Loke’s Singing Angels: the Kalam and abstract entities

0. Introduction

Readers of this blog are likely well aware of the Kalam cosmological argument, and its presentation by William Lane Craig. In particular, readers will probably be well aware of the line of defence for the second premise according to which the past must be finite.  This says that the past must be finite because, if not, then there would be an ‘actual infinite’ (or a ‘completed infinite’), and such a phenomenon could not exist without ushering in various purported ‘absurdities’ (such as those illustrated by Hilbert’s Hotel).

Craig and Wes Morriston have had an interesting exchange on this issue (a useful summary of which is found here). In short, Morriston argues that if the past cannot be beginningless because it would constitute an absurdity, then the future cannot be endless, or else it would constitute a symmetrical absurdity. Morriston imagines a pair of angels who take it in turns to sing praises to God forever. Theists, such as Craig, who hold that the future is everlasting, need to show that such a scenario is possible without resulting in the sorts of absurdities that apply to Hilbert’s Hotel, or to the beginningless past. They need a symmetry breaker between past and future.

Craig’s move is to argue that a beginningless past is an ‘actual infinite’, whereas an endless future is merely a ‘potential infinite’.

“No absurdity there, for the number of praises said by the angels will always be finite, even though increasing toward infinity as a limit” (Craig, “Taking Tense Seriously in Differentiating Past and Future: A Response to Wes Morriston,” Faith and Philosophy, 27:4 (2010), 451-456, p. 452)

The debate centres around what the notions of ‘actual infinite’ and ‘potential infinite’ refer to. Craig is right that the number of praises that have been said will always be finite. Yet, Morriston argues that this misses the point:

“But I was not asking for the number of praises that have been said. Instead, I was asking for the number of praises yet-to-be-said – that is, for the number of praises, each of which will eventually be said. In the world of my thought experiment, the series of praises yet-to-be-said is not growing, is never finite, and does not satisfy Craig’s definition of “potentially infinite.”” (Morriston, “Beginningless Past, Endless Future: A reply to Craig”, Faith and Philosophy 29 (4):444-450 (2012), p. 3)

Into this debate steps Andrew Loke, with his paper “On beginningless past, endless future, God and Singing Angels: an Assessment of the Craig-Morriston Dialogue“. In particular, Loke says:

“…the distinction between abstract and concrete infinities is helpful for responding to Morriston’s counter-argument based on the number of angelic praises yet-to-be-said” (p. 58).

It is to this response to Morriston that I want to focus here.

1. Loke’s argument (?)

Loke makes the central distinction between the number of praises that will have been said, and the number of praises that will be said. Somewhat confusingly, he refers to them as those ‘that-will-be-said’ and those ‘yet-to-be-said’, which sound like synonyms to me. However, he seems to be applying these labels in the right way, when he says:

“Craig’s response that the number of praises that-will-be-said is a potential infinite is inadequate, for Morriston is not asking about the praises that-will-be-said, but rather the praises yet-to-be-said, the series of which, Morriston argues, is not growing, is never finite, and does not satisfy Craig’s definition of potentially infinite, as noted above. Craig’s main point in reply to Morriston was about the praises that-will-be-said, not the praises yet-to-be-said, and thus he misses the thrust of Morriston’s counter-argument” (p. 63)

I agree with this much of Loke’s interpretation of the debate. Loke promises to respond in a way that does not merely fall into the same equivocation that Craig does, “In what follows, I shall (unlike Craig) focus on ‘the praises yet-to-be-said’” (p. 63).

The main thrust of Loke’s response is outlined in the following passage:

“In response to Morriston’s counter-argument, it seems that proponents of the Kalam Cosmological Argument do not have to follow Craig’s proposal to either deny Platonism with respect to propositions or deny that there are an infinite number of propositions. Rather, they can hold the view that the argument against the possibility of the actual infinite based on paradoxical implications is directed against the existence and actualization of an actual infinite number of concrete entities or events; it is not directed against the existence of an actual infinite number of abstract entities.” (p. 63)

Clearly, Loke is going to make his response to Morriston focused on the concrete/abstract distinction. Unfortunately, there is a slide between talking about an infinite number of propositions, and an infinite number of events. Also unfortunately, we are not given a definition of either ‘abstract’ or ‘concrete’. Loke appeals several times to J P Moreland, and a paper called “A Response to a Platonistic and to a Set-Theoretic Objection to the Kalam Cosmological Argument” (Religious Studies, Vol. 39, No. 4 (Dec., 2003), pp. 373-390) (henceforth ‘Response’). In that, Moreland offers a definition of ‘abstract object’ that we can only assume is close to what Loke has in mind. Moreland says:

“An entity is abstract just in case (a) it is not a person, and (b) it exists outside space and time in that it has no spatial or temporal location or duration” (Moreland, Response, p. 376).

Loke’s explicit reference to ‘Response’ is not to this, but to a similar distinction, according to which an actual infinite is absurd like the Hilbert’s Hotel example only if:

“(1) the members of the set are finite, located, moveable entities, which opens up the possibility of adding, subtracting, or rearranging the members of the set, and (2) the members of the set are spatially extended.” (Loke, p. 63)

Moreland’s idea is that what makes an actually infinite collection absurd is that it meets these two conditions. He explains that what the hotel example is supposed to be showcasing involves spatially located guests ‘moving about’:

• “Regarding Hilbert’s Hotel, the problem is that if we move the guests from one location to another, there just are no rooms available into which they can be moved. All of them are already filled. Moreover, there is no way to open up a new room by this procedure because there is no spatial region available into which they can be moved or new rooms can be added” (Response, Moreland, p. 379).

• “If one adds (or subtracts) members to an actual infinite set, then one has not increased the number of members of that set, but this is false since we have before us the new member who was added” (ibid)

Moreland is careful to identify the absurdity of an actual infinite with the ability to manifest this sort of behaviour. On page 380 of the same paper, Moreland makes the following careful statement of what is ruled out by Craig’s appeals to absurdities associated with the actual infinite:

“An actual infinite number of finite, contingent entities that i) can be added to or subtracted from a set and ii) are spatially (or spatio-temporally or temporally) extended cannot exist”

Moreland’s idea can be put like this: if an actual infinite meets both i) and ii), then it cannot exist. If it can exist, then (using modus tollens), we can infer that it does not meet both i) and ii).

Indeed, Moreland draws exactly such a conclusion, arguing that a platonist can affirm the existence of an actually infinite number of abstract entities. These entities fail to fulfil either of his conditions, in that they cannot be subtracted from or added to, and they are not spatially or temporally extended.

It should be noted that to avoid the absurdities, an actual infinite only need avoid satisfying one of Moreland’s two conditions, and not necessarily both (they are individually necessary and jointly sufficient conditions). We will come back to this.

So let’s make the distinction between two types of actual infinities; there are ‘active’ and ‘passive’ actual infinities. An actual infinity is ‘active’ iff meets both of Moreland’s conditions above, and it is ‘passive’ iff it is not active.

Loke shares Moreland’s insight, which is that actual infinities that are composed entirely of abstract objects are passive. For example, take the natural numbers, construed as platonic abstract objects. These constitute an actual infinity. These entities, it seems reasonable to think, cannot be ‘moved about’ in any way. It makes no sense to rearrange them, or subtract them. Of course, one can perform ‘subtraction’ using numbers, but one cannot remove a number from the total collection of the natural numbers. They are utterly unchangeable. Thus, some actual infinities, such as the natural numbers, are passive because they are abstract objects. As Loke says:

“Since abstract entities are not finite, located, moveable entities nor spatially extended, the argument against the possibility of the actual infinite based on paradoxes such as Hilbert’s Hotel does not apply to them” (p. 63).

So far, so good.

At this point, I think it will be helpful to codify this in a principle, which I shall call Loke’s principle (or ‘LP’):

Loke’s Principle) Actually infinite collections of abstract objects are passive.

 

Loke goes on to make what seems to me to be the central argument of the paper:

“Having made the point that argument against the possibility of the actual infinite based on paradoxical implications is directed against concrete rather than abstract entities, the proponent of the Kalam argument can reply to Morriston as follows: The number of praises yet-to-be-said – that is, the number of praises, each of which will eventually be said – is actual infinite, but the number of praises yet-to-be-said do not yet exist as concrete entities or events.” (p. 64)

Let’s try to be clear about what is being said here, because the passage is hard to read clearly. When he says “argument against the possibility of the actual infinite based on paradoxical implications is directed against concrete rather than abstract entities”, Loke seems to be merely articulating what is encoded in LP. If so, it is a statement against which I have no objection.

Loke makes the claim that the not yet uttered praises “do not yet exist as concrete entities”. There is an unfortunate scope ambiguity here. It’s not clear here which of the following Loke means to convey:

a) The future praises currently exist, but are not currently concrete

b) The future praises are currently concrete, but do not currently exist

Is the idea that they do not yet exist at all, or that they do exist, but merely as abstract objects? The phrase is ambiguous.

Loke goes on, but the ambiguity persists:

“Neither would it ever be the case that all of them have been actualised as concrete entities or events, for no matter how many praises have been said, an angel could still say one more. As noted earlier, Morriston agrees that ‘the number of praises that have been said by the angels in my scenario will always be finite,’ and that ‘the collection of praises that have been said will increase without limit.’ One might think it absurd that an actual infinite number of praises will never be said even though each of the praises will eventually be said, but given that ‘the praises that will eventually be said’ do not yet exist as concrete entities or events, this is a problem that remains in the abstract and does not cause problem in the concrete world.” (p. 64, emphasis added)

The phrase “this is a problem that remains in the abstract and does not cause problem in the concrete world” is even more difficult to understand. The implication seems to be that future praises are not concrete, and this quarantines the actual infinite in the abstract realm and not “in the concrete world”. Just like the way Moreland can have an actual infinite collection, just so long as they are abstract objects, Loke seems to be implying that the the praises that will eventually be said is an actual infinite, but one that is not a problem due to being a collection of abstract objects.

The question though is whether we should consider future praises to be abstract objects, or concrete objects.

Let us refresh our memories of the clear (enough) definition offered by Moreland:

“An entity is abstract just in case (a) it is not a person, and (b) it exists outside space and time in that it has no spatial or temporal location or duration” (Moreland, Response, p. 376).

Presumably, and entity is concrete iff it is not abstract. (Is there a third category?) If they are mutually exclusive, then we can ask ourselves wether future praises fulfil both of Moreland’s criteria or not. If so, then they are abstract; if not, then they are concrete.

Clearly, future praises are not people, so they meet condition (a). But do they exist outside space and time and have no spatial or temporal location or duration? There is a small head-scratcher about whether ‘a time’ itself is temporally located or not, but a a ‘future event’ seems to be clearly temporally located. It is in the future! The praise will also have a duration; it will take some definite amount of time to say the prayer. Thus, it seems like the future praises are not abstract by this criteria. Although they meet criteria (a), they do not meet criteria (b). They are quite obviously not ‘outside time’ being events in time (in the future) and they have duration. Unless there is a third category, future praises are concrete.

If that is right, then the ambiguity in Loke’s claim can be cleaned up. The future praises are currently concrete, but not currently existing. As time passes, they do not ‘become concrete’; rather, they begin to exist. Perhaps we can put it like this; as time passes the future praises ‘become actual’. Either way round, what is happening is that the merely potential future praises, which are always concrete, become actual and/or existing present praises, before becoming past praises.

But if that is right, then what has become of the original promise made by Loke in the start of the paper? Recall, he said:

“…the distinction between abstract and concrete infinities is helpful for responding to Morriston’s counter-argument based on the number of angelic praises yet-to-be-said” (p. 58).

Yet, future praises are concrete, just like past praises, given Moreland’s distinction. How is that a symmetry breaker? It is unclear how, if at all, the distinction between the abstract and concrete is supposed to be helping.

3. What do ‘abstract’ and ‘concrete’ mean for Loke?

If Loke were using Moreland’s definition of abstract, then he could not argue that future praises are not concrete. That would mean arguing that future praises are outside of time. This is clearly not the case. Thus, Loke must have something else in mind when he uses these terms. But what could that be?

One starting point is David Lewis’ book, Plurality of Worlds, chapter 1.7: Concreteness (p. 81 – 86). Lewis says:

“A spectator might well assume that the distinction between ‘concrete’ and ‘abstract’ entities is common ground among contemporary philos- ophers, too well understood and uncontroversial to need any explaining. But if someone does try to explain it, most likely he will resort to one (or more) of four ways” (p. 82)

These are:

1. The Way of Example: “concrete entities are things like donkeys and puddles and protons and stars, whereas abstract entities are things like numbers”, p. 82.
2. The Way of Conflation: “the distinction between concrete and abstract entities is just the distinction between individuals and sets, or between particulars and universals, or perhaps between particular individuals and everything else”, p. 83.
3. The Negative Way: “abstract entities have no spatiotemporal location; they do not enter into causal interaction; they are never indiscernible one from another”, p. 83.
4. The Way of Abstraction: “abstract entities are abstractions from concrete entities”. p. 84.

Could Loke mean one of these ways of construing the distinction?

If he took the Way of Example, then he would be saying that future events are “things like numbers”. Part of the problem here is what it means to be a number. Moreland gave an example of what it means to be a number, but that required that the entity does not exist at any time, which clearly future praises do not do. Moreland also thinks that numbers are “immutable” (‘Response, p. 380). Yet going from being abstract to being concrete certainly seems like a type of change. All these give us reasons to think that future praises are not “like numbers”.

I’m tempted to think that Loke is a member of the Way of Conflation. He doesn’t conflate the concrete/abstract distinction to the individuals/sets distinction though. If I was being mean, I would say that I think he conflates the concrete/abstract distinction with the future/not-future distinction. That would be entirely question begging. We are looking for a way of breaking the symmetry between past and future praises. Loke thinks that “the distinction between abstract and concrete infinities is helpful”, but if saying that future events are abstract just means that future events are future, then we have not been given anything at all. Mere tautologies cannot break the symmetry, surely.

More charitably, he may just mean to the distinction between the actual and non-actual. Future events are not actual (yet). They are mere potentials. As such they occupy a lower rung on the ontological ladder than present or past praises. It is hard for me to understand how this succeeds, although it is probably the most promising option. The future praises are ones which will happen. It is possible that our angel will curse his fate rather than praising God. It is possible for him to do so, even though he won’t do it. This is Craig’s view, as a Molinist. He thinks there is a course of contingent future events that God divinely foreknows will actually happen. It doesn’t matter whether God’s knowledge is propositional or not. What matters is that if there is an actual future, how can it be maintained that the future is merely potentiality? To hold that view, you should endorse open theism, according to which there are no truths about the future. One way or another, the set up of the thought experiment is that there is an angel who (actually) will say praises forever. Given that, the future is not mere potentiality, but is quite determinate (at least in this respect). Thus, this escape route doesn’t seem open. Or, at any rate, if we are arguing about whether there can be any truths about the future at all, we would be having a completely different conversation altogether. It has nothing to do with the abstract / concrete distinction any longer.

The Negative Way seems also no help to Loke. If abstract objects have no temporal location, then they cannot be future. Yet, future praises are future.

The Way of Abstraction also seems to be no help. Future praises are not the result of “somehow subtracting specificity” (Lewis, p. 84) from present or past praises. This Way treats the abstract / concrete relationship to be analogous to that between the map and the terrain. One is a less detailed version of the other. But in the set up, each praise is exactly the same.

4. Conclusion

At the end of all this, I cannot see what Loke is getting at. He wants future events to be non-concrete entities. But they don’t seem to be any good reasons for considering them to be abstract objects. They are a very bad match if we are considering Moreland’s view, which is the only thing Loke references, and he gives no explanation of what he means independently of that.