Stephen P. King (stephenk1@home.com)
Sat, 11 Sep 1999 15:54:17 -0400
Dear Bill et al,
If I could add my $.02 to this discussion. :-)
WDEshleman@aol.com wrote:
>
> Hitoshi,
>
> [WDE]
> objective:
> The nature or structure of an object independent
> of observation.
Would this not make the objective "nature or structure" of an object
indeterminate or probabilistic in-itself? I think that this is necessary
since we must allow for any consistent sequence of observations of the
object at the a priori level, e.g., independent of observation.
BTW, an observation is a measurement when we are using a mapping into
the reals, from what I have read. I think that an observation of general
would be best defined as a category theoretical morphism (an
"informorphism" to be specific) between objects which are to be
considered as members of equivalence classes prior to the particular
observation. These in turn give a natural meaning to the concept of
"Many-Worlds". :-)
> subjective:
> The nature or structure of an object as it appears to an
> observer.
Here we have the results of the mapping after the selection process has
occurred. It seems to be the consensus of experts looking into this type
of notion that the "nature or structure" is always tensed "after" the
act of mapping. Thus Pratt says "I think, therefore I was". BTW, I am
trying to define a generic example of Lance's "clocking" with the notion
of a mapping or "informorphism" between equivalence classes.
It is my belief that the commutativity of the informorphism diagram is
the best formal way of understanding this. This is discussed in Peter's
Section 9 in http://www.cs.brown.edu/~pw/papers/math1.ps, pg. 19-21.
> Light speed would then be subjective, and
> FTL (and/or infinite speed) an objective property?
I think so. If we consider the "objective property" as containing "all
possible" velocities, then to characterize it as "infinite speed" as an
average over these would be correct.
> [HK]
> Yes. In your definition of objective and subjective, I agree.
>
> [WDE]
> Yes, something is lacking in those statements; e.g.:
>
> 1) Where does objective end and subjective begin?
We must examine how it is possible to put a "cut" between the two
properties. What is the "mechanism" by which a specific subjective
property is selected from the equivalence class of all possible? I
propose that the mechanism of a Chu transform in general or the
informorphism specifically is the best candidate...
> 2) Can an objective mathematical model have both
> objective and subjective properties?
To be "objective" a model required that it can be encoded into a
"kickable" object. We can think of a stone sun dial as encoding the
various theorems of trigonometry and geometry, but note that without an
observer that can decode this information the "mathematical model"
concept is mute. Meaning is not inherent in objects, it is given only
within the context of interaction. We say that the meaning is "in our
minds not in the object itself", but actually it is in the interaction
between the two!
> 3) Mathematics doesn't "kick back" like objects do,
> so is there really no hope of obtaining a mathematical
> model that describes real phenomena? Does the ability
> to write it down with symbols demand that it is therefore
> objective?
This question reminds me of the problem of making a "true and complete
map" of a town. If we are going to encode in some kickable object
(matter or energy) the information that gives us a means of answering
any question about the town we must first ask if the town itself can
encode the information. There does seem to be a limit of how much
information (in bits) can be encoded in matter!
(http://pespmc1.vub.ac.be/ASC/Bremer_limit.html)
> 4) What is the degree of "distortion" experienced by
> an observer? Could the "distortion" sometimes be
> negligible? Or, always be zero?
We could think of this "distortion" as a measure of the noise inherent
in the interaction between object and observer. Frieden quantifies this
in his paper: Langrangians of Physics and the game of Fisher-Information
transfer. (Frieden & Soffer, Phys. Rev. E 52, Sept. 1995)
We could think of the distortion as a parameter \theta that we are
trying to estimate. This follow from the "imperfect observation", e.g.,
distorted experience of the object in question, y = \theta + x of
\theta, given the distortion x.
"The system comprising quantities y, \theta, and x is a closed one. No
other inputs effects are assumed present. It becomes apparent that the
closed nature of the measurement system implies an isolated physical
system as well."
"Consider the class of "unbiased" estimates, obeying <|theta^hat(y)> =
0; these are correct "on average". Then the "mean-square error e^2" in
the estimate \theta^hat obeys a relation e^2*I >/= 1 (1), where I is the
Fisher information: I = \integral dx p'^2(x)/p(x) (2).
The prime in "p'" denotes a derivative d/dx and the integration limits
are infinite. The quantity p(x) denotes the probability density function
for x and x is the noise [or in the case of your posed question, the
distortion]. We call equation (1) the Cramer-Rao inequality. "It
expresses the reciprocity between the error e^2 and the Fisher
Information I. The quantity I is therefore a quality metric of the
estimation procedure [or in the context of our discussion, the degree of
distortion in a given observation]. Since the quality increases (e
decreases) as I increases, I is called an "information"."
quotes from ibid. pg. 2274-5
If we try to imagine the conditions under which the "distortion" is
negligible or zero we would be constructing a model were infinite
computational resources would be required and an infinite amount of free
energy is available to be consumed by the computational process. Here I
am considering the act of observation as a computation of \theta.
This is similar "in spirit" to Matti's concept of NMP. :-)
> 5) Can the mind imagine a unique correct objective structure?
> Or will the mind always be restricted to contemplating
> multiple consistent objective structures? Optimistically,
> the unique correct objective structure, is the simplest
> model with the best explanation, is it not?
If we were to be able to affirm that a mind could imagine a "unique
objective structure", we would have be be able to defend it against all
possible contrafactuals, e.g., observations that imply a different
objective structure! How can we say that "the mind always be restricted
to contemplating multiple consistent objective structures", when where
is more than "one" mind in existence? A given mind can only contemplate
a structure that is "objective" (kickable) given its own standard of
"kickability" or \theta. (Umm, I had previously referred to this
quantity as \epsilon!
> 6) Would not an objective structure that allows instantaneous
> communication between its parts, be composed of rigid and
> incompressible objects that transmit displacements due to
> the direct contact between all of the objects?
Well, that would be the case if it were possible to construct such a
structure within the \theta of some observer! Such an observer would be
"omnipotent" and "omniscient", but completely lacking of the ability to
"change its mind", since its space of possible descriptions is a
singleton! Umm, this is exactly the situation that Matti is requiring
for the "Category of infinite-dimensional geometries"!
Onward,
Stephen
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