Alexander Pruss's Arguments Against Finitism, Part 2

I'm continuing my discussion of Alexander Pruss's Infinity, Causation, & Paradox (Oxford: Oxford University Press, 2018). Specifically, I am looking at his arguments against finitism.

3. The Argument from Infinitely Many Primes (see pp. 13–15)

3.1 Introduction

Pruss actually makes two arguments here, so I will need to talk about both of them. The first argument is very simple.

3.2 The First Argument

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

3.3 Response

In order for this argument not to be question-begging, we need to know what it means to say that "there are" (or "there exist") infinitely many prime numbers. In what sense do these numbers (or any numbers) exist, if they exist at all? The problem here is that there are different realist and antirealist views of mathematics. Pruss seems to think that acknowledging that the set of prime numbers has an infinite number of members commits us to saying that the set of prime numbers exists. But this is a controversial assumption and it needs to be justified. It may be that prime numbers do not exist, but that it's perfectly legitimate to treat them as if they do for the purposes of doing mathematics. In fact, if there are good arguments for finitism, they would seem to be good arguments for taking an antirealist view of numbers.

Thus, the finitist's response to the first argument will just depend on what Pruss means by "there are infinitely many prime numbers." If saying "there are infinitely many prime numbers" commits us to the metaphysical existence of numbers, then the finitist has the option of accepting premise (1) while denying premise (2). In that case, the argument leaves open the possibility that mathematical discourse does not require us to affirm the real existence of numbers (prime or otherwise). On the other hand, if saying "there are infinitely many prime numbers" is meant to convey that the set of prime numbers (which has an infinite number of members) is a legitimate mathematical concept, then the finitist can accept premise (2) while denying premise (1).

In short, Pruss needs to justify his claim that recognizing the mathematical legitimacy of infinite sets implies their existence. In fact, Pruss recognizes that he is interpreting "there are infinitely many prime numbers" in a Platonist way (Platonism being the view that abstract objects like numbers really exist), and he admits that there are other options besides Platonism. But he does not think that the finitist can avail herself of these antirealist options. This is where his second argument comes in.

3.4 The Second Argument

1. If finitism is true, then nothing with the structure of infinitude can exist.
2. If nothing with the structure of infinitude can exist, then mathematics involves the study of impossible situations and structures.
3. If mathematics involves the study of impossible situations and structures, then it has no applicability to the real world.
4. Mathematics has applicability to the real world.
5. Therefore, finitism is false. (1-4)

3.5 Response

Given that this argument is intended to directly address the possibility of combining finitism with an antirealist view of mathematical objects, we can assume that premise (1) is using the word "exist" in the sense of being instantiated in mind-independent reality (that is to say, reality that is independent of human minds). In that case, the finitist can accept the first premise.

Premise (2) slips in the word "impossible" without explaining what sort of possibility we're talking about. This is a crucial oversight on Pruss's part, because the success of his argument depends on what he means. If something can exist in mind-independent reality, then we would say that it's metaphysically possible. Anything that is metaphysically possible is also logically possible, but not everything that's logically possible is metaphysically possible. For instance, there's no logical incoherence involved in saying that my brother is a wooden desk, but this is not metaphysically possible. Or here's an example from a theistic perspective: There's no logical contradiction involved in saying that God wants us to abuse and torture children, but since God is morally perfect, such a scenario is metaphysically impossible (that is, if God exists, God cannot want us to do such things, because that would violate God's perfect goodness).

When it comes to mathematical possibility, we seem to be talking about logical possibility. A mathematical formula cannot be true unless it is logically coherent and strictly logically possible, but that doesn't automatically mean that the formula refers to objects that exist in the real world. To repeat a point I used in a previous post, 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.

Thus, the truth of premise (2) depends very much on whether we are talking about logical or metaphysical impossibility. If we are talking about logical possibility, then the premise is certainly false. Remember, some operations in mathematics are undefined precisely because they result in meaningless or logically contradictory results. This is why you would be prohibited from subtracting infinity in a mathematical context. So mathematics operates within certain guidelines specifically to stay within the realm of strict logical possibility.

However, if premise (2) is talking about metaphysical impossibility, then the premise seems true, at least more plausibly true than false. After all, prohibitions in mathematics do not result in metaphysical restrictions. Even if you are prohibited from subtracting infinity in mathematics, subtractions occur in reality, and if a hotel has an infinite number of guests, there wouldn't be any reason why all the guests couldn't check out. But this is precisely why a hotel with an infinite number of guests seems metaphysically impossible.

This problem carries over to premise (3). If we are talking about logical impossibility, then the premise is obviously true. Logically impossible situations have no bearing on reality. For instance, there is no possible version of reality where you can add only two kittens to a box already containing only two kittens and end up with a total of seven kittens, as delightful as that would be. But if we are talking about situations that are possible logically but not metaphysically, then it seems like those things could still have great applicability to the world. For instance, it is metaphysically impossible to subtract four kittens from a box containing two kittens and end up with negative two kittens. Yet this doesn't mean that the structure of reality is unaffected by the fact that two minus four equals negative two. So if premise (3) is talking about metaphysical impossibility, then the premise seems doubtful in light of the fact that premise (4) is undeniably true. As a result, neither of Pruss's arguments from infinitely many primes is successful.

4. The Argument from Potential Infinity (see p. 15)

4.1 Introduction

Finitists do not deny that a potentially infinite collection of things can exist. For instance, if a finitist believes in an afterlife that never ends, and if they adopt a presentist view of time, then the number of days they experience will continue to increase toward infinity as a limit, but at no point will they have experienced an infinite number of days. So the series of temporal events would be a potentially, but not actually, infinite collection. On the other hand, it is part of finitism to deny that actually infinite collections can exist.

4.2 The Argument

1. Since finitism can't affirm that infinite sets (such as the set of all natural numbers, or the set of all prime numbers) are actually infinite, then if finitism is true, it must be the case that infinite sets are only potentially (but not actually) infinite.
2. But some sets are actually infinite (e.g., powersets, like the set of all sets of natural numbers).
3. Therefore, since finitism can't affirm that infinite sets are actually infinite, finitism is false.

4.3 Response

Pruss seems to be interacting here with an approach to mathematical objects called intuitionism. If I have understood correctly, intuitionists would reject the claim that the concept of an actual infinite has mathematical legitimacy. Instead, they only affirm potential infinities. They would thus deny that the set of natural numbers is actually infinite. My understanding is that they opt for viewing natural numbers as a potentially infinite set or collection.

The problem with this argument is that it can't even get off the ground unless the earlier arguments work, and we have already seen why they do not. Pruss continues to write as if finitists have to accept Platonism (that is, mathematical realism), in which case it would be true that they would have to end up embracing something like intuitionism. But this is very strange, since in a later section he seems to be aware that finitism would imply the falsehood of Platonism: "Obviously, finitism undercuts mathematical Platonism" (15). So what's wrong with taking arguments for finitism as arguments against Platonism?

Before going further, I want to explain why I don't find the combination of finitism and mathematical Platonism to be plausible. It seems to me that either all numbers and mathematical objects exist as abstract objects, or none of them exist at all. Abstract objects are typically conceived as objects that exist beyond space and time and, most importantly of all, lack any causal powers. The number 3 doesn't cause anything. Now, if Platonism and finitism are both true, then we are left with a very puzzling scenario where some numbers exist as abstract objects but others don't. For instance, only a finite number of natural numbers could exist, even though the highest of these could always be followed by another natural number. There would just be no way to explain why some numbers existed and others didn't, which is why I say that it's more sensible to think that either all numbers exist as abstract objects, or none of them do. Thus, if some numbers exist as abstract objects, it is plausible to think that all of them do.

For this reason, we are justified in saying that if abstract objects somehow really exist, then it follows that actually infinite collections exist, and finitism is false. Alternatively, if finitism is true, then it plausibly follows that abstract objects don't exist. In that case, there doesn't seem to be much point in trying to demonstrate the implausibility of finitism given mathematical Platonism. The finitist will most likely just take an antirealist view of numbers and sets and such. Perhaps, on further reflection, some finitists will opt for intuitionism (by denying the mathematical legitimacy of the actual infinite), but they certainly don't have to do this, and it seems doubtful that many finitists would go that far. In short, premise (1) is false.

Four arguments down, and two more to go!