An Argument for Finitism, Part 2: Objections

In my last post I gave an argument for (metaphysical) finitism based on the absurdities and logical impossibilities that would result if an actually infinite number of things could exist in reality. I used the Hilbert's Hotel thought experiment. I also pointed out that while there are prohibitions in mathematics on performing operations like subtraction when dealing with actual infinities, those prohibitions would not apply in a real-world scenario. So it would seem that finitism is correct, or at least that it has a great deal of plausibility.

Now I would like to briefly consider some objections to the argument. I know of a number of objections that could be made, but I want to focus on two in particular because they seem to be the most common objections that come up, and other objections often turn out to be variations of these two. Each of the objections we will consider claims that finitism is false because we can point to examples of actual infinities that exist in reality.

Objection #1: The argument from mathematical realism

The first objection is based on the fact that there is an infinite number of objects involved in the realm of mathematics, things like numbers and sets. For instance, there are infinitely many natural numbers. But if there are infinitely many natural numbers, then finitism is false. For clarity, let's write this out like an argument:

1. If there are infinitely many natural numbers, then finitism is false.
2. There are infinitely many natural numbers.
3. Therefore, finitism is false.

The main problem with an argument like this is that it makes a very big assumption about what sort of existence is involved when saying that an infinite number of natural numbers exists. This is where it is important to distinguish between mathematical existence and metaphysical existence. Mathematical existence simply concerns whether the object in question—say, the set of all natural numbers—has a legitimate role to play in mathematical discourse. But this does not necessarily imply that the object exists in a metaphysical sense. The inference from mathematical legitimacy to metaphysical existence would need to be justified, especially in light of the fact that there are a variety of antirealist views about mathematical objects.

It is also important to be clear about what type of finitism we are talking about. It is possible to be a mathematical finitist, in which case one denies the mathematical legitimacy of the concept of an actual infinity, but here we are more interested in the status of metaphysical finitism. I am personally not interested in defending mathematical finitism, and I am inclined to be suspicious of it.

In short, the truth or falsity of premise (1) depends on what we mean when we say "there are infinitely many natural numbers" and on what type of finitism we are talking about. Metaphysical finitism does not seem to follow from the fact that infinity has a legitimate role to play in mathematical discourse. So I would simply reject the first premise. Then it's up to the mathematical realist to provide some reason for why we should accept the realist view.

One potential argument for realism is that an antirealist view based on the metaphysical impossibility of an actual infinity would imply that mathematicians are studying impossible scenarios or ideas. In that case, it is hard to explain how their work could have so much applicability to real life. Surely if there is such a strong correspondence between the mathematical realm and the physical world, then we can't deny the reality of mathematical objects. Doesn't this undermine the argument for finitism?

Every time I hear this kind of argument used in regard to infinities, I always think about negative numbers. I don't think it would be very wise to deny the mathematical legitimacy of negative numbers. It is hard to imagine trying to make sense of the physical universe without using negative numbers (think of the negative charge of electrons), and negative numbers also have great practical value (for instance, they help us to keep track of our debt, although we might wish they didn't). But in spite of the obvious mathematical legitimacy of negative numbers, it doesn't seem like a negative number of things could possibly exist in reality. An overdrawn bank account might show a balance of -$50, but that doesn't mean that there's a room somewhere containing negative fifty dollar bills. There are no real objects in the world corresponding to the negative number. We might use negative numbers in a temperature scale, but the scales are completely arbitrary (as indicated by the fact that there are different scales available). If a thermometer says it's -3 degrees, that doesn't mean there are negative three objects in the world to which those numbers correspond.

My point is that the mere fact that negative numbers have a valuable role to play in mathematical discourse is no proof that they exist in a metaphysical sense. In the same way, the legitimacy of the concept of the actual infinite is no proof that an actually infinite collection of things could exist in real life. So the objection from mathematical realism (which is really an objection from the applicability of mathematics) does not do anything to undermine the argument for metaphysical finitism based on Hilbert's Hotel.

Objection #2: The argument from points and instants

The second objection to metaphysical finitism is based on the claim that, in any given distance there is an infinite number of points, and in any duration of time there is an infinite number of instants. So say the length of a room is measured to be six meters. To walk across the room, you have to cross an infinity of halfway points. To get half way, you have to get a quarter of the way. But to get a quarter of the way, you have to get an eighth of the way. And so on.

It is the same with time. If time is composed of instants, then actual infinities do exist in reality, because an instant is just an infinitesimal moment in time, and between any two instants there will always be another instant. One way to prove this, again, is by dividing the length of time in half an infinite number of times. So the argument against finitism would go like this:

1. If finitism is true, then time could not be composed of instants, and space could not be composed of points.
2. But time is composed of instants and space is composed of points.
3. Therefore, finitism is false.

However, the finitist will simply reject the second premise of the argument. The skeptic of finitism cannot just assume that time and space are composed (respectively) of instants and points, otherwise they are just begging the question against finitism. After all, the argument for finitism would serve well as an argument against the truth of premise (2). Beyond that, there are good reasons to be skeptical of premise (2) anyway, because if we take it to be true, we encounter various paradoxes.

Consider the Grim Reaper paradox. You are alive at 12:00 PM. If you are still alive at 1:00 PM, a grim reaper will appear and strike you dead (tough break). If you are already dead, it will do nothing. However, if you are still alive at 12:30 PM, a second grim reaper will appear and strike you dead. And if you are still alive at 12:15 PM, a third grim reaper will appear and strike you dead. For each successive grim reaper, we assign them to kill you exactly halfway between 12:00 PM and the time assigned to the previous grim reaper.

What makes this such an interesting paradox is what happens if we keep doing this with an infinite number of grim reapers. Since grim reaper #3 will strike you dead at 12:15 PM if you're still alive, then grim reaper #4 will strike you dead at 12:07:05, grim reaper #5 will strike you dead at 12:03:52.5, and so on. The more grim reapers we add, the sooner after 12:00 PM your death should occur. And since there is an infinite number of grim reapers waiting to strike you down, the assigned time of your death will grow infinitely close to 12:00 PM.

But the end result is a contradiction: you cannot survive past 12:00 PM, and yet there is no grim reaper who can possibly kill you, because for any grim reaper we specify (grim reaper #n), we can always imagine another grim reaper (grim reaper #n+1) who should have already killed you halfway between 12:00 PM and grim reaper #n's assigned time. You can't be killed at any time after 12:00, and yet you can't survive past 12:00 either. Another way to say it is, you can't possibly survive to 1:00, but there is no time before 1:00 when you can possibly be killed.

Since the infinite grim reaper scenario results in a logical contradiction, it follows that it cannot be possible. But now we need to ask what makes it impossible. There is nothing logically incoherent about the idea of a grim reaper being assigned to kill someone at a specific time. Nor is there anything incoherent about the idea of a series of grim reapers being assigned to different times at which they will check if a person is still alive, and kill them if they are. With a finite number of grim reapers, you would simply be killed by the earliest grim reaper and then the rest of the reapers would not have to do anything. So the logical impossibility of the scenario described above must stem from the idea that time is composed of an infinite number of instants. And similar impossibilities arise from assuming that space is composed of points.

Now, I used to have trouble with this argument for the following reason. Take a physical object like a ruler and imagine cutting it in half. It seems perfectly possible to specify the exact halfway point on the ruler, right? And then you could specify a halfway point on one of the halves, and then you could specify a halfway point on one of those littler halves, and so on. And the thing that troubled me was that the point where you would make the cut already seems to exist before you get there, does it not? In that case, how can there not be an infinity of points on the ruler at which a cut might be made?

But this turns out to be misguided on two counts. First, to go on dividing something in half infinitely many times implies a potentially infinite process, not an actually infinite collection of points. If you're looking at it like a process, then the fact is that you will never end up making an "infinitieth" division, because you can't convert a potential infinity to an actual infinity one step at a time like that.

Second, the objection confuses mathematical divisibility with physical divisibility, and this is a crucial distinction. Mathematically, of course you could go on making divisions infinitely. But as we saw in our discussion of the first objection, this has no bearing on the question of whether the points at which you could make those divisions actually exist. Physically, it would be impossible to go on making divisions like this, not just because you could never complete an infinite number of divisions by doing one at a time, but also because it's not physically possible to do that with any object, even if there were enough time. It is conceptually possible, but this has no bearing on the question of whether there really are an infinite number of points at which you could make a physical division. So when we talk about making an infinite number of divisions at an infinite number of points, we are talking about a process involving an abstraction of a real thing. We imagine an infinity of points in our minds, but these points do not correspond to objects that exist in reality.

As a result, this second objection also fails. Not only does it just assume that time and space are composed (respectively) of instants and points, but the assumption turns out to be logically impossible in light of paradoxes like the grim reaper paradox.

Final Thoughts

It seems that the argument for finitism based on Hilbert's Hotel is a good one. The two objections based on mathematical realism and the reality of points and instants don't seem to do anything to overturn it, since they are each based on assumptions that are vulnerable to devastating criticisms. And what I find most interesting about these two objections is that neither of them even tries to resolve the Hilbert's Hotel paradox. Rather, it seems that we are just expected to bite the bullet and say that, actually, the absurdities of Hilbert's Hotel are possible after all. But this is very hard to take seriously.

The objector might reply that the argument for finitism based on Hilbert's Hotel relies too heavily on our own personal intuitions, and intuitions are not always reliable guides to truth. But the problem with this response is that, as we saw, in real life there is no way to prohibit operations of subtraction when dealing with infinities like there is in mathematics. People can check out of hotels if they want to! As a result, a rejection of finitism would force us to conclude that logical contradictions can exist in reality. If that's where the argument takes us, then something has gone very wrong, because then there would be nothing to stop us from saying that metaphysical finitism is both true and false at the same time. So I think we have a very powerful justification for embracing metaphysical finitism.