Thursday, September 8, 2016

Yeah, but is it actually actually infinite?


In response to my recent post about William Lane Craig’s kalām cosmological argument, several readers noted that Craig has replied to an objection like the one I raised, in several places, such as a response to a reader’s question at his Reasonable Faith website, and in his article (co-written with James Sinclair) on the kalām argument in Craig and Moreland’s Blackwell Companion to Natural Theology.  Let’s take a look at what he has to say.

First recall that examples like Craig’s famous Hilbert’s hotel thought experiment involve infinitely large collections of things (e.g. hotel rooms and guests) all of which exist at the same time.  Craig argues that various paradoxical consequences follow from these examples, which shows that the idea of an actually infinite collection of things is metaphysically suspect.   But a universe without beginning would entail an actually infinitely large collection – of days, years, or whatever other unit of time you care to take.  Hence (the argument concludes) the notion of a universe without a beginning is also metaphysically suspect.

My objection noted that Craig is (as I am) an adherent of presentism, i.e. the view that only present things and events exist, and thus that past and future things and events do not exist.  Given presentism, I argued, it is incorrect to say that a universe without beginning entails an actually infinite collection in the relevant sense.  For past things and events (including past days, years, etc.) do not exist.  Hence they do not form an infinite collection of things that exist all at once, as the hotel rooms and guests in the Hilbert’s hotel example do.  The only things and events that exist are present things and events, and they are not infinite in number.  The only day that exists is the current day, and a single day is obviously not an infinite number of days.  Something similar could obviously be said of weeks, months, years, etc.  Hence there is in the case of units of time nothing to parallel the rooms and guests of the Hilbert’s hotel example, so that the parallel fails even if we concede that the example succeeds in showing that an actually infinite collection is impossible.

As far as I can tell, Craig makes three points that might be thought to be relevant to my objection.  First, in both the Reasonable Faith Q & A and in the Blackwell Companion piece, he notes that the fact that past things, events, years, etc. do not exist does not prevent us from being able to count them.  We can correctly say, for example, that it has been fifteen years since 9/11, even though (given presentism) none of the years before the present one exist any longer.  And since we can count them (Craig seems to be saying) there must be a sense in which the years since 9/11 constitute a collection actually having fifteen members in it.  By the same token (so the argument seems to continue, if I understand it correctly) if the universe had no beginning, that would entail that there is an actually infinite collection of years.

Now, I’m not clear how this argument is supposed to constitute a reply to the objection.  No one denies that we can count past years even though they don’t exist anymore.  After all, we can count all sorts of things that don’t exist.  For example, Snow White knew seven dwarfs, and we can go through them, by name, and count them.  (First, Grumpy; second, Sleepy; etc.) But that we can count these dwarfs doesn’t entail that there is an actual collection with seven members in it, because the seven dwarfs, being fictional characters, are themselves not actual.  Similarly, that we can count past years – whether fifteen of them or an infinite number of them – doesn’t entail that they constitute an actual collection, since the past years themselves are also not actual in the relevant sense.

It seems to me that Craig’s argument here might be trading on an ambiguity between two claims:

1. The number of moments that have actually existed is infinite.

2. The number of moments that actually exist is infinite.

A beginningless universe would entail 1, but it would not entail 2, certainly not if presentism is true.  Yet 2, it seems to me, is what Craig needs for a beginningless universe to be relevantly like the Hilbert’s hotel example.  In the Hilbert’s hotel example, it is because we have infinite collections of things all of which exist at once that we get weird results, like wave after wave of infinitely large groups of guests arriving at the hotel and being able to check in even though the hotel is already full.  Precisely because all the past years, days, etc. do not still exist, it is hard to see how they constitute an actual infinite in the same sense of “actual.”  The collection of guests in the Hilbert’s hotel example is “actual” in the sense that the members all do exist; the collection of years in the beginningless universe scenario is “actual” only in the different sense that the members all did exist. 

Here’s another way to look at the problem.  Craig would not deny that it is legitimate for mathematicians to talk about infinite series in various ways, e.g. the infinite series of natural numbers.  The reason this is okay is that numbers (unlike hotel guests, hotel rooms, etc.) are not concrete objects.  Hence when talking about numbers we don’t get the bizarre results we get when considering scenarios in which an infinite collection of concrete objects exist.  But is a collection of units of time (minutes, days, years, etc.) more like a collection of concrete objects like guests and rooms, or is it more like a collection of abstract objects like numbers?  For Craig’s argument to work, it seems that we’d have to say that it is more like the former.  But in fact, this seems false.  It seems instead to be more like the latter.

This is especially plausible if, like Aristotle and Aquinas, we deny that time exists apart from change and the concrete objects that undergo change.  To speak of time apart from change is a bit like speaking of a universal like redness apart from actual red things—it is to engage in abstraction from the concrete conditions under which the thing in question (redness, or time) can actually exist.  Craig may be more inclined to think of units of time as relevantly analogous to concrete objects like hotel guests, etc. because he sympathizes instead with the Newtonian view that time can exist apart from concrete changing objects.

This brings me to a second remark Craig makes that might be thought relevant to my objection.  In the Blackwell Companion article, after developing the point about counting things that no longer exist, he asserts (contrary to what I just claimed) that “all the absurdities attending the existence of an actual infinite” do apply to a beginningless universe, despite past things and events no longer existing (p. 116).  For example, if the number of past events is infinite, he says, then the number of odd-numbered events is no smaller than the number of total events, since the series of odd numbers is of course infinite.  (He also rehearses other points along these lines.)

But the problem with this should be obvious from what has already been said.  Again, Craig does not have a problem with mathematicians talking about infinite series of natural numbers, despite the fact that the series of odd numbers is no smaller than the series of all natural numbers.  The reason is that numbers are abstract rather than concrete objects, whereas examples like Hilbert’s hotel are problematic because hotel rooms and guests are concrete rather than abstract.  A beginningless series of events will be problematic, then, only if it is more like the collection of rooms in Hilbert’s hotel than it is like the series of natural numbers.

But as I have said, the trouble with Craig’s position is that past things (whether events, years, or whatever) do not exist, at least not given presentism.  So they are not relevantly like the concrete objects which all exist together in the Hilbert’s hotel example.  Craig’s reply here thus seems to me to ignore the objection from presentism rather than answering it. 

A third remark from Craig which might be thought relevant to the objection I raised is also to be found on the same page of the Blackwell Companion article, where he cites Aquinas’s example of a blacksmith who has been working for an infinite amount of time and using one hammer after another.  The collection of hammers would constitute an actual infinite.  Now, Craig says that it would constitute an actual infinite even if the hammers did not still exist, and he cites the example only to illustrate his claim that past things need not continue to exist in order to constitute an actual infinite.  But it seems to me that what he should say, in order to try to make of this example a response to the objection from presentism that I have put forward, is this: Suppose such a blacksmith has been working for an infinite number of years, has used a new hammer each year, and has preserved each of these hammers.  Then we would have a collection that is actually infinite even by the presentist’s own lights.  And it would thus be relevantly analogous to the Hilbert’s hotel example.  So wouldn’t this show that the idea of a beginningless universe is paradoxical and metaphysically suspect in just the way Hilbert’s hotel is?

Perhaps this could be developed into a promising reply, but as it stands it too seems to me to fail.  For the most it would show is that there couldn’t be an actually infinite collection of hammers, for the same reason there couldn’t be (if the Hilbert’s hotel argument works) an infinite collection of rooms or guests.  But it doesn’t follow that there couldn’t be an infinitely old universe, precisely because days, years, etc., unlike hammers, don't stick around and thus don't lead to the existence of an “actual” infinite in the relevant sense.  The most the hammer example would show is that even in an infinitely old universe, you couldn't amass an infinite collection of things.  Why not?  Maybe because, though time itself needn't have had a beginning, types of material objects like hammers must have had one.  Nor could Craig easily dismiss this separation of what's true of time from what's true of material things, because, again, like Newton (and unlike Aristotle and Aquinas) he thinks that time can exist apart from the material things that change in time.

So, though I hate to disagree with Craig – I have nothing but respect for him, and have profited much from his work over the years -- it doesn’t seem to me that he has successfully rebutted the objection from presentism.  But maybe there’s another way to do it.  And as I’ve said, I’m not convinced that an infinitely old universe really is possible in principle, and thus I’m agnostic about the kalām argument.

135 comments:

Legion of Logic said...

Let's say we could time travel. If the universe was infinitely old, we could go back one year. We could go back 100 years. We could go back ten trillion years. We could keep going...and going...and never run out of years to hit.

Now, at the ten trillion year point, if we wanted to reach our starting point (present) at the natural time rate, it would take ten trillion years. That's because we can't simply ignore the past - the present is shaped by the past.

In the same sense that it would take ten trillion years to reach now from ten trillion years ago...how does one reach this place from...infinity ago? In which ten trillion years is as insignificant as one second?

hektikpecs said...

I think the infinitely old universe fails on less controversial grounds than this.

If time is in any way objectively real, then for a past event to be past, it must have once been present.

If any event is once present, there must have been a time when it had an age of zero. If we accept that an infinite set is analytically non-constructible, it's impossible for any past event to be infinitely old.

Philip Alawonde said...

Legion of Logic,

Why do you suppose the infinite case and ten trillion case are analogous: What do you mean by reaching now from infinity ago -- now pay attention -- as if infinity were some finite number?

See the problem? Such thinking already assumes what it wants to prove.

But let 's leave all that and attempt to transcend this question altogether thusly: What would it mean for the universe to exist an infinite time ago? I don't think this makes sense, neither is it the opposite of having a finite past. Also, what does it mean to say the universe had no beginning? Again I think this question is silly and betrays muddleheadedness, neither does it's denial entail holding that the universe is infinite.

In short, the point is just that -- contrary to what you and many others think -- denying that the universe couldn't have begun to exist doesn't necessarily imply that it is 'infinitely old', whatever that may mean; all it means is all it says, and it's pretty well compatible with saying...

Brandon said...

LoL & hektikpecs,

In an infinite past, there is no infinity ago, nor any event that is infinitely old. Every day is a finite distance from the present. There are infinite integers, but every single integer, without exception, is only a finite distance from zero.

Philip Alawonde said...

...that the universe has been in existence for a finite duration; in just the way that the segment (0,1] has no beginning (for any purported first number you give me, say f, I can give you a smaller, say f/2), yet it clearly has a finite length.

Thus, once again, the claim that the universe couldn't have begun to be doesn't entail the vague claim that it's infinitely old.

Philip Alawonde said...

*I should have mentioned parenthetically, if it wasn't already obvious to you, that that segment was one unit long.

Anonymous said...

LoL,

What does your time travelling device spend in order to travel in time? It will go as far as it can given what it has to spend. But, if your advice has infinite energy and thus infinite energy source, then such a device is absolutely imaginary. Which means, you're just asking a question about numbers crudely pinned on to some imaginary objects.

In short, your question leads one more to wonder about what a 'time travelling' device could possibly be more than it leads one to wonder about the physical universe. At which point we begin to think concretely about some kind of machine or we venture off into a Platonic realm of some sort. If we begin to think about some machine, then we begin to think about man's capacity for art; if we think about this, then we begin to think about finite man's finite relations to finite natural resources within a finite space or environment. At this point we are not thinking any more about infinite or the physical universe, but a small set of environmental factors converging on man's consciousness as aimed at a task of artificial construction and informed by whatever concepts are relevant for making something. The big vision becomes increasingly cramped.

Allen Hazen said...

From the second paragraph:
"First recall that examples like Craig’s famous Hilbert’s hotel thought experiment involve infinitely large collections of things (e.g. hotel rooms and guests) all of which exist at the same time. Craig argues that various paradoxical consequences follow from these examples, which shows that the idea of an actually infinite collection of things is metaphysically suspect. "
Respodeo: "Paradoxical" is said in many ways, as Professor Quine has argued in his "Ways of Paradox." It can mean antinomic, logically incoherent, or it can mean merely surprising, hard to credit until it is explained. And surely the experience of mathematicians since Cantor shows as much as any experience could, that the actual existence of an infinite collection -- a collection of coexistent spatial regions, say, if one is a presentist and doesn't like to think about collections of times -- is paradoxical only in this second, weaker, sense. But Craig needs to argue that it is paradoxical in the stronger sense.
(B.t.w.: Graham Oppy's book "Philosophical Perspectives on Infinity" ives excellent coverage of a wide variety of puzzles and arguments: recommended.)

Daniel Vecchio said...

Legion of Logic,

I don't think time travel to the past is possible on Presentism. There would be nowhen to go.


TheOFloinn said...

As Chad and Jeremy argued: "but that was yesterday, and yesterday's gone."
http://www.lyricsmode.com/lyrics/c/chad_jeremy/yesterdays_gone.html

Andrew2 said...

Perhaps Craig could give an argument like this:

1) If the universe is infinitely old, it is possible that an actually infinite collection of concrete objects exists

2) It is impossible for an actually infinite collection of concrete objects to exist

3) Therefore the universe cannot be infinitely old.

2) would obviously be supported by the Hilbert's hotel argument (which incidentally I find quite weak, but that's beside the point here). 1) seems pretty plausible. Suppose we have our blacksmith, and from eternity past, he's been making a new hammer each year, and storing them. But since there was no beginning to his making of hammers, at any given moment, there are an actually infinite number of hammers. So 1) seems quite plausible.

The conclusion should follow.

Anonymous said...

I am just going to leave this here.

http://alexanderpruss.blogspot.co.uk/2016/09/presentists-cant-reduce-time.html

Anonymous said...

Can a hierarchically ordered series of causes have an infinite number of members, even if there must be a "first" or "foundational" member, or, must all such series, unlike a temporally ordered series, have a finite number? (I agree the Kalam is not persuasive; a temporally infinite universe is conceptually possible IMHO). Further, if there can be an infinite number of members in a hierarchically ordered causal series, why "must" there be this "first member" to act a terminus of explanation...why couldn't there be an infinitely explainable universe? I know what I would say to answer that, but what say you all?

Anonymous said...

The most the hammer example would show is that even in an infinitely old universe, you couldn't amass an infinite collection of things. Why not? Maybe because, though time itself needn't have had a beginning, types of material objects like hammers must have had one.

Unless I am missing something, it seems to me that this worry can be easily dismissed. I mean, couldn't we modify the story just a bit more in saying that individual hammers are produced shortly before they're used? In that case, an actually infinite collection of used hammers will accrue over the course of an infinite past and it will also be true of each hammer that it had a beginning.

Brandon said...

Andrew2,

Since material objects need material from which to be made, (1) would already seem to imply that there is an actual infinite collection of material objects, since you would need suppose that there's also infinite materials -- if we are actually talking about things like hammers. (This doesn't eliminate all arguments along these lines, though. Scholastic objectors giving arguments similar to this tended to use souls as their examples, and I suspect that this is one reason why.)

Jonathan said...

@ Andrew2

"1) If the universe is infinitely old, it is possible that an actually infinite collection of concrete objects exists"
This may well be correct, but it does not support the conclusion. For argument's sake, let us grant the correctness of the second premise "2) It is impossible for an actually infinite collection of concrete objects to exist".
However the conclusion "3) Therefore the universe cannot be infinitely old" does not follow from these two premises.

In order for it to do so, the first premise would have to read:
"If the universe is infinitely old, it is necessary that an actually infinite collection of concrete objects exists"

The truth of this would require to be shown by a separate argument that has not been supplied.

Mikhail said...

Speaking of presentism, Pruss recently raised a strong objection to Aristotle's presentism:

"...to reduce time to something else requires giving a nontemporal account of the "wholly earlier" relation between events. But if presentism is true, there never exist two events one of which is wholly earlier than the other. For if one of them is present, the other is not... this means that Aristotle, who attempted to reduce time to change, cannot be consistent if he is a presentist."


from: http://alexanderpruss.blogspot.com/2016/09/presentists-cant-reduce-time.html

Anonymous said...

Brandon,

If the worry now is "material objects need material from which to be made." You could also say that God creates new hammers ex nihilo at discrete times over the course of an infinite past.

Jonathan,

I believe you are mistaken. Rewording Andrew2's syllogism for clarity:

(1*) If the past is infinite, then the existence of an actually infinite collection of concrete objects is metaphysically possible.

(2*) The existence of an actually infinite collection of concrete objects is not metaphysically possible.

(3*) Therefore, the past cannot be infinite.

The above syllogism is obviously sound since (2*) is just the negation of (1*)'s consequent. Hence, (3*) follows by contraposition. The consequent of (1*) does not need to say that the existence of an actually infinite collection of concrete objects is metaphysically necessary.

Anonymous said...

Pruss's reasoning in that blog post seems very suspect. We can take it as obvious that there has been series of prior states of affairs whose occurrence can be ordered by the "earlier than" relation. Not everything has to be argued for and/or developed from more basic concepts. But then the idea that the proponent of a reductionist theory of time who also thinks that it is only the present state of affairs that actually exists at any given moment doesn't have the metaphysical resources at her disposal to coherently employ the notion of an "earlier than" relation seems mistaken.

BB said...

One other problem with discussing infinite past time is that we need to specify what units we measure time in. There is nothing special about the second as a means to measure time. We are permitted to perform generalised coordinate transformations. Suppose that we had some time variable, t, which extends from - infinity to plus infinity, and we can also choose a standard unit of time arbitrarily, say the time that it took me to type this sentence, t_0. In that case, I can define a new measurement of time, t', such that t' = t_0 arctan(t/t_0), so t' has the range -pi t_0/2< t' < pi t_0/2. In this case, even though t extends infinitely into the past, we can construct a new unit which extends finitely in the past. The process is reversible, so that if the universe is finitely old, we can construct a different way of measuring time in which it is infinitely old. Given that the geometry of space time is Riemann rather than Euclidean or Minkowski, each of these different ways of measuring time is as valid as any other (although obviously if we change the coordinate system we change both sides of Einstein's field equation, and thus might introduce some additional pseudo-gravitational forces in the new coordinate system).

A discussion of this sort concerning time must include reference to general relativity (which requires the assumption that physical space/time can be represented by an Reimann geometrical space with a Mincowskian metric). It doesn't make sense to talk about a finite or infinite period of past time without first introducing some way of measuring past time, i.e. some coordinate system, and as soon as you have done that you need to think about how to convert from one measurement system into another, and that's the basis behind GR. Then you have to justify why God would treat your coordinate system as preferred over one where the past was either finite or infinite. Only after doing this are you in the position to ask the question behind this and the previous blog post. Until then, I don't think the question `Can the universe be infinitely old' can be answered, because the question is insufficiently defined.

Anonymous said...

BB,

Our concept of a finite duration of time is sufficient for the purposes of this discussion. With that concept in mind, it makes sense to ask whether the past history of the (temporal) universe is finite or infinite. Now, if we want to setup an experiment for the purposes of uncovering the laws of nature that arise from the final causality of physical bodies over time, then we need to specify some unit of time in which the necessary measurements will be made.

Brandon said...

You could also say that God creates new hammers ex nihilo at discrete times over the course of an infinite past.

This already assumes that there is no limit, in the nature of materiality, to how many material objects can be in a world, because if it's intrinsically impossible for a material universe to have an actual infinite collection of material objects, even God can't do this, regardless of whether the past is infinite or not. So for instance, if one thought (to give just a made-up example to see how the point works) that you can only have X number of material hammers in a universe before it necessarily collapses on itself, destroying everything in it, by the very nature of how material objects work, then when God creates X+1, the universe would necessarily collapse on itself, destroying everything in it. So on such a supposition even God can't preserve the universe and have more than X hammers in it.

It's related to a point I raised in the previous thread -- when creating a reductio, one needs to establish either that the supposition is a possibility to which one's opponents are (at least very probably) committed, or that the absurdity arises from the opponent's own principles rather than the supposition itself (i.e., that the supposition only makes more visible an absurdity that is already there without it). I don't think it's easy to show either with these hammer arguments. The reason the old scholastic soul arguments were more promising is that pure forms at least lack a lot of the restrictions material objects necessarily have by being material (they wouldn't take up infinite space, they wouldn't require infinite elemental parts, etc.).

In addition, bringing in creation changes the argument in two important ways. (a) It is harder to prove that something is impossible to divine omnipotence working miracles than it is to prove that something is impossible to something by nature; it's fairly easy to argue that no natural process can make infinite hammers, but it takes a clear and rigorous demonstration to establish that even God Almighty can't do it. (b) If the future is infinite, God can still make infinite hammers; so it turns the argument from an infinite collection argument to a traversal of infinite argument, since that's the only difference between the past and the future cases.

Philip Alawonde said...

Andrew2,

Valid argument, but it still assumes it makes any sense to talk about an infinitely old universe. So I ask, what's an infinitely old universe?

A. R. Diaz said...

“But it seems to me that what he should say, in order to try to make of this example a response to the objection from presentism that I have put forward, is this: Suppose such a blacksmith has been working for an infinite number of years, has used a new hammer each year, and has preserved each of these hammers.  Then we would have a collection that is actually infinite even by the presentist’s own lights.  And it would thus be relevantly analogous to the Hilbert’s hotel example.  So wouldn’t this show that the idea of a beginningless universe is paradoxical and metaphysically suspect in just the way Hilbert’s hotel is?

For the most it would show is that there couldn’t be an actually infinite collection of hammers, for the same reason there couldn’t be (if the Hilbert’s hotel argument works) an infinite collection of rooms or guests.  But it doesn’t follow that there couldn’t be an infinitely old universe, precisely because days, years, etc., unlike hammers, don't stick around and thus don't lead to the existence of an “actual” infinite in the relevant sense.  The most the hammer example would show is that even in an infinitely old universe, you couldn't amass an infinite collection of things.  Why not?  Maybe because, though time itself needn't have had a beginning, types of material objects like hammers must have had one.” (Prof. Feser)

Wait. What about a modal version of the response Craig should give? Craig could argue that an infinite, beginningless universe, makes it in principle possible (though not necessary) to amass an actual infinity of items (as the hammer example shows). But then, given Hilbert’s hotel, we know that to amass an actual infinity of items is in principle impossible (or conceptually incoherent or metaphysically suspect or what have you). Hence, an infinite, beginningless, universe would make possible in principle what is in principle impossible: an actual infinity of items. Hence, it makes no sense or its metaphysically impossible or what have you for there to be an infinite, beginningless, universe. It is sort of an indirect way of arguing against the possibility of an infinite, beginningless, universe, by showing that such a thing makes possible something that cannot be.

Anonymous said...

No Brandon, all that needs to be accepted is that it's always metaphysically possible to produce a hammer (or some such) in a period of time spanning a given finite duration. If that is accepted, then it follows that an infinite collection of hammers can be produced over the course of an infinite past -- just say that hammer N is produced at N units of time in the past, where the duration of a given unit is fixed. Hence, if someone also accepts an infinite past, then they will be forced to admit the metaphysical possibility of an actually infinite collection of hammers even on presentist assumptions. If this latter consequence is troubling, then they should abandon their commitment to an infinite past.

Anonymous said...

Philip Alawonde,

If it makes sense to talk about a finite duration of time, then it also makes sense to talk about an infinitely old universe for obvious reasons.

D.Baum said...

I tend to agree with this critique, but it's unfortunate from the perspective of an apologist whose audience is the general public and students. I can see why Craig puts so much effort into it. The Kalam is 1) simple to state, 2) does not require a page worth of background to understand it 3) has the rhetorical appeal of having "scientific support".

While the scientific support for that second premise is in depth, most modern audiences will be more conditioned to receive all that stuff about Bourd-Gouth-Valenkin theorems and General Relativity, than say... half a chapter on Parmenides and Heraclitus.

It seems that the Kalam has an easy learning curve, and more rhetorical force, but it lacks the rigor and power of a First Way or a Necessity/Contingency argument.

David M said...

"No one denies that we can count past years even though they don’t exist anymore." - Sure, but lots will deny that we can count past years that *never* existed, just like we can't count the number of possible but non-existent fat or bald men in the doorway.

"After all, we can count all sorts of things that don’t exist." - *Actually* count them?

"For example, Snow White knew seven dwarfs, and we can go through them, by name, and count them. (First, Grumpy; second, Sleepy; etc.) But that we can count these dwarfs doesn’t entail that there is an actual collection with seven members in it, because the seven dwarfs, being fictional characters, are themselves not actual." -
But they pretty clearly are actual - actual abstract artifacts (per Amie Lynn Thomasson). If they weren't, you couldn't actually count them (although you could of course actualize them yourself, qua abstract artifacts, ad libitum).

"Similarly, that we can count past years – whether fifteen of them or an infinite number of them – doesn’t entail that they constitute an actual collection, since the past years themselves are also not actual in the relevant sense." - That's the question though: what is the relevant sense (or which are the relevant senses)? In point of fact, Hilbert's hotel is just as much an abstract artifact as the seven dwarfs. It just involves more puzzling mathematical features. The point is its intentional bearing, that is, on concrete reality. So this claim seems wrong:

"The collection of guests in the Hilbert’s hotel example is “actual” in the sense that the members all do exist; the collection of years in the beginningless universe scenario is “actual” only in the different sense that the members all did exist."

The guests do not actually exist at all in the usual sense (the *actually exist* only as abstract artifacts) and the relevant contrast is not between "all do exist" and "all did exist" but between "all exist at the same time" and "all exist, but successively" where 'exist' is used in a way that is tense-neutral. So we get to the real question:

"But is a collection of units of time (minutes, days, years, etc.) more like a collection of concrete objects like guests and rooms, or is it more like a collection of abstract objects like numbers? For Craig’s argument to work, it seems that we’d have to say that it is more like the former. But in fact, this seems false. It seems instead to be more like the latter."

If we grant that Craig's use of Hilbert's hotel (an abstract artifact) appeals to (hypothetically) concrete objects, surely we should say the same about (hypothetically concrete) collections of units of (past) time. Certainly the past 15 years (for example) have a concrete reality (clearly born out in their perfectly real essential/causal connection to the actually existing present - unless we're Humeans) which is not at all analogous to the thoroughly abstract reality of numbers. So, for Craig’s argument to work, it seems that we’d have to say that past time is more like the hotel than like abstract numbers. And in fact, this seems true.

D.Baum said...

For all this talk about the possibility of amassing infinite hammers... aren't we forgetting that there are other necessary conditions for amassing infinite concrete objects?

Like, you would need an infinite amount of material, for example. Or you would need infinitely divisible material, or infinite energy to create infinite material... Infinite time is a necessary condition for making infinite (still existing) hammers, but it's not sufficient.

David M said...

(that is, the intentional object in each case is a concrete one)

Anonymous said...

...aren't we forgetting that there are other necessary conditions for amassing infinite concrete objects? Like, you would need an infinite amount of material, for example

As I said in an earlier comment, all that needs to be accepted is that it's always metaphysically possible to produce a hammer (or some such) in a period of time spanning a given finite duration. If you still can't shake the physicalist need for specifying some collection of preexisting material out of which a blacksmith might forge hammers, then suppose that God creates the hammers ex nihilo at discrete times over the course of an infinite past in topologically disconnected regions of space.

Brandon said...

No Brandon, all that needs to be accepted is that it's always metaphysically possible to produce a hammer (or some such) in a period of time spanning a given finite duration.

Positing an infinite past does not in any way commit one to such an possibility. Since the existence of hammers requires the conditions for the existence of hammers, and these are not purely temporal conditions, merely positing an infinite past does not establish that other conditions required for hammers allow unlimited hammers, as D. Baum and myself have both, entirely correctly, pointed out. You can't assume that the person you are arguing against is committed to the 'always'.

If you still can't shake the physicalist need for specifying some collection of preexisting material out of which a blacksmith might forge hammers...

This is an ignoratio elenchi. If an argument purports to establish that an infinite past is impossible it must *by the very meaning of that purport* establish that it is impossible under any consistent suppositions, not arbitrary ones that are simply made up to yield the impossibility. You aren't arguing that an infinite past is impossible for a world in which God creates things ex nihilo in topologically disconnected regions of space; you are supposed to be arguing that an infinite past is impossible.

D.Baum said...

" then suppose that God creates the hammers ex nihilo at discrete times over the course of an infinite past in topologically disconnected regions of space."

Oh, I see how you're using this now. As a modal objection... I suppose a response would be: all this needs to prove is that God could not do such a thing since it would lead to a logical impossibility, and omnipotence does not extend to the logically impossible.


In other news... I wonder if Dr. Feser has read Brentano's version?

Anonymous said...

Brandon and D. Baum,

The way I see it, we can always (if asked) build into our scenario the necessary material and spatial requirements needed for a blacksmith to produce hammer N at N units of time in the past -- or we can say that God provides these things ex nihilo as needed. In any case, nothing of metaphysical consequence turns on these materialistic sorts of considerations. And that's why philosophers who wrestled with these sorts of questions from John Philoponus on generally recognized that an infinite past would entail the metaphysical possibility of an infinite collection of objects in the present. So Feser's attempt to critique Craig's argument on presentist grounds is not very convincing. The real question that remains controversial is whether the sorts of consequences that Craig derives from his Hilbert's Hotel example rise to level of genuine metaphysical absurdity as opposed to being merely bizarre.

Philip Alawonde said...

Anonymous @ 9:46 AM,

Obviously your statement is not true -- my point is that obviousness comes in various subjective degrees.

Yes, so you want to demonstrate how talk of infinite time has to make sense if that of finite time does -- as it does. What's your 'obvious' minor premise?

PS. You may also want to define, inter alia, what you mean by infinite time!

Jason Wojtyla said...

We can show that it is impossible for the number of past events to be infinite without using collections of concrete particulars that simultaneously exist.

Consider a doorway. On one side of the doorway a particle is created and passes through the doorway. On reaching the other side of the doorway it is annihilated. Call this a Particle Event. Now if it were possible for there to be a past eternal universe it would be possible for there to be an infinite number of events in the past. If it is possible for there to be an infinite number of past events it would be possible for there to be an infinite number of Particle Events in the past. Call the most recent Particle Event PE1, the one before that PE2 and so on. If the number of past Particle Events could possibly be infinite then the number of Particle Events prior to number 3 is the same as the number of Particle Events prior to number 56. But this is wrong. Furthermore If the number of past Particle Events could possibly be infinite then the total number of Particle Events would be the same as all the even numbered particle events. Since it is impossible for the number of past Particle Events to be infinite the universe cannot be past eternal.

Note that the contradictions arise not in virtue of there being an infinite collection of concrete particulars existing simultaneously(since the particles are annihilated on reaching the other side) but rather in virtue of there being an infinite number of past events.

Anonymous said...

Philip Alawonde,

If we have a concept of duration as a measure of time, then it makes sense to ask whether the duration of the past history of the universe is finite or not. If it is not, then we say that the past history of the universe is infinite. Easy peasy lemon squeezy.

Anonymous said...

@Dr Feser
Robert Koons developed an argument for the finitude of the past that seems to circumvent your objection.

The paper is called "A New Kalam Argument: Revenge of the Grim Reaper" published in 2014 in Nous

To summarise briefly. You are alive at midnight. At 1Am grim reaper#1 materialises and, if you are alive, kills you. At 12:30 reaper#2 materialises and, if you are alive, kills you. At 12:15 reaper #3 materialises and, if you are alive, kills you. And so on out to infinity. Now it is obviously impossible for you to survive past 1AM. However it is also impossible for any grim reaper to ever kill you since for any grim reaper there will already have been an infinite number of grim reapers that materialise and killed you!

This would suggest that it is impossible for the to be an infinite number of past events. Robert Koons goes on in greater detail to show this

Anonymous said...

Call the most recent Particle Event PE1, the one before that PE2 and so on. If the number of past Particle Events could possibly be infinite then the number of Particle Events prior to number 3 is the same as the number of Particle Events prior to number 56. But this is wrong.

Why? If by "number" you are referring to the modern mathematical notion of a cardinal number then there is no problem.

Anonymous said...

Anonymous @ 11:05 AM,

FYI, Koons's argument presupposes the density of space, so those who favor an infinite past might circumvent it by supposing that space is not, in fact, dense. I see no way of modifying Koons's argument to get around this problem. Also, Koons forgets to rule out the possibility that there have always been an infinite number of "Fred particles" (those who read the paper will know what I am referring to here). I believe this defect in the argument can be repaired but it complicates things a bit. For what it's worth, I think that the inverse Tristram Shandy arguments that have been proposed (see, for example, here) are more promising.

Brandon said...

we can always (if asked) build into our scenario the necessary material and spatial requirements needed for a blacksmith to produce hammer N at N units of time in the past -- or we can say that God provides these things ex nihilo as needed.

You can't build anything into a reductio that is not one of the assumptions in the position to be reduced or something to which the person accepting the position is accepted, unless one can also establish that the absurdity arises only from those things, and not from the introduction of a foreign supposition to which the person is not committed. If someone says A&B, and I say, well, suppose C, then with A&B you get absurdity, and C is not something to which the person is committed, no problem has been created. Since positing an infinite past involves no assumptions about what's possible in terms of material or spatial possibilities, helping yourself to them at will corrupts the argument into an ignoratio elenchi.

In any case, nothing of metaphysical consequence turns on these materialistic sorts of considerations.

Nonsense. The scenario explicitly requires, in your own statement, metaphysical assumptions about production of material objects like hammers, so the nature of material objects, and thus what is metaphysically possible given them, is a major matter of metaphysical consequence here.

Anon said...

About the infinite hammer argument: if the argument is "if there was infinite time you could get an infinite number of hammers, but it's impossible to have an infinite amount of hammers, so you can't have infinite time", then the first premise seems to require a process of making hammers that requires time but nothing else, since if it required anything else (such as pre-existing matter to make the hammers out of) then you could be prevented from making hammers by a lack of that. Creation ex nihilo avoids that problem because it doesn't require matter, but doesn't it create a different problem because it doesn't take any time? The arguments talk about God creating a hammer a day or a hammer a minute, and then say if He has an infinite number of hammers or minutes He'd be able to create an infinite number of hammers (and therefore He must not have an infinite number of minutes), but if God can create a hammer instantaneously, then He can create an infinite number of hammers in an instant. I don't see how limiting the time He has to make hammers makes any difference since He doesn't need a set period of time to make hammers.

Anon said...

"infinite number of hammers or minutes" should be "infinite number of days or minutes".

Anonymous said...

Creation ex nihilo avoids that problem because it doesn't require matter, but doesn't it create a different problem because it doesn't take any time? The arguments talk about God creating a hammer a day or a hammer a minute...but if God can create a hammer instantaneously, then He can create an infinite number of hammers in an instant.

Not necessarily. In particular, if we already accept the premise that the existence of an actually infinite collection of objects is metaphysically impossible, then it follows immediately that God cannot instantaneously create an infinite collection of objects like hammers. On the other hand, the metaphysical possibility of God instantaneously creating a finite collection of objects is much less objectionable for obvious reasons.

Anonymous said...

Brandon,

Do you think it's metaphysically possible for God to have created hammers ex nihilo at a rate of one hammer per day for the entire past history of the universe?

Brandon said...

Do you think it's metaphysically possible for God to have created hammers ex nihilo at a rate of one hammer per day for the entire past history of the universe?

(1) It doesn't matter whether I do. What needs to be established is either (a) that it certainly is metaphysically possible, so that one cannot consistently hold otherwise, or (b) that people who would posit an infinite past would be committed to it already.

(2) There are obvious issues with merely assuming it, anyway. I've already given an indication of some of the kind of thing they are -- for example, hammers require infinite space or material and one may have reason to doubt that it's metaphysically possible for any material world to have infinite space. Maybe you're confusing two different candidates for metaphysical possibility and what is really metaphysically possible is only for God to have created hammers ex nihilo at a rate of one hammer per day up to any given finite number of hammers. Have you established that this is not, in fact, the case? No: ad hoc assumption.

Philip Alawonde said...

Anonymous @ 11:03 AM,

LOL, of course nothing easier than begging a question; I did not ask you to reaffirm your assertion, but to prove it.

Maybe you don't know how, although I gave a hint earlieron. I'll show you: What exactly would you mean by a finite or infinite duration? After explaining that, apply it to the universe and explain how it makes sense to say the universe has existed for an infinite duration.

All the best. :)

Anonymous said...

Brandon,

For whatever reason, I think you're just being difficult.

Philip Alawonde,

LOL back at you bro. :)

Brandon said...

For whatever reason, I think you're just being difficult.

Then I will cut it up into small pieces for you. The question explicitly on the table is: whether the hammer argument actually shows that the position, P, that it is possible for the past to be infinite even though there can be no actual infinity, is inconsistent.

I have already pointed out that

(a) given the logical structure required for a reductio, it can only do so if it derives the absurdity only from those things to which someone holding P is committed, or at least very probably committed, by holding P;

however,

(b) the hammer argument actually requires metaphysical assumptions about the production and nature of material objects and the material world to which P does not seem to commit anyone;

and moreover that introducing creation ex nihilo does not address the issue because

(c) it still requires assumptions about the production and nature of material objects and the material world to which P does not seem to commit anyone,

and

(d) it also introduces assumptions about divine omnipotence without establishing that holding P commits anyone to these assumptions.

You, on the other hand, have done very little more than to state that you can help yourself to any assumptions you please about material objects and creation ex nihilo to generate the contradiction, and in response I have pointed out that

(e) this approach turns any reductio into an ignoratio elenchi, because it makes it impossible, given (a), adequately to address the question on the table.

to which you have given no relevant reply.

D.Baum said...

Yeah, I have nothing much to add to what Brandon has said, only to remind the guys crying, "Ex Nihilo! Ex Nihilo!" that the person who really has motivation for claiming an infinite past is possible, namely atheists, have every reason to simply claim that "God" and "ex nihilo" are the impossible things in your syllogism.

Anon said...

Anonymous @12:21: "Not necessarily. In particular, if we already accept the premise that the existence of an actually infinite collection of objects is metaphysically impossible, then it follows immediately that God cannot instantaneously create an infinite collection of objects like hammers. On the other hand, the metaphysical possibility of God instantaneously creating a finite collection of objects is much less objectionable for obvious reasons."

Let's try a different way of putting it: You say God cannot create an infinite amount of hammers, but can create a finite amount. Your argument against an eternal universe has God performing the act of creating a finite number of hammers at a regular time interval, so that if He had an infinite amount of time He would create an infinite number of hammers; since it is metaphysically impossible to have an infinite number of hammers, then there must not be infinite time. But since the act of creating a finite number of hammers doesn't take God any time, why can't he infinitely repeat it in an instant? If you're right that it's metaphysically impossible (and I don't dispute that), then He couldn't, but whatever reason there was why He couldn't would not have anything to do with time (and would presumably therefore also apply to Him creating an infinite number of hammers over time).

Anonymous said...

Dear Brandon and D. Baum,

My contention is that nothing of metaphysical consequence turns on the limitations of the human form and the fact that, practically speaking, there is only so much material out of which hammers can be made. Hence, there is no metaphysical problem in supposing that a blacksmith has been producing hammers at a consistent rate for the entire past history of the universe even though such a scenario is not physically possible. In an effort to get you both to appreciate this rather basic metaphysical point, I've asked you consider the case of God producing the hammers (or the raw material that a blacksmith might use to produce the hammers) ex nihilo since there is no question of God being able to do this sort of thing for the Thomists who comment here. But alas!

Anonymous said...

but whatever reason there was why He couldn't would not have anything to do with time

Exactly! This is where Craig's Hilbert's Hotel argument comes in. God cannot instantaneously create an infinite array of hammers because (according to Craig) various metaphysical absurdities would result. Now, if we also think that God could have created a hammer ex nihilo for any given past day, then if we suppose for reductio that the past is infinite it follows that the existence of an actually infinite collection of hammers is metaphysically possible, which contradicts the conclusion of Craig's Hilbert's Hotel argument.

D.Baum said...

Well, now we're just repeating ourselves. My response to what you've said here is simply to point back to the last comment Brandon said.

A:P entails Q
~Q
Therefore ~P

B: I deny that P entails Q. Therefore P and ~Q. Support your claim.

A: Look, P entails Q because if we add X to P we get Q.

B: But now all you have is:
(P * X) entails Q
~Q
Therefore ~(P * X).... That is not distributive. ~X would still give you ~(P*X), therefore P may still be true.

Anon said...

Anonymous @3:00:
God cannot instantaneously create an infinite array of hammers because (according to Craig) various metaphysical absurdities would result. Now, if we also think that God could have created a hammer ex nihilo for any given past day, then if we suppose for reductio that the past is infinite it follows that the existence of an actually infinite collection of hammers is metaphysically possible, which contradicts the conclusion of Craig's Hilbert's Hotel argument.

OK, but what I'm saying is if we stipulate "God cannot infinitely repeat the act of creating a hammer", then I don't see why it matters whether he is given an instant, a finite amount of time, or an infinite amount of time to do it in; since He can perform the action instantaneously, whatever limitations exist on what He can do in an instant would apply to what He could do over time. So why can't we say "Since God can't create an infinite number of hammers in an instant, He can't create an infinite number of hammers over time, even if He had infinite time", in which case infinite time wouldn't allow for infinite hammers and therefore would not be shown to be impossible?

Scott Church said...

Great posts Ed! There is one thing though that unless I missed it hasn't been addressed yet here or in the previous one. As you mentioned, most, if not all of these argument assume presentism, which seems to be the main focus of the ongoing discussions also. Without getting too deep into the pros and cons of it here (a topic for another day), from a physics standpoint presentism doesn't fit very well with one of the most robustly demonstrated principles of relativity--the Lorentz boost. The Lorentz boost, which is a consequence of the equivalence of the speed of light in all inertial reference frames, is why spacial distances and the passage of time differ for observers that are in motion with respect to each other, which leads to the so-called "twins paradox." One of its less well-known consequences is that for space-like separated events (i.e. - ones that cannot reach each other without travelling faster than the speed of light), not only is the passage of time relative, but past and future are as well. Given two space-like separated events A and B, in some reference frames A will occur in B's future, and in others B will occur in A's. The difference is only a matter of perspective.

Reconciling this with a "flowing" present for which the past and future don't exist is to say the least, problematic. It can be done... sort of. From a relativistic standpoint, space-like separated events can't communicate with each other, so real events that can causally influence each other all have time-like world lines for which there is an absolute past and future, even if its flow is experienced differently in different reference frames. This can be used to define a preferred reference frame of sorts that's associated with the flowing present (which Craig and a few others have argued for). But this whole approach is kludged at best and goes against the most basic principles that underlie special and general relativity. Bring quantum non-locality into the mix (thereby blurring the isolating impact of space-like separation between events) and we have a whole string of complications that render B-theory ("block") time a lot simpler and more to the point. There aren't many physicists today who don't consider presentism a prime target for Occam's razor.

Now to be fair, you've rightly pointed out on many occasions that the most robust cosmological arguments are based on metaphysical principles that do not dependent on the current state of science. But given that the kalam argument is a notable exception, it's worth pointing out that science doesn't advance by rewriting current knowledge--it does so by building on itself and taking established theory to deeper layers of explanation. Future advances may show relativity to be emergent from something more fundamental for instance, but you can bet is isn't going to erase it and make the issues raised by the Lorentz boost conveniently go away.

In light of all this, it seems to me that if presentism is discarded many of the arguments against actually existing infinities are back on the table, and "beginning to exist" vis 'a vis accidentally vs. essentially ordered causality will need to be viewed in a different light as well. I'd love to hear your thoughts on all this. Thanks!

Tap said...

Lorentz boost has in no way been proven. It's a crackpot equation attempting to solve the difficulty of the Michelson Morley experiment. Time dilation and dimension contraction of an object due to motion has never been demonstrated.

Scott Church said...

With all due respect Tap, nonsense. The Lorentz boost has been tested and confirmed time and time again, as has gravitational redshift and related effects. The research is far too voluminous to cite here, but for a small sample one might start with Hafele & Keating (1972), Alley (1983), Allan et al. (1985), Bailey et al. (1977), or Vessot & Levine (1979). For that matter, you might want to spend some time in any undergraduate or graduate physics textbook. Furthermore, if the Lorentz boost is a "crackpot equation" then so is special relativity, which virtually requires it. And among other things, a special relativistic treatment of bound electron states also underlies the Pauli Exclusion principle, which makes chemistry possible. So... if you want to argue that special relativity is a crackpot theory, and chemistry is a crackpot science... including the biochemistry which keeps you alive and able to post here... well, good luck with that. :-)


REFERENCES

Allan, D. W., Weiss, M. A., & Ashby, N. (1985). Around-the-world relativistic Sagnac experiment. Science, 228(4695), 69-70. Available online at http://science.sciencemag.org/content/228/4695/69.

Alley, C. O. (1983). Proper time experiments in gravitational fields with atomic clocks, aircraft, and laser light pulses. In Quantum Optics, Experimental Gravity, and Measurement Theory (pp. 363-427). Springer US. Available online at http://link.springer.com/chapter/10.1007/978-1-4613-3712-6_18.

Bailey, J., Borer, K., Combley, F., Drumm, H., Krienen, F., Lange, F., ... & Flegel, W. (1977). Measurements of relativistic time dilatation for positive and negative muons in a circular orbit. Nature, 268(5618), 301-305. Available online at http://ivanik3.narod.ru/TimeLifeMezon/301-305Nature.pdf.

Hafele, J. C., & Keating, R. E. (1972, July). Around the world atomic clocks: predicted relativistic time gains. American Association for the Advancement of Science. Abstract online at http://adsabs.harvard.edu/abs/2000PhyEs..13..616K.

Vessot, R. F. C., & Levine, M. W. (1979). A test of the equivalence principle using a space-borne clock. General relativity and gravitation, 10(3), 181-204. Available online at http://link.springer.com/article/10.1007/BF00759854.

Tony said...

The Professor says
o one denies that we can count past years even though they don’t exist anymore. After all, we can count all sorts of things that don’t exist. For example, Snow White knew seven dwarfs, and we can go through them, by name, and count them. (First, Grumpy; second, Sleepy; etc.) But that we can count these dwarfs doesn’t entail that there is an actual collection with seven members in it, because the seven dwarfs, being fictional characters, are themselves not actual. Similarly, that we can count past years – whether fifteen of them or an infinite number of them – doesn’t entail that they constitute an actual collection, since the past years themselves are also not actual in the relevant sense.

I suggest, Professor, that your argument can be tightened up some. For example, the "counting" or enumerating of the 7 dwarves, it is true, does not mean that the "collection of 7 dwarves" is a collection of actual things, but this is not merely because "we can count things that don't exist" simply speaking. We can't. Not things that don't exist in ANY sense, because no THIS thing that doesn't exist can be enumerated distinctly from THAT thing that doesn't exist - precisely because nothing does not differ from nothing in ANY detail, not even in number. For example, you cannot count the number of Eniglots - for there is no fictional, metaphorical, or partial Eniglot in ANY sense to be enumerated at all. What can be counted, however, are "things" that are only "in a sense", but are not actual - like fictional dwarves. Each of the dwarves, though fictional, is real enough to be numbered distinctly, and thus counted as one of 7. It has the sort of imperfect, partial, fictional reality of the mind crafting a story: of things imagined - which operation is similar to things sensed, enough so to number them.

One might further observe that "a real collection" is not the same thing as "a collection of actual things". What Craig needs is for the collection to be a collection of actuals, for it is the actuality of the members of the collection that drives the absurdity of "an actual infinite", not the "reality of the collection". The "collection" that is "collected" solely by the operation of mind, has the kind of mental reality that also applies to the "collection of odd counting numbers". THAT kind of infinitude does not generate the sort of absurdity that Craig needs. One could generate a mentally infinite collection by naming the "collection of points at which I might slice this line segment", i.e. at 1/2, or 1/3, or 2/87, etc. The fact that this collection is real in the mind, and is infinite, does Craig no good.

A slightly different point should be raised about past events. Even if presentism insists only the present moment is real and actual in the fullest sense, this does not prevent ascribing a kind of reality to past things. Past things have a determinacy to them that exceeds "not being", even though they are not "being" simply speaking. But this too doesn't help Craig at all, because it is FULLY actual beings, in the infinite, that generate absurdities, not the degenerate past-tense sort of reality to past beings (at least, so far as has been shown forth).

Geremia said...

Do any modern philosophers make the categorematical infinity vs. syncategorematical infinity distinction?

David M said...

Tony:
"it is FULLY actual beings, in the infinite, that generate absurdities, not the degenerate past-tense sort of reality to past beings (at least, so far as has been shown forth)."

But is that right? How has that been shown? Based on Hilbert's hotel, which is no more fully actual than the seven dwarves (and possibly less so)?

Tony said...

David M, I suppose that in one sense you are right that the Hotel is no more actual than the dwarves: such a Hotel has never existed (to our knowledge). Hilbert just imagined it.

But for these purposes, the proof of the sort "and this is absurd" rests on the hypothesis "suppose a Hilbert Hotel REALLY existed, as fully and completely actual". The absurdity that results shows that the hypothesis supposed, the real actual existence, is itself absurd. This form of reasoning is exactly like all other instances of reductio ad absurdum. It is unnecessary for the Hilbert Hotel to BE actually real in order for the absurdity to be established as proven, just like it is unnecessary for the conditions of any other absurd hypothesis to ACTUALLY obtain in order for the reductio to be valid in any other proof. (And, if it were necessary, would defeat the reductio as a valid form of proof, of course.)

You are correct that I have not proven that the partial, degenerate past-tense sort of reality of past things is insufficient to kick in the absurdity (of the sort admitted in Hilbert's Hotel). Fair enough, I only asserted it.

I think, though, that the burden of proof actually rests with Craig, whose proof for God rests on it, to establish that it IS sufficient. The situation we have is that it is generally admitted (for Craig's interlocutors) that an "actual infinite" is impossible as it leads to absurdity, and Craig claims "an infinite past" gets to the SAME absurdities. He has to show that it does, not just REST when he gets to the point of "and the infinite past has some kind of partial, degenerate, past-tense sort of actuality". He has to then go on and prove that THIS sort of actuality generates the same absurdity. But so far as I can tell, he does just rest, on the presumption that for this purpose "actual is actual" and there is no significant difference once you claim that the past collection is actual.

I suspect that there may be, in some fashion, a proof of the sort Craig would actually need - if you delve enough in metaphysics and philosophy of nature and parse out what a past infinite would really imply. But Craig is - in my opinion anyway - generally adverse to exactly this sort of investigation into ethereal discussions of metaphysics / philosophy of nature with sufficient rigor to get there. He WANTS the claim to be as accessible as the absurdity that comes from a FULLY actual infinite, because among his interlocutors THAT is a more or less accepted claim already. If he were just as willing to get into the heavy-duty metaphysics, he would just go with St. Thomas's proof and be done with it. (Which doesn't make Craig's thesis - that God's existence can be proven from the kalam cosmological approach - false, it merely leaves one hook of it still left to be nailed down.)

Wesley P. said...

But are you proposing that it's impossible for me to specify a number of years it woukd take for these hammers to degrade? A trillion? A trillion trillion? A trillion trillion trillion? A hammer cannot be preserved infinitely. It will degrade. Meaning that even if an infinte number of hammers have existed, only a finite number is ever actual. You admitted that God may not be able to create an infinite number of hammers at once and substituted God making a new hammer each day, but we run into the same problem on the other side. God would need to preserve an infinte collection, but that would (if HH paradox is a true contradiction) not be possible for God and, on this side of the argument, you can't propose that God just preserve a finite number of one a day. In short, an actual infinite could not exist even given the starting premise. Hammers would not remain indefinitely actual, leavinv only a finite number evr actual.

And that which I've taken so long to write was actually already stated by Feser in the original post.

Francis Beckwith said...

Ed:

Does your analysis apply to the traversing argument? That is, if we think of the present as a consequence of the past series of events, and if the past series of events is infinite, would it not follow that any thing, X, in the present would have received its existence as a consequence of an infinite series of instrumental causes? The fact that the past series does not presently exist does not seem relevant, since, in a sense, the past series does "exist" in the being of X, just as the creator of an artifact, in a sense, "exists" in the artifact. (Or, as Aquinas would put it, the artifact "resembles" its maker.).

Frank

Anonymous said...

"the view that only present things and events exist, and thus that past and future things and events do not exist"

Except Popper's 3 Worlds argument makes a cogent counter example that both past and future do exist, if only in world 3.

Hal said...

Very interesting topic and discussion.
In my limited understanding, the concept of infinity is quite useful in mathematics.
But I wonder how one could ever determine whether or not an infinite set of some particular item exists? Is the concept of infinity of any use (i.e., does it have any meaning) when it comes to describing the world or the universe?

Tony said...

Even if one accepted Popper's 3 Worlds, as far as I saw, nothing in it requires the result that the past and the future exist in the same way that the present exists. And nothing requires that we accept Popper's 3 Worlds.

SK said...

Off topic

Will you ever make those posts about your conversion to Christianity?

Anonymous said...

Can we say this is the difference between a Platonic worldview vs. an Aristotelian worldview?

It's seems a copy of St. Bonaventure's argument on the same thing.

"If the world had existed from eternity it would follow that it is possible to add to the infinite. For instance, there would have been already an infinite number of solar revolutions, yet every day another revolution added. But it is impossible to add to the infinite. Therefore the world cannot have always existed."

Copleston, Frederick. "A History of Philosophy, Vol. II" Pg.263.

Just using hotel rooms instead of solar revolutions as the example.

Bonaventure was an Augustinian, Augustine was a Platonist (neo-platonist?). Basically, we are still arguing about 'do the forms exist outside the material itself.' Unless you agree on this, you are likely not going to agree about the finer details? It would be nice if people had name badges, like "Hello my name is So and So, I am a Platonist." It would be easier to understand everyone.

-Germanicus Vivat

Edward Isaacs said...

- Infinite amount of time has passed.
- Time cannot exist without concrete, changing objects.
- So at any "time" in which there is time, there are objects.
- So you at any "time" in which there is time, you can collect objects.

I thought it was a good argument because it seems to show that in principle, if you allow the possibility of an infinite time, you also have to allow the possibility in principle (not necessarily the actuality) of an infinite number of objects. Yet it's the impossibility of an actual infinite which the arguments from Hilbert's hotel etc. seem to show.

To argue the other side, though, instead of appealing to the beginning of the concrete objects, I would say that there has to be a limit to the size of any material universe (assuming you have already said there can be no actual infinites). In other words the blacksmith would, given Craig's premises, at some point run out of iron in the universe to keep making hammers.

Matthew Rodriguez said...

Dr. Feser, what do you think of Alexander Pruss' argument for thinking that an infinite past is impossible? He explains it in this blogpost:

http://alexanderpruss.blogspot.com/2009/10/from-grim-reaper-paradox-to-kalaam.html

Dianelos Georgoudis said...

Feser writes: “No one denies that we can count past years even though they don’t exist anymore. After all, we can count all sorts of things that don’t exist. For example, Snow White knew seven dwarfs, and we can go through them, by name, and count them. (First, Grumpy; second, Sleepy; etc.)”

This strikes as confusing. There exists a fairy tale that mentions seven dwarfs. After having read the tale there exists in our minds a picture of the seven dwarfs. There exist movies based on the tale where the seven dwarfs appear. But the seven dwarfs themselves do not exist and so we cannot count them. The expression “seven dwarfs” does not really refer to any “seven dwarfs” but is shorthand for some of the existents I mention above.

The concept of “existence” is fundamental in metaphysics and if one is not clear what one is meant by it then all philosophical hell is apt to break lose. I was thinking that a good definition is this: “X exists if and only if there are true propositions about X”. Now this definition shifts the problem to the meaning of “truth”, but the concept of “truth” is basic and admits of no definition. (Theories of truth strike me as epistemology.) After all without an accepted definition of truth how can one decide whether a definition of truth is correct or not? Incidentally the fact that “truth” admits of no definition is not problematic on theism, because theism entails that God has created us \s rational beings and thus possessing a sense of truth.

Coming back to the issue of the existence of past years, according to the above definition past years exist since there are true propositions about some properties of them, e.g. “The French Revolution started in the year 1789.” Now of course the year 1789 does not exist “anymore”, but from this it does not follow that it does not exist at all. Of course it exists. It is the time-span in which the French Revolution started.

Now Feser may disagree with my definition of “existence” and thus deny its implications. But then he must give an alternative definition of “existence” which implies that the year 1789 does not exist.

Tom Larsen said...

Ed,

// The most the hammer example would show is that even in an infinitely old universe, you couldn't amass an infinite collection of things. Why not? Maybe because, though time itself needn't have had a beginning, types of material objects like hammers must have had one. Nor could Craig easily dismiss this separation of what's true of time from what's true of material things, because, again, like Newton (and unlike Aristotle and Aquinas) he thinks that time can exist apart from the material things that change in time. //

Unless I've misunderstood, I don't think this response works. Let's take, as a simplified example, a steady-state two-dimensional Conway's-Game-of-Life universe, capable of infinite spatial growth, and containing a glider gun, iterating once every second. Given an infinite number of iterations, the universe will contain an infinite number of glider guns. If someone insists that a universe could be infinitely old but no infinitely-old universe could contain an infinitely-old process generating persistent objects, it's incumbent on them to give some non-question-begging reason as to why this is the case.

Wesley P. said...

Dianelos,

You wrote, "Now of course the year 1789 does not exist “anymore”, but from this it does not follow that it does not exist at all."


I believe this is all that's needed for Dr. Feser's objection to work. The mode of being as real and actual in the now does not apply to 1789, but it does apply to Hilbert's Hotel. HH, should it work, only applies to collections which are real beings now.

George LeSauvage said...

@Dianelos Georgoudis:

I was thinking that a good definition is this: “X exists if and only if there are true propositions about X”.

I have trouble with that. "Sherlock Holmes is a fictional detective created by Arther Conan Doyle. Therefore Sherlock Holmes exists." Why doesn't this work. Or worse, "'Married bachelors and square circles do not exist' is a true statement. Therefore married bachelors and square circles exist." Doesn't this send us off into Meinong-land?

David M said...

Tony:
"But for these purposes, the proof of the sort "and this is absurd" rests on the hypothesis "suppose a Hilbert Hotel REALLY existed, as fully and completely actual". The absurdity that results shows that the hypothesis supposed, the real actual existence, is itself absurd."

Certainly, and the same applies to an infinite past. The argument is "suppose an infinite past REALLY existed (not in the present moment (obviously!), but in the past, as the past), as fully and completely actual (i.e., just as 'fully and completely actual' as the finite past we are directly acquainted with from experience)." The absurdity that would result is analogous to the absurdity of Hilbert's hotel (Feser disputes this, but I don't think he provides cogent reasons for his claim that their is a relevant/significant disanalogy between the two cases).

"This form of reasoning is exactly like all other instances of reductio ad absurdum. It is unnecessary for the Hilbert Hotel to BE actually real in order for the absurdity to be established as proven. just like it is unnecessary for the conditions of any other absurd hypothesis to ACTUALLY obtain in order for the reductio to be valid in any other proof. (And, if it were necessary, would defeat the reductio as a valid form of proof, of course.)"

Well it's not that it's unnecessary, it's that it's (seemingly) impossible. Which is why I claim that HH has less reality than the seven dwarves.

"[Craig] has to show that it does, not just REST when he gets to the point of "and the infinite past has some kind of partial, degenerate, past-tense sort of actuality"."

I think this analysis gets the reality of the past wrong, though, as it at least *ought* to be presented in Craig's account. The past is the fully real past. It is not any kind of 'partial, degenerate' reality. It is the fully real past, which is as real and necessary and unchangeable in (relation to) the fully actual present moment as God himself, although it does not possess its being real (really past) and being necessary and unchangeable *per se*, but only by supposition of God's immutable eternal will (and its being actually past).

Brandon said...

If someone insists that a universe could be infinitely old but no infinitely-old universe could contain an infinitely-old process generating persistent objects, it's incumbent on them to give some non-question-begging reason as to why this is the case.

But the position that needs to be argued for is that an infinite past must generate the absurdity; the person positing that the past is infinite doesn't need to argue that this position is immune from absurdity regardless of what suppositions one might add on top of it.

Recall that the position to which the reductio is directed is (by the very set-up of the reductio) one that already independently accepts that an actual infinite is impossible. The only way to get an absurdity, then, is to argue that the infinite past requires an actual infinite. If it doesn't, the already-existent assumption that actual infinites are impossible blocks any arbitrary assumption that introduces an actual infinite.

Dianelos Georgoudis said...

@Wesley P

I agree with Feser on this. The Hilbert's Hotel argument applies to infinite sets of things that exist together and which we can handle at the same time, and thus is not relevant to the set of past years. My argument is that past years exist, and thus the set of past years (whether finite or infinite) also exists. And finally, given that the set of past years exist, I think there is a truth to the matter of whether it is finite or infinite.

I also agree on Feser's point about the Hilbert Hotel argument, namely that weird implications do not entail non-existence. In the human condition we do not handle actual infinities, and thus have no grounds to reason about what would be normal or weird in this case.

Tony said...

I thought it was a good argument because it seems to show that in principle, if you allow the possibility of an infinite time, you also have to allow the possibility in principle (not necessarily the actuality) of an infinite number of objects.

Edward Isaacs, I think that your notion of "possible in principle (not necessarily the actuality) needs to be fleshed out to be used in a proof like Craig's, for there are ways it can go wrong. For example, it is "possible in principle" to cut a line (or a thing whose edge is a line) at any location between 1" and 2", and that is an infinite # of locations. So you can "in principle" cut it at an infinite # of locations. But, in reality, you cannot cut it in an infinite # of locations, for each act of cutting takes an interval of time....(and for multiple other reasons it is not possible). So the "infinite in principle" in THIS case is an infinite of possible events, no one of which is necessary, but each one of which would be mutually exclusive of others of the infinite set.

Similar problems attend the "infinite in principle" claim of an infinite # of hammers. Just one is that hammers break down over time (if nothing else, protons decay over time, if you have 10^1000 years, you are going to get lots of proton decay).

Generically (within natural science, which is where we are with hammers), when you assert the "in principle" here, what seems (to me, anyway) to be true is that you are asserting it "at least from the standpoint of THESE 2 (or 3, or 20) principles, but not necessarily from the perspective of ALL principles of nature that also come into play". You have to actually specify the set of principles that you are including, and thus implicitly carving away all the other principles that are (for this) not being taken into account. In practice, we feel pretty good about it when we don't have to exclude the 4 or 5 or even 10 most obvious natural principles that would undermine the thesis we hope to be able to conclude. But it's still not a demonstrative way of proceeding. Is proton decay a fundamental aspect of material being as created, such that "assuming it away" would amount to eviscerating actual material beings like hammers? I don't know. Do you? We would have to know, in order to be able to assert with certainly that we "could in principle have hammers not subject to proton decay". Similarly, we would have to have a very good grasp of the totality of law of nature we are implicitly invoking in order to assert

you also have to allow the possibility in principle (not necessarily the actuality) of an infinite number of objects.

I just doubt that we have that level of comprehension of the principles of nature to say that with certainty.

Anonymous said...

The more I think about this, the more it seems to me that Feser's criticism just doesn't work.

If the universe had no beginning, then we have an infinite series of past years (say) Y1, Y2, Y3, etc. Now, even if Craig has to admit that those past years no longer actually exist on presentist grounds, he would still insist that it should be meaningful for us to determine the number of years that are represented by such a series, or the number of years indexed by the even numbers, or the number of years indexed by the integers greater than three, etc. And our answers to these sorts of questions will yield the very same metaphysical absurdities that are derived from the Hilbert's Hotel example. Hence, the universe must have had a beginning.

The point is that the same sort of reasoning that applies to actually existing concrete objects in the Hilbert's Hotel example can be easily adapted to the case of events/times that actually occurred in the past.

Anonymous said...

But is a collection of units of time (minutes, days, years, etc.) more like a collection of concrete objects like guests and rooms, or is it more like a collection of abstract objects like numbers? For Craig’s argument to work, it seems that we’d have to say that it is more like the former. But in fact, this seems false.

This is where Feser goes wrong. A series of times/events that actually occurred in the past is not a purely formal/abstract series in the sense of pure mathematics since, again, it involves times/events that actually occurred in the past. And that's why consideration of such a series should not result in metaphysical incoherence anymore than consideration of a collection of actually existing concrete objects.

Tony said...

I think this analysis gets the reality of the past wrong, though, as it at least *ought* to be presented in Craig's account. The past is the fully real past. It is not any kind of 'partial, degenerate' reality. It is the fully real past, which is as real and necessary and unchangeable in (relation to) the fully actual present moment as God himself, although it does not possess its being real (really past) and being necessary and unchangeable *per se*, but only by supposition of God's immutable eternal will (and its being actually past).

David M, for this point, it is unnecessary for me to maintain that the sort of "being" or "reality" that the past DOES (right now) have is lesser, as I said, "partial, degenerate reality". It is sufficient, for this, that I show that the past's "reality" is a different sort than the present. I.E., that when we say "is" of the past thing, and "is" of the present being, we do not use "is" univocally.

I agree with you that the past is determinate (I made that point originally), and that it cannot be changed. (I would not add the qualifier "due to God's immutable eternal will" because I think that this may confuse things for some: it also cannot be changed due to the fact that God cannot violate the laws of logic and do something metaphysically impossible.) These are attributes that lead me to AGREE that the past does indeed have a sort of "being" and "reality". But what is at issue is whether these lead to the conclusion that this sort is the same sort of "is" that applies also to my current computer right now, univocally. And I don't think so.

First, it is in no way manifest. You cannot argue "well, of course it is, it's the obvious truth of reality." What is apparent is that the past real is in some manner unlike things that are real now: things that are real now ARE CHANGING, and the past real thing cannot change. What is apparent is that the real thing that is now can be sensed, but the real past cannot be sensed, only remembered, which manifestly a different operation. What is apparent is that the real thing that is now is subject to contingencies, but (as you said) the real that is past is "necessary" in a sense not applicable to the real now. All of these apparent differences mean that in order to claim that the we say "is" of the real thing now in the same univocal sense we say it of the real thing past, you would have to argue away the apparent problems and show that they are only apparent, not true differences. Not only has that not happened, I don't think it can be done.

What I accept of your comment is that the past is wholly and fully "real", but I qualify: fully real in the special way that applies after the now of a real thing in the present, a kind of "actual" that is special to the past. It is, like the real of potency in the thing that CAN be made actual, clearly other than non-being, but other in a distinct way from what is actual simply. It is "being" in a non-univocal sense. It would be better, I think, to assert that the past real thing "was", and that this statement with the verb "to be" is itself an assertion of actuality that is true in the present. This current assertion of actuality shows that "being" is said of the past thing.

Tony said...

And, obviously, "was" differs from "is" even though both are forms of the verb "to be". We DON'T HAVE any other words with which to state the truth here, any more than we have better words to specify the "reality" of potency than to say that "it can be actual". The defining, referent form of "reality" is the actual, not the potency, the potency is potency in reference to the actual. Similarly, the defining, referent form of reality is "is", not "was", but the relationship of what was to "being, simply" is given by that word "was".

Dianelos Georgoudis said...

@George LeSauvage

““I have trouble with that. "Sherlock Holmes is a fictional detective created by Arther Conan Doyle. Therefore Sherlock Holmes exists." Why doesn't this work.”

Because the first proposition is not about a human being called Sherlock Holmes but about a fictional character named Sherlock Holmes. That fictional character certainly exists. And exists not only as a reference in the novel series, but also in our imagination, in movies, etc. Perhaps the confusion here is that “Sherlock Holmes” may refer to many different things and when using this expression we should clarify what we mean by it.

““Or worse, "'Married bachelors and square circles do not exist' is a true statement. Therefore married bachelors and square circles exist."”

Ah, I thought about this and considered writing “X exists if and only if there are true propositions about properties of X”, but then I thought this would be pedantic. After all “X does not exist” is not a proposition about X but a proposition about the set of existents, namely that X does not belong to it.

This last bit may sound like a sophistry, so let me explain. It seems obvious to me that we can only discuss truths about things that exist. Now “Married bachelors” is a contradiction in terms, so let's use instead X=”the largest prime number”. We know that X does not exist, but this knowledge is not really about X but about something that does exist, namely the set of prime numbers. It is because of the nature of prime numbers that X does not exist. Or take X=”the elephant that lives in the Oval Office of the White House in Washington DC”. The proposition “X does not exist” is not a proposition about X but about the Oval Office, namely the proposition that there is no elephant that lives there.

Why does it strike me so obvious that it makes no sense to discuss truths about things that do not exist? Because if something does not exist then there can't be any interaction with it, and therefore no way for it to affect me in any way whatsoever. But then how could I possibly know anything about it? For example the existing fictional character of Sherlock Holmes in my imagination can affect me (perhaps inspire me in some sense). But the non-existing human being Sherlock Holmes cannot affect me in any way. The non-existing largest prime number cannot affect me, but the existing idea about the largest prime number can affect me.

But let us push this matter a little further.

Consider the true proposition I just made “Things that do not exist cannot affect us in any way whatsoever”. Is this a proposition about things that do not exist? No, it is a proposition about things that exist, since what it really says is that “only things that exist may affect me in some way”. This is what we actually know. It is possible to describe what exists by describing what it is not, but this property of language should not confuse us.

Or consider the true proposition “5 is a prime number”, from which according to my definition it follows that “5 exists” (and also that prime numbers exist). From the knowledge that X exists it does not always follow that we know what X is. Indeed in many cases we know that X exists while having a very vague idea about what X actually is. So for example we know that electrons exist (not only because there are true propositions about electrons, but also because we have electron detectors). Indeed we know tons of things about electrons. But we deeply disagree about what electrons actually are (so for example physical realists who believe in the many worlds interpretation of quantum mechanics hold that every electron exists in a continuously growing and perhaps infinite number of copies in different parallel universes, physical antirealists believe that electrons are just stable patterns present in the physical phenomena we experience, and subjective idealists believe these patterns are created in the mind of God). The metaphysics of physical objects is as difficult as the metaphysics of numbers.

Wesley P. said...

@Dianelos Georgoudis

Yet 1789 cannot be said to be actual, to be in being. Truth propositions about 1789 exist. Mental conceptions do exist. But these exist only as mental beings, not real beings (in being actual). Even if you're hesitant about distinguishing between mental and real beings, we still have the fact that a truth proposition is not the same as its subject matter. Truth propositions also do not have their own act of existence as real beings, whereas Hilbert's Hotel and all of it's residence would.

Don Jindra said...

Dianelos Georgoudis,

Children are moved when watching Snow White. The monster under the bed affects them too. So I think your definition of existence is suspect.

The concept of 5 exists in our minds. But that does not prove 5 exists outside our minds. The fact that historical notions of 1789 exist in our minds does not prove 1789 exist outside our minds.

Suppose a poodle ate some of Marie Antoinette's cake. History has made no mention of this incident, nor of the queen's outrage about it. Maybe the incident happened, maybe it didn't. If it did happen, and all knowledge of it vanished when the queen lost her head, does the poodle now exist? Does the queen's outrage still exist?


Anonymous,

"A series of times/events that actually occurred in the past is not a purely formal/abstract series in the sense of pure mathematics since, again, it involves times/events that actually occurred in the past."

Our only record of Columbus's voyage of 1492 is found in abstract language and pictures. Where does it become more than abstract now?


Mr. Green said...

Philip: ...a picture that George Gamow half-jokingly made up in one of his popular books on mathematics. That imaginative experiment has nothing whatsoever to do with Hilbert save that he was one of the greatest mathematicians of the 20th century, so Gamow knew what he was doing.

Actually, the infinite hotel is apparently an example Hilbert himself did use, in unwritten lectures. See
The True (?) Story of Hilbert’s Infinite Hotel
.


Miguel: I would be wary against accusing Craig of getting his views confused there, provided that the man is very smart and has written *A LOT* on the subjects of God and time (he was, after all, president of the association of philosophers of time). I'm sure that you can find an answer from him on that subject.

I certainly don’t mean to be dismissive of Craig; of course, St. Thomas disagreed with him first, and he was a pretty clever bloke himself.


John D: [Alexander Pruss] claims that one (including Aristotle and Aquinas who try) cannot be presentists AND hold that time is reducible to something else. Perhaps he makes an error, but I'm not well versed enough in this issue to catch it.

I don’t know why he says Aristotle “reduces” time to change. Time is the measure of change, and a measure is not the thing measured, so I just don’t see how his argument applies to Aristotle.


Hektikpecs: If any event is once present, there must have been a time when it had an age of zero. If we accept that an infinite set is analytically non-constructible, it's impossible for any past event to be infinitely old.

But God doesn’t “construct” things, so I don’t think we need to accept that. (I wonder if this figures into Craig’s thinking in some way — doesn’t he believe that God is somehow, in some sense “in” time? In which case, it might follow that even God could create the past only in some sort of constructive, additive way. That would make it more plausibly analogous (or even equivalent) to an actual infinity.)

Dianelos Georgoudis said...

@Wesley P., Don Jindra

I am not discussing “actual existents” or “to be in being”, or “mental beings”, or “what exists outside our minds”. I am discussing the concept of existence, which is the fundamental concept in metaphysics. Clearly, things that exist have different natures, for example some have existed in the past, some exist now, some exist timelessly, some are physical, some are abstract, and so on. The point here is to clarify what we mean by “existence” so that we may reason about whether something exists or not, for example to reason about whether the year 1789 exists or not.

Now of course you may disagree with my definition of the concept of existence according to which the year 1789 in which the French Revolution started exists. But let me only point out that definitions should not violate the normal way we speak, and to say “The French Revolution started in the non-existent year 1789” sounds like nonsense. If a philosopher is not happy with the folk definition of words, then she is free to coin a new word and define it as she sees fit. But uuusing a common word after changing its usual meaning of a word is a recipe for confusion, not only in discussion with others but also in one's own thinking process. This problem by the way is very common and sometimes difficult to avoid - for example many physicists use the word “time” in a way grossly different than the usual meaning. In any case, since you seem to disagree with my definition I am curious to know what you mean when use the concept of existence. Could you give your definition?

Mr. Green said...

Tony: The past events and beings have a sort of reality that the future does not have, even though it doesn't have the same reality that the present has either.

Past events are not determinate to us, but they are to God, in the sense that He “already” knows the future, so is there really a difference here that is the right kind of difference?


David M: "Where would we even get this knowledge of past events to [try to] count them all? By reading an infinite number of pages of the Encyclopaedia Britannica? No human being could, in fact, know the infinite past to enumerate it in the first place."
Why is this? Surely there could be an infused knowledge of an infinite past. If the beatific vision is possible, why not a vision of the infinite past (assuming such a thing is indeed possible)?


Well, if such infused knowledge led to the same problem as an actual infinity, then there couldn’t be such a thing, but I don’t think that rules out the possibility of an infinite past per se. Perhaps you would insist that even if an infinite past were thus possible in itself, the eventuality of the beatific vision means we would have to come to that knowledge, and so God would never choose to create such a past. But not being in possession of the B.V. (although I hope the situation will one day be remedied), I’m a bit flummoxed by it, so it’s not clear to me that it would necessarily be the right kind of knowledge to provide a counter-example. Is knowing something through seeing God (in all His divine simplicity) equivalent to knowing it “successively”?


Francis Beckwith: if we think of the present as a consequence of the past series of events, and if the past series of events is infinite, would it not follow that any thing, X, in the present would have received its existence as a consequence of an infinite series of instrumental causes? The fact that the past series does not presently exist does not seem relevant, since, in a sense, the past series does "exist" in the being of X, just as the creator of an artifact, in a sense, "exists" in the artifact.

I think that insofar as we consider the past and present to form an essentially-ordered causal chain, that must be true. Aquinas’s position, then, is that the present — at least not all of it — need depend on (all of) the past. Now, I don’t think anyone has explicitly raised the point yet that according to our picture of physics, everything from the Big Bang on is in fact causally connected. So it is perhaps possible to argue from physics (as we know it) that events are too tightly intertwined for there be room for accidentally-ordered chains, and thus for there to be an infinite past. But that again ties us to our current understanding of the way physics happens to work, not metaphysical necessity — but since Craig is arguing from physics anyway, that would probably make a better argument.

Mr. Green said...

Matthew Rodriguez: Dr. Feser, what do you think of Alexander Pruss' argument for thinking that an infinite past is impossible?

A few people have mentioned the “grim-reaper” arguments, but I must admit those arguments have always left me flat. They seem to be nothing more than overly-complex versions of the hammer argument: sure, they end up describing a scenario that is impossible, but all that proves is that that particular scenario could never happen. There’s simply no good reason to conclude from that that an infinite past per se is a problem. (I also suspect that they all engender their contradictions by relying on essentially-ordered causal chains, which ignores the accidentally-chained past that is the only kind of infinite past up for grabs anyway.)


Anonymous: Now, even if Craig has to admit that those past years no longer actually exist on presentist grounds, he would still insist that it should be meaningful for us to determine the number of years that are represented by such a series, or the number of years indexed by the even numbers, or the number of years indexed by the integers greater than three, etc. And our answers to these sorts of questions will yield the very same metaphysical absurdities that are derived from the Hilbert's Hotel example.

But Craig doesn’t object to the purely mathematical concepts involved in Hilbert’s Hotel — he’s not claiming the math doesn’t work. An actually existing HH would indeed be bizarre, but if it in fact never existed, then all we have is abstractions about its hypothetical possibility. So it seems to me that the objection to Craig stands, because if the mathematical abstraction works, it doesn’t matter whether we got the idea from past events, or just by imagining it.


Daniel Vecchio: I don't think time travel to the past is possible on Presentism. There would be nowhen to go.

Yes; although, maybe God could travel me through time, since He exists outside it. Or rather, since it couldn’t really be “travelling”, perhaps God could create me in the year 1000 B.C., complete with memories of 2016, then pop me (or at least my body?) out of existence, and then pop me back into existence as a baby in 20th century (without my memories), until I reached the year 2016 and the moment of my “trip” into the past, at which point I would pop out of existence again…. Of course, I’m not sure that works, but I’m not sure that something along those lines couldn’t work, either. (It wouldn’t help finitise the past, because if even if God could put me an infinite number of years into the past, I simply would be unable ever to traverse it, successively, and reach the present. God could jump me around, but I’d only traverse a finite span at any given leg of my time-travelling journey.)

Mr. Green said...

The impossibility of an actual infinity is assumed for purposes of this discussion, but of course I cannot help pondering the matter anyway. Aquinas says that infinity is not a “species of multitude” — that is (as I understand it), he is pointing out that “infinity” is not actually a number. If I write “1, 2, 3… ∞”, that should not be read as though “∞” named a real number like 1 or π or googol. It’s merely a shorthand for “etc.”, “and so on”, “forever and ever, numbers without end”. But every number must be some number — if there were an infinity of anything, there would have to be some number of them, so there cannot be such a collection. Now St. Thomas does not argue from paradoxes: scenarios like Hilbert’s Hotel are indeed weird, but plenty of thing are strange or (literally) unimaginable and nevertheless real. What is it to say that mathematics can coherently cope with such infinite sets other than that it is possible (in itself). If the setup were intellectually consistent, then our failure to imagine would be irrelevant.

The interesting thing is that, even though mathematicians have done an impressive job of giving calculus a rigorous foundation based on limits, since the days of Leibniz and Cauchy and Weierstrass they have also figured out ways to define numbers that are infinitely big and yet really are numbers, not “merely” limits. It seems to me that we thus cannot rule out an actual infinity after all — which of course makes an infinite past defensible a fortiori. All the same issues of instrumentally-ordered causes and restrictions of the actual laws of physics apply, but I’m prepared to accept at least the metaphysical possibility of an actual infinity.



Anonymous: Further, if there can be an infinite number of members in a hierarchically ordered causal series, why "must" there be this "first member" to act a terminus of explanation...why couldn't there be an infinitely explainable universe?

Well, there can of course be an infinite number of members in whole causal conglomeration, as per the example of the blacksmith with an infinity of hammers. There are an infinite number of causes going on, but the essentially ordered part is only ever blacksmith→hammer→iron, three steps, at a time. However, you want to know why we couldn’t have a first item (the essential cause), a last item, and an infinity in between. But if there is a first and a last, then traversing between them must be a finite operation, by definition. Hence the usual conclusion that “one cannot proceed to infinity” in this matter.

But do we have to reach across the chain successively? Imagine the hand pushing the stick pushing the stone, and that stone pushing another, and so on, with the qualification that between the first and last stones are two smaller stones, and two smaller still between them, and so on. Suppose that the first and last stones are an inch in diameter, the second and second last half an inch each, then a quarter inch, etc. The distance between the first and last stones will be two inches. Of course we are pretending that we live in a world where matter is infinitesimally divisible, and let us also stipulate that forces are transmitted instantaneously. Then when I push the first stone, the entire chain of stones will move all at once… won’t it? There is an actual, identifiable source of the motion, so the “infinite cabooses with no engine” problem does not apply. Even if the in-between pebbles were not infinitesimally small (thus making the first and last stones an infinite distance apart), I don’t see a contradiction. (Well, other than the existence of an actual infinity of objects, which I am leaning towards accepting anyway.)

Mr. Green said...

David M: The *actually past* infinite series may not be (presently) 'real,' but it must have a robust foundation in (conceivable) reality that is not at all analogous to the strictly abstract mathematical notion of infinity.

What is not analogous (in a relevant sense), though? I guess the problem is this: if infinities are problematic (instrumental causation aside) because there would have to be some number of them, and “infinity” is not a number, then why does that not equally apply to ideas? Mathematicians can work around this by rephrasing all statements about infinite(simal)s in terms of limits, but an actual, instantiated infinity cannot by definition be a shorthand for something non-infinite. So doesn’t the same thing apply to the past? We’re not saying that the series of past events can be extended indefinitely as far as needed, because it all had to be actual at some point. Now, there was never an infinite collection of beings at any given point, but why isn’t the abstracted collection across history a legitimate collection? And thus it must have some number of members, which number must be infinite? Perhaps, as I suggested previously, the excuse is that we could never actually entertain such an infinite thought, so there is no infinite number of members to count after all. And certainly, we cannot think such a thing — in our present state… but what about the beatific vision? What about angels? And what about God? Surely He knows how many events happened in the past. In His eternity, are they not all “present” to Him [in some sense] as an infinite collection? But that may be attempting to apply an anthropomorphic, discursive picture to God’s knowledge, which is, after all, just His own act of Being.

Still, this isn’t entirely satisfying to me; it would be much easier if we could agree that the past simply did constitute an infinite collection (and that some actual infinities are perfectly acceptable in polite company). Yet if a past infinity — or abstractions of one — were enough like a present one, it would surely not have escaped Aquinas.

David M said...

Mr. Green wrote: "since the days of Leibniz and Cauchy and Weierstrass they have also figured out ways to define numbers that are infinitely big and yet really are numbers, not “merely” limits"

Can you explain this, and argue for its relevance?

David M said...

Mr. Green:
"Is knowing something through seeing God (in all His divine simplicity) equivalent to knowing it “successively”?"

You seem to be referring to mode of knowing, when what matters is the object of knowledge. The point I made was not that knowing an infinite past could only be a corollary of the BV, but that we can know about infinite things (possibilities or impossibilities) without needing (absurdly) to refer to an infinite book or have an infinitely big brain.

"We’re not saying that the series of past events can be extended indefinitely as far as needed, because it all had to be actual at some point."

I'm not sure I follow your meaning. Here's what I mean: The series of past events cannot be extended indefinitely, because each event had to be actual at some actual, definite point in the past. *If* we judge that an actual infinite collection is impossible, then that is what is relevant. Whether the actual infinite collection is simultaneous or successive makes no difference, because the successive past is not less determinate than the present, so it should not be less determinately conceivable. (Again, the issue isn't about successively running through each of an infinite number of finite members, it's about conceiving of them as an actual, determinate infinite whole.) We never get to an actual infinite collection in virtue of adding finite parts. An infinite past/successive collection is impossible if an infinite present/simultaneous collection is impossible, because the former no more determinately conceivable than the latter (it's not determinately conceivable at all).

David M said...

"Past [rather: future] events are not determinate to us, but they are to God, in the sense that He “already” knows the future, so is there really a difference here that is the right kind of difference?"

Yes, there is. Like us (and Aquinas), God knows the past as that which is determinately past, and the future as that which is indeterminately future. God knows that time began (in the beginning), and that it need not ever end.

Mr. Green said...

David M: Consider Robinson's hyperreal numbers, which define numbers larger than any real number, i.e that are infinite, but are not just a shorthand for "carry on this sequence indefinitely". So we have infinitely big numbers that are a species of multitude, and thus could presumably be used to count an actual infinity. And if the objection to actual infinities no longer stands, then an infinite past is certainly not a problem.


God knows the past as that which is determinately past, and the future as that which is indeterminately future.

What does "indeterminately" mean in that context?

Mihret Gelan said...

Question on infinity & finitude in general: What are the principles of infinity and finitude? What makes something potentially infinite in number (e.g. creatures) and something necessarily finite (e.g. Aristotle's categories of being - namely, ten in number i.e. finite)? By principle, I mean, the reason why something is

Don Jindra said...

Dianelos Georgoudis,

I consider myself just one of the folks, yet I don't use the word in the way you claim I do. When I say the French Revolution started in 1789, it never crosses my mind that 1789 still exists in some sense. In fact, I take some comfort in the belief that it no longer exists.

The French Revolution did start in the now non-existent year of 1789. I've never allowed this to confuse me. It started in a time and place that no longer exists. Almost daily I'm reminded of the fact that the year 1972 no longer exists along with that wonderful body I had then.

I do not agree some existence is physical, some abstract. For me, all existence is physical. The abstract exists only in the patterns of our brain, which is physical.

How do I define existence? I count it as what we can actually or in principle sense ourselves or through our instruments. All else has questionable claims of existence. That leads to the question, Does Time exist? We think we can sense it passing, and our timepieces convince us it does pass, but do we really sense something we could call time itself? I don't know, but I suspect not.

DAS said...

What is the definition of 'present'? How long does the 'present' exist for? Does it exist for a second? A nanosecond? A femtosecond?

If presentism requires the 'present' to be infinitesimal then it might contradict what we know of time with regard to the Heisenberg Uncertainty Principle, i.e. that we can tell when specific events occur with infinite precision. There is some minimum length of time during which each event may have occurred.

If presentism allows for a non-zero length of time, then even if all of the prior 'presents' no longer exist, then we would never have gotten to now, as an infinite number of non-zero 'presents' would have had to occur in order for us to be when we are.

bmiller said...

Mr. Green

Even if the in-between pebbles were not infinitesimally small (thus making the first and last stones an infinite distance apart), I don’t see a contradiction. (Well, other than the existence of an actual infinity of objects, which I am leaning towards accepting anyway.)

One of Aristotle's arguments against an actual infinity of material things is that material things actually take up space. No matter how small they actually are, they would still take up some amount of space. Then if the universe had limited space it would be impossible for them all to fit. Even if the universe had infinite space, then all possible nooks and crannies would be full, leaving no room for the hand or stick to move the stone and so on.

Had you considered this?

grodrigues said...

@Mr. Green:

"A few people have mentioned the “grim-reaper” arguments, but I must admit those arguments have always left me flat. They seem to be nothing more than overly-complex versions of the hammer argument: sure, they end up describing a scenario that is impossible, but all that proves is that that particular scenario could never happen."

I do not see what exactly are the problematic assumptions in Grim-Reaper scenarios. With some very mild technical assumptions on the particular accidentally ordered series, there is no need for infinite divisibility of time or an actually compresent infinite collection of Grim Reapers. Grim Reapers do not need super powers or infinite memory. Etc. and etc. The scenario is impossible, sure, but as far as I can see the impossibility follows squarely from assumption that the ordered series is infinite in the past and not from some totally alien supposition that the defender of an infinite past, or of its possibility, is not bound to accept.

Anonymous said...

grodrigues,

Technically, the GR scenarios presuppose either that time or space is infinitely divisible. In particular, the GR scenario that Pruss considers in the relevant blog post presupposes that time is infinitely divisible, while the GR scenario that Koons considers in his published work presupposes that space is infinitely divisible. So it would seem to be the case that some sort of divisibility assumption is needed for the argument to work.

Dianelos Georgoudis said...

@Don Jindra

“Children are moved when watching Snow White. The monster under the bed affects them too.”

Right. The character of Snow White in the movie the children are watching exists. Their idea that a monster may lurk under the bed also exists. Perhaps your point is that the children do not fear the idea of the monster but the monster, and this is so. But what *affects* them is not the monster but the idea.

The question here is whether something that does not exist can affect us in any way. I think the answer is obvious. After all, if something does affect us then it certainly exists. Conversely, things that do not exist cannot possibly affect us.

Finally, coming back to my definition, true propositions can be made only about the existing idea in the child's mind, not about the non-existing monster under the bed.

“Suppose a poodle ate some of Marie Antoinette's cake. History has made no mention of this incident, nor of the queen's outrage about it. Maybe the incident happened, maybe it didn't. If it did happen, and all knowledge of it vanished when the queen lost her head, does the poodle now exist? Does the queen's outrage still exist?”

I am not discussing epistemology. Surely there are a lot of existents about which we can't have any knowledge. Nor am I discussing “actual existence” or “past existence” - but what is common to all that has existed either in the past or now, either physically or abstractly, and so on. Still if the proposition “Marie Antoinette's poodle ate the cake” is true then (according to my definition) the poodle, the cake, the eating of the cake – all exist. Which seems to me to be right.

David M said...

Mr. Green:
I'm sure you have a better understanding of Robinson's hyperreal numbers than I do. Do you think they make HH intelligible as an actual material reality? I'm suspicious of the claim that they show anything about what is possible in material reality. Our claims about the latter must start from our experience and knowledge of the latter, not from ingenious mathematico-logical constructions - just as natural theology cannot simply be onto-theology (making the structure of our abstract thought about being to be the sufficient grounds for asserting the existence of a non-abstract God). So I wonder: In what sense are RHNs a(n actual) species of actual multitude?

"God knows the past as that which is determinately past, and the future as that which is indeterminately future." -- "What does "indeterminately" mean in that context?"

It means not determinate per se and in se, insofar, that is, as the future is not yet and contingent (even though it is determinate per accidens, insofar as it is known by God).

grodrigues said...

@Anonymous:

"Technically, the GR scenarios presuppose either that time or space is infinitely divisible. In particular, the GR scenario that Pruss considers in the relevant blog post presupposes that time is infinitely divisible, while the GR scenario that Koons considers in his published work presupposes that space is infinitely divisible. So it would seem to be the case that some sort of divisibility assumption is needed for the argument to work."

Yes, you are indeed correct. I was thinking of a variation of Grim Reaper scenarios where such an assumption is not needed; but there is a trade-off involved, so I have to think harder to see if indeed it can be made to work.

grodrigues said...

@DavidM:

"So I wonder: In what sense are RHNs a(n actual) species of actual multitude?"

This was my first thought when reading Mr. Green's answer. The actual construction of the hyperreal numbers is a piece of abstract mathematics needing fairly strong infinitary principles, including relatively strong choice principles like the ultrafilter axiom, so it is not at all clear why they are a species of multitude.

Anonymous said...

@grodrigues:

Yes, you are indeed correct. I was thinking of a variation of Grim Reaper scenarios where such an assumption is not needed; but there is a trade-off involved, so I have to think harder to see if indeed it can be made to work.

Fair enough.

For what it's worth, I think the real lesson we should takeaway from the GR paradox is that time cannot be locally dense, as if the series of times were akin to the real axis.

For an argument against an infinite past, I prefer the one given in this paper, which can also be found here, since no such assumptions are needed for this sort of argument to work.

Scott Church said...

All,

Regarding the GR paradox... For what it's worth, it's very unlikely that time is infinitely divisible. General relativity and quantum field theory both suggest that on the smallest spacio-temporal scales the two must account for each other. The field of quantum gravity is still in its infancy, but everything we currently know about it suggests that time is emergent and breaks down at scales comparable to the so-called Planck time, which is on the order of 5.4 x 10^-44 seconds.

Mr. Green said...

BMiller: No matter how small they actually are, they would still take up some amount of space. Then if the universe had limited space it would be impossible for them all to fit. Even if the universe had infinite space, then all possible nooks and crannies would be full, leaving no room for the hand or stick to move the stone and so on.

Well, I deliberately picked a convergent series for the size of my imaginary pebbles, so that they would (assuming such things could exist) fit into a finite space.
I wasn’t considering the idea that infinite objects would fill the universe densely, but clearly that’s not the case for a converging sum. What about an infinite number of pebbles all the same size? I imagine Aristotle’s conclusion here depends on the idea that everything has a proper place — for stones, that’s “down”, at the centre of the world, and of course if an infinite number of pebbles (with minimum size) were all conglomerated together, they would indeed fill all space. But if we have a physics that allows a line of pebbles to sit in place, one after the other, then an infinite universe could easily contain them with room left over.

Mr. Green said...

GRodrigues: I do not see what exactly are the problematic assumptions in Grim-Reaper scenarios.

I think the assumptions aren’t problematic (I don’t object to potential physical impossibilities, since we’re concerned with what’s metaphysically possible); but that itself is the problem: it’s only the given conjunction which results in a contradiction, not any individual premise on its own. That includes the infinity of time or substances, so all that is proved impossible is the particular scenario, not infinite time or an infinite collection per se.

as far as I can see the impossibility follows squarely from assumption that the ordered series is infinite in the past

Yes, so that ordered series is thereby ruled out. But a scenario where, say, all the Grim Reapers stand around having a cuppa on their tea break results in no contradiction, so on what grounds would I reject it? I previously mentioned that I think the contradiction results from the scenario’s being an essentially-ordered causal chain, but even then, it would be only an example of the problem, not a general disproof. (Although if someone wants to argue that it is a paradigmatic case that sufficiently illustrates the general case, I might not quibble. Accidentally-ordered infinities are still on the table, though.)

Mr. Green said...

David M: Do you think they make HH intelligible as an actual material reality?

Well, I wouldn’t call it “materially intelligible”, since of course matter is not intelligible: only form is. HH is intelligible if the abstraction or formalisation of it is intelligible, which of course it is: to say that we can describe it in a mathematically coherently way is just to say that it is formally consistent, it is logically possible. I think it’s simply wrong to conclude that something is impossible because we cannot imagine it. HH certainly seems strange to us, accustomed as we are only to finite collections, but what could it mean to say that this strangeness makes it impossible? Other than to express my unwillingness to accept the oddity of the results even though no logical contradiction can be demonstrated, so I’ll just call it “imaginary” instead?

I'm suspicious of the claim that they show anything about what is possible in material reality.

It may of course be materially impossible given the actual laws of physics. But since the only strange thing about HH is its size, the only reason I can see for metaphysical problems would be mathematical ones, and mathematically the logic works. If we abstract from our knowledge of matter to its most metaphysically broad scope, then there are no grounds for a contradiction; and if we are drawing from our experience of actual matter, then sure, given our present knowledge, it seems that the universe is finite in size, and that mass/energy is conserved, so certainly nobody could build HH (but then, that’s not the problem Craig or anyone else has with it).


GRodrigues: The actual construction of the hyperreal numbers is a piece of abstract mathematics needing fairly strong infinitary principles, including relatively strong choice principles like the ultrafilter axiom, so it is not at all clear why they are a species of multitude.

But why would that make them not be numbers? Or do you mean that “species of multitude” entails something else? Since we’re talking about bare metaphysical possibility, I take it that God can help Himself to whatsoever principles He wishes. If a system has a logical description, then God could create some kind of world modelled by that description. There’s always the possibility that actual physics (as far as we know it) requires something inconsistent with an arbitrary model (that may or may not be possible to shoehorn into the model even if we allow for miracles), but that doesn’t affect the metaphysical possibility.

Tony said...

Has anyone answered this problem, posed earlier: I think that Craig's argument rests on "whatever begins to exist has a cause." The arguments here going round and round center on Craig's argument that there CANNOT be an infinite past, and therefore time must have a beginning.

However, if we consider a line segment AB, and exclude the points A and B from consideration, we have the situation that the FINITE-length line has no first point or last point. It is an "open" line segment.

Yes, I hear you saying "well, when you 'exclude point A' you are merely demanding to exclude what is critical." But here's the rub: in considering motion, Aristotle shows in the Physics, in considering a change, that rest prior has a last moment, but the motion which follows does not. The interval of time during which the change occurs is bounded by rest before and rest after, there is a last moment of rest before the motion, and NOT a first moment of motion. Like the open line segment excluding A and B, the motion is finite but has no first moment or last moment. If you take any moment during the interval at which the motion IS occurring, there is always a prior moment before that moment in which the motion was occurring also. But the motion is finite, not infinite.

And it would seem this applies equally to the first motion(s) of the universe. Whatever is moving has been moving at an earlier moment, for there is no first moment of motion: motion occurs in intervals and applies to every interior moment of an interval. It does not pertain to the end-points of an interval, for those moments pertain to rest.

Time is the measure of motion, and if there is no first moment of motion, there is no first moment of time. We could have a situation where (granting Craig's thesis) time is finite, but it STILL does not have a first moment. Does this mean (as Craig needs) that time "begins to exist"? Or does it mean that time "has no beginning" because it has no first moment?

Whatever you want to say about that, let me throw out another bone of caution: does it even MAKE SENSE to talk of "time" beginning? Time is that of which OTHER THINGS are said to begin. For a house to have a beginning is for the house to have a first time in which it exists. What would it even mean to say that TIME has a first time in which it begins to exist? Is the idea even coherent? It's kind of like asking "how many basic units long is your basic unit?", only worse.

bmiller said...

@Mr. Green

I imagine Aristotle’s conclusion here depends on the idea that everything has a proper place — for stones, that’s “down”

I based my assessment on Physics Book 3 part 7. In context he is comparing and contrasting your example of infinite division, in which he agrees with you, with the alternative of infinite addition.

"With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can also actually be. Hence since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens."

I believe here he is assuming an finite universe, so an infinite number of sensible things would be bigger than that. I'll let you decide if up or down is in the mix conceptually.

But let me consider your example of pebbles in a line. Each pebble is extended in space and has a volume of V. Then the volume of 2 pebbles is V+V, 3 pebbles V+V+V...ad infinitum, then wouldn't the total volume consumed by the pebbles be infinite? Wouldn't the infinite universe you propose have to have more than an infinite volume to contain the pebbles with room left over?


Dianelos Georgoudis said...

@Don Jindra

“I do not agree some existence is physical, some abstract. For me, all existence is physical. The abstract exists only in the patterns of our brain, which is physical.”

If it is so, then my definition about the meaning of “existence” still holds. For if abstract objects exist only as physical patterns (of neuron structure and electrochemical activity) in our brain, then there are truths about such physical patterns. Which is all my definition requires. Similarly the year 1789 exists as a physical pattern in our brain about which true propositions exist. So I don't see what it is in my definition which you don't like.

But let me explain why I don't like your understanding of abstract objects.

Let me start by conceding that properly construed and inspired by the physical sciences one can describe a physical reality which accounts for all data we have. Therefore, since we have no datum that contradicts it, such a description of reality may be true. But the fact that a proposition may be true does not make it reasonable to believe that it is true. Solipsism also accounts for all data that we have and may therefore be true, but is not a reasonable belief. The proposition that God created the world 6.000 years ago (complete with much older looking dinosaur fossils) may be true, but it is not reasonable to believe it is. Similarly I find that what you suggest above is not reasonable, since it leads to many unreasonable implications:

A typical example of abstract object are numbers. Philosophers disagree about the ultimate nature of numbers, but most of them agree that numbers are abstract, and thus non-physical, objects, which exist timelessly. You suggest that numbers exist only as physical patterns in our brain. So let us consider the number 51. Your suggestion implies that the number 51 started to exist only when the first human thought of it thus forming the relevant physical pattern in her brain, and did not exist before that point in time. Or consider the fact that 51 has the property of being a prime. According to your definition being or not being a prime are properties that some physical patterns in or brain have. Another implication is that I have just created the number 3450917509480937148598388371987398479593911199, since nobody has thought of that particular number before me now. And what if I program a computer to randomly compute a similarly large number and print it on paper without anybody looking at it? Your definition implies that this printed number does not exist, since the respective brain pattern does not exist unless one looks at it. Or consider the quintillionth digit in the digital expansion of pi. Nobody has ever computed that digit, does it follow that it doesn't exist and that the proposition “the quintillionth digit of pi is 5” is neither true nor false? And suppose there is an archipelago in the Pacific Ocean with 51 islands. According to your suggestion it is nonsense to claim that this archipelago had 51 islands before the first humans evolved on earth, since the number 51 did not exist then.

[continues bellow]

Dianelos Georgoudis said...

[continues from above]

In conclusion I think the idea that abstract objects exists as physical patterns in our brain doesn't work – it turns out to be unreasonable to the point of being laughable. There is another solution which the physicalist should prefer, and which is suggested by the last example: That abstract objects of all kinds (and indeed all that is not clearly a physical thing) exist as a *property* of physical things, or, as physicalist philosophers like to put, *supervenes* on the physical. So, for example, the number 51 is a property of any physical agglomeration of 51 things, such as said islands in the Pacific, and thus of course the number 51 has existed as long as such agglomerations exist. Sticking to math (an age-long problem of metaphysics concerns the nature of math) it is interesting to note that all mathematical objects and truths can be interpreted in such a way, namely as properties of the physical world (assuming an infinite such world). For example the proposition “There is no largest prime number” describes a property of the physical world, namely that if one follows a particular set of rules pushing digits on a piece of paper one will never succeed in constructing such a number (many other such properties may exist). As does the proposition “pi is an irrational number”, or the proposition “the intersection of a plane with a cone may produce an ellipse” and so on.

Physicalism is the worldview of practically all atheists. Now after doing some clever thinking I find it is possible to make physicalism presentable, i.e. worthy of serious consideration in the philosophical discourse. Indeed many theists fail to realize the power of physicalism as an idea. But even then, in my judgment, if one compares the best version of atheism one can come up with with the best version of theism, then theism is found to be far more reasonable. Atheism (aka physicalism) suffers from too many conceptual problems, some of which by the way are produced by modern science.

Charles said...

Look, I haven't gone through the entire thread of comments here. So, this could've been dealt with ad nauseum and I am unawares. But I just don't think Feser is persuasive enough to make Craig's suspect. I get the difference between counting imaginary things and the fact presentism is, clearly, the best position to take. But, although the past is no longer in existence, it once was. It isn't purely imaginary to consider past units of time. I hardly see how the seven dwarfs can compare. Bottom line, has their been an infinite number of yesterday's? I think Craig's position is sound, not because past events no longer exist, but that they clearly did and can be quantified in a way that is like quantifying existing objects. The past may not exist, but it also isn't purely imaginary. That means it is quantifiable. And if it is quantifiable, then it falls into the same logic that applies to hotel rooms, brown verses black books, etc. I have almost all of Feser's books. Love his stuff. I also have tons of Craig's works too. Craig was the guy that got me interested in philosophy from a Christian POV, back when I became a Christian (had zero interest in it before then). But I think Craig has a firm grasp on this issue. And, besides, it's a small issue, in the scheme of issues, IMO.

Glenn said...

DJ,

"I do not agree some existence is physical, some abstract. For me, all existence is physical. The abstract exists only in the patterns of our brain, which is physical."

Some food for thought (which likely won't be as physically nourishing as, say, coconuts, seaweed or fish, yet may be more nourishing in some other way):

1. M&M's in a glass jar, because they're in a glass jar, are themselves made of glass?

2. While it is true that M&M's, like the glass jar, are physical, it is not likewise true that M&M's are physical because they're in something which itself is physical.

3. If it is true that the abstract exists only in the patterns of our brain[s], which [are] physical, and the abstract itself therefore necessarily is physical, then the house the architect is working on solely in his mind without any external aids necessarily must be physical.

3.1 Since the house the architect is working on solely in his mind without any external aids is in fact something necessarily physical, perhaps he ought to consider purchasing some sort of builder's risk insurance for it while he's working on it; after all, he may suffer a stroke, his brain may be damaged, and, consequently, the house he has been working on -- solely in his mind without any external aids, remember -- might never be completed.

3.2 Know of any insurance companies willing to assume the risk? Think Lloyd's of London might step in where no other insurer would dare to tread?

4. If it be granted arguendo that any existing thing is physical, including anything existing in the abstract, it would not follow that everything which exists necessarily exists in the abstract. This being so, it would follow that there are (at least) two modes of being for physical things: an abstract mode of being, and a non-abstract or concrete mode of being.

4.1 The house the architect is working on solely in his mind without any external aids would be an example of an existing thing with an abstract mode of being, while the house actually built outside his mind would be an example of an existing thing with a concrete mode of being.

4.2 The house the architect was working on solely in his mind without any external aid, which house had an abstract mode of being, may be said to be a model; and the house actually built outside his mind, which house has a concrete mode of being, may be said to be the thing modelled.

5. "The thing modelled must be like the model according to the form, not the mode of being. For sometimes the form has being of another kind in the model from that which it has in the thing modelled. Thus the form of a house has in the mind of the architect immaterial and intelligible being; but in the house that exists outside his mind, material and sensible being." ST 1.18.4.2

grodrigues said...

@Mr. Green:

"Yes, so that ordered series is thereby ruled out."

If the argument is that if an infinite past is possible then grim reaper scenarios are possible, grim reaper scenarios are impossible, then the desired conclusion follows by modus tollens.

"But why would that make them not be numbers? Or do you mean that “species of multitude” entails something else?"

If you had used your example using cardinal numbers I would have not interjected, because I can make sense of cardinal numbers as numbering multitudes, sets in this case, but I have no idea what a hyperreal number as a "species of multitude" means. What do you mean when using "number" and "species of multitude"? Any field, in the algebraic sense, counts as a "numbering" a "species of multitude"? Something stronger?

Mr. Green said...

Tony: does it even MAKE SENSE to talk of "time" beginning?

That doesn't worry me because I think in any reasonable context that's more or less equivalent to "change beginning". Strictly speaking, it is some substance that begins (in terms of being created or of starting to change), but since we aren't specifying what the substance is, we can talk abstractly about the beginning of time. (We can still be cautious of unreasonable contexts, though, in case someone smuggles in an illegitimate notion under cover of that way of speaking.)

Time is the measure of motion, and if there is no first moment of motion, there is no first moment of time. We could have a situation where (granting Craig's thesis) time is finite, but it STILL does not have a first moment. Does this mean (as Craig needs) that time "begins to exist"?

Interesting point. I wonder whether it's possible for something to be created without having a first moment of its existence. If an infinite past is possible, then why not? But existing is an essentially-ordered cause of being able to do anything else (such as change), so perhaps that does necessitate a first moment of existence. Unless there could be a substance that never caused any effect. Or if being in existence at any moment is sufficient to explain any effect happening at (or starting at) that moment — that is, why couldn't God create something "already" being in motion? If it had to start moving, then there would have to be a first moment at which it was not changing — the endpoint of the interval — followed by no first moment of change itself. (Or, conversely, no last moment of being at rest and a first moment of being in motion.)

Anyway, if something did not exist at the limit of the interval and does exist within the interval, that surely requires a cause just as much as something that has a first moment. The only question would be whether this applies even at the first moment of time (i.e. if the limit of the interval does not exist, there are no other beings relative to which we can say "this object did not exist, and then at some later point, it did exist"). I’m not sure how to deduce a cause without resorting to good old potency and act, essentially-ordered causal chains, etc., at which point we no longer need the Kalam-version of the argument.

Mr. Green said...

GRodrigues: If the argument is that if an infinite past is possible then grim reaper scenarios are possible, grim reaper scenarios are impossible, then the desired conclusion follows by modus pollens.

That’s what the argument is supposed to be, yes; but I would follow Brandon in arguing that the first premise is faulty. An infinite past is necessary for (certain) Grim-Reaper scenarios, but not sufficient: the contradiction follow from an infinite past plus some Grim Reapers plus a very particular arrangement of Reapers plus an essentially-ordered series of instructions they are stipulated to follow. But only the total combination is impossible, not any single ingredient on its own. It doesn’t follow that God couldn’t, say, create a universe where some small bodies continually rotate (without interacting) perpetually.

Any field, in the algebraic sense, counts as a "numbering" a "species of multitude”?

More or less (or maybe even only an abelian group)… my motivation was to sidestep the following potential objection: that if n is taken to number an (infinite) set, then adding or removing an element should yield numbers such that n-1 < n < n+1. I don’t think that is necessarily a good objection; just that it might “feel” more like a legitimate numbering system. But if I’m right that Aquinas means any number must be some number (not a shorthand for “and so on”) then cardinals would qualify and I shouldn’t try to go further than that. Do you think that is a sufficient response to I.7.4?

Mr. Green said...

BMiller: Each pebble is extended in space and has a volume of V. Then the volume of 2 pebbles is V+V, 3 pebbles V+V+V...ad infinitum, then wouldn't the total volume consumed by the pebbles be infinite? Wouldn't the infinite universe you propose have to have more than an infinite volume to contain the pebbles with room left over?

Yes, but that’s OK! The line of pebbles would stretch infinitely in one dimension, but not in height or width, so an infinitely big universe could still have lots of (infinite) room around the pebbles. In fact, a universe that was only finitely “tall” or “wide” could still have room around them (as long as the universe was infinitely long!).

bmiller said...

Mr. Green

If the series were arranged along the x axis with the first mover at the origin how could the entire series move anywhere other than along the x axis? Again if all possible positions along the x axis were consumed by pebbles, then none of the pebbles could move.

Or do you have a proposal of how to move the entire series in the y or z direction in a single motion?

Anyway, it seems to me that we are already in a scenario that is violating non-contradiction. We have infinite volume consumed but there is still more volume available. Sometimes things that work in mathematics cannot work in physics.

Don Jindra said...

Dianelos Georgoudis,

"Your suggestion implies that the number 51 started to exist only when the first human thought of it thus forming the relevant physical pattern in her brain, and did not exist before that point in time."

I've done more than suggest this. I've made this very argument here before. As far as we know, the number 51 has no existence outside human brains. It's a concept we find useful. It's descriptive, like "red" is descriptive. Our eyes happen to have receptors tuned to that band of light, so naming that band is useful. When we say there are 51 apples in the bucket, there aren't 51 apples. There are trillions of objects we call apples. It just so happens that we are interested in how many of those trillions are currently inside the bucket. The number 51 was invented to help describe that interest. So am I saying some really big number exists -- the total number of apples in existence? No. We call all of them apples, but in fact no two apples are alike. The word "apple" is an abstraction we find useful. It doesn't help to say there are 51 objects in the bucket because in fact there are countless "objects" in the bucket. We just happen to be interested in the big red ones we call apples. All of our symbols, including numbers, are a form of data compression. They filter out what we find insignificant to help us focus on what we find most significant. They help us describe and understand reality but they have no reality outside us -- as far as we know. I find no conceptual problems with that.


Don Jindra said...

Glenn,

"M&M's in a glass jar, because they're in a glass jar, are themselves made of glass?"

How would we know the M&M's aren't made of glass? They could be. I suggest we open the jar and make some tests. These tests would not be logical tests. They'd be "sensory" tests -- that is, if we hope to have confidence in the results.


"If it is true that the abstract exists only in the patterns of our brain[s], which [are] physical, and the abstract itself therefore necessarily is physical, then the house the architect is working on solely in his mind without any external aids necessarily must be physical."

I don't use the word "necessary" like many of you distinguished philosopher do. Things either are, or they are not (as far as we know). Nothing is truly necessary (even though I might speak of necessary effects from, say, an IF/THEN/ELSE instruction). But yes, the house plan the architect is working on solely in his mind is in fact physical. Just as no two houses on June Street are identical, the physical plan in the architect's head is not the same as the house standing on June Street. This is not two forms of being. It's two radically different physical things, like the house and the light reflecting off the house are two radically different physical things.

I can understand how Aquinas in the 13th century would argue models show few signs of being material. I have a hard time understanding how an informed person today could have the same confidence in that position.

Glenn said...

DJ,

1. How would we know the M&M's aren't made of glass? They could be.

Let's assume they are. So what? The question I asked had to do with whether (if they are) they are for the reason that -- i.e., because -- they're in a glass jar.

("M&M's in a glass jar, because they're in a glass jar, are themselves made of glass?")

2. I said nothing at all about a 'house plan'. But I will assume that you employ 'house plan' as shorthand for 'the house the architect (is working on)/(worked on) solely in his mind without any external aids'.

3. I can understand how Aquinas in the 13th century would argue models show few signs of being material. I have a hard time understanding how an informed person today could have the same confidence in that position.

Prior to this you acknowledged that the house the architect was working on solely in his mind without external aids is "radically different" from the house on (say) June street, so I find it hard to understand why a well-informed person -- an exemplar of which we take the liberty of assuming is one such as yourself -- might be wont to deny that the former house is immaterial relative to the latter house.

Don Jindra said...

Glenn,

"The question I asked had to do with whether (if they are) they are for the reason that -- i.e., because -- they're in a glass jar."

Maybe I was too subtle. I wouldn't assume the M&M's are glass. I would assume their composition was identifiable through the senses. I remember M&M's from my past. I can see these might be the same. I can pop the like-looking things into my mouth and chew. I might be unpleasantly surprised. But my senses would be the medium for that surprise. Everything I can learn about the composition of those objects is through the senses. I know you consider that an insufficient basis for the totality of what I can know. After all, why should I limit my knowledge to what can be sensed? But I'll stick to this limitation until someone convinces me he can deduce the M&M's are not glass without confirming all important details with his senses. I understand that this is a fundamental begging of the question. But I observe that those who stray too far from this begging of the question get nowhere. So I see no way out of the dilemma.

"I find it hard to understand why a well-informed person -- an exemplar of which we take the liberty of assuming is one such as yourself -- might be wont to deny that the former house is immaterial relative to the latter house."

For me that's a no-brainer (How's that for irony?). The house plan is an image in the mind, like the image of my wife's face. I have no doubt that such images are static, physical connections in the brain. The tough questions are about will, creativity, intuition, etc...

bmiller said...

@ Don Jindra,

Hope you don't mind a series of questions regarding 51.
1) What sense organ is used to determine the presence or absence of 51?
2) What are the component parts of 51? Would it simply cease to exist if one stopped thinking about it?
3) Is 51 extended in space and have mass? If so, what are the dimensions and mass?
4) If 51 is a merely a description, then what is a description?
5) Do the same answers to 51 apply to description?

Glenn said...

DJ,

Everything I can learn about the composition of those objects [i.e., the M&M's in a glass jar] is through the senses.

Let it be supposed, then, that the M&M's in the glass jar are subjected to a series of empirical tests, which tests meet with your approval, and that the reasonable and empirically sustainable conclusion after the conducting of those tests is that the M&M's are made of glass. How might your senses be utilized in answering the following question: is it true that the M&M's are made of glass because they're in a jar which itself is made of glass?

Glenn said...

DJ,

"I find it hard to understand why a well-informed person -- an exemplar of which we take the liberty of assuming is one such as yourself -- might be wont to deny that the former house is immaterial relative to the latter house."

For me that's a no-brainer (How's that for irony?). The house plan is an image in the mind, like the image of my wife's face. I have no doubt that such images are static, physical connections in the brain.1.


1. If it is a no-brainer that an examplar of a well-informed person such as yourself can deny that the former house is immaterial relative to the latter house, and thus indirectly or tacitly claim that the two houses are equally material, then it ought likewise be a no-brainer that the architect could order living room furniture from Hollywood's Blueprint Furniture, and tip the delivery guys $20 each for positioning the furniture in the living room of the house in his mind just as he wants it to be.

2. There is a difference between a house and a house plan. Another way to put it is to say that there is a difference between the plan for a house, and the house for which there is a plan. As indicated earlier, I had spoken of the architect working on a house in his mind, and not working on a house plan. If there is to be a house plan, then it would be intermediate between the house in the architect's mind, and the actual house constructed. That is, the house actually built would be based on the house plan, and the house plan would be based on the house the architect worked on in his mind.

3. You can claim that the house the architect worked on in his mind is physical just like the actual house, but curious readers will wonder why you seem loathe to test the composition of the former house by sensible means in the same manner you would happily test by sensible means the composition of M&M's in a glass jar.

4. The true irony here -- as it has been for years and years on this blog -- is that you're most confident in your assertions when empirical proof is absent or unattainable.

Glenn said...

Correction: Upon further review, I see that Blueprint Furniture is only near Hollywood, and not actually in Hollywood.

(Might as well revise my original rhetorical question while I'm at it: Since Hollywood, CA is in Los Angeles County, is Hollywood, CA for that reason itself a county?)

Don Jindra said...

bmiller,

There is no 51 like there is no tall. But just as I can sense tall objects I can sense (and count) 51 objects.


Glenn,

"it ought likewise be a no-brainer that the architect could order living room furniture..."

Does not follow. I can't fill up the photo of the house on June street with furniture. Are you going to deny the photo is material?

House plans are stored in a computer these days. That's physical. What in your opinion is the fundamental difference between the brain storing the plan and the computer? I see none.

I'm not quite following your distinction between the mental house plan and working mentally on the mental house plan. But maybe you mean what I mentioned above: will, creativity, intuition, etc... I agree these are tougher questions than a finished mental plan.

Neuroscience is an empirically based discipline, as is artificial intelligence.

Btw, I never implied contents must be composed of the same material as the container. My position is that in the relevant case, we know of no other "stuff", and can know of no other "stuff" than the container. It's not that the materials must be the same. It's that we have little evidence they are different in the relevant case. What little evidence there is, is based on what we don't yet know, not what we do know.

bmiller said...

Don Jindra,


There is no 51 like there is no tall. But just as I can sense tall objects I can sense (and count) 51 objects.

I don't mean to catch you up in anything, but this statement doesn't seem quite right. If 51 and tall do not exist, then how can they be related to sensible objects? Is there a different way to express your meaning?

But I think I understand your position from a high level. "Everything is physical in some way" ...right?

If this is your position, then the definition of "physical" is interesting to those who disagree with you. That's why I'm interested in how you understand 51 to be "physical". I think you can tell from my questions, I'm interested in comparing and contrasting your idea of how 51 is like and unlike other objects that everyone agrees are sensible.


Glenn said...

DJ,

"it ought likewise be a no-brainer that the architect could order living room furniture..."

Does not follow.


Yes, that's right -- it does not follow. And why does it not follow? It does not follow because the house in the architect's mind, unlike the house on June street, is neither material nor sensible, and one cannot place material and sensible furniture in a house which is neither material nor sensible.

I can't fill up the photo of the house on June street with furniture. Are you going to deny the photo is material?

Your question seems to imply that your argument is that the house in the architect's mind is material because a photo taken of the material house on, say, June street is itself material.

House plans are stored in a computer these days. That's physical. What in your opinion is the fundamental difference between the brain storing the plan and the computer? I see none.

One obvious indication that there is a fundamental difference is the fact that the magnet which can be used to wipe out or scramble representations stored in the computer isn't going to wipe out or scramble representations stored in the brain.

I'm not quite following your distinction between the mental house plan and working mentally on the mental house plan.

Well, I haven't made that distinction. (Which isn't to say that it isn't a distinction worth making, only that it is a distinction which I myself have not made.)

Neuroscience is an empirically based discipline, as is artificial intelligence.

As you no doubt will not grasp, artificial intelligence is intelligence ('intelligence') which is artifical.

Btw, I never implied contents must be composed of the same material as the container.

You have made it clear that, as far as you are concerned, they are in the case of abstractions. ("I do not agree some existence is physical, some abstract. For me, all existence is physical. The abstract exists only in the patterns of our brain, which is physical.")

My position is that in the relevant case, we know of no other "stuff", and can know of no other "stuff" than the container. It's not that the materials must be the same. It's that we have little evidence they are different in the relevant case. What little evidence there is, is based on what we don't yet know, not what we do know.

There is at least some evidence based on what we do know. For example, we do know that the architect is going to have a rather difficult time purchasing business risk insurance for that house he's working on solely in his mind without external aids, and we do know that that architect cannot place material and sensible furniture in that house.

Mr. Green said...

BMiller: If the series were arranged along the x axis with the first mover at the origin how could the entire series move anywhere other than along the x axis? Again if all possible positions along the x axis were consumed by pebbles, then none of the pebbles could move.

If they extended all the way up the "positive" direction of the x-axis, and all the way down the "negative" side, then there would be no end of the series to push, so I suppose there wouldn't be any way to move them. If they started at some point and extended away from you in only one direction, i.e. if there were a first pebble, then it would presumably be possible: each pebble would move from position x to position x+1. (Of course, there might be other factors preventing this in any particular world with any particular laws of physics; but I am considering only what must be metaphysically possible, in some possible world.)


We have infinite volume consumed but there is still more volume available.

I don't see why you would object to an infinite volume inside a "bigger" infinite volume. If the universe extended infinitely in all three dimensions, isn't it obvious that there would be infinite space above and below and left and right of a line of pebbles, even an infinite line?

Sometimes things that work in mathematics cannot work in physics.

And that is precisely what I dispute: mathematics is just abstract quantity; and if a given mathematical system is an accurate and complete model of some physical system, then to say something works mathematically just is to say that it works physically (as far as the quantities are concerned; physics is different from maths because it also has qualities, but surely nobody is claiming that an infinity of abstract shapes is possible, but not an infinity of grey, smooth shapes!)

bmiller said...

Mr. Green,

If they started at some point and extended away from you in only one direction, i.e. if there were a first pebble, then it would presumably be possible: each pebble would move from position x to position x+1.

Please excuse my exchanging your "up" with my "right" and your "down" with my "left" since I'm used to thinking of the x-axis in these terms. I will be confused if I don't keep this frame of reference.

I pictured your example as an infinite series of pebbles with a first mover with the series stretching to the right and the first mover at the origin. No pebbles to the left.
Now I picture the same series, but with a pebble at the origin instead of the first mover. Infinite pebbles to the right, still no pebbles to the left, but room for a pebble to move into a vacant position on the left.

But if all positions to the right of the origin are occupied by inert pebbles, what actual position on the right would the first mover occupy? Could I put another pebble on the right side of it?

Regarding the term infinite. There are different types of infinite number series in mathematics, so it's probably not useful to talk about an infinite series of numbers until the number series is defined.

if a given mathematical system is an accurate and complete model of some physical system,

I'm guessing your background is more mathematical right? I myself am a physical system but I doubt you can mathematically predict my movements.