Craig vs. Morriston on the Number of Future Events

Hey Andrew,

That seems like an odd objection to run as the objection appears to assume that you could construct a collection with an infinite number of members by successive addition. That seems like the kind of thing you'd have to argue for and would have a tough time doing that.

I'd be interested to know why Craig said that there must have been only a finite number of moments prior to this one. Was it Hilbert's Hotel again? I recall seeing a debate where he opened with something like this, "Philosophically, the idea of an infinite past seems absurd. Just think about it. If the universe never had a beginning then the series of past events in the universe is infinite. But, mathematicians recognize that the existence of an actually existing number of things leads to self-contradictions. For example, "What is infinity minus infinity?" Well, mathematically you get self-contradictory answers. This shows that infinity is just an idea in your mind, not something that exists in reality." I like Hilbert's Hotel better.

Andrew,

I missed the link to the discussion of Hilbert's Hotel. WLC writes:

But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds!

Maybe this is too quick, but isn't an answer to Craig's question that the mathematicians and the cleaning lady are both right in their own way? If the guest in room #1 is, say, Wes and he checks out at 9:00 a.m., then when the cleaning lady says at 10:00 a.m. "There is one less person in the hotel now than there was earlier this morning, his name was Wes" I think she speaks truthfully. No mathematician would deny this. If the mathematician says that "There are not fewer members in the set of guests in HH at 8:59 a.m. than there are in the set of guests at 10:00 a.m." I think the mathematician speaks truthfully. Here the mathematician is working from the idea that one set has fewer members than another only if you can't put the members of the first set into 1-1 correspondence with the members of the second. No cleaning lady would deny this.

We can do the same thing with numbers, right. If you have the set of primes and then kick out the smallest prime, there's a sense in which there is one less member in the first set than the second (i.e., 2 is contained in the first set only). There is a sense in which there is not one less member in the first set than the second because if you wanted to have a dance you could put the members of the sets into 1-1 correspondence.

I see no more absurdity in the first case than the second, but Craig doesn't deny (so far as I can tell) anything I've said about the second case. Indeed, I think he thinks everything I think about the second case is true. Strange, but true. So, help.

The Hotel was full before Wes checks in (i.e. all rooms are occupied by one and only one member). Then Wes checks in and enters a room. No members have left. It's still full (i.e. all rooms are occupied by one and only one member). That seems to me strange, and aren't we working with the same sense of "full" before and after Wes checks in?

Anyhow, not sure at all how this analogy helps when dealing with time, but I find a hotel with an infinite number of rooms pretty incredible.

I have always wanted to ask Craig if God knows an actual infinite number of things. It seems to me that if a theist is going to say that God is omniscient, then, they must also say that God occurrently knows every number. But, of course, there is an actual infinite number of numbers and, thus, God must occurrently know an actual infinite number of things, namely, numbers. But if it is as Craig claims, then, it is impossible that God occurrently know every number. However, it does not seem impossible that God occurrently know every number. Thus, it is not as Craig claims. That is, it appears that there can be an actual infinite number of things.

Here's the reductio:

(1) Suppose it is impossible that there be an actual infinite number of things.
(2) If (1) then God cannot occurrently know every number.
(3) God can occurrently know every number.
(4) Therefore, it is not impossible that there be an actual infinite number of things.

I suppose Craig could deny (3), but this seems a heavy price to pay just to sustain the Kalam argument.

In his 2003 article in Faith and Philosophy, Morriston argued (with regard to future events) that (1) if God has perfect foreknowledge; and (2) if hallelujahs will be sung ad infinitum; then the set of all the hallelujahs God knows will be sung is an actually infinite set. Because of God's foreknowledge, however, there is no fundamental asymmetry between future and past here; you couldn't, for instance, argue that this set doesn't count because future events do not yet exist, because one could turn around and say, likewise, that past events no longer exist.

Incidentally, I think Craig's rejoinder plays on a common ambiguity in the phrase "potential infinite"; one could mean an indefinitely growing finite, which suggests that it is growing toward the future. But in fact it can also mean that for any n additions or divisions there is an n+1 addition or division; in which case you could use the potential infinite to defend the possibility of a potentially infinite past just as easily as you could use it to defend a potentially infinite future. (Thomas Aquinas did this, in fact, in ST 1.46.2 ad6.)

The Hotel was full before Wes checks in (i.e. all rooms are occupied by one and only one member). Then Wes checks in and enters a room. No members have left. It's still full (i.e. all rooms are occupied by one and only one member). That seems to me strange, and aren't we working with the same sense of "full" before and after Wes checks in?

I think that's strange, but doesn't "full" typically mean that there is no room for an additional member? If that's what it means, then I can see why the idea is absurd! Perhaps you are working with some other conception of "full". Perhaps something along these lines. To imagine a "full" hotel of the right sort you should (i) imagine a hotel where the rooms can be put into 1-1 correspondence with the even numbers and (ii) imagine that for each of these rooms there is a guest. Now, imagine Wes shows up weary from his travels. Can we fit him in? Perhaps the idea is that it's absurd to say that we can and that if we concede that we cannot then we have to concede that there cannot be actual infinities?

So, let me ask: why is it absurd to say we can't find room for him? Suppose we have two sets, the set of evens and the set of odds. Call the members of the odds 'rooms' and call the members of the evens 'guests'. There's no absurdity in the suggestion that for every room there's a guest and for every guest there's a room. Now, suppose we construct a new set. It's the evens + 3. Call the members of this set 'new guests'. There's no absurdity in the suggestion that for every room there's a guest and for every guest there's a room. There's no absurdity in the suggestion that every room there's a new guest and for every new guest a room! Strange, but that's how it works.

Why does it change if by 'guest' we mean guest rather than even, by 'room' we mean room rather than odd, and by 'new guest' we mean new guest?

It seems like we have two senses of "full". One that corresponds to "no more room" and one that corresponds to "we can pair them off". I don't see much problem left over once we disambiguate.

Hi Andrew,

Do you recall if Morriston raised any of the criticisms he's made in print against Craig's a priori arguments against an actually infinite past? If so, I'd be delighted to hear about those arguments, and Craig's replies, since some of Morriston's criticisms on this head have been in print for a decade, and so far Craig has pretty much ignored them (note: I'm not talking about Morriston's criticisms of Craig's causal premise, or of his criticisms of Craig's big bang arguments -- just the a priori arguments about the possibility and traversability of actual infinites).

'Exapologist' has offered a number of links to various well-argued papers written by Morriston that extensively critique the Kalam argument.

cf.:

http://exapologist.blogspot.com/2009/03/william-lane-craig-and-wes-morriston.html

Andrew-

Morriston's point is that there are an actually infinite number of possibilities (given the temporal infinity of the future) NOW known to God. If God, as Craig admits, NOW knows the details of all that shall occur in the future, then God presently contemplates an actual infinite, viz., all the succeeding detail of events from here to eternity, otherwise God would not know "the future" exhaustively and definitely in the sense Craig believes. So if "actual infinites" are impossible in the sense Craig argues, wouldn't God's present knowledge of the infinite future constitute an actual infinite?

Tom

Even if the number of concrete future events is only potentially infinite, isn't it the case that in God's mind, in His knowledge, there are an actual infinite number of items? E.g., an item of knowledge concerning one future event, another concerning another future event, etc. And would not it be the case that as the present moves "forward," items of knowledge about the future are "checking out" of the "hotel" so to speak? That is, God ceases to believe (and hence know) that "event E *will* occur at time t" once t becomes present. So I guess this raises two concerns for Craig's position: (1) supposing that time, and events in it, will never stop happening, is not he committed to the existence of an actual infinite and not just a potential infinite; and (2) doesn't he have to accept that the Hilbert-hotel scenario characterizes reality?

If the guest in room #1 is, say, Wes and he checks out at 9:00 a.m., then when the cleaning lady says at 10:00 a.m. "There is one less person in the hotel now than there was earlier this morning, his name was Wes" I think she speaks truthfully. No mathematician would deny this.

I'm not sure she does. Let the rooms be numbered, so every room has some finite natural number, and let them all be occupied. Wes leaves room #1. Let all of the occupants thereafter shift down one room. The occupant of room #2 moves to #1, #3 moves to #2 and so on. She speaks truthfully when she says that 'Wes left'. She does not speak truthfully when she says 'there is one fewer person than there was'. There's the same number of persons as there was. After Wes leaves, all of the rooms are still occupied.

I've talked about Kalam with Wes (in fact, just before this debate). What is hard to see is why God could not himself produce an actual infinite. Here's a supertask. Take any countable set of instants between 0 (the beginning of the universe) and 10 seconds later. Let God count each of those instants of time. So for each instant there is some natural. You thereby have an actual infinite series of God's utterances.

But even that seems more elaborate than necessary. After all, an infinite number of temporal instants elapse in the first 10 seconds. Suppose, for instance, that no instants of time elapse between 0 and 1. No instant elapses between 1 and 2 and so on. The conclusion is that we can move from 0 to 10 without a single instant of time elapsing. That's false. But if a single instant of time elapses between 0 and 10, then infinitely many do. So again you have an actual infinite series.

It seems to me that the problem is between Craig's view of infinites and his Molinist view of divine knowledge. If God has comprehensive knowledge of every future state of affairs, then it doesn't seem to make sense that the future is only "potentially" infinite. For it's not as though God learns what is going to happen at some future time t+n, but he already knows what will happen at every future time.

I'm sure Craig has some response to this in the literature, but I'm not sure what it would be. It just seems odd to me to say that God has knowledge of an actually infinite future, when in reality it is always only potentially infinite.

[i]"She does not speak truthfully when she says 'there is one fewer person than there was'."[i/]

She does speak truthfully if she understands that in transfinite math "one less/fewer than before" means "one different than before." The minus ("-") symbol in transfinite math means something different than it does in finite math. Thus, the sentence "4-2" means "2 less than four"; while "infinity - 2" means "2 different than the first set of infinity."

Mike,

I don't have a view about the possibility of supertasks so while I wouldn't myself assert their possibility to challenge Craig's contention I wouldn't want to deny their possibility either in response to Wes' challenge. (Maybe angels rush in where fools fear to tread.)

I think we both agree that the philosophically important and interesting issue here has nothing to do with supertasks and the finitude of the past but with the semantics of "less" and "fewer". My original claim was that the cleaning lady speaks truthfully if she says:
(1) There is one less person in the hotel now than there was earlier this morning, his name was Wes.

I've not yet had my morning coffee, but I agree that this doesn't sound so good:
(2) There is one fewer person in the hotel than there was this morning.

Maybe "less" is easier to hear as having to do with something akin to diversity of the elements of the set whereas (although, it's not as if there are fewer differences or "diversities" before and after Wes leaves, there's just that we can construct one set that contains Wes as a member and contains all the other members of the Wes-less set) "fewer" is easier to hear as having to do with the comparable size. So, maybe the cleaning lady can say both (1) and (~2)? After all, this doesn't sound that bad to me:

(~2+) Although Wes did check out and no one else has checked in since, there isn't one fewer person in the hotel than there was this morning.

Nor does this (cleaning lady to management):
(1+~2) You'll be sad to hear that there is one less guest as that chap Wes you liked so much as decided to check out, but you'll be relieved that there isn't one fewer person in the hotel than there was this morning so there won't be empty rooms that aren't collecting rent.

On a somewhat unrelated matter, it seems that one of the reasons that Hilbert Hotel strikes some as absurd when presented by Craig (and maybe this is true in other presentations) is the way in which it suggests that the coherence of the story depends upon either the possibility of supertasks or the completion of an infinite number of tasks by means of something like successive addition. I don't have a problem with the idea that new guests can be "accommodated" by HH, but we're moving into murkier waters when we are asked to imagine that they are being "accommodated" but only after all the guests have been successfully moved into new rooms. (This is from Craig's presentation of it, "Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full!")

Clayton,

Suppose Wes leaves and his room is unoccupied. Forget about there being one less person; concentrate on their being one less room to clean. In a way, this brings out Craig's worries (incidentally, Craig's worries arise only for those who insist on representing actual infinities in the standard Cantorian way. If you move to non-standard representations of infinity (they're hardly non-standard anymore, but nevermind...) you get none of these problems). Here's the problem.

1. There's one less room to clean, viz. rm #1.

(1) is true in the sense that you have subtracted one room from the rooms that must be cleaned. But (2) is also true, since subtracting one room from the rooms that must be cleaned does not lessen the number of rooms that must be cleaned.

2. There is the same number of rooms to clean.

If there were a hotel like that, it would be pretty strange. I think that's his claim: viz. that there are no collections of actual objects that violate basic arithmetical laws.

One other obvious response to Craig is that numbers are actual objects, and they violate basic arithmetical laws. So what he says has to be restricted to concrete objects (and that looks ad hoc) or he has to deny that numbers are objects.

Actually, when Hilbert's Hotel is full, you can have infinitely many world each with infinitely many planets, each with infinitely many hotels chains each having infinitely many hotels, each having infinitely many guests, each taking infinitely many travel busses to take their baggage which can all arrive at Hilbert's Hotel at the same time and the hotel clerk can allow all of them to check in AND having infinitely many rooms left after they all checked in.

Bijections with Aleph Null are funny.

What I would like to know is:
What did they say about the personal cause of the beginning of metaphysical time?
Can time begin to exist or is it timeless?

Clayton,

Do you really think full means "we can pair them off"? "Admitting of no more"--that seems to me about right to me.

Anyhow, I'm not sure that I can go with Craig down the Hilbert Hotel path in the first place. I'm not sure I can imagine or conceive of a hotel with infinite rooms to get the party started. It's akin to an infinitely large island, although I think an infinitely large island is more obviously inconcievable since an island is bounded on all 4 sides by water (or some non-islandy stuff). The hotel has a leg up, in that (the one I'm picturing) has a front and a back, a roof, and a floor. But it has no sides and requires an infinitely large planet to rest on, which also has no sides. I suppose would could try to conceive one with a floor, four sides (or one cylindrical side, etc.), but with no roof. That's a little better, since it's darn near 6 sided. But then what's to keep out the rain?

Are we really conceiving of hotels and planets at all here? I doubt it. But I'd be interested to hear what others think about the "getting the parted started" business.

Arelius & Andrew,

I can't remember if this is Craig or this is me when I teach it, but I think the idea is that there cannot be an actual infinite number of concrete x's. So God could have knowledge of an infinite number of numbers, facts, sets, and so forth, but there couldn't be an actual infinite number of events which are typically thought to be concrete and not abstract.

Tully:

What justifies this distinction: There can be an actual infinite number of abstract entities, but there cannot be an actual infinite number of concrete entities? This seems to me to viciously ad hoc.

For what it's worth, I am entirely untroubled by Hilbert's Hotel. I think the source of the puzzlement there is that we have two competing intuitions about the "size" of sets:
1. If set A is a proper subset of set B, then A has fewer members than B.
2. Sets A and B have the same number of members iff there is a one-to-one correspondence between their members. A has fewer members than B iff A has the same number of members as some subset of B but A does not have the same number of members as B itself.

It seems to me that what we need to do is to recognize that we are dealing with two distinct notions of the size of sets, and then disambiguate our intuitions. Sure, there are fewer filled rooms in HH once the gifts shift by one. Fewer in sense (1)--the "containment" or "domination" sense. But there are the same number of filled rooms. The same number in sense (2)--the "counting" sense.

While I am untroubled by Hilbert's Hotel, or the existence of an infinite number of objects, or supertasks, or an infinite future, or even an infinite past (as long as it is not causally interrelated), I am troubled by the idea of backwards-infinite causal sequence. My worries have been awakened by three related things: (1) the grim reaper paradox; (2) Rob Koons' insistence on the well-foundedness of causation; (3) questions about whether God could create a world containing a backwards infinite sequence of causally interrelated libertarian-free creaturely actions.

Aurelius,

I don't see the nature of the ad hocness (and just to be clear, I'm not persuaded by Craig's examples either). I don't see what is ad hoc about trying to show that there cannot be any actually infinite concrete x's. What is ad hoc about that?
Hilberts Hotel, the infinite books in the library example, and so forth, are supposed to motivate the view that there are no actual infinites with respect to concrete things (or so I say). This seems perfectly consistent with allowing that there are actual infinites when dealing w/abstracta. In fact, I think it's pretty obvious (to me!) that numbers are infinite. But surely we can allow that there are an infinite quantity of numbers while questioning whether there could be an actual infinite number of concrete x's.

I don't see what is ad hoc about trying to show that there cannot be any actually infinite concrete x's. What is ad hoc about that?

The initial complaint was with their existing an actual infinite set of objects. It is ad hoc to allow an actual infinite set of abstract objects but no actual infinite set of concrete objects. Why? Because the very same problems that Craig fusses about occur as clearly for abstract objects as for concrete objects. The only reason not to complain about infinite sets of abstract objects is to avoid the embarrassment of having to deny that numbers are objects. Or worse, having to claim that there are only finitely many numbers!

Maybe Craig thinks that abstract objects exist in some second-class way?

Bill Craig denies abstract objects and is a mathematical conceptualist See below:

http://www.reasonablefaith.org/site/News2?page=NewsArticle&id=5985

http://www.reasonablefaith.org/site/News2?page=NewsArticle&id=6879

Bill Craig denies abstract objects and is a mathematical conceptualist. . .

If he were a conceptualist, then numbers would be concrete objects--thoughts in the mind of God. This won't help him avoid the problem of actual infinites, since it amounts to conceding that there is an actual infinite, the infinite set of numbers in God's mind.

In any case it looks more like Craig is not especially wedded to conceptualism. He writes,

My first inclination was to adopt some sort of Conceptualism which construes abstract objects as ideas in God’s mind.. . [B]ut the more I’ve studied the problem the more attracted I’ve become to various Nominalistic or anti-realist views of abstract objects which flatly deny their existence rather than re-interpret their existence in terms of conceptual realities.

But as I noted above, it's a pretty extreme (though not of course unprecedented) metaphysical move to deny that there are numbers.

I don't know that divine conceptualism requires an ontological plurality of objects in God's mind. Aquinas thought God had ideas, but also thought this was compatible with divine simplicity, so he at least didn't think it implies an ontological plurality.

I am not about to defend this view--I am not sufficiently confident of how the defense would go--but I will drop this suggestion: The individuation of even human thoughts is a tricky thing; if I believe that (p and q), then is that one thought, or two?

I don't know that divine conceptualism requires an ontological plurality of objects in God's mind. Aquinas thought God had ideas, but also thought this was compatible with divine simplicity, so he at least didn't think it implies an ontological plurality.

This is tricky. Here, consider Plantinga on possible worlds. For Plantinga, worlds are a species of states of affairs, and states of affairs are abstract simples. So naturally, worlds are abstract simples. They are not composed of simplier states of affairs. But, of course, what worlds represent is extremely complex. Similar considerations might make divine simplicity coherent. The divine mind is simple, but surely what it represents is not. Each of the ideas in the divine mind might well be concrete (how could they not be, since they are certainly not inefficacious). But these are no more than what is represented in God's mind. It does not follow that God is not simple.

Sorry, I still don't see what is ad hoc. If Craig is thinking in this way (and I don't see why he's not) this isn't ad hoc:

"There are infinite sets and numbers. God has an infinite number of beliefs. That seems reasonable. OK, well, can there be actual infinite sets of ANY OLD THINGS? How about events? Well, let's consider other concrete things, rooms in hotels, books in libraries and so forth. It looks like in these cases of concrete objects (or so Craig thinks), actual infinites are bizarre. Matching up sets of numbers one-to-one & adding the number 1 to a infinitely large set of numbers is one thing; adding one person to an infinitely large hotel is another...."

Perhaps the same sorts of problems arise with hotels and sets of numbers, but that doesn't mean that Craig's procedure is ad hoc.

Perhaps the same sorts of problems arise with hotels and sets of numbers, but that doesn't mean that Craig's procedure is ad hoc.

I'm afraid it does. That is, unless he can give a principled reason why (i) all the problems P that arise in the case of infinite concrete objects are serious but (ii) the very same set of problems P that arise in the case of infinite abstract objects are not. Either these are problems or they aren't. If they are problems relative to concrete objects only, he owes a principled explanation why. In the absence of one, restricting the problem to concrete objects is ad hoc: it is just to avoid claiming that there aren't numbers.

But if you'll look a few posts up, you'll see that Craig does not have a principled reason. He can't make the distinction in any general way. What does he do? He decides (incredibly, it seems to me) to just deny that there are numbers at all. It avoids the charge of being ad hoc. But it's an enormous cost to preserve the Kalam. Out of the frying pan, into the fire.

Mike,

In the links above, Craig does want to affirm a nominalist view with respect to numbers. That's surprising. I stand corrected.

Still, I wonder about the ad hoc claim nonetheless (at least applied to my hypothetical Craig). Suppose I can't give a principled reason (the one I owe to someone) why the hotel situation is different than the numbers without the hotel. The hotel situation still seems whacky--much more so than the numbers. Incredible even. I'm much less likely to believe it's possible than I am that there is an actual set of even numbers. I'm not even sure I can conceive of an infinitely large hotel. The same thing goes with an infinte number of books in an infinitely large library. Bizarre.
Suppose we think of these examples as intuition pumps, lopping off all arguments altogether. Get yourself a whole bunch of examples of actually infinite concrete F's. Note how bizarre the situations seems even if you can't figure out why. From this exercise, it doesn't seem ad hoc to claim that there are no actually infinite concrete x's while still holding there is are infinite numbers.

What if the hypothetical Craig who is a realist about numbers could perhaps said this: "I don't see how Hilbert's Hotel comes up in the purely abstract realm. What is odd about HH is that the hotel is full, but there is room for more--if people shift around internally. But this shifting is a temporal process, and you can't do that with abstracta, if you agree that abstracta are unchanging. That there is a hotel that is full but you can fit another person in--that is paradoxical. That there is a bijection between the naturals and the primes is not paradoxical in the same way, though it is initially a bit strange. Likewise, it is strange that there is a bijection between all the roooms in the hotel and the prime numbered ones, but it's not paradoxical. What is paradoxical is the consequence, in the case of the hotel, that one could, without kicking anybody out of the hotel and without doubling any guests up in a room, vacate all the non-prime-numbered rooms. This is paradoxical because it violates analytic consequences of our concept of fullness, an ordinary non-stipulative concept we have. The general concept of a bijection, however, is not such an ordinary concept--it is an entirely stipulative concept."

From this exercise, it doesn't seem ad hoc to claim that there are no actually infinite concrete x's while still holding there is are infinite numbers.

Craig tells us that there is something so wrong about "actual infinites" that there cannot be any. We agree about that I take it. If that's true, then it does not matter whether we are talking about abstract objects or concrete objects--if there is an infinite series of either, then there is an "actual infinite". Now you might say, there cannot be an infinite series of concrete objects but there might be an infinite series of abstract objects. That's fine, but then (contrary to Craig's initial arguments) the problem is not with there being actual infinites. Rather, the problem is with there being certain sorts of actual infinites--i.e., the concrete sorts. But suppose you can explain in some principled way why some actual infinites are ok and some are not. Great, problem solved. Or suppose you deny the existence abstact objects. Great again. The only thing you can't do, it seems to me, is offer no explanation for what looks like an invidious distinction. Failing an explanation we have a distinction that serves the purpose of not having to deny that there are numbers. That's the ad hoc aspect that I was referring to.

Aurelius & Daniel

You both discuss God’s knowledge as a counter-example to the claim that an actually infinite number of things cannot exist (and Mike uses a similar issue with respect to actions, "supertasks"). But I think people are missing something important about Craig’s argumentative approach. Craig uses two philosophical arguments to show the impossibility of an actually infinite number of past events. In the first, which is the only one discussed so far, Craig argues that the an actually infinite number of things cannot exist in the [physical] world. The limitation to “physical,” is important though I’m not sure if Craig uses that term. But he clearly intends such a limitation, as is evident by the use of only physical entities hotel rooms (as in Hilbert’s Hotel) and library books (which is a variation Craig has developed).

But most of the discussion about this first philosophical argument has centered around (i) abstract objects or (ii) God’s knowledge or acts. But the first philosophical argument isn’t designed to counter objections about (i) or (ii) The first philosophical argument (it seems to me) is designed to show that actually infinite number of physical events is impossible.

The discussions (and proposed counter-examples) of (i) abstract objects (like numbers) or (ii) God’s omniscience or supertasks are dealt with by the [i]second[/i] philosophical argument. There, Craig argues (along with Aquinas) that if an actually infinite number of events is possible (as Aquinas thinks it is), it cannot occur by successive addition. That is, the actually infinite number of successive events would have to be [i]created[/i] by God instantly.

Thus, the second philosophical argument answers the issues of God’s unlimited knowledge (which he wouldn’t acquire successively, but instantly, if we can even speak of God’s acquiring knowledge at all). It also deals with the supertasks issue (and, consequently, Alex’s infinite causal regress concern too) because it indicates that if some things can be actually infinite they can only be actually infinite due to God’s creative activity (supertasks) or nature (knowledge, and abstract objects on a conceptualist view).

Finally, this also answers the ad hoc issue that’s been floating around. It probably would be ad hoc to rely on the first philosophical argument (alone) when we’re dealing with abstract objects (where abstract means all non-physical objects). But that’s were the second philosophical argument comes into play. In short, if we’re talking about physical objects, we need to look to the first philo. arg. (against the existence of an actual infinite). If we’re talking about abstract objects, we need to look to the second philo. arg. (against the existence of an actual infinite formed by successive addition).

But most of the discussion about this first philosophical argument has centered around (i) abstract objects or (ii) God’s knowledge or acts. But the first philosophical argument isn’t designed to counter objections about (i) or (ii) The first philosophical argument (it seems to me) is designed to show that actually infinite number of physical events is impossible.

I'm not sure how this can be right. Craig expressly states that the problem is for actual infinites. The set of actual infinites is not a priori the set of concrete infinties. He recognises this. He knows he cannot allow actual infinites among abstract objects, either. Indeed, under the pressure of his commitments against actual infinites, he finally denies that there exist any numbers.

. . .if some things can be actually infinite they can only be actually infinite due to God’s creative activity (supertasks) or nature (knowledge, and abstract objects on a conceptualist view).

This claim I know is mistaken--if it is supposed to represent Craig's view--since he has denied to me directly that even God can perform a supertask. Wes can confirm this.

Mike

I’m not as interested in fleshing out Craig’s exact personal beliefs as I am in determining the truth of the two philosophical arguments. It seems to me, for the reasons I stated, that the two philosophical arguments (when taken together) adequately respond to the proposed counter-examples. Do you think so?

Alex: I don't see how your suggestion really helps Craig. For the Kalam argument to work, he needs an argument for the conclusion that "actual infinites are impossible,", not for "it's impossible to do X with an actual infinite amount of Y's."

While I'm not convinced that concrete actual infinites are impossible, J.P. Moreland has tried to give an account of why they're supposed to be impossible. I haven't read the article in a while, but Here's the link to the article.

Well, on the question of the soundness of arguments, nowhere in the first kinds of arguments do we get anything resembling a contradiction from the supposition of an actual infinite. We do get contradictions when we throw in commonplaces like:
1. If every room is full, then it would be impossible to fit in another person without some room having more than one person.
2. If no room in a hotel has more than two people, and a room is vacant, there will be no way of rearranging the guests to fill every room.

But these are commonplaces that the typical infinitarian will immediately say are true only when restricted to finite hotels. And that's no surprise because they are commonplaces we arrive at by thinking about finite hotels.

Alex,

When I asked Mike whether he thought the two philosophical arguments responded to the stated objections, I didn’t have in mind the point you raised about whether the actual infinite leads to contradictions at all. I think the point you raise must be handled separately from the issues of actually infinite abstract objects.

You’re much more math wiz than me, but it seems to me that the issue isn’t simply about “finite hotels.” Hilbert’s Hotel is an illustration of the absurdities that result in the physical world by performing basic *mathematical operations* with actual infinites.

That’s the locus of Craig’s discussion about the contradictions that result from performing identical operations with identical numbers and obtaining different results. In case anyone isn’t familiar with this, he’s the two situations Craig compares. In the first instance, all the rooms are full. At one moment, all the guests in rooms 4, 5, 6… check out. In an instant, the hotel is left with only three guests. In the second instance, all the rooms are full too. But now, all the guests in the odd numbered rooms simultaneously check out. That leaves all the guests in the even numbered rooms—still an infinite number of guests. When we compare these results, we see the contradictions that result from an actually infinite number of things existing in the world. In both instances we began with identical number of people (an actual infinite), subtracted identical quantities (an actual infinite), and reached different results (3 in the first case, and an infinite number in the second case). So it seems the issue is tacking a mathematical concept (the actual infinite) and placing it into the world in any way (hotels, books, whatever). The contradictions don’t result from being tied to finite concepts; they result from basic mathematical operations with an actual infinite.

Ryan:

This is the best I can do to formalize the argument.

If X and Y are collections (sets?) of objects, let X\Y be the subcollection of all the members of X that are not members of Y.

1. If A is a collection of concrete objects, and B and C are subcollections of concrete objects, and B and C have the same number of members, then A\B and A\C have the same number of members.
2. But if A is infinite, then (1) is false.
3. Therefore an infinite collection of concrete objects is impossible.

But surely the thing to do is just to deny (1).

Now, one might say that (1) is undeniable. After all, equals can be substituted for equals, so if you subtract equal numbers, you must get equal results by Leibniz's law. But that is mistaken. In the concrete case, we're not subtracting numbers. We're removing guests. And we're removing different collections of guests (A is not the same as B, even though they have the same number of guests).

It is a substantive claim that the number of members of A\B depends only on the number of members of A and the number of members of B. And why should we believe this substantive claim in the case of infinite sets?

A different way to formulate the argument would be this:

4. If B is a subcollection of A, then the number of members of A\B is equal to the number of members of A minus the number of members of B.
5. But if A is infinite, then there can be subcollections B and C that have the same number of members, but such that A\B has a different number of members from A\C.
6. Therefore, there cannot be an infinite collection.

But now (4) should be rejected. Subtraction is not well defined when both operands are infinite and equal, and the number of members of A\B depends not just on the number of members of A and the number of members of B, but on which of the members of A are in B.

In any case, I'm unable to get a valid argument to a contradiction from the assumption of an infinite collection, without assuming some disputable principle like (1) or (4). Granted, (1) and (4) hold in the finite case. But Craig of all people surely agrees that there are important differences between finite and infinite cases.

I wrote "A is not the same as B, even though they have the same number of guests". I should have written "B is not the case as C, even though they have the same number of guests".

It seems to me, for the reasons I stated, that the two philosophical arguments (when taken together) adequately respond to the proposed counter-examples. Do you think so?

For the reasons I've stated, no. Further, I was simply pointing out that Craig, who likely knows the arguments better than you or I, makes just the concessions one would make if it were actual infinites and not merely actual concrete infinites that were the problem.

Mike:

You wrote: "I was simply pointing out that Craig, who likely knows the arguments better than you or I, makes just the concessions one would make if it were actual infinites and not merely actual concrete infinites that were the problem."

I probably was a little hasty in making the strong split between the two philosophical arguments. I think the split might help things, but Craig clearly thinks that the argument against the existence of an actual infinite applies to abstract objects too. I've found a very interesting article on his website that specifically addresses the arguments presented here. Here's the link:

http://www.reasonablefaith.org/site/News2?page=NewsArticle&id=5163

Essentially, he argues that the only way the existence of numbers is a problem for the first philosophical argument is if Platonism is the *only* viable option: "As I explained in The Kalam Cosmological Argument, only if one is a Platonist is the admission of mathematical actual infinities incompatible with the claim that an actual infinite cannot exist.11 So long as Formalism, Conceptualism, or Nominalism remains a viable option, the kalam proponent need not deny the legitimacy of the mathematical actual infinite."

He also deals with the issue of God's knowledge as a proposed counter-example:

Does anyone know if there is an audio or video recording of the debate online -- or at least a transcript? If so, I'd be glad to get word on the link to it.

About my "debate" with Bill Craig... It was actually billed as a "dialogue," it was a friendly event, and there were no big surprises. Needless to say, we didn't have a meeting of the minds.

In view of the interest expressed by some people in this thread, and in view of the fact that one or two who heard the "dialogue" didn't understand what I was trying to say about an endless series of future praises, I have posted my response to Craig's opening statement here:

http://spot.colorado.edu/~morristo/DialogueWithBillCraig.pdf

There is also a series of powerpoint slides that go with my response. I tried to cover all of Craig's arguments in twenty minutes. It wasn't easy! I have included a link to the powerpoint as well.

In addition to the links to the opening statement and the powerpoint, I have included a few comments on what I take to be the inadequacy of Craig's claim that an endless series of future praises would be a merely "potential infinite."

Whether it's an actual or a potential infinite depends on how those terms are defined. All that matters, I claim, is whether the case as I described it is possible. I think it is. If I'm right about that, then all the paradoxical features of the actual infinite can be reproduced for an endless series of distinct, determinate, successive events.

I further argue that "presentism" (which Craig also appealed to) doesn't help his case.

Hi Wes,

Thanks very much for posting your opening remarks and your powerpoint slides. They're very helpful.

Here's a link to a more user friendly html file containing the same stuff mentioned above.

http://spot.colorado.edu/%7Emorristo/Morriston-Craig-kalam-dialogue-debate.html

Regarding Ryan's comment of March 19: if you click on the URL he offers, you get taken to the article only if you're registered on Reasonable Faith.

If you want to see the article Ryan is talking about, here's a link to it that you don't have to register for:

http://www.leaderu.com/offices/billcraig/docs/kalam_davis.html