Re: White Rabbits, Measure and Max

From: Alastair Malcolm <amalcolm.domain.name.hidden>
Date: Wed, 14 Jul 1999 22:02:00 +0100

----- Original Message -----
From: Russell Standish <R.Standish.domain.name.hidden>
To: <everything-list.domain.name.hidden>
Sent: 14 July 1999 03:23
Subject: White Rabbits, Measure and Max
> I was thinking some more on the physical laws issue, and on Wei Dai's
> point on what measure to apply to different universes. The solution
> given is to weight shorter strings over longer ones.
>
> One way of thinking about this is to state that there are _no_ finite
> strings at all in the everything world. Instead, the first n bits of a
> string contain information, and the remainder are "don't care"
> values. We also assume a uniform measure on all these infinite
> strings. In this picture, worlds who are entirely specified by strings
> with a smaller value of n will have higher measure than those with
> larger n. The measure will fall off exponentially with n, in fact
> precisely 2^{-n}, assuming the uniform measure above. This, then is
> the universal measure sought by Wei Dai. Clearly, the everything
> universe consists of all strings where we don't care what any bit
> is. These have zero information, and measure = 1.

I think this may have possibilities, but it drew me into a quagmire when I
looked at it a little while ago. If n is the information-relevant string
length of our (presumed) universe, m is an exponential mean of the
equivalent for all possible contrived universes subjectively similar to ours
(including flying rabbit universes), then it seems to me that if the number
of different possible such contrived universes is less than 2^(m-n), then we
should not expect to be in a contrived universe, with or without flying
rabbits - the rabbit paradox would be solved. This might seem to be
conceivable if we only had to concern ourselves with the single type of
contrived universe that mimics ours, plus visible deviations (such as flying
rabbits) from it - we can only perceive a limited number of such within our
lifetimes; however it would seem we also have to consider other larger
mathematical structures that would present to us contrived universes with
(additional) invisible deviations as well (rabbits under the floorboards, in
the Andromeda galaxy and so on). These have even longer, but finite-length,
information-relevant bit strings, so although making a small contribution
individually, their vast quantity may make them dominate - but I am not
sure.

Weighting shorter strings over longer ones may turn out to provide a useful
universal measure, but I haven't been able to see a way through to it
providing a solution to the flying rabbit problem.

Alastair
Received on Wed Jul 14 1999 - 14:18:54 PDT