1. Everything that begins to exist has a cause.
2. The universe began to exist.
3. Therefore, the universe has a cause.
Conceptual analysis of the notion of a cause of the universe enables one to recover a number of striking properties of this cause and to assess its significance for theism.
Graham Oppy has emerged as one of the kalam cosmological argument’s most formidable opponents.[ii] Now if Oppy claimed no more with respect to the kalam cosmological argument than what he says of theistic arguments in general, namely, that none of them is, in a strong sense, rationally compelling,[iii] then the contemporary mutakallim (or practitioner of kalam) might have little dispute with Oppy, choosing rather to challenge Oppy’s lofty standard of what constitutes a “good” argument.[iv] Fortunately for discussion’s sake, Oppy’s appraisal of the kalam cosmological argument is so abysmally low that that standard scarcely even comes into view in his discussion. In Oppy’s judgement, the philosophical arguments on behalf of the kalam cosmological argument’s key premiss are “very weak,” providing “no serious support” for the claim that the universe began to exist, while the scientific arguments in support of that claim are bereft of “any merit,” supporting at most the finitude of the universe in the past. Similarly, the argument’s first premiss is “not in the least bit obvious,” and the arguments offered on its behalf are “extremely weak.” If Oppy’s critique is correct, the kalam cosmological argument, even when assessed by standards more realistic and reasonable than Oppy’s own, falls short of being a good argument. In this response, I hope to show that the argument is, in fact, considerably stronger than Oppy claims, surviving even his trenchant critique.
ARGUMENTS FOR THE SECOND PREMISS
Let us turn first to an examination of the four supporting arguments I offer on behalf of premiss (2). Although Oppy characterizes these as two a priori and two a posteriori arguments, this characterization is mistaken, since there is no suggestion that the premises of the two philosophical arguments can be known to be true independent of experience, indeed quite the contrary. We should remind ourselves that even metaphysically necessary truths may in some cases be knowable only a posteriori. A more accurate classification of these supporting arguments would differentiate metaphysical arguments from physical arguments for the beginning of the universe.
First Supporting Argument
The first supporting argument for the finitude of the past is as follows (using Oppy’s numbering):
1.1 An actual infinite cannot exist.
1.2 An infinite temporal regress of events is an actual infinite.
1.3 Therefore, an infinite temporal regress of events cannot exist.
Oppy’s Worry about (1.2)
Oppy agrees with (1.2): “I am happy to grant that an infinite temporal regress is an actual infinite.” [v] Not content with that concession, however, Oppy raises the worry that the same arguments that show the past to be finite will require that the future be finite as well. This concern is curious, since it is irrelevant to the soundness of the argument before us. If the argument implies that the future as well as the past is finite, then (metric) time will someday come to an end (in a Big Crunch, perhaps?), just as it had a beginning. Oppy muses that we shall then have to “worry about the consequences of this conclusion for claims about the extent of life after death.”[vi] Such a worry is misplaced, however, for the kalam cosmological argument makes no claims about life after death, much less its temporal extension. It is hard not to see here a reference to claims of revealed religion which is out of place in this context.
But to consider the worry on its own merits: it is up to Oppy to show that the same arguments against the infinitude of the past imply that the future cannot go on forever. Oppy rightly discerns that on four-dimensionalism, according to which time is tenseless, the distinction between past and future is mind-dependent or perspectival, so that if an actually infinite number of events cannot exist, time must be finite in both the earlier than and later than directions.[vii] But the kalam cosmological argument presupposes from start to finish a theory, not of tenseless time, but of tensed time, according to which temporal becoming is an objective feature of the world. On such a view the future in no sense exists but is pure potentiality. This view seems to allow room for a third option: that while the universe has only a finite past, it has a potentially infinite future. With respect to the future infinity serves merely as an ideal limit which the series of successive events ceaselessly approaches but never achieves. By contrast a beginningless past cannot be merely potentially infinite, lest we commit ourselves to the incoherence of the temporal regress of events’ being finite in the past but ceaselessly growing in the earlier than direction! This fundamental asymmetry of tensed time subverts assertions that arguments against the infinitude of the past entail that the future must be finite and so come to an end.
Ironically, then, when Oppy asserts, “If there are reasons . . . for supposing that the past cannot be infinite, then surely those reasons will carry over to support the contention that the future cannot be infinite,”[viii] the mutakallim may greet Oppy’s claim with insouciance, for he, in a sense, agrees with Oppy’s contention that the future cannot be infinite, namely, if we take “infinite” univocally throughout to signify “actually infinite.” What he disputes is Oppy’s slide from the conclusion that the future is not actually infinite to the conclusion that the future is finite—an inference which holds only if one presupposes a theory of tenseless time. If we instead take “infinite” univocally to signify “potentially infinite,” then it is not the case that the argument against the infinity of the past, namely, the impossibility of backward continuing, carries over to support the contention that the future cannot be infinite.
Oppy concludes that while one may insist that the past is actual and the future is potential, it is “very hard to give non-question-begging content to this insistence. There are two perspectives—that of the presentist and that of the four-dimensionalist—from which there is no such distinction to be drawn.”[ix] Conspicuously missing here is the so-called “growing block” view of the past defended by C. D. Broad and Michael Tooley, according to which temporal becoming is a mind-independent reality but according to which events, once having become, do not cease to be. Perhaps Oppy does not mention this option because he aims merely to show that there are serious philosophical views according to which there is no metaphysical distinction between past and future. Granted; but then the claim that the future is metaphysically distinct from the past is no part of the kalam cosmological argument. It was Oppy who raised the dialectically irrelevant objection that arguments for the finitude of the past also imply the finitude of the future. The mutakallim agrees that this is the case if time is tenseless but can challenge Oppy to show that that conclusion follows if time is tensed. It is now Oppy’s responsibility to show that there are no philosophically tenable options according to which the future, unlike the past, is potentially infinite. Merely mentioning presentism and four-dimensionalism does not do the trick.
Moreover, it is far from clear that the presentist is unable to draw a meaningful distinction between the actuality of the past and the potentiality of the future. Future events have not as of yet been actualized, whereas past and present events have been actualized. Oppy rejoins that if we say something like this, then we “should be prepared to allow that the future is real in a way in which the past is not: the future is still to come in the real world, while the past is not, since it has already occurred.”[x] This rejoinder is odd because Oppy’s characterization captures nicely, in my mind, precisely the way in which the future is unreal and merely potential.
In sum, Oppy agrees with (1.2); and his attempt to show that the argument for the finitude of the past also implies the finitude of the future is irrelevant and, given a theory of tensed time, in any case unsuccessful. Therefore, the key premiss under dispute is (1.1).
Oppy’s Objection to (1.1)
Before we look at Oppy’s objection to (1.1), it is worth re-iterating that when it comes to the claim that an actual infinite cannot exist, I am not talking about what is often called “mathematical existence,” which basically amounts to freedom from proven, strict logical inconsistency. Logical consistency in this sense is usually taken to legitimize a mathematical notion, so that the relevant entities are taken to “exist” in the sense that such a notion plays a legitimate role in mathematical theorizing. This point is important because Oppy tends to blur the distinction between mathematical existence and what we might call pleonastically “ontological existence.” The latter is the subject of debate between Platonists and anti-Platonists with respect to mathematical and other abstract objects. This is a dispute of an ontological, rather than merely logical, nature. Hence, anti-Platonists, whose ontologies do not include mathematical objects, are typically quite content to accord mathematical legitimacy to, say, the set theoretical hierarchy, while Platonists, whose ontologies do include mathematical objects, include among their ranks Intuitionists who reject the legitimacy of higher mathematics.
Oppy classifies anyone who rejects the existence of an actual infinite as either a finitist or a potential infinitist.[xi] He applies this taxonomy to both pure and applied mathematics with regard to proponents’ views on experience, reasonable belief, and ontology respectively. The use of the term “ontology” in this connection is, however, misleading. Oppy is not, in fact, exploring the question of which, if any, numbers exist—otherwise, there is no room in his taxonomy for the anti-Platonist position that no numbers at all exist, however extensive the universe of mathematical discourse may be. Recognizing that Platonism is controversial, Oppy begs off a discussion of it; rather his concern is whether putative numbers are really numbers, that is, entities that would exist if all of classical mathematics were true and the quantifiers used in classical mathematics were ontologically committing.[xii]
What Oppy means by “ontology,” then, is really the question of the extent of legitimate mathematical discourse. The anti-Platonist need by no means be a finitist or potential infinitist in pure mathematics, since he denies either that the statements of classical mathematics are literally true or else that their truth commits one ontologically to the entities quantified over. When it comes to infinity in applied mathematics, Oppy’s concern is with the question of contingently instantiated infinities.[xiii] Thus the question of the existence of an actual infinite in the realm of mathematical objects like numbers has been passed over—unless one holds with Hartry Field that mathematical objects, if they exist at all, do so contingently. Oppy provides a similar taxonomy of finitism, potential infinitism, and actual infinitism in applied mathematics.
Oppy’s declining to deal with the debate over Platonism with respect to mathematical objects affects his final taxonomy of views of the infinite: he lists as the four contending views strict finitism, weak potential infinitism, strong potential infinitism, and (strong) actual infinitism.[xiv] The problem is that these same categories were used to classify positions both with respect to the extent of legitimate mathematical discourse and with respect to the number of contingently existing objects in any domain. Oppy characterizes the strict finitist as having “no proper use of the concept of the infinite;” as rejecting classical mathematics; as allowing only finite domains and magnitudes and only finitely many possible worlds. This conflation of positions with respect to two quite distinct questions leaves no room for someone who is a finitist ontologically but an actual infinitist about mathematical discourse, which he regards as fictional or as not ontologically committing.
Similarly, when Oppy concludes, “If we suppose that we understand classical mathematics, then either we shall be strong potential infinitists or we shall be strong actual infinitists,”[xv] he is conflating the debate over what constitutes legitimate mathematical discourse and the debate over ontology. Oppy must be supposing that Platonism is true and that therefore mathematical discourse is ontologically committing. Why else would he conclude, “To reject the suggestion that it is a contingent matter whether classical conceptions of infinity find application to the extra mathematical world, either we shall be intuitionists or constructivists—hence rejecting classical mathematics and, very likely, classical logic—or we shall be strict finitists”?[xvi]—in opposition to one who holds that it is metaphysically necessary that no actual infinite exist in the extra-mathematical world, just as it is metaphysically necessary that no mathematical objects at all exist, despite the quite legitimate use of the fictional or ontologically non-committing language of classical mathematics.
Now Oppy does not presume to deny the truth of (1.1), for he knows that finitism, whether mathematical or ontological, is a tenable position espoused by some of the most eminent philosophers and mathematicians. He seeks merely to undercut (1.1) by showing my defense of ontological finitism to be very weak. Unfortunately, his reconstruction of my defense misrepresents my case by treating sub-points as major points and then declaring them inadequate as probatory support of (1.1). In point of fact I present only one consideration in defense of (1.1), towit, “that while the actual infinite may be a fruitful and consistent concept in the mathematical realm, it cannot be translated from the mathematical world into the real world, for this would involve counter-intuitive absurdities.”[xvii] My exposition of Cantorian set theory is merely a praeparatio for the introduction of the puzzle cases elsewhere discussed by Oppy, and my animadversions on the ontological status of sets was a defensive move aimed at showing that ontological finitism does not imply mathematical finitism.
The key consideration, then, is the puzzle cases. Before we discuss some of these cases, however, we should do well to dismiss another theological intrusion into Oppy’s critique: if (1.1) is true, he claims, then there is no sense in which God can be actually infinite. He “can know only finitely many things, can perform only finitely many actions, and so forth.”[xviii] This theological consequence is irrelevant to the argument for (1.1) The kalam cosmological argument does not even aspire to prove that the personal Creator of the universe is omniscient or omnipotent. Moreover, we see again the slippage from denying that God is actually infinite in the quantitative sense at issue in (1.1) to the conclusion that God is finite in various respects. This does not follow (nor does Oppy present any argument for the implication), since the quantitative sense of infinity may be simply inapplicable to God. Omniscience need not entail knowing an infinite number of, say, propositions, nor need we think of omnipotence as entailing the ability to do an infinite number of actions.[xix] There is thus no reason to think that an orthodoxly conceived monotheistic God is susceptible to this sort of quantitative analysis.
Turn now to what Oppy calls the puzzle or problem cases. Because these involve, not strict logical inconsistencies, but, as I put it, “counter-intuitive absurdities,” whether one finds them troubling enough to embrace (1.1) will be to a considerable degree subjective. I find them sufficiently troubling, and I hope my readers will, too; indeed, I think they should, although I do not aspire to prove this.[xx] Benardete, who is especially creative and effective at concocting such thought experiments, puts it well: “Viewed in abstracto, there is no logical contradiction involved in any of these enormities; but we have only to confront them in concreto for their outrageous absurdity to strike us full in the face.”[xxi]
The “key point” that Oppy wants to make with regard to such cases is that they show at most that certain kinds of actual infinities cannot exist, but that this conclusion cannot be generalized.[xxii] This riposte strikes me as appropriate with respect to certain puzzles involving actual infinites; for example, those imagining completion of a so-called super-task, the sequential execution of an actually infinite number of definite and discrete operations in a finite time. But when it comes to cases involving the simultaneous existence of an actually infinite number of familiar macroscopic objects, then this sort of response seems less plausible. The difficulty here is two-fold: (i) nothing in the various situations seems to be metaphysically impossible apart from the assumption of an actual infinite; and (ii) the absurdities are not tied to the particular kinds of objects involved. If a (denumerably) actually infinite number of things could exist, they could be numbered and manipulated just like the guests in Hilbert’s Hotel or the books in an infinite library. In cases like these, any metaphysical absurdity is plausibly attributed to the existence of an actual infinite rather than to the particularities of the case. Thus, the problem cases of this sort do, it seems to me, call into question the possibility of the existence of an actually infinite number of things.
In any case, when we turn to Oppy’s detailed discussion of Hilbert’s Hotel,[xxiii] he does not, in fact, adopt the above strategy for dealing with the puzzle. Instead, he proposes simply to “outsmart” those who present such problem cases, where “outsmart” is defined as follows: “Outsmart, v. To embrace the conclusion of one’s opponent’s reductio ad absurdum argument.”[xxiv] He says that the finitist faces a dilemma: if a detailed physical account cannot be given of a hotel that would permit the transposition of guests envisioned in the story, then the metaphysical impossibility of Hilbert’s Hotel lies, not in its actual infinity, but in the envisioned maneuvering of guests. Although Oppy makes the point that the possibility of an infinite hotel with no vacancies does not commit one to the possibility of accommodating more guests by shifting guests about—maybe the hotel’s construction hinders the guests’ movements or the guests die off before their turn to move comes round—, nevertheless he remains fairly confident that the requisite account could be given. But if there is a way of giving such a detailed account, he asserts, then it turns out that Hilbert’s Hotel is possible after all. “There can, after all, be a hotel in which infinitely many new guests are accommodated, even though all the rooms are full, via the simple expedient of moving the guests in room N to room 2N (for all N).”[xxv]
But how does that follow? The ontological finitist does not think that the metaphysical impossibility of Hilbert’s Hotel lodges in mere physical considerations of layout and construction but in the supposition that an actual infinite can exist. Giving a detailed account of the construction of Hilbert’s Hotel on the assumption that an actually infinite number of things can exist provides no non-question-begging reason to think the whole scenario possible. Oppy justifies his recourse to the outsmarting strategy with the assertion, “these allegedly absurd situations are just what one ought to expect if there were . . . physical infinities.”[xxvi] This justification, however, falls short: for counterfactuals of the form “If a physical infinity of such-and-such a nature were to exist, then such-and-such a situation would obtain” are not in dispute. The problem cases would, after all, not be problematic if the alleged consequences would not ensue! Rather the question is whether these consequences really are absurd. The outsmarting strategy does nothing to alleviate one’s doubts that a Hilbert’s Hotel is absurd.
Moreover, what would Oppy say about scenarios involving inverse arithmetic operations regarding the guests in Hilbert’s Hotel?[xxvii] In transfinite arithmetic, inverse operations of subtraction and division with infinite quantities are prohibited because they lead to contradictions, but in reality one cannot stop people from checking out of a hotel if they so desire! In this case one does wind up with logically impossible situations, such as subtracting identical quantities from identical quantities and finding non-identical differences. We shall have occasion to discuss other problem cases below, but, in the meantime, I am far from satisfied with Oppy’s attempt to take the sting out of a problem case like Hilbert’s Hotel by simply “outsmarting” his opponent.
In sum, once the irrelevant theological considerations are set to the side, we find that Oppy actually agrees with the second premiss of the first supporting argument for the finitude of the past and fails to establish what he says is the “key point” with respect to the first premiss, preferring instead simply to “outsmart” his opponent, a strategy which falls short in showing why we ought not to be troubled by the apparent absurdity of the problem cases. Nor need we, pace Oppy, abridge classical mathematics in order to reject reasonably the metaphysical possibility of an actual infinite.
Second Supporting Argument
The second supporting argument for the beginning of the universe is as follows:
2.1 The temporal series of events is a collection formed by successive addition.
2.2 A collection formed by successive addition cannot be actually infinite.
2.3 Therefore, the temporal series of events cannot be actually infinite.
In Oppy’s view, this argument “fares even worse than the first,” for no clear and uncontroversial support can be given for either of its premises.[xxviii] Although the demand for uncontroversial support is unrealistic, Oppy’s claim that no clear support is forthcoming is important.
Oppy’s Objection to (2.1)
Although in my original exposition of the kalam cosmological argument I took (2.1) to be “obvious enough,” Oppy seeks to undercut (2.1) by demanding “some very substantial independent argument” before we are persuaded to accept it.[xxix] For, he points out, if time has the structure of the real numbers and if there are continuous processes taking place in time, then the collection of past events will not have been formed by successive addition.
It seems to me that this objection is based on multiple misunderstandings. In the first place, by “successive addition” I meant the accrual of one new element per (later) time through a process of temporal becoming. As we shall see, Oppy understands successive addition tenselessly or even timelessly. Second, I stipulate that all the events in the temporal series under consideration have equal, non-zero duration. Defining my terms, I wrote,
By an ‘event,’ I mean any change occurring within the space-time universe. Since any change takes time, there are no instantaneous events. Neither could there be an infinitely slow event, since such an ‘event’ would in reality be a changeless state. Therefore, any event will have a finite, non-zero duration. In order that all the events comprising the temporal regress of past events be of equal duration, we arbitrarily stipulate some event as our standard, and, taking as our point of departure the present standard event, we consider any series of such standard events ordered according to the relation earlier than. The question is whether this series of events is comprised of an actually infinite number of events or not.[xxx]
Even if time itself has the structure of the real numbers and there are continuous physical processes, nevertheless the series of events, as defined, will be formed by successive addition. If we take, for example, the collection of past seconds in the history of the universe, such a collection has been formed by the successive addition of seconds, even if those seconds can themselves be divided into infinitely many sub-intervals.
While I still think that (2.1) is obvious enough, nonetheless I do recognize that it is susceptible to powerful putative defeaters which will require substantial independent argument for (2.1) in response. For (2.1) assumes a theory of tensed time and the reality of temporal becoming. When I wrote The Kalam Cosmological Argument I was vaguely aware of theories of tenseless time but did not really take them seriously. Since then I have come to appreciate that tenseless time theorists like Oppy himself need to be taken very seriously, indeed, and so I have devoted two volumes to a defense of a theory of tensed time and of the objectivity of temporal becoming.[xxxi] While I still think that temporal becoming is as obvious as can be, even more obvious than the existence of the external world given us by sense perception, so that someone encountering (2.1) in ignorance of theories of tenseless time is quite within his rights in accepting it without substantial independent argument, nonetheless I recognize that a considerable defense must be mounted against detensers and defenders of the mind-dependence of becoming in order to defeat their proffered defeaters.
Moreover, I have myself argued elsewhere that if temporal becoming is an objective feature of reality, then time cannot have the structure of the real numbers.[xxxii] For temporal becoming would require the lapse of consecutive instants of time, which do not exist if time is continuous. Hence, if temporal becoming is real, as the kalam cosmological argument assumes, then Oppy’s view of time as a composition of instants is false. Since Oppy raises no other objection to (2.1), I hope by my above-referenced work to have met his demand for very substantial independent argument on its behalf.
Objections to (2.2)
In support of (2.2) I have argued that just as it is impossible to count to infinity, so is it impossible to count down from infinity. I take it that given the reality of temporal becoming the impossibility of counting to infinity is patent, since one cannot convert a potentially infinite series into an actually infinite series by successive addition of finite quantities. For, given any finite number n, n+1 equals a finite number. Hence, À0 has no immediate predecessor; it is not the terminus of the natural number series but stands, as it were, outside it and is the number of all the members in the series. While we can imagine an actually infinite series of events mapped onto a tenselessly existing infinite series of temporal intervals, such that each consecutive event is correlated with a unique consecutive interval, the question remains whether such a sequence of intervals can be instantiated, not tenselessly, but one interval after another. As remarked, the very nature of the actual infinite precludes this.
It is therefore surprising to find Oppy challenging the claim that an actual infinite cannot be formed by beginning at a point and successively adding members to the collection. I cannot help but think that his habit of thinking in tenseless categories misleads him here. For example, when Oppy, imagining a man running through empty space on a path of stone slabs so constructed that when his foot strikes the last slab another appears in front of him, says, “if the man runs for an infinite amount of time—that is, if for each [natural number] n, there is an nth slab that the man crosses—it is nonetheless true that infinitely many slabs are crossed: there is an actually infinite collection that is formed by successive addition,”[xxxiii] one is struck by the tenseless verbs employed throughout. Even when Oppy tries to take account of tense, commenting, “Craig will resist this way of characterizing matters: given his view that the future is not real, he will insist that it is at best true that infinitely many slabs will be crossed: the collection that is formed here by successive addition is at best ‘potentially infinite’,”[xxxiv] his inveterate habit of thinking in tenseless terms trips him up.[xxxv] For if temporal becoming is real, an infinite number of slabs will never be crossed: the finite series will just go on forever.[xxxvi]
How desperate, then, to attempt to refute the tensed time theorist’s position by turning the man’s task into a super-task, so that the slabs are crossed in progressively shorter intervals, the first in a half minute, the second in a quarter minute, the third in an eighth minute, and so on, so that at the end of one minute an actual infinity of slabs have been successively crossed! This is a fantasy that should not be taken seriously. That it is fantasy is evident in the fact that in all such scenarios the final state at w + 1 is causally unconnected to the successive states in the w series of states. Since there is no last term in the w series, the state of reality at w + 1 appears mysteriously from nowhere. The man (or a bouncing ball substituted for him) cannot reach the slab numbered w + 1 without having stepped there from the immediately preceding slab. The absurdity of such supertasks underlines the metaphysical impossibility of trying to convert a potential into an actual infinite by successive addition.
Oppy responds, “But, of course, the assumption that there must be an immediately prior instant is precisely what proponents of the possibility of this kind of supertask deny: if time is a continuum, then there is no instant that is immediately prior to a given instant.”[xxxvii] Here again we see the assumption that time may be adequately treated tenselessly as isomorphic to a line, an assumption which is false if temporal becoming is real. It is no accident that friends of super-tasks tend to be partisans of tenseless time. Moreover, their simply denying that there must be an immediately prior instant is hardly refutation of the claim that given a series formed throughout by successive addition the state of a physical object at w + 1 must be causally connected with an immediately preceding state. Why else is the lamp, after an infinite series of successive switchings, on rather than off (or off rather than on) at w + 1?
In any case, such super-tasks are not relevant to the argument under consideration, which concerns a collection of events which are all by definition equal in duration. Of course, the successive formation of the series of past events is not a case of beginning at some point and never ending but of the inverse, namely, never beginning but ending at some point. This strange case is reminiscent of Zeno’s Dichotomy paradox and is featured in the thesis of Kant’s First Antinomy concerning time, both of which I found insightful. Oppy, however, is considerably less enthusiastic: “there is nothing in either Zeno’s paradoxes or the first Kantian antinomy,” he says, to support (2.2).[xxxviii]
Curiously, however, Oppy in his discussion of the Dichotomy actually fails to say anything to resolve the paradox. He merely provides the customary mathematical analysis of the distance to be traversed during the time involved in terms of twin series of progressively shorter/briefer intervals converging to a point/instant at which Achilles is standing still prior to his run. Given that Achilles is standing still at the limit point/instant, “there is (of course) no first instant at which Achilles is in motion, and no first point that he moves to when he moves from the point at which he is at rest.”[xxxix] Zeno would agree! Oppy then simply declares, “But it would be a mistake to suppose that these considerations show that there cannot be motion—or change of state from being at rest to being in motion—if space and time have the structure of the real numbers.”[xl] This is settling the question by fiat. If we reject the Aristotelian analysis of the spatial interval as divisible into potentially infinitely many subintervals rather than as compounded out of an actually infinite number of subintervals, and if we are taking tense seriously rather than treating time as a tenselessly existing extension, then it is devilish hard to see how Achilles can even move. Be that as it may, in the case of an infinite past, my point was that the temporal sub-intervals traversed are not converging toward a limit but are of equal duration, so that the usual “solutions” to the Dichotomy paradox become irrelevant. How an enduring object could live through an actually infinite number of, say, hours to arrive at the present hour remains mysterious.
Oppy’s discussion of the thesis of Kant’s First Antinomy concerning time is even less adequate. The question raised in the thesis concerning time is how an infinite temporal series can have been formed by successive synthesis. Oppy agrees that “The infinite series 1, 2, 3, . . . , n, . . . cannot be completed by successive synthesis, if what is required is that there should be a last member of the series that is reached by adding units.”[xli] But then he makes the remarkable suggestion:
if we consider the very same series of elements in reverse order, . . . . n, . . . , 3, 2, 1, then we do have an infinite series that is completed by successive synthesis. Each member of the series is the successor of the immediately preceding element—reached by subtracting a unit—and there is a last element. If we understand the proposal that the world has no beginning in time to be the proposal that the series of states of the world is in one-one correspondence with the series . . . , n, . . . , 3, 2, 1, then Kant’s key assumption [viz., that there cannot be an infinite series with a last member in which each member is the unique successor of some other member] seems to be a very crude begging of the main point at issue.[xlii]
The proposed mathematical analogy is clearly inapplicable to tensed time. For the present event to have been reached by the successive subtraction of prior events would require that the temporal series of events exist tenselessly and yet suffer progressive diminution by the beginningless attrition of events from the earlier than direction until the past has been entirely deleted, leaving us at the present event! It is not enough to put the series of past events into a tenseless one-to-one correspondence with some series of numbers. The question is how the series has been formed, and subtraction is clearly maladroit.
Now Oppy says that it would be “a very bad objection” to insist that successive synthesis requires addition rather than subtraction, so that Kant’s argument is vindicated after all.[xliii] For “ ‘Successive synthesis’ requires no more than that each number of a series is derived in a law-governed fashion from the preceding member of the series.”[xliv] Once again, we see the tendency of detensers to strip tensed notions of all tense and even temporality. Rooted in the long tradition of kalam, Kant’s First Antinomy concerning time cannot be properly understood unless we recognize its use of irreducibly tensed concepts, particularly successive synthesis.
Finally, Oppy points out that if we do insist that an actually infinite series with a last member cannot be formed by successive addition as opposed to subtraction, then a different numerical series is available as a counter-example: . . . , -n, . . . , -3, -2, -1. “In this series, each member is obtained from the preceding member by the addition of a unit: Successive synthesis if ever there were such a thing.”[xlv] Oppy is correct in identifying the ordinal structure of the series of past events with that of the negative numbers, namely, w*; but there is no successive synthesis among the timelessly existing members of the negative number series. Kant’s worry about how the series of past events, having the ordinal type w*, could be formed by adding one member after another is not even addressed, much less resolved.
The formation of an infinite series of past events by successive addition would be like someone’s counting down all the negative numbers ending with 0 in the present. This happens to be one of the problem cases discussed by Oppy, so that it will repay effort to see what he has to say on that head. The bulk of his discussion, however, concerns the possibility of counting forwards to infinity, a discussion vitiated, as we have seen, by his assumption that if someone does not stop counting, then he does count to infinity. When he finally comes to the problem of counting backwards from infinity, Oppy is unusually concessive:
even if something very much like counting forwards to infinity turns out to be unproblematic, it remains highly doubtful that anything like counting backwards from infinity is similarly unproblematic. All of the kinds of objections that were raised in connection with the Tristram Shandy case will arise here as well; we shall need a detailed examination of principles of sufficient reason in order to determine whether we should allow that it is possible that there be a creature that does something much like count backwards from infinity.[xlvi]
Since the successive synthesis of an infinite past does, on Oppy’s own analysis, involve something very much like counting down from infinity, it is hard to understand why he treats Kant’s argument so dismissively.
Let us pursue Oppy’s trail further, then, by examining his response to the Tristram Shandy case, which involves a man who writes his autobiography so slowly that it takes him a whole year to record the events of a single day. Suppose that Tristram Shandy has been writing from eternity past at the rate of one day per year. Robin Small has shown that if Tristram Shandy hasbeen writing for an infinite number of years, then the most recent recorded day of his autobiography recedes to infinity, that is to say, to a day infinitely distant from the present.[xlvii] Nowhere in the past at a finite distance from the present can we find a recorded day, for by now Tristram Shandy has fallen infinitely far behind. The beginningless, infinite series of days which he has recorded are days which lie at an infinite temporal distance from the present. But it is impossible to traverse the temporal interval from an infinitely distant event to the present, or, more accurately, for an event which was once present to recede to a point infinitely temporally distant.
In response, Oppy cautions that by adding further assumptions to the Tristram Shandy story we can, as with any story, generate an inconsistency, but that does nothing towards showing that it is impossible for the series of past events to be actually infinite. It is simply inconsistent, he claims, to suppose that Tristram Shandy is now writing about any particular past day, treating the days as consecutive days. I am puzzled by Oppy’s response, since Small’s analysis did not try to identify which particular past day Tristram Shandy is presently recording but aimed to show merely that the recorded days lie at an infinite remove from the present. This is not in itself a contradiction. The infinite past must have in this case, not the order type w*, but the order type w* + w*, the order type of the series . . . , -3, -2, -1, . . . , -3, -2, -1. The problem rather is how one gets from the first series into the second by means of successive addition or temporal becoming. Oppy also rightly observes that it is the whole scenario which is impossible, which includes the requirement that consecutive days be recorded.[xlviii] But given that the task of writing one’s autobiography at the rate of one consecutive day per year seems obviously coherent, it seems that the blame must be placed on the assumption of the infinity of the past. What follows from the Tristram Shandy case, then, is that an infinite series of past events is absurd.
But suppose that such an infinite task could be completed by the present day. Here we encounter a problem that also arises with respect to the case of the person who claims to have been counting down from infinity and who is now finishing: . . . , -3, -2, -1, 0. We could ask, why did he not finish counting yesterday or the day before or the year before? If the person would have finished his countdown by today, then surely he would have finished it by yesterday, given that he has already had infinite time to complete the task.
Oppy’s initial response to this question is very odd. He says that Tristram Shandy’s writing (like the man’s countdown) has from eternity past always been converging on a certain endpoint T. “In order for him to put down his pen [or cease counting] at some other time T´ his writing [or counting] would need to have been converging on that other time.”[xlix] This response amounts to no more than saying that it has always been true that the relevant person will finish his task at time T, which is a truism. But why will he finish at T rather than T´? Oppy answers that is not clear that this is a serious difficulty, for, he asks, why not suppose that Tristram Shandy’s finishing when he does or the man’s completing his countdown when he does is just “a brute feature of the scenario, that is, a feature that has no explanation?”[l] It has always been the case that he will finish when he does, but why the man finishes when he does rather than at some other time is just inexplicable.
Resting with inexplicability may seem unsatisfactory, however, especially in light of the respectable role such reasoning plays in scientific cosmological discussions. Oppy justifies his insouciance on the grounds that Principles of Sufficient Reason requiring that there be an explanation are highly contentious. Oppy elsewhere presents objections to various versions of the Principle of Sufficient Reason such as the impossibility of providing an explanation of what has been called the “Big Contingent Conjunctive Fact” (BCCF), which is the conjunction of all the contingent facts there are, or of libertarian free choices.[li] The problem with this justification, however, is twofold. First, plausible defenses of the Principle of Sufficient Reason can be given.[lii] Oppy deems it “a delicate matter to discover a principle of sufficient reason that is both strong enough to yield the desired conclusion and yet not obviously in need of additional argumentative support.”[liii] Nonetheless, he himself thinks that it is “very plausible” that there are acceptable instances of the following schema for a principle of sufficient reason:
O (for every FG of kind K, there is an F¢G¢ that partly explains why the GFs rather than Q possible alternatives),
where O is an operator like “necessarily,” “it is knowable a priori,” etc., G is an ontological category such as a proposition, state of affairs, etc., F is a restriction such as true, contingent, etc., and Q is a quantifier like “any,” “every,” etc.[liv] But he thinks that it is not at all clear that there are acceptable instances of this schema that can be used to rule out scenarios like counting down from infinity. Although I am not sure what Oppy means by “GFs,” the following principle would seem to be an instance of his schema:
Necessarily, for any contingent state of affairs involving concrete objects there is a contingent state of affairs that partly explains why that state of affairs obtains rather than any other.
Such a principle would require that there be some partial explanation for why the man finishes his countdown today rather than at some other time. But, as we have seen, not even a partial explanation of why he finishes when he does can be given, for regardless of how we vary such factors as the rate of counting, they will be the same regardless of the time that he finishes and so do not furnish even a partial explanation of why he finishes today. So why is this instance of the schema not acceptable?
Second, and more to the point, there is no reason to think that requiring an explanation in the present case demands for its acceptability or plausibility the enunciation and defense of some general Principle of Sufficient Reason. Indeed, any such principle is apt to be tested inductively for its adequacy by whether cases like this constitute plausible counterexamples. The exceptions offered by Oppy, such as the inexplicability of the BCCF and libertarian choices, are simply irrelevant to the present case, for the BCCF is not at stake nor can a person counting from eternity at a constant rate choose arbitrarily when to finish his countdown. In the case under discussion we have a good reason to think that the man should have finished his countdown earlier than any time that he does, namely, he has already had infinite time to get the job done.[lv] If we deny that infinite time is sufficient for completing the task, then we shall wonder why he is finishing today rather than tomorrow or the day after tomorrow, or, indeed, at any time in the potentially infinite future. It is not unreasonable to demand some sort of explanation for why, if finishes today, he did not already finish yesterday. By contrast, if such a countdown is metaphysically impossible, then no such conundrum can arise. But clearly, there is no metaphysical impossibility in counting backwards for all time, unless time is past eternal. It follows that the past cannot be infinite.
Another problem case that arises in connection with (2.2) is what Oppy calls al-Ghazali’s Problem. The great mutakallim envisions our solar system’s existing from eternity past, the orbital periods of the planets being so co-ordinated that for every one orbit which Saturn completes Jupiter completes 2.5 times as many. If they have been orbiting from eternity, he muses, which planet has completed the most orbits? The correct mathematical answer is that they have completed precisely the same number of orbits. But this seems absurd, for the longer they revolve the greater becomes the disparity between them, so that they progressively approach a limit at which Jupiter has fallen infinitely far behind Saturn. Yet, being now actually infinite, their respective completed orbits are somehow magically identical in number. Indeed, they will have “attained” infinity from eternity past: the number of completed orbits is always the same.
Oppy’s discussion of al-Ghazali’s problem fails to connect with the problem as I understand it, since Oppy, construing the problem tenselessly, takes its point to be that there is a logical contradiction with respect to the number of orbits completed, so that he spends most of his space arguing that given Cantorian assumptions there is no unequivocal sense in which the number of orbits both is and is not same.[lvi] Temporal becoming, which lies at the heart of the puzzle, is left wholly out of account. The longer the planets rotate the more the numbers of their respective orbits diverge, yet having now revolved for infinite time their orbits are numerically identical, which seems absurd.
For all of these reasons the formation of an actual infinite by successive addition is a notoriously difficult notion, even more so than the static existence of an actual infinite. Philosophers typically deal with the problem cases Oppy discusses only by treating time tenselessly. If we take tense seriously, as this second supporting argument does, then it is very difficult, indeed, to see how the series of past events can be actually infinite. So while the two metaphysical arguments on behalf of the second premiss of the kalam cosmological argument may not approach Oppy’s standard of being rationally coercive, they are strong enough, in the absence of countervailing arguments for the infinitude of the past, to warrant belief that the universe began to exist.[lvii]
Third Supporting Argument
We come now to the first physical supporting argument for (1.2), the evidence of the expansion of the universe. Oppy rightly points out that while non-initial stages of the expanding universe are well-described by current cosmological models, we still lack the marriage of General Relativity and Quantum Mechanics requisite for a description of the earliest stage of the universe. It is to Oppy’s credit that he recognizes that such a description is not a necessary condition of having good scientific grounds for thinking that the universe began to exist. The history of twentieth century cosmogony has, in one sense, been a series of failed attempts to craft acceptable non-standard models of the expanding universe in such a way as to avert the absolute beginning predicted by the standard model. This parade of failures can be confusing to the layman, leading him mistakenly to infer that the field of cosmology is in constant flux, as new theories of the universe’s origin continually come and go, with no assured results. In fact, the standard model’s prediction of an absolute beginning has persisted through a century of astonishing progress in theoretical and observational cosmology and survived an onslaught of alternative theories. With each successive failure of alternative cosmogonic theories to avoid the absolute beginning of the universe predicted by the standard model, that prediction has been corroborated.
A watershed of sorts appears to have been reached in 2003 with Arvind Borde, Alan Guth, and Alexander Vilenkin’s formulation of a theorem establishing that any universe which has on average over its past history been in a state of cosmic expansion cannot be eternal in the past but must have a spacetime boundary.[lviii] Any universe which meets this condition cannot be extrapolated into the infinite past. Theorists intent on avoiding the absolute beginning of the universe could previously take refuge in the period prior to the Planck time, an era so poorly understood that one commentator has compared it with the regions on the maps of ancient cartographers marked “Here there be dragons!”—it could be filled with all sorts of chimaeras. But the Borde-Guth-Vilenkin theorem does not depend upon any particular physical description of the universe prior to the Planck time, being based instead on deceptively simple physical reasoning which will hold regardless of our uncertainty concerning that era. It single-handedly sweeps away the most important attempts to avoid the absolute beginning of the universe, in particular the hypothesis of an eternal inflationary multiverse and higher dimensional brane cosmologies. Vilenkin pulls no punches: “It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. With the proof now in place, cosmologists can no longer hide behind the possibility of a past-eternal universe. There is no escape, they have to face the problem of a cosmic beginning.”[lix] The Borde-Guth-Vilenkin theorem is now widely accepted by cosmologists. As a result, theorists who would avert the beginning of the universe are forced to deny the single assumption of that theorem: that the universe’s history has been one of cosmic expansion. Although speculative models of the universe have been crafted on the assumption that that condition is not met, such models encounter daunting difficulties, both observationally and theoretically.[lx]
Nevertheless, Oppy expresses four reasons to be “cautious.” Now the admonition to be cautious is one with which no one would disagree. But an examination of Oppy’s four reasons suggests that he is not being cautious but is rather making a determined effort to avoid the drift of the evidence. His first two reasons concern the initial cosmological singularity found in the standard model. In dependence upon John Earman’s study of singularities in General Relativistic spacetimes,[lxi] Oppy notes that whether the spacetime metric is extendible through an initial singularity depends on certain mathematical conditions placed on that metric. This hardly constitutes a reason for caution, however, since such conditions are virtually universally accepted by physicists as part of physically meaningful cosmogony. The point of Earman’s discussion is to show that “under plausible constraints on what is to be counted as a physically meaningful extension, there are no physically meaningful extensions through the big bang of the standard models.”[lxii] The presence of an initial cosmological singularity is uncontroversially taken to preclude spacetime’s extendibility to prior times.
Second, Oppy seeks to capitalize on Earman’s remark that “Even if my argument succeeds, it remains open that there is some mathematically meaningful extension—involving lower continuity/differentiability conditions than those required for a physically meaningful extension—and that God or some other metaphysical cause operates in this mathematical time.”[lxiii] Earman does not take this idea seriously,[lxiv] but Oppy suggests that the hypothesis that God operates in a prior metaphysical time to create the universe leaves room for the suggestion that there is some other cause of the universe which “is part of an infinite regress of contingent causes.”[lxv] Then there is nothing in the empirical evidence for a singular beginning of spacetime to rule out the claim that there is an infinite regress of prior contingent causes. I must confess that I have no idea at all what Oppy is talking about. Such a purely mathematical extension is physically impossible, so that the entities of that prior regime are presumably abstract objects which cannot by their very nature stand in causal series. In any case, Oppy’s postulation of a prior regime of temporally ordered contingent causes is an exercise in metaphysics, and to advert to metaphysics at this point just is to admit that the scientific evidence supports (1.2).
In the end Oppy’s real point in mentioning these exotic possibilities is to say that in light of such possibilities we are “far from having good reason” to suppose that quantum gravitational replacements of the standard model will feature an absolute beginning of the physical universe, or at least of the contingent universe. This inference is multiply confused. First, any hypotheses on which quantum gravity models might be extended into the infinite past have absolutely nothing to do with the two possibilities mentioned by Oppy. Indeed, quantum gravity models typically have non-singular origins, so that there is no question of trying to extend spacetime metric through a singularity. Second, the suggestion that the physical universe may have an absolute beginning but that the contingent universe (presumably that series of contingent causes prior to the singularity) does not draws a distinction foreign to physical cosmology and therefore plays no role in quantum cosmology. Any such speculation is metaphysical. Third, most of the models of the three major research programmes being pursued in quantum gravity today—string theory, loop quantum gravity, and semi-classical approaches like the Hartle-Hawking and Vilenkin models—are not past eternal but involve an absolute beginning of the universe, as in the standard model. Finally, fourth, the Borde-Guth-Vilenkin theorem gives us, as we have seen, good reason to think that tenable quantum gravitational replacements of the standard model will not be past eternal.
The third reason Oppy gives for exercising caution is that “in the standard Big Bang models, for every time t , there is an earlier time t´, and the state of the universe at t´ is a causal determinant of the state of the universe at t. Thus, it turns out that even in the standard Big Bang models, there is no ‘absolute beginning’ of the physical universe.”[lxvi] The salient point is that since the spacetime metric cannot be extended all the way to t=0, the universe lacks a beginning point. Hence, the universe did not, despite its past temporal finitude, begin to exist.[lxvii]
The fundamental shortcoming of this objection is its assumption that having a beginning entails having a beginning point. This is not, in fact, how the locution “begins to exist” is typically understood. For one thing, the proposed definition commits us to the reality of points, which surely overloads the expression “begins to exist” with unintended ontological commitments. Moreover, the use of the expression in astrophysical cosmogony belies the supposed entailment. Contemporary cosmologists frequently “cut out” the initial cosmological singularity as a merely ideal point on the boundary of spacetime so that the universe has no beginning point of its existence; but they do not therefore think that the universe no longer begins to exist or that the mystery of the origin of the universe has thereby been solved. Rather the key idea in having a beginning is past metrical finitude. Time may be said to begin to exist just in case for any non-zero, finite interval of time that one picks, there are only a finite number of congruent intervals earlier than it. Or, alternatively, time begins to exist just in case for some specified non-zero, finite interval of time, there are no congruent intervals earlier than it. On either explication beginning to exist does not entail having a beginning point.[lxviii] Oppy’s third worry therefore need not trouble us.
Oppy’s fourth reason for caution is that if we concede, as seems right, that the physical universe is finite in the past and so in this sense “begins to exist,” then the question is whether the first premiss of the kalam cosmological argument is true under this interpretation of “begins to exist.” Is it true that everything with a finite past has a cause? Having little confidence in his foregoing worries, Oppy now agrees that the physical universe probably is finite in the past but thinks that the universe might have come into being uncaused.
Why does he think that? Oppy later explains that if we adopt Grünbaum’s explication of “object x begins to exist at time t,” then the kalam cosmological argument “is in ruins.”[lxix] The mutakallim will be unfazed by this conclusion, however, since Grünbaum’s explication is patently inadequate, not to say irrelevant. For, according to Grünbaum, x begins to exist at t just in case (i) x exists at t, (ii) there are times prior to t, and (iii) there is no time prior to t at which x exists. This would exclude by definition that time began to exist, a negative conclusion which is contrary to physical cosmogony and which ought not in any case to be settled by mere definition.
Oppy argues that on my own explication of “x begins to exist at t” the beleaguered kalam cosmological argument still “is in ruins”[lxx]—that is, if we assume that an object begins to exist if and only if there is some time (that is, instant) at which it begins to exist. For I amend Grünbaum’s account in such a way that “x begins to exist at t” just in case (i) x exists at t and (ii) there is no time prior to t at which x exists. If we do not accord any reality to t=0, observes Oppy, then the universe on this account does not begin to exist. But, of course, I simply reject the gratuitous assumption that something’s beginning to exist implies its beginning to exist at t, at least if t ranges only over instants. If t ranges over non-zero finite intervals as well, then the explication is both adequate and unproblematic.[lxxi] The question, then, to be taken up below, is whether the causal premiss of the kalam cosmological argument is more plausibly true than false under such an understanding of “begins to exist.”
In sum, Oppy’s treatment of the scientific evidence for the universe’s beginning from the expansion of the universe reveals a determined scepticism. He draws sceptical conclusions from worries that the community of contemporary cosmologists does not share. He assumes, contrary to normal and scientific usage, that beginning to exist entails having a beginning point. Even conceding that the physical universe probably is finite in the past, he is sceptical that there need be a cause of the universe’s coming into being. In short, he does not give good grounds for resisting the evidence from contemporary cosmology for the second premiss of the kalam cosmological argument.
Fourth Supporting Argument
The second supporting scientific argument for the universe’s beginning is based on the thermodynamic properties of the universe. In a certain respect, the evidence of thermodynamics is even more impressive than the evidence afforded by the expansion of the universe, for while an accurate physical description of the universe prior to the Planck time remains and perhaps always will remain unknown, thereby affording room for speculations aimed at averting the origin of time and space implied by the expanding cosmos, no such uncertainty attends the laws of thermodynamics and their application. Indeed, thermodynamics is so well-established that this field is virtually a closed science.[lxxii]
Already in the nineteenth century, physicists realized that the application of the Second Law of Thermodynamics to the universe as a whole implied a grim eschatological conclusion: given sufficient time, the universe will eventually come to a state of equilibrium and suffer “heat death.” But this apparently firm projection raised an even deeper question: if, given sufficient time, the universe will suffer heat death, then why, if it has existed forever, is it not now in a state of heat death? The advent of relativity theory and its application to cosmology altered the shape of the eschatological scenario predicted on the basis of the Second Law but did not materially affect the fundamental question. In contrast to their nineteenth century forebears, contemporary physicists have come to question the implicit assumption that the universe is past eternal. P. C. W. Davies concludes, “The universe can’t have existed forever. We know there must have been an absolute beginning a finite time ago.”[lxxiii]
Oppy gives short shrift to the evidence of thermodynamics. This is unfortunate because this field continues to generate a good deal of discussion in contemporary cosmology. Very recent discoveries provide strong evidence that there is effectively a positive cosmological constant which causes the cosmic expansion to accelerate rather than decelerate. Paradoxically, since the volume of space increases exponentially, allowing greater room for further entropy production, the universe actually grows farther and farther from an equilibrium state as time proceeds. But the acceleration only hastens the cosmos’s disintegration into increasingly isolated material patches no longer causally connected with similarly marooned remnants of the expanding universe. Each of these patches faces, in turn, thermodynamic extinction. Thus, the same pointed question raised by classical physics persists: why, if the universe has existed forever, is it not now in a cold, dark, dilute, and lifeless state?
If we postulate the finitude of past time and space, such problems are avoided. The reason for the observed disequilibrium state is that spacetime had an absolute beginning in a low entropy condition a finite time ago and is on its way towards states of increasing disorder.
Oppy dismisses the evidence of thermodynamics for two reasons: (i) “thermodynamical considerations can establish only that the physical universe is finite in the past: they cannot establish that there is no infinite regress in the contingent universe”; and (ii) “neither can they establish that there was an initial state of the universe at t=0.”[lxxiv] These reasons are very weak. With respect to (i), insofar as Oppy draws a distinction between the physical universe and the contingent universe, he is introducing a distinction which is foreign to astrophysical cosmology, not to say altogether mysterious. It just is to admit that the evidence of thermodynamics shows that the universe is not past eternal. As for (ii), there is neither interest in nor necessity of showing that t=0 was an initial state of the universe. Oppy implicitly admits that thermodynamical considerations do show that the universe began to exist in that past time is metrically finite, and so he is forced to deny, not the second, but the first premiss of the kalam cosmological argument.
In sum, whatever we think of the metaphysical arguments for the finitude of the series of past events, we have good scientific evidence for the beginning of the universe. Of course, by its very nature scientific evidence is always provisional and so never rationally compelling, as Oppy requires; still it cannot be said that contemporary cosmology gives “no good reason” to suppose that the universe had an absolute beginning.
It seems to me therefore that the case for the crucial premiss of the kalam cosmological argument is considerably stronger than Oppy would have us believe. No reason, much less good reason, has been given to think that the universe is infinite in the past. On the contrary, the idea of a temporal series of events infinite in the past and formed by successive addition is deeply problematic. 0Physical cosmologists themselves are beginning to recognize the force of these metaphysical problems.[lxxv] For example, Ellis, Kirchner, and Stoeger ask, “Can there be an infinite set of really existing universes? We suggest that, on the basis of well-known philosophical arguments, the answer is No.”[lxxvi] Similarly, noting that an actual infinite is not constructible and therefore not actualizable, they assert, “This is precisely why a realised past infinity in time is not considered possible from this standpoint—since it involves an infinite set of completed events or moments.”[lxxvii] These misgivings represent endorsements of both of the kalam arguments which I defended above. Ellis and his colleagues conclude, “The arguments against an infinite past time are strong—it’s simply not constructible in terms of events or instants of time, besides being conceptually indefinite.”[lxxviii]
Arguments for the First Premiss
So if Oppy’s scepticism concerning the kalam cosmological argument is to be justified, he must fall back onto his critique of the first premiss, that everything that begins to exist has a cause. This premiss strikes me as obviously true—at least, more so than its negation. First and foremost, it is rooted in the metaphysical intuition that something cannot come into being from nothing. To suggest that things could just pop into being uncaused out of nothing is to quit doing serious metaphysics and to resort to magic. Second, if things really could come into being uncaused out of nothing, then it becomes inexplicable why just anything or everything does not come into existence uncaused from nothing. Finally, the first premiss is constantly confirmed in our experience. Non-theists who are scientific naturalists thus have the strongest of motivations to accept it.
Experiential Support for the First Premiss
Oppy’s scepticism extends to this first premiss as well. First, he questions what it means to “begin to exist.” I take it that he intends here to resume the discussion of the fourth caution concerning the physical evidence for the beginning of the universe. There, we recall, Oppy conceded that the universe began to exist in the sense of being finite in the past but expressed doubt that a universe which began to exist in that sense must have a cause. Adverting to my explication of “x begins to exist at t,” Oppy questions my claim that we have strong empirical support for premiss (1.1). Remarkably, he says that the answer is “plainly negative.”[lxxix] How does he justify this surprising verdict? He explains, “In experience, we only ever meet with objects whose coming into existence is preceded by times at which those objects do not exist. Nothing in experience bears on the question of the causal antecedents of objects that begin to exist at t=0.”[lxxx]
This objection to the evidentiary support for (1.1) is multiply confused. First, if, following Oppy’s lead, we substitute for (1.1)
1.1´. Everything that begins to exist at t has a cause,
then the kalam cosmological argument becomes plainly invalid, since I deny that “x begins to exist” entails “x begins to exist at t.” If we try to make the argument valid by substituting for (1.2)
1.2´. The universe began to exist at t,
then (1. 2´) is plausibly false, since t=0 is plausibly at most an ideal point. The question rather is simply whether everything that begins to exist has a cause, and experience uniformly supports this truth. We never experience things popping into being without a cause. Oppy is willing to concede that the universe began to exist in the sense that it is finite in the past. Our experience also uniformly supports the proposition that everything that is finite in the past has a cause.
Second, in any case, surely experience does support the judgement that if something exists at a time t and there is no time prior to t at which the thing exists, then the thing has a cause. My explication of “x begins to exist at t” leaves it an open question whether there are times prior to t or not. For all we know from (1.1), there may be no times not preceded by other times. So Oppy has no grounds for denying that the evidence does support the truth of (1.1) so understood. What that implies about the temporal status of the cause of the universe inferred from the conjunction of (1.1) and (1.2) is a question for subsequent discussion. Of course, we have no direct experience of t=0, nor, for that matter, of most other times in the history of the universe; but if Oppy wants to defeat the evidence for (1.1), he needs to present some reason to think that the difference between those times and the times we do experience is not merely an accidental feature of those times irrelevant to whether things can pop into being uncaused at those times or whether there is a difference that renders those times somehow such that things can come into being without causes at those but not at other times. He has said nothing to make it plausible or even credible that t=0 is essentially different from other times in this respect. Hence, Oppy is mistaken when he concludes that the experiential support for (1.1) is “extremely weak.”[lxxxi]
Obviousness of the First Premiss
Oppy then turns his critical eye toward my claim that (1.1) is obviously true. First, Oppy objects that if conceivability is a good guide to metaphysical possibility, then it seems undeniable that it is possible for there to be a universe without a cause. He does not respond, however, to my (or, really, Anscombe’s) point in The Kalam Cosmological Argument that there is a difference between mere imaginability and conceivability.[lxxxii] I should say that the picture of a universe popping into being without a cause is mere imagination. But notice here that I presuppose once more that in beginning to exist the universe literally came into being. I suspect that Oppy, like certain other tenseless time theorists who have objected to (1.1), finds the notion of an uncaused beginning unproblematic because, given their background beliefs about the mind-dependence of becoming, something’s beginning to exist does not entail something’s coming into being but its merely having finite, tenseless extension in the earlier than direction. Perhaps this also bears on why Oppy thinks that something which begins to exist in the sense that it is finite in the past needs no cause of its beginning to exist.
Second, although Oppy thinks that “hard questions” need to be asked about the meaning of “cause” in (1.1),[lxxxiii] his real concern turns out to be what is meant by a “thing.” If we understand things to be individual particulars and the like, he says, then it “is worth asking” why we should suppose that (1.1) is worthy of belief, when more extensive Principles of Sufficient Reason are not.[lxxxiv] I have already given three reasons why (1.1) is worthy of belief, and Oppy has given reasons for challenging more extensive Principles of Sufficient Reason. The counterexamples to the more extensive principles are not counterexamples to the more modest (1.1). While we could argue a fortiori from the stronger principles to (1.1), there is no reason to think the falsity of the more ambitious principles implies the falsity of the more circumspect (1.1). Oppy objects that the reasons I give for believing (1.1) support just as strongly the more extensive principles, so that if the reasons fail to make those principles worthy of belief, neither do they render (1.1) worthy of belief. I think it is evident, however, that the reasons I gave for accepting (1.1) are not equally strong reasons for supposing that, for example, the BCCF has an explanation or that the state of affairs of my freely choosing to perform some action has an explanation. It would be reckless to infer that because these states of affairs do not have any explanation that concrete objects can pop into being uncaused.
Third, Oppy envisions a scenario according to which there might be different products of indeterministic particle decay. While it is clear that there would be a material cause of those products, it is not obvious, he opines, that they would have an efficient cause. Oppy does not claim that it is clear that such products would have no efficient cause (maybe there are indeterministically operating efficient causes), but he does think that before we can assent to the claim that everything that begins to exist has an efficient cause for its coming into existence, we need to be told “a lot more about the analysis of efficient causation.”[lxxxv] Absent such an analysis, it is “not the least bit obvious” that everything that begins to exist has an efficient cause of its beginning to exist.[lxxxvi]
I myself think that it is far more obvious that (1.1) is true than that Oppy’s scenario constitutes a bona fide counterexample. It seems to me that the particle which decays is an indeterministic efficient cause of the products of decay. Indeed, I should venture to say that while one might have efficient causation without material causation, as, for example, in mental acts of creation, it is impossible to have material causation without efficient causation, since the thing originally constituted by the matter is the efficient cause of the effect. Be that as it may, if one does find Oppy’s objection compelling, one could avoid his objection by substituting a less ambitious premiss for the causal principle enunciated in (1.1), for example,
1.1.´´ If the universe began to exist, then the universe has a cause.
In that case Oppy’s counterexample would be irrelevant, and an efficient cause of the universe would be required, if by “cause” we understood “efficient cause.”
In point of fact, however, (1.1) does not state that the cause of something which begins to exist is an efficient cause. For all we know, it could be either an efficient or a material cause. It is only with argument’s conclusion that the mutakallim will argue that the cause of the universe must by its very nature be an immaterial being, since it created all of physical reality. Oppy takes cognizance of this point in a footnote but objects that if “cause” in (1.2) means “either an efficient or a material cause,” then “one could not argue for the existence of an immaterial God on the basis of this premise.”[lxxxvii] This intrusion of theological concerns is not only irrelevant to the obviousness or truth of the premiss, but also wrong-headed, for one does not argue on the basis of (1.1) for the immateriality of the First Cause, since many efficient causes are material. Rather it is on the basis of a conceptual analysis of “cause of the universe” that one is able to deduce many of the significant properties of the First Cause reached in the argument’s conclusion, including its immateriality.[lxxxviii] Oppy notes that if one draws merely the conclusion that there is either an efficient or a material cause of the universe, then one will need further supporting argument to get to the claim that the universe has an efficient cause. This is the case, however, only if one presupposes that material causation can exist without efficient causation; but if it can, then a simple disjunctive syllogism will do the trick, since the universe cannot have a material cause. In any case, what should not go unnoticed is that Oppy has quietly abandoned his claim that the causal premiss, so understood, is not obviously true. This puts Oppy in the awkward position of having now conceded, at least tacitly, that the second premiss of the kalam cosmological argument is probably true and its first premiss obviously true.
The above response answers Oppy’s third objection; but I think that there is something more to be said here about Oppy’s claim. Oppy’s strategy throughout his book seems to be to raise so many philosophical conundrums that the sceptic can take refuge in unanswered questions. At the end of his chapter on cosmological arguments alone Oppy has no less than six pages of questions that need to be answered in the analysis of cosmological arguments, including eight philosophically difficult, multi-faceted questions about the nature of causation (and there are more, he assures us, not mentioned!). There is something perverse about this way of doing philosophy. If we had to have all our philosophical questions answered first, we should scarcely be warranted in believing anything. A principle so perspicuous as “Everything that begins to exist has a cause” can be rationally accepted prior to the resolution of every philosophical conundrum about causation. Of course, if specific defeaters are brought against it, then one needs to rebut or undercut them; but that is a quite different project than requiring answers to profound philosophical questions prior to our justifiably believing it. Of course, Oppy might retort that he is not denying that the causal principle is rationally acceptable on the grounds that have been given but merely insisting that it is not rationally obligatory. But then we come back full circle to the unrealistic standards Oppy sets for what constitutes a good argument.[lxxxix]
Quentin Smith reports, “a count of the articles in the philosophy journals shows that more articles have been published about Craig’s defense of the Kalam argument than have been published about any other philosopher’s contemporary formulation of an argument for God’s existence” (Quentin Smith, “Kalam Cosmological Arguments for Atheism,” in The Cambridge Companion to Atheism, ed. Michael Martin, Cambridge Companions to Philosophy [Cambridge: Cambridge University Press, 2007], p. 183).
[ii] See his Arguing about Gods (Cambridge: Cambridge University Press, 2006), pp. 137-54, which presupposes his Philosophical Perspectives on Infinity (Cambridge: Cambridge University Press, 2006).
[iii] See Oppy, Arguing about Gods, pp. 7-13.
[iv] See, e.g., William Lane Craig, critical notice of Arguing about Gods, by Graham Oppy, Philosophia Christi 10 (2008): 435-42.
[v] Oppy, Arguing about Gods, p. 142. Cf. “This seems right: an infinite number of events stretching back into the past would form an actually infinite set, as would an infinite number of events stretching into the future” (Ibid., p. 141).
[vi] Ibid., p. 141.
[vii] Ibid., p. 142.
[viii] Ibid., p. 141.
[ix] Ibid., p. 142.
[x] Ibid., p. 141.
[xi] Oppy, Philosophical Perspectives on Infinity, pp. 261-4; cf. pp. 244-5
[xii] Ibid., pp. 242-3.
[xiii] Ibid., p. 260.
[xiv] Ibid., pp. 291-3.
[xv] Ibid., p. 293.
[xvii] William Lane Craig, The Kalam Cosmological Argument (London: Macmillan, 1979), p. 69.
[xviii] Oppy, Arguing about Gods, p. 140; cf. p. 139.
[xix] See William Alston, "Does God Have Beliefs?" Religious Studies 22 (1986): 287-306; Thomas P. Flint and Alfred J. Freddoso, “Maximal Power,” in The Existence and Nature of God, ed. Alfred J. Freddoso (Notre Dame: University of Notre Dame Press, 1983), pp. 81-113.
[xx] Here, again, we touch on a very subtle point concerning what constitutes a “good” argument. Oppy considers only arguments which are rationally compelling in a strong sense to be any good. But he confuses an argument’s being good in this sense with the argument’s proponent’s being able to prove that his argument is good. The claim that one’s argument is good is a meta-claim about the argument’s claims. One can coherently present one’s argument with the conviction that others ought to accept it without thinking that one can show that they ought to accept it. If someone’s intuitions were so defective, for example, that he failed to see that something cannot be simultaneously red all over and green all over, then his denial would be irrational even if we could not prove that it is irrational.
[xxi] José A. Benardete, Infinity: An Essay in Metaphysics (Oxford: Clarendon Press, 1964), p. 238. He has in mind especially what he calls paradoxes of the serrated continuum, such as the following:
“Here is a book lying on the table. Open it. Look at the first page. Measure its thickness. It is very thick indeed for a single sheet of paper—1/2 inch thick. Now turn to the second page of the book. How thick is this second sheet of paper? 1/4 inch thick. And the third page of the book, how thick is this third sheet of paper? 1/8 inch thick, &c. ad infinitum. We are to posit not only that each page of the book is followed by an immediate successor the thickness of which is one-half that of the immediately preceding page but also (and this is not unimportant) that each page is separated from page 1 by a finite number of pages. These two conditions are logically compatible: there is no certifiable contradiction in their joint assertion. But they mutually entail that there is no last page in the book. Close the book. Turn it over so that the front cover of the book is now lying face down upon the table. Now—slowly—lift the back cover of the book with the aim of exposing to view the stack of pages lying beneath it. There is nothing to see. For there is no last page in the book to meet our gaze” (Ibid., pp. 236-237).
To my mind this conclusion itself is evidently metaphysically absurd. Although Oppy, following A. Hazen, offers expansions of the story so that someone opening the book will have some sort of visual experience, rather than as it were, a blank (Oppy, Philosophical Perspectives on Infinity, pp. 83-5), that does not negate the conclusion that there is nothing there to see, since there is no last page. Benardete imagines what would happen if we tried to touch the last page of the book. We cannot do it. Either there will be an impenetrable barrier at w + 1, which seems like science fiction, or else our fingers will penetrate through an infinity of pages without first penetrating a page, which recalls Zeno’s paradoxes in spades, since the pages are actual entities. What makes paradoxes like these especially powerful, as Benardete points out, is that no process or supertask is involved here; each page is an actual entity having a finite thickness (none has the measure of a degenerate interval) which could be unbound from the others and all the pages scattered to the four winds, so that an actual infinity of pages would exist throughout space. If such a book cannot exist, therefore, neither can an actual infinite.
[xxii] Oppy, Arguing about Gods, p. 140.
[xxiii] As Oppy observes, my illustration of an infinite library does not raise any issues not involved in the more engaging illustration of Hilbert’s Hotel.
[xxiv] Oppy, Philosophical Perspectives on Infinity, p. 48.
[xxv] Ibid., p. 53.
[xxvi] Ibid., p. 48.
[xxvii] Oppy suggests using J. Conway’s recently developed constructions called surreal numbers to define operations of subtraction and division of transfinite numbers (Oppy, Arguing about Gods, p. 140); but he explicitly denies that such non-canonical theories can be applied “to real-world problems, if one wishes to treat one’s models with full ontological seriousness” (Oppy, Philosophical Perspectives on Infinity, p. 272). Oppy does not show, nor does he think, that the results of operations on surreals would be any less counter-intuitive when translated into the concrete realm.
[xxviii] Oppy, Arguing about Gods, p. 144.
[xxix] Ibid., p. 143.
[xxx] William Lane Craig, “The Cosmological Argument,” in The Rationality of Theism, ed. Paul Copan and Paul K. Moser (London: Routledge, 2003), p. 120.
[xxxi] William Lane Craig, The Tensed Theory of Time: a Critical Examination, Synthese Library 293 (Dordrecht: Kluwer Academic Publishers, 2000); idem, The Tenseless Theory of Time: a Critical Examination, Synthese Library 294 (Dordrecht: Kluwer Academic Publishers, 2000).
[xxxii] William Lane Craig, “The Extent of the Present,” International Studies in the Philosophy of Science 14 (2000): 165-185. Consider also in this connection the many variations on the Grim Reaper Paradox (Benardete, Infinity, pp. 259-61; Oppy Philosophical Perspectives on Infinity, pp. 63-6, 81-83; Jonathan Hawthorne, “Before-effect and Zeno causality,” Noûs 34 : 622-33). There are denumerably infinitely many Grim Reapers (whom we may identify as gods, so as to forestall any kinematic objections). You are alive at 12:00 p.m. Grim Reaper 1 will strike you dead at 1:00 p.m. if you are still alive at that time. Grim Reaper 2 will strike you dead at 12:30 p.m. if you are still alive then. Grim Reaper 3 will strike you dead at 12:15 p.m., and so on. Such a situation seems clearly conceivable but leads to an impossibility: you cannot survive past 12:00 p.m. and yet you cannot be killed at any time past 12:00 p.m. Oppy’s solution to a similar paradox concerning infinitely many deafening peals, viz. that there is no particular peal responsible for your deafness but that the collective effect of infinitely many peals is to bring about deafness, not only involves a most bizarre form of retro-causation but is in any case inapplicable to the Grim Reaper version, since once you are dead no further Grim Reaper will swing his scythe, so that collective action is out of the question. The most plausible way to avert such paradoxes is by denying that time and space are constructions out of an actually infinite number of points. (My thanks to Alexander Pruss for drawing my attention to this version of the paradox.)
[xxxiii] Oppy, Arguing about Gods, p. 143.
[xxxv] Oppy’s understanding of the potential infinite is not the customary notion of a limit concept; rather he construes it modally. A potential infinitist in the realm of ontology, he says, is one who is committed to the truth of claims of the form "à$, that is to say, claims to the effect that for any natural number there is a possible world in which that number of objects exists, but who denies the truth of any claim of the form à"$, that is to say, any claim to the effect that there is a possible world in which there are as many objects as all the natural numbers. A major shortcoming of these characterizations is that they are tenseless and so incapable of handling views of time which regard tense and temporal becoming as objective features of reality and, hence, worlds in which the future is potentially infinite in the sense of growing toward infinity as a limit.
[xxxvi] Similarly, Oppy’s earlier discussion of counting to infinity is predicated upon Dretske’s assumption that if one never stops counting, then one does count to infinity (Oppy, Philosophical Perspectives on Infinity, p. 61). Oppy fails so much as to mention, much less take account of, the difference between an actual and a potential infinite in this case. One who, having begun, never stops counting counts “to infinity” only in the sense that one counts potentially infinitely.
[xxxvii] Oppy, Arguing about Gods, p. 144.
[xxxviii] Ibid., p. 144.
[xxxix] Oppy, Philosophical Perspectives on Infinity, p. 97.
[xli] Ibid., p. 116.
[xlii] Ibid., pp. 116-7.
[xliii] Ibid., p. 117.
[xlvi] Ibid., p. 63.
[xlvii] Robin Small, “Tristram Shandy’s Last Page,” British Journal for the Philosophy of Science 37 (1986): 213-16.
[xlviii] Oppy, Philosophical Perspectives on Infinity, p. 57, n. 3.
[xlix] Ibid., p. 59.
[l] Ibid.; cf. p. 63; Oppy, Arguing about Gods, pp. 141-2.
[li] Oppy, Philosophical Perspectives on Infinity, pp. 279-80.
[lii] See Alexander Pruss, The Principle of Sufficient Reason: A Reassessment, Cambridge Studies in Philosophy (Cambridge: Cambridge University Press, 2006).
[liii] Oppy, Arguing about Gods, p. 141-2.
[liv] Oppy, Philosophical Perspectives on Infinity, p. 285, cf. pp. 275-6.
[lv] Notice, too, that if there is any probability of his finishing in infinite time, then he will have already finished.
[lvi] Oppy, Philosophical Perspectives on Infinity, pp. 8, 49-51.
[lvii] It is noteworthy that Oppy agrees that if the temporal series of past events is not actually infinite, then the conclusion follows that the universe is itself finite in the past (Oppy, Arguing about Gods, p. 142), which fact implies that the universe began to exist.
[lviii] Arvind Borde, Alan Guth, and Alexander Vilenkin, “Inflation Is Not Past-Eternal,” http://arXiv:gr-qc/0110012v1 (1 Oct 2001): 4. The article was updated in January 2003.
[lix] Alex Vilenkin, Many Worlds in One: The Search for Other Universes (New York: Hill and Wang, 2006), p. 176.
[lx] See discussion in William Lane Craig and James Sinclair, “The Kalam Cosmological Argument,” in Blackwell Companion to Natural Theology, ed. Wm. L. Craig and J. P. Moreland (Oxford: Blackwell, 2009), pp. 125-82.
[lxi] John Earman, Bangs, Crunches, Shrieks, and Whimpers: Singularities and Acausalities in Relativistic Spacetimes (New York: Oxford University Press, 1995), chap. 7.
[lxii] Ibid., p. 207.
[lxiv] Ibid., p. 210.
[lxv] Oppy, Arguing about Gods, p. 146.
[lxvi] Oppy, Arguing about Gods, p. 147. Oppy overlooks the fact that Earman, from whom he borrows this point, confines his discussion to classical General Relativistic spacetimes only and so takes no cognisance of quantum physical effects.
[lxvii] But see the different take on the objection by Quentin Smith: he holds that the initial singular point of the universe is not real and that therefore the sequence of instantaneous states of the universe is a beginningless series converging toward zero as a limit. Each state is caused by its predecessor and there is no first state. But any initial non-zero interval or state, such as the first second of the universe’s existence, “is not caused by any or all of its instantaneous states and is not caused by any external cause” (Quentin Smith, “Kalam Cosmological Arguments for Atheism,” in The Cambridge Companion to Atheism, ed. Michael Martin, Cambridge Companions to Philosophy [Cambridge: Cambridge University Press, 2007], p. 189). Smith takes “the beginning of the universe” to refer to the Planck era, that state which lasts until 10-43 second after the singularity. As a state of non-zero duration, the beginning of the universe therefore has no cause of any sort. The universe therefore comes into being uncaused out of nothing.
[lxviii] Earman tacitly concedes the adequacy of this reply, objecting instead that in that case the premiss “Whatever begins to exist has a cause” is not an obvious “metaphysical truth,” since it is not a consequence of “Every event has a cause” (Earman, Bangs, Crunches, Shrieks, p. 208). I do not think our confidence in the former truth is in any way based upon the less obvious entailing claim.
[lxix] Oppy, Arguing about Gods, p. 149.
[lxxi] Oppy himself provides an explication that takes account of non-zero finite intervals of time by suggesting that x begins to exist at t just in case x exists at all times in some open or closed interval (t, t´>t) and x exists at no times in any open interval (t´´<t, t). In that case, even if t=0 is unreal, the universe began to exist. But he thinks that we have no empirical evidence for the premiss that everything that begins to exist in this sense has a cause.
[lxxii] One recalls Eddington’s remark:
“The second law of thermodynamics holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations—then so much the worse for Maxwell's equations. If it is found to be contradicted by observation, well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but collapse in deepest humiliation” (Arthur S. Eddington, The Nature of the Physical World [New York: Macmillan], p. 74).
[lxxiii] Paul Davies, “The Big Questions: In the Beginning,” ABC Science Online, interview with Phillip Adams, http://aca.mq.edu.au/pdavieshtml. Cf. P. C. W. Davies, The Physics of Time Asymmetry (London: Surrey University Press, 1974), p. 104.
[lxxiv] Oppy, Arguing about Gods, p. 148.
[lxxv] Besides the paper by Ellis et al. cited below, see also Rüdiger Vaas, “Time before Time: Classifications of universes in contemporary cosmology, and how to avoid the antinomy of the beginning and eternity of the world,” http://arXiv.org/abs/physics/0408111 (2004).
[lxxix] Oppy, Arguing about Gods, p. 149.
[lxxxi] As Oppy notes, I also floated an argument for the causal principle based on a Kantian a priori category. I have not given any more thought to this suggestion and so shall not pursue it here.
[lxxxii] Craig, Kalam Cosmological Argument, pp. 144-5.
[lxxxiii] Oppy, Arguing about Gods, p. 151. I shall therefore ignore Oppy’s misconstruing the causal principle in terms of event/event causation, since he does not seem to exploit the analysis. Suffice it to say that (1.1) does not require that every event have a cause.
[lxxxiv] Oppy, Arguing about Gods, p. 152.
[lxxxvi] Ibid., p. 153.
[lxxxviii] Craig, “Cosmological Argument,” pp. 128-9.
[lxxxix] I am indebted to Graham Oppy for discussion of several issues discussed in this exchange.