On Mon, Sep 19, 2011 at 01:14:04PM -0400, Stephen P. King wrote:
>
> Exactly why are there not a continuum of OMs? It seems to me if
> we parametrize the cardinality of distinct OMs to *all possible*
> partitionings of the tangent spaces of physical systems (spaces
> wherein the Lagrangians and Hamiltonians exist) then we obtain at
> least the cardinality of the continuum. It is only if we assume some
> arbitrary coarse graining that we have a countable set of OMs.
I do not assume an arbitrary coarse graining, but do think that each OM
must contain a finite amount of information. This implies the set of
OMs is countable.
>
> >
> >The problem with this argument is that all rational numbers, when
> >expressed in base2, ultimately end in a repeating tail. In decimal
> >notation, we write dots above the digits that repeat. Once the
> >recurring tail has been reached, no further bits of information is
> >required to specify the rational number. Another way of looking at it
> >is that all rational numbers can be specified as two integers - a
> >finite amount of information.
>
> I must dispute this claim because that reasoning in terms of
> 'two integer' encoding of rationals ignores the vast and even
> infinite apparatus required to decode the value of an arbitrary pair
> of 'specified by two integers' values.
Both the human brain, and computers are capable of handling rational
numbers exactly. Neither of these are infinite apparatuses. If you're
using an arbitrary precision integer representation (eg the software
GMP), the only limitation to storing the rational number (or decoding
it, as you put it) is the amount of memory available on the computer.
The amount of information needed to represent any rational number is
finite (although may be arbitarily large, as is the case for any
integer). Only real numbers, in general, require infinite
information. Such numbers are known as uncomputable numbers.
> The same applies to the
> notion of digital information. Sure, we can think that the observed
> universe can be represented by some finite collection of finite bit
> strings, but this is just the result of imposing an arbitrary upper
> and lower bound on the resolution of the recording/describing
> machinery. There is no ab initio reason why that particular
> upper/lower bound on resolution exists in the first place.
>
It rather depends what we mean by universe. An observer moment, ISTM,
is necessarily a finite information object. Moving from one observer
moment to the next must involve a difference of at least one bit, in
order for there to be an evolution in observer moments. A history, or linear
sequence of observable moments, must therefore be a countable set of
OMs, but this could be infinite. A collection of such histories would
be a continuum.
A world (or universe), in my view, is given by a bundle of histories
satisfying a finite set of constraints. As such, an infinite amount of
information in the histories is irrelevant ("don't care bits"). But if
you'd prefer to identify the world with a unique history, or even as
something with independent existence outside of observation, then
sure, it may contain an infinite amount of information.
> >
> >I notice this paper is an 02 arXiv paper, so rather old. It hasn't
> >been through peer review AFAICT. There was a bit of a critique of it
> >on Math Forum, but that degenerated pretty fast.
> >
> >Cheers
>
> Ideas are sometimes like vine or a single malt whiskey that must
> age before its bouquet is at its prime.
>
Partly I was wondering how much effort to put into it. Unfortunately,
it appears that the author's email addresses are no longer valid, as
it would be very interesting to have him engage in our discussions.
> Onward!
>
> Stephen
>
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Visiting Professor of Mathematics hpco...@hpcoders.com.au
University of New South Wales http://www.hpcoders.com.au
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