Some of you may remember the commenter 'spur' from the old Powerblogs incarnation of this weblog. His comments were the best of any I received in over ten years of blogging. I think it is now safe to 'out' him as Stephen Puryear of North Carolina State University. He recently sent me a copy of his Finitism and the Beginning of the Universe (Australasian Journal of Philosophy, 2014, vol. 92, no. 4, 619-629). He asked me to share the link with my readers, and I do so with pleasure. In this entry I will present the gist of Puryear's paper as I understand it. It is a difficult paper due to the extreme difficulty of the subject matter, but also due to the difficulty of commanding a clear view of the contours of Puryear's dialectic. He can tell me whether I have grasped the article's main thrust. Comments enabled.
The argument under his logical microscope is the following:
1. If the universe did not have a beginning, then the past would consist in an inﬁnite temporal sequence of events.
2. An inﬁnite temporal sequence of past events would be actually and not merely potentially inﬁnite.
3. It is impossible for a sequence formed by successive addition to be actually inﬁnite.
4. The temporal sequence of past events was formed by successive addition.
5. Therefore, the universe had a beginning.
Premise (3) is open to a seemingly powerful objection. Puryear seems to hold (p. 621) that (3) is equivalent to the claim that it is impossible to run through an actually infinite sequence in step-wise fashion. That is, (3) is equivalent to the claim that it is impossible to 'traverse' an actual infinite. But this happens all the time when anything moves from one point to another. Or so the objection goes. Between any two points there are continuum-many points. So when my hand reaches for the coffee cup, my hand traverses an actual infinity of points. But if my hand can traverse an actual infinity, then what is to stop a beginningless universe from having run through an actual infinity of events to be in its present state? Of course, an actual infinity of spatial points is not the same as an actual infinity of temporal moments or events at moments; but in both the spatial and the temporal case there is an actual infinity of items. If one can be traversed, so can the other.
The above argument, then, requires for its soundness the truth of (3). But (3) is equivalent to
3*. It is impossible to traverse an actual infinite.
(3*), however, is open to the objection that motion involves such traversal. Pace Zeno, motion is actual and therefore possible. It therefore appears that the argument fails at (3). To uphold (3) and its equivalent (3*) we need to find a way to defang the objection from the actuality of motion (translation). Can we accommodate continuous motion without commitment to actual infinities? Motion is presumably continuous, not discrete. (I am not sure, but I think that the claim that space and time are continuous is equivalent to the claim there are no space atoms and no time atoms.) Can we have continuity without actual infinities of points and moments?
Some say yes. William Lane Craig is one. The trick is to think of a continuous whole, whether of points or of moments, as logically/ontologically prior to its parts, as opposed to composed of its parts and thus logically/ontologically posterior to them. Puryear takes this to entail that a temporal interval or duration is a whole that we divide into parts, a whole whose partition depends on our conceptual activities. (This entailment is plausible, but not perfectly evident to me.) If so, then the infinity of parts in a continuous whole can only be a potential infinity. Thus a line segment is infinitely divisible but not infinitely divided. It is actually divided only when we divide it, and the number of actual divisions will always be finite. But one can always add another 'cut.' In this sense the number of cuts is potentially infinite. Similarly for a temporal duration. In this way we get continuity without actual infinity.
If this is right, then motion needn't involve the traversal of an actual infinity of points, and the above objection brought against (3) fails. The possibility of traversal of an actual infinite cannot be shown by motion since motion, though continuous, does not involve motion through an actual infinity of points for the reason that there is no actual infinity of points: the infinity is potential merely.
We now come to Puryear's thesis. In a nutshell, his thesis is that Craig's defence of premise (3) undermines the overall argument. How? To turn aside the objection to (3), it is necessary to view spatial and temporal wholes, not as composed of their parts, but as (logically, not temporally) prior to their parts, with the parts introduced by our conceptual activities. But then the same should hold for the entire history of the universe up to the present moment. For if the interval during which my hand is in motion from the keyboard to the coffee cup is a whole whose parts are due to our divisive activities, then the same goes for the metrically infinite interval that culminates in the present moment. This entails that the divisions within the history of the universe up to the present are potentially infinite only.
But then how can (1) or (2) or (4) be true? Consider (2). It states that an infinite temporal sequence of past events would be actually and not merely potentially infinite. Think of an event as a total state of the universe at a time. Now if temporal divisions are introduced by us into logically prior temporal wholes such that the number of these actual divisions can only be finite, then the same will be true of events: we carve the history of the universe into events. Since the number of carvings, though potentially infinite is always only actually finite, it follows that (2) is false.
The defense of (3) undercuts (2).
So that's the gist of it, as best as I can make out. I have no objection, but then the subject matter is very difficult and I am not sure I understand all the ins and outs.