Monday, August 29, 2011

Actual infinity

I am occasionally asked what I think about actual infinities.  Mainly because of a Grim Reaper argument, but also because of the bob argument here, I do think there is a problem about certain kinds of actual infinities.  But not all of them.  The Grim Reaper argument, as well as the bob argument, only tells against one kind of actual infinity: an actual infinity of causal influences on a single event.  So while one could conclude that actual infinities are impossible, a more conservative move is to conclude that it is not possible for an infinite number of causal influences to be in the past history of any event.

This limited anti-infinitism means that I do not need to worry about arguments in favor of actual infinities such as the following:
  • The actual existence of an infinite future.
  • The actual existence of mathematical infinities.
  • The intuitively very plausible possibility of a simultaneous infinity of objects.  At least, it's plausible to me.
In fact, my worry is not so much about an actual infinite as such, but about infinitely many causal influences coming together.  

Initially, when I came to the Grim Reaper argument against an infinite past, I found the anti-infinitist conclusion very counterintuitive.  But now it has become clear to me that it's not: for it's not that counterintuitive to think that absurdities can result if an infinite number of causal influences can work together.  My pro-infinitist intuitions were based on non-causal mathematical considerations.  But I can, if I wish, retain those intuitions.  

In fact, I can even retain the intuition that there could have been an infinite past, as long as this does not imply a backwards-infinite causal chain.  Consider, for instance, a world that consists of a multiverse of universes.  The first universe is one year old.  The second universe is two years old.  The third is three years old.  And so on.  The world as a whole has an infinite past.  But as long as there isn't the wrong kind of causal dependence between the universes, such an arrangement need not imply the kind of infinity in causal influence that my arguments lead me to think is problematic.

This strengthens the Kalaam argument by showing that the premises can be weakened: the Kalaam argument only needs the kind of causal anti-infinitism that I now cautiously accept.


Leonhard said...

Doesn't your Bob and rabbit paradox assume processes that are asymmetrical in time? If its possible that whatever Bob decays into can (though it might be extremely improbable) turn into another Bob again then what's the problem? As far as I know it there aren't any time-irreversible processes known to Science. So for an infinitely old universe, any Bob-chain would eventually die because as long as there's a finite probability for Bob to choose to not spawn, its then inevitable that this will obtain eventually. However given enough time quantum-bob-tunneling will start a new Bob-chain that can propagate. The density of Bob chains at any given time will depend on the processes involved when Bob dies and the history of the cosmos in which Bob-chains propagate. If quantum-tunneling upsets your stomach, then we can imagine a biological species that evolves to be like this. Arguably it wouldn't propagate for long. If the universe is cyclical, and/or fecund, and/or inflationary then even though the Bob-chain dies out its possible to find new environments where chance events will produce another Bob. Likewise if we go back far enough in time, we'll find the start of each Bob chain.

I think something similar would go for your rabbits. We can follow trees of ancestry back to a single breeding pair, and given no one (in the immediate area?) they are either produced by the girl, or we'd follow the tree back to the common ancestor between the rabbit and the hare. Like in our universe. In either case as time goes on, all the rabbits die in the heat death of the universe. And depending on the mechanics of the universe there's either parallel worlds (inside black holes, or in inflationary space) that will provide conditions for rabbits to exist, or the universe might somehow refreshen itself like has been suggested, most recently by Penrose, so that eventually a state where rabbits arrive (via magic-girl, god or evolution).

I'm not sure that the word "Always" makes sense in an inflationary cosmology. This universe, it seems, will inevitably die. Other universes in either the fecund or inflationary models aren't "parallel" to ours. What does would the word "Always" mean if time is more like a tree, than a line?

Alexander R Pruss said...

I don't think we should assume that all processes are metaphysically necessarily time-reversal symmetric. In fact, the processes in our universe aren't time-reversal symmetric. They are charge-parity-time reversal symmetric. It would be weird to think that CPT symmetry is metaphysically necessary!

It seems that with a branching time, a reasonable sense of "always" is "along all future branches".

David J. Houston said...

I've been wrestling through the issue of God's relationship to time and this argument - that depends substantially on the idea of actual infinities - has given me some trouble:

(B1) If God is timeless then the B-Theory of time is correct.
This seems to follow pretty easily since if God is timeless he cannot experience temporal becoming which entails that all of his acts that we experience in time were performed in one divine eternal action and the only theory of time that can support such a view is the B-Theory.
(B2) If the B-Theory is correct then reality is a space/time block that neither shrinks nor expands.
The B-Theory is often said to present time as another dimension to reality so that all reality would be a space-time block. If the space/time block were to shrink or grow it would entail either the ‘shrinking block ‘theory or the ‘growing block’ theory and neither account is compatible with the B-Theory. At this point a problem arises for Christian theists since,
(B3) Christianity teaches that the ‘eternal state’2 will consist in an everlasting succession of events.
The Bible seems to indicate that this is true. Some have been willing to say that Hell is not everlasting but few (if any) have been willing to say that Heaven is not. In any case,
(B4) If God created an everlasting succession of events this entails that either God actualized either an actually infinite number of events or a potentially infinite number of events.
(B5) God cannot actualize an actually infinite number of events since this is logically impossible.
The truth of (B5) was illustrated by the ‘Hilbert’s Hotel’ example earlier. (B5) rules out the first disjunction from (B4) and it follows that God actualized a potentially infinite number of events but,
(B6) A potentially infinite number of events would require the space/time block to grow everlastingly but this is incompatible with the B-Theory.

At this point we must choose between rejecting (B3), which amounts to accepting a finite view of time, or accepting it thereby rendering the consequent of (B1) false and by modus tollens we would have to conclude that
(B7) God cannot be timeless.

My belief in classical theism leads me to reject (B7) but I don't know how to stop the argument from going through. I think that my problem might be with my limited understanding of actual infinities. How would you attack it?

David J. Houston said...

PS: I just lifted this out of a paper I was working on so forgive my reference to the Hilbert's hotel illustration that I didn't include. I'm sure you are familiar with it!

Alexander R Pruss said...


Why not deny B5? I don't see any logical contradiction in the Hilbert's Hotel story, nor even any metaphysical impossibility.

Heath White said...


Here's another reason to doubt B5: "events" is ill-defined. If you slice it finely enough, any finite period of time already contains an infinite number of events. And obviously God can create that.

Alexander R Pruss said...


I think that folks who don't like actual infinities deny that time is infinitely subdivided. It's either discrete or merely infinitely subdivisible.

Heath White said...

I am willing to be corrected, as I am out of my area of expertise here, but:

From 10:00am to 11:00am this morning, how many events were there? And what is the argument (other than arguments for the impossibility of an actual infinite, which would be begging the question) that there were not an infinite number of events in this stretch of time?

David J. Houston said...

So my mistake was accepting B5... do you think a guy like Bill Craig would accept it? Looking at it now I'm wondering if I found the HIlbert's hotel example persuasive not because of a genuine contradiction but because of its weirdness! I'll have to spend some more time reading up on actual infinities. Could you recommend a resource or two?

Anonymous said...

I'm not sure why the Grim Reaper examples simply don't show that that particular combination of scenarios is impossible. At any rate, I believe that an infinite series of causes is possible, although ultimately you need to ground it in something outside the series (God) — but maybe you would count that as "short circuiting" the causal chain, and thus not really infinite in the relevant way.

(Consider the infinite series of mirrors: I don't have a problem supposing that the image could be reflected an infinite number times, but there still has to be an original object to get reflected an infinite number of times.)

Alexander R Pruss said...


Well, the Grim Reaper scenarios play off an intuition that certain kinds of recombinations are possible. Rob Koons has a paper developing the point better than I've been able to.

As for the mirrors, I think that's a forwards chain--there is nothing that has an infinite number of causes there.


Well, the Grim Reaper cases might move you there--you might think that what's wrong with the GR scenario is that it assumes that there are infinitely many times between noon and 1pm, or something like that.

Thomas Henry Larsen said...

David, you are working on the assumption, of course, that the B-Theory of time is correct. I'm just not familiar enough with the arguments for and against it, but it's something to think about.

Alex said...

Hello Mr. Pruss. My question is, can the grim reaper paradox show that an infinite causal series which is simultaneous,i.e where the cause and effect are simultaneous with each other, is impossible? I understand you take the paradox to prohibit an infinite causal series which extends through time. Since that wouldnt be the case in a infinite simultaneous chain, would the paradox affect its actual possibility?

Alexander R Pruss said...

I think what is ruled out is an infinite number of causes for one effect. So if the simultaneous infinity of entities could all conspire to have a single effect, that's ruled out. But if not, it's not ruled out.

mjm said...

The idea of an "actual physical infinite" is incoherent.
If an acutal physical infinite is possible, then there could be an infinitely long road along which
two cars A and B have been driving, A at 60 mph, B at 120 mph, forever, throughout an infinite past. A more realistic scenario might be two particles A and B, each in uniform rectilinear motion on parallel paths, B at twice the speed of A, throughout an infinite past. Each must be at some point on the road (path). Any two points on the road are at a finite distance apart. Cars A and B cannot be at the same point P. That is where they would be if A had been 60 miles from P one hour ago, and B had been 120 miles from P one hour ago. But any argument that would put them at the same point now would put them at the same point then. And if they were at the same point then, they cannot be at the same point now.

B must have covered twice the distance since then. The point where A is cannot be less than the point where B is, nor can it be greater. For each point, if B were at any such point, there would be some *finite* time and distance such that B would have been separated from A by that distance at that time, after driving 120 mph for that *finite* time. Car B cannot be "at infinity" because that is not a point on the infinite line. It is to be nowhere at all. If, per impossible, B were "at infinity", A would be as well, and so neither would be at any point on the line (i.e. each would be nowhere).

Whatever "being at infinity" could mean, it must mean at least that A and B are not at any points on
the line. Therefore B cannot have been traveling at twice the speed of A for an infinite time along an
infinite path. If the actual physical infinite is possible, that scenario would be possible. Therefore there could not be an actual physical infinite.

Here, "traveling" (motion through space and time) is the critical element whose addition to the pure
mathematics of infinity leads to contradiction. "Traveling" is not to be found in the pure mathematics of infinity.

The assumptions of Newtonian space and time and rectilinear paths are for simplicity of argument only,
and are assumptions normally admitted in any case by the disputants over the matter of actual physical infinity. If actual physical infinity is possible, it is possible in Newtonian space and time. Also,
if it is not possible in Newtonian space and time, then the defense of actual physical infinity has an
a priori argument against the possibility of Euclidean space and Newtonian space and time, which is of a kind assumed impossible by most parties to the defense.