Angra Mainyu's blog: A Reply to the Kalam Cosmological Argument (v. 7.4.1)

The KCA is defended by some theist philosophers in combination with other arguments intended to bridge the gap between the conclusion that the universe has a cause and a conclusion that the cause of the universe is a being with certain properties, in particular a personal agent of great power.

I will call the combination of the KCA with those other arguments, “KCA+”.

In this essay, I will assess the KCA+, and argue that it provides no support for the conclusion that there is a powerful personal agent who created the universe, both by challenging the premises of the KCA itself, and the creation account proposed in usual versions of the KCA+.

b. I will focus mostly on William Lane Craig’s version of the KCA+, but I will also assess some alternative arguments.

c. I will use the word '‘argument’ to denote (depending on context) either formal arguments (i.e., premises and conclusion), or the arguments (in the sense of ‘arguing a case’) given in defense of said premises; these usages are common and shouldn’t cause confusion, given context.

d. I will call the parts of this essay ‘sections’ and ‘subsections’, without using "sub-subsections" or similar terms. Context and some links between different parts of this essay should prevent ambiguity.

e. By 'incoherent' I mean that a claim is either meaningless, or meaningful but contradictory.

2. An analysis of the proposed creator, and the proposed creation account.

According to William Lane Craig, the creator proposed by the KCA+ is a personal being who is timeless without creation, and temporal with creation.

But that raises questions like: ‘What does the word ‘timeless’ mean, in this context?' 'Is such claim coherent?' ‘Is it compatible with the premises of the KCA?', etc.

In this section I will assess some of those issues, and generally Craig’s creation account.

2.1. Timelessness.

The word ‘timeless’ is not a word colloquially used, in the relevant sense.[1 ]

Yet, Craig does not seem to provide a clear definition, which leaves the matter of the meaning of the claim that God is timeless without creation but temporal with creation, obscure at best.

In order to try to understand what Craig means by ‘timeless’, or at least to approximate the concept, one potential approach would be to look at a list of entities that Craig calls “timeless” and at a list of entities that he does not call “timeless” - and much better yet, entities that he clearly would categorize as not timeless -, and try to grasp the meaning of the term ‘timeless’ from that.

Now, entities that are not called “timeless” by Craig are easy to find. Moreover, he claims that God is temporal with creation, and given his argumentation, it seems clear that his position is that, say, horses, planets, humans, etc., are not timeless.

On the other hand, entities that Craig calls “timeless” are not so easy to find. Craig gives the example of a number[r2 ], but he’s not a Platonist, and furthermore, questions the existence of numbers. [r3 ] Still, we may understand his example as implying that if “numbers exist” were true – as an ontological claim -, then a number would be an example of a timeless entity.
However, that example alone does not suffice to grasp the meaning of the word in Craig’s usage. Nor does the combination of the example of a number and the very different example of God.

Still, perhaps we may add to the list some examples of entities – or alleged entities – called “timeless” by some other philosophers. Maybe Craig and those other philosophers are all using the word ‘timeless’ in the same sense.

If so, on the list of timeless entities – or candidates to timeless entities, if they were entities and existed -, we would have, say, the number 5, the set of natural numbers, the proposition that water is H2O, etc.

On the list of non-timeless entities, we would have trains, planets, humans, mosquitoes, viruses, telephones, apples, oranges, human souls if they existed, etc.

However, if one tried to grasp the meaning of ‘timeless’ based on those lists, it would seem obvious that personal beings are not timeless.

If it’s not obvious that personal beings are never timeless, then that approach to grasping the meaning of the word ‘timeless’' in this context does not work at all – at least, not for me.

Perhaps, given Craig’s relational theory of time, it might be said that being timeless is at least equivalent to not standing in temporal relationships, even if the meaning is not the same. That does not seem to help much in my view, but it’s something. Moreover, there is another way of at least approaching Craig’s concept of timelessness, namely considering some of the consequences of timelessness according to Craig, which one may deduce from some of his statements.

So, based on that, and even though what Craig means by ‘timeless’ remains obscure, I will proceed to analyze some of his claims.

2.1.1. Timelessness and change.

Craig claims that in his timeless state, God is changeless, or unchanging. However, Craig also claims that God can change and changed and ceased to be timeless.[r 4 ]

But how can we make sense of those claims?

We can understand the idea that an object O remains changeless for a while and then changes. That means that O remains unchanged as time goes by, during some period, and after that period ends, O changes.

However, clearly that is not applicable here, since that would imply that God’s changeless state obtains during a temporal interval through which God remains unchanged, yet Craig is claiming timelessness in addition to changelessness, and while Craig’s usage of the word ‘timeless’' is not clear, it is at least clear that a timeless state cannot be one during which time goes by.

So, the claim of changelessness is also very obscure.

Moreover, in the context of his defense of the KCA, Craig claims that instantaneous changes are impossible, and that every event/change has a finite, non-zero duration. [r 5 ]

So, let’s consider the change in God from being timeless to being temporal. That is surely a change, in the usual sense of the word ‘change’ - which is Craig’s usage -, and so it follows from Craig’s claims that it has a finite, non-zero duration, say d > 0.

Thus, there is a temporal interval of non-zero duration d that ends at the first temporal state of the world, namely the first state at which God is temporal. That entails there was a temporal interval of non-zero duration d before there was time. But that is absurd.

Craig actually addressed this objection or an essentially similar one on his website[r 6 ], and contended that:

a. When he defines what “event” means[r 5 ], in the context of his defense of the KCA, he’s trying to rule out instantaneous changes, and in that technical sense of ‘event’', anything that is instantaneous would not count as an event.

b. The creation of the universe by God or, for that matter, God’s changing from timeless to temporal, would not qualify as an event in that sense, since it’s instantaneous.

c. When he said that God changed and ceased to be timeless, he was using the word ‘change’ in a different sense from the sense in which he was using that word in the context of the Kalam Cosmological Argument. Furthermore, when he said that God changed, in that context, he merely meant that God’s properties at his timeless state are different from his properties at his first temporal state. [r 6 ]

d. In any case, any problem can be resolved simply by stipulating, in the context of the defense of the KCA, that one is talking about changes of equal, non-zero and arbitrary duration, so the alleged contradiction can be resolved simply by a clarification of one’s terms. [r 6 ]

However, all four claims a-d. above are mistaken, for the following reasons:

a. Craig defined '‘event’ to mean ‘any change’, and then claimed that because any change takes time, then under that definition, there cannot be any instantaneous events, which is another way of saying that because any change takes time, there cannot be any instantaneous changes.

However, it’s not the case that anything instantaneous would not count as an '‘event’ in that sense of ‘event’ if that instantaneous thing were a change. Any change, instantaneous or not, would be an event, by definition. Rather, in the context of that defense of the KCA, Craig claims that any change takes time [r 5 ], and from that and the definition of '‘event’ as 'any change’ - rather than from the definition of ‘event’ alone – he concludes that there are no instantaneous events.

In that context, Craig does not define ‘change’. Rather, he uses '‘change’ in the usual sense, and defines ‘event’' in terms of ‘change’'.

The objection here is that the change in God from his timeless state to his first temporal state would be, well, a change, and as such, by Craig’s own claims, it would have a finite, non-zero duration, resulting in a contradiction.

b. As before, since ‘event’ means ‘any change’ in the sense in which Craig uses the term ‘event’' in the context of his defense of the KCA, God’s becoming temporal does qualify as an event, since that is clearly a change in God. If said change in God were instantaneous and had no finite, non-zero duration, then that would contradict Craig’s claim that any change takes time, and has a finite, non-zero duration.

c. If by saying that God changed and ceased to be timeless, Craig had merely meant that God’s properties at his timeless state are different from the properties God has at his first temporal state, then for that matter Craig might as well had said that God changed and ceased to be temporal, becoming timeless. But clearly, in context, that would be a very different claim.

So, considering context, one can ascertain that Craig did not merely mean that God has different properties at those two states.

Moreover, we can tell that when he said that God changed, Craig meant that...God changed, using the word ‘change’' in the usual sense of that word in English.

The usual sense of ‘change’ is also the way in which Craig used ‘change’ in the context of the defense of the KCA as well[r 5 ], since he defined ‘event’ in terms of change, but gave no definition of ‘change’ or made any suggestion that he was using ‘change’ in a non-standard, technical fashion.

d. Stipulating that one is going to talk about those events in particular does not change the fact that the claim that any change takes time and has a finite, non-zero duration, plus the claim that God changed from a timeless state to a first temporal state, entails a contradiction. The contradiction still follows from Craig’s claims.

In my assessment, Craig’s contradictory claims plus the fact that in his reply to the objection mentioned above he misunderstood the claims he had made earlier, only compound the problem of the obscurity of his creation account, raising serious doubts about his own understanding of the position he’s defending, and of the coherence of said position.

Still, we may consider whether there is a way out for a defender of a position similar to Craig’s.

If there were a way out of this particular difficulty, it would require denying that any change takes a positive amount of time, since time before time makes no sense, and God clearly changes, on Craig’s description or similar ones, from a state at which he’s timeless, to a state at which he is not timeless.

However, denying that any change takes a positive amount of time would raise another difficulty, namely that instantaneous changes seem to be a problem under a theory of time that maintains that time is discrete, which is the kind of theory defended by Craig and usually espoused by defenders of the KCA+.

More precisely, if time is dense, and – for instance – there are two temporal intervals A=[a, b) and B=[b, c), then the temporal distance between A and B is zero. If P obtains all through A, and ¬P obtains all through B, there is a sense in which it makes sense to say that the change from P’s obtaining to P’s not obtaining is instantaneous. Yet, if time is discrete, this option is not available, and it’s hard to see how one can make sense of instantaneous changes in discrete time.

But let’s consider other difficulties, independent from the previous objection.

2.1.2. Timelessness, changelessness and quiescence.

According to Craig, the universe cannot have existed in an “absolutely quiescent state”, like God’s – alleged – initial timeless state. [r 5 ]

However, as in the case of changelessness, that raises the question of how we can make sense of such a claim[2 ]. Usually, we would understand that a quiescent object is one that is at rest as time goes by, but given Craig’s claim of timelessness, the claim of quiescence is puzzling.

In the rest of this subsection, I will continue to analyze that claim, as well as the claim of changelessness in the context of his creation account.

2.1.2.1. A brief analysis.

Let’s consider first two scenarios, ordering states of the world causally.

Scenario a:

First state of the world:

Timeless state S. The only object that exists is O. [3 ]

Second state of the world:

Temporal state. t=0. The objects are O and U, and O is the cause of the existence of U.

Third state of the world:

Temporal state; t=r > 0. The objects are O, U and, perhaps some other objects.

Scenario b:

First state of the world:

Initial temporal state; t=0. The only object is O.

Second state of the world:

Temporal state; t=r>0. The objects are O and U, and O is the cause of the existence of U.

Third state of the world:

Temporal state; t=s > r. The objects are O, U and, perhaps some other objects.

So, allegedly and going by Craig’s statements, the [causally] first state of the world would be changeless and/or absolutely quiescent in scenario a, but neither changeless nor quiescent in scenario b. [2 ]

But that seems clearly false, since the first scenario and the second seem to be exactly the same in terms of changes and lack of quiescence. In particular, the entity O at the first stage of the world changes from that state to the second state of the world in the two scenarios.

Moreover, if, in scenario b, object O begins to exist, then it seems so does object O in scenario a, and so in particular, Craig’s God would be an entity that begins to exist but has no cause of his existence, contradicting premise 1 of the KCA.

On the other hand, if it is not the case that in scenario b, object O begins to exists, then for that matter it might be, for all we know, that if the past is finite, it is not the case that the universe begins to exist, contradicting premise 2 of the KCA.

In brief, those ‘two’ scenarios appear to be ontologically identical, despite the label ‘timeless’ in the first one, and rather than two scenarios, it seems to be one with different labels.

At this point, someone might suggest that scenario a might not properly represent Craig’s view of God’s being timeless without creation and temporal with it, and that perhaps, on his view, there might not be a first temporal state at t=0, but instead an open temporal interval of the form (0, u] for some u>0, and that there are times arbitrarily close to zero, but not an initial moment. However, Craig’s position entails that such open interval is impossible because it would be an actual infinity, so this objection would fail.

In the following subsections, I will analyze this matter in greater detail.

2.1.2.2. Some terminology and assumptions for the rest of subsectio n 2.1.3.

1. I will use use bold, italic, uppercase letters (e.g., J, K, etc.) to denote finite temporal intervals, and bold italic lowercase letters (e.g., t, u, etc.) to denote temporal instants. I will also number them (e.g., t(1), t(2), etc.).

2. When I talk of temporal intervals or instants, I’m talking about actual features of the world – whatever those happen to be -, not about mathematical models.

3. When I speak of temporal intervals, I’m assuming those intervals are of finite duration, though not necessarily of equal duration.

4. By '◄' I mean ‘earlier than’, in the usual, temporal sense of ‘earlier’'. For instance, J◄K means that J and K do not overlap in any interval of non-zero duration, and J happens earlier than K.

5. When I prove lemmas, unless otherwise stated, I’m assuming any parts of Craig’s position, as required. In other words, those lemmas are statements that follow from Craig’s position. I will explain why that is his position when such explanation is needed.

2.1.2.3. Changes and infinity.

Lemma 1: Infinitely many actual changes in the world are impossible, regardless of whether their duration is equal.

Proof:

This can easily be seen in Craig’s defense of the KCA. For example, he claims that an infinite temporal regress is an actual infinity. [r 7 ]In context, Craig is talking about changes of equal temporal length, but clearly, his claim that they constitute an actual infinity in the sense of cardinality does not depend on their length, and he claims that such infinities are impossible.

2.1.2.4. Discrete time

Definition: A temporal interval J has property Q if and only if there is some positive integer N(J), such that J does not have more than N(J) temporal subintervals different from each other. Equivalently, J has property Q if and only if it is not the case that there are more actual different temporal subintervals of J than any natural number.

Lemma 2: If there are temporal intervals, then every temporal interval J has property Q.

Proof:

Let’s assume there is a temporal interval J that does not have property Q.

By lemma 1, there are only finitely many actual changes that happen in the world during J. So, let m > 2 be a natural number greater than the number of actual changes during J.

Since J does not have property Q, there is some natural number r > (2* m) (for instance), such that J actually has r different temporal subintervals, J(1), J(2), J(3).., J(r).

Let us consider the following changes, for k in {1,...,r-1}

E(k): God changes from not knowing that J(k) is past to knowing that J(k) is past.

F(k): God changes from not knowing that J(k) is present to knowing that J(k) is present.

Note that those are actual changes in the world, more precisely in the mind of God. Given that the intervals are different from each other, there are at least r-1 > m distinct changes, contradicting the assumption. That proves lemma 2, under the assumption that God exists, which is clearly Craig’s position.

Lemma 3: If there are temporal intervals, every such interval J is divided in finitely many ordered temporal intervals, each of which has no actual proper subintervals.

Proof:

Let n(J) be the total number of actual temporal subintervals of each subinterval J; by lemma 2. n(J) is finite for all J.

Then, given some interval K, if n(K)=1, we’re through, since K has no proper subintervals.

Suppose that lemma 3 is true for any J such that n(J) < n, for some n > 1, and let K be such that n(K)=n.

Then, let K(1) be a proper subinterval of K. Then, by hypothesis, the lemma applies both to K(1) and to any part of K before and after K(1). Then, by induction, the lemma is proved.

Lemma 4: If there are temporal instants, then for every two instants t and u such that t ◄ u, there is some nonnegative integer N(t, u), such that there are no more than N(t, u) instants between t and u.

Proof:

Similar to the proof of lemma 2, with the required changes to address instants instead of intervals.

Lemma 5: Either there are temporal intervals that do not have actual divisions (i.e., no actual proper subintervals), or there are instants.

Proof:

On a tensed theory of time, temporal becoming is a real feature of the world, so if there are no intervals, it seems clear that there are instants. If there are intervals, there are intervals that don't have actual divisions in smaller intervals, by lemma 3.

2.1.3.5. An absolute temporal beginning

Lemma 6: If there are temporal intervals, there is an absolutely first such temporal interval K(0), which has no actual proper subintervals. Moreover, if there is more than one interval, then there is a second such interval K(1) with no proper subintervals. Generally the past is composed of finitely many such temporal intervals, each of which has no proper subintervals.

Proof: Let’s assume there are infinitely many past intervals {K(n)}, for all natural numbers n.

Let’s consider the events

G(n): God changes from not knowing that K(n) is past to knowing that K(n) is past.

H(n): God changes from not knowing that K(n) is present to knowing that K(n) is present.

Then, there are infinitely many actual past changes, contradicting lemma 1.

That proves that if there are temporal intervals, there are finitely many past ones, assuming that God exists – which is Craig’s position -, and other parts of his position.

Since each such interval is divided in finitely many subintervals each of which has no proper subintervals by previous lemmas, then it follows that if there are temporal intervals, then there are finitely many past intervals with no proper subdivisions, and a first such interval K(0).

If there is some other interval J such that begins earlier than J than K(0), then by lemma 3 we can divide J in intervals that are not further divided, and one of them would be earlier than K(0), a contradiction.

By a similar procedure, we can construct a second one, and so on, and the total number up to the present time must be finite by lemma 1, since otherwise we could construct an infinite sequence of temporal changes.

That proves lemma 6.

Lemma 7: If there are temporal instants, there is a first temporal instant t(0). Similarly, there is a second instant t(1), a third one t(2), and generally the past is composed of a finite number of such instants.

Proof:

Like the proof of lemma 6, removing superfluous parts and making the necessary changes to address instants instead of intervals.

Lemma 8: Either there is a first temporal instant t(0), or a first interval K(0) that has no proper temporal subintervals.

Proof:

It follows from lemmas 5, 6, and 7.

Lemma 9: There is an absolute first, temporal state of the world T(0), with no temporally proper states contained in it. Moreover, there is a second such state T(1), a third T(2), and generally the past if composed of finitely many such states.

If there are instants, then let T(n) be the state of the world at t(n), for all different past instants, taking t(n) ◄ t(n+1). It’s clear, then, for k>n, T(k) is different from T(n), though we may as well prove that too, as follows: For k>n, at T(k) God knows that t(n) is past and that there are at least n past instants, whereas at T(n), it is not the case that God knows that t(n) is past and it is not the case that God knows that there are at least n past instants.

So, if k>n, then the states T(k) and T(n) are actually different from each other. Also, since each T(n) corresponds to a single instant, it has no proper temporal subdivisions.

If there are no instants, then there are intervals with no proper subintervals by lemma 5. So, let T(n) be the state of the world at K(n), for all past intervals with no proper temporal subintervals, and taking K(n) ◄ K(n+1).

Then, for k>n, at T(k) God knows that K(n) is past and that there are at least n past intervals with no proper subintervals, whereas at T(n), it is not the case that God knows that K(n) is past and it is not the case that God knows that there are at least n past intervals with no proper subintervals. So, if k>n, then the states of the world T(k) and T(n) are different from each other.

Also, since each T(n) corresponds to a single interval with no proper subintervals, there are no actual temporal divisions in each state T(n).

This establishes lemma 9.

2.1.3.6. No ontological difference

Let’s consider now the following two scenarios, ordering the states in terms of causal priority.

Scenario 1:

First state of the world:

Timeless state S.

The only object is God, which exists timelessly at S and without a cause of his existence.

Second state of the world:

First temporal state T(0).

God exists temporally, and the universe[4 ] exists. The change from the first to the second state of the world, including the creation of the universe and God’s own change from timeless to temporal, takes place because God intends to bring it about.

Third state of the world:

Second temporal state T(1). The objects are God, the universe, and perhaps some other objects.

Scenario 2:

First state of the world:

First temporal state T(0).

The only object is God, who exists temporally at T(0) and without a cause.

Second state of the world:

Second temporal state T(1).

God exists temporally, and the universe exists. The change from the first to the second state of the world, including the creation of the universe, takes place because God intends to bring it about.

Third state of the world:

Third temporal state T(2). The objects are God, the universe, and the same other objects as in scenario 1 (if any), with the same causes.

It appears that even though the word ‘timeless’ is used in the first scenario, the first causal state in scenario 1 is ontologically no different like the first temporal and causal state T(0) in scenario 2. In other words, it seems those aren’t really two scenarios, but one scenario with two different labels.

This is so because:

i. In both cases, what we have is a first state of the world changing into the next, then the next, and so on.

ii. In both cases, God exists without a cause of his existence at the first state of the world.

iii. In both cases, the universe exists at the second state of the world, and exists because God intends to bring it about, and so on.

iv. Generally, one is not able to find a distinction on those states by inspection of the description, other than the difference in the words that are used.

Also, saying that the difference between the two scenarios is that one state – namely, state S in scenario 1 – is timeless and changeless and the other – namely, T(0) in scenario 2 – is temporal and changing, or that in one of them there at least one tensed fact, whereas in the other one there are no tensed facts, etc., would fail to address the objection, since that would just amount to denying that there is no ontological difference, without explaining how or why that is so, which is a burden on the defender of the KCA +, since it seems intuitively very plausible that the states are ontologically identical just by looking at the description of those states, regardless of the word ‘timeless’.

In particular, one can tell that despite the claim that the first state in scenario 1 is said to be timeless, it is not changeless. In fact, ‘both’ states are not changeless, and for the same reason, namely that just as God changes from his first to his second state in scenario 2, he does so in scenario 1.

So, it seems it remains the case that there is an entity – i.e., God – that exists at a first state of the world and without a cause of his existence, changes to a second state, then to a third one, etc., regardless of whether the first state of that entity is called ‘timeless’.

Thus, if the concept of timelessness is coherent and entails changelessness, then scenario 1 is improperly described as timeless, since the first state is not changeless, and hence not timeless. It appears, then, that scenario 1 is just scenario 2, plus a false claim of timelessness.

Let’s consider now potential objections, claiming ontological differences between scenario 1 and scenario 2.

Objection 1: An ontological difference between scenario 1 and scenario 2 is that temporal states of God must change as time goes by, whereas the timeless state S in scenario 1 could have remained unchanged.

In other words, if God exists at temporal state T(0) – as in scenario 2 –, then necessarily, there will be a second temporal state T(1), and necessarily, God will change from T(0) to T(1). On the other hand, if God exists in a timeless state S (as in scenario 1), then it is possible, given that first timeless state S of God, that God never changes.

The problem with that objection is that it follows from any creation hypotheses posited by defenders of the KCA + – i.e., given what God’s first state actually would have been, based on the implications of their claims –, that it is impossible that God never changes given that particular first allegedly timeless state S, which shows both scenario 1 and scenario 2 are indistinguishable in that regard as well, as the following reasoning shows (as usual, the states of the world are ordered causally):

Scenario 3:

First state of the world:

Timeless state S. The only object is God, who exists timelessly at S and without a cause.

Second state of the world:

Temporal state T(0). God exists temporally, and the universe exists temporally. The change from the first to the second state of the world, including the creation of the universe and God’s own change from timeless to temporal, takes place because God intends to bring it about.

Scenario 4:

First state of the world:

Timeless state S'. The only object is God, which exists timelessly at S' and without a cause.

There are no temporal states of the world.

In scenario 3, if God does not intend at the allegedly timeless state S to change, then at T(0) God just found himself altered.

In other words, if God’s intent to change does not exist at the allegedly timeless state S, then the change is not something brought about because God intends to bring it about, but something that happened to God, beyond his intent.

That is so because the change under consideration is a change in God’s state, and from the causally first state S. In other words, it’s a change from S to T(0).

So, if God does not intend at S to bring about the change, then said change cannot have been caused by God’s intent at T(0) or at any later state of the world, given that S is causally prior to T(0) and to any other state.

However, that contradicts the hypothesis that the change in scenario 3 happens because God intends to bring it about, rather than being something that happens to God.

Thus, at S, God intends to bring about the change. Hence, given God’s state at S in scenario 3, it is impossible that God does not change, since God can't fail to bring about what he intends to bring about.

Moreover, just as in scenario 3 it is impossible that God does not change, the same is true and for the same reasons in scenario 1. Also, the state S in scenario 3 is different from the state S' in scenario 4, in which God does not change, and so he does not intend to change.

So, if the concept of timelessness is coherent and state S' in scenario 4 is a timeless state, then that state is ontologically different from any initial state of the actual world posited by defenders of the KCA+, which – despite claims of timelessness – remains ontologically indistinguishable from a first temporal state, as far as one can tell.

At this point, someone might suggest that, in scenario 3, God exercised his libertarian free will at T(0) to bring about the change, but at S, it is not the case that he intends to change.

However, leaving aside issues about the coherence of libertarian free will, that is impossible for the same reasons I gave above, namely that since the change under consideration is God’s change from S to T(0), it was not decided at T(0) or later, due to the causal priority of S.

Thus, as shown above, at S God intends to change, and given S, necessarily God changes.

If that conclusion is incompatible with God’s having libertarian free will at S, then that would not block the conclusion I proved above – since the argument goes through just as well -, so that would only mean that Craig’s description is incompatible with God’s having libertarian free will at S, creating a new problem for Craig’s account, rather than resolving one.

On the other hand, if that conclusion is compatible with God’s having libertarian free will at S, the point remains that given S, necessarily God changes, and so objection 1 fails.

Objection 2. An ontological difference between scenario 1 and scenario 2 is that given the first temporal state of God in scenario 2, it is not only necessary but causally necessary that the state changes, whereas given timeless state S in scenario 1, it is necessary that God changes, but not causally necessary.

Actually, even if sometimes a state of affairs can determine but not causally determine following states, this is not the case of the state S and God’s being in a different state later, since given S, it is necessary and causally necessary that God changes.

The reason for that is that the change in God from the timeless state S to the first temporal state T(0) is a change from the causally first state of the world, and brought about by God. So, given that what causes God to change from the first state S is that God intends to change, and nothing at T(0) or generally later than S could have caused a change from S, then God intends to change at S, and that is causally sufficient to bring about that God changes.

Objection 3. While it’s true that there is no ontological difference between scenario 1 and scenario 2, the initial state in both scenarios is timeless, not temporal. The scenario improperly described is scenario 2, rather than scenario 1.

Assuming for the sake of the argument that the first state of God is somehow is properly called ‘timeless’ in scenario 1 for some reason, then it seems that the same is true of the first state of the universe in the following scenario, understanding that metaphysical time begins with the universe.

Scenario 5:

First state of the world:

The universe exists without a cause of its existence. Nothing else exists – there may or may not be some things in the universe, but nothing beyond the universe.

Second state of the world:

The universe has changed and exists in its second state. Nothing else exists.

Just as God exists without a cause in scenario1, scenario 2, and scenario 3, the universe[4 ] exists without a cause in scenario 5. And just as God changes from the first to the second state of the world in scenario 1, scenario 2, and scenario 3, the universe so changes in scenario 5.

After that, God in those scenarios continues to change from one state of the world to the next, and the same is true of the universe in scenario 5.

Also, when the universe changes from its first state to its second state, there is no previous change, and no period during which the universe remains unchanged. But that is exactly what happens in the case of God in scenario 1.

Given the above, there seems to be no sense of ‘quiescent’ or ‘changeless’ in which the universe would fail to be quiescent or changeless in scenario 5, but God wouldn’t fail to be so in scenario1.

So, there seems to be no ontological difference that would justify denying that the first state of God is temporal in scenario1, but wouldn’t justify denying that the first state of the universe is temporal in scenario 5.

Someone might suggest that in scenario1 and scenario 2, the change from the first to the second state of the world, including the creation of the universe and God’s own change from timeless to temporal, takes place because God intends to bring it about, whereas in scenario 5, the cause of the change is not specified.

However, that would be unrelated to the issue of changelessness, quiescence, etc., and in any case, we may as well further specify the scenario and add that, in scenario 5, the change from the first to the second state of the universe is brought about by the universe’s conditions at its first state (including, of course, any particles if there are any, etc., and generally the universe’s causal powers and/or causal powers of the particles, etc.), regardless of whether we put that in terms of substances and causal powers and liabilities, or in terms of substances and laws, or in some other way.

Thus, the universe, which has certain conditions in its first state, brings about the change to its second state, just as God and his intent at the first state brings about the change in scenario1.

We may also stipulate if we so choose – though we don’t need to – that the change indeterministic in scenario 5, so that given the first state of the universe, there is more than one possible second state, but only one actually obtains; or we may alternatively make it deterministic, etc.

In any case, that would not make any difference in terms of changelessness or quiescence.

2.1.4. Timelessness sans creation. More on Craig’s description.

In this subsection, I will address a more elaborate description of God’s alleged timelessness sans creation, which Craig gives in one of his articles[r 8 ].

In that article too, he claims that sans creation, God is changeless.

However, as I argued above, that claim would be false. More precisely, in the account of creation that Craig gives, God is not changeless, but changes just as he or the universe would change in scenarios in which there is no allegedly timeless state of affairs.

In addition, Craig claims that God is immobile. How can one make sense of such a claim?

It makes sense, of course, to say that an object remains immobile for a while, so as time goes by, the object does not move with respect to some frame of reference. But here the claim does not seem to be about spatial movement, so plausibly the claim is equivalent to the claim of changelessness, which has the shortcomings already addressed.

Else, Craig would have to explain what he means.

That aside, Craig gives an argument in support of the view that in his changeless state, God is also timeless. A first and decisive problem is that, as argued above, the state in question – i.e., the first state of God – would not be changeless. But let’s consider Craig’s argument in greater detail:

He claims that there is a possible world – say, W' - in which God refrains from creating anything, and claims that in that case, the state in question can be plausibly and coherently conceived of as “timeless”.

Then, he argues that similarly, the first state of God in the actual world is timeless, since there is no intrinsic difference between the first state at the actual world and the first state at W'; Craig also claims that the initial segments of the actual world TW and the other world W' would be identical.

It’s still not clear to me what Craig means by '‘timeless’ - if anything -, but leaving that aside, the claim that there is no intrinsic difference between the two states is false, at least in any relevant sense of ‘intrinsic’, as the following argument shows:

Let S be the first state of the actual world TW, according to Craig’s description. Then, God exists at S, and nothing else exists at S. Let S' be the only state of the world W', at which God exists alone.

Then, given S, necessarily God changes, as the analysis of scenario 3 above shows.

On the other hand, given S', it is not the case that necessarily, God changes, as the analysis of scenario 4 above shows. Furthermore, given S', necessarily God does not change – though the fact that given S', it’s not the case that God necessarily changes, is sufficient to establish that the states are not intrinsically identical, as Craig claims.

Similar considerations apply to causal necessity, also as argued above.

As for the claim that the initial segments would be identical, it is also false, since any initial segment of TW on Craig’s account would contain the first state S, which is a state at which God and only God exists, and such that given such state S, necessarily God changes, whereas that is not true of any initial segment of W'.

So, if one assumes Craig’s account of creation, then given any initial segment of the actual world, necessarily change occurs. Whether which changes occur are determined by such initial state is another matter, but there is no need to discuss it in this context, since the objections succeed either way.

In light of the fact that given any initial segment of the actual world assuming Craig’s account of creation, change necessarily follows, whereas it’s not the case that given any initial segment of W', change necessarily follows, we ought to conclude that Craig’s claim is not true, and the initial segments are not identical.

Alternatively, and to make the point shorter, any initial segment of TW contains S but not S', whereas any initial segment of T' contains S' but not S. Hence, they’re not identical.

Similar considerations can be made with regard to casual necessity, also as argued earlier.

Let’s consider now two potential objections:

Objection 4: The difference between S and S' described above is not intrinsic, whereas the difference between T(0) in scenario 2 and S in scenario 1 or scenario 3 is intrinsic.

That objection claims that there is a difference between T(0) and S, but does not explain how there is that difference, or what kind of difference that would be, or why it would be intrinsic. Saying that S is intrinsically like S' does not seem to address the problem, because by the descriptions, S seems indistinguishable from T(0), but relevantly different from S' - even assuming S' is coherent, which is not clear, either (e.g., how can we make sense of a person who does not change at all, not even in his states of mind? It seems he wouldn’t even think, feel, or do anything, not even in his mind; if so, how can he be conscious? But Let’s let that pass).

Objection 5: God’s state of mind at S resembles more God’s state of mind at S' than it resembles the state of mind that God would have at an initial temporal state, like the state T(0) presented in scenario 2.

Leaving aside that it’s not at all clear that it’s coherent to posit any state of mind – or any person, for that matter – in a purportedly timeless changeless state like S', objection 5 makes a claim of a difference without any explanation as to how the state of mind of God in S and T(0) would be different, and further, different in a sense that is connected to the issue of time.

Given that S and the first state at T(0) in scenario 2 appear not to be different at all, based on the descriptions – as argued above -, there appears to be no good reason to accept this objection, either, at least until a defender of a KCA+ can explain that difference coherently.

2.1.5 Metrically amorphous time.

While Craig rejects Swinburne and Padgett’s account of undifferentiated time before creation [r 8 ], he contends that a variant of it would be compatible with God’s creation.

So, someone might try to combine the KCA with an account like that, instead of timelessness. However. the account in question is obscure as well, and those defending it would have the burden of explaining what they mean. But moreover, for that matter, if the account is coherent, they would have to explain why God would be the only or most plausible candidate for existing in this kind of amorphous time, rather than some weird quantum thing, or space, or something along those lines.

2.2. Conclusions based on the analysis of the proposed creator.

Based on the analysis so far, in my assessment one ought to reject the KCA+ as defended by Craig, or relevantly similar versions, since:

a. It’s not clear that the idea of a timeless agent is coherent.

b. Even if a timeless agent is coherent, it’s not clear that the idea of an agent that is timeless sans creation but temporal with creation is coherent.

c. There are strong reasons to think that the creator proposed by Craig actually would be an entity that begins to exist but has no cause of his existence, contradicting premise 1 of the KCA.

d. If the creator proposed by Craig for some reason can be properly called ‘timeless’ sans creation, but temporal with it – which seems very implausible given the previous subsections -, it seems that plausibly, the same would apply to the universe if it had a first state, which on its own would block the KCA, as argued earlier.

It might be suggested that, perhaps, some versions of the KCA+ defended by other philosophers fare better.

However, as long as they claim that there is a personal creator that is timeless sans creation and temporal with creation, or make any similar claims, it seems that the same problems will likely arise, and such claim seems to be a key part of the KCA+, at least in all usual variants.

In any case, any defender of the KCA+ ought to explain what they mean when they say that the creator is timeless without creation and temporal with it, or by any alternative but also obscure claims they make etc., and in particular, explain how it’s not the case that God begins to exist in their model of creation.

All that said, I will address the premises of the KCA and the arguments offered in support of them in the rest of the essay, and I will raise a number of objections that do not depend on any of the previous ones.

3. The first premise of the KCA.

The first premise of the KCA states:

P1. Everything that begins to exist has a cause.

In this section, I will assess first the meaning of the first premise, and then arguments given in support of it.

3.1. Beginning to exist and coming into existence. An alternative principle.

In his defense of the KCA, William Lane Craig assumes an understanding of ‘begins to exist’[r 9 ] which I will call 'C-begins-to-exist', and which can be stated as follows:

An entity X C-begins to exist at t if and only if the following conditions obtain:

CBTE-i. X exists at t, and it’s a tensed fact that X exists at t.

CBTE-ii. There are no states of the world at which X exists timelessly.

CBTE-iii. Either there is no t' < t such that X exists at t', or any t' < t at which X existed is separated from t by an interval of positive duration.

In that definition, X ranges over entities, and t over instants or intervals of finite, non-zero duration.

Let’s briefly analyze the conditions:

CBTE-i. X exists at t, and it’s a tensed fact that X exists at t.

The requirement that X's existence at t be a tensed fact, in particular, rules out that any objects begin to exist if a tenseless theory of time is true.

It seems odd to me that Craig would include tense in the definition of "begins to exist", but he argues that, under a tenseless theory of time, a universe with a first event did not begin to exist just as a meter stick does not begin to exist just because it has a first centimeter.[r 9 ]

That argument sounds odd to me as well, since while a meter stick does not begin to exist in virtue of having a first centimeter, that's not relevant, since having a first centimeter is a spatial, not a temporal claim, while 'begins to exist' - at least, in this context – is clearly a claim about time, not space.

In fact, the stick in question may properly be said to have a beginning in space because it has a first centimeter - at least if there is no demand for an arbitrarily precise spatial beginning -, as long as we – explicitly or by context – pick a direction in space to say which centimeter is first and which last, or which endpoint is the spatial beginning and which one is the spatial end.

But similarly, even if a tenseless theory of time is true, the stick does have a beginning in time if there is, for instance, a first year at which it exists – at least if there is no demand for an arbitrarily precise temporal beginning, and a fuzzy beginning will do -, and in the case of time, we needn't specifically pick the direction, since the direction from past to future is already implicit in our language.

That does not appear to be a difficulty for a temporal beginning of existence under a timeless theory of time. Intuitively, to say that the stick begins to exist seems to be the same as to say that the stick has a temporal beginning, or a beginning in time, and then it follows that the stick does begin to exist, regardless of whether a tenseless theory of time is true.

CBTE-ii. There are no states of the world at which X exists timelessly.

This is an even odder condition, since the term ‘timeless’ is extremely odd on its own, to say the least. It’s not clear that it’s even being used coherently by Craig, as I argued earlier. But I've already addressed the matter of timelessness, so I will go no further on this point, and assume from now on and for the sake of the argument that the concept of C-begins-to-exist is a coherent concept.

CBTE-iii. Either there is no t' < t such that X exists at t', or any t' < t at which X existed is separated from t by an interval.

This condition, together with the fact that t ranges over instants or intervals of positive finite duration, entails that if there is a finite open interval (0,t] at the beginning of the universe, then the universe began to exist. It’s a debatable matter whether that respects our usual terminology, given that it’s arguable whether the metric of time is intrinsic. But I will no go further on this, either, and will grant for the sake of the argument that the metric is intrinsic.

In addition to giving that definition, Craig understands 'begins to exist' and 'comes into being' as semantically identical. In fact, he actually defines 'begins to exist' to mean the same as 'coming into being', and then defines "comes into being". [r 9 ]

I don't think that that captures the meaning of 'comes into being' or 'comes into existence'.
The idea of coming into existence – or coming into being; I see no semantic difference between the two expressions – seems to imply a change in the state of the world, plausibly from a state at some time t' at which some entity X does not exist, to a later state at a time t > t', at which X does exist.

Granted, someone might say that the idea of coming into existence does not require that the previous state of the world at which X does not exist be temporal, and a causally prior state suffices, even if it’s a timeless state. For the reasons I explained earlier, there are serious questions about the coherence of a timeless state that is causally prior to a first temporal state. We may assume here for the sake of the argument that it is coherent in order to assess the first premise and some variants, but it remains the case that I cannot make any sense of such a state, so it’s hard for me to make any intuitive assessment on that.

Still, in any case, we may Let’s leave the issue of the coherence of a timeless cause of a first temporal state and define temporally coming into existence, or coming into existence in time, as follows:

An entity X temporally comes into existence or into being at t if and only if the following conditions obtain:

TCIE-i: X exists at t.

TCIE-ii: There is a time t' < t such that X does not exist at t'.

TCIE-iii: There is no time t'' such that X exists at t'', and t' < t'' < t.

In this context, X is also any entity, and t any instant or any temporal interval of positive duration.

Given those definitions, Let’s consider another potential principles, as an alternative to premise 1.

P1': Everything that temporally comes into existence has a cause of its existence.

I do not claim that the principle is true. It may well be challenged. However, I offer it in order to compare it with the principle offered in premise 1 of the KCA, since I find P1' to be a lot more intuitively plausible, at least if one assumes that there might be a beginning of time – a matter I will consider in the next subsection.

3.2. Arguments in support of the first premise.

In this section, I will consider arguments in support of the first premise given by William Lane Craig in his defenses of the Kalam Cosmological Argument [r 10].

3.2.1. Nothing comes from nothing.

Craig maintains that the principle that nothing comes from nothing is intuitive and supports the first premise. In this context, he points out that objects don't just pop into existence without a cause, and provides examples by considering hypothetical scenarios in which certain objects come into existence without a cause, and pointing out that those scenarios are counterintuitive.

However, the examples he provides as obviously counterintuitive are all examples of objects temporally coming into existence without a cause, in the sense in which I defined 'temporally com es into existence' earlier.

It is true that those scenarios are also examples of objects that would begin to exist without a cause, and even C-begin-to-exist without a cause if a tensed theory of time is true.

However, it may very well be that the feature that makes it intuitive that those objects probably have causes is that they temporally come into existence, not that they begin to exist – or maybe it’s some other feature.

On that note, there is evidence against the view that what makes it intuitive – at least to me; readers are invited to use their own intuitions and check for themselves, of course – that the objects in those hypothetical scenarios probably have causes is not the fact that those objects begin to exist, or that they C-begins-to-exist. The evidence in question comes precisely from considering a hypothetical scenario in which time begins to exist at t=0. In that scenario, I find it particularly counterintuitive that any of the objects that exist at t=0 would have efficient causes. As for other, non-efficient causes, it’s less clear, but at least, it’s not intuitive that they all do have such causes. Rather, it’s intuitive that some of those objects do not have causes.

So, after considering that, premise 1' still looks intuitive[5 ], but premise 1 does not.

So, on intuitive grounds, one may reject premise 1.
Incidentally, the defender of the KCA is appealing to intuitions in support of premise 1, and then arguing that time has a beginning as a means of supporting premise 2. But if time has a beginning, then premise 1 becomes counterintuitive. So, if the appeal to intuitions in the context of the KCA is acceptable – as the defender of the KCA maintains -, then the KCA ought to be rejected – at least, going by my intuitions.

Incidentally, also, even Craig’s examples – like, say, a horse coming into existence without a cause – are counterintuitive as examples of what might actually happen. I do not see any good reason to think it’s metaphysically impossible for any of that to happen. At least, if conceivability is a guide to metaphysical possibility, they seems conceivable. The reason that those events do not actually happen may very well have to do with the causal structure of the actual world, but not with what’s metaphysically possible. Still, the other considerations I’m giving are enough to reject the KCA on intuitive grounds, regardless of the issue of metaphysical possibility.

3.2.2. Another alternative principle.

After further considering my intuitions on the matter by contemplating more scenarios (but which don’t involve a beginning of time), it also seems intuitively plausible to me, after reflection, that for every object O that exists at some time t, there is some time u < t, and some object U at u which is a cause of O’s existence at t.

If that is so, then perhaps it’s the fact that E exists at t which makes it intuitive that there is an object E' at some earlier time t' which is a cause of E’s existence at t, and thus that there are causes of E's existence at any time t.

Given that, we may consider the following alternative premise:

P1'': For every object O that exists at t, there exists a time u < t, and some object U that exists at u, such that U at u is a cause of O’s existence at t.

Here, t and u might be instants or finite intervals, and U might or might not be O at a previous time.

P1'' does seem intuitively plausible to me. But P1'' would be a problem for the defender of the KCA, since P1'' is in conflict with the second premise of the KCA, since it leads to an infinite past.

That said, P1'' is problematic, since some people have intuitions against infinite regress. On that note, personally I used to have conflicting intuitions on the matter, and I used to think infinite regress was impossible, though upon further reflection, now regress seems more intuitive to me than a beginning. [6 ]

In any case, if P2'' is rejected, if one considers a scenario in which time has a beginning, then P1 appears intuitively very implausible, whereas premise P1' remains unaffected, and appears still intuitively plausible – at least, at the actual world -, so we’re back to choosing P1' instead of P1. [5 ]

So, based on those considerations, I would be inclined to tentatively accept P1' as plausibly actually true, but reject P1, at least as long as one considers the hypothesis that time had a beginning as a live option.

Still, even the acceptance of P1' is tentative, since it’s not clear that our intuitions are so reliable in cases involving all of time, a small dense universe, etc. [5 ], and leaving aside the issue of metaphysical necessity.

Of course, someone might say that her intuitions are different. I invite readers to make their own assessments, but I would suggest that premise 1 is intuitive under the assumption that there is no beginning of time, and counterintuitive under the assumption that there is a beginning of time, whereas premise 1' is intuitive either way – as always, as long as the premise is considered a claim about the actual world, or similar worlds, etc.

As claims of metaphysical necessity – which is what Craig is claiming in the case of premise 1 -, I see no significant intuitive support for either P1 or P1'. But still, even as a candidate for a metaphysically necessary principle, P1' seems at least a lot more intuitively plausible to me than P1, even if P1' is not particularly plausible, either.

3.2.3. Is the universe an unjustified exception?

Another argument that Craig makes is that making an exception for the universe would be ad-hoc. He asks why wouldn't ordinary objects come into existence all around us, without causes, if universes could do that.

But leaving aside for now the vagueness and/or obscurity (at best) [4 ] of the word ‘universe’, if time had a beginning, it may well be that the relevant difference is that the universe did not temporally come into existence; in other words, assuming that time had a beginning, the universe began to exist or even C-began to exist, but did not temporally come into existence. That seems a lot more intuitive to me that Craig’s model in which there is a beginning of time and yet what exists at the beginning has efficient causes – not to mention that the coherence of Craig’s model is doubtful.

At this point, someone might object to the previous considerations and suggest that my intuitions are unusual. [5 ]

In particular, they might raise questions like: ‘Haven’t scientists continued to look for a cause of the Big Bang? Why haven’t they stopped looking for causes? Did and/or do they have different intuitions?'

However, the fact is that the Big Bang model does not provide an understanding of the universe beyond a certain point in the past, where effects from forces other than gravity should be taken into consideration, and some scientists are trying to figure out the causes of a very hot, dense, and small universe that existed about 13.7 billion years. But those scientists seem to be asking the question: 'What caused the state in which the universe was dense, small and hot?' (or similar questions), on the understanding that before the first state of the universe that can be reliably described with present-day models, there were other states of the universe that are beyond the descriptive capabilities of current scientific understanding.

Moreover, while it may be that if some scientists assess that our universe began to exist, in some limited sense of ‘universe’, that would not entail an assumption that time had a beginning, and so they might look for causes earlier in time – for instance.

In the end, readers will make their own intuitive assessments of course, but I don’t think my intuitive assessments with regard to premise 1 and premise 1' are unusual. In any case, I invite readers to assess the matter by their own intuitions: As I see it, premise 1 is intuitive under the assumption that time does not have a beginning, and very counterintuitive under the assumption that time does have a beginning. Premise 1' is intuitive either way.

That said, there are difficulties with appeals to our intuitions about time and space in contexts such as the purported beginning of time, such as:

a. There may be significant differences between the intuitions of different people. There may well even be differences between their pretheoretical intuitions. So, perhaps, and at least on some the issues under discussion, there may not be one single human normal pretheoretical intuition on a matter.

b. The intuitions of a person on some of these matters may well change over time, as that person learns more about physics, cosmology, philosophy, etc., and sometimes it’s difficult to figure out what a person’s pretheoretical intuitions are.

c. There seems to be no particularly good reason to think that pretheoretical intuitions are more reliable than intuitions developed later – let alone more than the intuitions of specialists, like cosmologists.

d. In any case, and plausibly more importantly, modern physics shows that in unfamiliar environments like something very small or massive, things are ‘weird’, and our intuitions about time, space and related matters do not seem to work well. But the universe a long time ago was both very small and massive.

Of course, none of that represents a problem for use of our intuitions in daily life or relevantly similar cases, but it raises questions about their reliability in cases like those involved in the context of discussions of the KCA, since in that context, those intuitions are implicitly used in all of the universe.

All that said, if our intuitions are a reliable guide in this case – as defenders of the KCA claim or assume -, then for the reasons I explained above, I would be inclined to tentatively accept premise 1', but reject premise 1 unless one has good reason to believe that there is no beginning of time.

Moreover, if the premises are meant to be metaphysically necessary –which is what Craig maintains in the case of premise 1 -, I do not see either premise 1 or premise 1' to be intuitively plausible. At least, it seems perfectly conceivable that objects temporally come into existence without a cause, so if conceivability is a good guide to metaphysical possibility, then that gives us a reason to think that premise 1' is not metaphysically necessary, either. On the other hand, if conceivability is not such a guide, I do not know how else to assess metaphysical possibility in those particular cases.

Regardless, leaving aside other possible worlds, the previous considerations are enough to reject P1. Moreover, if for some reasons one principle or another ought to be accepted as metaphysically necessary – but that would have to be argued for -, I would say that P1' is a lot more intuitively plausible than P1.

3.2.4. Empirical evidence.

In addition to the question of which principle or principles are intuitive, another question is whether empirical evidence supports the first premise. Craig claims that that is so, and points to a vast number of examples. However:

a. The empirical examples that Craig give would also support P1'', which would contradict the second premise of the KCA.

b. In any case, none of those examples favor premise 1 over premise 1'. In fact, those examples are exactly what one would expect to see if P1' is true, but time has a beginning and P1 is false.

Also, premise 1' does not become less intuitive even under the assumption that time has a beginning.

c. Empirical evidence is usually a very poor guide to metaphysical possibility. For that matter, there is plenty of empirical evidence that, say, people are not capable of flying like Superman. But I do not see any good reason to think that it’s metaphysically impossible for that to happen.

Granted, someone might try to run a KCA claiming only that the first premise is true, rather than necessarily true. But in that case, the objections that I’m raising and which do not involve the issue of metaphysical possibility are not affected.

In addition to all of that, using a similar argument from empirical evidence, we may find support to hypotheses such as:

EE1: Actually, every intelligent being has at least one non-intelligent cause.

EE2: Actually, every agent has at least one cause that is not an agent.

EE3: Actually, every personal being has at least one non-personal cause.

And so on. Granted, a defender of the KCA + might give other arguments against those inferences, but that’s a matter they would have to deal with when defending their proposed cause of the universe – i.e., when trying to go from ‘The universe has a cause of its existence’ to the conclusion that the cause is God -, even if they managed to establish that the premises of the KCA are correct, and even leaving aside the problems with the concept of the agent that they propose.

4. The Second Premise of the KCA.

The second premise of the KCA states:

P2. The universe began to exist.

In addition to the question of the meaning of ‘begins to exist’, an important issue is the meaning of ‘universe’'.

Craig stipulates that in the second premise of the KCA, the universe is the whole of material reality.[4 ] [r 1 1 ]

While it’s not clear what ‘material’ means, assuming it’s coherent and precise enough in the context of the KCA, it’s clear enough that at least the “whole of material reality” contains the (or a) multiverse if there is a multiverse, and contains any universes older than our universe, using here the word ‘universe’' in a sense in which it’s sometimes used in astronomy and cosmology, but which is more restrictive than the sense in which the word ‘universe’' is used in the second premise of the KCA.

Hence, arguments showing that the universe, in that more restrictive sense of the word ‘universe’, began to exist, would fail to provide support for the second premise of the KCA.

I will come back on this point later, when I assess Craig’s arguments allegedly based on science. But first I will address other arguments, made by Craig, Pruss, Koons and Waters.

4.1. Hilbert’s Hotel

The first argument I will consider is the ‘Hilbert Hotel’ argument, defended by Craig intended to show that an actual infinity is metaphysically impossible. Based on that conclusion, Craig argues [r 1 2 ] that an infinite past is metaphysically impossible, since it would be an actual infinity. [7 ]

According to Craig, the ‘Hilbert Hotel’ argument shows that an actual infinity is counterintuitive, and based on that, he claims that an actual infinity is plausibly metaphysically impossible – apparently, according to Craig, it’s counterintuitive from the perspective of some intuitions of metaphysical possibility.

So, let’s assess the argument:

The Hilbert Hotel is a hotel with a denumerable number of rooms, [8 ] and Craig maintains that it’s the actual application of the concept of infinity to the real world, rather than the consideration of abstract sets, what brings the counterintuitiveness of an infinity to one’s attention.

Of course, a hotel like the one he proposes is counterintuitive, since (for example) we would never be able to build it, we wouldn’t be able to communicate with the rooms in real time as in the proposed scenario, etc. However, all of that counterintuitiveness seems to have everything to do with an infinite hotel, the capabilities of humans or similar beings, etc., and nothing to do with an actual infinity.

On the other hand, the alleged counterintuitiveness claimed by Craig seems to result from a misunderstanding of the meaning of the words, and disappears once the meaning of the words in the scenarios he brings up are clarified.

To see this, let’s consider some of the scenarios Craig proposes as supporting his claim of counterintuitiveness:

First scenario:

Craig presents a scenario in which all of the rooms are occupied, and a new guest arrives. According to him, somehow the fact that there are no more guests after the arrival is a problem, or counterintuitive. However, that is not the case, once it’s clear what one means by ‘more guests’, for the following reasons:

1.a. In the sense of cardinality, there are no more guests after the new guest arrived than there were before. However, that merely means that there is a bijection between the set of guests before the new arrival, and the set of guests after the arrival, which not only not counterintuitive, but is actually as clear as is the fact that there is a bijection between the set of positive integers and the set of positive integers plus zero.

1.b. On the other hand, there are seem to be senses of ‘more guests’ in which there are more guests after the new arrival than there were before, like the sense that all of the previous guests are still there, and there is also a guest who wasn’t there before.

In terms of sets, the set of guests before the new guest arrives is strictly contained in the set of guests after the new guest arrives in this case.

Perhaps, someone might object to this second way of understanding the expression ‘more guests’, since – for instance – this would not allow to conclude that the set {1, 2} has more elements than the set {4}.

But that objection would miss the point, which is that there is nothing problematic about the matter once the meaning of the expression ‘more guests’ (and related ones, like ‘identical number’, etc.) is clarified. For that matter, we may consider another way of speaking of numbers of elements, as follows:

1.c. Let’s say that set A and set B have the same number of elements if and only if the sets A\B[9 ] and B\A have the same cardinality, and that A has more elements than B if and only if the set A\B has greater cardinality than the set B\A.

So, is it true that there are no more guests after a new guest arrives, in the first scenario?

In the sense of cardinality, it’s true.

In the senses considered in 1.b and 1.c, it’s false.

Once one has clarified what the question is about, that alleged counterintuitiveness disappears.

Now, Craig also says that it’s counterintuitive that the new guest is accommodated even if all the rooms were full. Of course, shifting infinitely many guests from one room to the next would not be doable for any human being, or for that matter any alien from another planet, etc., but that’s not relevant.

So, leaving practical considerations aside, I do not see any problems here for the metaphysical possibility of the Hotel, since the fact that the rooms were full does not change the fact that the guests may hypothetically be told to move to the next room, etc.

If the shift is gradual, actually the shift will never end – i.e., there will always be one guest moving from one room to another one, or getting ready to move, etc. -, but again, I wonder what the problem is supposed to be – again, leaving aside the obvious fact that neither we nor some aliens from another planet will ever be in a position to built such a thing.

However, in any case, if the issue of accommodating a new guest were somehow a problem for the metaphysical possibility of the Hotel – and quite frankly, I do not see any good reason at all to think that that would be so -, then that would be a problem for the metaphysical possibility of Craig’s hotel, but there appears to be no good reason to think that it would be a problem for actual infinities in general.

For example, the hypothesis that there are infinitely many galaxies, stars, planets, etc. - which is an alternative considered seriously in present-day science – is not affected.

Second scenario:

On this scenario, not just one but denumberably infinitely many new guests arrive at the desk, and the proprietor talks to them, etc.

As in the case of the first scenario, Craig raises the issue of the number of guests, and the fact that all of the rooms were full before the new guests arrived, and yet all of the new guests are accommodated.

But that does not work, either, for the following reasons:

2.a. It might be that the meaning of the word 'desk' is such that it’s not coherent to say that infinitely many show up at the desk. However, that infinitely many guests can arrive at the desk is not is not a condition included in the definition of the Hotel in question, so if the arrival of infinitely many new guests turns out to be an incoherent scenario due to the meaning of the word ‘desk’' (and ‘guests’, implying something like humans, etc.), that does not tell us anything about the metaphysical possibility of the Hotel.

2.b. On the other hand, if there is nothing in the meaning of the word ‘desk’ and the other terms involved in the scenario making that scenario incoherent, then there seems to be no problem.

Of course, things like building such a hotel, or communicating at arbitrarily fast speed with infinitely many people, etc., are counterintuitive if someone were to suggest that such a thing actually exists.

However, as long as the beings are sufficiently different from humans in terms of powers, etc., that does not seem to be a problem for the metaphysical possibility of the Hotel.

Moreover, as before, any such counterintuitiveness is not a consequence of assuming an actual infinity, but of assuming the infinite hotel that Craig and Sinclair describe, so even if that were a good objection to view that the Hotel is metaphysically possible, that would not affect, say, the hypothesis that there are infinitely many galaxies.

2.c. The issue of the number of guests before and after the infinitely many new guests arrive is handled as in the case of the scenario in which one guest arrives, which was already considered.

2.d. The issue of the accommodation of the infinitely many new guests even though all of the rooms were already full is also handled as in the case of the first scenario, only that in this case, the new guests might never be all accommodated, if the shift is gradual. If it’s instantaneous, then all are accommodated. In any case, basically the same considerations given in the case of the first scenario apply here as well.

Third scenario:

On this scenario, one guest departs. According to Craig, it’s counterintuitive that there is no one fewer guest. But there is no counterintuitiveness, as long as one considers what is meant by ‘fewer’.

In other words, there are as many guests as before in the sense of cardinality, but there is a sense in which there were more guests before the departure, namely that all of the guests that are in the hotel after the departure were in the hotel before the departure, but there was also one guest before the departure that is no longer in the hotel after she departed (obviously).

In terms of sets, the set of guests after the departure is strictly contained in the set of guests before the departure. Once again, all of this is unproblematic, and any issues related to the number of guests do not appear to be counterintuitive at all as long as one keeps in mind what one means by ‘more guests’, ‘fewer guest’, etc.

Fourth scenario:

Like the third scenario, but in this case, all of the guests in the odd-numbered rooms check out.

As before, Craig claims that it’s counterintuitive that there are no fewer people in the Hotel.

But as before, there is no such counterintuitive as long as one keeps in mind what is meant by ‘fewer’. The previous considerations suffice.

Fifth scenario:

This time, all of the guests in all rooms except for #1, #2, and #3 leave, and even though the number of guests that left is the same as the number of guests that left in scenario 3, the number of remaining guests is only 3.

Craig seems to believe that such argumentation is a powerful argument against the metaphysical possibility of the Hotel, and as a result allegedly against the metaphysical possibility of actual infinities. But any alleged counterintuitiveness with regard to the numbers disappears ones one considers what the words mean, and basic math.

For that matter, the set of odd natural numbers and the set of natural numbers have the same cardinality, but if N is the set of natural numbers, A=N\{odd natural numbers}[9 ], and B=N\N, then A is an infinite set, whereas B is the empty set. That is not a problem at all.

As for other features of the Hotel itself, if any of them is a problem for the metaphysical possibility of the Hotel – and I see no good reason to believe so, as explained earlier; the metaphysical possibility of it does not seem in any way counterintuitive to me, at least as long as the people involved are not human or similar to human in powers, etc. -, then there seems to be no good reason to think that that is a problem for actual infinities in general, or in particular – for example -, for infinitely many galaxies, planets, stars, etc.

Given the previous considerations, the conclusion is that the ‘Hilbert Hotel’ argument fails to provide any evidence against the metaphysical possibility or even the existence of actual infinities.

That said, I would like to also comment on a particular claim he makes. Despite the fact that Craig distinguishes between three types of possibility in this context (and impossibility, and necessity) - namely, strict logical possibility, strict logical possibility augmented by the meaning of terms within the scope of modal operators, and metaphysical possibility –, and he makes it clear that his claim is that actual infinities are metaphysically impossible, he does claim that there is a contradiction, apparently in subtracting equal quantifies from equal quantities and have different numbers in the end.

In particular, he gives the example that the set of even numbers E has an identical number of elements as the set A of natural numbers greater than four, and yet if we subtract E from the set of natural numbers N, we get an infinite set, but if we subtract A from N, we get a set with only 3 elements.

So, allegedly, it would be contradictory that identical numbers from an identical number and we did not get the same number.

But there is no contradiction, because A and E (and N) have an identical number of elements in the sense that there is a bijection between them. But there is no contradiction in saying that there is a bijection between A and E (and/or between each of them and N, for that matter), but there is no bijection between N\A and N\E. In fact, not only is that not contradictory, but it’s clearly true.

Nor is any of that counterintuitive, for that matter.

So, in short, there is nothing to the “Hilbert Hotel” argument that Craig defends.

4.2. Grim reapers.

The ‘grim reapers’ argument (or GR argument) has different forms, and it might be given in support of the hypothesis that actual infinities are impossible, and/or that infinite temporal regress is impossible, and/or that time is necessarily discrete, among others.

In this subsection, I will focus on one of Alexander Pruss’s versions of the argument. [r 1 3]

The argument is as follows:

Pruss: [r 1 3]

"1. If there could be a backwards infinite sequence of events, Hilbert’s Hotel would be possible.

2. If Hilbert’s Hotel were possible, the GR Paradox could happen.

3. The GR Paradox cannot happen.

4. Therefore, there cannot be a backwards infinite sequence of events."

Briefly, the GR Paradox is as follows:

Fred is alive at t₀– which is 11.00 am in Pruss’s scenario -, and then, there is a grim reaper (say, GR_n), set to kill Fred at t_n=(t₀+(1/n) seconds), if Fred is alive, and to do nothing if Fred is dead. Fred cannot survive a grim reaper attack.

The reasoning used in defense of premise 2 is that if the Hotel were possible, it would be possible to make one reaper in each room (GR_nin room #n), and the staff could program it to act at t_n=(t₀+(1/n) seconds).

So, what to make of the argument?

First, let’s mirror the reasoning, and make an argument against infinitely many galaxies, for example, by making one grim reaper per galaxy.

Should we then conclude that infinitely many galaxies are not possible?

I do not think so. In fact, there are hypothetical scenarios that have infinitely many galaxies and are consistent.

For example, there seems to be no contradiction in a scenario with infinitely many galaxies but in which time is relative and in which no entity has the power to send information from one galaxy to another faster than the speed of light. In such a scenario, there seems to be no way to derive the paradox, since in such a scenario, no one would be able to make reapers who can check on Fred like that.

Granted, someone might try to derive the paradox in some other way, but that can be blocked by conditions on the causal structure of the hypothetical scenario – or world if one accepts talk of possible worlds.

Moreover, there are scientific, non-contradictory models involving infinitely many galaxies. There seems to be no contradiction in adding the stipulation that no entity has the power to send information from one galaxy to another faster than the speed of light – which may be a problem for theism, but not for the scenario’s immunity to the GR paradox.

So, if infinitely many galaxies are possible, then clearly it’s not possible that a contradiction obtains. And there is no good reason to believe they’re not possible.

Granted, someone might posit a variant in which the infinitely many reapers are in the same place, or they’re spaceless unembodied entities that send signals to one another, etc.

But the key point in this reasoning is that this would not be a problem for the possibility of infinitely many galaxies, but for the problem of infinitely many reapers with certain powers in a certain scenario.

This seems to work against premise 2 in Pruss’s argument as well, because it seems it is not the case that if Hilbert’s Hotel were possible, then a contradictory GR scenario would be possible. Rather, if Hilbert’s Hotel were (or is) possible, there would be (or there are) possible worlds/scenarios at which it exists, and then no contradiction would arise (or arises) in any of them, plausibly because of the causal structure of each of the worlds/scenarios, which would vary (or varies) from one possible scenario to another possible scenario.

This seems to block the argument for premise 2, as well as an argument directly from an infinite past to a contradictory GR scenario.

If an infinite regress of past events is possible, then no paradox arises in those possible scenarios, and what blocks the formation of the paradox plausibly varies from scenario to scenario.

At this point, someone might raise the following objection:

‘If it’s metaphysically impossible for there to be infinitely many galaxies, then that explains why a contradictory GR scenario involving infinitely many GR – one per galaxy – would be prevented.

But if it were metaphysically possible for infinitely many galaxies to exist, then a contradictory GR argument would seem to be possible. Else, what would prevent the formation of a contradictory GR scenario, in a possible world or scenario with infinitely many galaxies?'

However, I will argue that the objection is misguided, on the following grounds:

a. The question ‘what would prevent the formation of a contradictory GR scenario in a possible world or scenario with infinitely many galaxies?’ may be interpreted as:

a.1. A question about a general necessarily true principle that somehow would in some sense ‘prevent’ the formation of the GR scenario.

In that case, one may point to the necessarily true principle that contradictions are impossible; in particular, one does not need a further principle that actual infinities are impossible, or that infinitely many galaxies are impossible.

a.2. A causal question, namely, ' what would causally prevent a GR scenario from coming to be, in a possible scenario with infinitely many galaxies?'

In that case, the answer is that it depends on the specific possible scenario or world in question.

If the scenario is complete – like possible worlds, if one accepts talk of possible worlds – or at least sufficiently specified, then in different possible scenarios (or worlds), there are different causal structures of the scenario/world, and in some cases, different things would fail if some entity or entities attempted to bring about the contradictory GR scenario, or if each of them attempted to perform a single task that together would be a contradiction, etc.

b. The question raised in the objection may be mirrored by a similar question, in reply to the suggestion – made by the person arguing against actual infinities, or at least against infinitely many galaxies – that an explanation as to why a contradictory GR scenario involving infinitely many galaxies wouldn’t happen is that actual infinities – or at least infinitely many galaxies – are metaphysically impossible.

The mirror question would be: ‘What would prevent the formation of infinitely many galaxies – not one at a time, of course, but in block so to speak – in a possible scenario in which there are finitely many objects?'

Then, we may consider options:

b.1. If the person positing the metaphysical impossibility of actual infinities, or at least of infinitely many galaxies, responds that what would prevent the formation of infinitely many galaxies in a possible scenario in which there aren’t infinitely many objects is that actual infinities – or at least infinitely many galaxies – are metaphysically impossible, then that would not seem any stronger than a reply to the original question in the objection that says that what would prevent the formation of a contradictory GR scenario in a possible world or scenario with infinitely many galaxies is that contradictions are not possible.

More precisely, there is of course a clear difference between the two replies: we do know contradictions are impossible, but we do not know that infinities are, or that infinitely many galaxies are.

However, and leaving that aside, the two replies are similar in that, in both cases, the person answering the question is appealing to a principle they hold is necessarily true – in one case, the impossibility of contradictions; in the other, the [alleged] impossibility of an actual infinity, or of infinitely many galaxies -, as an answer to a question as to what would prevent the formation of a certain scenario from a previously given scenario.

This of course generalizes to other replies to the mirror question that are based on some [allegedly] necessarily true principle, as an answer to the question ‘What would prevent the formation of infinitely many galaxies – not one at a time, of course, but in block so to speak – in a possible scenario in which there are finitely many objects?'

Such replies are also appeals to some [allegedly] metaphysically true principle as explaining what blocks such a formation, and as such they seem no stronger than the appeal to the metaphysically true principle that contradictions are impossible as a reply to the question of why a contradictory GR scenario would not be formed in a possible scenario in which there are infinitely many galaxies, if infinitely many galaxies were possible.

b.2. If the person positing the metaphysical impossibility of actual infinities, or at least of infinitely many galaxies, gives a causal answer as to what would prevent the formation of infinitely many galaxies in a possible scenario in which there are finitely many objects, then that does not seem to be any better than the causal reply above.

To be more precise, I think point a . above suffices to show that the objection is misguided, but b. is a way of showing that the person arguing against an actual infinity is at least in no better position.

4.3. Aristotelian-discrete time, infinite regress and more grim reapers.

Pruss also distinguishes between the following two types of discrete time:

1. Time is rigidly discrete if there necessarily is a minimum temporal unit.

2. Time is discrete in an Aristotelian sense if there are in fact finitely many moments of time between any two given times in the finite past, but each interval can be subdivided infinitely many times.[r 1 3][r 1 4 ]

Then, he gives an argument against infinite regress if time is discrete in an Aristotelian sense.

The reasoning is basically as follows: if there were an infinite sequence of past events D(-n), for all natural n, such that D(-n) precedes D(-m) if n>m, then D(-n) could cause something at time t₀+1/n for some t₀, contradicting the hypothesis that time is discrete in an Aristotelian sense. According to Pruss, there is no reason to rule out all of those happening together.

But the difficulty seems to be the same as in the previous case. In fact, what Pruss seems to be doing is going from the finite to the infinite case, including in the infinite case scenario some entity or entities with the causal power to bring about all of those things at time t₀+1/n, and who exercise such powers successfully. But that’s precisely not what happens in a consistent scenario in which time is discrete in an Aristotelian sense, and in which there is infinite temporal regress. For that matter, one might make an argument against infinitely many galaxies if time is discrete in the Aristotelian sense – programming one reaper per galaxy -, but that fails as the argument above fails.

At this point, someone might ask questions like.

a. What would prevent that scenario from happening, if an infinite temporal regress is possible?

b. What would prevent that scenario from happening, if infinitely many rooms, or galaxies, are possible?

If the question is about some general metaphysical principle, we know that contradictory states cannot come to exist. If it’s a causal question, we may properly ask, in the case of a specific hypothetical phenomena H, what would causally prevent H from happening.

But if the hypothetical scenario is sufficiently determined – like a possible world, presumably -, then different possible scenarios have different causal structures.

So, essentially, the reply to this kind of objections is along the same lines as the reply to the objection in the immediately previous subsection.

4.4. Grim placers and/or grim signalers.

In this subsection, I will address a variant of the argument from grim reapers, defended by Robert Koons[r 1 5 ].

On this variant, each grim reaper – or rather grim placer, given his job – checks whether there is some Fred particle at a specified location, and if there isn’t, he places one at some specific location. Otherwise, he keeps the particle where it is.

According to Koons, his argument shows that time is not dense, and furthermore that infinite temporal regress is not possible.

4.4.1. Possibility of a grim placer.

Among other hypotheses, the argument assumes the following one:

1. There is a region R of duration d (for some finite d>0) in a possible world W, and a grim placer G_d, such that GP_d has the intrinsic power and disposition to do as follows:

a. If there is no Fred particle at any distance y<d from a fixed plane P, then G_d creates and places a Fred particle at a designated location exactly d meters from P.

b. Otherwise, GP_d keeps any Fred particle that is closer to plane P in its position, and does nothing more.

It’s not clear to me what criterion or criteria Koons is using, as a guide to metaphysical possibility, in order to assert that one such scenario is possible.

It might be argued that it’s intuitively clear that such entities are possible. But for that matter, it seems no less intuitively clear to me that it’s possible that no agent exists, so if I were to accept that criterion, that alone would make Koons’s premise no stronger than an intuitive assessment that there is no necessary being.

In any case, Let’s grant hypothesis 1 for the sake of the argument.

4.4.2. Compressibility of spacetime.

Another assumption of Koons’s argument – though this one is an assumption for a reductio – may be stated as follows:

2. If there is some object A with an intrinsic property Q in a region R of finite duration e in a possible world W, then:

a. There is a function f from the parts of R to the parts of some region R' of some possible world W', such that f is topology-preserving and compresses time and space by half. For instance, if the duration of R is e, the duration of R' is e/2

b. There is a counterpart A' of A and a counterpart Q' of Q, such that A' intrinsically has property Q' in R'.

Hypothesis 2 is used for a reductio against temporal density. Koons argues that hypothesis 2. is reasonable under the assumption that time is dense.

If there is an intrinsic metric of time, it’s not clear to me that no properties are incompressible, no processes require at least some amount of time, etc.

But Let’s grant hypothesis 2 as well, for the sake of the argument.

4.4.3. Infinitary patchwork and binary patchwork.

A key principle of Koons’s argument is what he calls “infinitary patchwork” [r 1 5 ] (or IP), and which can be stated as follows: [10]

Let’s suppose the following conditions obtain:

IP(1): W = {W_n} is a countable series of possible worlds, and R={R_n}, is a countable series of regions of those worlds, such that for all n, R_n is a region of W_n.

IP(2): f is a function from R into the class[1 1 ] of spatio-temporal regions of some world W, such that f preserves the metric and topological structure of each of the R_n, and such that, if n≠m, then f(R_n) does not overlap f(R_m).

Then, there is a possible world W’, and an isomorphism g from the spatio-temporal regions of W to the spatio-temporal regions of W’, such that the following obtains:

IP(C): The part of W_n within R_n is exactly like the part of W’ within g(f(R_n). [r 1 5 ].

Koons also proposes binary patchwork, which is the binary counterpart of infinitary patchwork. In other words, in the case of binary patchwork, only two regions are patched.

Here, two key questions are:

i. Should we accept both principles?

ii. If we do, does Koons’s conclusion follow?

Let’s address ii first.

As Koons points out when he considers what he calls “The Amazing Vanishing Particle”, his argument requires that each grim placer – or grim signaler, as he renames them – have the power to send a signal to a successor, and the power to receive a signal from a predecessor.

In particular, even though the placers or signalers have only powers intrinsic to each interval, they have the power to send a signal beyond their own interval.

Without that power, the argument would fail, as the following scenario shows:

1. There is a sequence of temporal intervals {I_n=(t_n+1,t_n]}, for all natural numbers n.

2. Each interval I_n has length d*2^-n

3. During interval I_n, or at least during the last portion of it, there is a Fred particle at the designated position d*2^-nmeters from the designated plane P. There is no other Fred particle during interval I_n

4. During interval I_n, there is a grim placer #n, GP_n.

That’s not contradictory, and is consistent with Koons’s premises, as long as the powers do not involved sending signals beyond the interval in which they’re exercised.

If the scenarios to be patched were like that, there would be no contradiction.

So, a key assumption in Koons’s argument is that the powers of the grim placers of signalers, even if intrinsic to each spatio-temporal region in the sense defined by Koons, include powers to act beyond such spatio-temporal region.

Now, Koons says that there is no action at a distance, since the intervals may be contiguous, like in the previous example I_n=(t_n+1,t_n].

However, it seems that wouldn’t be enough, as the following example shows:

Let’s say that grim signaler#n, or GS_n, exists in I_n=(t_n+1,t_n], whose length d*2^-n

Let’s further stipulate that GS_nhas the power to send signals that persist within I_n, but not beyond her interval. At any time in I_n-1, it is not the case that the signal sent by GS_npersists, so there is nothing that GS_n-1, who only acts in I_n-1=(t_n, t_n-1] might detect.

That is, of course, unless I_n-1has the power to detect signals that exist at exactly t_nor earlier, but the problem is that the use of infinitary patchwork does not entail that I_n-1even exists at t_n, or earlier.

So, Koons’s argument requires that each grim signaler has a power that, while intrinsic to her interval, involves sending a signal that will persist for some time – even if a very small time, and even if the amount of time may vary from possible world to possible world – into a temporally later interval.

Yet, even that would not be enough.

On that note, one of the objections that Koons considers is that powers and dispositions can fail.

In order to reject this objection, Koons assumes that whether a disposition is followed and whether the exercise of a power is successful is a matter intrinsic to the spatiotemporal region in which the power in question is exercised.

So, Koons’s argument uses not only that each grim signaler has the intrinsic power in her interval to send a signal that endures for at least some time into the next interval, but also that she exercises that power successfully, and that the successful exercise of that power is a matter intrinsic to her own spatio-temporal region, even if it involves the persistence of her signal in another spatio-temporal region.

In fact, Koons’s argument assumes at least the following conditions:

a. When applying binary or infinitary patchwork, one may stipulate that the powers intrinsic to a spatio-temporal region R₁₁ whose temporal component is an interval I₁ involve powers to send signals that exist in a region R₂ whose temporal component is an interval I₂ that does not overlap with I₁ and I₂is later than I₁, provided that the temporal distance between I₁ and I₂ is zero (like (a, b] and (b, c], or [a, b) and [b, c)).

b. Moreover, when applying binary or infinitary patchwork, one may stipulate that the matter of whether an exercise of such powers is successful is also a matter intrinsic only to the interval at which they were exercised, even in cases in which the stipulation described in condition a. above is also made.

Conditions a. and b. would imply that when both the stipulations described in a. and b. are made, the matter of whether a signal – whether the signal is a Fred particle or something else – exists at some time in a spatio-temporal region R₂is a matter intrinsic to a spatio-temporal region R₁that is disjoint from R₂.

However, under that very weird notion of intrinsicality, there is no good reason to accept either of the patchwork principles. Such principles are extremely counterintuitive upon reflection, and after Koons’s conditions are analyzed as above, and there is no reason to reject our intuitions on the matter and come to believe that the principles are true.

4.4.4. Binary patchwork suffices.

Let’s grant in this subsection for the sake of the argument that the principles of infinitary and binary patchwork are true – including conditio ns a. and b., even though as I explained in the previous subsection, there seems to be no good reason to believe that they are true.

Given the kind of powers and dispositions that Koons patches – as seen in the previous sections -, infinitary patchwork is not required to conclude that temporal density is metaphysically impossible. In fact, binary patchwork suffices to establish that for every two non-overlapping temporal intervals I₁ and I₂, the temporal distance between them is non-zero. But that rules out temporal density, because if time were dense, the temporal distance between some temporal intervals like (a, b] and (b, c], or between [a, b) and [b, c) would be zero.

So, let’s prove from binary patchwork that there are no temporal intervals such that the temporal distance between them is zero:

Let’s stipulate the following conditions:

A: There are two temporal intervals (a, b] and (b, c], in world W₁. In (a, b], there is one entity E₁₁with the power and disposition to send a signal of type T₁ into (b, c].

B: In W₁, E₁₁exercises her power successfully.

C: In another world W₂, there are also intervals (a, b] and (b, c], and there is no signal of type T₁ in any of those two temporal intervals, and no entity receiving or sending any signals.

D: All of the powers and dispositions in those intervals are intrinsic to those intervals, and whether the exercise of powers is successful is also a matter intrinsic to the intervals in which the powers are exercised.

We may pick the spatial intervals to cover all of space at those times, or some other stipulation of our choosing; there are many options. I will leave the spatial condition aside to simplify, but nothing hinges on that.

Also, we may pick temporal intervals that are open into the future and closed into the past instead of open into the past and closed into the future. That makes no relevant difference, either.

So, by binary patchwork, we patch interval (a, b] from W₁and interval (b, c] from W₂ into some world W₃.

In W₃ there is no signal of type T₁at any time in (b, c], since there is no such signal in W₂and W₃is an exact duplicate of W₂in interval (b, c].

Similarly, in W₃there is in (a, b] one entity E₁₁with the power and disposition to send a signal of type T₁ into (b, c], and who exercises her power successfully in W₃as well. But given that the power was exercised successfully, there is a signal of type T₁at least at some time in (b, c] in W_3.

That is a contradiction.

A potential objection would be that even if there is no signal of type T₁ at any time in (b, c] in W₂and even if W₃ is an exact duplicate of W₂in interval (b, c], it is possible that there are signals of such type in (b, c] in W₃, and that the condition of exact duplication should not be understood as ruling out other entities. But such an objection would seem to use the expression ‘exact duplication’ in a way that does not seem to resemble the meaning of the words, and there appears to be no good reason to accept a modified principle based on the usage of ‘exact duplication’ suggested in this objection, either.

Still, I will give an alternative argument to the conclusion that binary patchwork suffices to establish that time is not possibly dense:

Let’s stipulate that there are two intervals (a, b] and (b, c], in world W₁. In (a, b], there is one entity E₁₁with the power and disposition to send a signal of type T₁ into (b, c]l; no signal of type T₂ is sent. In (b, c], there are two entities, E₁₂ and E₁₃, with the following powers and dispositions:

i. E₁₂ checks whether a signal of type T₁ reaches (b, c]. If it does, then E₁₂ sends a signal of type T₁ into a later interval. Else, she does nothing

ii. E₁₃ checks whether a signal of type T₂ reaches (b, c]. If it does, then E₁₃ prevents any signals of type T₁ from reaching any later interval. Else, she sends a signal of type T₁ into a later interval.

In W₁, both E₁₂ and E₁₃ exercise their powers and dispositions successfully: E₁₂ detects a signal of type T₁, and sends a signal of type T₁ into a later interval, whereas E₁₃ detects nothing and so, according to her dispositions, she also sends a signal of type T₁ into a later interval.

On the other hand, in world W₂, in (a, b] there is one entity E₂₁ with the power and disposition to send two signals into (b, c]: one of type T₁, and one of type T₂. There are no entities checking for, sending or blocking such signals in (b, c] in W₂. Moreover, E₂₁ exercises her powers and dispositions successfully in W₂.

All of the powers and dispositions involved in the scenario, as well as whether they’re successfully exercised, are intrinsic to the intervals in which the powers and dispositions are exercised, regardless of whether they involve making signals persist into another interval. This stipulation mirrors exactly Koons’ stipulation that the powers and dispositions of the GR in his argument, as well as whether they’re successfully exercised, are intrinsic to the intervals in which the powers and dispositions are exercised, in a situation in which his grim reapers have the power to send signals into a later interval.

Then, just by binary patchwork, we can paste (a, b] from world W₂ with (b, c] from world W₁. In other words, by binary patchwork we obtain a world W' such that E₂₁ exercises her powers and dispositions successfully in (a, b] W', and so do E₁₂ and E₁₃ in (b, c], which is a contradiction, since that would involve that a signal of type T₁ is successfully sent into a later interval, and no such signal is reaches any later interval.

A potential objection would be that arguably, there is a problem with the conditions in the interval (b, c] in W₂, since the two entities E₁₂ and E₁₃ cannot jointly exercise their powers in a world at which both a T₁ and a T₂ signal reach the interval. However, that does not seem to be a problem if one assumes that powers can fail, as Koons seems to. One may posit that they both have the powers in question, and powers possibly fail. So, it seems to me that the objection fails.

Still, I will give another alternative argument in support of the conclusion that binary patchwork suffices to establish that time is not possibly dense:

Let’s stipulate that there are two intervals (a, b] and (b, c=b+1 hour], in world W₁. In (a, b], there is one entity Lexwith the power and disposition to send a kryptonite signal into (b, c].

In W₁, Lex exercises his power and disposition successfully, sending a kryptonite signal into (b, c]

On the other hand, in world W₂, in (b, c] there is one entity Clark with the power and disposition to fly faster than a speeding bullet for all of the duration of the interval, that is for an hour. Clark, however, is vulnerable to kryptonite signals. Any kryptonite signal will kill him, and his death will happen in less than a second, regardless of how long the signal lasts for. This vulnerability to Kryptonite is one of Clark’s intrinsic properties in (b, c]. Clark exercises his powers and dispositions successfully in W₂.

All of the powers and dispositions involved in the scenario – and generally other properties, like liabilities -, as well as whether they’re successfully exercised, are intrinsic to the intervals in which the powers and dispositions are exercised, regardless of whether they involve making signals persist into another interval. This stipulation mirrors exactly Koons’ stipulation that the powers and dispositions of the GR in his argument, as well as whether they’re successfully exercised, are intrinsic to the intervals in which the powers and dispositions are exercised, in a situation in which his grim reapers have the power to send signals into a later interval.

Then, just by binary patchwork, we can paste (a, b] from world W₁ with (b, c] from world W₂, into some other world W₃. Then, in W₃, Clark flies faster than a speeding bullet for an entire hour in (b, c] - i.e., for the whole duration of the interval -, but on the other hand, Clark dies in no more than a second into (b, c] due to the kryptonite signal successfully sent by Lex. But that is impossible.

In this construction, if required, we may stipulate that in the beginning of (b, c], Clark is located in a particular place, and that the kryptonite signal reaches that particular place.

So, the conclusion is that binary patchwork suffices to show that any two temporal intervals are at a positive temporal distance from one another, and that time is not dense.

A potential objection might hold that perhaps when using the patching principles, powers and dispositions and their successful exercise can be intrinsic to spatiotemporal regions, but some other properties, like liabilities, cannot, blocking the kryptonite argument. But if that is what the principles hold, then why should anyone accept ad-hoc patching principles like that?

That aside, and for the reasons given in the previous subsection, one should not accept the patchwork principles, even leaving aside the arguments given in this subsection.

4.4.5. Infinite past, undefeated.

Let us now grant again Koons’s assumptions for the sake of the argument, let’s take into consideration the conclusions – from binary patchwork, as explained in th e previous subsection – that time is not possibly dense, and also that there is a positive temporal distance between any two given temporal intervals such that one precedes the other, and let’s see why the conclusion that an infinite past is impossible, is blocked:

Let’s suppose that there are infinitely many past temporal intervals of positive duration {I_n}, for all natural n, such that I_n+1 precedes I_n._.

The infimum of the distances between pairs of intervals may or may not be zero, but in any case, the distance between each two intervals is always positive, and thus conditions a and b are not enough to specify that the signal persists from one of those intervals into the next, blocking Koons’s argument.

It might be objected that Koons’s assumptions are not limited to conditions a and b, and in particular, that the condition of zero distance can be replaced by another condition.

More precisely, it might be posited that Koons’s conditions are something like:

a’. When applying binary or infinitary patchwork, one may stipulate that the powers intrinsic to a spatio-temporal region R₁₁ whose temporal component is an interval I₁ involve powers to send signals that exist in a region R₂ whose temporal component is an interval I₂ that does not overlap with I₁ and I₂is later than I₁, provided that there is no temporal interval between I₁ and I₂that is disjoint from at least one of them [alternatively: from both of them].

b’. Moreover, when applying binary or infinitary patchwork, one may stipulate that the matter of whether an exercise of such powers is successful is also a matter intrinsic only to the interval at which they were exercised, even in cases in which the stipulation described in condition a’ above is also made.

The problem is that in that case, binary patchwork alone entails a contradiction regardless of whether the past is finite, whether time is dense, etc., as the following argument shows:

A. By binary patchwork, as in the previous subsection, we establish that the temporal distance between two temporal intervals is not zero.

B. Using Koons’s argument against temporal density, we establish that there cannot be infinitely many temporal intervals between two given intervals.

C. Using B and A, we pick two temporal intervals I₁ and I₂that are at non-zero distance, but such that there is no other temporal interval between them.

D. Applying the alternative conditions, and using an argument like that given in the previous subsection but without the stipulation that the temporal distance between the two intervals is zero, we get a contradiction.

Granted, someone might suggest that Koons’s implicit assumptions go beyond a. and b., but not as far as a ’ and b’. But if there is no requirement that the intervals be at zero temporal distance between each other for the signal to persist, one may ask the person positing this alternative what other condition can be given that does not go as far as a ’ and b’, how that would entail that necessarily there is no infinite number of past non-overlapping temporal intervals, and why should one accept such conditions?

As it stands, Koons’s argument fails to show that temporal density is impossible, and even assuming for the sake of the argument that it shows that, then it fails to show that infinitely many past days – for instance – are impossible.

There seems to be no way of fixing those shortcomings.

Since the arguments I give in this subsection use some of the results established in subsection 4.4.4, if objections to all of the arguments I gave in that subsection were to succeed, that would block the arguments I give in this section. However, I reckon that those objections failed, as argued there, and while someone might raise further objections, they would need to be argued for.

In any case, and for the reasons given in an earlier subsection, one should not accept the patchwork principles, even leaving aside the arguments given in this subsection and the immediately previous one.

4.5. An infinity by successive addition?

In this subsection, I will address another one of Craig’s arguments. According to Craig, on a tensed theory of time, the reality of temporal becoming makes it impossible for there to be an infinite past series of events of equal duration, since those changes would have to happen by successive addition, but by addition of one change or event at a time, it would never be possible to reach an infinity. [r 1 6 ]

While it seems clear to me that it would be impossible for there to be such an infinite series with a beginning point, a key question here is whether a beginningless series is impossible.

There appears to be a significant difference between a case with a beginning point and one that does not have a beginning point, namely that in the case in which there is no beginning point, there are no two events and/or times separated by an infinite temporal distance, or by infinitely many intervals of equal duration, so the temporal distance to be traversed from any specific time to another is always finite, and so is the number of intervals of equal duration traversed by anyone or anything from any specific time to another.

As long as traversing finite distances is not a problem, it seems that would block the argument.

Granted, someone might still object to the infinite series of events without a beginning and ask how the infinite series of past events was formed in the first place, claiming that it would never form under a tensed theory of time.

However, a reply to that line of argument would be to point out that precisely, since there is no beginning point, there is no need for an infinite series to be formed from a certain time on, and also no time at which the infinite past hadn’t already happened.

Given the previous considerations, and after reflection, I conclude that the argument fails.

In the following subsections , I will address some other arguments defended by Craig, and which purportedly support his conclusion that it’s impossible for there to be an infinite series of past events on a tensed theory of time. I will also address an argument based on a modification of one of Craig’s scenarios, and defended by Ben Waters.

But before I address those arguments, I would like to point out that, while Craig defends those arguments against an infinite past in the context of a tensed theory of time, my replies to those arguments do not require a tensed theory of time, though of course they do not reject it, either. Thus, if someone defended arguments essentially like those defended by Craig but without assuming a tensed theory of time, the replies I give below would be equally applicable to them

4.5.1. Tristram Shandy.

One of Craig’s arguments against an infinite past is based on the story of Tristram Shandy. [r 1 6 ]

Shandy is a man who writes his autobiography, at a rather slow pace: it takes Shandy a year to write the events of a single day. Also, he writes about the events of one day, then the following day, and so on.

According to Craig, we should reject an infinite past because it’s obviously coherent to write an autobiography at that pace, but if Shandy had been writing from infinity, that would lead to absurdities.

However, what is obviously coherent is to write such an autobiography starting at a specific day.

On the other hand, to write an autobiography counting from infinity and meeting Craig's conditions is logically impossible.[r 1 7 ]

To see why this is impossible, let’s suppose otherwise, let’s suppose the number of past years has the order type of the non-positive integers, and let’s enumerate the past years in the following way: [12]

Last year is 0, the previous year is -1, and so on.

For instance, if the current year is 2013, then 2012 is 0, 2011 is -1, 2010 is -2, and so on.

Now, let F be a function from the set of non-positive integers into itself such that for all nonnegative integers r and n, F(-r) = -n if and only if -n is the most recent year Shandy wrote about during the year -r.

For instance, if, in the year -2000, Shandy wrote about a day in the year -300001 and about a day in the year -300000, then F(-2000) = -300000.

Given the rate at which Shandy writes, and given also that, when writing his autobiography, Shandy never writes about his future, we have the following conditions:

1. F(-r) ≤ -r.

2. F(-r-365) = F(-r) - 1.

By induction:

3. F(-r-2*365) = F(-r-365-365) = F(-r-365) - 1=F(-r) - 2

4. F(-r-k*365) = F(-r) - k, for all nonnegative k.

So, in particular, taking r=0.

F(0) - k = F(-k*365) ≤ -k*365.

Hence, for every nonnegative integer k,

5. 364*k ≤ -F(0)

That’s contradictory, as easily seen by taking (for instance) k = 1 + (F(0)*F(0)).

So, what’s logically impossible is Tristram Shandy scenario itself, but that does not have anything to do with whether an infinite past is possible.

Craig acknowledges that the scenario is logically impossible, yet claims that because it seems “obviously” coherent to write one’s autobiography at a rate of one day per year, it seems to them that the problem is the infinite past.

I have to confess that I find his reply puzzling. Clearly, the task of writing one’s autobiography at a rate of one day per year ‘from infinity’ - and meeting the conditions stipulated by Craig – is contradictory, and so not coherent, let alone obviously so.

Incidentally, there is a contradictory scenario about the future that seems similar to the ‘Tristram Shandy’ scenario, in the sense that accepting it would involve making the same kind of error as accepting the Tristram Shandy scenario.

Alice Shandy writes a novel starting in the year 2000, and the story is set in her future. She always writes about future days. Also, it takes Alice a year to specify what happens in one day in her novel, and she always writes her novel in sequence. In other words, she writes about what happens on some day d₁, and after she finishes writing about d₁, she moves to the day after d₁, without jumping to any later day.

Also, for any number n, Alice will spend more than n years writing her novel.

The ‘Alice Shandy’ scenario is contradictory, but that does not warrant a conclusion that an unbounded future is metaphysically impossible.

Someone might insist that the task of writing a novel about the future, writing about consecutive days at a rate of one day per year, is obviously coherent, and so the fault must be with the idea of a future with an unbounded future number of years, but it seems apparent to me that such a reply would be very mistaken.

Incidentally, in the Alice Shandy argument, whether presentism is true makes no relevant difference in this context, for at least the following two reasons:

a. Even on presentism, the past is not real, even if it was. But the future also will be real. Granted, someone might insist on the actuality of temporal becoming in the past, or something like that, but that does not make a difference to whether there is a contradiction.

b. In any event, and leaving point a. aside, the fact is that the contradiction in the Alice Shandy case is obtained based on what she will do according to the scenario. One may very well reach contradictions using the future tense, and that does not make the argument dissimilar in a relevant way.

The ‘Tristram Shandy’ scenario is similar to the ‘Alice Shandy’ scenario in the sense that Craig constructs a contradictory scenario involving tasks that might intuitively strike some people as possible, and the contradiction is the stipulation that Shandy has been writing at that pace for all of the past, in a similar way to the Alice Shandy scenario, in which the contradiction is that Alice will be writing at that pace for more than any specific finite number of years.

In any case, the analogy with the Alice Shandy scenario is only meant to illustrate the problem, and of course not needed. I would say that we should in any case reject the ‘Tristram Shandy’ scenario because the task of writing one’s biography at a rate of one day per year ‘from infinity’ and under the stipulated conditions is a contradiction, and the fact that the assumption that it’s possible results in absurdities does not tell us anything about whether an infinite past in which the set of past years has the order type of the negative integers is metaphysically possible.

4.5.2. Methuselah.

Ben Waters defends an argument that is similar to Craig’s “Tristram Shandy” argument, but with some modifications, intended precisely to avoid some of the objections raised against Craig’s construction. [r17b]

However, even if Waters’s argument avoids some of the mistakes in Craig’s argumentation, it makes an unjustified assumption, and as a result, it provides no good reason to think that an infinite past is impossible or even not actual – Waters claims it’s not actual, though if his reasoning were correct, it seems it could be extended. Still, the main point is that Waters’s argument does not support the claim that the past is actually finite.

Basically, Waters:

a. Defines D as the set of all days prior to today, and DF as the set of all days in D at a finite distance from today, and uses '≤' for the total relation on D in the obvious manner.

b. Argues that if there is a function f from DF to DF such that f(d)≤d for all d in DF and such that f(d+2)=1+f(d) for all d, d+2 in DF, then DF is finite.

c. Argues that if DF is finite, then the set D of all days previous to today is also finite.

d. Argues that a function f with the properties in question does exist.

Waters’s claims in b and c are true. The problem is d.

On that note, Waters’s description is a lot more detailed than Craig’s in the Shandy argument, but in the end, the error is similar. More precisely, Waters stipulates that:

A. Methuselah has been alive for all days d in DF, and for every day, he has a perfect memory of what he was doing in the immediately previous day.

B. He has been writing his memoirs at a pace of half an entry per day, for all days in his life. That is, it takes him two days to write down the activities of one previous day.

C. If, on a certain day d, he remembers that on (d-1) he was writing about the events on (d-m) for some positive integer m, then he will continue working on the entry for (d-m) if that entry is not complete; else, he will start writing the entry for (d-m+1).

D. If, on the other hand, on d Methuselah does not remember that on (d-1) he was working on an entry for (d-m) for some positive integer m, he will start working on an entry for d.

E. Methuselah is in a logically possible world that is like the actual world in all relevant temporal facts.

While stipulations about what Methuselah will do under such and such conditions might sound innocent at first glance, the fact is that Waters assumes that in a world with the same relevant temporal facts as the actual work, Methuselah did perform the tasks in question for every past day that is at a finite temporal distance from the present, and so the function in question – allegedly – exists.

But that’s obviously impossible if the cardinal of DF is infinite, so that implicitly but quite transparently makes the assumption that it is actually not the case that for every day d in the actual world, and for every natural number n, there is a day d(n) in the actual world such that d(n) is n days earlier than d.

Granted, it might be argued that there are independent reasons for believing that such a task is possible in a world with the same relevant temporal facts as the actual world. But that would have to be argued for. Waters’s argument does not show it, or gives any good reasons to suspect so.

In particular, pointing out that the task carried out by Methuselah on each single day is possible in a world with the same relevant temporal facts as the actual world, and then appeal to a general patching principle, wouldn’t succeed it without explaining what the patching principle is and why one should believe it.

Moreover, given that the patching that would be required in the Methuselah case would involve the memories of a single person about past activities, etc, it seems that any patching principle that might be applied to this case would need to make it intrinsic to one temporal interval whether something persists into some other, disjoint temporal interval – or something akin to that -, which is at least one of the why reasons the patching principles defended by Koons ought to be rejected.

To be clear, I’m not suggesting that Waters is using a patching principle, but rather, that if someone attempted to fix Waters’s argument by positing such a principle, the burden would be on them, and furthermore, given the previous considerations, it seems that would fail as well.

Perhaps, another way of trying to defend Waters’s argument would be to suggest that Methuselah’s powers and dispositions are possible, and so allegedly the burden to show that it is not possible that a being exercises them successfully in a world with the same relevant temporal facts as the actual world would be on the person objecting to Waters’s Methuselah argument.

This defense fails as well, as the following points illustrate:

1. Let us stipulate, only to simplify terminology and for the rest of this subsection, that world W has an infinite past – or equivalently by definition, that in W the past is infinite – if and only if there is some day d in W such that for every natural number n, there is or there was another day d(n) in W, such that d(n) is or was at least n days earlier than d.

Similarly, let us stipulate that W has a finite past – or equivalently by definition, that in W the past is finite – if and only if for every day d in W, either there were no days in W prior to d, or the number of days in W that happened prior to d is a positive finite number.

Those are only stipulative definitions applicable to the rest of this subsection, which are sufficient for the purposes of my reply to Waters’s argument. I make no claim that the past cannot be finite or infinite in some other sense, or that those definitions match common usage, but at least they capture enough cases for the purposes of this part of the reply to Waters.

Also, Waters uses logically possible worlds instead of metaphysically possible worlds, so let’s stipulate that for the rest of this subsection we’re talking about logically possible worlds unless otherwise specified.

However, I would like to point out that an argument like Waters’s but using metaphysical possibility instead of logical possibility would not succeed, either. Some of the objections would have to be modified to some extent, but the argument against an infinite past would still fail because of its unwarranted stipulation about the relevant temporal facts in the actual world.

2. Let us now consider the powers and dispositions attributed to Methuselah in Waters’s paper:

2.1. For every pair (d-1,d), the power to remember perfectly on d what one did on (d-1).

2.2. The power and disposition to write a diary of one’s past activities at a rate of half an entry per day.

2.3. The disposition that, for each d, if on day d one remembers that on a day (d-1) one was working on an entry for day (d-m) for some positive integer m, then one continues working on that entry if the entry for (d-m) is incomplete, and starts working on the entry for (d-m+1) if the entry for (d-m) is complete.

2.4. The disposition that, for each d, if on day d it is not the case that one remembers working on an entry as described in 2.3, then one begins working on an entry for d.

2.5. Additionally if needed, at least all powers of an adult, healthy and young human being of average intelligence and strength, when no stronger capability is specified. This particular condition is not required by Waters, but it might be implicit, and in any case it’s obviously possible and is not a problem to add it.

3. The powers and dispositions described in 2 above are possible powers and dispositions. Moreover, it is possible that an entity exercises them successfully on every day in its life.

For example, let us stipulate that in world W1 Bob begins to exist on day d, and that in W1, Bob has the powers and dispositions described in 2, and also exercises them successfully on every day in his life. This is unproblematic, but does not tell us anything about whether the past is actually finite or infinite, or even whether the past is finite or infinite on W1.

4. Let us also stipulate that in world W2, Bob has and exercises the powers and dispositions described in 2 successfully, on every day in his life, and furthermore, that there is some day t such that Bob is alive in W2 on every day d earlier than t and at a finite temporal distance from t.

Then, that implies that on W2, the past is finite. While it seems that W2 is plausibly a logically possible world, that does not give us any good reason to think that the past is actually finite. It would be improper to stipulate that W2 is exactly like the actual world with regard to all relevant temporal facts, and then from that conclude that the past is actually finite.

But that is what Waters is doing in the case of Methuselah.

While points 2 – 4 above suffice, I will give two analogies.

1’: Same stipulations as in 1 above.

2’: Let us consider the following powers and dispositions:

2.1’: The power to come to know, on every day d in one’s life, that at least some event E(d) happened on some day p(d) earlier than d.

2.2’: The disposition to write down, on every day d in one’s life, at least one of the events that happened on p(d).

3’: The powers and dispositions described in 2’ above are possible powers and dispositions.

Moreover, it is possible that an entity exercises them successfully on every day in its life.

For example, let us stipulate that in world W3, Alice begins to exist on some day (d(0)+1), and that on W3, Alice has the powers and dispositions described in 2’, and exercises them successfully on every day in her life. This is also unproblematic, but does not tell us anything about whether the past is actually finite or infinite, or even whether the past is infinite or finite in W3.

4’: Let us now stipulate that world W4, Alice is alive on every day earlier than some day t and at a finite temporal distance from t, and that on W4, Alice has the powers and dispositions described in 2’, and exercises them successfully on every day in her life.

Then, it follows that the past is infinite in W4.

However, this does not give us any good reason to believe that the past is actually infinite, even though it seems that W4 is logically possible.

In particular, it would be improper to stipulate that W4 is exactly like the actual world with regard to all relevant temporal facts, and then from that conclude that the past is actually infinite. The stipulation in the case of Bob or Methuselah – and the corresponding conclusion – is improper as well.

As a reply to the Alice analogy, it might be argued that some of the differences between the Alice scenario and the Bob/Methuselah scenario are relevant at least some of the matters at hand for some reason, but that would need to be argued for. In particular, if the allegedly relevant difference would require the application of some sort of patching principle, that would seem to fail too, for the reasons I gave earlier in this subsection.

In any case, I would like to stress that the Alice scenario is not required to establish the point that the reasoning behind the Methuselah argument is flawed. It’s only an analogy that I hope will illustrate a point better, but if it doesn’t, one may just ignore it, and focus on points 2 – 4 above, which suffice.

Let us now consider a second analogy, this time involving the future:

1’’. Let us stipulate, only to simplify terminology and for the rest of this subsection, that world W has an infinite future – or equivalently by definition, that in W the future is infinite or will be infinite – if and only if there is some day d in W such that for every natural number n, there is or there will be another day d(n) in W, such that d(n) is or will be at least n days later than d.

Similarly, let us stipulate that W has a finite future – or equivalently by definition, that in W the future is finite or will be finite – if and only if for every day d in W, either d is the last day in W, or the number of days in W that happened, happen and/or will happen after d is a positive finite number.

Those are only stipulative definitions applicable to the rest of this subsection, which are sufficient for the purposes of my reply to Waters’s argument. I make no claim that the future cannot be finite or infinite in some other sense, or that those definitions match common usage, but at least they capture enough cases for the purposes of this part of the reply to Waters.

2’’. Let us now consider the following powers and dispositions, mirroring those attributed to Methuselah in Waters’s paper, namely:

2.1’’. For every pair (d-1,d), the power to remember perfectly on d what one did on (d-1).

2.2’’. The power and disposition to write a novel set in an imaginary future at a rate of half an entry per day, where an ‘entry’ for some day d is an account of the events that take place in the fictional story on day d.

The condition that the novel is set in an imaginary future means that if one is writing on some day d1, and one is working on an entry for some day d2, then d1 is earlier than d2 in the usual order - or would be so if d1 existed, if one prefers -, and one is writing fiction. Of course, since it’s a fictional story, it is not required that d2 exists or will exist.

2.3’’. The disposition that, for each d, if on day d one remembers that on a day (d-1) one was working on an entry for day (d+m) for some positive integer m, then one continues working on that entry if the entry for (d+m) is incomplete, and starts working on the entry for (d+m+1) if the entry for (d+m) is complete.

2.4’’. The disposition that, for each d, if on day d it is not the case that one remembers working on an entry as described in 2.3’’, then one picks any positive integer m of one’s choosing, and begins working on an entry for (d+m). [As an alternative analogy, the positive fixed integer is fixed and is m=1 trillion).

2.5’’. Additionally, at least all powers of an adult, healthy and young human being of average intelligence and strength, when no stronger capability is specified.

3’’. The powers and dispositions described in 2’’ above are possible powers and dispositions. Moreover, it is possible that an entity exercises them successfully on every day in its life.

For example, let us stipulate that in world W5. Lilith begins to exist on day d, and that in W5, Lilith has the powers and dispositions described in 2’’, and also exercises them successfully on every day in her life. Also, her life lasts for 36500 days, and if she has to pick a number m of her choosing as specified in 2.4’’ above, she picks m=1 trillion [in the variant in which m is fixed, she does not need to pick].

This is unproblematic, but does not tell us anything about whether the future is actually finite or infinite, or even whether the future is finite or infinite on W5.

4’’. Let us also stipulate that in world W6, Lilith has and exercises – and/or had and exercised, will have and will exercise, etc. - the powers and dispositions described in 2’’ successfully, on every day in her life, and furthermore, that there is some day t such that Lilith is or will be alive in W6 on every day d later than t and at a finite temporal distance from t.

Then, that implies that on W6, the future is finite. It seems that W2 is plausibly a logically possible world. Whether it’s a metaphysically possible world is another matter, but there seems to be no contradiction in – say – stating that a relational theory of time is true, and also that all changes will eventually cease in W6. No contradiction seems to follow.

However, even if W6 is a logically possible world, that does not give us any good reason to think that in the actual world the future is or will be finite.

It would be improper to stipulate that W6 is and will be exactly like the actual world with regard to all relevant temporal facts, and then from that conclude that the future is or will be finite in the actual world.

But that is relevantly similar to what Waters is doing in the case of Methuselah.

As an objection to the Lilith analogy, someone might suggest that perhaps W6 is not logically possible after all. However, since there appears to be no contradiction in the description and there appears to be no good reason to suspect that there might be a hidden contradiction, the burden of showing that there is any serious doubt about the logical possibility of W6 would be on the person raising that objection. Moreover, that would be a distraction anyway, since it should be clear that stipulating that W6 has and will have the same relevant temporal facts as the actual world and then concluding from that and the Lilith scenario that the actual world has or will have a finite future would be improper regardless of whether W6 is logically possible. Similarly, the stipulation is improper in the Methuselah scenario in Waters’s paper.

Alternatively, someone might object to the Lilith analogy on the basis that – allegedly – presentism is true, and for some reason that's a problem for scenarios involving the future. However, leaving aside the facts that presentism would not be logically necessary even if it were metaphysically necessary, and that even on presentism, one can properly make stipulations and reason about what will happen, an objection based on presentism would miss the point as well, since regardless of any issues involving presentism, the conclusion that the future is or will be finite based on the Lilith scenario ought to be rejected because of the unwarranted stipulation that the Lilith scenario happens in a world that has and will have the same relevant temporal facts as the actual world. That stipulation is relevantly similar to Waters’s stipulation in the Methuselah scenario.

In any event, and as in the case of the Alice analogy, the purpose of the Lilith analogy is to better illustrate why the Methuselah argument does not succeed. But the Lilith analogy is not required, and if it does not make matters more clear, one may just ignore it and focus on points 2 – 4 above, which suffice.

4.5.3. Orbits and parity.

Another argument against an infinite past defended by Craig [r18] is based on orbits and the alleged parity of some numbers.

According to this argument, it’s absurd that if Jupiter completes 2.5 orbits for each one Saturn completes, they would both have completed the same number of orbits, if they have been orbiting the Sun for an infinite number of years.

However, and leaving aside the fact that any planet, star, etc., only lasts for finitely many years in the actual world, the fact is that there appears to be no absurdity at all. As in the case of the Hilbert Hotel argument, any alleged problem seems to be a confusion about the meaning of the words.

More specifically, that the number of completed orbits would be the same if they had been orbiting forever and there were an infinite past composed of denumerably many years merely means that there would be a bijection between the set of orbits completed by one of the planets, and the set of orbits completed by the other planet, and that’s as clear as the fact that, say, the set of natural numbers that are multiples of two has the same cardinality as the set of natural numbers that are multiples of five.

In addition to the previous argument, Craig also says that if a planet had been orbiting for eternity, the number of completed orbits would be both even and odd, and provides an unusual definition of '‘even’ and ‘odd’' that would apply to infinite cardinals, and under which infinite cardinals would turn out to be both even and odd.

However, that is not a problem, either, but rather, a consequence of Craig’s unusual choice of definitions and the fact that there is a bijection between any two denumerable sets.

Of course, once again in reality planets don’t last indefinitely, but that is not at all relevant.

4.6. Scientific cosmology.

According to Craig, modern scientific cosmology supports the second premise of the Kalam Cosmological Argument. In this section, I will assess his arguments on the matter, and raise some objections.

4.6.1. The Friedmann–Lemaître Model.

Craig contends that what he calls the “Standard Hot Big Bang Model”, or Friedmann-Lemaître model [r 1 2 ], supports a beginning of the universe that is an absolute origin ex-nihilo, and space and time themselves come into being at an initial singularity.

However, a singularity in that mathematical model indicates that the equations of General Relativity don’t properly describe some phenomena in the distant past, not that there is some actual thing of infinite density – that would not even be defined.

In fact, General Relativity only takes into consideration one force – namely, gravity -, ignoring the rest, and while that’s usually be good enough for large objects, but in a very small universe like the universe in a distant past, other forces and the possibility of quantum effects should be taken into account as well. [r 1 9 ] More precisely, what was small was the universe in the sense in which the word ‘universe’ is used in some scientific models, but whether the universe was small in the sense in which the word ‘universe’ is used in the KCA is another matter.

In any event, the proper conclusion here is that we’re not justified in applying those equations beyond a certain time in the past, which of course provides no support for the second premise of the KCA.

In addition to that, the model in question entails an infinite past sequence of changes, since as we move further back in time, the density of the universe tends to infinity, and so there are more than n consecutive changes in which the density of our universe decreases, for each n, even if the past in that model is metric-finite.

So, one of the models that Craig offers in support of his claims against an infinite past is incompatible with his stance on the possibility of actual infinities, as long as an infinite regress of changes is an actual infinity.

Granted, Craig or another defender of the KCA may simply point out some of what I mentioned above, namely that the model in question is not applicable beyond a certain time in the past, and so their stance on infinities is not undermined or defeated. But that only underscores the fact that this cosmological model is not applicable beyond a certain time in the past, and thus for that reason of no use for a defender of the KCA, even if the were ‘universe’ were being used in the relevant sense here - which is not the case.

Also, while someone might suggest that such a regress is not an actual infinity if presentism is true because only the present exists, that objection is incompatible with Craig’s position, since for the same reason, even infinitely many past years would not be an actual infinity on presentism.

4.6.2. The Borde-Guth-Vilenkin Theorem.

In addition to the previous model, Craig often appeals[r 2 0 ] to a paper by Guth, Borde and Vilenkin[r 2 1 ], and offers that paper and the theorem proved in it as evidence in support of the second premise of the KCA.

In that paper, the authors reach the conclusion that if some reasonable assumptions obtain, then the past boundary of the inflating universe is reached in a finite past, and some new physics is required to described that boundary.

However, they make no claims in that paper about what the new physics would be, beyond mentioning that several alternatives have been discussed, including a quantum event in which the universe originates.

But even if such quantum event actually happened a finite time ago, that would be part of the universe in the sense in which the word ‘universe’ is used in the KCA.[4 ]

Would that imply that the universe, in the sense in which the word ‘universe’ is used in the KCA, began to exist?

That would need to be argued for. Showing a beginning of the universe in a narrower sense of ‘universe’ would not suffice.

4.6.3. Other arguments based on scientific cosmology.

Craig offers a number of other arguments, based on different hypotheses in modern science, basically arguing that all of the hypotheses that aren't too implausible lead to the conclusion that the universe had a beginning.

A crucial issue here is what is meant by ‘universe’ in the context of those hypotheses, and what is meant by ‘universe’ in the context of the KCA. [4 ]

What a defender of the KCA would have to show in order to properly support premise 2 is not only that the object called '‘universe’ in the context of those scientific hypotheses began to exist, but that all of those hypotheses support a beginning of the universe in the sense of the word ‘universe’' that is relevant in the KCA. [1 3 ] In particular, they would still need to show that, plausibly, either:

i. There is no time in some realm (say, an older universe in a restrictive sense of ‘universe’, or a multiverse, etc.) that is prior to the existence of the object called ‘universe’ in those scientific hypotheses, or

ii. If there is time in some realm that is prior to the existence of the object called ‘universe’ in those scientific hypotheses, that realm would plausibly not qualify as ‘universe’ in the context of the KCA.

But Craig and other defenders of the KCA have not shown that, so the burden remains on them.

Still, we may speculate about what kind of argument would be required to support the second premise of the KCA using empirical evidence.

Perhaps, a defender of the KCA might try a probabilistic argument like the following argument:

If the object called ‘universe’ in scientific models had an infinite past, then so would the universe in the sense in which the word ‘universe’ is used in the KCA.

So, if, based on the empirical evidence available to us, we should revise our prior probabilistic assessment of the hypothesis that the object called ‘universe’ in the context of scientific models has an infinite past, assigning higher probability to that hypothesis than before the empirical evidence was considered, then we should also reduce the probability assigned to the second premise of the KCA.

In other words, empirical evidence that increases the probability that the object called ‘universe’ in the context of scientific models has an infinite past, also decreases the probability that the second premise of the KCA is true.

But then, reciprocally, empirical evidence that decreases the probability that the object called ‘universe’ in the context of scientific models has an infinite past, also increases the probability that the second premise of the KCA is true.

A first difficulty here is that different models might use the word ‘universe’ somewhat differently.

But Let’s assume that ‘universe’ means the same in all of the models, to simplify.
Even then, the probabilistic argument above would not be enough to establish that the second premise of the KCA is true, or even probably true.

In order to justify an assessment that the second premise of the KCA is probably true, based on empirical evidence, a defender of the KCA would also have to show that:

a. Based on empirical evidence, we should conclude that the object called ‘universe’ in scientific models probably has a finite past. He would not have to show what the proper probabilistic assignment to P is, but show it’s P > 0.5.

b. Given the probability P in a., it is probable that the universe, in the sense in which the word ‘universe’ is used in the KCA[4 ], has a finite past.

However, Craig and other defenders of the KCA have not shown that, even assuming P=1, then it’s probable that the universe, in the sense in which the word ‘universe’ is used in the KCA, has a finite past.

Moreover, even if we granted for the sake of the argument that if P has a high value (say, P > 0.8), then b. is true, still Craig and other defenders of the KCA have not shown that we ought to assign such a high value to P.

4.6.4. Cyclic models and other options for an infinite past.

Some of the models of a universe with an infinite past that Craig considers and rejects[r 2 2 ] in his defense of the KCA are cyclic models of the universe.

For instance, in particular, he considers a model proposed by Frampton. [r 2 3 ]

In his assessment, Craig mentions some of the objections to the model and the replies by Frampton, and in the end concludes that the field is too young to make a full judgment, but that nevertheless it seems that some of the problems of older cyclic models remain.

So, even though Craig says that he’s not passing full judgment, he does give the impression that he’s making some kind of probabilistic assessment, implicitly saying that some of the objections that defeated some earlier cyclic models probably will defeat this one as well, and further, will defeat all cyclic models. But Craig does not provide any adequate support for such an assessment.

In any case, and regardless of Craig’s claims about that particular cyclic model, a question in this context is whether we should assess that cyclic models of a universe with an infinite past probably fail. I do not see any sufficient reason to reach that conclusion at this point.

Granted, there are objections to such models, but no model, cyclic or otherwise, is generally accepted at this point. Rather, scientists are working on the development of different hypotheses, and a number of different options remain open.

In particular, the option of a cyclic universe with an infinite past continues to be a live one, and there are several recent papers discussing a number of variants of it as potential options.[r 2 4 ]

Moreover, even leaving aside cyclic models, there appear to be non-cyclic options of a universe with an infinite past still on the table.[r 2 5 ]

So, it seems that whether the past is infinite is an open question in science, and there appears to be no good reason to take a stance at this point.

5. Conclusions.

Neither the first nor the second premise of the KCA is properly supported by the arguments given by Craig and others who propose them, and I see no good reason independently of those arguments to accept said premises, either.

Moreover, the arguments in support of the second premise are arguments for a beginning of time, which, if correct, would be good evidence against the first premise.

All of the above would give us enough reasons to reject the KCA, but additionally, there are very serious questions about the coherence of the creator and/or the creation account proposed by Craig – i.e., the view that God is timeless without creation and temporal with it -, and generally by any similar versions of the KCA+.

6. Notes.

[1] In some contexts, people might talk about, say, a timeless work of art, but that’s clearly not the sense of ‘timeless’ that is relevant here.

[2] It’s not clear to me whether Craig uses '‘changeless’ and ‘quiescent’' to mean the same. But either way, the problems for his position remain, as argued above.

[3] Or the only concrete object that exists is O, if abstracta are part of a correct ontology. That issue is not relevant to the argument under consideration, so we may stipulate anything in that regard if we so choose. So, if a reader includes abstracta in an ontology, then the scenarios should read ‘concrete object’ instead of ‘object’', but other than that, the arguments are the same.

[4] According to Craig, “the universe” in the context of the KCA is defined as “the whole of material reality”. [r 1 1 ]

It’s not clear to me what “material” means in this context, or even if Craig using the word in a manner that is both coherent and sufficiently precise for the whole Kalam Cosmological Argument.

But assuming that his usage is coherent and precise enough, what is clear is that if there were a multiverse, infinite past cyclic universes, etc., all of that would be contained in the universe in the sense of ‘universe’ intended by Craig.

[5] Our intuitions about space and time, while very useful in daily life, do not work so well in some of the situations modern physics deals with, like the proximity of a black hole, or the early universe, or subatomic particles, etc. It’s not at all clear, to say the least, that those intuitions are reliable in contexts like a purported beginning of the universe, etc.

However, given that defenders of the KCA propose intuitions in support of the first premise, we may offer rebuttals based on intuitions as well, among others.

[6]

My intuitive assessment of infinite past vs. a finite past has changed over the years – while I used to find both of the alternatives counterintuitive (so, I had conflicting intuitions, and surely, one of my intuitions was misleading), I used to find the former more counterintuitive, whereas now it’s the latter –, my intuition that if time and space began to exist together in a finite past, then there is no efficient cause of time or space, has not changed.

[7] The claim that an infinite temporal regress would constitute an actual infinity is at least debatable on presentism, which is precisely the theory of time that Craig espouses. But Let’s grant for the sake of the argument and unless otherwise specified that an infinite temporal regress would constitute an actual infinity, even on presentism.

[8] Greater infinities would not make a relevant difference, so one may assume a larger infinity for that matter, and then make a similar analysis and raise similar counterarguments.

[9] By A\B I mean the set of elements of A that are not elements of B.

[10] Incidentally, such a principle could provide grounds for arguments against theism. For example, one can consider a scenario in which a being with the mind of a four-years-old human suffers horribly for a long period – which surely is possible -, and then by means of infinitary patchwork, cover all of spacetime with such beings like that suffering horribly, and nothing else, which can be used to strengthen even the argument from suffering – though I do not believe it requires strengthening -, at least when it comes to objecting to versions of theism that posit that God exists necessarily.

[11] Koons uses the word “set”, but it’s not entirely clear to me that that would be a set.

For instance, for all I know, there might be a possible world W with spatial structures of cardinality x, for any cardinal x. If so, the class of spatio-temporal regions of W would not be a set. In any case, this is a side issue.

[12] I’m stipulating 1 year = 365 days and ignoring leap years, for the sake of simplicity; a more complicated proof would include leap years, but it’s clear that the contradiction does not depend on whether we take into consideration leap years.

Also, the assumption that the set of past years has the order type of the set of non-positive integers is acceptable in this context, since that is the main hypothesis that Craig claims ought to be rejected due to the Tristram Shandy argument, and since in any case, I intend to show that the 'Tristram Shandy' argument does not show that such a past is impossible.

[13] I’m taking into consideration the fact that defenders of the KCA are actually also defenders of the KCA+, and that rules out the use of the word ‘universe’ in the KCA in a narrow sense, such as the use that it might be given to the word ‘universe’ in some scientific models.

At any rate, using ‘universe’ in a narrow sense in the second premise of the KCA would on its own make the argument irrelevant in the context of philosophy of religion, since that conclusion would not rule out the that there was some older realm before the universe, and more precisely a realm that does not entail or suggest a personal creator.

7. References.

[r1] William Lane Craig and J. P. Sinclair, “The Kalam Cosmological Argument”, in “The BlackWell Companion to Natural Theology”, Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6
Page 102.

[r2]

http://www.reasonablefaith.org/causation-and-spacetime

[r3]

http://www.reasonablefaith.org/current-work-on-god-and-abstract-objects

[r4] http://www.reasonablefaith.org/site/News2?page=NewsArticle&id=5971

[r5] William Lane Craig and J. P. Sinclair, “The Kalam Cosmological Argument”, in “The BlackWell Companion to Natural Theology”, Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6
Page 106.

[r6] http://www.reasonablefaith.org/site/News2?page=NewsArticle&id=9269

[r7] William Lane Craig and J. P. Sinclair, “The Kalam Cosmological Argument”, in “The BlackWell Companion to Natural Theology”, Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; pages 115, 116.

[r8] William Lane Craig, “God and the Beginning of Time”.

http://www.reasonablefaith.org/god-and-the-beginning-of-time

[r9] William Lane Craig and J. P. Sinclair, “The Kalam Cosmological Argument”, in “The Blackwell Companion to Natural Theology”, Edited by William lane Craig and J. P. Moreland; pages 184, 185.

[r10] William Lane Craig and J. P. Sinclair, “The Kalam Cosmological Argument”, in “The BlackWell Companion to Natural Theology”, Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; page 182.

http://www.reasonablefaith.org/defenders-2-podcast/transcript/s4-6

[r11] http://www.reasonablefaith.org/causal-premiss-of-the-kalam-argument

[r12] William Lane Craig and J. P. Sinclair, “The Kalam Cosmological Argument”, in “The BlackWell Companion to Natural Theology”, Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; page 108-115.

[r13]

http://alexanderpruss.blogspot.com/2009/10/from-grim-reaper-paradox-to-kalaam.html

[r14] http://pros blogion.ektopos.com/2009/10/02/from_grim_reape/

http://prosblogion.ektopos.com/2009/10/02/from_grim_reape/#comment-23553

[r15] Koons, R. C. (2012), A New Kalam Argument: Revenge of the Grim Reaper. Noûs. Doi: 10.1111/j.1468-0068.2012.00858.x

Also: http://www.robkoons.net/media/83c9b25c56d629ffffff810fffffd524.pdf

http://onlinelibrary.wiley.com/doi/10.1111/j.1468-0068.2012.00858.x/abstract

[r16] William Lane Craig and J. P. Sinclair, “The Kalam Cosmological Argument”, in “The BlackWell Companion to Natural Theology”, Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; page 117-124.

[r17] Josh Dever, in “Worlds Apart”, Taiwanese Journal for Philosophy and History of Science, 10 (1998), pointed out that the scenario is contradictory.

This was also pointed out by Graham Oppy, in “Arguing about Gods”, Cambridge University Press (2006).

[r17b] Ben Waters, in Philosophia Christi Volume 15, Number 2 (2013)

[r18] William Lane Craig and J. P. Sinclair, “The Kalam Cosmological Argument”, in “The BlackWell Companion to Natural Theology”, Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; page 120.

[r19]

1. Matts Roos, “Expansion of the Universe – Standard Big Bang Model”. http://arxiv.org/abs/0802.2005

2. http://csep10.phys.utk.edu/astr162/lect/cosmology/planck.html

3. http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_13.pdf

4. http://preposterousuniverse.com/writings/dtung/

[r20]

William Lane Craig and J. P. Sinclair, “The Kalam Cosmological Argument”, in “The BlackWell Companion to Natural Theology”, Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; page 141.

http://www.reasonablefaith.org/contemporary-cosmology-and-the-beginning-of-the-universe

[r21] Guth, Borde, and Vilenkin, “Inflationary spacetimes are not past-complete”, Physical

Review Letters 90, 151301.

http://arxiv.org/abs/gr-qc/0110012

[r22] William Lane Craig and J. P. Sinclair, “The Kalam Cosmological Argument”, in “The BlackWell Companion to Natural Theology”, Edited by William Lane Craig and J. P. Moreland, © 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6; pages 150-157.

[r23] Paul Frampton, “Cyclic Universe and Infinite Past”.

http://arxiv.org/abs/0705.2730

[r24] There are many examples. I will mention a few, but one might as well choose others.

To be clear, I’m not suggesting that any particular model is likely, but merely pointing out that the matter of an infinite cyclic past is open in modern scientific cosmology.

So, for example, one can mention – among others:

Yun-Song Piao, “Proliferation in Cycle”. http://arxiv.org/abs/0901.2644

Yun-Song Piao, “Design of a Cyclic Multiverse”. http://arxiv.org/abs/1001.0631

Hua-Hui Xiong, Yi-Fu Cai, Taotao Qiu, Yun-Song Piao, Xinmin Zhang, “Oscillating universe with quintom matter”. http://arxiv.org/abs/0805.0413

Kazuharu Bamba, Kuralay Yesmakhanova, Koblandy Yerzhanov, Ratbay Myrzakulov, “Reconstruction of the equation of state for the cyclic universes in homogeneous and isotropic cosmology”. http://arxiv.org/abs/1203.3401v2

Yi-Fu Cai, Emmanuel N. Saridakis, “Cyclic cosmology from Lagrange-multiplier modified gravity”. http://arxiv.org/abs/1007.3204

Kazuharu Bamba, Ujjal Debnath, Kuralay Yesmakhanova, Petr Tsyba, Gulgasyl Nugmanova, Ratbay Myrzakulov, “Periodic Cosmological Evolutions of Equation of State for Dark Energy”. http://arxiv.org/abs/1203.4226.

[r25] For instance:

Pierre-Henri Chavanis, “A simple model of universe with a polytropic equation of state”- http://arxiv.org/abs/1208.1192

9 comments:

Fox ITK said...: Hi Angra

I am involved with a debate with a Christian Theist who argues that the sorts of changes God seems to though highlighted above, are not changes in God's essential properties, rather they are changes in his accidental extrinsic properties.

He therefore claims God is still 'unchanging'

My response to this is- this is a vacuous account of change, if essentialism is true, then it follows any X with an essential property Y always possesses Y as long as X pertains. Everything tha exists can change only in its accidental properties, that's what they mean. Of course essential properties might not always obtain but that just means X no longer obtains.

In addition, that God intends to create, that he was sans creation wholly timeless, these seem to me like they wouldn't be extrinsic properties but intrinsic even if not essential.

But am I wrong on this? It sounds like a desperate attempt to maintain the notion that God is unchanging in the face of similar objections posted in your paper. How would you respond to this? Many thanks.; June 6, 2016 at 2:01 PM
Angra Mainyu said...: Hi Fox in the Know,

You're right about that if essentialism is true, then it follows any X with an essential property Y always possesses Y as long as X pertains, so everything that chances and still exists only changes accidentally.
So, it follows from the claim that God exists necessarily that his essential properties do not change. Craig claims that God is not changeless with creation, but does change, so he's talking about accidental changes.
In any case, I would be inclined to reply that the issues of accidental vs. essential changes or intrinsic vs. extrinsic properties, just don't raise any trouble for the arguments I make.
For example, in 2.1.2.1 and 2.1.3.6, I compare two scenarios, and argue that the relevant states appear ontologically identical; one might grant that the changes are accidental, and grant that the properties are either extrinsic or intrinsic (whatever the theist prefers), and my point seem to go hold just as well.
But perhaps the theist is raising a specific objection to one of my arguments on the basis of one of such distinctions, or you have some other part of my post in mind? If so, please let me know.; June 6, 2016 at 7:45 PM
Fox ITK said...: Thanks for the reply Angra.

The theist I am in discussion with wants to hold that God's state is not and does not change in essential or accidental properties, and even when he is no longer wholly timeless or when he acts on his intention, that these are just extrinsic changes with relation to a change in some other state (like Cambridge Change). That seems wrong to me, the loss of these properties seems like a change in God- even if non essential. Regardless, the same could hold for presumably anything if this is his account of change, making the statement 'God is unchanging' somewhat trivial.

And those changes still need to be accounted for- he seems to think that the change from being timeless sans creation to in time with creation was purely instantaneous despite being a presentist of the kind Craig is, where the 'present' is not made up of durationless instants. I find the notion of the creation event (the creation of time) bizarre if considered simultaneous with its cause- being simultaneous means 'at the same time' are they really saying that time began to exist at the same time that it was caused to exist?

In addition, he seems torn between the idea of 'timeless sans creation' and 'in meta time prior to creation' what are your thoughts if the Craigstian takes this route rather than the former?; June 7, 2016 at 4:51 AM
Angra Mainyu said...: I agree that the theist's position is mistaken: clearly, on the view he (or Craig) describes, God does change from his timeless state to his first temporal state: If God is timeless at state of the world S1 and temporal at S2, then that's a difference between the two states, and so it's a change (and a change in God).
Moreover, God also changes in that at his timeless state, he knows that he is timeless. At his first temporal state, he knows that he is temporal, and that the time is prior to any other time, etc. So, it follows from the description given in the theistic account that God changes (because he is different at different states). If those changes are extrinsic, accidental, very minor, minuscule, the smallest possible changes, or whatever, those are still changes.
Now, if at least one of those changes in God is not caused at all, then it follows from the theistic account that causeless changes are possible. Why then, should one believe that everything that begins to exist has a cause? Moreover, if there is no cause of that change, then God found himself changed - he didn't bring about his own change -, which is a problem in its own right.
On the other hand, if all of the changes in God are brought about (or caused, if you like) by God's decision at the first temporal state of the universe, the the whole change from the timeless state (which is allegedly causally prior to the first temporal state) to the first temporal state (which is allegedly causally later) was in fact brought about by something at the first temporal state, contradicting the alleged causal priority of the timeless state.

So, it seems that the theist is committed to the conclusion that some of the changes were brought about by God's decision at the timeless state. But then, what's the ontological difference between a first timeless state as the causally first state, and a first temporal state (with no timeless state at all) as the causally first state. It seems that despite the use of the word "timeless", the description supports no difference between the alleged timeless state and a first temporal state. Moreover, a change caused at the first temporal state is in turn problematic on the theist's radical presentism.
Side note: I think this sort of presentism can be challenged on other grounds (I think decisively).

Regarding the meta time issue, I would need more details. For example, if he finds that route acceptable for the theist, why might the universe not exist in meta time? How is metatime described?; June 7, 2016 at 7:34 AM
Fox ITK said...: Thanks again for the reply.

The move to a timeless state is being argued to not be a change in God as you state and I concur with, but a change in the world- changes like this are extrinsic apparently (despite this not being defined) in the sense described in this example:

Bill has the property of being a grandson. His grandfather dies and he loses that property, but this isn't a change in Bill, but a change in something else in the world, so despite losing a property, Bill hasn't changed in his being.

All apparent changes are being described as this- he believes the entering into time was simultaneous with creation, but not a change just the loss of a property, or the gaining of a new one in virtue of time existing- this isn't a change in God, so there is then no need to account for how this change was instantaneous.

I find this just bizarre, because clearly a great many changes we want to speak of (not just of God) as changes, aren't just this kind of Cambridge Change, but changes in those beings themselves.

On 'meta time' I agree- I think he risks many of the criticisms leveled against so called 'cosmic time' being raised. Alas he hasn't replied to my attempt for clarity.

As an aside, my discussion with this theist has led me to start reading this article by Craig he has linked to regarding Divine timelessness and personhood ( a case he hasn't made, but he scrambled for this link). I have only just begun to see if Craig is successful, any thoughts on it you might have (or links to good replies on it) would be much appreciated:

http://www.reasonablefaith.org/divine-timelessness-and-personhood; June 7, 2016 at 2:01 PM
Angra Mainyu said...: Regarding Bill's example, I agree that when his grandfather dies and Bill loses that property, that isn't a change in Bill (it's actually a relation between Bill and another person what changes).
But let's say that Bill now learns that his grandfather just died. Bill changes from believing that his grandfather is alive to believing that his grandfather is dead. Bill's mind has changed (his knowledge changed to, but I prefer to avoid issues with the epistemic status of his belief). The change may be small, accidental, etc., but it's still a change in Bill's state of mind.
Granted, the theist may think that God doesn't have beliefs, and that if we talk about God's beliefs, we're speaking by analogy, because we cannot comprehend the mind of God enough to describe it more accurately, but God has an immediate perception of truths, etc. (or something along those lines).
However, that doesn't affect the point about change: in the timeless state, God believes, intuitively and/or immediately perceives and/or apprehends in a godly way (or whatever we call it; let's say "believes*" for short) that the universe does not exist. At time 0, he believes* that the universe does exist, and that time t=0 is present; at time 1 (the next moment in the presentist's account), God believes* that t=1 is present, t=0 is past, and the universe exists. And so on. So, God changes from his state of mind at the timeless state of the world to his state of mind at t=0, and then keeps changing.
By the way, does the theist believe that God does not change even after t=0, t=1, etc.? (I'm asking because Craig believes God does change later).

Regarding your question, thanks for the link. I'm afraid I don't have any links with good replies at hand, and some commitments in meatspace are keeping me too busy these days to make a commitment to analyze his article in sufficient detail to post a reply. So, sorry I can't be of help.; June 7, 2016 at 3:43 PM
Fox ITK said...: Angra, the conversation descended into a complete failure to accept that whatever label you put on non essential properties there are those that are mere relational properties, and those that are non essential properties had by things themselves. But I'll chew on your response regarding God knowing- this particular theist does indeed think God lacks beliefs, and thoughts, but has awareness of 'propositions' I asked if this was conscious awareness, because I have real trouble making sense of that, and again no response came (consciousness without duration?) But still, yes there looks to be problems however it plays out.

On Meta time, I intend to ask more questions, as it has been adopted to avoid many of the problems associated with timelessness and I think he believes that meta time will avoid timelessness and avoid the problem of 'cosmic' temporality. I would think his notion is similar to one Craig is 'sympathetic' to, a sort of fifth dimension orthogonal to the 4th dimension of time (like hypertime). I guess this is undifferentiated dense time or something, to avoid the idea of an actual infinity of discrete states.

I don't think Craig though thinks this would be immune to powerful objections that totally undermine the KCA.

On the 'timeless intention' to create, Craig says this is 'freely chosen'. But if God has always had prior to creation this intent, and never began to exist, at what state did he choose? At the moment the intent was chosen did he create simultaneously, and there was time? But then in what sense was the intention ever timeless? Craig seems to indicate there is indeed a timeless state of intention, but it seems he's committed to two distinct timeless states if he wants there to be a free choice 1) prior to choice 2) when choice is made.

Apologies if these replies take up too much of your personal time- but your articles are very much appreciated and are an excellent rebuttal to the Kalam, I would love more and hope you can find time in the future.; June 8, 2016 at 1:52 AM
Angra Mainyu said...: Regarding God's knowledge, I use it only because it indicates a change in God's beliefs* (whatever those are), i.e., in his mental states.
Knowlegdge per se wouldn't do be enough, since a person's knowledge might change without any actual changes in that person.
For example, Bill knows that his car is in his garage. If the car is stolen while he's at work, Bill no longer knows it, but he hasn't changed.
But when Bill finds out, his beliefs change - and that's a change in Bill, even if accidental, small, etc.
However, God's beliefs* (whether they're beliefs/immediateperceptions/whatever they are) cannot be false, and moreover, he knows everything, so when something changes, so do God's beliefs*.

Btw, the theist you're talking to seems to be denying (even if implicitly) that God is temporal with the universe; he seems to be implying that God is timeless at every state.
That wouldn't work on presentism (as Craig himself argues), since God knows that (say) 1935 is past and 2016 (or rather, a moment in it) is present, but in a century God will know 2016 is past, etc., so his knowledge (and his beliefs*) change. That is an intrinsic change in God, not just in the relations between God and other entities. Craig argues for intrinsic change, for example see this article, though Craig's argument is based on God's knowledge, while I would base it on God's beliefs* corresponding to that knowledge, since knowledge sometimes might change without intrinsic change in some cases (not in God's case, but I think that needs an explanation, which is I why I point to God's beliefs*).
Even if God has no beliefs but awareness of propositions, he is aware at some times that a certain proposition (like "1945 is present") is true, and at some times, he's aware that it is not true, so that's an intrinsic change (i.e., what changes is his state of mind, this "awareness", whatever it is).; June 8, 2016 at 9:45 AM
Angra Mainyu said...: With regard to the timeless free choice, I think Craig would probably reject two different timeless states, one prior to another, since God in his timeless state is allegedly changeless. In the essay above, I challenge that alleged changeless on the grounds that it seems indistinguishable from a first moment in time, etc., but his position seems to be that God always intended to create in the timeless state; he probably would deny that a free choice needs a time or state at which it hasn't been made.
I don't have time for a full argument, unfortunately, but here's a consequence of some of his points: in the timeless state, according to Craig, knows the proposition "Ganssle types on December 14, 2006." (which is tenseless); see this IEP entry. That requires a certain (even if timeless) mental state (God's timeless mental state at a world where he doesn't create anything would be different from that in the actual world), and at every world with the same initial segment as the actual world (in the causal order), God has the same mental state (since the timeless state is the causally first state of the world), and so God knows that Ganssle types, and so Ganssle then types. It follows that the actual world is the only possible world with the same initial segment as this world, in the causal order (even though according to Craig, the world is not causally determined). I suppose that someone might say that God's mental state in the timeless state does not count as part of the initial segment in the causal order, but I'd say there is no good reason not to count it.
At any rate, even if the argument from the causally same initial segment, we have the following argument: any world that shares a temporal initial segment with the actual world, is the actual world (that follows from God's mental states and his infallible foreknowledge; there is an asymmetry in the case of humans, so I think a similar case wouldn't work).

P.S.: thanks for the appreciation of my articles. I don't think I'll be able to write anything long and detailed for the foreseeable future (I'm also focusing on other things), but maybe some day I will.; June 8, 2016 at 9:45 AM

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Sunday, January 6, 2013

A Reply to the Kalam Cosmological Argument (v. 7.4.1)

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