Math Reflections and Resonances - Part 4
Several species of bijective mapping
One possible
use of the Jordan curve theorem in the context of hermeneutics is, as indicated
previously, to illustrate the problem of confusing methodology and ontology. Essentially,
this problem arises when the individual fails to realize that a given methodology
(consisting of a latticework of neighborhoods of varying hermeneutical character) does not
provide a basis for bijective mapping with some given noumenal latticework on the 'other'
side of the boundary structure of the closed curve of the phenomenology of the
experiential field. The methodology produces, instead, a mapping onto some other aspect of
the phenomenology of the experiential field within the boundary structure that
circumscribes the field.
Usually,
this 'other aspect of the phenomenology' is a conceptual or hermeneutical structure which
one takes to be reality when it is actually a projection or mapping of certain aspects of
the methodology onto the phenomenological manifold. Such a mapping may be
phenomenologically bijective, but it does not establish a bijective mapping which would
allow one to show a homeomorphic relationship between phenomenal and noumenal structures
or figures.
Thus, there
are two different kinds of bijective mapping which are possible. One kind of bijective
mapping is between: (1) a neighborhood of experiential field or phenomenal point and (2) a
given neighborhood of noumenal points along the boundary structure which separates
phenomenological neighborhood point-sets from noumenal neighborhood point-sets. One must
keep in mind here, however, that the idea of "separation" is dialectically
complex. As a result, one is not always in a position to distinguish where one leaves off
and the other begins. This first kind of bijective mapping emphasizes the role of the
merging of horizons and the removing of methodological veils that interfere with the
establishing of such bijective mappings.
The other
kind of bijective mapping is between two different neighborhoods of experiential or phenomenal
points such that one of these phenomenal neighborhoods is mapped onto the other by
means of imagination, with little, or no, contact with noumena. In other words, the
hermeneutical operator is in its projecting/construction mode rather than in its merging
mode.
Naturally,
there can be various kinds of combinations of the two sorts of bijective mapping. However,
the more the ratio of the two kinds is dominated by the projection mode rather than the
merging mode, the greater the individual will be removed from a true understanding of
ontology and that which makes ontology of such structural character possible.
Consequently,
all methodology constitutes a mapping process that attempts to establish various degrees
of homeomorphism between phenomenal and noumenal point-set neighborhoods. Difficulties
arise when methodology identifies a phenomenal neighborhood as a noumenal neighborhood,
and, therefore, assigns an incorrect set of boundary parameters to that phenomenal
neighborhood of points (i.e., designates a structure as being outside of, and independent
of, the focal/horizonal boundary manifold, when, in point of fact, that structure really
is inside of, and dependent on, the focal/horizonal boundary manifold.
Although the
ideal case in hermeneutics would exist when a homeomorphic relationship held between two
structures, one is not likely to achieve this in very many, if any, cases. One reason for
this is that, with the exception of all but the simplest issues, the ontological context
tends to have an inherently richer structural character than does the hermeneutical
context. In effect, this means congruence involves a special, limited case of
homeomorphism - namely, the existence of a latticework of neighborhood point-sets focusing
on key or central or fundamental themes (which correspond to the idea of topological
properties) as the criteria for determining whether or not one can say that one structure
has a character which is reflective of the structural character of another latticework of
neighborhood point-sets with which it is being compared.
The
foregoing observation suggests one will have to differentiate between peripheral and
essential neighborhood point-sets. As long as certain essential, hermeneutical
neighborhood point-sets display homeomorphic properties with respect to certain
ontological neighborhood point-sets, one may be able to tolerate the fact that various
peripheral neighborhood point-sets do not display such homeomorphic properties.
Nonetheless, one still is confronted with the task of distinguishing between the essential
and the peripheral since being able to establish homeomorphic mapping relationships
between peripheral structures, in the absence of mapping relationships with respect to
essential structures, may serve little, or no, purpose.
The nature of a neighborhood in topology and hermeneutics
The
E-neighborhood of a point 'p' [UE (P)] can be defined as the set consisting of all those
points having a distance from p which is less than some arbitrarily chosen positive number
E. If p is part of a line, then, the E-neighborhood is an open interval of
distance 2E.
If p is part
of a plane, then, the E-neighborhood is an open disc of radius E, although the
circumference of the disc is not considered to be part of the disc and, therefore, falls
outside the radial distance of E. Finally, if the point p is a part of some
three-dimensional space, then, the E-neighborhood will be a sphere of radius E. Again,
however, the surface of the sphere is not considered to be part of the sphere and,
consequently, the sphere's surface falls outside of the E-neighborhood of the point.
In any given
figure F, one can differentiate between boundary points, ' b ', and interior points, ' i
'. Boundary points have an E-neighborhood in which some of the points in that set fall
outside of the figure. On the other hand, interior points have an E-neighborhood that
falls totally within the figure.
One thought
that arises in relation to the foregoing concerns points along the complex boundary
structure that separates/links the manifold of the phenomenology of the experiential field
with the manifold of 'external' ontology or reality. Seemingly, for some, if not all,
points along this boundary structure, the E-neighborhood would have some points falling
outside of the boundary structure and, therefore, overlapping part of the external,
ontological manifold.
On the other
hand, the E-neighborhood of the same point would have some points that overlap with the
phenomenology of the experiential field. Looked at in this way, one might want to speak of
the phase relationships among different points of the E-neighborhood which tie the
ontological manifold to the phenomenological manifold. Presumably, these phase
relationships are the loci of transduction activity.
Consequently,
at least in terms of the phenomenological/hermeneutical context, the E-neighborhood should
not be thought of as merely a static structure that establishes a distance relationship
with a given point p. The E-neighborhood also may constitute a sort of dynamic envelope
which contains phase relationships linking manifolds on both sides of a given point of the
boundary structure. The nature of these phase relationships would vary with the identity
of the boundary structure point being considered.
A second
idea to consider is the possibility that inference may be conceived of as a complex
topological relationship involving a series of boundary points and interior points
connected by a dialectical relationship of some sort. For example, one could consider an
interior point to be an expression of focus, whereas a boundary point would be an
expression of horizon. As such, these two points mark the structural 'distance' - as
measured by a particular inference gauge- covered by the E-neighborhood surrounding some
aspect (i.e., 'point') of the structural character of the latticework of a premise in a
given inference network.
Both horizon
and focus could be considered as essential features of the complex boundary manifold
structure that like a macrophage has surrounded or engulfed a 'piece' of reality and begun
the process of 'digesting' that chunk of ontology - or, if not digesting, carrying on a
dialectical relationship with the reality structure. Seen from this perspective, the
phenomenology of the experiential field is a sort of complex membrane (or boundary
manifold) that is engaged by reality from a number of different directions and fractal
levels, including from "within'.
For example,
the body impinges on that boundary manifold just as much as the 'external' world does. As
a result, both focus and horizon can have E-neighborhoods that overlap with various
aspects of 'internal' as well as 'external' reality.
Furthermore,
one should keep in mind that horizon is part of the E-neighborhood of focus, just as focus
is part of the E-neighborhood of horizon. As such, when memories come forth without being
sought and engage different aspects of the structural character of focus, this is an
example of focus being part of the E-neighborhood of horizon.
On the other
hand, when one seeks out particular information one has learned in the past, this is an
example of horizon being part of the E-neighborhood of focus. Thus, in any given case, the
determining factor of which is to be considered to be the E-neighborhood of the other will
depend on the direction of the hermeneutical vector or tensor to which identifying
reference is being made - that is, whether the dominant orientation of the dialectical
engagement is from focus to horizon or from horizon to focus.
One could
continue along the same line of thought as outlined above and think of "normal"
consciousness as a relatively simple, closed, curve-structure or manifold of n-dimensions
which is part of a fractal network in which there are different levels of scale in this
network which fall beyond the perimeter of the parameters of normal consciousness. Indeed,
as difficult as it may be for normal consciousness to come to grips with, normal
consciousness actually may be quite near the bottom of the fractal network.
All of this
fits in with the work of, among others, Gazzaniga. However, one is not necessarily
required to share Gazzaniga's position that there is no essential, unitive potential in
consciousness capable of ordering, directing, orienting, shaping and
integrating the various tensor contributions of different fractal levels of modular
conscious structures.
Adherency, Continuity and methodology
Open sets
refer to sets consisting of only interior points, without any boundary points.
Thus, for instance, when considered as an open set, a disc on the plane does not include
the boundary points which form the perimeter of the disc. Furthermore, the point-set that
constitutes a ball in three-dimensional space does not include the boundary sphere which
surrounds it when that ball is construed as an open set.
If every
E-neighborhood of some given point 'p' contains at least one point of a set X, then, the
point p is adherent to the set X. Essentially, what adherency means,
is that if one has a set of points which is adherent to some given point-set X, then, when
the adherent points are not members of the set X, they are considered to be infinitely
close to the set X.
Thus, in the
case of a set of points adherent to a ball, B, of radius r about some point 'p', then,
aside from the points of the set which are part of the ball itself, the adherent points
will involve those points of the set which are on the boundary sphere surrounding the
ball. However, if it should be the case that the only points adherent to the point-set of
the ball, B, are those points of the ball itself, then, the point-set is said to be
closed. Therefore, any open set of points can be converted into a closed set
merely by adjoining all the boundary points surrounding the open set to that set.
This concept
of adherency may connect up with the ideas of inference and entailment. For example, one
might speak in terms of degrees of adherency in which there is a dialectic between, or
among, the points of an adherent set which are not members of some given point-set X and
those points of the adherent set which are members of the point-set X. The stronger, more
multi-faceted and more nuanced the dialectic, the greater the degree of adherency, and the
more plausible would be the inference or entailment relationship being considered. As
such, closeness would not be a matter of distance but of inferential or entailment
adherency.
A
point-structure which was adherent to some other point-set structure X would have
inferential or phase relationship ties with X. The greater the degree of adherency, the
more dense would be the network of phase relationships linking the E-neighborhood of the
adherent point-structure with the point-set structure X.
A further
consideration surrounds the issue of whether one should consider the hermeneutics of the
phenomenology of the experiential field to be a matter of an open set or a closed set. A
further possibility is that, under some circumstances, the hermeneutics of the
phenomenology of the experiential field may be closed in some respects but open in other
respects. A lot may depend on the structural character of one's understanding at a given
time.
Moreover, as
odd as it may sound, a hermeneutical open set would be one that did not include horizonal
input, whereas a closed set would be one that did include horizonal input. Consequently,
if considered in these terms, a closed set would be more receptive to horizonal input than
an open set.
An
additional consideration is the following. One might treat degrees of adherency as a sort
of measure of the openness or closedness of a point-set. If one were to do this, then, the
dialectic between a point-structure 'p' and the point-set structure X could be affected by
how open or closed the point-set structure is since it would be a direct reflection of the
extent to which points of the E-neighborhood of 'p' were part of the point-structure X, as
well as the character of the closeness of those aspects of the E-neighborhood of 'p' which
were not part of X.
A map from a
point set X to a second point set Y is said to be continuous only if for every point 'p'
of X, as well as for every E-neighborhood N of f (p) of Y, one can find a second
neighborhood S of 'p' in X that can be mapped by f to a subset of N. If the inverse image
of every open set in Y is also open in X, then, the map f from X to Y is said to be
continuous.
The
foregoing characterization seems to make our understanding of continuity dependent on the
methodological procedures one used in relation to any given 'p' in X and in relation to
any given subset of N in Y. For example, how one goes about choosing or selecting
point-structure candidates, or how one goes about deciding on a mapping structure from 'p'
to a subset of N, or how one goes about selecting a given subset of N, could all affect
whether or not one will find the mapping to be continuous. Even if the decisions
concerning the construction of the structural character of the mapping process are the key
factors in establishing continuity (given that point-sets X and Y have certain structural
properties that establish an envelope of constraints and degrees of freedom within which
the mapping process must operate), nonetheless, continuity still would depend on the
process that led to the construction of a map which could show one how to link any given
point in X with its image in Y.
Moreover,
while there might be a point'p' in X for which there is no mapping f by means of which one
can locate an image counterpart in a subset of N, nevertheless, until one establishes such
a non-homeomorphic relationship, one still would have an evidential basis for treating the
mapping as continuous. In other words, our understanding of continuity again would be
dependent on the underlying methodological procedures and concomitant capabilities of
showing that a given mapping is homeomorphic or non-homeomorphic.
Having said
the foregoing, one last caveat is in order. A distinction must be made between our
understanding of continuity in any given set of circumstances and the actual structural
character of those circumstances independent of our understanding of them.
What we take
to be continuous, on the basis of our methodological procedures for constructing or
generating maps, may not, in fact, be continuous. However, our procedures may not, yet,
have been able to establish this or are inherently incapable of doing so.
Alternatively,
the actual structural character of a given aspect of the 'fabric' of ontology may be
continuous, but that structural character may not be continuous in the way that our
methodology indicates (or fails to indicate) is the case. As a result, there is a
confusion between our conception of continuity in a given case and the actual structural
character of continuity in such a case.
A further
possibility is that our concept of continuity in a given case may be an analog for the
actual structure of continuity in that set of circumstances. However, unless we properly
understand the nature of that analog relationship, we may have a distorted understanding
of what makes continuity possible in that case.
As far as
the notion of continuity is concerned, what may be fundamental to any given 'space' or
latticework is not a series or set of points. Point-structures, themselves, may be a
function of an underlying fractal set of ratios of constraints and degrees of freedom.
This fractal set would have resulted through the dialectic of dimensions that has been
structured and arranged by the order-field which penetrates and permeates the different
levels of scale on which these fractal sets are given manifestation.
If this is
the case, one need not accept the idea that there are any elementary, simple
point-structures such as those posited in modern physics. Thus, for example, electrons,
photons, quarks and gluons may not be 'simple' in structural character. They may have a
complex structure which is dependent on a more fundamental set of dimensional-order
themes.
All
so-called point structures may merely be 'openings' on a given level of scale. Through
such openings a given ratio of constraints and degrees of freedom may be given expression.
These ratios
may, themselves, be manifestations of a deeper fractal dialectic of dimensionality which,
in turn, is dependent on an underlying order-field. This underlying order-field determines
what the character of each dimension will be as well as determines how, when and under
what circumstances such a dialectic of dimensions will occur.
The
character of the continuity which is given expression through the unfolding of an
order-field is a function of a shifting ratio of constraints and degrees of freedom as one
moves about a given latticework or neighborhood that is the manifestation of such an
order-field. In other words, the ties or links binding a latticework/neighborhood together
are the phase relationships that give expression to the spectrum of constraints and
degrees of freedom which the underlying order-field makes possible. This occurs in the
form of a latticework within which different aspects of that spectrum are manifested in
the form of point structures that represent dominant themes, on a given level of scale, of
the intersection of the dialectic of dimensions.
Each point
structure in the latticework or neighborhood is, in effect, a ratio of constraints and
degrees of freedom drawn from the spectrum of possibilities in the underlying order-field.
Consequently, as one moves about a given latticework or neighborhood, one will encounter
different manifestations of the set of constraints and degrees of freedom ratios which
constitute the envelope of values that govern or regulate the ontology of the given
latticework in question on a given level of scale.
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