Math Reflections and Resonances - Part 2
Three types of tensors and their hermeneutical counterparts
If one
generalizes the concept of a vector space in linear algebra, one arrives at the idea of tensor
space. Just as the elements of vector space are referred to as vectors,
the elements of tensor space are known as tensors.
Tensor
algebra is a means of, among other things, describing the curvature of surfaces (and even
the curvature of space itself) which is manifested in some region of consideration. The
curvature tensor is a term that refers to the operations used to generate these sort of
description.
The energy-impulse
tensor is an idea, often used in the theory of relativity, which relates how the
energy of a particle and the impulse of a particle cannot be separated from one another.
Indeed, this tensor combines these two components (i.e., the energy and impulse of a
particle) together into one inextricable structure whose overall character is shaped by
the dialectic of these two components.
Finally,
there is another kind of tensor known as the tension or deformation tensor. This
tensor describes the deformation or tension occurring in an elastic medium.
All three of
the foregoing terms (curvature tensor, energy-impulse tensor, and the tension or
deformation tensor) appear to be very pregnant with possibility with respect to
hermeneutics and phenomenology. For example, the hermeneutical operator may, in fact, have
a structural character somewhat akin to the energy-impulse tensor in the sense that the
former consists of six components which cannot be separated.
Thus, (a)
the process of making identifying reference; (b) the turning of reflexive consciousness to
the aspect of the experiential field to which identifying reference has directed our
attention; (c) the characterization of that aspect now being attended to; (d) the
application of the interrogative imperative; (e) the use of mapping/inferential
operations; and, (f) the establishing of congruence relationships, all dialectically
interact together. As a result, they cannot be isolated from one another.
These
components come together and shape how one another proceed. These components also
contribute, each its own way, to the shaping, organizing and orienting of the latticework
of understanding which emerges through the collective efforts of these components or
facets of the hermeneutical operator.
In addition,
the ideas of a curvature tensor and a deformation tensor also lend themselves well to the
notion of a phenomenological manifold which gives expression to waveform structures that
are functions of a number of interacting and, often times, inseparable components.
Sometimes the hermeneutical operator is a means of exploring the character of curvature or
deformation of a given aspect of the experiential field created by the presence of
sensory, emotional, conceptual or spiritual waveform latticeworks. On the other hand,
sometimes the hermeneutical operator is itself the cause of the curvature or
deformation/tension in the phenomenological manifold as a result of the waveform
latticeworks which it generates.
One further
idea occurs in relation to vectors and the hermeneutical context. Hermeneutical vectors
and tensors can be added together. Indeed, one can conceive of the link between the
focal/horizonal structure now being operated on through reflexive consciousness and the
focal/horizonal structure which preceded it (in the form of a memory), or the
focal/horizonal structure which will follow it (in the form of learning or insight) as the
hermeneutical transformations that are necessary in order to generate and link different
structures. In other words, the latticework of focal/horizonal structures constituting a
given understanding and which is given expression over time, or comes into being over
time, is linked by a set of phase relationships that are a series of hermeneutical
operations. This series of operations permits one to travel or map a functional link
between one focal/horizonal structure and other such structures occurring prior to, or
subsequent to, any given point of reference.
Lattices, phase relationships, and latticeworks
In the
1930s, Garrett Birkhoff, along with John von Neumann, introduced the concept of a lattice.
This concept was originally developed with the intention of generalizing and organizing
some of the relationships existing between, and among, the subsets of a given set.
Subsequently, the lattice concept was extended to encompass various relationships existing
between certain mathematical structures (such as groups and topological spaces) and their
substructures.
Formally
speaking, if a set S has two operations known as intersection and union ,
then, that set is a lattice provided that the structural character of the set is able to
preserve three axioms for any elements of the set which may be selected:
(a)
associativity
(a
‘intersection’ b) ‘intersection’ c = a ‘intersection’ (b
‘intersection’ c);
(a
‘union’ b) ‘union’ c = a ‘union’ (b ‘union’ c).
(b)
commutativity
a
‘union’ b = b ‘union’ a;
a
‘intersection’ b = b ‘intersection’ a.
(c)
absorption
a
‘union’ (a ‘intersection’ b) = a
a
‘intersection’ (a ‘union’ b) = a.
The subsets
of a set constitute a lattice under the operations of union and intersection.
One of the
things which is appealing with respect to the idea of a lattice is the way it focuses on
the relationships between, or among, different subset components of a given set, or the
relationships between, or among, the various sub-structure components of a given
structure. However, in the context of hermeneutics, the operations describing such
relationships need to be provided with a far greater dialectical and dynamic character
than is possible with the operations of intersection and union.
This is
where the notion of a phase relationship will be, potentially, much more powerful than the
rather static or fixed ideas of intersection and union which key in or the theme of
membership or inclusion/exclusion. At the same time, the dialectic among phase
relationships is much more difficult to try to grasp hold of since the very nature of a
dialectic is that it often tends to be multifaceted, nuanced, and complex in character.
In any
event, use of the term latticework in this article, is intended to retain the
emphasis on relationships in Birkhoff and von Neumann's original idea of a lattice.
Simultaneously, the notion of a latticework directs attention toward the
dynamic/dialectical complexities of phase relationships and away from the rather static
character of the inclusion/exclusion relationships associated with lattices.
The issue of continuity: a neighborhood perspective
Representation
theory focuses on issues surrounding the mapping of a group, ring or algebra
homomorphically into either a linear transformation of a vector space, or a group or a
ring of matricies of a vector space. The vector space into which a group or ring or
algebra is being mapped is known as the representation space. To map something
homomorphically requires that one do so without destroying the structure of what is being
mapped.
In topology,
a figure is a point set. Topological figures which are homeomorphic have the same connectivity.
More
specifically, suppose one has two figures, A and B. Suppose, further, that A is
transformed into B such that the transformation does not involve any tearing, or pasting
together, of A during the course of the stretching, bending and other deformations that
are a part of the transformation.
In order for
the two figures to be homeomorphic, there must be associated with each point p of A, a
unique point f(p) in B. Furthermore, there must also be associated with each point p of B,
a unique point of A. In short, the map f which relates each point p of A with its
transform f(p) is a bijection of A onto B.
The
no-tearing and the no-pasting requirement means that in any mapping f, if the mapping is
bijective, then, for any two points, 'a' and 'b', of A which are sufficiently close
together, one must find that the images of these two points- namely, f(a) and f(b) also
will be close together. In essence, this means that the two points, a and b (along with
their images), satisfy the condition of continuity.
From the
perspective of the present dissertation, what makes any two given points
"sufficiently close together" is the character of the latticework of phase
relationships in which the two points are embedded and to which they give partial
expression. This latticework ties the two points together through a web or set of phase
relationships.
Among other
things, this web of phase relationships establishes the constraints and degrees of freedom
that determine the neighborhood surrounding the points and through which the points are
linked together. When the integrity of the set of constraints and degrees of freedom
establishing the boundaries and parameters of the neighborhood are preserved, despite
undergoing various kinds of transformations (such as bending, twisting and deformations),
then, the phase relationships that tie two points to a neighborhood (and, therefore, to
each other) remain viable and operative.
Under such
circumstances, the two points can be said to have been "sufficiently close
together" for connectivity or continuity to have been preserved. Notice, however,
that the meaning of the term "sufficiently close together" may vary from one
situation to another, depending on: (a) the structural character of the neighborhood
involved; (b) the nature of the web of phase relationships which tie together any two
given points within that neighborhood; (c) the structural character of the forces which
are brought to bear on the neighborhood, together with (d) any dialectic ensuing from the
engagement of the neighborhood with such forces.
Although all
of these four factors shape the character of the meaning of the term "sufficiently
close together" in any given situation, obviously, the bottom line on this (and that
which demonstrates whether two points are, or are not, sufficiently close together) will
be whether or not one can show, in some way, that there is at least one phase relationship
which still links the two points together after the two points have undergone one or more
transformation processes. In short, one must have a demonstrable way of getting from one
point to the other to show that connectivity or continuity has been preserved.
Moreover, in
the hermeneutical and phenomenological contexts, connectivity or continuity are not the
all-or-none phenomenon which they seem to be in many areas of mathematics. In other words,
in order for connectivity or continuity to be preserved in the hermeneutical context,
every link existing prior to a given transformation may not need to exist after the
transformation.
The
integrity of a neighborhood can be minimally preserved if at least one route or path ties
the points of a neighborhood together. In effect, this means that at least one set of
phase relationships must be the same both before and after a given hermeneutical
neighborhood undergoes one or more transformations.
Thus, the
neighborhood of a point could be seriously disrupted and, yet, still maintain the
character of its structural integrity. Therefore, under circumstances of extensive
disruption, in order for one to be able to say of any two given points in that
neighborhood that they were sufficiently close together, those two points must fall within
the perimeter of the set of phase relationships which mark a minimal path or route of
connectivity through the latticework of the neighborhood.
On the other
hand, if, due to any combination of the four factors cited above, a hermeneutical
neighborhood is pushed beyond its limits such that even the minimal structural integrity
of the neighborhood is lost or breached, then, the property of connectivity or continuity
may disappear along with the neighborhood. One cannot be absolutely sure that all links
between the two points have been ruptured since it becomes difficult, and sometimes
impossible, to carry out the cross-checks and cross-referencing which are normally
possible when the two points are rooted in a neighborhood and which are necessary for one
to be able to trace whether certain phase relationships are still intact within the
latticework of the hermeneutical neighborhood ( much as an electronics expert might test
to see whether different circuits are still operative or viable).
Indeed,
sometimes one doesn't have the methodological means to gain access to the two points in
which one is interested. Under these circumstances, one only can make inferences about any
links which might exist between the two points by looking at the structural properties of
the surrounding neighborhood and trying to determine if any observed structural
differences could be accounted for by the absence of a link between the two points in
question. Consequently, when the surrounding neighborhood has been disrupted to the point
that its integrity may have been breached and, as a result, any connectivity which might
exist has fallen into the interstitial crevices which lie beyond the resolution
capabilities of the methodological capabilities available to us, one has lost the context
against which one can methodologically push in order to be able to arrive at hermeneutical
determinations about two points in particular.
Homeomorphic mapping in hermeneutical contexts
Bijective
mapping involving homeomorphic figures may be thought of as a special case of analog
structures. Or, approached from a slightly different angle, analog structures can be
considered to be a more general and complex version of homeomorphic figures. In both
cases, the essential issue is the preservation of a set of links or phase relationships
across one or more transformations.
In the case
of topological figures, transformations concern stretching, bending and other sorts of
deformations. However, in the case of hermeneutics and phenomenology, the transformations
involve different kinds of transduction in which waveforms of one sort become translated
into waveforms of another kind- such as sounds into sensations, or sensations into
conceptual structures, or sensations into emotional structures, and so on.
Furthermore,
whereas the points being discussed in a topological context are geometric points which
hold position without occupying space, the points being discussed in a hermeneutical or
phenomenological context are, or can be, structurally complex, involving a variety of
fractal levels of manifestation. As such, the "hermeneutical point" or the
"phenomenological point" tend to be 'point-latticeworks or
'neighborhood-latticeworks' occurring within a larger latticework at junctures of
intersection of interacting forces or dimensional dialectic within that larger
latticework.
The
dialectical forces operating through that juncture of intersection provide a relatively
stable - or, at least, temporarily stable - neighborhood that helps shape the structural
character of the larger latticework. This is done by giving expression to the phase
relationships which contribute to the set of constraints and degrees of freedom that
collectively constitute the larger latticework.
A structure
consists of a set of neighborhoods whose internal dynamics, together with the dialectics
of these neighborhoods with one another along their common boundaries, give expression to
a set of constraints and degrees of freedom that is capable of preserving the integrity of
the neighborhoods and their dialectics over time. Almost by definition, a "common
boundary" occurs whenever two neighborhoods interact with one another. Therefore,
'distant' neighborhoods can share a common boundary if the phase relationships of the
given latticework within which the neighborhoods occur permit such interaction.
Structures
are the end result of a process involving the placing of constraints on some aspects of
various dimensions, as well as the permitting of the expression of some aspects of the
degrees of freedom and constraints of various dimensions. This means one cannot really
speak of the idea of dimensional dialectics being a reductionistic position. In point of
fact, structures constitute a veiling, narrowing down or restricting of the dimensions
that help give expression to such structures, just as the dimensions constitute a veiling
or restricting of the underlying order-field which makes dimensions and their dialectic
possible.
As indicated
previously, two figures, A and B, are said to be homeomorphic if there is both a
continuous bijective (i.e., one-to-one- correspondence) map m of A onto B, as well as an
inverse map m-1 which is also continuous. Such a map is referred to as a topological
map or homeomorphis.
Those
characteristics of sets revolving around issues of connectivity are called topological.
Furthermore, all homeomorphic images of a given set will possess the same topological
characteristics as the set for which it is a homeomorphic image.
In view of
the above, latticeworks and analog relationships seem to deal with certain
topological-like properties a great deal since both are vitally concerned with, among
other things (and in their own way), issues of connectivity. However, the latticeworks and
analog relationships of hermeneutical/phenomenological contexts involve n-dimensional
topological properties given that the kind of connectivity issues with which they are
concerned involve multiple levels of inferential mappings and congruence relationships.
This aspect
of n-dimensional topological properties would seem to raise some further issues. For
example, if the structural character of chaotic systems encompasses multiple levels in
which the character of each level is self-similar with other levels of the structure but
not self-same, what does this do to the issue of bijective mapping?
On the other
hand, is it not conceivable, and even plausible, to suppose that if the structural
character of a latticework or neighborhood retains its integrity (and, therefore,
preserves the property of connectivity in some minimal fashion) across the
transduction/transformation process, then, the structures (i.e., phenomenal and noumenal)
which are analogs for one another will be homeomorphic since one can still show that they
are minimally bijective? In other words, connectivity (in the form of a set of phase
relationships reflecting the same kind of inferential/dialectical links, constraints and
degrees of freedom as the image structure) still exists between them.
Therefore,
one not only can map, in a continuous and bijective fashion, from one structure onto the
other, but there is an inverse continuous map which exists as well.The very notion of two
structures being minimally bijective suggests that some set of central, crucial,
fundamental, essential or critical set of phase relationships, as well as constraints and
degrees of freedom, have been preserved during the process of transduction or
transformation. As a result, there exists a neighborhood in the hermeneutical structure
which is capable of being homeomorphically linked with a neighborhood in the noumena
structure.
The total
integrity of such neighborhoods will not be preserved completely from one fractal level to
the next (or even across the transformation process on the same level). Nonetheless, there
may be sufficient preservation of sets of phase relationships, constraints and degrees of
freedom within such neighborhoods that one will be able to see how one level connects with
another in a self-similar fashion which never strays beyond certain parameters of
structural character. As such, self-similar structures or latticework figures manifest
minimal, continuous bijective properties and, thereby, establish a basis for connectivity
to be preserved - albeit not as completely as would be the case with self-same structures.
In terms of
the relationship of phenomena and noumena, the foregoing considerations can be construed
in several ways. Either the noumena is mapped onto the phenomena and, thus, the latter is
the image of the former, or, the phenomena is mapped onto the noumena, and, in this case,
the noumena would be the image of the phenomena. Quite conceivably, both possibilities may
occur in the sense that during the process of sensory transduction, the noumena is mapped
onto the experiential field, whereas during the process of hermeneutical determination,
the phenomena are mapped onto the noumena as mediated by, or represented in terms of, the
experiential field.
Here, of
course, one would be working on the assumption that the transduced, phenomenal
representation of the noumena can be shown to satisfy criteria of bijectivity and,
therefore, be homeomorphic with the aspect of the noumena to which identifying reference
is being made. The latter often occurs (at least in non-mystical cases) in an indirect
manner since it is really a matter of mapping phenomena onto a conceptual latticework (or
theory or model) which attempts to account for why given phenomena have the structural
character they do.
On the other
hand, one could not automatically eliminate the possibility that the hermeneutical
operator is capable of engaging the noumena at the boundary manifold where the noumena and
phenomenal meet and interact. Under such circumstances, the individual seeks out (that is,
one tries to determine) those structural aspects of the boundary manifold which are
contributed by noumena and which are congruent with the structural character of one's
understanding.
Naturally, a
key problem here is whether the hermeneutical operator is merely projecting itself, in the
form of constructions of imagination, onto the character of ontology. The other
possibility is whether the hermeneutical operator actually has established a legitimate
bijective mapping, from the structural character of the understanding which it has
generated in the phenomenology of the experiential field, to the structural character of
some aspect of the noumena with which it is concerned. Nevertheless, the realization that
such a problem exists is not at all the same thing as saying the problem cannot be solved
- such as Kant seems inclined to do.
There is a
complex, fractal boundary which forms between noumena and phenomena. One explores the
structural character of boundary systems in order to be able to come to terms with the
different latticework systems which are at work shaping, organizing, and orienting that
boundary.
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