Holographic Images - Part 5
Curvature, dimensionality and transform space
Using the
basic ideas of Riemann, Pietsch attempts to construct a holographic theory of mind.
For example, Pietsch treats any instance of periodicy in perceptual space as a set of
coordinates which can be given transformational expression in an appropriate counterpart
coordinate system in the mind. Moreover, such a transform will be an expression of an
operation revolving around the basic notion of least curvature.
Thus, the
constructs of perceptual space will be built from the units of least curvature that are
inherent in perceptual space, whereas a corresponding construct in mental space will be
built from the units of least curvature that are appropriate to mental space. However, in
each case, the units of least curvature of perceptual space are transforms of the units of
least curvature of mental space and vice versa.
One of the
problems with Pietsch's foregoing position is the assumption that mental space actually
has units of least curvature. This assumption geometrizes the mind and makes it a function
of geometric conceptions of, and approaches to, ideas concerning the identity of the
structural character of basic building blocks in the mind (assuming, of course there are
such things as basic building blocks). With this geometrization of the mind comes the
spatialization of the mind.
When one
tries to represent other, non-spatial dimensions through the perspective of spatial
coordinate systems, then, irrespective of how many coordinate axes one uses to construct
this representation, the representation will always be problematic in its presentation.
This is because each of the additional spatial axes being used is constructed from points
whose structural character is peculiar to the spatial dimension and may not be
translatable into, or reflective of, the structural character of the 'points' of the
dimension being represented - assuming, of course, that non-spatial dimensions can be
analyzed in terms of points of any kind whatsoever.
At the very
best, the relationship between the spatial axis and the non-spatial dimension which that
axis purports to represent might be an analog one. However, even if the spatial axis could
have an analog relationship with the dimension being represented, one needs to understand
the non-spatial dimensional significance of the structural character of the complex
function to which each point on the spatial axis will give expression.
In other
words, the 'points' of another dimension - to the extent that they can legitimately be
referred to as points at all - will have a significance and meaning peculiar to that
dimension's latticework nature. As such, these 'points' give expression to that
dimension's unique set of constraints and degrees and freedom which describe what can and
cannot occur through, or within, such a dimension.
What is
expressed as curvature in the spatial dimension may not be expressed as curvature in the
other dimension in question. In fact, the idea of spatial curvature may have no meaning or
significance or counterpart - analog or otherwise - in a non-spatial dimension.
Curvature is
but one instance of structural character, and an important question to ask oneself in this
regard is this: Is any function based on curvature - no matter how complex that function
may be - capable of generating a model that is congruent with the structural character of
a non-spatial dimension being represented through a spatial axis system? The answer to
this question will depend on whether or not an analog relationship between the spatial and
non-spatial dimensions can be generated.
The capacity
to plot the graph of a function in a spatial context is a very fruitful procedure. It
provides a way of helping one to visualize and see relationships which may not be readily
apparent in the functional form of those relationships.
This
heuristic component carries over into the realm of transforms in which a transform of a
structure in perceptual space may permit one to interact with the underlying set of
constraints and degrees of freedom to which the perceptual structure gives expression, in
a way that would not be otherwise possible. Nonetheless, if the initial functional
characterization of something - in this case, some dimension 'x' - is problematic, this
will carry over into the graph of that function.
When it
comes to the representation of non-spatial dimensionality through the use of n-axes of a
spatial coordinate system, people seem to forget that such systems are expressions of the
constraints and degrees of freedom characteristic of the geometrization of space.
Consequently, the point-structures of spatial systems, whether considered in Euclidean or
non-Euclidean terms, have the potential for distorting, if not totally obscuring, the
actual structural character of the non-spatial dimensions being represented. In short, the
structural character of points in the spatial dimension (and, again, Riemann views this
structural character as a matter of curvature) may not be capable of capturing, or be
translatable into (as a transform operation of some sort), or be an analog for, the
structural character of some other non-spatial dimension.
The
geometric perspective assumes, in principle, that a spatial transform or spatial analog or
a function based on the spatial property of curvature inherent in the basic building
blocks of space - namely, points - can be found for any and all other non-spatial
dimensions. More specifically, in the case of the mental realm, the geometric perspective
assumes: (a) that the mind is continuous in the same way that such a perspective claims
space is continuous (i.e., as an infinite set of infinitesimal points); (b) that the
mental realm is constructed from basic unit points in the same way that space is thought
to be constructed from basic unit points; (c) that such points give expression to the idea
of least curvature in the generation of lines, surfaces/contours and solids which occur in
both physical and mental space and that the structures generated in these respective
'spaces' are transforms of their corresponding counterparts in the other mode of space;
and, finally, (d) that there is a one-to-one correspondence between the structures
generatable in physical space and the structures which are generatable in the mental mode
of space.
The
foregoing assumptions should be questioned very closely, if not abandoned altogether. A
tremendous amount of distortion, error, problems and biases enter into the idea of
dimensionality as a direct result of a failure to examine the assumption which underlies
the geometrization and spatialization of dimensionality.
To be sure,
where analogs or transforms or functions can be established that permit one to develop a
heuristic dialectic between non-spatial dimensions and spatial coordinate systems, then,
one should pursue this opportunity. However, one also should approach such a dialectic
with a healthy amount of circumspection and reflect, from time to time, on what one is
doing and what is meant when one uses the structural units of the spatial dimension to
construct representations of non-spatial dimensions.
The methodology of tensors
According to
Paul Pietsch, all forms of feeling, thinking, motivation and so on which occur in the
mental realm constitute least curvature structures capable of being expressed in transform
space as a particular kind of phase spectrum. As such, behavior-whether in an explicit
form or in the form of thoughts, feelings and so on- is a mental transform of sensations
and perceptions.
The
'mechanism' making transformations, of whatever sort, possible is rooted in the idea of
tensors. Tensors were developed after Riemann's introduction of curvature into the
vocabulary of geometry. Just as Riemann had discussed the manner in which the
relationships about a point ( relationships that constitute curvature) remain invariant,
even under transformation, tensors also describe a set of relationships that remain
invariant across transformation operations.
One might
argue, however, that tensors constitute a methodology for handling the dynamics or
dialectics of the ways in which the points of a region or neighborhood interact with one
another. Thus, whereas curvature represents a sort of static kind of look at the
structural units of which geometric figures are constructed, tensors appear to involve a
dynamic exploration of how the structural units of space interact with one another under
various conditions of stress and strain. In short, tensors are used to represent and
explore the idea of change.
Tensor
relationships are very much like relative phase relationships in the way in which they
behave when subjected to transform operations. For example, the absolute values of change
being described by tensors may be quite different in various situations to which
identifying reference is being made.
Moreover,
these absolute changes are often not accessible to measured determination, any more than
absolute phase relationships are accessible to measured determination. Nonetheless, the
relative aspects of change occurring in the context of such absolute changes tend to
transform in the same way from situation to situation.
Tensors have
the capacity to capture the structural character of the relative relationships in
conditions of change or transition and to be able to preserve that structural character
(usually in the form of complex ratios) as one moves from one coordinate system to another
by way of transform operations. This capacity goes to the very heart of the idea of a
tensor.
Because of
the capacity of tensors to preserve the structural character of relationships across
coordinate systems, Pietsch argues that tensors actually define the coordinate system into
which they are transformed. In other words, most mathematical operations
presuppose the existence of an already defined coordinate system of given structural
character and are, then, introduced into a given coordinate system in terms of the basic
structural properties of that system.
Apparently,
however, tensors actually determine the character of the structural properties out of
which the coordinate system is constructed. As such, rather than being thrown into a
pre-defined coordinate system and adapting itself to conform to that pre-defined
coordinate system, a tensor actually gets a coordinate system to conform to the invariant
properties of the tensor.
In other
words, a tensor shapes a coordinate system from the bottom up rather than merely being
grafted onto that system in an adapted form. Therefore, a tensor imposes its own invariant
infra-structure on a coordinate system and, in a sense, forces that coordinate system to
observe or respect that invariance.
A coordinate
system is relative or derived in the sense that it constitutes a representation of some
other previously manifested reality- of a physical, material, mental or spiritual nature.
A coordinate system, at the very least, presupposes a hermeneutical orientation toward, or
approach to, certain aspects of the phenomenology of the experiential field.
In effect, a
coordinate system constitutes an expression of this orientation in the form of a
geometrization of an aspect of the phenomenology of the experiential field to which
identifying reference is being made. Therefore, to argue, as Pietsch does, that tensors
define a coordinate system by virtue of the way they impose their invariant, relative
relationships onto a coordinate system does not necessarily really say something about the
structural character of ontology apart from, or beyond, the character of the interaction
of a given tensor with a given coordinate system.
As
geometrizations of various aspects of the phenomenology of the experiential field, a
coordinate system is generated from a certain arrangement of basic geometric units -
namely, points. In geometry, of whatever sort, straight lines, curves, surfaces, contours,
solids and dimensions are all generated by ordering points in a prescribed fashion. This
prescribed fashion is the methodological process which is required to produce a geometric
figure of a given structural character.
Tensors also
are about points. More specifically, tensors describe the structural character of the
relative relationships which occur during processes of change or transition involving
these points. In this sense, tensors presuppose the existence of points.
In fact, one
might suppose that points represent something like the simplest possible structures one
can imagine which are capable of undergoing processes of transition and change. If there
were no points undergoing transitions, then, there would be nothing for tensors to
describe.
One cannot
have relationships in the abstract which do not relate to, or are not linked to,
interacting structures, of some sort, that undergo change. The very concept of
relationship, especially of a relative nature, presupposes the existence of some sort of
structure (or structures) which is ( or are) being explored in terms of the character of
the network of relationships linking two or more aspects of the structure (or structures).
These "aspects" which are being identifyingly referred to, and which are being
studied in terms of the character of their linkages, are geometrically represented by
points.
To be sure,
one can drop these points or aspects from consideration once one has a handle on the
structural character of the relationships among them and, thereby, derive an abstraction
or abstract representation of the original context of change. However, one must not forget
that a tensor - as an example of one kind of possible abstraction of such a context of
change - is derivative, ultimately, from a context in which the structural character of
relationships is a function of the structures being related, together with the dialectic
that is made possible by the spectra of ratios of constraints and degrees of freedom
encompassed by those structures.
Relationships
are not independent of structures being related. Relationships are not autonomous,
self-sustaining entities. The character of a relationship is colored by the structures
which it ties together.
The very
character of a tensor's unique manner of abstraction is the way such an abstraction zeroes
in on the character of relative changes in various contexts and eliminates all other
properties from consideration. What colors the character of those relationships is very
much a function of the structural character of the aspects or points which are being
studied vis-a-vis the character of their relationships.
As indicated
previously, Riemann argued that the measurable relationships in the neighborhood of a
given point are exactly the same as the measurable relationships in the neighborhood of
any other point, irrespective of the coordinate system in which the point exists.
Similarly, in the case of tensors, the argument seems to be that the measurable
relationships of change in the neighborhood of a given point are the same as the
measurable relationships of change in the neighborhood of any other point irrespective of
the coordinate system in which such change occurs. Thus, the structural character of the
relationships involved in relative change remains the same irrespective of the kind of
coordinate system one uses to give representational form to the character of that change.
In the
foregoing sense, the structural character of the relationships which are captured and
preserved by tensors actually represent a set or envelope of constraints and degrees of
freedom that specify how any given coordinate system can give expression to the structural
character of that change in the context of the properties of that coordinate system.
Therefore, tensors do not so much define a coordinate system as they are a means of
guiding, orienting, and ordering a coordinate system in terms of the structural character
of the relationships of relative change which the system is attempting to capture and
preserve vis-a-vis some other coordinate system.
In general,
from the perspective of tensor analysis, there are two kinds of relationships which can be
used to describe the structural character of the dynamics of change: covariation and
contravariation. Covariation refers to relationships of transition having the same
directional character; that is, they proceed in the same direction. Contravariation, on
the other hand, refers to the sort of contrary relationship which the opposite ends of a
stretched rubber sheet or rubber band have with one another.
Tensors are
able to give representation to either of these sorts of change relationships individually,
as well as both of them together in whatever combination suitably captures the structural
character of the change to which the tensor is making identifying reference. These latter
form of tensors are known as mixed tensors.
Tensor
transformations consist of a set of rules for translating a given tensor, R, into a
different coordinate system. If a given tensor R in one coordinate system
does not equal a given tensor counterpart, R, in another coordinate system after the rules
of tensor transformation have been applied to the first tensor (or vice versa), then, the
changes being described do not constitute a true tensor - that is, they are not invariant
changes.
Such changes
are, instead, fluctuations of a local nature and reflect, at best, conditions of local
constancy in the relationships of change which are manifested in the system in question.
In other words, these sort of fluctuations are thought of as being empirical in nature.
They are not analytical as supposedly is the case in instances of true tensors.
This
empirical/analytical distinction seems a little odd in light of the fact that the
structural character of a given tensor is derived originally from examining the nature of
change in some region of the phenomenology of the experiential field. To be sure, to the
extent that a tensor is supposed to capture and preserve, in abstracted form, the
structural character of a given instance of changing conditions, then, a given tensor,
ONCE it has been determined, should remain invariant across coordinate systems. In this
sense, of course, the tensor is somewhat analytical, but this quality or property of
analyticity is predicated on, and presupposes, an empirical context. As a result, thinking
in terms of such an analytic/empirical distinction, may be somewhat misleading.
Seemingly,
what really is being referred to in the foregoing is a distinction between: (a) conditions
of change manifesting relative relationships that are invariant across coordinate systems,
as opposed to (b) instances of change manifesting properties of relative relationships
which do not remain invariant as one moves from one coordinate system to another via the
agency of transformation operations. In essence, the distinction between tensors and
relationships of change restricted to localized, coordinate contexts is that the former
exhibit the quality of symmetry, whereas the latter do not.
Symmetry
relationships in a given coordinate system reflect, or are alleged to reflect, the
structural character of some aspect of ontology or some aspect of the phenomenology of the
experiential field or both, to which the coordinate system is making identifying
reference. Consequently, when one seeks to understand something, there will be tensors on
each side of the hermeneutical equation which purports to reflect congruence between
ontological and hermeneutical/phenomenological structures.
One side of
the hermeneutical tensor equation consists of the aspect(s) of ontology which help make
possible an experience of a given structural character to which identifying reference is
being made through the focal/horizonal character of a given aspect of on-going
phenomenology. The other side of the hermeneutical tensor equation consists of the aspect
of understanding/orientation which the individual has with respect to, or has toward, the
aspect of the phenomenology of the experiential field to which identifying reference is
being made.
The tensors
on each side of the hermeneutical equation must have the same character in order for that
equation to have epistemological status or meaning. In other words, such an equation needs
to give expression to a tenable, if not accurately reflective, relationship between, on
the one hand, certain aspects of the ontology and, on the other hand, certain aspects of
the hermeneutics of the phenomenology of the experiential field which are being linked
through the hermeneutical tensor equation.
Thus,
hermeneutical applications of the idea of tensors is a matter of seeking symmetry - that
is, relationships of invariance - which are preserved across different contexts. In the
hermeneutical frame of reference, these contexts do not necessarily represent geometric
coordinate systems. Nonetheless, one needs to discover tensors whose structural character
remains invariant as one moves from the context of the phenomenology of the experiential
field to the context of ontology to which that phenomenology is making reference but which
is, to some extent, independent of that phenomenology.
In effect,
hermeneutical field theory can be construed as involving an attempt to establish
hermeneutical equations that contain tensors displaying the same character. When the
tensor components on each side of the hermeneutical equation display the same character
this indicates that some feature of invariance concerning the structural character of
change has been preserved in both ontology as well as phenomenology. The existence of such
symmetries permits the structural character of an aspect of phenomenology to reflect the
structural character of an aspect of ontology.
Seen from a
slightly different perspective, hermeneutics involves, among other things, a study or
exploration of the structural character of the properties of change occurring in and
around the neighborhood(s) of one or more aspects of the phenomenology of the experiential
field. This exploration is done in an attempt to determine the structural character of the
forces of stress, strain and vectoring being exchanged with different aspects of ontology
and which together (that is, as a dialectical function of both phenomenology and ontology)
generate a focal/horizonal 'point'-structure of an observed experiential character.
There are
many aspects of the holographic process that cannot be easily, if at all, subsumed under
the structural wing of ordinary transformations. Use of tensor transformations renders the
idea of decoding the data of transform space into the structures of perceptual space much
more tractable than do ordinary transformations.
From the
perspective of tensor transformations, the transition from transform space to perceptual
space can be described in terms of how a given set of relative values concerning the
structural character of certain changes is preserved as one moves from one kind of space
to the other. Through the maintaining of symmetry with respect to the property of the
relative values of structure to which a given set of changes give expression, tensors are
able to show how the underlying structural character of change is able to manifest itself
across coordinate systems.
In short,
tensors can be used to represent phase relationships in a way that is independent of any
specific coordinate system. Because Pietsch believes tensors actually define, through the
rules of tensor transformation, the character of the coordinate system into which the
tensors are introduced, he maintains that when tensors are used to represent relative
phase relationships, then, in effect, phase relationships can be said to define the
coordinate system into which the phase relationships are introduced by means of tensor
transformations. Pietsch believes this would be the case irrespective of whether one was
talking about memory, perceptions, thoughts, and so on.
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