1. Introduction
Issues concerning the ultimate origins of the universe have generated huge public interest recently. It began with the publication of The Grand Design by Stephen Hawking and Leonard Mlodinow, in which the authors claim that, because there is a law like gravity, the universe can and will create itself from nothing, and thus there is no role for God.Footnote 1 This has prompted heated responses from scientists, philosophers, and religious leaders in the media, many of them asking, in effect, ‘where that law came from’. Subsequently, Roger Penrose released his new book Cycles of Time, arguing that what came before the Big Bang was the end of another universe,Footnote 2 prompting the question whether there is an infinite temporal regress of universes or not. These discussions make it very timely to examine afresh whether an infinite temporal regress of events is possible.
Proponents of the Kalam Cosmological Argument (KCA) for the existence of God have for a long time answered in the negative. This argument has been used for centuries by Christian, Jewish, and Islamic theologians as a support for theism; the argument as formulated by its noteworthy recent proponent, William Lane Craig, is as follows:
1. Everything that begins to exist has a cause.
2. The universe began to exist.
3. Therefore, the universe has a cause.Footnote 3
For premise 2, one of the arguments typically offered is that the absurdities which result from paradoxes such as Hilbert's Hotel show that physical infinities cannot exist, and since an infinite temporal regress of events is a physical infinity, it follows that an infinite temporal regress of events cannot exist.Footnote 4
‘Friends of infinity’ (i.e. those who think that physical infinities can exist) have replied by arguing that the opponents of physical infinities have not demonstrated what the problem is supposed to be. As Graham Oppy argues,Footnote 5 ‘friends of infinity’ can take the following strategies in response to opponents such as Craig:
Strategy 1: Oppy calls this ‘outsmarting’ the opponent, that is, to embrace the conclusion of the opponent's reductio ad absurdum argument. He explains: ‘In many cases, these allegedly absurd situations are just what one ought to expect if there were large and small denumerable, physical infinities.’Footnote 6 For example, with regards to Hilbert's Hotel's ability to accommodate new guests by shifting rooms even though it is full, a friend of infinity will say that this is what ought to be expected of a hotel with an infinite number of rooms. In an earlier article, Oppy writes, ‘As far as I know, Craig nowhere explains what the difficulty is supposed to be.’Footnote 7
Strategy 2: Oppy writes,
It is very easy to tell a story about any subject matter that is inconsistent. It is very easy to tell a story about any subject matter in which there are events for the occurrence of which no sufficient reason is provided within the story. So one cannot argue from the mere fact that one has described an inconsistent scenario involving the occurrence of large and small denumerable, physical infinities to the conclusion that it is impossible for there to be large and small denumerable, physical infinities. At the very least, one has to be able to show that the inconsistency arises only because of the presence of large and small denumerable, physical infinities (and hence, cannot be removed by varying other features of the story while holding constant the fact that the story involves large and small denumerable, physical infinities).Footnote 8
With regards to Hilbert's Hotel, Oppy suggests that the problem might be related to the difficulty of constructing a hotel that will permit the checking out of infinitely many guests in a finite amount of time rather than the presence of large and small denumerable, physical infinities.Footnote 9
Strategy 3: Oppy writes, ‘Furthermore it is very easy to make the mistake of supposing that, because certain kinds of large and small denumerable, physical infinities really are impossible, it follows that all kinds of large and small denumerable, physical infinities are impossible.’Footnote 10 So even if a hotel with an infinite number of rooms is impossible, it does not follow that there cannot be other kinds of physical infinities such as an infinite temporal regress of events. Wes Morriston suggests that, even if successful, the argument from Hilbert's Hotel would only entitle one to conclude that physical infinities with elements that co-exist and bear a changeable physical relationship to one another cannot exist. However, events in a temporal series do not co-exist according to the dynamic theory of time, rather they exist one after another. Furthermore, events in a temporal series cannot be changed, ‘manipulated’ or removed in a manner analogous to checking people out of hotel rooms.Footnote 11 Thus, it cannot be concluded from Hilbert's Hotel that an infinite temporal regress of events cannot exist.
In response to these objections, I will give an argument against the existence of physical infinities and then discuss it in respect to the strategies proposed by Oppy. Before I start, here are some preliminary remarks.
First, the argument that I will be presenting is directed against the existence of an actual infinite number of physical (and not abstract) entities (due to limitation of space, I will not discuss whether my argument would work against abstract infinities); it is not directed against the mathematical legitimacy of the infinite. Concerning mathematical legitimacy, Craig and Sinclair explain,
Historically, certain mathematical concepts have been viewed with suspicion and, therefore, initially denied legitimacy in mathematics. Most famous of these are the complex numbers, which as multiples of √ –1, were dubbed ‘imaginary’ numbers. To say that complex numbers exist in the mathematical sense is simply to say that they are legitimate mathematical notions; they are in that sense as ‘real’ as the real numbers. Even negative numbers and zero had to fight to win mathematical existence. The actual infinite has, similarly, had to struggle for mathematical legitimacy. For many thinkers, a commitment to the mathematical legitimacy of some notion does not bring with it a commitment to the existence of the relevant entity in the nonmathematical sense.Footnote 12
Second, it is important that neither the opponent nor friend of infinity should beg the question in the dialectic. The opponent should not argue that physical infinities are impossible because they entail x, without arguments for thinking that x cannot possibly exist other than x being merely ‘strange’ or intuitively absurd. If, however, the opponent does have independent argument(s) (independent in the sense that the argument does not presuppose whether a physical infinite can exist or not) for thinking that x cannot possibly exist, then a friend of infinity should not counter by merely presupposing that physical infinities can exist, and insist that the entailment of x serves as a counter-example to the opponent's argument. Rather, a friend of infinity would have to rebut the independent argument(s) which the opponent offers.
How, then, can we know whether physical infinities exist or not? What kind of evidence or arguments do we need? Here I suggest that for any y, if it can be shown that certain metaphysically or logically necessary truths contradict y or what is entailed by y, then y does not (and indeed cannot) exist. My argument therefore proceeds as such:
1. Whatever entails the violation of metaphysical or logical necessities cannot exist.
2. Physical infinities entail the violation of metaphysical necessity.
3. Therefore, physical infinities cannot exist.
The above argument is obviously valid. The crucial question is whether premise 2 is true. In what follows, I shall demonstrate how physical infinities do entail the violation of metaphysical necessity. My argument shall proceed as follows: In Section 2, it will be shown that numbers do not have independent causal power, and that this metaphysical fact F is independent in the sense that it does not presuppose whether a physical infinite can exist or not. In Section 3, I will provide an illustration with two scenarios, one involving a finite number of entities and the other an infinite number of entities. In Section 4, a metaphysical analysis of these scenarios will be given, and it will be shown that the scenario involving an infinite number of entities entails the violation of metaphysical fact F. I will then discuss this conclusion with respect to Oppy's strategies in Section 5.
2. Numbers and independent causal power
I shall begin by demonstrating that numbers do not have independent causal power. Now it might be claimed that the number of things does make a difference to the physical world. For example, twenty 10-pound weights would make an accurate scale register two hundred pounds under normal circumstances, while ten 10-pound weights would not be sufficient. However, it should be noted that the reason why that is so is because each of the weights has causal power with respect to the scale, in virtue of their mass. To illustrate, let us stipulate that there are twenty weights (call them A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T), each of 10 pounds. Under normal circumstances, ten weights (A, B, C, D, E, F, G, H, I, J) would be insufficient to make an accurate scale register two hundred pounds, but twenty weights would be sufficient. One might argue that this example illustrates that the number of entities that make up a collection is directly relevant to the causal powers attributable to the collection. In reply, it can be agreed that this is the case. However, the reason why this is the case is because the causal powers (in virtue of the mass) of K, L, M, N, O, P, Q, R, S, and T act together with the causal powers of A, B, C, D, E, F, G, H, I, and J on the scale register; it is not because of the causal power of the number twenty, nor the number twenty changing the causal power of any of the weights. Fundamentally, it is only the causal power of each of the things in the set (in this case, each of the weights), and not the number of things, which makes a difference to the physical world. The ‘number’ of entities is causally inert with respect to the metaphysical quality of the entities, in the sense that it does not change the causal power of each of the entities; rather, it is a mere abstraction of the entities which exist. Since that is the case, it cannot be claimed that the ‘number’ of entities in conjunction with the entities' causal powers would make any difference with respect to the presence or absence of causal powers. In this sense, numbers by their nature do not have independent causal power.
To illustrate further that the number ‘twenty’ does not have independent causal power apart from the weights, suppose that a certain entity U has zero mass. In this case, either ‘twenty’ or ‘ten’ would not make a difference to the reading on the weighing scale. For 20 × 0 = 0 and 10 × 0 = 0. The ‘number’ of a set of entities is not the sort of entity which in conjunction with the entities in the set would have a certain causal power that the entities would not have had.
In short, whether a set of things has causal power or not ultimately depends on the things in the set, and not the number in conjunction with the things. Let us call this metaphysical fact F. It should be noted that this metaphysical fact is not based on whether a physical infinite can exist or not; all that it is based on is the abstract nature of numbers. Hence, this metaphysical fact does not express a ‘prejudice’ against the existence of physical infinities.
3. The Christmas present illustration
Having established that numbers do not have independent causal powers, let us consider two scenarios. Suppose that there is a group of finite number of people with each person carrying one packet of Christmas presents, and suppose that all the packets are similar, i.e. they all contain the same things, are packed the same way, etc. Suppose that these people put these Christmas presents into a bag, and each person subsequently grabs just one present from the bag. Let us define ‘leftover presents’ as presents that remain after (i) each person has grabbed one present, and (ii) there is no person who has no present. Now with respect to whether there are any leftover presents, it should not matter whether the presents had been placed in a bag and each person grabs a present from the bag, or whether the presents had been placed side by side in a line and each person grabs a present from the line, or whether the particular present each person grabs had been previously carried by any particular person, for all these considerations are causally irrelevant with regards to the presence or absence of leftover presents. In other words, no particular position, present, or grabbing of any particular present from any particular position is metaphysically privileged over the other where the presence of leftover presents is concerned. The presents are all similar, and there is nothing ‘magical’ about their particular spatial locations, be it in a line or in the bag, and there is nothing ‘magical’ about the grabbing of any particular present from any particular position. So long as each person just grabs one present, there should not be any leftover, regardless of where each present is grabbed from. To grab one present from one position rather than another is clearly not a process of making extra presents. Therefore, taking into account that no particular grabbing of any particular present from any particular position is metaphysically privileged over the other, the following principle follows:
M: The presence or absence of leftovers should be independent of where each present is grabbed from.
It is important to note that what grounds M is this metaphysical principle P.
P: Each person subsequently grabbing one present from one position rather than another has no causal power with respect to the presence of leftover presents.
It is uncontroversial that principles M and P are metaphysically necessarily true for finite sets. Now suppose that there is a group of an infinite number of people, with each person carrying one packet of Christmas present, and suppose that all the packets are similar, i.e. they all contain the same entities, are packed the same way, etc. Suppose that these people stand in a line next to each other and put down their presents in a line in front of them, and then grab the presents in this manner: person 1 in the line grabs the present that had been previously carried by person 2, person 2 grabs the present that had been previously carried by person 4, person n grabs the present that had been previously carried by person 2n … as illustrated by the following figure:
What would happen is that each person would walk away with one present, and there would be an infinite number of presents left over. (One might raise the worry whether all the presents could be picked up at the same time. In reply, the line of presents need not be a straight one. Rather, it could be a curving criss-crossed line such that the present previously carried by person 2n in the line is always well within reach of person n [alternatively, one can stipulate that all the persons have infinitely long arms; this stipulation is consistent with the assumption of this scenario {which involves an infinite number of people and presents} that physical infinities could exist], such that all the presents could be picked up at the same time).
But if they put their presents back in the line, and each person grabs one present in this manner: person 1 grabs the present that had been previously carried by person 1, person 2 grabs the present that had been previously carried by person 2, person n grabs the present that had been previously carried by person n … as illustrated by the following figure:
In this scenario, there would be no more presents left. What this shows is that, if there were an infinite number of presents, then each person grabbing a present from one position rather than another would make a difference as to whether there are leftovers. (This would not have been the case if the presents were finite. If there were six presents, and person n were to grab the present that had been previously carried by person 2n, then there would be three presents left with three persons who have no presents, but these would not be considered ‘leftover presents’ because, as noted previously, leftover presents are defined as presents that remain after (i) each person has grabbed one present and (ii) there is no person who has no present). Indeed, one would think that it would be better for each person n to take the present that had been previously carried by person 2n rather than the present that had been previously carried by person n, since there would then be an infinite number of leftovers for which they can come back repeatedly, and each person would have more than one present. Unending supply of infinites upon infinites of leftover presents would be generated, simply by picking the presents from the right positions! But one wonders, where did those extra leftover presents come from? One would suspect that such a scenario is impossible and that there is a violation of metaphysical necessity somewhere. I shall now demonstrate that that is the case.
4. A metaphysical analysis of the illustration
For the Christmas present illustration, I argued that
M: The presence or absence of leftovers should be independent of where each present is grabbed from.
is grounded by
P: Each person subsequently grabbing one present from one position rather than another has no causal power with respect to the presence of leftover presents.
I argued that P (and hence M) is metaphysically necessarily true for finite sets of physical things, and this is uncontroversial. The crucial question to ask is whether P (and hence M) is metaphysically necessarily true only for finite sets, or is it metaphysically necessarily true for ‘any set with any number of members’ that can exist. Consider the latter possibility, formulated as follows:
4. Let z = a set of physical things (e.g. person-present), and let n = the number of physical things in z; P (and hence M) is metaphysically necessarily true for ‘any z with any n’ that can exist.
If premise 4 is true, then the opponent of physical infinities can proceed to argue as follows.
5. As shown by the Christmas present illustration, M is not true for a physical infinite (i.e. each person grabbing a present from one position rather than another would make a difference as to whether there are leftovers).
6. Therefore, a physical infinite cannot exist. (From [4] and [5])
A friend of infinity would have to deny premise 4; he would have to object to P being metaphysically necessarily true for ‘any set with any number of members’. Now a friend of infinity cannot deny premise 4 by arguing that:
4′. P is true of finite sets.
5′. Therefore, P is true of any z if and only if n is finite (denial of premise 4).
The obvious problem with the above argument is that 5′ does not follow from 4′. From the fact that P is true of finite sets, it does not follow that P is true of sets if and only if n is finite. To deny premise 4, a friend of infinity would have to claim that the number of person-present in conjunction with each person subsequently grabbing one present from one position rather than another would make a difference concerning the presence or absence of causal power with respect to leftovers. He would agree that a finite number in conjunction with each person subsequently grabbing one present from one position rather than another would not have any causal power with respect to leftovers. However, he would have to claim that, acting collectively, an infinite number of people can bring about different results – whether there are leftover presents or a one-one distribution of presents – by the way in which they collectively act. What this amounts to is the claim that infinite collectives can have collective causal powers which finite collectives cannot have.
However, it should be clear by now that such a claim is false. As demonstrated previously, the ‘number’ of a set of things is causally inert and therefore irrelevant with respect to the metaphysical quality of the set of things, in the sense that it does not affect the causal powers of the set of things. Rather, it is a mere abstraction of the things that exist in the set. As shown above, the ‘number’ of a set of things is not the sort of entity which in conjunction with the things in the set would have certain causal power that the things would not have had. Whether a set of things has certain causal power or not ultimately depends solely on the things (in this case, a thing = ‘each person subsequently grabbing one present from one position rather than another’), and not the number in conjunction with the things. Since that is the case, it cannot be claimed that ‘the number (whether finite or infinite) of person-present’ in conjunction with ‘each person subsequently grabbing one present from one position rather than another’ would make a difference concerning the presence or absence of causal power with respect to leftovers. Rather, the presence or absence of such causal power would ultimately depend solely on ‘each person subsequently grabbing one present from one position rather than another’, and the number n would be irrelevant. Hence, the number of physical things in a set should not matter where the range of P over z is concerned. Now we know that P ranges over any z where n is finite (both friends and opponents of infinity are agreed on this). However, since n is irrelevant, it is not the case that P ranges over any z only where n is finite; on the contrary, it is the case that P ranges over any z for any n. Hence, P is metaphysically necessarily true, not merely ‘for finite sets’, but for ‘any sets with any number of members’ that can exist.
It should be emphasized that my argument is not based on the presupposition that physical infinities cannot exist, which would be begging the question. Rather, my argument is based on the independent metaphysical fact F that numbers are causally inert. It should also be stressed that, at this stage of the dialectic, a friend of infinity should not beg the question.
He should not simply argue that ‘physical infinities do not obey principle P, therefore premise 4 is false’. For to argue that would be to presuppose that physical infinities can exist, which is precisely what is being denied by the opponent. Additionally, a friend of infinity should not simply claim that ‘it is unproblematic that there would be causal capacities present in infinite collections that are not present in finite collections, thus if metaphysical fact F really does entail the rejection of this unproblematic claim, then it would be question-begging to insist on metaphysical fact F in the present context.’ For to claim that ‘it is unproblematic that there would be causal capacities present in infinite collections that are not present in finite collections’ would be to beg the question against the opponent who affirms that this claim is problematic. As noted in the Introduction, it is important that neither the opponent nor friend of infinity should beg the question in the dialectic. A friend of infinity should not counter my argument by merely presupposing that physical infinities can exist, and insist that the entailment of x (where x is understood here as the violation of P) by a physical infinite serves as a counter-example to the opponent's argument. Rather, he would have to rebut the independent reason the opponent offers, and the independent reason in this case would be the metaphysical fact that numbers are causally inert in the sense explained above. Additionally, a friend of infinity should not merely claim that it is question-begging to insist on metaphysical fact F simply because it entails the rejection of physical infinities. Rather he would have to rebut the independent reasons supporting metaphysical fact F (which have been discussed in previous sections) and to claim that an actual infinite number does have independent causal powers. If a friend of infinity should choose to make such an ad hoc claim, then I would like to see him/her defend it. But here is my preemptive response: to claim that an actual infinite number has causal powers such that it can actually produce infinite leftovers by acting on the persons' actions in grabbing the presents from certain positions, which otherwise do not have causal power with respect to leftover presents, is really to claim that such numbers are concrete particulars with highly active causal powers of their own. If this claim were true, such a number would not be abstract anymore, but it would be something that is concrete and existing alongside the sets of presents and people! But of course, numbers are not concrete particulars which exist alongside the concrete particulars that they are of. Hence, it is metaphysically necessarily true that the causal powers of any set of entities ultimately depend on the entities in the set and not the number in conjunction with the entities, but this principle would be violated if physical infinities exist.
5. An assessment of Oppy's strategies
We shall now discuss the strategies proposed by Oppy and show why they fail. With regards to strategy 1, it would be problematic for a ‘friend of infinity’ to embrace the reductio of my argument by claiming that these ‘absurd situations’ in my Christmas present illustration are just what one ought to expect if there were large and small denumerable, physical infinities. This is because it has been shown that the reductio of my argument actually involves a violation of metaphysical necessity, and with regards to Oppy's question ‘what the difficulty is supposed to be’, the violation of metaphysical necessity is exactly what the difficulty is. If one does indeed expect such a violation to be entailed by physical infinities, then based on premise 1 physical infinities cannot exist.
With regards to strategies 2 and 3 and the sort of objections Oppy and Morriston offer against Hilbert's Hotel, we can easily adjust the details of the Christmas present illustration such that the violation of metaphysical necessity arises only because of the presence of an infinite temporal regress of events. Suppose there is a ‘Christmas present generator’ which has been generating similar Christmas presents at fixed temporal intervals as long as time existed.
Suppose there is also a ‘person generator’ which has been generating persons at the same fixed temporal intervals as long as time existed. Suppose that the presents and the persons continue existing after they have been produced. Now it should be clear that with respect to whether there is any leftover present should each person grab one present, no grabbing of presents produced at any particular instant is metaphysically privileged over the other. This is because the presents are all similar, and there is nothing ‘magical’ about the particular instant at which they are produced. In accordance with premise 4, each person grabbing one present produced at one instant rather than another is not causally active where the presence of leftover presents is concerned, and this is true for any n that can exist, where n represents the number of temporal regress of events.
Therefore, it is metaphysically necessary that
M1: The presence or absence of leftovers should be independent of the grabbing of presents produced at any particular instant for any n that can exist.
Now suppose at time t 0 the person who was generated at time t -1 picked up the present generated at t -2, the person who was generated at t -2 picked up the present generated at t -4, the person who was generated at t -n picked up the present generated at t -2n, as illustrated by the following figure:
Now if there were an infinite temporal regress of events, what happens is that each person would walk away with one present, and there would be an infinite number of presents left! But if they had grabbed the presents this way: person who was generated at t -1 picked up the present generated at t -1, the person who was generated at t -2 picked up the present generated at t -2, the person who was generated at t -n picked up the present generated at t -n … , as illustrated by the following figure:
The result is that there would not be an infinite number of presents left! Therefore, there would be a violation of metaphysical necessity truth M1.
Note that such a violation remains even if (as Oppy suggests) we vary other features of the story. For example, we can vary other features such as stipulating dogs instead of persons and bones instead of presents, and the violation will still persist. Thus, nothing in my argument depends on persons or Christmas presents; the illustration using persons and Christmas presents are merely conceptual tools which help us to understand the metaphysics involved. However, it is obvious that in this scenario, so long as the temporal regress of events is infinite, the violation will persist, but if the regress were finite, no such violation would occur. Therefore, the violation of metaphysical necessity is necessarily related to the assumption of an infinite temporal regress of events. Hence, an infinite temporal regress of events is metaphysically impossible.
6. Conclusion
This paper advances the discussion concerning the ultimate origins of the universe by showing what exactly the problem with physical infinities is. The problem is not merely that intuitively absurd situations would result if physical infinities exist. Rather, the problem is that physical infinities entail the violation of metaphysical necessity, in particular the violation of the metaphysically necessary principle that the causal powers of a set of entities ultimately depend on the entities in the set and not the number in conjunction with the entities. It has also been shown that in certain scenarios such violations would follow only if physical infinities such as an infinite temporal regress of events exist. Since whatever entails the violation of metaphysical necessity cannot exist, physical infinities such as an infinite temporal regress of events cannot exist.Footnote 13
