Tuesday, May 20, 2014

The completed infinite

It is very hard to deny that it is logically possible that every rabbit has at least one offspring and there are no loops (no rabbit is its own ancestor). But in that situation, there will be infinitely many rabbits.

"Not so fast!" say the defenders of the distinction between the potential and the completed infinite. This is a case of a potential but not a completed, or actual, infinite. But why? On the scenario in question, there are infinitely many humans. A standard answer is to embrace a theory of time, like growing block or presentism, on which there are no future entities, and then to say something like this about the scenario:

• Infinitelymany rabbits will come into existence, but there are only finitely many rabbits and at any given future time there will only be finitely many then.
Not only does this require abandoning eternalism, which the correct theory of time, but it requires further work to explain why it couldn't also be the case that every rabbit has the property that its offspring take half as long as it did to produce offspring.[note 1] But in that case, after a finite amount of time there would be infinitely many rabbits. Further, this approach requires either nominalism or a special story about why concrete objects like rabbits can't form a complete infinity, while mathematical entities like prime numbers can (not a big problem for me, since I'm not a Platonist).

My proposal is that we should see the denial of a completed infinite differently. Rather than seeing it ontologically as saying that there are not infinitely many of anything, we should see it causally. A student has completed a class provided that the class is available for her to build on, either in her future thinking and work or as a prerequisite for other classes. Likewise, a process is completed provided that its product is available for other processes to build on.

A completed infinity, I propose, is the sort of infinity that can be causally built upon. The rabbits in my initial scenario cannot be built upon: they aren't all causally available to anyone. In that initial scenario, there is a plausible explanation about this in terms of time: there is no time at which there are infinitely many rabbits, so there is no time at which you can build on all of them.

But my causal finitism suggests that the same is true on my modified scenario where the rabbits breed faster and faster. Maybe that scenario can produce an infinite number of rabbits in a finite amount of time. But nonetheless, only finitely many of the rabbits will irreducibly work together causally. (I wonder whether irreducibility rules out overdetermination. Worth thinking about...) Let's say you cast a glance at that infinity of rabbits. You will only see finitely many at a time—your field of view is only finitely large and finitely sharp. Only finitely many of them will eat up your garden. And so on.

If we see a "completed infinity" as a causal notion, then we have no worries about Platonist mathematics. For mathematical entities are typically taken to be causally inert, and even if for some epistemological reason we do not take them so, we could still think that only finitely many are involved in any one causal interaction.