Math Reflections and Resonances - Part 3
Brouwer's fixed-point theorem and reflexive consciousness
Brouwer's
fixed-point theorem is considered to belong to topology because it deals with continuous
mappings. Essentially, this theorem states that if one has a closed circular disc D (and
this encompasses the perimeter of the disc as well), then, every continuous map of a
figure F into itself has at least one point which will be mapped into itself, and this
point is referred to as a fixed-point. Stated in a slightly different way,
the theorem provides a basis for arguing that if the disc D is distorted in such a way
that the figure F continues to lie wholly inside the disk after the distortion, then,
there will be at least one point of the figure which will remain the same both before and
after the distortion- that is, the point will occupy the same position across the
distortion.
Perhaps, an
excellent phenomenological counterpart for the fixed point theorem is reflexive
consciousness. Sometimes, of course, distortions or transformations occur which disrupt
reflexive consciousness, and one becomes either temporarily dislocated or disoriented
(e.g., short-term amnesia), or one becomes chronically disoriented and/or dislocated as in
the case of schizophrenia (although even here one might want to argue that some points in
the intensely distorted landscape of the schizophrenic may retain their original positions
throughout the periods of deformation). On the whole, however, awareness of awareness is a
kind of stabilizing feature that allows one to locate oneself in phenomenology despite the
tremendous numbers and varieties of transformation and deformation which are occurring
there.
More
specifically, suppose: (a) one takes the horizon as describing a complex
phenomenological/hermeneutical n-dimensional analog for a closed circular disc; (b) one
treats focus or intentionality as constituting a structurally complex point within the
n-dimensional "space" defined by the horizonal perimeter (which is actually a
set of dimensional parameters or set of constraints and degrees of freedom), and (c) one
takes the relation between focus and horizon to be a dialectic describing a
phenomenological/hermeneutical process of transformation generating hermeneutical
structures or figures that lie wholly within the n-dimensional parameters of the closed
disk that are defined by the horizon. Given the foregoing three conditions, then, for most
deformations of the medium or "fabric" of the phenomenology of the experiential
field in which such transformations occur, there will be certain points along either the
horizon or focus or their dialectic which will remain fixed. That is, such
point-structures will occupy the same position within the phenomenology of the
experiential field after the transformation as they did prior to the transformation.
The
fixed-point aspect may be especially relevant with respect to the dialectic between focus
and horizon since a fundamental feature of being able to demonstrate the fixed-point
character of certain aspects of the phenomenology of the experiential field will revolve
around the issue of connectivity. In other words, one must be able to establish a viable
or plausible mapping between: (a) the neighborhood of a point in the pre-transformation
phenomenology of the experiential field and (b) a given neighborhood of a point in the
post-transformational phenomenology of the experiential field.
This mapping
must be such that the relationships of the points in question with their respective
surrounding neighborhoods will manifest minimal bijective mapping properties between the
two neighborhoods. In effect, one is talking about the structural character of the
identity of a given pre-transformational point/neighborhood relationship and the
congruence of such a relationship with the structural character of the identity of another
point/neighborhood relationship - namely, a post-transformational relationship.
A few
caveats with respect to the foregoing are in order. First, to speak of the horizon as
being a closed curve in any simplistic sense would be very misleading. The fact of the
matter is, the horizon of the phenomenology of the experiential field intersects with the
structure of the rest of ontology to form a complex boundary-manifold structure.
The
boundary-manifold is capable of changing its range of constraints and degrees of freedom
as a function of the dialectic occurring along that boundary. The boundary-manifold is
also a function of the way in which the internal dynamics of a given individual's
hermeneutics of the phenomenology of the experiential field help lend shape to the
dialectic along the aforementioned boundary.
As a result,
the set of constraints and degrees of freedom describing that boundary dialectic can
expand or contract over time. Nonethless, at any given time, at least under most
"normal" circumstances, the n-dimensional boundary which sets-off the
phenomenology of the experiential field from the rest of ontology represents an analog for
a closed circular disc in which any given level of scale represents a complex structural
slice of that n-dimensional boundary.
A second
caveat involves the issue of mystical states which seem to dissolve the boundary or
barrier between the individual's phenomenology of the experiential field and the rest of
ontology. These boundaries actually may be dissolved in some absolute sense. Or, the
boundary/barrier "merely" may undergo an incredible change in the structural
character of the level of scale.
Alternatively,
the structural character of the dialectic across the boundary may undergo a transformation
in which horizons merge such that one cannot separate one from the other even while the
two remain distinct on some level of scale of sufficient refinement. Or, some combination
of the second and third possibilities may be involved.
In this
event, the individual's normal veils of perception and understanding, that establish the
set of constraints and degrees of freedom through which we normally interact and respond
with the rest of ontology, are rendered inoperative. When this occurs, a less distorted,
more deeply penetrating and more intense communion may occur than is usually described by
our "normal" rationalistic dialectic with ontology.
Memory and
learning constitute complex structural "point/neighborhoods" which are the
result of hermeneutical operator activities. Among other things, these activities generate
mappings from a given focal/horizonal dialectic to those aspects of the phenomenology of
the experiential field which are sensitive to receiving such mapping (e.g., the memory
field).
These
mappings may be bijective, non-bijective, or somewhere in between, depending on the extent
to which congruence can be established between the memory point/neighborhood structures
and the original focal/horizonal dialectic through which the hermeneutical operator
mapping arose (i.e., a learning process). In fact, one might think of memory as a mapping
onto, or into, reflexive consciousness which satisfies Brouwer's fixed-point theorem to
varying degrees, ranging from a minimum of one point (in the case of an extremely
distorted memory) up to a bijective mapping (as in the case of, say, eidetic imagery) in
which the present awareness of a memory structure is, despite a variety of distortions
that may have occurred in the interim, homeomorphic with the latticework which was
originally laid down at the time of experiential learning when the memory structure was
first created or generated.
Giving the Jordan curve theorem an interpretative twist
The Jordan
curve theorem states that a simple closed curve, (i.e., one that does not intersect
itself) divides the plane on which it occurs into two parts. This theorem has two aspects.
One aspect
concerns the idea of a curve which does not intersect itself. The other aspect revolves
around the idea of a plane which has been divided into two parts. Both of these aspects
have topological properties.
The first
aspect is topological because any given homeomorphic image of a simple closed curve will
also be a simple closed curve since any given simple closed curve is a homeomorphic image
of a circle. The second aspect is topological because the complement to a
simple closed curve involves two disconnected parts, and this is also a topological
property.
Although the
theorem appears to be quite simple, it is often quite difficult to prove. One
of the reasons for this is that one can run into considerable difficulty trying to
determine whether any given point falls inside or outside of a given closed curve,
provided that such a curve is sufficiently complex.
One possible
application of a modified version of the Jordan curve theorem to the context of
phenomenology and hermeneutics might concern the noumena/phenomena boundary-manifold
structure. This boundary structure could be construed as representing a complex closed
figure of n-dimensions.
Moreover,
one also could consider the way in which the boundary-manifold tends to separate the plane
of existence into two parts. More specifically, this would involve: (a) those aspects of
noumena which fall outside the boundary and, therefore, beyond the perimeter of the
interface of ontology and the phenomenology of the experiential field; (b) those aspects
of the phenomenology of the experiential field, including the points of noumena/phenomena
interaction, that fall within the perimeter of the boundary-manifold.
As is
generally the case with respect to the Jordan curve theorem, one also has trouble in
hermeneutics trying to determine if a given point falls inside or outside the
aforementioned boundary-manifold structure. This is an important point to establish.
It concerns
one's ability to determine whether one is merely talking about some projection of
imagination, or whether one is talking about some aspect of ontology that lies beyond the
boundary structure and, yet, which helps make a boundary-manifold of such structural
character possible. In other words, the point being raised concerns the issue of whether
one has constructed something which is shaping the boundary structure in a distortive or
veiling manner, or whether one has merged horizons with an aspect of the ontology that
lies on the other side of the boundary structure and which is responsible for that aspect
of the boundary structure having the character it does.
One possible
source of difficulty with respect to the application of the Jordan curve theorem to the
hermeneutical and phenomenological contexts concerns the following question: How does one
deal with the fact that reflexive consciousness seems to constitute a case of a manifold
intersecting itself, and, therefore, appears to fall outside the purview of the Jordan
curve theorem which is about curves that do not intersect themselves?
By
definition, the meaning of "simple" is non-intersecting. If a curve intersects
itself, it cuts the plane into more that two sections and, therefore, falls outside of the
purview of the Jordan curve theorem. However, in relation to the issue of reflexive
consciousness, the property of intersection need not necessarily cause the manifold of the
phenomenology of the experiential field to be compartmentalized into more than two
sections.
If this is
so, then, in some cases (reflexive consciousness being one of them) one may be able to
separate the issues of simplicity, on the one hand, and, on the other hand, the property
of generating more than two sections in a manifold surface during the process of
intersection. Thus, conceivably, in special cases a curve that intersects itself may not
be simple and, yet, still qualify as a context for which the Jordan curve theorem has
applicability.
The
focal/horizonal structure could be considered as a closed curve which intersects itself
under the condition or operation of reflexive consciousness. Although the focal/horizonal
structure compartmentalizes the manifold of the phenomenology of the experiential field
into two sections - namely, that which falls within the focal/horizonal structure and that
which falls outside the parameters of the boundary of that structure, nonetheless, when
the operation of reflexive consciousness is introduced, no further compartmentalization
occurs.
There are
still just two compartments such that the boundary structure which separates 'inside' from
'outside' remains intact, the same as it was before. What has been added is an added
dimension of awareness of the separation and the structural character of the boundary as
having a certain set of latticework properties.
In fact, the
focal/horizonal structure can be construed as a fractal version of the relationship which
the manifold of the phenomenology of the experiential field has with the larger field of
ontology within which it resides. Indeed, one even can take any given focal/horizonal
structure and take it down to another level of scale by examining the details of any
aspect of the larger focal/horizonal structure. This changing of perspective through
switching levels of scale can be repeated as many times as one's methodology and ontology
permit.
In some
cases reflexive consciousness seems associated with instances in which the manifold of the
phenomenology of the experiential field is compartmentalized into more than two sections.
For example, phenomena such as fugue states, multiple personality, or the sorts of things
which Gazzaniga and Fodor talk about in relation to the idea of modular consciousness,
all seem to lead to the compartmentalization of the manifold of the phenomenology of the
experiential field in a way that yields more than two sections of the manifold.
However,
reflexive consciousness may not necessarily be the cause of such compartmentalization.
Other kinds of forces or transformational elements may have introduced fissures into the
manifold. When the operation of reflexive consciousness is introduced so that these
compartments become aware of themselves as a structure of one sort rather than some other,
everything is left as it was before, and no new compartments are introduced.
Consequently,
considered in terms of any given compartment of the manifold, reflexive consciousness
doesn't introduce new compartments. It highlights the character of the inside/outside
distinction of the boundary structure of the compartment which gives expression to the
focal/horizonal structure that has come into play.
The
compartment's focal/horizonal structure is the closed curve, and everything which is not
included in the latticework of that structure is, by definition, outside of the structure.
There may be many compartments beyond the boundary parameters of the focal/horizonal
structure, but they are all part of the same 'outside'. As a result, the Jordan curve
theorem still holds in as much as we are still talking about a closed curve that divides
the manifold into two sections despite its intersecting with itself through the operation
of reflexive consciousness.
The reason a
geometric closed curve which intersects itself divides the plane into more than two
sections is due to the nature of the structural character of the points that generate the
curve or from which the curve is constructed. Geometric points are said to have position,
but they do not occupy or take up any space.
The idea of
"position" refers to the relationships of proximity (as well as, 'before' and
'after) which points have with respect to another. Leaving aside considerations of whether
one can speak intelligibly of points that do not occupy space, one can still locate points
by describing their relationships with one another in the context of a given curve, figure
or structure. Thus, if one is proceeding in a given direction along a curve, different
points will be before other points, and other points will be after those points. Moreover,
some points will be proximate to certain points and distant, relatively speaking, from
still other points.
Considered
in terms of these relationships, when a curve intersects itself, it cannot but help to
affect the character of such relationships. By crossing the boundary structure of
the curve, it cuts through one or more of these relationships.
Thus, for
example, one point which was next to another point is no longer next to that point. Each
of those points is now next to a new point that has come between them. Or, whereas prior
to the intersection, one point may have preceded another point in terms of when,
respectively, one encountered those points as one traveled in a given direction along the
curve, after the intersection, the point which previously had occurred after another point
may now be encountered before that point as one traces a path along the curve.
If none of
these sorts of changes in relationship occurred, then, one would have to question just
what the meaning of the idea of geometric intersection involved. Indeed, in geometric
contexts, inherent in the very nature of the process of intersection seems to be the fact
that the relationships among certain points are disturbed or affected in some discernible
way. If there is no trace or evidence that such a disturbance of point-relationships has
occurred, then, in the geometric context, one has a prima facie case that intersection has
not occurred.
Of course, a
curve does not have a relationship with just itself. It also has a relationship with the
plane on which it exists as a curve and which makes its existence as a geometric structure
possible. This is where the aspect of compartmentalization of the plane comes into effect,
for when a curve intersects itself, it alters the relationship which the curve has with
the plane since the curve/plane relationship goes from (in the case of a closed curve)
being one of two compartments, to one of being more than two compartments.
In the
context of phenomenological structures, reflexive awareness of such a structure can be
seen as a closed curve that intersects with itself without, at least as far as the issue
of compartmentalization is concerned, altering the relationship which the focal/horizonal
structure has with the rest of the manifold of the phenomenology of the experiential
field. There was an inside/outside relationship before the operation of reflexive
consciousness was introduced, and there is still the same inside/outside relationship
after the operation of reflexive consciousness is introduced.
Such a
structure may, or may not, be simple in the geometric sense, but it does not generate or
introduce any new compartmentalizations. Therefore, the operation of reflexive
consciousness preserves the logical character of what seems, essentially, to be meant by
the idea of simple curve - i.e., that which does not alter the relationship of the
structure with the larger manifold context as far as to the issue of compartmentalization
is concerned. In effect, one has one interior compartment (formed by the focal/horizonal
dialectic) whose contents are constantly changing with shifts in the character of the
focal/horizonal dialectic.
Thus, there
is a sense in which boundary structures are crossed (focally and horizonally) as one goes
from pre-reflexive awareness to reflexive awareness, and, therefore, the minimal
conditions for intersection have been satisfied. Nonetheless, the change in boundary
perspective does not lead to compartmentalization as in the case of geometric
intersection, although the former does lead to a change in the character of the
relationship which the focal/horizonal structure has with itself.
The
geometric curve's intersection with itself affects the character of the pre-intersection
relationships which points had with one another. On the other hand, in the case of
reflexive consciousness, the change in the character of the relationship concerns the
dimension of self-awareness and what that relationship makes possible. Such reflexive
consciousness serves as a 'doorway' though which the hermeneutical operator can be
introduced, dynamically and dialectically, in a conscious rather than in an unconscious
manner, and in a directed rather than a haphazard manner.
Thus, the
key issue in the whole aspect of intersection, as far as the Jordan curve theorem and
reflexive consciousness are concerned, is a matter of compartmentalization. As Gazzaniga
has demonstrated, on certain levels of scale, reflexive awareness can be
associated with a compartmentalization of the phenomenology of the experiential field.
However, as was suggested in the foregoing discussion, the nature of the intersection of
the manifold of the phenomenology of the experiential field need not necessarily affect
this facet of things.
In general,
reflexive consciousness is a phenomenological manifold which intersects itself in, at
least, two sets of neighborhood points. One set of neighborhood points is called the
"horizon". The other set of neighborhood points is called the "focus".
The structural character of the latticework which links focus and horizon is a complex
vectoral or tensoral expression of the hermeneutical operator. This tensoral expression
often highlights the distinction between inner and outer. In this sense the effect of
reflexive consciousness is congruent with certain aspects of the Jordan curve theorem.
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