Alexander Pruss's Arguments Against Finitism, Part 3

In this series of posts I am looking at a number of arguments against finitism given by Alexander Pruss in his book Infinity, Causation, & Paradox (Oxford: Oxford University Press, 2018). So far I have looked at four arguments (although it's technically five, since one of them involved two different versions). In this post, I will look at the final two.

5. The Argument from "If-Thenism" (see pp. 15–17)

5.1 Introduction

In this section, Pruss considers a view of mathematics that does not involve affirming mathematical realism. Instead, this view (which he calls "if-thenism" but which could also be called deductivism) conceives of mathematics as discourse about what follows logically from different axioms without making judgments about whether the axioms are true. He sees this as a plausible alternative to mathematical Platonism, but he claims that, like Platonism, deductivism is incompatible with finitism.

As Pruss points out, p entails q provided that in every possible world where p is true, q is true. Another way to say it is that there is no possible world where p is true and q is not. From this it follows that "an impossible proposition entails every proposition, since there is no world where an impossible proposition is true" (p. 16). An impossible proposition would be something like, "Some squares are circles." There is no possible world where this statement is true. It is a logical contradiction, and any proposition can be proven from a contradiction. This is called the principle of explosion. It is not Pruss's invention.

5.2 The Argument

1. Any axiom entailing a logically impossible proposition entails that all propositions are true.
2. If finitism is true, then the axioms of arithmetic entail a logically impossible proposition.
3. Therefore, if finitism is true, then the axioms of arithmetic entail that all propositions are true. (1, 2)
4. The axioms of arithmetic do not entail that all propositions are true.
5. Therefore, finitism is false. (3, 4)

5.3 Response

Premise (1) is simply a statement of the principle of explosion, so I don't see much wisdom in challenging it.

Jumping ahead a bit, Pruss does not explicitly state premise (4), but he strongly implies it by calling attention to the fact that if all propositions are true, "then there are square circles and round triangles" (p. 16). If it were really the case that the axioms of arithmetic entailed the truth of square circles and round triangles, then they would have to be rejected, and this seems like an obvious dead end, so premise (4) is also granted.

The key premise, then, is going to be premise (2). Pruss speaks generally about "the axioms of arithmetic," by which I take him to be referring to the axiom of infinity and any other related axioms. The axiom of infinity entails the existence of at least one infinite set, which is the set of natural numbers.

The metaphysical finitist is not committed to saying that the existence of an infinite set is logically impossible, but only that it is metaphysically impossible. So it seems to me that the finitist could just reject premise (2). The only logical impossibilities that arise from infinite collections are those that occur when we perform operations like subtraction in contexts where those operations are undefined. Since they are undefined, they would not seem to have any bearing on the logical coherence of the axioms themselves. Otherwise Pruss himself would have to concede that the axiom of infinity entails a logical impossibility. Since he does not, I take it that undefined operations with logically contradictory results do not undermine the logical coherence of the axioms of arithmetic. In that case, I don't see why this should pose a problem for the metaphysical finitist.

To recap, the axiom of infinity does not commit the finitist to the literal existence of an infinite set, but only to its logical coherence and mathematical legitimacy. It should also be noted that the finitist does not need to deny that mathematics is a tool for discovering truth. All the finitist needs to be committed to is saying that certain mathematical structures can't be instantiated in mind-independent reality, even though they're logically consistent. Again, negative numbers come to mind as another example of this. There is nothing logically incoherent about saying that someone has a negative number of, say, baseballs, but it does seem metaphysically impossible.

6. The Argument from Future Infinities (see pp. 17–18)

6.1 Introduction

Pruss opens this discussion by saying that the finitist has to allow for the possibility of an infinite future. Since I agree with Pruss that an infinite future is plausible (say, if the universe continues to expand forever, or if there is an eternal afterlife), I think that the most plausible version of finitism needs to contend with the likelihood of an infinite future. This rules out the eternalist view of time, since that view would then entail that an actually infinite series of future events exists. As a result, finitists will want to embrace either presentism or growing-block theory, since both of these views deny the existence of future events. I myself prefer presentism.

Pruss's argument here is based on a thought experiment. The section is a little confusing because he presents the thought experiment in a series of steps, but these steps do not constitute the argument itself. So I will explain the thought experiment here and then try my best to formulate an argument based on the point that Pruss is making.

If the future is endless, then Pruss contends that it is possible that an infinite series of events will take place. For instance, someone could flip a coin an infinite number of times. It's important to keep in mind, however, that presentists and growing-block theorists would both deny that future events exist, which means that, at no point in the future will there have been an actually infinite number of coin tosses (assuming that there is a first coin flip). At any time in the endless future, it will still be the case that the coin has only been flipped a finite number of times. This is a potential infinity, not an actual infinity.

Pruss is aware of this, but he thinks there is an argument to be made against finitism here that has nothing to do with whether the future events exist or not. Rather, it has to do with whether the future events can be counted. An infinite set is said to be countable if its members can be placed in a one-to-one correspondence with the natural numbers. Pruss provides a proof for his method of counting, which is very technical and over my head, but I don't think it matters for my purposes.

To illustrate that future events can be counted, Pruss gives an illustration. Imagine that the future is endless, and that every day a new piece of paper will be produced that is green on one side and red on the other. The green side will have the nth natural number written on it, and the red side will have the nth prime number written on it. (So the numbers will increase sequentially: on day 2, the green side says 2 and the red side has the 2nd prime number.)

Over the course of an infinite future, Pruss claims that "[a]ll the positive integers and all the primes will be written down" (p. 17). From this, we can construct an argument against finitism.

6.2 The Argument

1. If an infinite number of future events can be counted, then an infinite number of events will happen over an infinite future.
2. If an infinite number of events will happen over an infinite future, then finitism is false.
3. An infinite number of future events can be counted.
4. Therefore, finitism is false. (1–3)

6.3 Response

Let's consider the argument's premises in reverse order. Premise (3) is only true if (as Pruss claims) events don't have to exist in order to be counted. I am inclined to agree with this idea, especially because, as a presentist, I don't want to say that past events cannot be counted, even though they don't exist. But it's crucial to remember that counting, in this context, refers to the way that countability is defined in mathematics with infinite sets. Future events don't exist (again, Pruss concedes this to the finitist), but we can still imagine an infinite series of events. It's important to realize, then, that this infinite series of future events is fictional. It literally does not exist, and it never will. But we can still speak coherently about it the same way that mathematicians can speak coherently about actually infinite sets. Any finitist who rejects intuitionism can agree with this.

For that reason, it seems to me that the events in this fictional series of future events can obviously be placed in a one-to-one correspondence with the set of all natural numbers. So I am inclined to accept premise (3).

The main problem I see with premise (2) is that the phrase "an infinite number of events will happen over an infinite future" is ambiguous. Let's suppose that there will be an everlasting afterlife. In that case, the future will never end. But the finitist denies that the future consists of an actually infinite number of events, because there will never be a time when an infinite number of events has occurred. To make this point more clear, consider the following two statements:

5. Each event in an endless future will occur.

6. All events in an endless future will occur.

The mistake Pruss makes is thinking that (6) can be inferred from (5). This is simply false. If presentism is true, and if the future is endless, then for any event E that we can specify in the series of future events, it is true that E will occur. But it does not follow that all events will occur, because this would imply that a potentially infinite series has somehow been converted into an actually infinite series, which is impossible. You cannot complete an infinite series of tasks by performing one task at a time, nor can you form an actually infinite collection by adding one member at a time. This is true even if we allow for the possibility that an actually infinite number of things could exist.

This means that Pruss is wrong to infer that, if each positive integer and each prime number will be written down, then "all of the positive integers and prime numbers will be written down." Given the truth of either presentism or growing-block theory, this is impossible. Again, this has nothing to do with finitism, but with the relevant theories of time. Nobody can ever succeed in writing down all the positive integers and prime numbers if they start today, even if finitism is false.

If "an infinite number of events will happen" is supposed to mean that all future events will happen, then the finitist will accept the second premise but reject the first one. Premise (1) is rejected because, as Pruss himself admits, being able to count an item does not mean that it needs to exist, and counting all the future events (which is necessarily a non-existent series) does not entail that all of those events will happen.

On the other hand, if "an infinite number of events will happen" is supposed to mean that each future event will happen, then the finitist will simply deny premise (2), since the fact that each member of a potentially infinite series will be realized does not entail that an actually infinite set will be formed.

Either way, the argument fails.

Conclusion

I do not find any of Pruss's arguments against finitism to be convincing. In fact, reflection on these arguments has actually strengthened my conviction that metaphysical finitism is correct. I just see no good reason to think that an actually infinite collection of things could really exist any more than I think that a negative number of things could really exist.

In the next post, I will offer a few final reflections on Pruss's preferred theory of causal finitism.