April 27, 2008

Hilbert and Kalam

In his debate with Kirk Durston, atheist Jeffrey Shallit claimed (during his first rebuttal) that Hilbert never said an infinite regress of causes was mathematically impossible. He then picked up a book by Hilbert and said,

I have this paper by Hilbert you (Durston) refer to here: David Hilbert, "On the Infinite." You can come up and look at it afterwards. There's no proof in there. Sorry. You've been fooled by William Lane Craig who's not exactly the most reliable source. He's a Christian apologist. There's no proof in here. This article was written in 1925. What Hilbert was talking about was the *then* understanding of the physical universe. The *then* current '1925-82-years-ago' understanding of the universe. He said in *that* understanding there's no infinite quantities in the universe. But this is 2007! We've learned a heck of a lot about the universe since then. And I'm not a physicist. But my physicist colleagues assure me that there are very respectable physical theories in which in fact there are infinite quantities in nature. So, uh, let's see if I can find the slide for that. Yeah, so, here it is. In fact, I would assert that there is no logical reason to rule out an infinite regress of causes. We could have a singularity times zero. And an event at time one over 'n' plus one, causing an event at time one over 'n' for all 'n.' So, we have an event at time of 4th that causes an event at time of 3rd. And an event at time of 5th that causes that event at time of 4th and so forth. You get an infinite regress of causes. This is actually very similar to some of the claims made about the singularity of the big bang. That there are an infinite number of states after the singularity of the big bang. So, in fact, there's no logical reason and please don't believe him when he says Hilbert proved it. He didn't do any such thing.

Unfortunately, Durston was given a limited amount of time during his conclusion to rebut Shallit (although, I believe, Durston still won the debate), but I figured since Shallit made a snide and disingenuous remark towards you, it would only be fair that you respond. So, was Shallit right about Hilbert? And given Shallit's argument would an infinite regress be possible in the universe?



I don't know Jeffrey Shallit, but I'm afraid that this is ignorance on parade. I have not at any time made the claim that Hilbert offered a proof that an infinite regress of causes is mathematically impossible. Rather I cite Hilbert as an example of a great mathematician who, though enthusiastic about the mathematical existence of the infinite, denied that the actual infinite exists in reality. What the example of Hilbert shows is that one need not restrict classical mathematics in order to deny that the actual infinite exists in the mind-independent world.

What's really peculiar is Shallit's "that was then, but this is now" move—as though views of mathematical existence are tied to the times! The use of infinitary mathematics in scientific theories doesn't commit us to the existence of an actually infinite number of things. For example, we can model spacetime as an uncountable infinity of points, but that doesn't imply that points actually exist.

Now consider the example Shallit gives. If I understand him correctly, he imagines the initial singularity at some time t = 0 and then imagines a series of fractions converging toward 0 as a limit. For example, we could imagine the first second of time to be divided into intervals: . . . , 1/8, 1/4, 1/2.

Now what is the import of this exercise supposed to be? I'm not sure what he takes it to prove. The universe on this view is still finite in the past. On Shallit's view the universe still came into being a finite time ago and therefore requires an external cause.

Is this supposed to be an argument for the existence of an actually infinite number of things? But then why should we regard the series of intervals converging toward zero as anything more than a mathematical idealization? Such a series of intervals can be plausibly regarded as potentially, not actually, infinite, as the subdividing goes on without limit.

Does it show that if we "cut out" the point 0, we would then have no beginning to the universe? No. Time begins to exist if and only if for any finite interval of time you pick there are only a finite number of equal intervals earlier than it. Having a beginning does not entail having a beginning point.

Is it supposed to show that you could traverse an infinite regress of causes? But the kalam argument against traversing an actual infinite concerns a series composed of intervals which are of equal duration, not progressively shorter duration. So what is the point supposed to be?

For similar speculations on such convergent series, see my "J. Howard Sobel on the Kalam Cosmological Argument" under Scholarly Articles: Existence of God.