Final Reflections on Alexander Pruss's Discussion of Finitism

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.

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.

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.

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.

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.

An Argument for Finitism, Part 1

Finitism is the view that only finitely many things exist. Different types of finitism can be identified based on what we mean by "exist." Metaphysical finitism holds that an infinite number of things cannot be instantiated in mind-independent reality.

Instantiation of something means that there is an instance (or an example) of that thing. Mind-independent reality is whatever exists "out there," in the real world, independently of human thought (it leaves open the question of whether reality is ultimately dependent in some way on a divine mind). Anything that can be instantiated in mind-independent reality is said to be metaphysically possible.