RE: White Rabbit vs. Tegmark

From: Brent Meeker <meekerdb.domain.name.hidden>
Date: Thu, 26 May 2005 10:11:37 -0000

>-----Original Message-----
>From: Patrick Leahy [mailto:jpl.domain.name.hidden]
>Sent: Thursday, May 26, 2005 10:21 AM
>To: Alastair Malcolm
>Cc: EverythingList
>Subject: Re: White Rabbit vs. Tegmark
>
>
>
>On Thu, 26 May 2005, Alastair Malcolm wrote:
>
>> An example occurs which might be of help. Let us say that the physics of
>> the universe is such that in the Milky Way galaxy, carbon-based SAS's
>> outnumber silicon-based SAS's by a trillion to one. Wouldn't we say that
>> the inhabitants of that galaxy are more likely to find themselves as
>> carbon-based? Now extrapolate this to any large, finite number of
>> galaxies. The same likelihood will pertain. Now surely all the
>> statistics don't just go out of the window if the universe happens to be
>> infinite rather than large and finite?
>>
>> Alastair
>
>Well, it just does, for countable sets. This is what Cantor showed, and
>Lewis explains in his book. Cantor defines "same size" as a 1-to-1
>pairing. Hence as there are infinite primes and infinite non-primes there
>are the same number (cardinality) of them:
>
>(1,3), (2,4), (3,6), (5,8), (7,9), (11,12), (13,14), (17,15), (19,16) etc
>and so ad infinitum
>
>You might say there are obviously "more" non-primes. This means that if
>you list the numbers in numerical sequence, you get fewer primes than
>non-primes in any finite sequence except a few very short ones. But in
>another sequence the answer is different:
>
>(1,2,4) (3,5,6) (7,11,8) (13,17,9) etc ad infinitum.
>
>In this infinite sequence, each triple has two primes and only one
>non-prime. Hence there seem to be more primes than non-primes!
>
>For the continuum you can restore order by specifying a measure which just
>*defines* what fraction of real numbers between 0 & 1 you consider to lie
>in any interval. For instance the obvious uniform measure is that there
>are the same number between 0.1 and 0.2 as between 0.8 and 0.9 etc.
>Why pick any other measure? Well, suppose y = x^2. Then y is also between
>0 and 1. But if we pick a uniform measure for x, the measure on y is
>non-uniform (y is more likely to be less than 0.5). If you pick a uniform
>measure on y, then x = sqrt(y) also has a non-uniform measure (more likely
>to be > 0.5).
>
>A measure like this works for the continuum but not for the naturals
>because you can map the continuum onto a finite segment of the real line.
>In m6511 Russell Standish describes how a measure can be applied to the
>naturals which can't be converted into a probability. I must say, I'm not
>completely sure what that would be good for.
>
>Paddy Leahy

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
Received on Thu May 26 2005 - 14:11:04 PDT