Room Service

A Response to Loke and Haitov

Recently, Andrew Loke and Eli Haitov published an article in the journal Analytic Philosophy in which they present an ostensibly novel argument involving Hilbert’s Hotel against the possibility of Past-Eternalism (PE)—the view that there are infinitely many past events.¹ This is another philosophical argument in a series of arguments that have been used to support the Kalam premise that the universe had a beginning.²

However, as I will argue, Loke & Haitov’s reasoning is fundamentally flawed. In §1, I outline their argument. In §2, I demonstrate why it fails, and in §3, I conclude with some final remarks.

1. No free lunch

The argument concerns a variation of Hilbert's Hotel. We are to imagine a hotel with infinitely many rooms, each having one meal in it. We can label the rooms r₁, r₂, … for all natural numbers. Infinitely many guests arrive at the hotel, and we can label them g₁, g₂, … for all natural numbers. Each guest gₙ is assigned room rₙ (that is, g₁ is assigned r₁, g₂ is assigned r₂, and so on). When a guest checks into a room, they then eat the meal in that room. Accordingly, if each guest checks into their assigned room, each will eat a meal and there will be no meals left over. However, we can suppose that each guest has the power to check into another room instead. We can express their argument as follows:

1. Necessarily, the causal power of a group of x for causing extra y depends on individual x having the causal power to cause extra y.³

2. Necessarily, if x has no causal power for causing extra y, then an infinite number of x would have no causal power for causing extra y. (From 1)

3. A guest who checks into one room rather than another has no causal power to cause extra meals.⁴

4. Necessarily, an infinite number of guests, each of whom checks into one room rather than the other, would have no causal power for causing extra meals. (From 2,3)

5. If Hilbert’s Hotel is possible, then it’s possible that guest n checks into room 2n, for all of the infinitely many guests.

6. If it’s possible that guest n checks into room 2n, for all of the infinitely many guests, then the infinitely many guests would have the causal power for causing extra meals.

7. Hilbert’s Hotel is not possible. (From 4-6)

8. If Past-Eternalism is possible, then Hilbert’s Hotel is possible.⁵

9. Therefore, Past-Eternalism is not possible. (From 7,8)

The authors proceed to argue that while their reasoning is sufficient to rule out the possibility of Past-Eternalism, it leaves open the possibility of an endless future (or what we might call “Future-Eternalism”). While it might be argued that their argument does admit of future-oriented parodies, I will not explore that possibility here. In the following section, I will discuss why Loke & Haitov’s argument fails.

2. Making a meal out of it

2.1 Contributory causes

According to the first premise, if a group has the power to cause y, then each member of that group must also have the power to cause y. However, there is an ambiguity in what is required for an individual to have that causal power. After all, we often talk of contributory causes, where any particular contribution does not produce the effect on its own, but does in coordination with other contributory causes. For example, I might contribute causally to the movement of a large boulder by pushing it, but the boulder ultimately moves because of the concerted effort of myself and others. In other words, the individual acts of pushing are contributory causes of the boulder moving, and the pushings jointly are a sufficient cause to produce the effect. This is a simple case of distributed causation.

If we understand causal powers in this way, the individual guests do have the causal power to cause extra meals.⁶ After all, individual guests might contribute causally to the result that each gets a meal and that meals remain by coordinating their room selection in the way Loke describes. Each guest was a contributing cause here in the same way that each of the people pushing on the boulder were contributing causes of the boulder moving. While no individual guest can guarantee extra meals, each guest’s room selection might contribute to a room selection pattern that results in extra meals, and so contribute to causing extra meals.⁷ Accordingly, premise (3) is false, and so the argument is unsound.⁸

2.2 Rejoinder 1: the causal power requires more than contribution

It might be responded that contributory causes aren’t really causes in the intended sense. For example, we might stipulate that x having the relevant causal power requires x be able to bring about the given effect by itself rather than merely as part of a coordinated effort. In this sense, I would not have the relevant causal power to move the boulder since, we suppose, I could not move it on my own.

However, there is an obvious problem with this rejoinder: the premise understood this way is plainly false, since it admits of clear counterexamples. Groups routinely produce effects that their members cannot individually cause (in the relevant sense). In the boulder case, for example, while none of the individual pushers have the relevant causal power to move the boulder, the group together is able to achieve that result. In the HH case, while none of the individual guests have the relevant power to cause extra meals, the group does, since the group could pattern its room selection to achieve that result. On this way of understanding causal powers, premise (1) is false, and so the argument is unsound.

2.3 Rejoinder 2: the guests aren’t even contributory causes

The authors might instead respond that being a contributory cause is enough to have the relevant causal power, but that the guests in HH do not have even that power. While people pushing in the boulder case is causally relevant to the boulder moving, guests swapping rooms in HH is not causally relevant to causing extra meals. In the boulder case, the pushers individually contribute some non-zero amount of force to make the boulder move. As such, there’s a sense in which they contribute a non-zero percentage to producing the resultant effect. In HH, however, the guests contribute exactly 0% to causing extra meals. And no matter how many people we have each contributing 0% to causing extra meals (even infinitely many), they cannot cause extra meals. Rather, the swapping is as relevant as dreaming about extra meals; that action has no bearing on whether there are extra meals.

Based on remarks from their paper and subsequent replies from Loke, I suspect that this is how they would reply to my critique. While I find this talk of percentage causal contribution a bit problematic,⁹ there is a more basic error in this reply. Something having 0% causal contribution does not entail that it has no causal contribution. Indeed, the HH case is one where the individual guests contribute 0% to causing extra meals but still contribute to causing extra meals.

This may sound incoherent, but it is not. Consider the HH scenario where each guest checks into their assigned room (i.e., nobody swaps) and eats the corresponding meal. The guests collectively have caused all of the meals to be consumed, but how much did each guest contribute to this result? Since each guest ate one meal of infinitely many meals, each contributed 0%.¹⁰ However, consuming a meal is causally contributing to the result of consuming all of the meals, and this is true no matter how many meals are consumed in total. And of course, it would be wrong to infer from the fact that each guest contributes 0% to this result that the group couldn’t collectively consume all of the meals.

More generally, we can coherently suppose that causation is distributed among infinitely many contributions such that each contribution is effectively 0% of all of the contributions. As this example shows, that does not require that the individual contributions are not contributions or that the contributions together could not produce the effect. In Loke & Haitov’s HH scenario, each guest contributes 0% to producing a room selection pattern that results in extra meals (and so contribute 0% to causing extra meals), but they still contribute to producing a room selection pattern that causes extra meals (and so contribute to causing extra meals).¹¹ In the same way that consuming one of the infinitely many meals contributes to consuming all of the meals, selecting a room that’s part of a room selection pattern that results in extra meals contributes to causing extra meals. In both cases, this is so even though each contribution contributes 0% to the respective effect. Accordingly, (3) is false and the argument is unsound.

If proponents of the argument still resist this conclusion, I would then question whether we have a shared understanding of the relevant terms, and would seek clarity in order to try to reach resolution. Regardless, if they’re employing a sense of “causal power” given which the guests in HH do not have the relevant causal power (but that contributory causes in other cases do), I think that premise (1) would be false. After all, the group is able to cause extra meals by patterning its room selection in the right way, and this is so regardless whether on some sense of “causal power” the individual guests count as having the power to cause extra meals. If they do not count as having that causal power, then the HH case is a counterexample to (1).¹² In that case, (1) is false and the argument is unsound.

3. Conclusion: better Loke next time

In sum, while there’s a potential ambiguity in how we’re understanding the relevant causal powers, however that ambiguity is resolved, the argument fails. If we count contributory causes as causes of the relevant sort, then premise (3) is false since the guests in HH do have the relevant causal power, and this is so even though each contributes 0% to the final result. If we do not count contributory causes as causes of the relevant sort, then premise (1) is false since it admits of clear counterexamples (e.g., the boulder case). If causal powers of the relevant sort are somehow understood in a way given which the guests in HH don’t have the relevant causal power but the other counterexamples do not apply, then (1) is still false since the group nevertheless has the relevant causal power even if the individual guests do not. As such, either (1) is false or (3) is false, and so the argument is unsound.

If you enjoy posts/content like this, consider checking out my YouTube channel, www.youtube.com/@Friction. If you’d like to contribute to making more content like this (and getting some perks), you can check out my Patreon (much thanks to the current patrons!). All of my links can be found here. Anyway, thanks for reading, and I hope you got something out of it!

1. This is a slightly strange way to define “past eternalism” since it’s compatible with the past being finite in temporal extent. Regardless, we can proceed with this understanding.

2. It may be objected that the impossibility of PE would not entail that the universe had a beginning since, for example, it would remain possible that metric time was preceded by non-metric time involving finitely many events but not a strict beginning.

3. Loke & Haitov stipulate that “x” is an action. However, this is a bit strange and doesn’t align well with their later remarks. Instead, it makes more sense to let x denote individuals, since individuals—rather than token actions—have causal powers. Also, I have dropped the word “relevant” where it appears in the argument, as I find it superfluous.

4. Here, they understand the causal power to cause extra meals as the power to make it the case that otherwise unavailable rooms/meals are available after each guest has consumed a meal.

5. They offer more detail here, but it’s not essential to my analysis.

6. Recall, “causing extra meals” does not require cooking a new meal, or anything like that. All that it requires making it such that meals remain after each guest has selected a room and consumed the corresponding meal.

7. It might be observed that talk of extra meals is unnecessary, and that we could focus instead on extra rooms. This is right, although we need to be clear that causing extra rooms in the relevant sense requires making it the case that although each guest has their own room, empty rooms remain. I will leave my analysis in terms of causing extra meals, although note that nothing relevant is lost if we frame the discussion merely in terms of causing extra rooms.

8. This is premise 1.3 in Loke & Haitov’s article.

9. I think there are cases where contributory causes cannot be meaningfully quantified in terms of percentage contribution, although I will not discuss that here.

10. Someone may wish to invoke infinitesimals to model the contributions here, although that isn’t necessary to make the point.

11. Technically, the room selection pattern alone doesn’t produce that result, but that with the fact that each guest eats the meal in its selected room.

12. Premise (1) may be suspect even on the most charitable interpretation, since it’s not obvious to me that in order for some collection to be able to produce some effect, individual members of that collection must have the causal power (even in a contributory sense) to produce that effect. Why not suppose that, at least sometimes, the causal power exists only at the collective level and not at all in the individuals? I will not explore this possibility further here, but it’s worth noting that this is another potential shortcoming of Loke & Haitov’s argument.