Ronald Keith Tacelli, S.J.: Whichever Way You Slice It

Whichever Way You Slice it A Response to Baldner and Staley

Whichever Way You Slice It
A Response to Baldner and Staley

Ronald Keith Tacelli, S.J.

In a recent issue of the Lyceum (Spring '91), I argued that if the world is without beginning, the past must comprise an actual infinity of events.  Certain thought-experiments, like the following, helped propel the argument along.  Suppose the universe is everlastingly old.  Suppose, further, that on each past day God created an immortal soul.  This means that there would exist right now an actual infinity of souls—i.e., that there would currently exist a plurality of souls more numerous than any finite quantity.  Now in the past history of this universe there must have occurred as many days as there are presently existing immortal souls.  Since there is an actual infinity of immortal souls, there could have occurred no  fewer days.  And so there must in the past have occurred a succession of events more numerous than any finite quantity.  Therefore the past of this universe must be actually infinite.

                It surprises me that anyone would raise even an eyebrow about this conclusion, let alone a question; and yet two able philosophers[1] have argued in these pages that it is not merely questionable but false.  Why?


                Kevin Staley says that “a beginningless past entails only that there has been a potential infinity of past events” (p. 17).  He concedes that past events are actual, in the sense that each past event “has really occurred.”  But still the past events of a beginningless universe do not comprise an actual infinity, merely a potential one.

                What is a potential infinity?  According to Staley, it is “an indeterminate collection, which, because it is ever in the process of increasing or decreasing, fails to be a completed whole or totality.  As soon as this process ceases, it finds itself to be actually finite.  It is infinite only with respect to process . . .” (p. 20).  Staley proposes the  following illustration: 

Suppose that matter, a Thanksgiving turkey for example, is infinitely divisible.  Suppose also that an immortal master carver exists, who is so skilled in his craft that he is capable of cutting any slice of turkey, no matter how thin, in half again.  On Thanksgiving morning, he is slicing turkey and placing these slices in a pile on a platter.  He cuts a nice thick slice, and he cuts it in half.  He places one half of the slice on the platter, and cuts the remaining half in half again.  He places this half of the half on the platter, and slices the remaining half in half again.  He continues in this fashion for eternity. (p. 19)

                Now neither the turkey nor the carver nor the original slice is actually infinite.  All three are wholes with determinate limits.  But the first thick slice is potentially infinite, says Staley, because it can be cut into smaller and smaller pieces without end.

                Staley notices that there is a problem with applying his example to the case of the beginningless universe.  For everything in the example is actually finite[2] until the  process of slicing begins.  But in the case of the beginningless universe, what is it that corresponds to the first slice, and what to the act of slicing?  Is the first slice the entirety of the past, or a bit of the past?  If the entirety of the past, then, since the slice is actually finite, Staley would have to say that the past is actually finite, that it contains only so many events.  (And in that case what does the act of slicing correspond to?)  If the first slice corresponds to a part of the past (say the last billion years), the turkey from which it was taken would represent the whole of the past.  But then is the turkey of limited or unlimited size?  If limited, then the same problems arise; if unlimited, then the example involves the actual infinity of the past.

                So nothing Staley has proposed in his example corresponds to the potential infinity of the beginningless past.[3]  We might try to make his example correspond by changing it a bit.  We might suppose that the Master Carver has been carving for all eternity.  In that case, the platter would be piled with infinitely many slices.  This quantity would at any point in time already have resulted from the process of carving, and would therefore be actually infinite.  It would not be actually finite and infinite with respect to process only.

                Now if every slice resulted from an act of slicing, could there have been in the history of this Master Carver fewer acts of slicing than there are actual slices?  Clearly the answer is No.  And so all the acts of slicing that have terminated in this (present) slice cannot comprise a potential infinite; for infinitely many acts of slicing must have led up to this act and this slice.

                That is what I was trying to point out in my original article.  If quantitative predicates can apply to such a past at all, then it cannot be spoken of as a potential infinite, because the kind of finitude proper to the potential infinite is lacking here.  There is another kind of infinity involved in our apprehension of a beginningless past—a kind that is not merely capable of indefinite extension or division.

                I spoke of this quantitative infinite as “actual” and as a “completed” set.  Why?  Because the members belonging to it are infinitely many, and because, since the set is formed by successive addition, this infinite plurality has been achieved:  no further successive additions are needed to reach it.  At any point the succession of days has already accumulated to infinity; just as, to get back to our Master Slicer, at any point his acts of slicing, and the slices resulting, have already reached infinity.

                But Staley's dissatisfaction goes deeper.  He thinks that my analysis of a beginningless past involves the claim that the past is an infinite totality.  But no such totality is possible, he insists; the very “notion of an infinite totality is an incoherent notion, analogous to the notion of a square circle” (p. 22).

                Perhaps it is.  But this in no way counts against my claim that an actual infinite follows from the hypothesis of a beginningless universe; it counts instead—if it counts at all—against the hypothesis from which an actual infinite follows.  Let me illustrate:

                Three people are standing before you, discussing the meaning of life.

                A:  All reality is illusion.

                B:  Then it would follow that at least some illusions are real.

            C (rebuking B sharply):  That can't be right! If you claim that something is real and an illusion, you are claiming that something is real and not real—which is absurd!

                B:  Well, sure; then so much the worse for A's hypothesis; but not (surely?) for my claim that this really follows from it.

Just so here.  I never claimed that an actually infinite past is possible.  In fact, I believe with Staley that it is not.  I merely claimed that if the universe is without beginning, then the past is actually infinite.  Staley's argument seems to run:  an infinite whole or totality is impossible; but an actually infinite past means some sort of whole or totality; therefore the hypothesis of a beginningless universe does not entail an actually infinite past.  But no:  whether or not the notion of an infinite whole or totality is incoherent, the hypothesis surely entails it.  And that is my claim.

                Staley says that his position is

born out by the way in which we ordinarily use the terms ‘whole’ and ‘infinite’.  If one reflects carefully on those sorts of things within one's experience that one considers to be wholes of a certain sort, note that each of these wholes has certain limits and is a whole precisely in virtue of those limits.  A story is a whole story because of its beginning, middle, and end.  A wall is a whole wall because of its top, sides, and bottom.  A wall which had no top simply would not be a whole wall.  That which is infinite is that which lacks a limit, and so fails to be a whole in some respect . . . .

                I want to avoid talking about “the set” of past events, “the series” of past events, or “the past” considered as the collection of all past events, because phrases such as “the set,” “the series,” and “the past” falsely suggest that we can talk about past events as some sort of totality—which is just what cannot be done in a beginningless universe. (p. 22-3)

                With much of this I sympathize.  But as a way of dealing with the implications of a beginningless past it seems pretty forced and unnatural.

                Suppose, for example, you see what looks like a wall, but stretching up, up until the clouds take it from your sight.  Nearby stands a philosopher.  “Gosh!” you say to him. “It doesn't seem as though that wall even has a top!”  He answers:  “If it's a whole wall, of course it would have a top.  But it's not a whole wall.  In what you see, there is a brick above any brick you care to name.  And in that sense the wall is potentially infinite.  There really or truly are infinitely many actual bricks here.  But we don't have an actually infinite wall.  We've got to avoid talking about “the wall,” that means we can talk about what's here as some sort of totality—just what we can't do with all these vertically arranged bricks that have no top limit.”

                Now this seems to me a very strange sort of answer.  For if the structure—in deference to the philosopher, we can call it that instead of ‘wall’ —if the structure is not being extended continuously upward, then all the bricks that make it up already have their places in the vertical arrangement.  And if the structure has no top, the bricks that make it up must be infinitely many.  So in some sense the structure is infinite.  But not just potentially infinite.  If it were, it would have a finite number of bricks, and would merely be capable of unending extension.  We mean by its infinity something more than that.  If there cannot be an infinite multitude, then we know the wall must have a top; but we also know that it cannot be just potentially infinite in lacking one.

                I will even make Staley a present of the word ‘actual’.  I would ask him only to admit that when we consider the hypothesis of a beginningless universe, we are faced with a kind of infinity which is other than potential.  Since that infinity has at any and every point in the history of the universe already been achieved, I called it ‘actual’.  If Staley can think of a better word, so be it.  But ‘potential’ is emphatically not that word.


                For all my criticism of its main conclusion, much of Staley's article strikes me as sound.  But Stephen Baldner's is another kettle of fish—or, to keep with the metaphor, another platter of turkey.

                Baldner will not admit that the past of a beginningless universe is any sort of infinite at all.[4]  He writes that “the past of this universe is, if eternal, neither a potential nor an actual infinite; because it has no actuality at all it is wrong to characterize it as any sort of infinite” (p. 3).

                Why wrong?  Because by ‘actuality’ Baldner means ‘present existence’.  Since the past is not present, it cannot be actual in any sense; and therefore can be neither an actual infinity nor a potential one—since the potentially infinite must be actually finite.

                Now by ‘actual’ I could not have meant ‘presently existing’.  The argument is, after all, about the past.  And my point is simply this:  the hypothesis of an everlasting world leads us to affirm some kind of infinity about the past—but a kind that is not merely potential, since this event's happening depends on there having actually occurred infinitely many successive events before it.  This infinite succession must already have happened, already been achieved.  And so ‘actual’ seemed the most natural word to describe it.  But I have no wish to quibble about words.  Let Baldner limit the term ‘actual’ to ‘presently existing’.  This merely leaves us without a name for what we notice.  And clearly Baldner does notice that my examples point to a kind of infinity that the word ‘potential’ will not cover—but a kind that cannot in his sense be called ‘actual’.  Thus: 

Fr. Tacelli has raised problems about counting successions of Popes in an eternal Church . . . .  He uses this as an example to show that an eternal past would necessarily entail the existence of an actual infinity.  Now I might concede to him that the Church could not have existed eternally in the      past . . . but such considerations are only accidental to the question . . . .  Whether the eternal past of this universe, if it existed eternally in the past with all of its currently existing species (including man), would imply an actual infinity, is a secondary question.  The prior question for the debate . . . is whether the material universe in some form could have existed eternally in the past without resulting in an actual infinity. (pp. 3-4)

                But such considerations are in no way accidental to the question I was asking:  Does the eternity of the world—of  this material universe—entail an actual infinite?  Baldner says he “might concede . . . that the Church could not have existed eternally in the past.”[5]  But why?  The Popes would not constitute an infinite set of presently existing members; and so the problem of an actual infinite (given Baldner's understanding of ‘actual’) should not—could not—arise.  Why then the concession?  Because he sees that a kind of infinity more than merely potential is entailed by the affirmation of a beginningless Church.  But clearly an infinity of that very kind is entailed by the same affirmation about the world.

                I had supposed in my example that the Church were as old as the beginningless universe.  Well, suppose the universe did begin—say, 2000 years ago.  Would this assumption cause Baldner to question the possibility of there being a present Pope?  No.  And why not?  Because he would know ahead of time that there could have been only so many of them, i.e., that the succession of papal reigns leading to this present reign must be a finite succession.  And if we believe that this universe began, then we believe that the realm of spatio-temporal being has a finite history; i.e., we believe that there was a first event, and that only so many events could have occurred from that first to this present.  Never mind that the events in happening cease to be present; all of them have happened (just as all previous papal reigns have occurred), and their having happened is a necessary condition for the universe's reaching this point in its history (just as all papal reigns before the present one are necessary conditions for this succession to the throne).  But ‘all’ means ‘as many as there were’.  And in an eternal universe this could have been no fewer than infinitely many.  And so the affirmation of an eternal universe involves the affirmation that the universe must have suffered infinitely many successive transformations in order to arrive at this point in its history.

                Granted:  I assume that there is a unity underlying all the changes taking place in the universe; that history is not a mere punctiform substitution of one thing for another.  If it were, we could not even say that one thing really follows after another; and in that case, I agree, there could not have been an infinite succession.  There could be no real history at all.  But that is not the sort of mad universe Baldner believes in, is it?  Could his willingness to avoid an unwelcome conclusion drive him to embrace even that?  If so, then he needs not an argument but a prayer.[6]



Boston College

Boston, MA

[1] Kevin Staley in “Infinity and Proofs for God's Existence” (Fall '91); Stephen Baldner in “The Past Just Ain't What it Used to be” (Fall '92).  References to these articles will be given in parentheses in the body of the text.


[2] Except perhaps the age of the Master Carver!


[3] And Staley himself admits as much:  “[S]ome . . . speak about the past of a beginningless universe as an actual infinity, since there actually have been infinitely many past events in a universe without beginning (quite unlike the pile of turkey slices).  But,” he continues, “to say that there actually have been infinitely many past events is to say something like there have really been or truly been infinitely many events.” (p. 24).


[4] Baldner qualifies this:  “There is, however, a sense in which an  eternal past might be said to be potentially infinite:  if the  past were eternal, we could always count more and more past days.  We could never count an actually infinite number of them, but we could always (potentially) count more.  In this sense only I would agree that the past would be, if eternal, a potential infinity.” (pp. 2-3)  He would probably also agree that, if the universe were not eternal, there is a sense in which the bounded past might be said to be actually finite.


[5] Just for the record:  The point of my example was not to show the impossibility of an infinitely old Church (or even of an infinitely old universe); it was to show that questions of actual infinity arise in sets or series only one of whose members is presently existing.  As I put it then:  “[S]ince it is the series of Papal elections that [this present] election is part of, the possibility of [this present] election stands or falls with the possibility of there having been infinitely many Papal elections (and completed terms) before [it].”  (p. 20).


[6] I am deeply grateful to Kelly Clark, Norris Clarke, SJ, Michael Pakaluk, and Sharon Yannaccone, who criticized an earlier draft.