Alexander Pruss's Arguments Against Finitism, Part 1

I have been slowly making my way through Alexander Pruss's Infinity, Causation, & Paradox (Oxford: Oxford University Press, 2018). This book is interesting and at times very technical, so it is often difficult for me to understand, but paradoxes related to infinity are one of my favorite topics, so I find myself going back to this book every so often and giving it another try.

Pruss's main goal in the book is to defend a position called causal finitism, which holds that an event (or "output") cannot be preceded by an infinite series of causes. Another way to say it is that no event can have an infinite causal history. In order to set the stage for his discussion, Pruss attacks a more general view simply called finitism (pp. 10–18). Finitism is the view that there can only be finitely many things, whether they be causes or anything else.

I find myself quite attracted to metaphysical finitism, as I've explained in a couple of recent posts, and I have always been puzzled by Pruss's animosity toward the view. In the next couple of posts or so, I will try, to the best of my ability, to articulate his various arguments in a step-by-step fashion. Listing the steps of each argument is a helpful way for me to think more carefully about it, so I have tried very hard not to misrepresent Pruss's claims, since I don't want to attack any strawmen. After presenting each argument, I will explain why I do not find any of them to be persuasive.

In this discussion it will be important to keep in mind the distinction between actual infinity and potential infinity. A set is actually infinite if it contains an infinite number of members (or, better yet, if the members of any of the set's subsets can be placed in a one-to-one correspondence with the members of the whole set). A collection is potentially infinite if it is finite but increasing in number successively and indefinitely. The set of all natural numbers is actually infinite, but if you are counting the natural numbers one at a time without stopping, then at any time, the collection of numbers you've counted can only be potentially infinite.

1. The Argument from Time (see pp. 10–11)

1.1. Introduction

In order to understand this argument, we need to differentiate between three different views of time. These three views are called eternalism, presentism, and growing-block theory. 

Eternalism holds that all events in the temporal series are equally real, whether these events exist in the past, present, or future. For this reason, eternalism is compatible with the view that everything in time and space exists as part of a four-dimensional "block." On this view, the whole block exists, and our differentiation between which events are past, present, or future is simply a matter of where in the block we are located. Another way of saying this is that the block exists "tenselessly." Temporal becoming (in which things come into being and cease to exist) is not an objective feature of reality. 

Presentism holds that the only things that exist are those that presently exist. Past and future events do not exist, and the present is constantly changing. Temporal becoming is thus an objective feature of reality, and reality is tensed rather than tenseless. 

Growing-block theory affirms the existence of past and present events but denies the existence of future events. On the one hand, this view agrees that everything in time and space exists as part of a four-dimensional block. On the other hand, this view also agrees that temporal becoming is real and that the present is constantly changing. The combination of these two ideas implies that the spacetime block is constantly growing as new events are added.

1.2 The Argument

1. If finitism cannot be combined with eternalism or presentism, then if finitism is true, the growing-block theory of time is true.
2. Finitism cannot be combined with eternalism or presentism.
3. Therefore, if finitism is true, then the growing-block theory of time is true. (1, 2)
4. Growing-block theory is not true.
5. Therefore, finitism is not true. (3, 4)

1.3 Response

Since eternalism, presentism, and growing-block theory are the only available views of the ontological nature of time (at least that I know of), premise (1) seems correct.

Jumping ahead, I am also inclined to agree with premise (4), though maybe not for the reason Pruss gives. Part of the reason I don't find growing-block theory to be plausible is that I find it somewhat incomprehensible. I can understand to some degree a block universe that just exists tenselessly, and I can understand the idea that temporal becoming is real, but the hybrid view of the growing block seems fundamentally incoherent. How can there be real temporal becoming that results in a (mostly) tenseless block? If most of the block is tenseless, why is it constantly in flux at one edge?

Furthermore, it seems to me that growing-block theory is susceptible to one of the biggest criticisms I have of the block universe view in general: the idea that people exist as four-dimensional "worms" (not my word, for the record) rather than as three-dimensional persons. On a four-dimensional view, I am not a person enduring consciously through time; rather, I am a four-dimensionally extended thing that lacks consciousness (i.e., I am not aware of myself as a four-dimensionally extended thing), and the part of me that I perceive as existing "now" is not really a full person, nor is it identical to the part of me that I perceived as existing "now" five minutes ago, which implies that the person I am now, to whatever degree it exists, is literally not the same person that existed five minutes ago, much less twenty years ago. Technically, it's not even really a person. And I'm not even sure who this "I" is that seems to perceive "myself" as a person who exists now. There's a lot more that could be said about all of this, but suffice it to say, I do not affirm the four-dimensional view of time because it seems to be at stark odds with things that we know to be true about ourselves.

It seems that the only hope for finitism, then, will be to reject premise (2). But I think this premise is very easy to reject. In fact, it seems to me that finitism can very easily be combined with either eternalism or presentism.

Pruss argues that finitism and eternalism are incompatible because the conjunction of the two would suggest that the future is finite, whereas it is "surely" possible to have an infinite future, "say with a new toy soldier being produced every day forever" (p. 10). Pruss simply asserts this, but I'm not sure why a finitist could not just believe that there will be a last event, bringing the series of temporal events to an end. It would at least be a perfectly coherent view. Nevertheless, since I do not accept eternalism anyway, it makes no difference to me.

Obviously, then, I affirm a presentist view of time. So why doesn't Pruss think that finitism can be plausibly combined with presentism? Pruss is largely concerned with various paradoxes that arise if we try to affirm an infinite series of causes, things like the Thompson's Lamp paradox (where a switch for a lamp is toggled an infinite number of times in an hour; at the end, will it be on or off?), and the Grim Reaper paradox, which I talked about in my last post. And his main complaint here is that "finitism plus presentism does nothing to rule out the infinitely many togglings of the switch in Thompson's Lamp." Why not? Because he says, "Given presentism. . . finitism is compatible with infinite sequences of causes, as long as at no particular time are there infinitely many causes" (p. 10). This, I think is where Pruss completely misses the mark.

On a presentist view of time, every past event is an event that has already been instantiated in reality. In other words, even though past events do not presently exist, each past event once did exist (i.e., there was a time when it was present). For the presentist, this is what distinguishes past events from future events, because future events have never existed, whereas past events have existed. So if the series of past events is infinite (i.e., by having no beginning), then finitism is false, because there have been an actually infinite number of events.

On the other hand, if it's metaphysically impossible for an actually infinite number of things to exist, as the finitist believes, then an actually infinite number of events cannot be instantiated in reality, even in a sequential order. It has nothing to do with which events are presently happening. So if finitism and presentism are both correct, then it follows that the number of past events can only be finite. There is no logical incompatibility here. And it also follows that an infinite series of causes cannot exist, which means that there couldn't be an infinite series of switch togglings or grim reapers.

2. The Argument from Hilbert's Hotel (see pp. 11–13)

2.1 Introduction

I did a very recent post about Hilbert's Hotel, so I won't repeat the whole discussion here. I'll just summarize the thought experiment. Imagine a hotel with an infinite number of rooms. Even if all of the rooms are occupied, the proprietor could accommodate any new guests by shuffling the current guests around in different ways. For instance, if one new guest shows up, the proprietor can just have everyone move to the room with the number that comes directly after their current room number. The guest in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on, and the new guest takes room 1. Now suppose that an infinity of new guests shows up. The proprietor can accommodate them too, by having all the current guests move to the room with the number that is twice their current room number. The guest in room 1 moves to room 2, the guest in room 2 moves to room 4, the guest in room 3 moves to room 6, and so on. In this way, all the odd-numbered rooms are cleared, and all the new guests can check in. The proprietor could do this an infinite number of times. But this seems strange because prior to the arrival of each new guest, all the rooms in the hotel are occupied.

2.2 The Argument

1. If finitism is true, then Hilbert's Hotel is impossible.
2. Hilbert's Hotel is not impossible.
3. Therefore, finitism is false.

2.3 Response

Hilbert's Hotel has often been used as a way of showing how it would be impossible for an actually infinite number of things to exist, because then you could end up with a hotel where new guests can always be accommodated even though all the rooms are full, and no matter how many new guests check in, there will always be the same number of guests staying in the hotel. Pruss objects that, while Hilbert's Hotel is strange, it may not be absurd. After all, the platypus is strange, but it exists! In order for Hilbert's Hotel to be absurd, we would need to show that it's impossible. But all the thought experiment shows, according to Pruss, is that "infinity is roomier than we previously thought" (p. 12).

But this response will not do. First, the problem with always being able to accommodate new guests without increasing the total number of guests in the hotel is not that it's logically impossible, but that it seems to be metaphysically impossible. Granted, this part of the argument cannot be proven since it appeals to our intuitions about what is real and possible, but it does seem like there is an obvious difference between the strangeness of the duck-billed platypus and the strangeness of always being able to accommodate new guests in a fully occupied hotel.

Second, and more importantly, there is a much bigger problem with Hilbert's Hotel that Pruss doesn't really contend with: what happens when the guests start to check out? Pruss briefly acknowledges that "[y]ou could even have infinitely many people vacate the hotel and still have it full. If all the people in the odd-numbered rooms leave, you can tell each person in the even-numbered rooms to move to a room whose number is half of their room number" (p. 11). But what Pruss doesn't even mention is that operations like subtraction are undefined when it comes to infinite values precisely because it would otherwise lead to logically contradictory conclusions. For instance, if you have an infinite number of guests, and all the guests in even-numbered rooms check out, you are left with an infinite number of guests (infinity minus infinity equals infinity). If all the guests check out, you will have no guests left (infinity minus infinity equals zero). And if all the guests in the rooms numbered 4 and higher check out, you will be left with three guests (infinity minus infinity equals three). In each case, we subtract infinity from infinity but end up with a different result. This is a real contradiction, and while you can prohibit operations like subtraction when dealing with infinities in a mathematical context, in a real life hotel, guests have a tendency to check out.

Since a hotel with an infinite number of rooms would lead to logical contradictions, it follows that the hotel can't exist in real life. Why not? The answer seems to be that an actually infinite number of things cannot exist in real life, since it would lead to logical contradictions. So a finitist could simply say that premise (2) of the argument is false.