god does not exist’. We snipped this off just to simplify the argument, but now we should bring it back in:

1. If Darth’s God exists, then you would know that he exists
2. You do not know that Darth’s God exists & you do not know that Darth’s God does not exist
3. Therefore, Darth’s God does not exist

The problem is that so far the conclusion is not the negation of this conjunct. To make it the negation, the conclusion would have to be ‘you know that it is not the case that Darth’s god exists’.

But the modus tollens we applied originally does not get us to this new conclusion. That is, the following is invalid:

1. If Darth’s God exists, then you would know that he exists
2. You do not know that Darth’s God exists
3. Therefore, you know that Darth’s God does not exist

Why is this invalid? Well, simply put, we can imagine the premises true and the conclusion false. Here is one example. Let’s just grant premise 1, as it is basically a definition anyway. Let’s suppose the subject in question does not know that this God exists (making premise 2 true). All we have to further suppose is that he doesn’t realise that this entails that Darth’s God doesn’t exist (which would make the conclusion false). This would mean that the premises are true and the conclusion is false. And there is nothing logically contradictory or incoherent about this supposition; there could easily be someone who fits the bill. Therefore, it is possible (i.e. logically consistent) for the premises to be true and the conclusion to be false, and that is what it means for an argument to be invalid.

The wider point is just that it is possible to not know everything that logically follows from what you know. When I pointed this out to Darth, I used a mathematical example. Suppose there is some conjecture in mathematics that is currently unproven. Either the conjecture is true, or it is false (it has to be one or the other). But as it is unproven, I don’t know which it is. But I do know the basic axioms of mathematics and the inference rules. So technically the truth or falsity of the conjecture (whichever it is) follows logically from stuff that I know. So this is an example of how you can not know what logically follows from what you know. And that means that you have to do more than just show that something is implied by what someone knows to conclude that they know the implication as well; often we are ignorant of the implications of what we know.

3. What about people who do know the conclusion?

But let’s suppose that if Darth walks his agnostic interlocutor through the reasoning, then he is highlighting the consequences of their belief to them. They might not have been aware of the consequences of their belief beforehand, but now they are, because Darth has helpfully demonstrated it to them.

Indulge me with a little dialogue:

Darth: ‘You believe neither that God exists, nor that he doesn’t exist, right?’

Agnostic: ‘Sure’

Darth: ‘Well, my God is such that if he existed, then you would know about him’.

Agnostic:’Ok, sure’

Darth: ‘Ok, Good. So it follows from you not currently knowing that my God exists that he doesn’t exist, by modus tollens. Right?’

Agnostic: ‘Oh yeah, I see what you mean. My mental state of not believing in him is logically incompatible with him existing.’

Now that Darth has raised to the level of consciousness how it follows from her beliefs that Darth’s God doesn’t exist, shouldn’t we say that our agnostic now knows the conclusion?

Well, maybe that’s fine. I mean, what if the agnostic person simply says something like: ‘Well, I guess I’m not agnostic about your version of God then. I am agnostic generally about the notion of god, but now you have spelled out the logical consequences of your particular God existing, I guess I am an atheist about that God; I positively believe, even know, that your God doesn’t exist.’

And once we spell it out like that, it seems perfectly reasonable. I mean, it is fine to not have exactly the same attitude towards every god concept. You might be more sceptical about the Mormon God than the Islamic God, or whatever. You might be an atheist about the Mormon God, but only agnostic about the Islamic God, etc. Why think we should have an absolutely universal attitude towards all god concepts?

Yet, this move is dismissed by Darth in this video (timestamped). Ask Yourself says that although he is generally an agnostic, he is an atheist with respect to Darth’s conception of God. Darth calls that a “childish response”, and laughs at it. But Darth’s dismissal here is itself a silly thing to say.

The problem for Darth here is that it is obviously unproblematic to take different attitudes towards different God concepts, especially if we are allowed to do what Darth does and tack on properties that God has, like ‘being such that you would know if he existed’, etc. As a particularly trivial example, consider the following:

A) The god such that it doesn’t exist

Obviously, if anyone bothered to think about this god concept, they would likely come to believe that it doesn’t exist (it doesn’t exist by definition). We can generate a slightly less trivial example as follows:

B) The god such that if I exist, then it doesn’t exist

Presuming you know that you exist, then you can easily conclude that this god doesn’t exist either. It is easy to come up with examples of this sort of thing. The god such that if I am having a sensation of blue right now (while looking at the sky or whatever) then it doesn’t exist; the god such that if I am thinking about arguments like this right now then it doesn’t exist, etc, etc.

So anyone who says they are agnostic will almost certainly caveat that claim somewhat, such as “…but obviously I do not mean that I have no opinion about trivially non-existent god concepts, such as the god such that it doesn’t exist, or the god such that if I existed then it wouldn’t exist, etc. About those types of god concept I do have an opinion, I believe that they do not exist.” It’s not childish or irrational to make that move at all.

But imagine we were to insist, as Darth seems to, that the terms ‘agnostic’ and ‘atheist’ could only be used to indicate an absolutely uniform attitude towards every god concept. Surely, then the term ‘theist’ would also fit that pattern. But if so, we could generate just the sort of trap that Darth thinks he has set. Consider the following god concept:

C) The god such that if you believe in it, then it doesn’t exist.

There is going to be something contradictory about believing in such a god. If you know C), and believe in this god, you can conclude (with a helpful interlocutor who will walk you through the steps) that such a being doesn’t exist. Therefore, there is precisely the same sort of contradiction in claiming to believe in this sort of god.

So imagine the following dialogue:

Interlocutor: So Darth, you believe in God, correct?

Darth: I am a theist, yes.

Darth: How so?

Interlocutor: Well, believing in the god such that if you believed in it then it wouldn’t exist entails that it doesn’t exist. If you grasp that inference, but continue to believe in it, then you believe in two incompatible propositions; you believe that it exists and also believe that it doesn’t exist.

Darth: But I am a Christian theist. That means I believe in the Christian God, not the God that is such that if I believed in it then it wouldn’t exist. I don’t believe that god exists at all.

In this dialogue, the attempt by the interlocutor to trap Darth into being committed to believing that the god such that it wouldn’t exist if you believe in it is obviously disingenuous. When Darth says he is a theist, he doesn’t mean he believes in that god. He means he believes in the Christian God.

This whole trap requires a kind of bait and switch, in getting Darth to commit to ‘theism’, but then to insist that he means to assert that he believes in a god concept that cannot be rationally committed to. The way out of this, which is a perfectly reasonable way out, is for Darth to insist that he has a non-universal attitude towards the family of god concepts; one of them he believes in, but the others he positively disbelieves in. That is to say, he is a theist with respect to Christian theism, but atheist about all the other god concepts (such as the one on the table). Yet, this move, nuancing the meaning of theism towards specific god concepts, is exactly the move that Ask Yourself made with respect to atheism and agnosticism. This is the very move that Darth decried as childish. But unless he allowed himself to make the same move, he would be caught by this version of his own problem.

Let’s summarise where we are:

• Darth’s argument is invalid for the simple reason that it trades off the false notion that everyone knows the logical consequences of the things they know. That is false, as the mathematical example shows. So even if you are someone who knows that if Darth’s god existed you would believe in him, and that you do not believe in him, all it takes is to be unreflective enough about the consequences of this to not form the belief that he does not exist for your mental states to be consistent. Not only is this possible, but this sort of thing happens to all of us all the time. Nobody knows all the consequences of the propositions they believe. Even Darth.
• But then if we consider someone who has actively and explicitly considered the propositions and implications in question here, they should just accept that they are not universally agnostic; about some god concepts they do actively believe that there is no such god. Not only is there nothing silly about this sort of move, because of trivial god concepts (like the god such that it doesn’t exist), unless Darth made use of the same move he would be caught in his own argument.

It might be that you believe something without realising that you believe it. Perhaps people believe things but their psychology forces themselves to deny that to themselves, like if someone witnesses something traumatic as a child and represses the memory, etc. Perhaps our sinful nature has a similar psychological effect, forcing us to repress our inate belief in the Christian God. Wouldn’t this undermine the agnostic’s claim to not believe in God?

Not really. Let’s make a distinction between gods that if they existed then you would explicitly believe that they existed, and ones where you would either explicitly believe that they exist or have a suppressed belief that they exist. The first is the version of Darth’s God that we have dealt with already. Simply not being aware of belief in this God is enough to entail its non-existence. But what about the new version? Let’s rewrite the argument from above:

1. If Darth’s God exists, then you would either explicitly believe that he exists, or have a suppressed belief that he exists
2. You do not explicitly believe that Darth’s God exists
3. Therefore, Darth’s God does not exist

This argument is invalid. It has the form ‘if p, then (q or r)’, ‘not-q’, therefore ‘not-p’. It is invalid precisely because we have to rule out both q and r in order to derive not-p; just ruling out q is not enough.

What this means is that agnosticism is not contradictory with this version of God, even if someone like Darth was to walk them through the steps to bring it to their attention. From their own lack of explicit belief, all they could conclude is that either that god doesn’t exist, or they have a belief that they are not aware of. Nothing else follows. So it makes sense to remain agnostic in such a circumstance. The trap Darth is trying to set doesn’t even spring if the type of belief involved can be suppressed. It only gets off the starting blocks if the condition associated with the god existing is such that you can determine if it holds or not. In this case, the agnostic cannot introspect and tell which of the two options (god not existing, or them having an unconscious belief) is true.

5. Conclusion

Darth’s argument here only seems like it works because he presents it in an aggressive way. It is all rhetoric and no substance. When people try to talk to Darth, it often seems like he gets the better of them. But if he were to drop the hyper-aggressive style and talk like an equal with someone, it would be clear under fair logical analysis that the argument is hopelessly flawed.

Tom Jump’s moral theory

0. Introduction

Tom Jump is an atheist YouTuber with a prolific output, making videos several times a week, mostly debating Christian philosophers. Recently, he did a debate with Ask Yourself in which they discussed whether there are moral facts or not, with Jump arguing that there are. Jump’s position is that there is something like a ‘moral law’, which has similarities to physical laws. Their debate descended into a squabble over whether a specific statement of Jump’s expressed a proposition or not. I don’t want to follow that part of the discussion, but I do want to look at Jump’s theory as (I think) he intends it, and to point out some of the problems it has.

1. What is the theory?

As I said above, Jump thinks that morality is objective, in a similar way to physical laws. Objective in this context means that it exists independently of any minds. We take physical laws, like the law of gravitation, to obtain in the universe just as much whether there are any minds present or not. If a moral law is objective, then it too would obtain just as well without any minds in existence.

Jump’s idea is that objective morality is defined in relation to the notion of involuntary impositions. I think we can have a go at reconstructing his idea as follows:

Action x is a morally wrong iff x is an involuntary imposition on A, for some agent A.

Part of the problem here is that we need to get clearer on what it means for something to be involuntary. This is because on various ways of understanding this term we run into trouble.

2. First go.

On one way of thinking about it, involuntary means something like ‘not actively consented to’. When things happen to someone but they have not specifically chosen that they happen, these are immoral. And sometimes that is right; sometimes things we haven’t actively consented to are immoral.

But this definition of ‘involuntary’ cannot be  what it means to be immoral, because if it were then it would classify things as immoral that are obviously not. For example, surprise birthday parties are not immoral, yet the recipient has ‘not actively consented to’ them happening. So it isn’t the case that the definition immoral is an involuntary imposition on someone, if involuntary means ‘not actively consented to’.

3. Second go

One might think that the problem with the surprise birthday case is that it isn’t involuntary unless you have actively stated that you do not want a surprise party. So maybe we could improve things by changing the definition of ‘involuntary’ from ‘not actively consented to’ to something like ‘against stated preference’. So before, the surprise party was immoral just because you didn’t say anything about the party, but now it would only be immoral if it went against your actively stated preferences. Assuming you have never actively expressed a preference for not having any surprise parties, it would not be an involuntary imposition on you for your friends to throw one for you, and so not immoral. This is an improvement, because it doesn’t misclassify surprise birthday parties.

And there is something fairly intuitive about this idea. Certainly sometimes things that go against our actively stated preferences are immoral. If someone tells you they do not want to have sex with you, but you continue to try to have sex with them then this would be a case of sexual assault, and clearly immoral.

But again, despite this partial alignment, this definition of involuntary cannot be the what it means to be immoral. That’s because there are obvious cases where things go against our stated preference, and are thus ‘involuntary’ in that sense, but that are not immoral. Imagine I go into a bar and order a beer. After I have finished it I state that my preference is for it to be on the house. If the bartender insisted that I have to pay for it, this would make it an ‘involuntary imposition’, because it is against my stated preference. But it is not an immoral thing for the bartender to do; he is quite within his moral rights to charge me for the beer, regardless of whether I stated that I would prefer not to pay for it. So there are obvious cases of things that are against my stated preferences which are not immoral.

4. Third go

As another try, we might say that something is involuntary if it is ‘against my desires’. We might think that the problem is that our previous two tries to define ‘involuntary’ were about whether we do, or no not, say something in particular. In contrast, we might think that it is about whether we have a desire or not, and not about what we say at all. So let’s define ‘involuntary’ as ‘against our desires’.

This would help with the surprise party example, as follows. Assuming I am the sort of person who enjoys surprise parties, then even though I haven’t actively stated that I consent to it, it wouldn’t be involuntary as such, because it wouldn’t be against my desires. It is the sort of thing I would have consented to had I known about it, because I desire that sort of thing to happen. So it is not involuntary, and so not immoral. So far, so good.

However, this is no help in the bartender case. I might just order a drink and desire for it to be on the house, but not say anything out loud. Is the bartender doing something immoral by charging me for the beer? No, clearly he is not. So this has the same problem here. Sometimes things happen that we don’t want to happen which are not immoral. Too bad.

This version also has problems from the other direction too. The problem is that sometimes people have immoral desires. Take a heroin addict who asks for his doctor to prescribe him some heroin. Clearly, the addict desires the heroin. Prescribing it to the addict wouldn’t be an involuntary imposition on him. But it is at least of dubious moral value for the doctor to do, if not outright immoral, even if the doctor wants the money being offered. Take another example: maybe some unstable (North Korean?) dictator asks an advanced country (the UK?) if he can buy some tanks from them. Clearly, it wouldn’t be an involuntary imposition on him to sell him the weapons, and maybe the other country wants the money. Still, just because both parties desire it doesn’t mean it is not immoral. We can easily iterate these examples.

5. Fourth go

One standard way to respond to these sorts of objections is to retreat from what people actually desire, to what they would desire if they were in some idealised state; if they were perfectly rational, etc. We might think that the heroin addict happens to desire another hit, but that he is just suffering from a lack of rationality. If he were being perfectly rational, then he would not desire to have more heroin; he would desire to get clean instead. And there is something intuitive about this particular example.

However, I think it is not so straightforward. The connection between rationality and desires is surprisingly complicated, and something debated at length by philosophers. One simple view, known as ‘Humeanism’ in the literature, is that someone is rational when their actions efficiently realise their desires. If I desire not to get wet, then knowingly walking into the rain without an umbrella is irrational. But, change the desire and the very same action becomes rational – if I want to get wet, then leaving my umbrella behind is rational.

The problem with this simple theory is that if you change the desire to, say, wanting to do something immoral, then the rational thing becomes whatever efficiently realises that desire, which would be to do something immoral. So if you want to kill someone, it might be rational to hit them over the head with a spade. Clearly, there is no guarantee that a perfectly rational person would have no immoral desires on this theory.

We could avoid this problem by abandoning the simple Humean theory. Instead of saying that only desires can motivate us, we could include beliefs too. Being rational might mean something like doing whatever realises your desires but is not believed to be immoral. So take someone who believes that it is wrong to murder people, but desires to kill you. He would be irrational if he hits you on the head with a spade because, although his actions realised his desires, they contradicted his beliefs about what is immoral.

But if someone had Jump’s starting point, then this option would collapse the whole project into circularity. We would have been led down the following path: the definition of immorality involves voluntariness, which in turn involves rationality, which itself involves the notion of beliefs about immorality, and the whole thing becomes a circle. We were offered a definition of immorality which in turn used the notion of immorality.

Thus, Jump is left with a dilemma: either tacitly include the notion of immorality in the definition of rationality, leading to circularity, or stick with Humeanism, and the problem of immoral desires.

6. Final thought

Even if this huge problem were somehow avoided, there is another one that is perhaps even more pressing. The whole point of this theory was supposed to be that it was a theory of objective morality. That means that the moral law that Jump was trying to express (which was supposed to be a bit like a physical law), doesn’t depend on minds to be true. But that is not the case here. If something is immoral when it is involuntary, then it depends on the person having some kind of intentional state, some desire or ‘will’, for it to be in contrast to. In a world where there were no people, there would be no wills for any action to be in contrast with, and so nothing would be immoral. There would be no true proposition, such as ‘x is immoral’, just because there would be no person on whom x would be an involuntary imposition. Thus, this theory is blatantly a variety of subjectivism and not a version of objectivism at all.

Does the impossibility of Hilbert’s Hotel rule out an infinite past?

0. Introduction

I recently went on the Unbelievable podcast and debated with a Christian philosopher called Andrew Loke. Following our conversation, we have had a lively email exchange where we have been continuing to debate the same issues. This has been a helpful exchange for me, as it has focused my attention on one or two very specific things and made me get clear about them. Here, I want to get some of it down into a blog post.

Hilbert’s Hotel is a hotel with infinite rooms in it. For convenience, I will talk of ‘a Hilbert’s Hotel’ (or HH) to refer to any infinite object like this (such as a library with infinite books in, or a book with infinite pages in, etc). An object, x, is ‘a HH’ if and only if both of the following are true:

a) proper parts of x can be put in a one-to-one correspondence with the whole of it, and

b) each part of x concretely exists in the present.

What I want to know is, does the impossibility of a HH in this sense mean that the past is finite? In order to answer this, let us assume that there is no metaphysically possible world with an HH in it; HH’s are not possible. Let’s call that our ‘original assumption’. Now the question for this post is, if we make the original assumption, is it possible that the past is infinite?

1. New Kalam

To put it the other way round: if a HH is impossible in the present, does that rule out an infinite past? If it did, then we could formulate a new version of the Kalam as follows:

1. If is possible that the past was infinite, then a HH would be possible (assumption)
2. HH’s are impossible (original assumption)
3. Therefore, it is not possible the past is infinite. (1, 2, modus tollens)

I think that this argument is valid, but unsound; the first premise is false. I will explain why I think that here.

2. a1-worlds and a2-worlds

Before we get there, I will motivate why someone might think differently to me. Here is one reason for thinking that the argument is sound, which Andrew endorsed. It requires thinking of two different ways that the past can be infinite, which I will call a1-worlds, and a2-worlds.

Imagine that every ten minutes God creates a grain of sand ex nihilo, and adds it to a heap of sand. This seems intuitively possible. Whenever the past is infinite, and also contains a continuous cumulative process, like adding grains of sand to a heap, or adding rooms to a hotel, we shall all this an ‘a1-type infinite past’, or an ‘a1-world’ for short.

If God had been doing this once every ten minutes forever, and if time had no beginning, then there would now exist an actually infinite object, i.e. a heap of sand with an actually infinite number of grains. So there would exist an HH in the present. Thus, it seems like if it was possible that the past were infinite like this, then it would also be possible that there is an HH in the present.

This motivates premise 1.

But now premise 2 comes in. It says that HH’s are not possible. If the antecedent of premise 1 (time is infinite) is true, then the consequent is true (a HH is possible). But because of premise 2 (a HH is not possible), the consequent of premise 1 is false. Therefore, the antecedent of premise 1 is false (by modus tollens). Thus, the past is finite. This is how the new Kalam works.

However, we can think of a different type of infinite past. Imagine God creates a grain of sand ex nihilo every 10 minutes (just like above), but instead of adding them to a pile, he immediately annihilates them. In this scenario, there is no infinite heap of sand, no HH, in the present. The past is infinite, but it involves no continuous incrementally increasing cumulative process. Call such a world an ‘a2-world’.

Because an a2-world doesn’t involve an HH existing in the present, the impossibility of an HH cannot rule out the possibility of such an infinite history. This makes it different to an a1-world in a directly relevant way. The impossibility of an HH in the present is compatible with the possibility of an infinite past (contra the new Kalam), just so long as it is an a2-type infinite past and not an a1-type infinite past (i.e. just so long as there is no continuous incrementally increasing cumulative process).

3. Disambiguating the first premise

Given that there are two types of infinite past (ones with a continuous incrementally increasing cumulative process, and ones without) the original premise was ambiguous. It said:

1. If it is possible that the past was infinite, then a HH would be possible

It doesn’t distinguish between the two ways that the past can be infinite (i.e. a1- or a2-type). If 1 is true on both its disambiguations, then it is a distinction without a difference. However, if it is true on one but false on another, then we need to disambiguate to know which one is which.

What we need to do is consider each as a separate interpretation of the above premise, to see how it plays out on both disambiguations. So we have:

i) If the past is an a1-world, then a HH would be possible

ii) If the past is an a2-world, then a HH would be possible

If both of these are true, then the new Kalam holds. If one is false, then its first premise is also false. No surprises, it is true on the first, but false on the second. Let’s go through them one by one to see how this works.

Given that the definition of an ‘a1-type’ history includes as a clause that a HH is present, then it is obvious that whenever there is an a1-type history there is also a HH. That much is just a matter of definitions. But, because there are no possible worlds at all where there is a HH (because of premise 2 of the new Kalam), there are no a1-type worlds either. Both antecedent and consequent are necessarily false. And (somewhat counterintuitively for those not familiar with classical logic) that means that i) is a tautology, and so necessarily true.

The truth of ii) though is not so easy to determine. a2-worlds don’t themselves involve HH’s, so on their face it is not so obvious that they imply the possibility of HH’s. In order to rule out a2-worlds, the defender of the Kalam must find a way of arguing that a2-worlds are impossible. How might they do that?

Here is how. God can make a grain of sand, and he can then either destroy it or put it on a heap. Both creating grains to add to piles, and creating grains to destroy, are possible for God. If he can do one, he can also do the other. And the thought is that if he could make 1 grain and put it in a heap (which he surely can), then he could make 2, and if he could make 2 he could make 3, etc. By induction, it seems, he could make an infinite pile. So if an a2-world is possible, then so is an a1-world.

And that just means that if he can make and destroy grains of sand, then he can also do something impossible (make grains and put them in an infinite heap). In classical logic, anything that implies an impossibility is necessarily false.

If the possibility of an a2-type history implies the possibility of an a1-type history, and an a1-type history is impossible (which it is by the original assumption), then an a2-type history is also impossible (by modus tolens):

1. If a2 is possible, then a1 is possible
2. a1 is not possible
3. Therefore, a2 is not possible (1, 2, modus tolens)

Premise 1 of this little argument is what I want to focus on. If it succeeds, it would collapse the disambiguation between a1-type and a2-type histories. It would mean that the first premise of the original argument (‘If the past was infinite, then a HH would be possible’) would be true. And if that were the case, then the impossibility of a HH would mean that time was finite, and it would be game over.

So the salient question becomes:

Does the possibility of an a2-type history imply the possibility of an a1-type history? In other words: if it is possible for God to spend forever creating grains of sand and then destroying them, does that mean that it is also possible for God to spend forever creating grains of sand and piling them up?

I think I can make a pretty strong case to say that the answer is ‘no’. It involves the realisation that if a HH doesn’t already exist, it cannot exist, and why that’s the case. If I’m right, then the impossibility of a HH doesn’t tell us whether past is finite or infinite. It might be, for all the premises of the new Kalam say, that the past is infinite.

4. Life in an a1-world and an a2-world

To understand why the possibility of an a2-world doesn’t imply the possibility of an a1-world, it helps to consider what is true in a1-worlds and in a2-worlds.

Imagine, first, that an a1-world is actually the case. In that setting, an HH exists in the present. Maybe an infinite heap of sand, or an infinite hotel, whatever. The past in that world involves a continuously increasing incremental cumulative process, like God adding grains of sand to a heap.

The number of grains of sand there are in the infinite heap right now is at least equal to ω1. But how many grains were there in the pile just before God added the most recent grain? However many there were, it cannot be any finite number, because there is no finite number, x, such that ω1 – 1 = x. So before God added the latest grain of sand, there was already more than a finite number of grains of sand in the pile.

This point generalises. Before God added the latest, say, 1,000,000 grains of sand, there were still more than infinite grains of sand in the heap (because there is no finite number, x, such that ω1 – 1,000,000 = x). What this shows is that if there is a HH in the present, then there has always been a HH throughout the infinite past. If the heap is infinite now, then there is no point at which the pile had finite members.

We can also generalise this into the future too. If the heap has ω1 grains in it now, then if God started removing grains of sand from it one by one, there would always remain an infinite number of grains of sand left in the pile. It wouldn’t decrease below infinite, no matter how long he continued to take grains of sand away.

So on any a1-world, where there is a HH in the present, it follows that there must also be a HH throughout the past and throughout the future as well.

If there is a HH in an a1-world, it follows (trivially) that it is possible that there is a HH there too. Yet, our original assumption was that a HH is not possible. Thus, the a1-world is inconsistent; in it a HH is both possible (because it exists at every point in time), and also impossible (by the original assumption). They are not just inconsistent, but they are ‘full’ of contradictions, in the sense that there is a contradiction true at every time in every a1-world. Thus, a1-worlds are inconsistent, thus are not metaphysically possible worlds at all.

Now consider an a2-type history. In this case, there is no HH in the present. We know from the above reasoning that had there been a HH in the past at any point, then there would still be one now. If one existed any finite number of days ago, then not even God could have removed all the grains from it one by one at regular intervals such that there are none left today. And that is true no matter how long ago God started to remove grains.

Not only would there still be grains left, there would still be infinite grains left, no matter when he started to remove grains. Needless to say, if there was a HH at some point in the past, it would still exist now.

So, if there is no HH today, there never has been.

There is a future tense version of this too. God could start putting grains of sand in a heap now, but no matter how long he continues to do that there will never be an infinite number of them.

So, if there is no HH now, then there never will be.

So on any a2-world, there is no HH; there never was, and there never will be. This makes a2-worlds the mirror image of a1-worlds. Rather than being full of contradictions, a2-worlds are ’empty’ of them (there is no contradiction at any time in any a2-world). Unlike in the case of the a1-worlds, the original assumption that HH’s are impossible does not cause any contradiction here, because it is always false that there is a HH in all a2-worlds.

5. Objections

Imagine that an a2-world is actual. That means that there is no HH in the present, and no continuous incremental process throughout the past. However, is it possible for God to make a HH in this world? If there is no HH in the present, is it possible for there to be one?

The answer is: not if he has to start with nothing and can only add grains one at a time at regular intervals. No matter how long he does that for, he will never get to there being more than finite grains. An a2-world cannot change into an a1-world. Not even God can do that, assuming he can only add them one at a time.

That’s also true on the assumption that he adds any arbitrarily large finite number of grains to the pile at regular intervals. It doesn’t matter if he adds one grain, or a million, or a billion billion billion; so long as it is at regular intervals, like every 10 mins, there will always only be a finite number of grains, no matter how long he has been doing it. Thus, it is not possible for God to make an infinite heap of sand if there isn’t already one there.

There are two possible objections to this.

We might question the condition that God can only add finite numbers of grains to the heap at any one time. Maybe God could create all the grains at the same time, instantaneously. Maybe he could just click his fingers and make an infinite pile of sand ex nihilo. If so, then it wouldn’t be impossible for God to change an a2-world into an a1-world.

Alternatively, we could question the condition that God can only add grains at regular intervals. Maybe, God could do a supertask, and get infinite grains added in a finite amount of time. Instead of adding them at regular intervals, he could speed up the time it takes him to add them to the pile. If so, then it wouldn’t be impossible for God to change an a2-world into an a1-world.

Let’s take these one by one.

Firstly, let’s suppose that God could make infinite grains of sand appear instantly as one big heap of sand. The proposal is then that: if God existed, then a HH would be possible. But that would then contradict our assumption that a HH is metaphysically impossible. From that we could conclude that God doesn’t exist:

1. If God could make a HH all at once, then a HH would be metaphysically possible (new assumption)
2. A HH is not metaphysically possible (original assumption)
3. Therefore, God cannot make a HH all at once (1, 2, modus tollens)

So the problem with the idea that God could have just made a Hilbert’s Hotel all in one go, is that it is metaphysically impossible to do that (by the original assumption). Hilbert’s Hotels are impossible, so it is impossible to make one in one go.

What about if God did a supertask? A supertask is where an infinite sequence is squeezed into a finite space or time. For example, imagine if you took one step to the door that got you half way, and your next step was only half the remaining distance, and the next half the remaining distance, etc. In this way, you could fit an infinite number of steps into a finite distance (between here and the door).

Similarly, maybe God could create the first grain of sand and add it to the heap in 10 minutes, and the next one in 5 minutes, and the next one in 2.5 minutes, etc. By the time 20 minutes had passed, he would have made an infinitely big heap of sand.

This would be an incremental process, but with an ever shorter interval between iterations. We might think that this is better than the previous idea, because it doesn’t require God doing anything infinite in extent; each iteration he adds a finite amount of sand to the heap (one grain), unlike in the last one where he did an infinite task in one jump.

However, the counter-argument is the same. If a supertask was possible (whether for God or for anyone else), then a HH would be possible. If our original assumption is that a HH is not possible, then a supertask is not possible either.

1. If God could make a HH by a supertask, then a HH would be metaphysically possible (new assumption)
2. A HH is not metaphysically possible (original assumption)
3. Therefore, God cannot make a HH by a supertask (1, 2, modus tollens).

So the problem with the idea that God could have just made a Hilbert’s Hotel by a supertask, is that it is metaphysically impossible to do that (by the original assumption). Hilbert’s Hotels are impossible, so it is impossible to make one by a supertask.

The wider lesson is this: if HH’s are impossible, then there is nothing that God could do to make one.

What this shows is that if we hold fast to our assumption that HH’s are metaphysically impossible, then all God could do is add or subtract finite amounts of grains at regular intervals. He couldn’t do a supertask to build one, or make one with the snap of the fingers. And that means that if there is no HH now, then even God couldn’t make one.

And this means that if an a2-world is actual, then an a1-world (with an HH in it) is impossible.

6. Conclusion

Let’s circle back. Earlier, we had looked at the following implication:

1. If a2 is possible, then a1 is possible

We saw a reason to think that this is plausible. After all, if it is possible for God to spend an infinite past making and then destroying grains of sand (i.e. if an a2-type history is possible), then it is possible for him to not destroy them, but pile them up instead (i.e. an a1-type history would be possible). Creating grains of sand is not beyond God’s power, and neither is putting grains of sand in a heap. Thus, surely, if an infinite past is possible, then God could have spent that whole time making an infinite heap of sand, as it just involves him doing things he can do. This is a compelling point. This is the best argument for thinking that an a1-type world is possible.

The problem, as we saw, is that it runs into our original assumption, that a HH is impossible. That’s because if God did accumulate infinite grains of sand, like in an a1-type world, then there would be a HH now. If the original assumption is true, then there can’t be a HH now. So there can’t be an a1-world.

And that feels right to me. An a1-world has a HH; but HH’s are impossible; so a1-worlds are impossible too. The impossibility of a1-worlds just is the impossibility of the HH’s that exist in them after all. The impossibility of a1-worlds is a logical consequence of the impossibility of HH’.

We looked at an inference, which was that the possibility of a2-worlds implied the possibility of a1-worlds. But if there is no HH in the present, and HH’s are impossible, then there is nothing God can do to make one. Not even God can change an a2-world into an a1-world, if HH’s are impossible. That makes the inference false. The possibility of an a2-world does not entail the possibility of an a1-world. Nothing does (because they are impossible).

This leads us all the way back to the disambiguation we started off looking at. There, we had disambiguated the following:

1. If it is possible that the past was infinite, then a HH would be possible

The first disambiguation was a tautology, and so true. The remaining disambiguation to consider was the second one:

ii)  If it is possible that the past was an a2-world, then a HH would be possible

We know that the consequent is false (because of the original assumption). But it seems like the antecedent is true; it is possible that the past was an a2-world. That would make ii) false.

And that means that the original premise is ambiguous, and on one disambiguation, (the second one) it is necessarily false. Here is the premise again:

1. If it is possible that the past was infinite, then a HH would be possible

If the past is infinite, then that means that it is an a2-world (because a1-worlds are impossible). At all a2-worlds, HH’s are not possible. Therefore, the antecedent is true and the conclusion is false. This means that the impossibility of HH does not mean that the past is finite, and our new Kalam argument is unsound.