Robert A. Larmer: Does a Beginningless Universe Imply an Actual Infinity of Past Events?

Does a Beginningless Universe Imply an Actual Infinity of Past Events?

Robert A. Larmer

Recent discussions of the kalam cosmological argument have focused on the issue of whether an actual infinity of past events is impossible.  Settling this question, it is thought, will settle the question of whether a beginningless universe is conceivable.  Unfortunately, this focus on the issue of whether an actual infinity of past events is impossible, has obscured the question of whether a beginningless universe does, in fact, entail an actual infinity of past events.

                It is to this largely unexamined question that Ronald Tacelli, Kevin Staley and Steven Baldner have turned their attention in a recent series of articles.[1]  Tacelli has argued that a beginningless universe does entail an actual infinity of past events:  both Staley and Baldner have denied this, albeit for different reasons.  My own view, which I will defend after summarizing the argument to date, is that Tacelli is correct in his claim and that neither Staley nor Baldner provide any reason to doubt that a beginningless universe entails an actual infinity of past events.

                Tacelli's thesis is that the existence of an everlasting, i.e. beginningless, universe entails the existence of an actually infinite set of past events.  His argument for this is that “the present motion would not be the present motion of an eternal [beginningless] universe unless infinitely many motions preceded it.”[2]  On Tacelli's view, it is precisely our realization that a beginningless universe implies an actual infinity of past events that explains our insistence that in such a universe there must always exist an event which occurred before any past event.[3]  If there were a first event, i.e. an event not preceded by another event, this would imply not only that the universe had a beginning, but that there is a finite number of events in its past.  Only if there exists no first event and thus no finite number of events in its past is it possible that the universe be beginningless.

                Staley has criticized Tacelli on the basis that the existence of a beginningless universe implies not an actual, but rather a potential infinite.  He contends that “in a universe which lacks a beginning the past is infinite, but only potentially so.”[4] Tacelli, he argues, fails to distinguish between the notions of actual and potential infinity and is thus led into error.

                Baldner's criticism of both Tacelli and Staley is that neither has come to grips with the ontological status of the past.  Both Tacelli and Staley think that the existence of a beginningless universe implies the existence of an infinity of past events; whether that infinity be conceived as actual or potential.  Baldner's view is that “the past of . . . [the] 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.”[5]  Since there is no sense in which past events actually or potentially exist, the question of whether the number of past events in a beginningless universe is actually or potentially infinite cannot be raised.

                My own view is that Tacelli's thesis is correct and that the criticisms of Staley and Baldner are ill-founded.  This can be made clear by a closer examination of their criticisms.

                Staley's criticism is that Tacelli has failed to distinguish clearly between an actual and potential infinity and that once this is done it will become evident that a beginningless universe implies not an actual, but potential, infinity.  Staley argues that a beginningless universe could not imply the existence of an actual infinity of past events, since “when applied to the notion of infinity . . . actual means complete in the sense of constituting some sort of whole or totality.”[6]  Since a necessary condition of a set of events being a whole or totality is that it have “a beginning, a middle and an end,”[7] a beginningless universe could not imply the existence of an actual infinity of past events.  Thus Staley is not only prepared to say that the set of past events in a beginningless universe does not constitute an actual infinite, but that an actually infinite set cannot exist.  He comments, “to put my argument differently, even in a beginningless universe, the past cannot be an actual infinity, that is, an infinite totality, simply because the notion of an infinite totality is an incoherent notion, analogous to the notion of a square circle.”[8]

                In saying this, Staley agrees with proponents of the kalam argument who argue that an actually infinite set of past events cannot exist.  Where he disagrees is that rejecting the possibility of an actually infinite set of past events commits one to the view that the universe had a beginning.  The usual strategy of those defending the kalam argument is to argue that, since a beginningless universe implies an actually infinite set of past events, and since an actually infinite set of past events cannot exist, the universe must have had a beginning.  Staley does not think this inference is well-founded, since he thinks that a beginningless universe implies not an actual, but a potential, infinity of past events, and that a potential infinity of past events, unlike an actual infinity of past events, is conceptually possible.

                Staley's claim that a beginningless universe implies not an actual, but a potential, infinity of past events seems the product of a misunderstanding and misapplication of the concept of potential infinity.  First, by definition, a potential infinite is always finite.[9]  It was for this reason that Cantor referred to it as a false infinite.[10]  To say, therefore, that the number of past events in the universe is potentially infinite is to admit that only a finite number have occurred and that the universe had a beginning.

                Staley shows some awareness of this difficulty when he comments that the potential infinity of a beginningless universe is not completely analogous to the potential infinity of sub-dividing a finite quantity.[11]  He goes on to note that “a beginningless universe is . . . infinite at every moment of its existence     . . . .”[12]  This, however, is to give away the game.  If, by definition, a potential infinite is at every moment finite, it makes no sense to call the past of a beginningless universe, which is infinite at every moment of its existence, a potential infinite.

                Second, Staley seems to confuse the process of counting the members of an actually infinite set with defining such a set.  From the fact that it is impossible to reach by counting backwards any first event in a beginningless universe, he wants to deduce that such a universe implies only a potential infinity of past events.[13]  It is true that the process of counting is potentially infinite, in the sense that, although at any point in the counting process one will only have counted a finite number of events, one could keep counting indefinitely.  This, however, is only to emphasize that one can neither traverse or form an actual infinite by counting.  What Staley fails to realize is that the condition of being able to continue counting indefinitely is the existence of an actually infinite number of past events.  As Tacelli comments in his original article,

 

before we begin to count, we know (on our hypothesis) that the events which have preceded this one cannot be counted.  We know that this succession really did precede the present.  Our question is:  Must it comprise a completed infinite series?  Our inability to count is another way of answering the question Yes.  It is our apprehension of the quantitative infinity involved in the hypothesis that makes us realize we cannot count the members.[14]

A large part of Staley's confusion arises from the fact that he fails to realize that the concept of potential infinity has no application in set theory.[15]  Inasmuch as the concept of a potential infinite is not a set theoretic idea,[16] to describe the set of past events as a potential infinite is fundamentally mistaken.

                Staley is not oblivious to this difficulty, but his response is not adequate.  At one point in his article, he comments that he wants to be taken as making a claim about past events taken individually, not as a collection or series.  Unfortunately, he is not consistent in this.  At numerous points he is quite prepared to talk of the set or series of past events.[17]

                Neither will it do, having emphasized the impossibility of referring to past events in a beginningless universe as a set or collection, to introduce the notion of a “set, loosely conceived”[18] of past events.  Not only is it difficult to understand what is meant by a “loosely conceived set,” it is puzzling how, if it is nonsense to talk of the set of past events in a beginningless universe, loosely conceiving such a set makes it any less nonsense.  What is clear is that the concept of potential infinity cannot be applied to a set, be that set loosely conceived or not.  There is, therefore, no third option:  the set of past events must be either finite or actually infinite.

                Turning to Baldner's criticism of Tacelli, we see that he is prepared to rule out an actual infinity of past events on the basis that past events no longer exist.  Because there is no sense in which past events are actual, there can be sense in raising the question of whether there exists an actual infinity of them.  Strictly speaking, “the past of . . . [the] universe is, if eternal, neither a potential nor an actual infinite; because if has no actuality at all it is wrong to characterize it as any sort of infinite.”[19]  Baldner allows that

 

there is . . . 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 . . . the past would be, if eternal, a potential infinity.[20]

                Baldner's argument, depending upon how it is read, proves either too much or too little.  On one reading, he might be taken as claiming that, since past events do not exist, they cannot be enumerated.  If there is no sense in which the past exists there is no sense in attempting to count past events.  There is no sense, therefore, in raising the question of whether the number of past events is actually infinite.

                On this reading, Baldner's argument proves too much.  It rules out not only the possibility of raising the question of whether the number of past events is actually infinite, but the possibility of raising the question of whether the number of past events is finite.  If, in principle, it makes no sense to enumerate past events we must dismiss as nonsense any attempt to ask whether the number of events that has transpired in the history of the universe is finite or infinite.  Unfortunately, we must also dismiss as nonsense a host of other questions.  On this line of argument it will make no sense to ask how many Popes there have been or how many birthdays one has had, since these are past events and cannot, therefore, be enumerated.  In short, the consequence is to rule out any attempt to enumerate past events.  Not only does it become impossible to discuss infinite series in the past; it becomes impossible to discuss finite series in the past.

                A more charitable reading would be to take Baldner not as denying that past events can be enumerated, but as claiming that the fact that past events are no longer actual precludes them forming any kind of actually infinite set.  Put somewhat differently, this amounts to the claim that an actually infinite set would have to be composed of actual things or events.

                Baldner, if this be his argument, confuses the issue of how past events are to be conceived with the issue of how many past events have occurred.  He seems to think that, because past events are no longer actual, a beginningless universe does not imply that an actual infinity of past events has occurred.  This is a mistake.  It is true that past events are no longer actual events, but this scarcely negates the claim that a beginningless universe implies that an actually infinite number of past events has occurred.  What is at issue is not whether a beginningless universe implies an actual infinity of presently existing events, but whether it implies that an actual infinity of past events has taken place.  Tacelli makes essentially this point in his original article when he writes,

 

[Events] have indeed passed away.  That is part of what we mean by calling them past.  But their being past does not mean that there were not many of them and it emphatically does not mean that we cannot know how many there were.  For if the universe is everlasting then we can know.[21]

He goes on a little later to say that the disanalogy between the past and the present

 

is not relevant to the question of actual infinity . . . .  [Past events] may not be there for us to see, but that does not make them irrelevant to  our understanding of (or possible interest in) what . . . [now occurs].[22]

                I conclude that neither Staley nor Baldner has provided any reason to doubt that the existence of a beginningless universe implies the existence of an actually infinite number of past events.  If, as proponents of the kalam argument insist, an actual infinity of past events could not have taken place this implies that a beginningless universe is inconceivable.

 

 

University of New Brunswick

Fredericton, N.B.

Canada



[1] See R.K. Tacelli, “Does the Eternity of the World Entail an Actual Infinite,” Lyceum 3 (Spring 1991), pp. 15-22, K.M. Staley, “Infinity and Proofs for the Existence of God,” Lyceum 3 (Fall 1991), pp. 15-26, S. Baldner, “The Past Just Ain't What it Used to be:  A Response to Kevin Staley and Ronald Tacelli, S.J.,” Lyceum 4 (Fall 1992), pp. 1-4.

 

[2] Tacelli, p. 21.

 

[3] Tacelli, p. 18.

 

[4] Staley, p. 23.

 

[5] Baldner, p. 3.

 

[6] Staley, p. 21.

 

[7] Staley, p. 22.

 

[8] Staley, p. 22.

 

[9] See, for example, J.P. Moreland, Scaling The Secular City (Grand Rapids, Michigan:  Baker, 1987), p. 22.

 

[10] See Robin Small, “Cantor and the Scholastics,” The American Catholic Philosophical Quarterly Vol. LXVI, No. 4 (Autumn 1992), pp. 407-428, p. 408.

 

[11] Staley, p. 24.

 

[12] Staley, p. 24.

 

[13] Staley, p. 17.

 

[14] Tacelli, p. 17.

 

[15] Moreland, pp. 21-22.

 

[16] Moreland, p. 21.

 

[17] Staley, pp. 23-4.

 

[18] Staley, p. 23.

 

[19] Baldner, p. 3.

 

[20] Baldner, pp. 2-3.  Baldner, no less than Staley, misapplies the notion of potential infinity.

 

[21] Tacelli, p. 18.

 

[22] Tacelli, p. 20.