Thursday, December 31, 2009

Brief Jotting on Infinite Regresses

Sometimes when infinite regress arguments come up, especially in the Intro Phil classroom, the following sort of argument (sometimes in a more crude form) is given by a student as a possible reason why they might not work -- that is, as a possible reason for the conclusion that an infinite regress is possible. Let's confine our attention to the case of an infinite regress of causes, and let's take some effect. We can assign that effect 0, and its cause -1, and the cause of that -2. Now the negative numbers are an infinite set; so it's possible that the series could be stretched back. So it's entirely consistent for there to be an infinite regress of causes: no contradiction in the numbers.

This sort of argument, which I've heard from students in several different forms, clearly doesn't work against infinite regress arguments, and in fact is wholly irrelevant to them, but I think it's worthwhile to stop a moment and consider the precise logical misstep involved, if only because it's useful for explanations.

Suppose I want to see whether it's possible for me to have infinite apples. I assign numbers to my apples, and recognize that for each apple that I have there is a higher possible number of apples. Aha! says Tom, thus it's at least in principle possible for me to have infinite apples, because for any number of apples I have I could have a greater number of apples! But Tom is mistaken; the fact that for any number of apples I could have a greater number of apples is entirely consistent with saying that it is impossible to have infinite apples. For what Tom has proven is not that I can have infinite apples but that for any finite number of apples I could have a greater finite number of apples.

So it is with the students' objection to infinite regress. Assigning numbers in this way only shows that, as far as the numbers go, any finite series can be exceeded by another finite series. But this is entirely consistent with there being only finite series. It would be entirely possible for this to be true and for there to be no infinite series at all, because we never actually got around to talking about infinite series, just as in the apples example we never actually got around to talking about infinite apples. We just talked about how there was no limit to the size of finite series, or to finite collections of apples. From "For any finite number of apples a greater number of apples is possible" one cannot infer "There is a possible collection of apples greater than any finite number of apples". It is the latter that you would need in order to have shown that an infinite collection of apples was possible; the former is consistent with the impossibility of having infinite apples, because it only talks about finite numbers of apples.

Of course, in the end, it's obviously the case that whether I can have an infinite number of apples has nothing to do with numbers and everything to do with apples and what's needed in order to have apples. You can't have infinite apples, not because infinite numbers are impossible, but because you couldn't possibly have the apple trees to produce more than a very large finite number of apples. And likewise, whether I can have an infinite regress of a certain type of cause has nothing to do with numbers and everything to do with those causes and what you need in order to have those causes. This is true generally of disputes like this -- whether the world can actually be eternal with an infinite past or future, whether there can actually be infinitely many objects in the universe, whether there can be actual infinitesimals, and so forth.

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