Saturday, April 29, 2006

The Kalam Argument, Zeno's Paradoxes, and Omniscience

In my previous post on this topic I argued (a) that Craig's argument against the possibility of actually infinite collections of real things doesn't work given presentism, and (b) Craig's argument against the possibility of traversing an actually infinite series in finite, step-wise fashion doesn't work given a B-theory of time.

My post generated quite a discussion among three of my commentators (Tom, HammsBear, and Don Jr). In this follow-up post I'd like to address a couple of the issues they raise.

1. The first issue concerns (a). Both HammsBear and Tom suggest that, given my proposal that presentism works best when combined with a version of theism in which God's memories provide truthmakers for truths about the past, Craig's first argument would work against a presentist who held that time had no beginning for that would mean that God has an actually infinite number of memories of past events.

Frankly, I don't see this as a big issue. It seems to fare no better or worse than another challenge to Craig's argument, namely, that an omniscience God would have to know an actually infinite number of propositions (e.g., 1+1=2, 1+2=3, etc.). Hence, either Craig's argument works, in which case it implies that God cannot be omniscient, or we have to conclude that the argument doesn't work. But this seemingly nasty dilemma has, I think, a straightforward reply: God's knowledge is a single unified gestalt - in one cognitive act he grasps all there is to know about all there is to know. Thus, we shouldn't think of God's knowledge as built-up piecemeal from atomic propositions but rather as a continuous field that contains all true propositions virtually, from which particular truths may be distinguished by abstraction. In much the same way, a continuous geometrical plane is not built-up out of discrete points but is a field within which endless numbers of points may be picked out by abstraction. This is not a new proposal, by the way, but the classical way of thinking about God's omniscience.

2. The second issue concerns (b) by way of Zeno's paradoxes of motion. Do these paradoxes imply that there is, say, an actually infinite number of spatial points between any two locations, or an actually infinite number of events between any two times? If so, then any sort of change or motion would require traversing an actually infinite collection in stepwise fashion.

One response (proposed by HammsBear) is finitism - space and time come in finite indivisible quanta. This view (defended by A.N. Whitehead) deals nicely with Zeno's paradoxes - if correct, then between any two places there can only be a finite number of places and between any two times there can only be a finite number of events. On the other hand, finitism is very counter-intuitive. First, even if Planck time shows that there is a physically minimal quantum of time, it is metaphysically possible that that quantum be smaller, indefinitely. So in some other possible world the quantum is smaller, and so forth for any non-infinitesimal quantum. What, then, makes the quantum the size it is? Second, imagine two quantum-sized particles travelling in a parallel line in the same direction, with the first going exactly twice as fast as the second. Suppose they both start at (t0,x0). Clearly, when the first particle is at (t4,x4), the second one will be at (t2,x2), but where is the second particle when the first is at (t3,x3)? It seems that it would have to be between (t1,x1) and (t2,x2). By hypothesis, however, no such location exists. That's weird.

A second response to Zeno's paradoxes, and the one I favor, is the one proposed by Aristotle. Basically, Aristotle introduced a distinction between actual and potential infinites (the same distinction Craig uses in the kalam argument). The number of points between any two places, said Aristotle, is potentially infinite in that it is endlessly divisible, but not actually infinite. In other words, continuity is metaphysically prior to discontinuity - discontinuities can only exist in a more fundamental continuity. Here's a mathematical analogy: What is more basic, the line or the points on the line? Aristotle would say the line. Points have zero dimension, so you can string them together end to end and never build up a line of any length whatsoever. You could, of course, pick a point and drag it, thereby defining a line segment, but to drag it in the first place there has to be some kind of continuous field in which to drag it.

Of course, this doesn't remove all the perplexity behind Zeno's paradoxes, but either approach, Aristotle's or the finitist's, would give us a way to avoid countenancing an actual infinity of spatial places or temporal events.


At 4/30/2006 10:48 AM, Blogger Don Jr. said...

Very nice post, Alan. I thought I had things clear (in my own head), but this clears things up a whole lot more.

Although it wasn't specifically addressed in your post, one thing that perplexes me about people using Zeno's paradox to argue for the possibility of traversing an infinite, besides the fact that it begs the question, is that Zeno himself didn't argue that way. Zeno basically argued: There is an infinite amount of points between any two points; it is not possible to traverse an infinite; therefore, motion is illusory. Others, though, append the first premise (which I say is faulty) with it is possible to traverse the space between two points; therefore, it is possible to traverse an infinite. The problem with all this is that it presupposes that points actually exist, or as you put it, that the points are more basic than the line.

At 5/01/2006 1:57 AM, Blogger Ocham said...

I'm still struggling with the premiss of this argument. I found the following reference:

"The Traversal of the Infinite. An argument for the finitude of the world's past which originated with Philopinus (490?-575?). An infinite series cannot be completed (the infinite cannot be traversed). But if the world were infinite in past time, then 'up to every moment an eternity has elapsed' (Kant) and thus an infinite sequence would have been completed. Therefore the world is finite in past time. This argument has been offered by, among others, al-Ghazali, St Bonaventura, and Kant. It was, however, decisively refuted by Aquinas and, somewhat more subtly, by Ockham. Aquinas pointed out that traversal requires two termini: a beginning and an end. But any past time which could count as a beginning is only a finite time ago. Consequently we do not, in the required sense, have a traversed infinity." (J. MckIntosh, U of Calgary).

See also Kretzmann 'Ockham and the Creation of the Beginningless World', Franciscan Studies, 1985.

At 5/01/2006 12:38 PM, Blogger HammsBear said...

Alan, Tom, ocham, don jr, the articles you guys have been linking have all been good reads.

At 5/01/2006 3:08 PM, Blogger Tom said...

I'm in a freefall. I don't see at all how Alan's (Craig's/Alston's) comments save omniscience. But that might be because I'm not sure I even understand them. I keep reading this same argument by different people and approaching it from different angles, but I just don't see that it works.

More later,

At 5/01/2006 4:11 PM, Blogger Alan Rhoda said...

Thanks, Don. I agree with you concerning the relation of Zeno's paradoxes to the kalam argument.

Occam, Aquinas didn't come close to refuting the "traversal of the infinite" argument. On the contrary, the fact that traversal requires two termini supports the kalam argument. It's the absence of a beginning terminus given an infinitely old universe that creates the problems, exactly as the kalam arguer contends.

Tom, hang in there, bro. I understand your concerns. There's no easy answer, but the goes for all really fundamental questions. We have competing intuitions about how to model time, space, omniscience, and so forth that seem to get us stuck in perplexities no matter how we answer. The trick is to strike a balance between our conceptual models and the reality they are supposed to model. Cantorian set theory, infinitesimals, modal logic and so forth are neat conceptual tools, but they are just that - tools. We should neither fear nor worship them. And we should leave open the option of dropping, replacing, or upgrading them when they no longer prove useful for the job. Omniscience is very tricky to articulate without running into paradoxes, that's for sure. But denying (the possibility of) omniscience isn't an attractive option either. I say the better option is to retool as best we can, while realizing that our finitude is a fact we can't escape. At some point or other everyone has to punt to mystery. That may not be entirely satisfying, I grant, but it's the truth and we need to learn to live with it.

At 5/02/2006 12:06 AM, Blogger Ocham said...

>>the fact that traversal requires two termini supports the kalam argument. >>

Baffled as always. What is the argument meant to be? If a traversal requires a start and end point, there can be no traversal.

Again, what was the original argument that this was about? I've searched up the thread, but no success. Thanks.

At 5/02/2006 12:15 AM, Blogger Ocham said...

I looked at another argument earlier, and it reminds me of Cantor's point that all arguments against the actual infinite involve a petitio. They begin by assuming that number has certain properties, which include properties intrinsic to finite numbers, then 'prove' that all numbers are finite. E.g.

"To suppose, then, that the actually infinite collection of past events has been traversed in step-wise fashion is to suppose that it's possible to get from -∞ to 0 one number at a time. But that's impossible, since -∞+1 = -∞. Hence, one could never arrive at 0 or the present moment."

To suppose that every series can be traversed step wise is to suppose that every series is finite. Ergo there is no actual infinity, on the assumption that every series can be traversed step wise. Petitio!

At 5/02/2006 12:29 PM, Blogger Brandon said...

Like Ocham, I'm baffled by the claim that Aquinas doesn't refute the traversal argument. If every traversal requires a beginning and an end, and an infinite past has no beginning, this is a problem only if we already assume that traversal of an infinite past would require traversal of infinite days. But on the infinite past view, every day in the past is finitely distant from the present; it's just that for every finitely distant day there's a day that is more distant. Thus this is true: For every day in the past, traversal of the days from that day to today is traversal of a finite number of days. The fact that there are infinite such days doesn't change this. This is true just as much as it is true that the fact that every integer is a finite distant from 1 is not affected by the fact that there are infinite integers.

At 5/02/2006 2:49 PM, Blogger Ocham said...

Agree with Brandon. In addition, Cantor's positing of the existence of infinite integers involves positing 'limit ordinals' A limit ordinal o is such that for every number x that is 'after it' there exists a number y that is between x and o. Thus it is easily shown that there is no way of 'stepping' from the limit ordinal to any point in the infinite series 'after' it. To any such point there would have to be, by definition, a point between it and the limit. Hence no 'step' is possible: that's what a limit ordinal is, and the definition of infinite integers involves the position of limit ordinals: numbers such that you cannot step from any of them directly into the finite series.

IN summary: a point in the past that is such a limit ordinal, is such that we could not, by definition, 'get' from there to now in step-wise fashion.

And a further point, as Brandon correctly notes, there can be infinitely many points which lie a finite distance from the present, without there being any 'limit' ordinal. That traditionally is the difference between a potential infinity (infinitely many, but no object acting as limit), and actual (at least one limit).

At 5/05/2006 1:44 AM, Blogger Ocham said...

I think there is something to the objector's argument here. Their point as they have expressed it is wrong, I think, but there is something I think they are trying to say which I have commented on at Brandon's place. He has left a link.

At 7/19/2006 10:25 PM, Blogger SMG said...

There are two things I would like to contribute to this discussion. First, Aquinas' statement that the concept of "traversal" entails a starting point, or two termini, is false. Precisely put, to traverse some thing x means to cross over or pass through every proper part of x. So understood, the concept of traversal does not imply a starting point. Even Richard Sorabji, in his book Time, Creation, and the Continuum, has acknowledged this.

Second, William Lane Craig has responded to the charge that a pure A-theory (i.e., presentism) nullifies the argument from the impossibilty of an actual infinite when it is applied to the series of past events. See his "A Swift and Simple Refutation of the Kalam Cosmological Argument?" (available at I believe Craig also discussed the issue in his book, The Kalam Cosmological Argument.

At 7/24/2006 12:53 PM, Blogger SMG said...

Ah, here is an article that deals directly with the question of whether the A-Theory invalidates the argument against the actual infinite when it is applied to the series of past events:


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