On Thu, 26 May 2005, Brent Meeker wrote:
I agree with all you say. But note that the case of finite sets is not
really any different. You still have to define a measure. It may seem
that there is one, compelling, natural measure - but that's just
Laplace's principle of indifference applied to integers. The is no more
justification for it in finite sets than infinite ones. That there are
fewer primes than non-primes in set of natural numbers less than 100
doesn't make the probability of a prime smaller *unless* you assign the
same measure to each number.
Brent Meeker
I'll answer both Brent and Hal (m6556) here.
Yup, I hadn't thought through the measure issue properly. Several
conclusions from this discussion:
* As Brent says, you always have to assume a measure. Sometimes a measure
seems so "natural" you forget you're doing it, as above. Another example
is in an infinite homogeneous universe, where equally a uniform measure
seems natural, and also limiting frequencies over large volumes are well
defined, as per Hal's message.
* As Hal points out, it *is* possible to assign probability measures to
countably-infinite sets.
* Alternatively, you can assign a non-normalizable measure (presumably
uniform) and take limiting frequencies. But then as per Cantor the answer
does depend on your ordering, which is something extra you are adding to
the definition of the set (even for numerical sequences).
* Different lines of argument can easily lead to different "natural"
measures for the same set, e.g. Hal's "Universal Distribution" vs.
Laplacian indifference for the integers.
* For me, the only way to connect a measure with a probability in the
context of an "everything" theory is for the measure to represent the
density of universes (or observers or observer-moments if you factor that
in as well).
* Since the White Rabbit^** argument implicitly assumes a measure, as it
stands it can't be definitive.
* But the arbitrariness of the measure itself becomes the main argument
against the everything thesis, since the main claimed benefit of the
thesis is that it removes arbitrary choices in defining reality.
Paddy Leahy
^** "This is a song about Alice, remember?" --- Arlo Guthrie
