In a recent series of posts, I have been talking about Alexander Pruss's discussion of finitism in his book Infinity, Causation, & Paradox (Oxford: Oxford University Press, 2018). Pruss maintains that it is impossible for an infinite series of causes to precede an event, so he embraces a sort of finitism called causal finitism. Most of his book is aimed at showing how causal finitism is capable of resolving numerous paradoxes related to infinity.
As part of his discussion, Pruss devotes a small section to arguing against a more general finitism, which holds that it is impossible for an actually infinite number of things to exist. He presents six arguments against finitism, which I have discussed in the previous three posts. Perhaps the main failing of Pruss's discussion is that he does not differentiate between mathematical finitism and metaphysical finitism. My preferred position is metaphysical finitism. As for mathematical finitism, I have no position to stake out. In that sense I am agnostic toward it, but I am inclined to be skeptical of it because it seems unnecessarily radical.
Although I have been critical of Pruss's dismissal of finitism, I don't want to give the impression that I dislike his book. I am finding it to be a very stimulating and interesting read. There's much that Pruss says that I find helpful and commendable. I have chosen to focus on Pruss's arguments against finitism, but that discussion only represents a tiny portion of the book. So in spite of my disagreements with Pruss, I do think the book is a worthwhile read for anyone who takes an interest in this topic.
Strangely enough, I think that my own acceptance of metaphysical finitism has only been strengthened by reflecting on Pruss's arguments. Going into the book, my position was that metaphysical finitism seemed plausible enough, but it was also too contestable for me to feel comfortable with building much on it. I've never been drawn to mathematical realism or Platonism, but I never felt easy with dismissing realist or Platonist views on finitist grounds, because that seemed to run the risk of reasoning in a circle. So my preference for an antirealist view of mathematical objects has been somewhat tentative.
This means that, going into Pruss's book, I was actually quite open to his thesis of causal finitism, since it is a more modest position. Paradoxes like the Grim Reaper paradox certainly seem to undercut the idea that there could be an infinite series of causes, and causal finitism seems to resolve these paradoxes nicely without forcing us into difficult discussions about the nature of mathematical objects. So it seemed to me that there could be great practical value in opting for causal finitism over a more general, metaphysical finitism.
However, there were two things in Pruss's book that changed my mind about this. First, his arguments against finitism turned out to be unsound. There seem to be two main assumptions driving much of what Pruss says, and both of them are very questionable. One is that mathematics could not have such a successful application to physical reality unless it was possible for infinitudes to exist. The other is that (metaphysical) finitism is incompatible with a presentist view of time. Both of these assumptions turn out to be easily defeated, as I have tried to show.
But the second factor that helped change my mind about causal finitism was a complete surprise to me. Since Pruss rejects finitism simpliciter, I naturally wondered what explanation he would give for why there cannot be an infinite series of causes. This is important because finitism (by which I mean metaphysical finitism) could easily resolve the different paradoxes explored by Pruss throughout his book. Why can't you have an infinite number of grim reapers? Why can't a switch be toggled an infinite number of times in an hour? Metaphysical finitism has a straightforward answer: it is not possible to instantiate an actually infinite number of things in reality, because it would result in absurdities and contradictions. But Pruss rejects finitism, so his answer is different. He simply shows that these paradoxes can all be resolved by supposing that an infinite series of causes is impossible. Why can't a switch be toggled an infinite number of times in an hour? Because this would violate causal finitism. So his argument is that causal finitism should be embraced because it offers a good way to resolve these paradoxes.
But obviously this raises a question: why can't an infinite series of causes exist? Pruss addresses this question in a very short section entitled, "Why is causal finitism true?" (see pp. 161–62). In it Pruss writes:
If I am right, then causal finitism is true. But why is it true? What 'metaphysical force' prevents infinitely many causes from congregating in the causal history of some event? After all, surely, for any finite number n, it is possible to have n causes working together. Why only for finite n? . . . If the arguments so far succeed, then we are justified in thinking only finite numbers of items are possible in causal histories, but we do not have an explanation of why there is such a restriction. (p. 161, italics in original)
Naturally, I was curious to see what Pruss would have to say about this. I was absolutely shocked by what I read. Continuing on:
The explanatory question is deeply philosophically interesting. But we need not answer it here. . . . It is likewise not necessary to give an explanation of why causal finitism is true to uphold the arguments in this book. (p. 161)
Pruss does very quickly gesture in the direction of finding an answer "in the nature of the causal relation" (p. 162), but he does not make the argument. He just sort of leaves the reader to figure it out for herself.
As I read this section, I couldn't help but feel that a very obvious answer could be given here, namely, that metaphysical finitism is true! In other words, an infinite causal series cannot exist because it is impossible for an actually infinite number of things to be instantiated. I find this to be a much more plausible account of why an infinite series of causes cannot exist than anything we could say about the nature of the causal relation itself. After all, Pruss himself points out that there is nothing absurd in having a finite series of causes. The absurdities only seem to arise when we bring infinity into the equation. The connection seems blindingly obvious to me, and the only reason Pruss ignores it is on the basis of several arguments that are surprisingly easy to defeat.
I am at least glad that Pruss does not contend that the truth of causal finitism is just a brute fact. That wouldn't work for him anyway, since he has devoted quite a lot of time to defending the principle of sufficient reason, and so he would be contradicting himself if he claimed that there's no explanation for why causal finitism is true. But if his arguments against finitism fail, as I think they do, then the most reasonable explanation for causal finitism is metaphysical finitism. At any rate, it would certainly be the simplest.