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.

Nicely done, Bill. You've got it just right.

Anyone without access to AJP who would like a copy of the paper can download a pre-print draft from my website or email me for the official copy.

Posted by: Stephen Puryear | Wednesday, February 11, 2015 at 04:32 PM

Hi Bill,

You've got a very insightful blog here. Keep that lantern of yours filled with oil!

Not having access to Puryear's article, I can only interact with some of the references you have made here.

Firstly, let me just say that it seems to me that the argument in question is likely predicated upon a tensed theory of time, inasmuch as any accumulation of durative events seems to rely upon a temporal view that accounts for events that are both actual and ephemeral.

I'm not sure whether Puryear speaks to this issue, but it appears plausible to accept that a universe with an infinite tensed history would also be comprised of an infinite series of past durations that are of equal length (not just successively infinitesimal ones). If this is the case, then I would also think that one could move to undercut the philosophical arguments that employ paradoxes to buttress an infinite past by virtue of the material change that occurs throughout duration of any length (not in the abstract sense of Thomistic change per se, but as through the phenomenal conditions of matter itself). As such, if the duration of material change can stand in for past events then I think the thesis of causal finitude can weigh in upon this problem as well; and as Robert J. Spitzer has deftly brought out, an infinite regression of conditioned realities is equal to exactly nothing.

Best regards,

Nathan

Posted by: Nathan | Thursday, February 12, 2015 at 04:28 AM

With the caveat that I have not read S. Puryear's article, there are two different senses of infinity at play. An infinite temporal line has an infinite number of temporal parts, in the same sense as a finite spatial extension has an infinite number of spatial parts (assuming infinite divisibility), but an infinite temporal line is infinite in yet another, stronger sense, a metric sense to borrow mathematical terminology, which is the sense that seems to me to be relevant to the argument. In other words, W. L. Craig could retort that his defense may not have been stellar, but changing it a little bit the argument stands.

S. Puryear could retort back, assuming I have understood the summary correctly, that my response is all well and good, but it still does not solve the conundrum. For W. L. Craig bases his solution of Zeno's paradox on a rejection that space is actually, infinitely divisible. But if that is true, what possible basis could he have for rejecting traversal of the infinite, if his rejection is based on an analysis of the seemingly absurd consequences that entail from dividing time in an infinite number of temporal parts? But then my response would be (1) that time and space are not parallel (2) what do we mean that the universe is past infinite? Here is a first response, that seems to evade any talk of temporal parts and divisibility: there is an infinite sequence of (t_n) of events placed at times t_n such that for every n the time distance d(t_n, t_{n + 1}) is greater than some fixed, positive constant e. This last condition prevents the sequence of time events to "pile up" or, once again in mathematical terms, to have an accumulation point, and thus it must "run off to infinity". And this condition *must* enter the analysis in *some* way, shape or form, for otherwise, and for W. L. Craig's chagrin (full disclosure: mine as well), S. Puryear's is correct and W. L. Craig's argument could be turned against him by having the absurd conclusion that not only an infinite time distance cannot be traversed, neither can a finite one.

Someone could retort that my response cannot be quite correct, for it seems to go through independently of whether the A-theory or B-theory is correct, and Craig is quite explicit that the successive-addition argument against the past-infinitude presupposes the correctness of the A-theory of time. To which I would respond that the specific quoted argument does rely on the correctness of the A-theory of time, and so does my response. What W. L. Craig does say (besides arguing against the B-theory of time) is that the past infinitude of the universe can still be rejected on the basis of a rejection of actual infinities.

Posted by: G. Rodrigues | Thursday, February 12, 2015 at 07:23 AM

Nathan,

Thanks for the kind words. "the argument in question is likely predicated upon a tensed theory of time" Yes. The rest of yor comment I couldn't follow.

Rodriguez writes >>in the same sense as a finite spatial extension has an infinite number of spatial parts (assuming infinite divisibility)<< I don't understand. If a line segment is infinitely divisible, it does not follow that it has an infinite number of spatial parts.

>>For W. L. Craig bases his solution of Zeno's paradox on a rejection that space is actually, infinitely divisible.<<

I could be wrong, but I think what Craig rejects is that space is actually divided infinitely. He rejects actual infinities. Nevertheless, a line segment, say, is infinitely divisible. As I am sure you know, there is a difference between potential infinity and actual infinity.

Posted by: BV | Thursday, February 12, 2015 at 11:41 AM

Very interesting. Thanks for the precis of the article. I do not have any critical comments right now, though I have doubts that the view that wholes are ontologically prior to their parts entails that a temporal interval is a whole whose partitions depend on our conceptual activities.

I think readers who are interested in this particular post might have some interest in a related topic, in particular, the topic of whether wholes are more fundamental (in some sense) than their parts. It sounds like Craig is sympathetic to this view, and Jonathan Schaffer gives a very interesting defense of this view in his 2009 paper, "Monism: The Priority of the Whole." Probably many of your readers already know about this, but here is a link to the paper (many of his papers can be found on his website):

http://www.jonathanschaffer.org/monism.pdf

I wonder if it might be open for Craig to save his argument by saying that a temporal interval's parts are not dependent on our conceptual activities; but, they are in some sense less fundamental than the whole. He'd have to say how parts being less fundamental could save his view though. I don't know if it could be done or not.

Posted by: Shields | Friday, February 13, 2015 at 11:08 AM

You're welcome. I recommend you not use your surname. How about 'Spear'?

It seems that there is a logical gap between

1. A temporal whole is logically/ontologically prior to its parts

and

2. A temporal interval is a whole whose parts are due to our conceptual activities.

Can you tell a story according to which (1) is true but (2) false?

Puryear is a careful thinker and apparently he saw no gap.

Posted by: BV | Friday, February 13, 2015 at 11:53 AM

Bill,

A simple question. If an infinite past would only be a potential infinite, then what would a bona fide example of an actual infinite be?

Posted by: Crude | Saturday, February 14, 2015 at 11:19 PM

Crude,

The issue raised in Puryear's paper is not whether an infinite past is potential or infinite, but whether, GIVEN that an infinite series of past events must be actually infinite, that actual infinity can be traversed or not.

You want an example of an actual infinity. The set of natural numbers is an example that most philosophers today would accept. If there is such a set, then its cardinality cannot be finite. That is to say: the number of its elements cannot be finite. The set in question has aleph-nought elements, aleph-nought being the least transfinite cardinal.

Of course, one can reasonably question whether the natural numbers form a set. If they do, then they form an actually infinite set. If they do not, then the infinity of the natural numbers is only potential.

Posted by: BV | Sunday, February 15, 2015 at 04:49 AM

Bill,

"The issue raised in Puryear's paper is not whether an infinite past is potential or infinite, but whether, GIVEN that an infinite series of past events must be actually infinite, that actual infinity can be traversed or not."

Alright, but then I'm confused about something else you said. Earlier:

"This entails that the divisions within the history of the universe up to the present are potentially infinite only."

I took this to mean that the infinite past of the universe was nevertheless potentially infinite.

Posted by: Crude | Sunday, February 15, 2015 at 02:02 PM

What I try to do above is merely summarize Puryear's argument. His claim is that Craig's defense of (3) undercuts (1), (2), and (4). If you go to Puryear's site, you can get a copy of his paper. That may clear things up.

Posted by: BV | Sunday, February 15, 2015 at 02:59 PM

Hi Bill,

It's nice to drop in on your blog every now and then. Always interesting stuff to interact with.

Regarding Puryear's argument, though, it seems to me that the two cases of continuous motion and a beginningless past are not exactly parallel and that this affords Craig a line of response.

The first point is that continuous motion takes place over a metrically finite interval, say, [0, n], whereas a beginningless past takes place over an metrically infinite interval, say, (-∞, 0]. Given any partition of these intervals into discrete, finite parts, the measure of the first is finite and the measure of the second is infinite. The second point is to *deny* that it is only "our conceptual activities" that can partition a continuous interval. Rather, and in line with the earliest formulations of kalam-style arguments going back to John Philoponus, the beginningless past can be *naturally* partitioned simply by pointing to recurring events like the lunar cycle and the diurnal and annual motion of the sun. (Like most ancients, Philoponus would have assumed that heavenly bodies are not susceptible to generation and corruption.) So the kalam argument as the ancients would have formulated it would have said that if the past were beginningless, then an actually infinite number of days, years, lunar months, etc. would have been traversed, which is impossible, etc. It's never been a standard assumption of kalam-style arguments that temporal divisions can only be introduced by us.

For this line of response to work, of course, it must be the case that ordinary cases of continuous motion are *cannot* be naturally partitioned into an actually infinite collection. But isn't this plausibly the case? The smallest natural divisions in continuous physical motion are probably at the quantum or Planck-length level, which would at most yield a *finite* collection of parts for any continuous motion that has both a beginning and an end.

In sum, then, Puryear's objection is smart, but I think Craig has a plausible rebuttal.

Posted by: Alan Rhoda | Wednesday, February 18, 2015 at 07:16 AM

It's good to hear from you, Alan. Things appear to have worked out well for you and I'm glad to see that you have time to continue your philosophical work.

As for the first point, why must continuous motion take place over a metrically finite interval? Suppose a beginningless universe goes through cycles of expansion and contraction endlessly. Such expansion/contraction would count as continuous motion, I should think, but it would take place over a metrically infinite interval.

Your second point is very interesting, but since we know that the heavenly bodies are subject to generation and corruption, it is not clear how their motion could naturally partition a continuous interval.

And the same would go for Philoponous' assumption that the human race always existed -- we know that that is false.

Nevertheless, as I pointed out above, there seems to be a logical gap between (1) temporal wholes are logically prior to their parts and (2) the partitioning of temporal wholes is due to our conceptual activities. Puryear slides from (1) to (2). So it looks as if the mere possibility of Philoponous types example does show a failure of entailment.

I couldn't think of a way to make that failure of entailment graphic, but I think you have supplied the lack.

Best to you and your family,

Bill

Posted by: BV | Wednesday, February 18, 2015 at 12:39 PM

Thanks, Bill.

It's nice to finally have stable, full-time work. It's not full-time philosophy work, but hey, it pays the bills, provides me access to research materials, and affords me the opportunity to philosophize in my spare time. Plus I don't have to grade tall stacks of mostly mediocre student papers—never my idea of a fun time.

I like how you express the logical gap between "(1) temporal wholes are logically prior to their parts and (2) the partitioning of temporal wholes is due to our conceptual activities." That seems to be the crux of the matter as far as Puryear's critique of Craig is concerned. I would agree with Puryear that Craig can be read as endorsing (2), but I don't think that Craig has to read that way. And even if Craig does endorse (2), it's not clear why defenders of the Kalam argument need to endorse it.

Warm regards,

Alan

Posted by: Alan Rhoda | Wednesday, February 18, 2015 at 01:02 PM

Hi Alan,

Thanks for the interesting comment. In the paper, I take myself to be criticizing one of two basic replies to what I call the anti-finitist objection, which is, in one guise, that motion across a finite distance involves traversing an actual infinite. The first reply, the one I target in the paper, is Craig's, which is that space and time have parts only insofar as we specify those parts in thought. The second reply is that space and time are atomic, so that any finite magnitude of either has only a finite number of parts. When you suggest that one could deny that it is only our conceptual activities that divide a continuous interval, and instead hold that there are "smallest natural divisions" in physical motion, I take this to be equivalent to suggesting that one could opt for a version of the second reply rather than the first. But that's not really a rebuttal to my objection, since my objection only targets the first reply.

I have not yet decided whether the second reply is viable, but it does seem to me to commit one to the view that space and time are discrete, not continuous. Given that space and time have been widely thought to be continuous, this would be a bullet one would have to bite.

Bill, regarding your last point, I agree that there's a logical gap between (1) and (2). (2) entails (1), but (1) does not entail (2). But my argument does not assume that (1) entails (2). Craig explicitly affirms (2), and properly speaking that is the target of my criticism.

Posted by: Stephen Puryear | Wednesday, February 18, 2015 at 02:19 PM

I base my ascription of (2) to Craig on remarks such as these:

"If one thinks of a geometrical line as logically prior to any points which one may care to specify on it rather than as a construction built up out of points (itself a paradoxical notion), then one’s ability to specify certain points, like the halfway point along a certain distance, does not imply that such points actually exist independently

of our specification of them... By contrast, if we think of the line as logically prior to any points designated on it, then it is not an ordered aggregate of points nor actually infinitely divided.Time as duration is then logically prior to the (potentially infinite) divisions we make of it." (Craig and Sinclair 2009: 112–3, emphasis added; cf. Craig and Smith 1993, 27–30)It seems to me that the bold bits cinch it for the interpretation that the divisions are due to our conceptual activity.

Posted by: Stephen Puryear | Wednesday, February 18, 2015 at 03:08 PM

Hi Stephen,

Thanks for the thoughtful response.

I agree that your interpretation of Craig is probably correct. And I agree that your argument correctly identifies a serious problem with Craig's way of handling what you call "the anti-finitist objection".

My proposed response falls between the two responses you consider in your paper. I'm suggesting (pace finitism) that there are real continua (infinitely divisible manifolds) and (with Craig) that continua are more fundamental than their parts, but (pace Craig) that parts of a continuum can be specified independently of conceptual activity on our part. For a beginningless temporal continuum to have an actually infinite number of parts all there has to be is a finite-size measuring stick (so to speak), like a periodic natural process, such that if it were applied to the continuum would yield a measure of infinite cardinality.

Cheers.

Posted by: Alan Rhoda | Wednesday, February 18, 2015 at 04:05 PM

Hi Bill,

I've got a couple of comments. First,

"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.)"

It's unclear to me why Puryear thinks this entailment holds, and why you think it's at all plausible. Whether or not the points are posterior to the whole, you've got an infinite number of them, an actual infinity. How is it supposed to follow that the partition depends on our mental activities? I assume that there is more reasoning here that has been omitted for the sake of brevity.

Also, this is not the point of Puryear's argument, but I think the following premise is vulnerable:

4. The temporal sequence of past events was formed by successive addition.

What, exactly, is meant by "successive addition" in this context? I guess I take a beginningless past to mean that for every event, there was some prior event (though I suppose looped time would count by this definition, leave that aside for now). Does that qualify as being "formed by successive addition?" At what point was the formation complete? The only thing I can think to say is that the formation has always been complete. There has always been an actual infinity of past events. In that sense, the infinite series of events was never "formed" by successive addition or any other way.

Finally, I recommend Wes Morriston's work arguing against the Kalam argument. Although I am working on a paper arguing that he has overstated his case in his responses to Craig, I generally accept his reasoning:

http://philpapers.org/rec/MORMTB

Best,

Spencer

Posted by: Spencer | Friday, February 20, 2015 at 02:17 PM

Hi,

This is my first time to comment on your blog, though I have enjoyed reading it for some time. I am teaching Craig's argument this week and have found this entry and subsequent comments particularly helpful.

I take the difficulty with Craig's defense of (3) to be that there is nothing to prevent the opponent of the kalam cosmological argument from replying that the beginning-less history of the universe up to the present moment is not an actual infinite sequence of events but instead a continuous singular event that is only potentially comprised of infinite shorter intervals.

It seems to me that Craig could reply that unless the history of the universe has a beginning it cannot be a whole that we merely conceptually divide into parts. This is because (to put it crudely) a whole (spatial, temporal, or otherwise) must be "bounded" in order to be a unity. We identify and discriminate events in virtue of beginning, middle, and end. If events must have a beginning, then the beginning-less history of the universe could not be a singular event. As a consequence, either the universe has a beginning or the history of the universe consists in an actually infinite sequence of events.

Or maybe the beginning-less history of the universe is continuous but not an event. I wonder if this alternative would incur its own problems of traversability. Can there be a flow of time from past to present if time is continuous and unbounded?

Posted by: mangibus | Sunday, February 22, 2015 at 11:53 PM

mangibus,

Thanks for your comment. By the way, I like what you say on your faculty page about the ethical pre-conditions of philosophical insight. This is a very important, and nowadays too often neglected neglected, topic and it is good you are pursuing it. Bravo!

You make a point that I think Puryear should consider. Your point, I take it, is that if we divide a whole, then there must be a whole we divide, but in the4 case of a beginningless universe there is no whole. The line segment AB is a whole we can divide as many times as we like. But consider a ray (in the geometrical sense) that has its origin at point A and then extends infinitely to the left. I take it you are saying that in a case like this there is no whole to be divided because there is no bound to the left, but only the bound at A.

And similarly for temporal intervals and sequences of events.

Puryear needs to consider whether this undercuts his undercutting.

Posted by: BV | Monday, February 23, 2015 at 04:09 AM

Spencer,

You appear to be just begging the question. You seem to be just assuming that every infinity is an actual infinity. Consider the line segment AB. Suppose that it has no proper parts except those we introduce by dividing it,and that it is in this sense logically prior to its parts. Then it would not have an actual infinity of proper parts: the infinity would be potential only. Why must an infinitely divisible line segment be actually and infinitely divided?

Similarly for temporal intervals and sequences of all kinds.

Furthermore, what would it mean to say that a line segment is logically/ontologically prior to its parts if it was composed of them?

Posted by: BV | Monday, February 23, 2015 at 04:25 AM

mangibus,

Thanks for your comment. Let us suppose for the sake of argument that the universe had no beginning in time, and that there are no divisions within time apart from the ones we make in thought. Your point, if I understand it, is that under these assumptions, the history of the universe up to the present could not be a whole, thus could not be a whole we divide into parts, contrary to the assumption. (Nor could it be said to be a unity or a singular event.) Well, suppose I grant the point that a whole must be bounded, and thus that the history of this universe is not a whole. I see no problem for the assumed view. What matters is not that that history is a whole, but that it can be divided, and a thing need not be bounded in order to be divided. In Bill's ray example, I of course cannot divide it at the midpoint, since there is no such thing, but I can still divide it into parts. For instance, I can divide it somewhere near the bounded end, and then further divide that finite part, etc.

Posted by: Stephen Puryear | Monday, February 23, 2015 at 05:53 AM

Thank you, Stephen and Bill, for your replies. I should be getting ready for class so this will be quick and off the cuff.

Stephen, you reply:

"I see no problem for the assumed view. What matters is not that that history is a whole, but that it can be divided, and a thing need not be bounded in order to be divided."

I would have thought that what matters, especially, is that history can be a beginning-less continuum rather than an actually infinite sequence of discrete events or a continuum with a beginning. The fact of the potential divisibility of history, I think, does not show this (an actual infinite would also be potentially divisible). Examples I can easily grasp of actual continua involve bounded wholes with a beginning, middle, and end (e.g., the discrete motion of reaching for the coffee cup). I am open to the possibility that there can be actual unbounded continua but remain unsure how to think about such things and the impact on Craig's argument.

Back to Bill's ray example. Is the sort of division you explain is possible supposed to show how the ray could be traversed in step-wise fashion or is your point about divisibility supposed to show something else? Thanks for your patience with an amateur metaphysician.

Posted by: mangibus | Monday, February 23, 2015 at 08:51 AM