Mormonism and the KCA

by metacognizant

Introduction

Blake Olster’s critique of the kalam cosmological argument has, on the face of it, much force. Olster is an obviously intelligent individual, skilled in areas as diverse as rhetoric and mathematics. His critique of the kalam is vicious. I suppose that it is because the success of this argument has as many negative repercussions for Mormon theology as it does positive repercussions for a version of theism that espouses aseity in God’s nature that Olster writes such a blistering critique of the kalam. However, while the argument has some force on the face of it, it breaks down under criticism.

On a metaphysically extant actual infinite

First, I’d like to point out that mathematical legitimacy does not imply actual (that is, metaphysical) legitimacy. To say that simply because a concept has been successfully dealt with in mathematics, it successfully exists in reality is a non sequitur. In order to get away with this claim, one has to adopt—and successfully argue for—a Platonic view of numbers, which is itself subject to a variety of difficulties. Indeed, there are a variety of Nominalist and Conceptualist alternatives to Platonism, and they are much more popular. This is why the mathematician David Hilbert can remark that set theory is “One of the most vigorous and fruitful branches of mathematics… a paradise created by Cantor from which nobody shall ever expel us… the most admirable blossom of the mathematical mind and altogether one of the outstanding achievements of man’s purely intellectual activity,” (Hilbert 1964, p. 141) but nevertheless conclude, “Finally, let us return to our original topic, and let us draw the conclusion from all our reflections on the infinite. The overall result is then: The infinite is nowhere realized. Neither is it present in nature nor is it admissible as a foundation of our rational thinking – a remarkable harmony between being and thinking” (Hilbert 1964, p. 151). Likewise, an article in the Proceedings of MACAS 2, Second International Symposium on Mathematics and its Connections to the Arts and Sciences concludes, “actual infinity…has no application outside of mathematics at all…. We have investigated actual infinity in all domains which reasonably could be suspected to sustain it. But the result is such that we can give the last word to D. Hilbert, who, at the end of his famous paper praising actual infinity of set theory, comes to a plain and astonishing conclusion,” and Hilbert’s aforementioned conclusion is then quoted (2008, page 271: accessible online). An additional justification—aside from mathematical legitimacy—must be given in order to demonstrate that the concept of an actual infinity actually exists in reality (namely, reason, evidence, etc.).

The Hartle-Hawking model of quantum cosmology unifies general relativity and quantum physics where it matters the most: in the beginning of the universe. However, the search for a unification between quantum mechanics and general relativity continues. Why? Precisely because this model uses mathematics that have no application in the real world—multiples of the square root of negative one, if you’re interested. Even though this cosmological model is mathematically legitimate, and it solves an enormous problem in contemporary cosmology, these numbers don’t exist in the real world, and so this model is not heralded as the true solution to this cosmological problem. I’m saying this to illustrate a point: what is mathematically legitimate may not be metaphysically actual. Thus, an argument beyond, “it’s consistent mathematically,” must be given to solve the absurd implications of an actual infinite.

Finally, before I go on to critique Olster’s argument, it must be noted that, should an actual infinite be accepted as extant in the real world, set theory actually proves a past-finite physical reality. Credit for this argument goes to a professor of mathematics that visits a forum I frequent. He goes by the name of yggdrasil. I won’t link to the forum, as it doesn’t have a moderator, and I’m afraid of possible trolls. I’m simply saying this to give credit where it is due. Here’s his argument:

(Definition 1)  A process is defined as a couple {E, o(E)} where E is an enumerable collection of events such that each individual event takes the same amount of finite time and o(E) is a well-ordering of E that represents the enumeration of E.  The process {E, o(E)} is realizable if E can be realized as a sequence of events according to the enumeration o(E).

(Definition 2) A well-ordering X is realizable if there exists a realizable process {E, o(E)} such that X = o(E).  Let R denote the set of all realizable well-orderings.

(Definition 3) Let W represent the well ordering {0 < 1 < 2 < 3 < …} and let W* represent the well ordering {0 > -1 > -2 > -3 > …}

(1) Everything that has a finite past must have an external explanation for its existence
(2) The physical universe has a finite past
(3) Therefore, the physical universe must have an external explanation for its existence (by 1 and 2).

(2.1) If X and Y are well-orderings such that X is in R and Y has the same order type as X then Y is also in R.
(2.2) W* has the same order type as W.
(2.3) W is not in R
(2.4) Therefore W* is not in R (by 2.1, 2.2, and 2.3).
(2.5) If the physical universe has an infinite past then its past can be realized as a process {E, W*}.  Hence, an infinite past implies that W* is in R.
(2.6) The physical universe must have a finite past (by 2.4 and 2.5)

Justification for Premise (2.1): The capacity for a well-ordering to be realized is unchanged after renaming the well-ordering relation, and isomorphisms of ordinal type are no different from such a renaming.

Justification for Premise (2.2): This is a mathematical fact.  The order isomorphism from W* to W is given by sending 0 to 0, -1 to 1, -2 to 2, -3 to 3, etc.

Justification for Premise (2.3): The most important premise. W is not in R for the same reason that it is not possible to complete the task of counting all of the non-negative natural numbers (0, 1, 2, 3, etc.) while taking a fixed amount of time to count off each number.  Clearly, a process of this kind cannot be realized.

Justification for Premise (2.5): If the physical universe did have an infinite past then, starting from the present, we could construct a realizable process {E, W*} where E is constructed by partitioning the universe’s past into contiguous one second intervals endowed with the natural temporal ordering.

What does this mean? It means that even if Olster’s arguments prevail, set theory has already shown that an actual infinite cannot apply to temporality—that is, it still follows that the universe is finite in existence. Nonetheless, let’s continue; I’m writing a review of Olster’s argument, not a defense of the kalam in general.

The First Infinity Argument

Olster seeks to show that the absurd implications of an actual infinite don’t apply to an infinite past-time. He does this in a few different ways. To begin, let’s note William Lane Craig and Paul Copan’s (C&C hereafter) argument:

1.1) An actual infinite cannot exist.
1.2) An infinite temporal regress is an actual infinite.
1.3) Therefore, an infinite temporal regress of events cannot exist.

Olster correctly notes that—mathematically speaking—“an infinite temporal regress is not the same as a beginningless series of events in time. An infinite temporal regress constitutes a ‘well-formed’ infinite series. It has a beginning term, 0 (or -1 depending on how the set is set up), and then counts backwards {0, -1, 2-, -3….}. An infinite temporal regress has the same mathematical properties as the set of real numbers beginning with 0 (or 1) and counting forward. However, because a beginningless series of events is a ‘not-well-formed series,’ it has different mathematical properties.” Because of this, Olster seeks to replace (1.2) with:

1.2*) A beginningless series in time is an actual infinite.

However, the distinction that Olster makes is not applicable outside of abstract mathematics. When transposed to the real world, note that, from our perspective—as well as metaphysically—, an infinite temporal regress is the same thing as a beginningless series of events. This is a mathematical distinction that is not applicable to the situation of a reality that has always existed, given our perspective. The reason for this is because the present moment exists, and we can view the past from our current perspective (therefore allowing an eternally extant temporality to be the well-formed set: {0, -1, -2….}). Both an infinite temporal regress and a beginningless series of moments in time refer to the same set of events; one simply starts from our perspective and looks to the past infinity, and the other has no starting point, but has reached us. Therefore, there is actually no distinction in reality. In fact, it is arguably necessary to maintain that this argument must take the form of (1.2), as our perspective is what is certain.

After Olster substitutes (1.2) for (1.2*), he correctly notes that the absurdities of Hilbert’s Hotel do not apply to a not-well-formed infinite set; however, as we have seen, an eternally extant temporality is equally construed as a well-formed infinite set, and thus the absurdities of an actual infinite do apply to this set. (Interestingly, this also helps to illustrate the absurdities of an actual infinite extant in reality: the absurdities of Hilbert’s Hotel both do and don’t apply to an actually infinite past-time, as the set can be construed both as a well-formed set and a not-well-formed set.)

In addition, even if one were to somehow prove that we cannot look backwards in time from our perspective—thereby forcing an infinite past-time to take the properties of a not-well-formed set—extraordinary absurdities still exist. Take al-Ghazali’s thought experiment involving two beginningless series of coordinated events. (Infinite past-time in this thought experiment takes the form of a not-well-formed set.) Envision our solar system existing from eternity past, with the orbital periods of the planets being so coordinated that for every one orbit which Saturn completes, Jupiter completes 2.5 times as many. If they have been orbiting from all of eternity, which planet has completed the same number of orbits? The correct mathematical answer is that they have completed exactly the same number of orbits. But this is obviously absurd, as the longer they revolve, the greater becomes the difference between them, so that they progressively approach a limit at which Jupiter has fallen infinitely far behind Saturn. Yet, with an actually infinite past, their respective completed orbits are somehow identical. Indeed, they will have attained infinity from eternity past—that is, their number of completed orbits is always identical. Moreover, Ghazali asks, will the number of completed orbits by either or odd? Either answer seems absurd; it’s tempting to deny that the number of completed orbits is either even or odd. But post-Cantorian transfinite arithmetic gives quite a different answer: the number of completed orbits is both even and odd! For a cardinal number n is even if there is a unique cardinal number m such that n = 2m, and n is odd if there is a unique cardinal number m such that n = 2m + 1. In the aforementioned thought experiment, the number of completed orbits is (in both cases) an actual infinite (hereafter, let Mi be the mathematical symbol of the actual infinite), and Mi = (2 x Mi) = ((2 x Mi) + 1). So Jupiter and Saturn have each completed both an even and an odd number of orbits, and that number has remained equal and unchanged from all eternity, despite their ongoing revolutions and the growing disparity between them over any finite interval of time. This is obviously absurd, and, again, this absurdity results from a not-well-formed set (paraphrased from BTNC, p. 120).

Olster also asserts that the absurdities of Hilbert’s Hotel do not apply to an infinite past because “the rooms in the Hotel all exist at once and are actual in the same moment. That is not true of the infinite past. Only the present moment is actual or ontologically real assuming an A-theory of time (which both C&C accept). Thus, the past events do not actually exist to be transposed and reordered as the story of Hilbert’s Hotel requires.” But this is simply irrelevant to the absurdities of an actual infinite. Here I will quote William Lane Craig and James Daniel Sinclair in their article for the Blackwell Companion to Natural Theology (hereafter BCNT),

The question, then is whether events’ temporal distribution over the past on a presentist ontology precludes our saying that the number of events in a beginningless series of events is actually infinite. Now we may take it as datum that the presentist can accurately count things that have existed but no longer exist…. He knows how old his children are and can reckon how many years have elapsed since the Big Bang… The nonexistence of such things or events is no hindrance to their being enumerated. Indeed, any obstacle here is merely epistemic, for aside from considerations of vagueness there must be a certain number of such things…. [I]f we consider all the events in an infinite temporal regress of events, they constitute an actual infinite…. Because the series of past events is an actual infinite all of the absurdities attending the existence of an actual infinite apply to it. For example, if the series of past events is actually infinite, then the number of events that have occurred up to the present is no greater than the number that have occurred at any point in the past. Or again, if we number the events beginning in the present, then there have occurred as many odd numbered events as events. If we mentally take away all the odd-numbered events, there are still an infinite number of events left over; but if we take away all the events greater than three, there are only four events left, even though in both cases we took away the same number of events (pp. 115-116).

Therefore, regardless of whether or not these events still presently exist is irrelevant; absurdities still result. In addition, if the past does not presently exist, then mentally altering it is as real as physically altering it should it presently exist.

Olster then states,

“Further, the past cannot be jumbled around like the persons in Hilbert’s Hotel for reasons quite unrelated to the problems of infinities – the past is fixed and unchangeable once it occurs. Year 351,067 B.C. cannot be exchanged for year 465,789 B.C. Thus, we cannot take away all of the odd years. We have the infinite series of past events just as they have occurred and in the very order they occurred and we cannot alter them in the way C&C suggest for Hilbert’s Hotel to create a supposed absurdity.”

First, one must ask, why can’t they be exchanged? If infinite time has elapsed, then the dates of 351,067 B.C. and 465,789 B.C. are arbitrarily designated dates given by us because of our frame of reference. Because if time had no beginning, ultimately the same amount of time had passed in the date of 351,067 B.C. as it had in 465,789 B.C. Since these dates, then, are simply our arbitrary designations, why can we not manipulate them? I see no reason to assert that we cannot do so. In fact, if infinite time has elapsed, I would say that these dates are more changeable than any physical event is. Finally, to assert that simply because the past events do not presently exist they are mentally immutable—and again, mentally manipulating events that have existed is as real as physically manipulating events that currently exist—is even more absurd than an actual infinite.

Olster summarizes,

“For these reasons, the supposed “absurd” stories used by C&C to demonstrate that an actual infinite is absurd simply have no application to the type of infinite order involved in the past without a beginning. All of the supposedly absurd stories, like Hilbert’s Hotel, or the Tristram Shandy autobiography, all depend critically upon properties of the order that the infinite past does not possess. The past events are not like an infinite number of guests in an infinite number of existing rooms all of which actually exist in the same moment that can be shuffled around and transposed and still maintain the same order of infinite numbers.”

But as we have seen, this is simply not the case. Besides the fact that Olster did not actually interact with the Tristram Shandy autobiography (I should note that this thought experiment’s absurdities do not necessitate any shuffling or transposing of things at all. Also, absurdities in this thought experiment result from viewing a past-time infinite both as a well-formed set and as a not-well-formed set [BCNT, pp. 120-121]. I imagine that Tristram Shandy’s autobiography was not in C&C’s original argument to which Olster was responding, as this story brings absurdities in a variety of ways; he most likely skimmed it in an article, and so he can’t be faulted for not interacting with it.), we have seen that the absurdities of an actual infinite do apply to an infinite past.

Nonetheless, Olster then proceeds to depict these absurdities as if they were not absurdities:

Further, the supposed absurdity is contrived by C&C. Take the first supposed absurdity — that the number of occupied rooms equals the number of rooms plus one for a new occupant and there are no “more” occupied rooms than before the new occupant arrived. Absurd? Not really. C&C illicitly use the concepts of “number” and “more” to trade on our intuitions about finite numbers and then apply them to transfinite numbers where such intuitions do not apply…. In the context of infinite set logic, all infinite collections can be put into a one-to-one correspondence with proper subsets of themselves and so our ordinary expectations about the way finite numbers behave do not apply in this new context. Is it absurd to suggest that we can have a Hotel that is full and then move all of the occupants to even numbered rooms and leave an infinite number of odd numbered rooms for an infinite number of new guests? Hardly…. C&C suggest that it is absurd that the “number” of occupied even and odd numbered rooms can correspond to just the odd numbered rooms: but that is the way that transfinite numbers work. In fact, once we state that even numbered rooms can be put into one-to-one correspondence with even and odd numbered rooms taken together, the assertion becomes quite ordinary and mundane. Indeed, one could not reject such a statement without simply objecting to Cantor’s theory of transfinite mathematics altogether. (My emphasis.)

Now, C&C do not deny the mathematical legitimacy of transfinite arithmetic. However, as has been shown earlier (by the quoting of mathematicians, at the top), it is possible to reject the metaphysical legitimacy without rejecting mathematical legitimacy. One reason is this: set theory operates within axioms, and these axioms do not exist outside of the theory itself—notice the bolded part of the quote. Because of this, while it may be mathematically legitimate to let Mi + 1 = Mi, it is obviously not metaphysically legitimate. I’ll use human beings as an example. It is a metaphysical absurdity to say that one can remove half of the people from a group, Mi, and still have the same amount of people, Mi, remaining in the total group, Mi, if and only if one can remove the same amount of people, Mi, from the total amount of people in the group, Mi, and end up with no people left in the original group. The absurdity is compounded by the fact that one can take the same amount of people away from the group, Mi, and end up with arbitrary numbers, such as 4, 300,000, 28, etc. Craig and Sinclair summarize this nicely,

[T]he contradiction lies in the fact that one can subtract equal quantities and arrive at different answers. For example, if we subtract all the even numbers from all the natural numbers, we get an infinity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the identical number of numbers and yet did not arrive at an identical result. In fact, one can subtract equal quantities from equal quantities and get any quantity between zero and infinity as the remainder. For this reason, subtraction and division of infinite quantities are simply prohibited in transfinite arithmetic—a mere stipulation which has no force in the nonmathematical realm (BCNT, p. 112. Emphasis in the original).

A careful reader of Olster’s argument will have noticed previous declaration: “Grah[a]m Oppy has pointed out that there are a number of other theories of transfinite numbers that have been developed that deal with inverse operations with transfinite numbers without contradiction.” However, what Olster does not mention is that Oppy explicitly denies that one can apply “non-canonical theories to real-world problems, if one wishes to treat one’s models with full ontological seriousness,” and then states, “The use of nonstandard analysis may yield numerical answers, but it does not yield insight into philosophical problems” (Oppy 2006, 272-273). So subtraction and division are still prohibited within any transfinite arithmetic that hopes to be taken with ontological seriousness.

Again, then, while these implications may not be absurd within the axioms of set theory in mathematics, they are when dealing with concrete items that exist in reality. However, Olster counters this claim by stating,

C&C claim that they have no objection to the abstract mathematical notions of an infinite series of numbers; they only object that such things cannot exist in reality. But the objection made by C&C is valid only if the theory of abstract transfinite numbers is wrong as well. When we subtract all odd numbers from all counting numbers, we have a set that has the same number of terms. Why is this not absurd when applied to abstract numbers of rooms but somehow becomes absurd and unthinkable when applied to real rooms? C&C’s objections are really objections to the very notion of infinite numbers and not merely to whether such numbers can be mirrored in reality.

Now we have already seen that a variety of mathematicians disagree with Olster here. Indeed, a large variety of quotations can be given of mathematicians appreciating set theory but claiming that the actual infinite has no bearing in reality. I’m not sure Olster intends to do this, but he makes a Platonic assumption of the reality of numbers. Philosophies of mathematical objects can be construed in two broad ways: realism (exist) and anti-realism (do not exist). Realism views of mathematical objects include the views of mathematical objects as abstract objects (Platonism), and as concrete objects (Physicalism, Human Conceptualism, and Divine Conceptualism). Anti-realism views of mathematical objects include Conventionalism, Deductivism, Fictionalism, Structuralism, Constructibilism, and Figuralism. It is only if the Platonic account of numbers is correct that Olster’s argument—that we cannot object to absurdities of an infinite in reality if we allow their mathematical legitimacy—holds. However, Platonism is a philosophy that few actually embrace due to its implications, and with such a wide variety of (all equally plausible) alternatives to Platonism, one cannot dogmatically state that it is correct.

Olster then concludes his argument against 1.1-1.3 as follows:

This first argument commits the fallacy of equivocation in the sense that it imputes the properties of individual members of a finite set to infinite sets as a whole. Finite sets obey Rule R1. However, it is a mistake to impute this same rule to the properties of infinite sets. Sets do not necessarily obey the same rules that apply to individual members of sets.

But this is entirely begging the question. It is no doubt that infinite sets work on different axioms than their finite members do. However, the whole purpose of Olster’s argument is to refute the idea that an actual infinite cannot exist in reality. The rules and axioms of set theory are bound up in set theory, and, again, a further rationale for a belief must be given other than “it’s mathematically legitimate.”

In conclusion then, Olster’s first argument fails to show that absurdities from actual infinites do not exist in reality. Paradoxes like Hilbert’s Hotel and others do, in fact, apply to an actual infinite past-time. Finally, it is possible to deny the metaphysical legitimacy of a mathematical entity without denying the mathematical legitimacy of entity.

The Second Infinity Argument

Olster’s second argument is much worse. His entire argument is devoted to knocking down a straw man, and he quote mines to set that straw man up. For time’s sake, I won’t deal with much of his argument here. I simply instruct the curious reader to order the BCNT (well worth the money) and read Craig’s argument in here. I’ll point a few things out, though. To review the argument:

2.1 The temporal series of events is a collection formed by successive addition.
2.2 A collection formed by successive addition cannot be an actual infinite.
2.3 Therefore, the temporal series of events cannot be an actual infinite.

Olster states,

Premise 2.1 is not true of all temporal series and certainly is not true of a beginningless past series that terminates in the present. An actual past infinite collection is not “formed by successive addition,” as if we could add a finite number of terms together and somehow they sum to an infinite number. Rather, for any term added to the past at any given point in the past, the past is already an infinite collection at that past time and therefore is not formed as an infinite collection by such addition. Indeed, C&C have assumed in premise 2.1 that the “temporal series” of past events has the same properties as a potential infinity rather than an actual infinity, for they assume that an infinity is open ended and completed by adding to it rather than being a completed infinity without a beginning term. One cannot form a collection by adding to it if the collection already exists before the addition. The infinite past already exists as an infinite temporal series before the addition of any term and therefore cannot be formed as an infinite temporal series by addition. I accept that one cannot by beginning with any one member of an infinite set complete an infinite set by successively adding new members to the set…. [A]n infinite collection can be added to by successive addition if it has a core that has always existed as an eternal past…. [T]he collection is not formed as an infinite by successive addition, but exists as an infinite past without some prior explanation for some first event (my emphasis)

But is this true? What Olster actually declares is that each temporal moment in this infinite past has not actually passed; rather, some temporal moment came into being with a whole and entire infinite behind it, yet it did not temporally follow from any previous moment (the first moment has no “prior explanation for some first event”). But surely, if temporality is infinite, then by definition, every member in the set has passed. If, in fact, a series of events did not actually pass by successive addition, this series of events would be timeless. It would be timeless because the definition of what is temporal is that it passes by successive addition; that is, any temporal member of a set either has elapsed or will elapse. However, if some member of a set simply exists—it does not ever come to pass nor has it ever elapsed—then it is, by definition, timeless. But if temporal succession began to exist from a state of timelessness before, isn’t that the conclusion of the argument already? Indeed, what Olster argues for ends up entailing precisely what he argues against.

However, this is not the structure of William Lane Craig’s (WLC’s hereafter) usual argument. I’ve read this argument in Reasonable Faith, 3rd. ed, and the BCNT. To quote the BCNT,

Although the problems will be different, the formation of an actually infinite collection by never beginning and ending at some point seems scarcely less difficult than the formation of such a collection by beginning at some point and never ending. If one cannot count to infinity, how can one count down from infinity? (p. 118, emphasis in the original)

(A note: one might assert that construing the argument in this way assumes a beginning point—“from” infinity. However, it must be noted that any arbitrary point in the infinite, , can be construed as a beginning point in the infinitely distant past, but not the beginning of the infinite past. Again, if the past is truly actually infinite, then any arbitrary point in Cantor’s Mi can function as a beginning point from the infinitely distant pass to traverse to the present, without saying that the selected point is the beginning of the whole set.)

And this is precisely how the actual argument takes place: from our frame of reference, proving the impossibility of an infinite past. Admittedly, I have not read C&C’s argument in The New Mormon Challenge, so I don’t know if he structures the argument differently. However, as presented, almost all of Olster’s argument is a straw man against WLC’s usual argument. Indeed, if you look at most of the quotes given by WLC, you’ll see that he’s answering objections wholly different than the ones Olster is throwing at him. Because of this WLC looks obviously absurd, but the fact is he’s simply not answering the same objections Olster has. Instead, he answers them with thought experiments. You can read them in the BCNT; I don’t want to type this all up.

For an example of quote mining, take Olster’s quote of WLC:

If we were to ask why the counter would not finish next year or in a hundred years, the objector would respond that prior to the present year an infinite number of years will have already elapsed, so that by the Principle of Correspondence, all the numbers should have been counted by now. But this reasoning backfires on the objector: for, as we have seen, on this account the counter should at any point in the past have already finished counting all the numbers, since a one-to-one correspondence exists between the years of the past and the negative numbers.

Olster then makes WLC out to look like a fool:

Yet Craig makes a modal error here. Craig (and C&C in their article in NMC) asserts that the infinite counter necessarily must have finished counting “all” of the negative numbers prior to today. The fact is that a person counting for an infinite number of years could have reached zero yesterday, but it is not necessarily the case…. [I]t is simply false that the counter must have finished counting all of the negative numbers prior to today. C&C confuse “all negative numbers” with “infinitely many negative numbers.” The two sets are not necessarily the same. Yet Craig’s argument works only if, necessarily, the counter must have counted all negative numbers by today. The argument is simply modally confused.

But let’s look at WLC’s sentences in the paragraph prior to the quotation:

[T]he question I am posing is not whether there is a logical contradiction in such a notion, but whether such a countdown is not metaphysically absurd. For we have seen that such a countdown should at any point already have been completed. But Sorabji is again ready with a response: to say the countdown should at any point already be over confuses counting an infinity of numbers with counting all the numbers. At any given point in the past, the eternal counter will have already counted an infinity of negative numbers, but that does not entail that he will have counted all the negative numbers. I do not think the argument makes this alleged equivocation, and this may be made clear by examining the reason why our eternal counter is supposedly able to complete a count of the negative numbers ending at zero. In order to justify the possibility of this intuitively impossible feat, the argument’s opponent appeals to the so- called Principle of Correspondence used in set theory to determine whether two sets are equivalent (that is, have the same number of members) by matching the members of one set with the members of the other set and vice versa. On the basis of this principle the objector argues that since the counter has lived, say, an infinite number of years and since the set of past years can be put into a one- to-one correspondence with the set of negative numbers, it follows that by counting one number a year an eternal counter would complete a countdown of the negative numbers by the present year.

So WLC has already addressed this in full. It is simply quote mining by Olster that seems to indicate otherwise. Also, note that WLC is not actually replying to Morriston, as Olster declares that he is. Indeed, WLC’s article that’s quoted was written 8 years before Morriston’s. Now, I’m skimming through most of this argument. As I’ve said, it’s almost entirely a straw man. I’ll review Olster’s conclusion to this section:

Now it is clear that the task of counting does not begin at some point if it has been taking place from all eternity…. But surely the problem with a beginningless eternity cannot be the fact that it cannot have a beginning. Zeno’s point from the race paradox is that Achilles cannot complete the journey because he cannot begin it. He cannot begin it because to do so he must complete one of the tasks that make up the journey, he must first complete another task and thus have already begun. But the claim that the past has no beginning cannot be refuted by arguing that a beginningless past could never begin – for that is just what the proponent of an infinite past claims.

We’ve already seen that this isn’t Craig’s argument (though admittedly, I have not read C&C’s specific argument, and it may be different), but this is the first time Olster brings up Zeno’s paradox, so I want to address this. Zeno actually has a few paradoxes, and his most notable are his Stadium and Dichotomy paradoxes. In the Stadium paradox, one begins at some point but can never reach the end. In the Dichotomy paradox (which is the paradox Olster attempts to address), Zeno argued that before Achilles could cross the stadium, he would have to cross halfway; but before he crossed halfway, he would have to cross a quarter of the way; but before he could cross a quarter of the way, he’d have to cross an eighth of the way; and so on, ad infinitum. However, in this paradox, Zeno did not argue that Achilles could not begin, but that he could not arrive at any point, as traversing from the infinite to the finite is impossible. The reason why Achilles can reach a point though, is because these distances are both potential—if one accepts the Aristotelian dictum that a line exists conceptually prior to any divisions we make of it—and unequal, so they add up to a finite length (paraphrased from BCNT, p. 119). And this is precisely the line of argument the defender of a finite past takes: that it is impossible to bridge the gap between the infinite and the finite, not that a beginningless series of events is impossible because it doesn’t begin. For the actual argument, please just read the BCNT.

On a beginningless multiverse and the kalam

Olster begins this section by asserting that “the infinity arguments made by C&C do not apply to a beginningless universe of the type posited by the chaotic inflationary or quantum vacuum theories of cosmology,” even though these theories (attempt to) show that the past is beginningless—that is, it’s an infinite set. Now, this may (and probably should) strike some readers prima facie absurd, so let’s look at Olster’s rationale for this assertion:

Let’s take the chaotic inflationary theory first…. It is possible, and in fact predicted by the chaotic inflationary theory, that our local universe may not be the only “pocket universe” or “space-time bubble” that exists. From this theory it follows that if the universe contains at least one inflationary domain of a sufficiently large size, then it begins unceasingly producing new inflationary sub-domains…. Moreover, each separate pocket universe begins with different initial conditions that give rise to different constants of Nature…. Each bubble universe within the multiverse constitutes an epoch of a discrete space-time continuum. However, there is no continuous time metric between any two space-time epochs. Because this theory predicts that each inflationary sub-region does not have the same initial constants as our local universe, it is possible that there is no continuous time metric that is shared between any two epochs…. Thus, it is possible that the multiverse has always existed even though each of the bubble universes has only a finite past…. The quantum fluctuation theory of cosmology also entails the possibility of an eternal spacetime manifold. The quantum fluctuation theory posits that prior to the big bang event that gave rise to our local universe, there had always existed a quantum vacuum of the type conceived in many current inflationary theories of cosmology. In this vacuum there are innumerable events that occur within the limits of the Heisenberg Uncertainty Principle. None of these events are causally or temporally related to one another. They occur at random and there is no time-metric to measure their proximity to one another. Indeed, each of these events constitutes its own space-time universe in the sense that there simply is no causal or space-time continuum obtaining to place them in relation to one another. There is no beginning to this vacuum condition – it is simply the lowest energy state possible in the physical world.

Olster also states that, regarding this quantum universe, “Such a reality must be regarded as quiescent in the sense that no events give rise to a series of events until the vacuum decays into a false vacuum creating the energy from which the Big Bang derived. Craig has admitted, correctly in my view, that the infinity arguments do not apply to a quiescent universe (i.e, a physical reality having no events).” However, this is based on a misunderstanding of WLC. A quantum vacuum with spatially disjointed events occurring would not be a quiescent universe in the sense that WLC concedes that kalam arguments don’t apply to it. WLC is conceding that kalam arguments do not apply to “‘relative timelessness,’ according to which God, prior to creation, exists in an undifferentiated, non-metric time” (Craig 2001, 10). WLC’s position is that if no events have occurred—as in a truly quiescent universe—, time has yet to occur. This view is held by a variety of philosophers of religion, and WLC is simply conceding that these arguments do not apply to this state: it is conceived, identically, as timelessness. It should be obvious that kalam arguments do not apply to timeless states. But if any actual quantum fluctuation occurs, time has begun, and WLC’s position would be that kalam arguments apply to this state.

What I don’t think Olster knows, though, is that this understanding of time is specific to a B-theory of time. This understanding of time is based off of the Einsteinian interpretation of special relativity (hereafter SR) that time is not independent, in any way, of space, and an Einsteinian interpretation of SR actually necessitates a B-theory of time. On this interpretation it is true that time is causally disconnected at singularities because new space, laws of nature, and matter are created. However, Einstein’s interpretation has been criticized as far too positivistic (he was, after all, a positivist), in that it is possible that time as it plays a role in physics is simply a pale reflection of true—metaphysical—time. Indeed, Kant showed that time is a form applicable not only to the external world, but to consciousness as well (Critique of Pure Reason, A34; B51). However, quite a few readers of mine might not want to posit something that is not necessarily physical, so I’ll explain a few physical reasons why Olster’s assertion isn’t true (or at the very least, necessary).

First, objective “temporal becoming is compatible with relativity theory if we reject space-time realism in favor of a neo-Lorentzian interpretation of the formalism of the theory” (Craig 2001, 138). I cannot stress it enough that a neo-Lorentzian interpretation of SR is in every way empirically identical (Craig 2001, 182). Additionally, a variety of arguments can be given for the simplicity and beauty of a neo-Lorentzian interpretation, and it is no more ad hoc than its rival (Craig 2001, 181-196). Furthermore, as WLC explains, on a neo-Lorentzian interpretation of SR,

[C]lock retardation, the Twin Paradox, and asymmetrical aging become comprehensible phenomenon. The adequacy of such causal explanations is not the primary point here. Rather the salient point is that a neo-Lorentzian theory involves appeal to some sort of causal explanation for the physical effects predicted by SR. By contrast, in Einstein’s theory, these effects simply follow…from Einstein’s denial of the existence of a fundamental frame and his two postulates… Given his postulates, a new kinematics follows according to which reciprocal length contraction and clock retardation occur, but without any dynamical explanation. As a theory of principle rather than a constructive theory, Einstein’s SR is based on postulates[,] which are characterized by their very non-empirical character…. As a constructive theory, the neo-Lorentzian approach promises to enrich our understanding of the causal structure of the world in a way that Einstein’s cannot. (Craig 2001, 185)

So a neo-Lorentzian interpretation is arguably more enlightening to us. Finally, a neo-Lorentzian interpretation of SR has been continued and researched to this day by physicists, including notable figures such as H. E. Ives, Geoffery Builder, and S. J. Prokhovnik. In fact, the debate between which interpretation is correct has intensified in recent years, due to the viability of a Bohmian—or de Broglie Bohm—interpretation of quantum mechanics, which requires absolute simultaneity and, hence, absolute time (Craig 2001, 182). Quantum physicist Henry Stapp states, “The simplest picture of nature compatible with quantum theory is the model of David Bohm. It explains all of the empirical facts of a relativistic quantum theory, including, in particular, the impossibility of transmitting ‘signals’ (i.e., controlled messages) faster then light” (quoted in Craig 2001, 231). Quantum physicists Callender and Weingard go one step further and state, “when cosmological factors are considered, the de Boglie-Bohm interpretation remains the only satisfactory interpretation of quantum theory” (quoted in Craig 2001, 233). Again, should a Bohmian interpretation of quantum mechanics hold, a neo-Lorentzian interpretation of SR is necessary.

What does this mean? It means that one cannot dogmatically assert that an Einsteinian interpretation is correct, and there is no possible temporal continuity between the bubble universes in the multiverse or a quantum vacuum and its spawned universes—as Olster has done. Should a neo-Lorentzian interpretation hold, time is actually independent of space, though it is modified by it. A neo-Lorentzian interpretation of SR is arguably more powerful, as shown above, and so this argument of Olster’s fails, because on this interpretation the arrow of time continues as a whole despite a variety of singularities and new cosmological occurrences with different laws of nature.

Additionally, as mentioned before, an Einsteinian interpretation of SR necessitates a B-theory of time. The A-theory of time can be demonstrated as being much more explanatorily powerful than the B-theory (see: The Tensed Theory of Time: A Critical Examination and The Tenseless Theory of Time: A Critical Examination). Here are a few arguments for the A-theory and against the B-theory:

For the A-theory: (i) Tensed sentences, which can neither be translated into synonymous tenseless sentences nor be given tenseless, token-reflexive truth conditions, correspond, if true, to tensed facts; (ii) the experience of temporal becoming, like our experience of the external world in general, is properly regarded as veridical.

Against the B-theory: (i) In the absence of objective distinctions between past, present, and future, the relations ordering events on the tenseless theory are only gratuitously regarded as genuinely temporal relations of earlier/later than; (ii) the claim that temporal becoming is mind-dependent is self-defeating, since the subjective illusion of becoming involves itself an objective becoming in the contents of consciousness; (iii) the tenseless theory entails perdurantism, the doctrine that objects have spatio-temporal parts, a view which is metaphysically counter-intuitive, incompatible with moral accountability, and entails the bizarre counterpart doctrine of transworld identity.

All this together entails that Olster’s argument fails; the arrow of time continues as a single arrow despite being warped by space at various occurrences.

Nonetheless, Olster’s assertions that either of these universes are plausibly past-infinite is unfounded—even if kalam arguments do not hold. Olster is excused of this with regards to a chaotic inflationary multiverse—absolute proof of this did not come about until 2003, and as far as I can tell, Olster wrote this review sometime in 2002. In 1994, Arvind Borde and Alexander Vilenkin were able to show that any space-time eternally inflating toward the future cannot be geodesically complete in the past–that is to say, there must have existed at some point in the indefinite past an initial singularity. In 2003, Borde and Vilenkin in cooperation with Alan Guth were able to strengthen their conclusion by crafting a new theorem independent of the assumption of the weak-energy condition (accessible online here). They note, “Our argument can be straightforwardly extended to cosmology in higher dimensions,” (Borde, Guth, and Vilenkin, p. 4). What this means is that there cannot be any past eternal universe–inflationary or not. Indeed, Vilenkin pulls no punches: “It is said that an argument is what convinces reasonable men and a proof is what convinces 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” (Vilenkin 2006, 176). Their theorem does not apply just to the universes and multiverses of chaotic inflationary theory individually (as Olster notes that Linde’s model is self-reproducing), but to the entire multiverse as a whole—that is, even though it can reproduce itself, it still must originally be past-finite. So this assertion by Olster is false.

However, while Olster is excused regarding this multiverse, he has no excuse regarding quantum fluctuation models of a multiverse. He seems to have read a variety of WLC’s articles, judging by his sources cited, but he has altogether skimmed over one of WLC’s most popular articles. WLC’s 1993 article for the British Journal for the Philosophy of Science, titled, “The Caused Beginning of the Universe: A Response to Quentin Smith” (accessible online here) specifically addresses quantum fluctuation models. A notable part of this article is the following:

Vacuum fluctuation models are incompatible with observational cosmology. As Isham ([1990], p. 10; [1992], sec. 2) points out, there is in such models simply no way in which the mathematics can select one particular moment within the pre-existent, infinite, and homogeneous time at which a fluctuation should occur which will spawn a universe. Similarly, no way exists for specifying a certain point in space at which such a creation event should occur. Rather vacuum fluctuation theories tend to predict a creation event at every time t, or more precisely, as quantum theories they predict a non-zero probability of a creation event within any finite time interval, with an infinite number of creation points distributed evenly throughout space. This leads at once to an infinite number of creation events within the wider spacetime. But then the fluctuation-formed universes would inevitably collide with each other as they expand, which contradicts the findings of observational cosmology, since we do not see such “worlds in collision,” to borrow a phrase.

This alone suffices to disprove Olster’s assertion that a past-infinite vacuum fluctuation model is possible. Note that even if the vacuum was spatially infinite, given infinite time a universe would have spawned at every point in the vacuum and expanded, thereby providing the same results that we do not observe.

So to summarize, given a neo-Lorentzian interpretation of SR, time still holds a single arrow—even through singularities—and kalam arguments would apply to any cosmological model. Furthermore and finally, no cosmological model exists that is coherently past-infinite (for more on this, see WLC and James Daniel Sinclair’s article in the BCNT).

On Logical Possibility of Infinite Time

Olster’s entire section here can be summarized in four sentences: “C&C don’t claim that a contradiction in first order logic can be derived from the proposition that the universe is not created and thus without a beginning. What they claim is that the idea is absurd. They thus claim that the notion of an infinitely old universe is metaphysically impossible – that is, there is no possible world in which such a universe can exist…. My argument does not attempt to establish that infinities actually do or do not occur in the real world; but merely demonstrates that it is logically possible that the world has always existed.”

WLC concedes this all over the place. He concedes this in an article both quoted by Olster and myself earlier in this argument. What is necessary, though, is showing that it is metaphysically possible. Olster also misunderstands modal logic as dealing with brute logic, rather than metaphysics. Possible world semantics necessitates metaphysical legitimacy. That is, p exists in some possible world, x, iff p can be or could have been actualized in reality. If something cannot exist in reality, it cannot exist in any possible world, and what is metaphysically absurd cannot exist in reality. Let’s review Olster’s argument, though:

Sp = the set of past times at which it is logically possible that the universe actually existed set:
3.1) The members of the set Sp can be placed into a one-to-one correspondence with the members of the set of real numbers.
3.2) Sets whose members can be placed into a one-to-one correspondence with one another have the same number of members.
3.3) The set of real numbers has Mi members.
3.4) Therefore, set Sp has Mi members.

A keen observer notices that this argument begs the question. (3.1), (3.2), and (3.3) simply assume that numbers exist as abstract, platonic objects, and therefore, the axioms of set theory exist in reality and actually apply to numbers. However, we have no reason to believe this. I myself hold to a constructivism with mathematical objects, and I therefore deny (3.3), as (3.3) is simply unproven if platonism is disregarded. As I have repeated again and again, an argument beyond logical consistency within the axioms of set theory must be given for something’s metaphysical possibility.

Furthermore, an actual infinite has additional problems when applied to the past. We have already seen that set theory shows that an actual infinite cannot apply to past-time. We have also seen that paradoxes such as Hilbert’s Hotel both do and don’t apply to an infinite past-time, as the same events can be construed as both a well-formed series and a not-well-formed series—given that they have occurred and we have our frame of reference. I contend, then, that because this violates the law of the excluded middle, a violation in first order logic is derived from an infinite past. Additionally, an actually infinite time is modally self-contradictory. This is an argument I’ve made, and it may need some revisions. Nonetheless, I’m reasonably sure that it holds:

1) An actually infinite amount of time is possible.
2) It is a modal property of infinite time that, given infinite time, all broadly logical states of affairs will—and must—obtain.
3) If a state of affairs must obtain, that state of affairs necessarily obtains.
4) It is a broadly logical state of affairs that a sentient being will arise.
5) Therefore, given infinite time, necessarily a sentient being will arise (2, 3, and 4).
6) It is a broadly logical state of affairs that nothing ever comes into existence.
7) Therefore, given infinite time, necessarily nothing ever comes into existence (2, 3, and 6).
8) It is impossible for a sentient being to arise if nothing ever comes into existence. Likewise, it is also impossible for it to be true that nothing comes into existence if a sentient being arises.
9) But it is a modal necessity of infinite time that both necessarily nothing will ever come into existence and necessarily a sentient being will arise (5, 7).
C) Therefore, (1) is false (8, 9).

Premise (4) is self evident, as we are here. Premise (6) is true because we can imagine a possible world where nothing exists or comes into existence; this is what gives meaning to the philosophical question, “why is there something rather than nothing?” Premise (3) and (8) are true by definition. What might surprise some people is that (2) is also true due to the definition of infinity as it applies to time. Given infinite time, no sufficient reason can be given why a state of affairs does not obtain if it is broadly logically possible. If no sufficient reason can be given why something does not obtain, it will obtain–given infinite time. If it will eventually obtain, it inevitably obtains, and if it inevitably obtains, it must obtain. Finally, premises (5), (7), and (9) follow from the truth of the other propositions. Therefore, (1) is false. This argument does not say that both of these states of affairs will obtain, given infinite time (as they both obviously can’t obtain), just that both are modally required to obtain, if infinite time exists.

This argument demonstrates that a violation of first-order logic is derived from an actually infinite time—should the argument hold.

Conclusion

In conclusion, Olster has mustered quite a defense of an infinite time and an argument against the kalam. However, upon close inspection, his arguments do not hold. A large part of his argument assumes a platonic view of mathematical objects, but asserting that this establishes metaphysical legitimacy simply begs the question—as evidenced by the quotation of mathematicians early in this essay. We have seen that the absurdities of an actual infinite do, in fact, apply to an infinite past-time. We have also seen that Olster’s argument against WLC’s usual presentation of the second argument is misguided and invalid. Furthermore, we have seen that kalam arguments apply to all cosmological models, should a neo-Lorentzian interpretation of SR hold, and we have a variety of reasons to believe that it does—or at least, can, and so Olster’s argument must be taken as inconclusive at best. We have also seen that no cosmological model has been shown to be plausibly past infinite (again, for more, see WLC and Sinclair’s article in the BCNT). Additionally, we have seen that Olster’s argument for the logical possibility of the infinite past begs the question against all who do not hold to a platonic view of mathematical objects. Furthermore and finally, we saw that it appears that first order logic is violated in the case of past-time with the law of the excluded middle and Hilbert’s Hotel, in addition to the possible validity of my argument.

I conclude by quotation of David Hilbert: “Let us draw the conclusion from all our reflections on the infinite. The overall result is then: The infinite is nowhere realized. Neither is it present in nature nor is it admissible as a foundation of our rational thinking – a remarkable harmony between being and thinking.”

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