Reply to William Lane Craig

Two readers of this blog have pointed out this post at William Lane Craig's blog. In the post, he responds to a question about my debate with Kirk Durston. Craig says I exhibit "ignorance on parade".

Well, there's a lot of ignorance to go around. My debate was with Durston, not with Craig. I was responding to Durston's claim (made at 05:37) that "Mathematics dictates that time itself would have had to have a beginning at some point in the past." In the debate that Durston took part in just a few days earlier at McMaster University, he claimed that Hilbert, in his 1925 paper, "On the Infinite" had proved mathematically that there could not be an infinite regress of causes." But this is not true. All Hilbert did in that paper was claim that then-current consensus about the physical universe was that no infinite quantities existed in it. That's a far cry from any kind of mathematical proof. William Lane Craig, like anyone else, can go read Hilbert's paper and verify that this is the case.

I pointed out that in fact, there is nothing mathematical that rules out an infinite regress of causes. For example, you could have an event at time -(n+1) causing an event at time -(n) for all positive integers n. Thus, an event at time -2 causes an event at time -1, an event at time -3 causes an event at time -2, etc. There is nothing logical or mathematical to rule this out. You can even have an infinite regress of causes if time has a beginning. If time begins at time 0, then you can have an event at time 1/(n+1) causing an event at time 1/n for all positive integers n. Thus, for example, an event at time 1/3 causes an event at time 1/2, an event at time 1/4 causes an event at time 1/3, etc. Again, nothing logical or mathematical rules this out.

Now you might say that once we bring our current state of physical knowledge into the picture, the first scenario is ruled out. But even modern physicists consider the possibility of infinite time-like curves that occur in the past of some other point; for example, in their study of Malament-Hogarth spacetime. Thus, I would contend that apologists like Durston and Craig have a really naive view of spacetime, one that is essentially based on the understanding of 100 years ago, not modern physics.

When I called Durston on this at the debate, his response was really comical. Here it is as I have transcribed it, beginning at 1:06:48:

"First, regarding Hilbert. He [Shallit] pulled a mathematical trick
there. Those of you who are used to summing infinite series
will know that if the x decreases exponentially, it comes to a
finite value. So let me explain how this really works.

Let's assume... now I don't know whether he's saying that.
Has he dodged the issue here, as to whether or not the past is
infinite or not? So let's assume the past is infinite. So
let's call this debate time 0, this hour here of the debate is
time 0. The next hour after this will be time 0+1, time 2, and
so forth. And in the past, we'll go to, the last hour before
this debate will be negative 1 hour, hour negative 2, and so forth.
Now if you want to assume, and this is to illustrate why there's
a problem of traversing an actual infinite series in
reality. Let's say that each step in the series is one hour
long. Now what he seems to be arguing, or what he's insisting
here, I'm not sure, is that Hilbert, that the past can be
infinite, that is, there's an actual hour infinitely separated
from this one here. So let's call that -infinity. We'll never
get there by getting in a time machine and going back, so let's
just take a quantum leap back into the past, we're now at minus
infinity. Now some of those of you who are familiar with
infinite set theory might be a little uncomfortable at this
point because if the past is, if you're saying it actually is
infinite, what you mean is that there actually is an hour back
there that is infinitely separated from this one. So let's
count our way down now. Infinity minus 1, infinity minus 2.
It's one hour each, not an exponentially decreasing amount of
time like that little equation he put up there, but just a
steady hour each time.

Or you could go with a multi-universe, this universe is a
product of another universe, and we're working our way down
from -infinity to the present. At what point in time will you
arrive at 0? You will never traverse an infinite series in
reality if you must stop at a discrete amount of time for a
constant amount of time in between. And that pretty much lays
to rest this notion that time itself can be infinite as far as
the past goes. It can be potentially infinite. you can do
lots of mathematical things, I can hold my hands together and
say there's an infinite number of mathematical points, no
problem, those are imaginary. But the moment you have a
discrete amount, that occupies a discrete amount of time, like
a minute, or a second, or an hour, and it is not decreasing
then suddenly you have a problem, if you want to actually
traverse that infinite series in reality."


After Durston's casual slur on my character at the beginning, this is completely incoherent. My equation was not "exponential", so that criticism is nonsensical. Secondly, it is perfectly possible to have an infinite past without having any point at infinite distance from the current time; for example, we could define times -1, -2, -3, etc., without having to define an actual point called "-infinity". (In exactly the same way, the negative integers are an infinite set that does not contain an integer called "-infinity".) This is not exactly a controversial point, but it is a misconception common to undergraduates. Mr. Durston, a graduate student, should know better.

Craig doesn't seem to understand what the debate was about. He says, "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!", thereby entirely missing the point. Hilbert's claim was about the 1925 understanding of physical existence, not mathematical existence. And anyway, views of mathematical existence do change through time. Consider, for example, the views of people like Brouwer.

Craig says, "On Shallit's view the universe still came into being a finite time ago and therefore requires an external cause." No, I didn't say that at all, and I don't hold that. In my second example, I said that "you can have an event at time 1/(n+1) causing an event at time 1/(n) for all positive integers n". This doesn't say anything about time 0, and it is logically possible to have an infinite chain of causes stretching back in this way, with nothing happening at time 0 at all - an uncaused beginning.

In general, Craig seems to have an extremely naive, almost childish view of infinity. Read Craig's reply to Sobel. On page 9 he says, "Imagine an actually infinite regress of past causes terminating in the present effect. In this case, the regress of causes terminating, say, yesterday, or, for that matter, at any day in the infinite past, has exactly the same number of causes as the regress terminating in the present. This seems absurd, since the entire regress contains all the same causes as any selected partial regress plus an arbitrarily large number of additional causes as well. Or again, if we number the causes, there will be as many odd-numbered causes as there are causes, which seems absurd, since there are an equally infinite number of even-numbered causes in the series in addition to the self-same odd-numbered causes."

It seems that what bothers Craig is perfectly understandable to any mathematician: namely, that the set of positive integers has the same cardinality as the set of integers greater than n (for any positive n), and the same cardinality as the set of even positive integers. All this was well understood 125 years ago, but it seems the Christian apologists haven't caught up.

Altogether, I would say these arguments by Durston and Craig are embarrassingly naive.