Holographic Images - Part 6
Pietsch
summarizes his position in the following way:
(a) mind can
be treated as a species of complex information- namely, information concerning phase;
(b) as a
methodological starting-point, one approaches the phase information of (a) by
characterizing and exploring it in terms of the geometry of a Riemannian universe in which
the basic unit of structure is that of curvature in a continuum of indefinite dimensions;
(c) the
relative phase values that are used to describe different aspects of mind are expressed as
ratios of curvature;
(d) tensors
can be used to represent the ratios of curvature;
(e) the
activities of mind can be treated as instances of tensor transformation in which the same
underlying structural character of relative change is preserved as one moves from one
mental modality or operation to another;
(f) due to
the manner in which tensors allow one to consider the structural character of change
independent of any given coordinate system, one has no need to specify whether one is
dealing with perceptual space or some other transform of perceptual space such as Fourier
transform space;
(g) the
structural character of coordinate systems are a function of tensor transformations rather
than tensor transformations being a function of the structural character of a given
coordinate system.
Relationships
involving relative phase values in perceptual space are said to be time-dependent. This
time dependency is translated into a spatial dependency in the transform space of, say,
Fourier analysis.
However,
both the time-dependent, as well as the space-dependent, relationships of relative phase
values are governed by a set of constraints and degrees of freedom that are manifested in
each coordinate context. In other words, in the case of ordinary transforms, the
coordinate axes don't expand or contract. As a result, the ordinary transforms give
expression, in each coordinate context, to an inherent structural framework on to which,
respectively, the time-dependent or space-dependent relationships are grafted - a
structural framework to which these relationships must accommodate themselves.
Thus, there
is an analogical relationship between perceptual space and transform space in the sense
that phase relationships in transform space are required to obey a set of rules or
principles that are comparable to, or analogs for, the sort of rules or principles which
the phase relationships in perceptual space are required to obey. Furthermore, in each
case, these rules or principles are reflections of the fixed character of the coordinate
structure of the respective spaces.
Tensors, on
the other hand, are independent, supposedly, of the sort of rules and principles which the
structural character of any given coordinate system imposes on ordinary transformation.
Therefore, the distinction between, on the one hand, perceptual space, and, on the other
hand, various kinds of transform space becomes empty.
There is
only the underlying structural character of relationships which are undergoing transition.
If these relationships in transition are expressions of true tensors, then, that
underlying structural character will remain the same from one coordinate system to
another.
For all
practical purposes, the structural character of different coordinate systems ceases to
have primary importance as a shaping force. In other words, from the point of view of
tensor symmetry relationships, the structural character of any given coordinate system
becomes derivative from, and predicated on, the character of the shaping force which the
form of a given tensor has on such coordinate systems.
Seemingly,
on the basis of what has been said above, a tensor would appear to be a fundamental
shaping force in determining the structural character of curvature. After all, curvature
is said to be at the heart of the geometry of any coordinate system.
Since
tensors are said to be the defining determinant of the shape of a given coordinate
context, presumably, curvature is really giving functional expression to the structural
character of some underlying dialectic among a set of changing - relative to one another-
phase relationships. The feature of capturing the structural character of symmetry (i.e.,
invariance) in an underlying dialectic among a set of changing phase relationships is
precisely what constitutes a tensor.
Therefore,
in view of the foregoing considerations, tensors represent the internal dialectics of
curvature dynamics. This is the case since tensors establish the set of constraints and
degrees of freedom which will regulate how a coordinate system must manifest itself if the
structural character of the conditions of change being undergone by a set of relative
phase relationships in one coordinate system are to be preserved in some other coordinate
system.
According to
Pietsch, subjective constructions concerning the structural character of space and time
represent information transforms of those aspects of ontology being gauged by various
modes of operationalizing methodology such as rulers, clocks, and so on. However, whereas
the methodology of measurement is rooted in issues of physical structure, the character of
subjective constructions are rooted in the realm of ideas. Both, however, are said to be
expressions of nature.
The above
position seems to be somewhat shaky since one could easily argue that the methodology of
measurement is, in fact, a subjective construction and, therefore, squarely rooted in the
realm of ideas and the mental. As such, the methodology of measurement is as much an
expression of information transforms as are other modes of subjective constructions.
To be sure,
the methodology of measurement tends to focus on how to establish congruence between the
structural character of a given mode of measurement and the structural character of a
given aspect of reality that is assumed to be independent of subjective constructions and
which is referred to as being physical/material. Nonetheless, the characterization of
something as being physical/material is itself a subjective construction that may or may
not reflect the actual character of the aspect of ontology to which identifying reference
is being made, depending on what one means by the idea of 'the physical' or 'the
material'.
The
distinction between, on the one hand, subjective constructions, which are inclined to
focus on so-called non-physical aspects of experience, and, on the other hand, modes of
measurement, which tend to explore the properties of supposedly physical aspects of
reality that are encountered and engaged through experience, is really a matter of what
sorts of things each mode of engagement is inclined to focus in on and emphasize. However,
both constitute instances of subjective construction seeking congruence between a
structure of experience and that aspect of ontology which would make experience of such
structural character possible.
The search
for, and attempt to establish, congruence relationships marks the dialectic of the
hermeneutical realm. This realm consists of an overlap of structures-namely, those
structures which are rooted in the phenomenology of the experiential field and those
structures of ontology which are, to a certain extent, external to the phenomenology of
the experiential field but which touch upon, engage, interact with, shape, affect, or are
affected. as well as shaped and engaged by, the phenomenology of the experiential field.
When operating properly, this realm gives expression to the merging of horizons.
In search of hermeneutical tensor equations
A
fundamental part of the hermeneutical challenge is the need to search for, and struggle to
determine, the precise nature of the appropriate hermeneutical tensor equation in a given
context of ontological/phenomenological interaction. The nature of what is appropriate in
any given situation will be a matter of what permits one to grasp the structural character
of that aspect(s) of ontology which helps make a given aspect of one's phenomenology of
the experiential field have the character it does.
Hermeneutical
field theory involves the problem of how one goes about identifying, reflecting on,
characterizing, questioning, and mapping the character of the 'point-structures' of the
phenomenology of the experiential field so that one can try to establish congruence
relationships with the character of the 'point-structures' in the fabric of ontology that
are of the same tensor character as the point-structures of the phenomenology of the
experiential field. The dynamics/dialectics of point-structure interactions and the use of
point-structures to generate configurations, not merely in the form of geometric lines,
contours, surfaces, solids and so on, but also in the form of hermeneutical latticeworks
of varying degrees of complexity, non-spatial dimensionality and discrete continuity,
etc., become extremely important components of the process of understanding.
This all
could go under the rubric of the manifold problem introduced in a previous chapter in
relation to a brief discussion of some of Kant's ideas. In other words, the foregoing
makes reference to the problem of determining the structural character of both the
phenomenological manifold as well as the ontological manifold.
Furthermore,
questions are raised about what these two manifolds have to do with one another, as well
as what principles of dialectic govern the interaction of these two kinds of manifold
under different circumstances. Here, of course, one enters the realm of hermeneutical
tensors and hermeneutical tensor equations.
Brillouin
speaks of tensor density and tensor capacity. Capacity and density are not
the same thing.
Density
concerns the ratio of how tightly a given magnitude, quantity or substance is packed into
a given context that constitutes an independent magnitude from the first magnitude.
Capacity refers to the maximum magnitude to which a given degree of freedom of a
latticework can be extended before it is constrained by other aspects of the structural
character of either that latticework, or before it is constrained by the structural
character of other latticeworks with which it interacts.
Brillouin
maintains that a true tensor is "the product of a density and a capacity".
Under normal circumstances, density and capacity are independent of one another. However,
when one is dealing with a true tensor, Brillouin contends that the respective operations
of density and capacity cancel the features which make them independent under normal
circumstances.
In this
sense, capacity becomes a set of constraints and degrees of freedom which shape the way in
which density can be manifested in the context of that capacity's structural character. Of
course, density is also a set of constraints and degrees of freedom, but it is the
expression of a dialectic that occurs within the context of, and is encompassed by, the
structural character of capacity.
Every
capacity has its own unique density. Density is an expression of how that capacity's
latticework distributes the set of constraints and degrees of freedom to give expression
to that latticework's structural character.
Moreover,
every density has its own unique capacity. Capacity marks the parameters or limits within
which, and through which, a given density of constraints and degrees of freedom can be
distributed in order to give expression to a latticework's structural character.
Capacity and
density represent two facets of the dialectic of structural character, either with itself
or with some other, independent latticework. As such, every structural character
constitutes a tensor.
This tensor
determines the shape or form of the 'point-structures' giving expression to the manner in
which a given capacity and a given density engage or encounter one another in the region
of intersection. This is the case irrespective of whether: (a) the region of intersection
is a function of the way a given latticework spontaneously distributes its own set of
constraints and degrees of freedom; or, (b) the region of intersection is an induced
function of the way two or more latticeworks dialectically engage one another to generate
interference patterns that re-distribute and shape and vector their respective sets of
constraints and degrees of freedom.
In short,
every structural character is a product of, at a minimum, a capacity (which is the
thematic woof and warp that establishes the envelope of possibilities constituting a
latticework) and a density (which is a distribution pattern of relative phase
relationships within the set of constraints and degrees of freedom which give expression
to capacity's structural themes). Furthermore, every structural character is a true tensor
as long as the integrity of that structural character is preserved across coordinate
systems - that is, as long as the spectrum of ratios of density to capacity characterizing
a given structure retains its essential integrity across transformations
The rules of
transformation permitting one to move from one kind of representational space to another
kind of representational space (e.g., from perceptual space to Fourier transform space -
both of which are, actually, species of representational space) are the various
hermeneutical operations. These operations seek to establish or discover the identity of
the tensor character which might permit one to treat one species of representational space
as an analog for the other species of representational space.
From this
search, one hopes to establish a tensor equation. This equation needs to show that,
despite the differences of 'curvature' in the two species of representational space,
nonetheless, the structural character of the latticework in question has been preserved,
both with respect to its thematic characteristics (i.e., its capacity) as well as with
respect to its dialectical characteristics (i.e., its density), as one moves from one
representational space to another such space. Thus, one can say that a true tensor is an
analog structure whose properties are independent of the curvature medium (including
hermeneutical, phenomenological and ontological mediums) through which they are given
expression or into which they are introduced.
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