May 18, 2009

Pruss on Forming an Actual Infinite by Successive Addition

Alexander Pruss has suggested a counterexample to a priori arguments for the finite age of the universe on his blog.

An Alpha-widget (AW) spends a year playing the violin, and then make a copy of themselves. Each AW takes only half the time to make a copy of itself that was spent in it's own creation. So each generation of AWs produces the next in half the time it took for it's own creation. It is possible for an alpha-widget to be made over the period of a year. So there would be a potentially infinite but never completed sequence of alpha-widgets coming into existence: the first at the beginning of year zero, the second at year 1.5, the third at year 2.75. this seems to be logically possible.

However if Alpha Widgets are possible so are Beta Widgets (BW). A BW does the same thing as an AW but in opposite order: it first makes the almost-duplicate in half the time it took to be made itself, and then plays the violin for a year. if a beta-widget is made in a year, then half a year later, it makes another beta-widget. That one, then, makes another in a quarter of a year. And so on. By the time two years are up, we have an infinite number of beta-widgets produced by successive addition. Or so Dr Pruss claims. (BWs aren't producing potential infinites, so far as I can see).

I can't see any reason for denying that a Beta Widget is possible - but a real actual infinite seems absurd.

Now is it simply better to claim that actual infinities are not realizable in space-time? But that one may exist in thought (God could count up to one, or hold an infinite number of thoughts in his head). For it does seem that it is the physicality of Hilbert's Hotel that jars us.

I'm confused. Dr Pruss and you both seem to be correct - but you both can't be. Please help, before I start believing in Kantian antinomies.


When I first read your question, Graham, I thought to myself, “Surely, he’s confused! This can’t be an accurate rendering of Pruss’s argument.” So I emailed Alex to obtain the original, and to my surprise I found that you have, indeed, faithfully rendered his objection!

It seems to me that Alex’s argument is just a very confusing and needlessly complex elaboration of a simple objection already made by detractors of the kalam cosmological argument.

The Alpha widget thought experiment is terribly confusing, not simply because of its superfluous details about violin playing and so forth, but because as described it just makes no sense. (If you doubt this, just try to draw a timeline of what is described.) In the first place, the story requires that there be a year zero. But in our calendar system we move from 1 B.C to A.D. 1, and there just is no year zero. So we might think that by “the beginning of year zero,” Alex is speaking hypothetically of a timeline which does feature a year zero, at the beginning of which the first Alpha widget comes into being. But then we’re told that an Alpha widget has the property of making another widget in half the time in which it was made. That immediately raises the question as to how much time it took to make the first Alpha widget. Well, Alex says the second one was made during the interval 1.0 to 1.5. Since that’s half the time it took to make the first widget, the first one must have been made in a year’s time. That implies that the first widget must have been created between -1.0 and the beginning of year 0. Then all the days of year 0 it played around until December 31 of year 0. But then what happened? It’s supposed to create another widget, right? But it didn’t! Instead it played around for another year from 0 to 1.0 before starting to create, which contradicts the story conditions.

That might make us think that Alex doesn’t, in fact, envision a year 0 but thinks that God created the first widget instantaneously at the beginning of year 1, so that the point designated by 0, at which the first widget came into being, is the first point of year 1 and perhaps even the beginning of time itself. Then that widget played around for a year before beginning to create at 1.0. But then, since the first widget came into being instantaneously, the second should be created in half that time, which, if this makes any sense at all, must mean instantly at 1.0, not 1.5, as the story conditions require.

The only sense I can make of this story is that the first widget was made between 0 and 1.0, so that it comes to exist at the end of year 1. Then between 1.0 and 2.0 it did nothing. Then between 2.0 and 2.5 it created a second widget. This one waits around until 3.5, and then it creates a third widget between 3.5 and 3.75. And so on.

So the idea is that we have a series of intervals, each separated by a year, converging toward zero as a limit: 1, 1/2, 1/4, 1/8, 1/16, . . . . It’s the constant one year intervals separating these progressively shrinking intervals that’s supposed to make this sort of series unobjectionable, for it can never be completed. I’m not sure that this feature of the story does serve to render the scenario metaphysically possible; but never mind: the question at issue is whether the possibility of such a sequence implies that the Beta widget scenario is metaphysically possible.

Now the Beta widget scenario, which is also confusingly described, basically removes the one year intervals separating the creation periods. The first widget comes to exist at 1.0, the second comes to exist at 1.5, the third at 1.75, and so on. But then we see that, stripped of its window dressing, this scenario is just the old supertask objection all over again. The first task takes one minute to perform, the second takes a half-minute, the third a quarter minute, and so on. At the end of two minutes an infinite supertask has been magically completed. There’s nothing new here.

Nor is there any reason to think that the metaphysical possibility of the Alpha scenario implies the possibility of the Beta scenario. Indeed, removing the one year intervals plausibly makes all the difference in the world!

Perhaps the intuitive force of Pruss’s illustration is that Alpha and Beta widgets both seem to be essentially the same sort of machine: a machine that makes a successor in half the time it was made. Such a machine seems possible in the one case but not in the other; but surely, the metaphysical possibility of such a machine doesn’t depend on the contingent fact of whether it waits around for a year before starting to make a successor! It seems to me, however, that so to reason is to forget that it is the metaphysical possibility of the whole scenario that is at stake, not some isolated fact about a machine. And then the difference of the one year intervals does make the scenarios quite disanalogous.

You should realize that Alexander Pruss has a mind that is never at rest and is constantly toying with new arguments. He likes to throw them up against the wall and see if they stick. You’ll notice that he himself doesn’t say that he finds this proffered argument convincing. Indeed, if you’ll look at his exchange with Wes Morriston at http://prosblogion.ektopos.com/archives/2009/03/how-god-could-c.html you’ll find a stimulating defense on Pruss’s part of kalam arguments against forming an actual infinite by successive addition. For example, he says,

why can't Craig say this:

(*) Any stretch of the process starting at the beginning but ending short of the two hour mark is possible, but the process as a whole is not possible.

There are, after all, processes satisfying (*). For instance, consider the following process. During the first hour, God says: ‘I promise that next year I will create finitely many horses, at least one in number.’ During the next half hour, God says: ‘I promise that next year I will create finitely many horses, at least two in number.’ During the next 15 minutes, God says: ‘I promise that next year I will create finitely many horses, at least three in number.’ And so on. The process truncated at any length of time short of 2 hours is possible. But the process going for the whole two hours is impossible, since then God would have made an incompossible set of promises (to satisfy all the promises, he’d have to create a finite number of horses which exceeds every integer), and an essentially morally upright God can’t do that.

He also observes that Morriston’s sort of objections seems to make one vulnerable to so-called Grim Reaper Paradoxes (see brief discussion in my and Jim Sinclair’s article in the new Blackwell Companion to Natural Theology.) I think Pruss’s mind is still unsettled on this whole question. (In fact, as this Answer was about to go online, I received the following email from him: “Bill, On the other hand, I just posted an argument on the other side: http://alexanderpruss.blogspot.com/2009/05/is-time-continuum.html.”)

Finally, I want to add that this sort of objection is in a sense quite besides the point with respect to the possibility of the sequential formation of an infinite past. For these so-called supertasks all envision tasks featuring infinite progressions of ever shorter intervals. But the formation of an infinite past by successive addition involves the completion of tasks featuring an infinite number of intervals of equal duration. Achilles may be able to cross the stadium, but how he can traverse an actually infinite past remains, to my mind, deeply problematic.