On 9/21/2011 6:41 AM, Russell Standish wrote:
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.
[SPK]
Umm, how does the finiteness of the elements of a set X induce finiteness of X? I may have missed this in my studies of set theory.


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.
[SPK]
True, but that misses my point. Brains and Computers are not entities existing in an otherwise empty universe; we have to consider a multiplicity of mutually observing and measuring entities and the internal interpretational and representational structures thereof. Consider a simple digital camera. The images that the camera can capture are limited by the pixel resolution of the camera, this is a constraint induced by the physical design of the camera. The camera itself, as a physical object, is not limited in the detail of its properties by those intrinsic constraints. We must take care to not assume that the limits of the observational or measurement process is not assumed to be that of the system that is making the observations/measurement.

While a the measured properties of an object A as determined by object B is limited to the resolving abilities of B, this in no way is a constraint on the properties of A. To consider the properties of A one at least might consider the set of all possible measurements of A and one might notice that these involve real valued variations, say of relative position, and thus the total set of measurable information of A is infinite, at least in principle. BTW, this is one reason why the Hilbert space of a realistic QM system is infinite, even modulo linear superposition!

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.
[SPK]
Surely Reality is not limited to the rationals! Are we to be crypto-Pythagoreans, claiming to believe that only the rationals exist, yet still using pi, e and other irrationals without question??? If Nature is computational, does it not make sense that its computations /information accessing and processing might not be limited to the rationals?

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.

[SPK]
I consider an Observer moment to be the content of experience on an ideal non-anthropomorphic observer that might obtain in a minimum quantity of time, thus there is a maximum quantity of energy involved, as per the energy-time uncertainty relation (which is controversial as time is not an observable per se!). If we assume that this observer is constrained by the laws of QM then its ability to communicate its information/knowledge to another via emission and/or absorption events is finite, it is "quantized", but its observational content is only constrained by the Heisenberg Uncertainty relation, a relation that does not put an upper bound on any single observable, it only constrains simultaneous measurement of pairs of canonically conjugate variables.

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.
[SPK]
I like the Kripkean notion of a world, where W is the non-empty set. SInce W is not constrained by some prior notion of elementhood, one is free to use any self-consistent set theory to define W. One can even adopt the anti-foundation axiom and have some real fun!

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.
[SPK]
I did a search and believe that his email is now *eastm...@yahoo.com.* I am sending a CC of this post to him in hope of a response.

Onward!

Stephen



Onward!

Stephen

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