Where the Conflict Really Lies : Science, Religion, and Naturalism by Plantinga Alvin

Author:Plantinga, Alvin [Plantinga, Alvin]
Language: eng
Format: epub, pdf
Publisher: Oxford University Press
Published: 2011-10-16T05:00:00+00:00

Ideally, we should like m (the measure) to have the following properties: that m is defined for every set of real numbers, that the measure of an interval is its length, that the measure is countably additive, and that it is translation invariant. Unfortunately, as we shall see … it is impossible to construct a set function having all of these properties.25

These problems with probability and infinite magnitudes arise at a more basic level. Suppose we think about logical probability in terms of possible worlds. Clearly, if there are only finitely many possible worlds, there’s no problem: the logical probability of a proposition A will be the proportion of A worlds; the conditional probability of a proposition A on a proposition B will be the proportion of A worlds among B worlds, that is, the quotient of the number of worlds in which both A and B are true by the number of worlds in which B holds. If there are infinitely many possible worlds, however, there will be infinitely many mutually exclusive propositions (for any possible world W, for example, there will be the proposition that W is actual). And now problems rear their ugly heads. For example, suppose propositions form a countable set; each of the possible worlds is presumably as likely (on no contingent evidence) to be actual as any other; but (given countable additivity) clearly it won’t be possible to assign each proposition of the form W is actual the same nonzero probability in such a way that these probabilities sum to 1.26 And of course problems are only exacerbated if there are more than countably many possible worlds. Here as elsewhere infinity presents serious problems. One possibility, obviously, is to follow Leopold Kronecker and a host of finitary mathematicians and stoutly declare that there aren’t any actual infinities. There may be quantities that approach infinity as a limit, but there aren’t and couldn’t be any actually infinite quantities. Given the various paradoxes of infinity (for example, Hilbert’s hotel), this has a certain ring of sense. Whether it is actually true, however, is of course a monumentally contentious question.

With respect to our current topic, the McGrew et al. objection to the FTA: if we don’t reject infinite magnitudes, perhaps the most sensible way to proceed is to give up countable additivity. The velocity of light could fall within each of infinitely many mutually exclusive and jointly exhaustive small intervals; the probability that it falls within any particular one of these intervals is zero, but of course the probability that it falls within one or another of them is one. This seems to fit well with intuition, or at any rate as well or better than any other proposed solution. For example, suppose space is in fact infinite, and suppose it is divided into infinitely many mutually exclusive and jointly exhaustive cubes one cubic mile in volume. Suppose you know that exactly one of them contains a golden sphere of radius one-half mile. You will of course

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.