Math Reflections and Resonances - Part 5
Tensor matrices
One can
generalize descriptions for the points of E3 Euclidean space to an
n-dimensional context in the following way. Each and every point 'p' of E3 can
be designated in terms of a triplet of real numbers (r1, r2, and r3)
since there is a one-to-one correspondence mapping which can be established from the point
set of E3 onto the set of real numbers (and, therefore, the mapping is
bijective). Similarly, one can characterize En in terms of the set of n-tuples
(n1, n2, ..., nx) in which each n-tuple constitutes a
point structure which gives expression to, and is shaped by, n different elements or real
numbers.
The
foregoing ideas could be applied to a hermeneutical context if one were to make a few
adjustments. In any given latticework structure (which has a role somewhat comparable to
the fixed Cartesian coordinate system of Euclidean space), a designated point-structure
within that latticework gives expression to an n-tuple of dimensions which intersect or
engage one another at that juncture to give expression to the point structure of observed
character.
As such,
this kind of n-tuple is a way of summarizing the structural character of the spectrum of
nodes contained in the latticework in question. However, this n-tuple is not an array of
real numbers. It is an array of what might be referred to as a dimensional tensor matrix.
Each
dimensional tensor matrix constitutes an envelope of constraints and degrees of freedom
that give expression to a complex set of shaping forces. Just as a mathematical tensor
allows one to take into account the different ways a given set of forces twists, stretches
and distorts a given aspect of a manifold, so too, a dimensional envelope of constraints
and degrees of freedom constitutes a complex shaping force, represented in the form of a
matrix. This matrix weaves together a much larger set of components than is the case for a
'normal' tensor which gives expression to, say, three components.
A matrix
is described as a rectangular array of numbers. Matrices can be of any size, but when they
are square matrices, in which the number of the rows and columns are the same, this number
is usually referred to as the order of the matrix.
When one
adds two matrices of the same order together, the rules for combining them are fairly
simple. One just adds the corresponding entries or cells of each matrix.
Multiplication,
on the other hand, is somewhat more involved. One has to multiply the rows of the first
matrix by the columns of the second matrix, a cell at a time, and then, one adds together
the resulting products of each such multiplication. Matrices are
non-commutative under the operation of multiplication.
In the
context of hermeneutics, a matrix is not a rectangular array of numbers. It is a
latticework array of cells. These cells are capable of giving representational expression
to a wide variety of possibilities, including: experiential point-structures,
neighborhoods, phase relationships, various vectors and tensors forces, as well as
hermeneutical operations.
Generally
speaking, the hermeneutical operations that combine different matrices together will be
more dialectical in character than is the case with their mathematical counterparts, such
as addition and multiplication. In effect, in a hermeneutical matrix, any cell is capable,
at least potentially, of interacting with any cell of the matrices which it engages, as
well as the other cells of its own matrix environment.
Consequently,
although one can write down a general form for the idea of a hermeneutical dialectic
between (among) two (or more) matrices or latticework arrays, the structural character of
the dialectic will be affected by the nature of the values one substitutes into the
various cells of the matrices. In other words, rather than being able to encompass the
possibilities for interaction within the framework of a rule (such as exists for
the multiplication of matrices in mathematics), the dialectic of
hermeneutical/phenomenological matrices is rooted in a principle involving the
hermeneutical operator that often forms a chaotic attractor basin, resulting in
self-similar but not self-same products.
The simplest
case of the hermeneutical dialectic involves focus and horizon. Each cell of the matrix
giving expression to this simplest-case hermeneutical dialectic constitutes a interaction
or current between focus and horizon.
As such, a
cell describes a point-structure of a determinate set of constraints and degrees of
freedom. How these cells are filled out in any particular case - that is, the values they
will assume - will depend on the individual and the circumstances being engaged.
In effect,
the above matrix is a product matrix which gives expression to the general form of the way
two matrices - namely, a focal matrix and a horizonal matrix -dialectically interact with
one another. One also must keep in mind that the foregoing matrix is merely a slice of an
n-dimensional manifold in which there are a variety of vectoral currents or forces of
dialectical interaction that are operating on each cell of a given matrix from other
dimensions.
Furthermore,
the angle of engagement or orientation of these dimensional currents may touch on, or
interact with, certain cells but not others. In other words, some cells are susceptible to
such dimensional currents, while others are not - or, at least, may not be under certain
circumstances.
In addition,
each of these cells is capable of establishing phase relationships with other cells within
the matrix array. However, those cells which are connected through phase relationships
need not be contiguous. Yet, whether or not they are contiguous, the phase relationships
which link a series of cells form a neighborhood.
The
hermeneutical operator is the simplest expression of a hermeneutical tensor matrix. It
indicates that each cell of the matrix is being shaped by a variety of forces (in this
case, the various components of the hermeneutical operator) that are stretching,
squeezing, and, in general, altering the structural character of the phenomenological
fabric of the experiential field being given expression during the operator's dialectical
engagement of various aspects of that fabric.
Functional analysis, structure and abstract space
Functional
analysis has arisen largely during the last fifty years. It is predicated on two facts:
(a) an extremely diverse collection of mathematical operations share a remarkable number
of similar features; (b) when such operations are performed on a variety of mathematical
objects, these objects manifest properties in relation to those operations which are
extremely similar to, if not the same as, one another, despite the dissimilarities among
these mathematical objects. As such, functional analysis is concerned with the exploration
for, and determination of, the structural character of those properties which seem to be
most essential and fundamental to mathematical operations and mathematical objects in
general.
The idea of
a structure in mathematics can be described in the following way.
First, there must be a set of objects whose character gives expression to the structure of
that set. These objects are manifestations or carriers of the structure of the set in
question.
Secondly,
there must be some manner in which these objects are related to, or interact with, one
another. This mode of interaction is usually defined in terms of operations, functions,
relations and so on. Finally, there must be a set of distinguished elements in the carrier
which serve as indicators or indices for the structure carried by the set of objects to
which identifying reference is being made.
The set of
carrier objects, together with the operational processes and the set of distinguished
elements contained in the carrier, all are said to constitute the signature of the
structure. When one takes a given system of axioms and applies that system to
a particular signature in a way that establishes the constraints and degrees of freedom
within which the elements of that signature are to manifest themselves, then, a
mathematical structure is said to be generated.
In its own
way, hermeneutics, at least as envisioned in this article, shares many of the same
concerns as do functional analysis and general structure theory. Among other things,
hermeneutics seeks to discover those structures which seem to be most fundamental to
hermeneutical operations and phenomenological objects. For example, one could think of a
latticework as the most fundamental hermeneutical object, and the simplest form of a
latticework would be a dialectical phase relationship which links two point structures-
namely, focus and horizon.
Furthermore,
one also could think of the hermeneutical operator as being the most fundamental
expression of a hermeneutical operation. In general, all rational operations are a
function of some combination of, or series of, constraints and degrees of freedom as
shaped, organized, oriented, and structured by a recursive use of the hermeneutical
operator on one or more latticeworks within the phenomenology of the experiential field.
In
functional analysis the idea of space is far removed from any geometrical sense of the
word. Moreover, the space of functional analysis is quite different from the idea of space
in the normal day-to-day sense of the term. However, because there are a number of aspects
of the concept of space in functional analysis which bear a sort of family resemblance to
the concept of space as used in linear algebra and analytic geometry, the term
"space" has been retained for use in relation to the objects of functional
analysis. Similarly, although terms such as "length", "distance" and
so on are still used in functional analysis, they no longer carry the meanings which they
have in a geometrical context.
As
understood in functional analysis, the term "abstract space" is used in
reference to a given set of elements for which a limiting process has been given a
well-defined meaning. Thus, for any given sequence of elements e1, e2,
e3, ..., en which tends toward some limit y = lim yn,
with n --> infinity, such a sequence constitutes an abstract space.
Sometimes,
when studying the relationship among a number of elements of an abstract space, one would
like to establish whether the elements are 'close together' or 'far apart'. In order to do
this, one requires a distance function. More specifically, a distance function is a
real-valued function, d(x,y), which is greater than or equal to zero, and which is defined
for all pairs of elements within the abstract space to which it is applied.
Any space for which a distance function has been defined is known as a metric space.
Normed
spaces refers to those spaces in which there is a procedure for assigning a non-negative,
real number to each of the elements of the space. This assignment process serves as an
index or measure of the magnitude of the element involved in that process, and the
numerical index is known as a norm.73 This is written in the following
way: ||x||, and so on.
Moreover,
for any given ||x||, there must be certain properties which are present:
(a) ||x||
> 0, for x not equal to 0 and ||0|| = 0;
(b)
||(lambda) (x)|| = ||lambda|| X || x||, and ‘lambda’ is any given real or
complex number;
(c) ||x +
y|| is less than or equal to ||x|| + ||y||.
One can
derive a metric from a norm in a relatively simple way. This can be done by specifying
that the distance function between any two given elements of a space is to be the norm of
their difference: d(x,y) = ||x - y||.75
A linear
space is a space for which: (a) the operation of addition for any two elements of that
space must be defined; and, (b) the operation of multiplication involving elements of that
space together with real and/or complex numbers must also be specified. More specifically,
in the case of (a), for each pair of elements (x,y) of a space, there is unique element x
+ y in that space. In the case of (b), there is a unique element Lx associated with each
number L and each element x of that space. In addition, such operations must satisfy a
variety of conditions involving commutative, associative and distributive properties.
Hilbert
spaces are actually special cases of normed linear spaces. In Hilbert spaces a
complex-valued function (x,y) is defined for every pair of elements x and y in that space.
This function is referred to as a scalar product, and it must have certain properties in
order to qualify as being an example of Hilbert space.
The notions
of "abstract space", "distance", "magnitude", and
"Hilbert space" all seem to have implications for the hermeneutics of the
phenomenology of the experiential field. However, appropriate modifications and
alterations need to be introduced.
For
instance, the limit process, which is at the heart of the idea of abstract space in
functional analysis, could be construed in terms of the recursive hermeneutical process
through which one approaches reality as a limit by means of the hermeneutical operator
acting on the latticework objects of the phenomenology of the experiential field. If
successful, such a process generates an understanding that is similar to, or reflective
of, the original structural character of reality which it reflects.
However,
since the two are not self-same, the understanding approaches reality as a limit but does
not ever quite become one with it. This is especially the case in view of the fact that,
for the most part, any given hermeneutical understanding is largely restricted to certain
levels of scale, whereas reality cuts across innumerable levels of scale. Therefore, as
far as rational hermeneutics is concerned (and, leaving aside the issue of trans-rational
or mystical hermeneutics), such understanding only approaches, as a limit, one structure
of reality on a given level of scale.
Furthermore,
latticework objects are a set of elements which are to be ordered and shaped and organized
both within themselves, as an individual latticework, as well as among themselves, as a
collection of latticeworks, that are linked together by various phase relationships. This
ordering aspect is comparable to the sequential feature of the elements of an abstract
space in functional analysis.
In addition,
the idea of order-space may stand behind (in the sense of being a more essential,
fundamental source of) the general notion of space as a non-geometrical concept. As
characterized in mathematics, abstract space, linear space, metric space, normed space and
Hilbert space, all seem to be about certain kinds of structural and structured
relationships. Although such relationships do not occupy space in any geometric or
everyday sense of the term, they do presuppose some sort of context within which, and
through which, the relationships can be expressed, operated on, organized, shaped,
oriented and shaped. Therefore, one might characterize order-space as that which:
(a) makes
expression of such relationships possible,
(b)
specifies the structural character or properties or set of constraints and degrees of
freedom which such relationships may assume under different circumstances or conditions,
and (c)
designates the structural parameters which any dialectic may have that occurs between, or
among, elements occupying this sort of space.
As such,
abstract space, linear space, metric space, normed space and Hilbert space are all special
cases of the more essential and fundamental expression of order-space.
Indeed, all
of the foregoing varieties of spaces are derived by using the hermeneutical operator on
latticework objects of the phenomenology of the experiential field. The experiential field
is a more general structural form than any of the various structured spaces which may
arise in it. So, the hermeneutics of the phenomenology of the experiential field is the
more general structural form underlying the mathematical notion of abstract space, while
the hermeneutics of the phenomenology of the experiential field is, itself, made possible
by an underlying order-space.
Nonetheless,
in all of these cases, the space being talked about is not geometric in any sense, nor is
it extended in any way such that it can be said to occupy or constitute a
physical/material medium. Order, in and of itself, need not be extended in any way. It
specifies the parameters, constraints, degrees of freedom, and so on which any operation,
relationship, dialectic, condition, event, process, state dimension or object may have as
an expression of what such order makes possible.
Seen from
this perspective, a dimension is a specialized structural expression of order-space which
is unique in the set of constraints and degrees of freedom to which it gives expression.
That is, no two dimensions possess the exact same profile of constraints and degrees of
freedom. Therefore, each dimension leaves its own particular signature or trace in any
dialectical engagement in which it is involved.
In a way, a
dimension is like a gene on a chromosome which has characteristic DNA sequences specifying
the constraints and degrees of freedom associated with that gene. Like a gene on a
chromosome, the gene does not activate itself but must be moved to action by something
which operates on it - namely, other aspects of order-space. These aspects of order-space
assume the role of activating forces that stipulate which dimensional genes will be
activated at what time and under what circumstances. This whole dialectic between
order-space and a dimension assumes the shape of an 'ontological operon' which governs the
expression of the dimensional gene.
Finally, to
take one last term of functional analysis and transplant it to the hermeneutical/phenomen
ological context, consider the concept of distance. In the hermeneutical/phenomenological
context, the idea of distance or a distance function may be about the structural character
of the phase relationships between two point structures, especially with respect to the
'closeness' or 'distance' of the inferential link between the two structures.
In this
sense, any given point structure is inferentially closer to some point structures, while
being inferentially farther away from other such point structures. This holds true whether
one is discussing inferential relationships within a latticework or among latticeworks.
Although the meaning of 'closer and 'farther' may be construed in terms as simple as how
many inferential mapping steps does it take to get from one point structure to another
point structure, there is no reason why these terms couldn't be expressed in ways which
involve other hermeneutical structural properties such as homeomorphism, continuity,
connectivity, neighborhood, analog features, fractal dimensional character, or ratio of
constraints and degrees of freedom.
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