More on the nature of infinity
[go back to Raven’s previous argument]
Date: Thursday, July 23, 1998 12:35:21 PM
From: Fr. Ronald Tac sj
To: Greg Raven
Hello from Boston. It’s ironic that this correspondence touches on infinity. If I waited any longer with my reply, I'd be in the position of illustrating what I was trying to explain! Ah, well. In fact there has been (at least) one major reason for my delay: I can't seem to send mail out from my own computer. So I decided not to wait for BC’s technicians; instead, here I sit in the basement of Gasson Hall, hoping that this computer is not as bug-ravaged as mine. Now on to business.
You say in your reply, "The nature of infinity," that infinity "is not a number, but rather a representation of some unspecified, arbitrarily large number. Therefore, [you add,] counting to infinity or counting backwards from infinity (!) is nonsensical."
Now, Greg, I agree with you that infinity is not A number, in the sense that it is a definite so many. But that doesn't mean that it is unspecified or arbitrarily large. In fact it’s not arbitrary at all. What you mean by "infinity" is a quantity greater than any definite so many. Certain things follow from this with respect to addition, subtraction, multiplication, and division; so the notion can't be arbitrary, right?
But you're right to question whether it’s possible to count to infinity or backwards from infinity. And that’s just the point. If the universe always existed, then it has achieved, through an infinite succession of finite changes, its PRESENT state. All of these (and not merely some of these) successive motions have accumulated to this point. This means that the steps through which the universe had to go to reach this point were literally infinite in quantity. But then there were no fewer steps it must have gone through to reach last year, or the beginning of the last millennium, or one hundred trillion BC. The Kalam argument is merely pointing to a flaw, or apparent incoherence, in the atheist world-view — perhaps I should say the traditional atheist world-view (since you don't seem to share it) — that pictures the universe as matter changing endlessly in endless time. If someone holds to that view, then he’s got to hold that the universe has, so to speak, in its concrete existence, counted to infinity — in fact, has always ALREADY counted to infinity, because any point you care to take, however far distant in the past, must, if the universe always existed, have been preceded by infinitely many concrete successions. (But though the universe has, so to speak, successfully, at every concrete point in its history, already "counted to" infinity--since every present moment, on this hypothesis, is infinity-plus-one — it could never, even in an infinity of time!, reverse the process of counting to the point that there was nothing further to count! Doesn't that give you pause? Think of it like this: There’s an angel ticking off events within a beginningless universe until the birth of Greg Raven. Once Greg is born the angel stops ticking. But if the angel tried to reverse the process it would never stop ticking. This would mean that one and the same infinite quantity could be traversed — but in one direction only! Does that sound right?)
All the Kalam argument is trying to show is this: the world of space and time cannot be infinitely old and therefore must have begun. Since it must have begun its cause cannot be spatio-temporal. Since its cause cannot be spatio-temporal it cannot be material. Therefore the world must have been caused by an eternal, non-material cause. That causation was either emanation or something akin to choice. But if emanation, then the effect would have to be as eternal as the cause. But the argument has just shown that the effect (the universe) cannot be eternal. So the causation cannot be emanation. Therefore it must be something akin to choice. The causation of the universe by an eternal non-material agent is what religious people call God’s creation of the world.
Now there are questions to raise against this argument; but, really, Greg, I don't think your own objections are very damaging.
You mention the infinity of the circle (or the square: or for that matter any enclosed geometrical figure). But the circle is a finite figure (unless you want to argue that it’s "composed" of an infinity of points). True enough, one could travel in a circle forever. But notice, if you went round the circumference once, your next circuit would be covering the same ground a second time; then a third time. And this is where the Kalam is helpful. Could you have achieved this point in your making the rounds if in order to reach it you must have gone round infinitely many times before? This is what I cannot see. And this is why Hilbert and other great mathematicians have claimed that infinity is just a mathematical fiction — once you try to apply it to the concrete physical world, you end up with nonsense.
The Koch snowflake is, as you say, an example of the potential infinite. But I wonder whether you see why the notion of the potential infinite is so important. It gives us a way of speaking about a process that really is without end — but in such a way that we don't have to assume that the endless process has come to an end! If the growth of something is potentially infinite, it is concretely at every stage ACTUALLY finite. This so-called infinity merely denotes increase or decrease without end.
The Mandelbrot curve — great graphics, by the way! — is something else. Let’s assume that such a curve really exists. It would be subject to all the paradoxes that any actually infinite quantity is subject to (cf. Hilbert’s Hotel). But there is a (for you) fatal disanalogy here to the supposed infinite universe. The Mandelbrot curve is a figure and so all its parts are present in space simultaneously. But the history of the universe is something that is made by succession. And the Kalam agrument is saying: if history is made up of the successive accumulation of events, then the history of this universe, which has after all reached the present, cannot be infinite. If it were, it would be as if you said to me: Hey, I just finished drawing a Mandelbrot curve! My reply would be: No you didn't. At best you drew a finite equation; or maybe you drew an actually finite figure and affixed to it the label "Mandelbrot Curve — actually infinite." Because if you really had drawn a Mandelbrot curve you would have had to go through an infinity of steps, would have to have completed a quantitatively infinite task — and that can't be done.
Your end is intriguing. You say that maybe God did create the universe; but then again maybe he didn't. Maybe the universe "came into being because the potential was there for it to come into being." Well, I guess I agree with that, Greg. But in order for there to be potential, there has got to be something ACTUAL. The Kalam agrument is just thinking through what can and cannot be true of such an actuality. In all of this we're hampered by our human limitations; but as with all ultimate questions, we've got to make do with what we've been given and strain forward into the unknown.
Thanks for your patience, Greg. The next one won't be so long in coming.
Ron Tac sj
read Raven’s response