Holographic Images - Part 4
On the Structural character of a point
In June of
1854, Georg Friedrich Bernhard Riemann, gave a lecture entitled: "On the Hypotheses
which lie at the Foundations of Geometry". In this lecture he said:
"...geometry
presupposes not only the concept of space but also the first fundamental notions for
constructions in space as given in advance. It gives only nominal definitions for them,
while the essential means of determining them appear in the form of axioms. The relation
(logic) of these presuppositions [postulates of geometry] is left in the dark, one sees
neither whether and in how far their connection [cause-effect] is necessary, nor a priori
whether it is possible."
In essence,
what Riemann was getting at in his lecture is that philosophers and mathematicians had
imposed an Euclidean order on the ontology of space without bothering to determine whether
or not such an imposition was warranted. Furthermore, the imposition had occurred without
anyone having a fundamental and clear grasp of the extent to which the logical
relationships among the set of postulates that have been imposed on ontological space are
necessary.
Riemann felt
one of the fundamental problems with geometry was that its foundations had been left in
shadows. Instead of having started from true first principles, Riemann claimed Euclidean
geometry had emerged from certain kinds of presuppositions that were somewhat removed
from, and beyond, the realms of defensible foundational considerations.
One of the
shadows which had been cast across the foundations of geometry concerned the idea of a
point. Riemann believed the same fundamental principles governed the properties of points
in both curves as well as straight lines, but this set of common principles
could not be elucidated as long as one approached geometry in the traditional manner of
Euclid. Consequently, Riemann proposed to construct a multi-dimensional concept of space
using the idea of quantity as the basic building block in his construction process.
Riemann's
starting point was an intuition about the nature of quantity. This intuition revolved
around the idea that one encountered quantity through measurement.
In other
words, whatever quantity is, it is something that is measurable or to which the process of
measurement is applied. For Riemann, measurement involved the superimposing of two
magnitudes: one magnitude was the quantity whose magnitude was not currently
known; the other magnitude was the mode of measurement which was to be used to determine
the character of the first magnitude.
The key to
this process of superimposing was locked within the idea of continuity. Superimposing
could only occur, according to Riemann, when one magnitude is part of the other magnitude
with which it is being compared. In other words, one magnitude only could be
superimposed with another magnitude when the two were, in some way, continuous.
The
aforementioned feature of being 'part of consists of a very precise and exacting sense of
the notion of continuity. More specifically, in order to demonstrate that two magnitudes
are continuous in the way which would be necessary to make superimposing possible, one had
to show that, at a minimum, at least one of the elements of a magnitude had the capacity
to affect at least one of the elements of the other magnitude.
Suppose one
had two elements x and y. Suppose, further, that a one unit change in element x brought
about a one unit change in element y.
If one
constructs a graph of x versus y based on the foregoing relationship, one will get a
straight line. A straight-line graph describes a linear relationship between the elements
being graphed. In such a relationship, the ratio of x to y remains constant irrespective
of the size of the values involved.
A curve can
be described as the envelope of its tangents. When dealing with the very
prototype of curvature - namely, a circle, tangents can be constructed for each and every
point of the circle.
Of
importance here, as far as Riemann's project is concerned, is the fact that the tangent is
linked to a single point. A tangent is also a function of an angle, and this angle can be
construed as being a sort of indicator of directionality.
In the case
of a straight line, all the points on that line are considered to have the same direction.
As a result, any attempt to construct a tangent for the points on a straight line
would not be able to reveal any information about changes in directionality.
On the other
hand, in the case of a circle, neighboring points along any aspect of the circle's
curvature will display slightly different directional characteristics. These directional
characteristics are revealed in the differences manifested in the unique nature of the
tangent which can be constructed for each of these neighboring points. When these tangents
are altered, as one moves from point to neighboring point along the curvature of the
circle, the transitions in the value of the tangent inform one about how the aspect of
directionality is affected by shifting from one point to the next.
If one
assigns a tangent to any given single point on the x-y curve, the curvature of the point
at that juncture will establish the slope of the assigned tangent. Furthermore, if one
could actually examine a single point on the curve, the direction of that point would
coincide with the slope of the tangent which had been constructed for that point.
In
actuality, however, one could never really examine such a single point. This is the case
since the points on the line supposedly have position without occupying space, and are,
therefore, infinite in a way which does not permit any individual point to actually be
identified in a concrete manner. The points exist as neighboring
relationships of relative position without size.
Nonetheless,
one can increasingly reduce the values of x and y so that they approach the hypothetical
point on the curve to which a tangent has been drawn. As the values of x and y get closer
and closer to this hypothetical point, the discrepancy between the value of curvature and
the slope of the tangent becomes increasingly smaller.
In short,
one approaches the limit of changes in y in relation to x. The process of
locating such limits is the task of differential calculus.
Through the
operation of differentiation, which is one of the basic operations of differential
calculus, one attempts to establish those limit-approaching ratios of x and y (known as
derivatives). These ratios permit one to identify the juncture where the curvature of a
single point on a curve is synonymous with the slope of the tangent which can be drawn to
that point.
Supposedly,
the derivative acts as a guarantee of the continuity between x and y at a given point.
Theoretically, this limit ratio or derivative is capable of satisfying Riemann's
requirements for the process of superimposing of magnitudes such that y becomes part of x.
Derivatives
have an important link to 'e' - the base of natural or Naperian logarithms. 'E' links y to
x as a function: namely, y = ex. Thus, when x = 1, then,: y = ex = e1
= 2.71821...; when x = 2, then,: y = ex = e2 = (2.71821...) x
(2.71821...) and so on. The plotting of the graph of this function yields a smooth,
regular sigmoid curve. The uniqueness of 'e' lies in the fact that the value of the
function y = ex in any given case yields the same value as the derivative in
that case.
Although, as
indicated previously, one never actually can see the relationship of points being referred
to in the limit ratio that constitutes the derivative, in the instance of 'e, the graph of
y = ex gives a macro depiction (i.e., a structure in perceptual space) of the
structural character of curvature on the micro scale of infinitesimal points. The
implication here is that if one actually could see what the structural character of a
derivative is like on the infinitesimal scale of neighboring points along a curve, one
would see what one sees when one plots the graph of the function y = ex -
namely, a smooth, regular sigmoid curve.
A possible confusion between methodology and ontology
There may be
a confusion in the foregoing between the idea of a derivative which serves as an index of
relationship in a given region of space and the actual point itself. In other words, the
derivative associated with e designates a limit-area or region near to, or in the
neighborhood of, a given point that is part of the graph of y = ex. When this
derivative is translated into graph form, it yields a smooth, regular sigmoid curve.
However, this sigmoid curve may not so much capture the structural character of a given
point as it captures the structural character of the relationship of a set of neighboring
points when the property of directionality undergoes transition as one moves through
curvature.
The
derivative is always relational and contextual. The derivative never concerns a single
point in isolation. It focuses on how one point relates to another point in terms of
alterations of directionality as one moves from one point to another along a curve.
Similarly,
the function of y = ex is always relational. As such, although one can isolate
points on the curve that are described by this function, these points are indices for
relationships between x and y. In this sense, they are special kinds of points -
relational points.
Relational
points link together two or more values or magnitudes in the form of a juncture which can
be static, dynamic or dialectical, depending on the character of the things which are
being linked. Therefore, neither the graph of the derivative associated with 'e', nor the
graph of the function y = ex, actually isolate or identify or make reference to
a single point.
One might
suppose, nonetheless, that the reason why Riemann's intuition works is due to the way it
allows one to explore the structural character of relationships among points and values in
regions of space which can be made arbitrarily small to suit one's current needs for
precision and rigor. The fact one has not captured the actual fundamental unit of space
(assuming, of course, there is such a fundamental unit) doesn't really matter since one
has found a unit which is small enough to help one to explore and capture the structural
character of what one is studying.
In this
sense, what is important in Riemann's methodological process of superimposing is not that
one element, y, becomes part of some other element, x. What is important is that one's
units of measurement provide a means of capturing the relationship among a set of points
that are fundamental to the structural character of the magnitude or quantity being
measured.
The better
one's mode of measurement, the more congruent will be the structural character of the
fundamental relationships in one's mode of measurement with the structural character of
the fundamental relationships in that to which identifying reference (through measurement)
is being made. The key lies in congruence (broadly construed) and not in Riemann's notion
of superimposing.
Continuous
relationships are a matter of discrete continuity in which discrete features, aspects,
properties, etc., are linked together by a set of inter-locking and overlapping
relationships. The continuity is provided through these facets of inter-locking and
over-lapping properties which provide a means for certain aspects of a structure to
continue to manifest themselves despite the fact other aspects of that structure no longer
are expressed. This is like the way in which the handing on of the baton in a relay race
permits the race to continue despite the fact that a new, discrete entity (i.e., a runner)
has entered the picture, while previous discrete participants in the race no longer
continue to play a role.
Curvature: a unit of relational measurement
In an
attempt to elucidate Riemann's thinking, Paul Pietsch, in his book Shufflebrain,
asks us to suppose that 'e' is the only metering device available to one. In addition,
Pietsch suggests one consider the visual image of a string of pearls made up of e-units.
These units can be increased or decreased in number and which together can be used to form
different kinds of curvature. However, the string of e-units can neither be stretched nor
broken.
If one had a
flat surface on which there were two points x and y and one wished to determine the
shortest distance between them, one could use the string of e-units as a measuring device.
Seemingly, the shortest path between the two points would be that one which contained the
least number of e-units. If the shortest path were represented as being 'x' e-units in
length, this length would not change if one were to curve the string by putting it around
a person's neck.
Thus, flat
surfaces and curved surfaces can be related through the notion of least curvature of the
path which links any two points on either surface. As indicated previously,
least curvature is defined in terms of determining the least number of e's that can link
the two paths.
Next,
Pietsch asks one to imagine a triangle which is to be measured by the string of e-units.
because the string is very loose relative to the rigidity of the lines of the triangle,
there is considerable difficulty in getting an accurate measurement of the length of the
triangle's sides. Yet, if one decreases the size of both the triangle and the string of
e-units, then, the accuracy of the measuring device becomes increasingly more accurate
when any given side of the triangle and the length of the string of e-units approach one
another as a limit.
Supposedly,
at infinity, at least one of the points of the string of e-units can be superimposed on at
least one of the points of the sides of the triangle. When this occurs, then, at least at
one point, one magnitude (i.e., the measuring device) becomes part of another magnitude
(i.e., the structure to be measured).
In this way,
the measuring magnitude, consisting of e-units, is said to have one feature in common with
the quantity magnitude being gauged by the measuring device. The feature which they hold
in common is said to be curvature.
One cannot
actually argue that a given length of a string of e-units is the same as any given side of
the triangle. This would give rise to an apparent contradiction in which the straight line
of a triangle has the same smooth, regular sigmoid character that a graph of e-values has.
Nonetheless, at least at one juncture, the relationship between the string of e-units and
a side of the triangle manifests the property of superimposing in which both have the same
degree of curvature and, thereby, one becomes a part of the other.
The
foregoing account seems to create a problem. If one cannot say that any given side of a
triangle, taken as a whole, has a superimposable relationship with a string of e-units,
taken as a whole (i.e., a straight line is not the same as a sigmoid curve, then, just
what becomes of the idea of measurement?
Presumably,
in order for one magnitude to be able to measure or gauge another magnitude, then, one of
the magnitudes taken as a whole must be superimposable on the other magnitude taken as a
whole. Whenever and wherever there is deviation from a relationship of superimposing of
the two magnitudes, one introduces a degree of error or inaccuracy into the measuring or
gauging process.
If one is
uncertain as to the number of points at which superimposing holds, then, one is really
uncertain about the actual gauge of the magnitude being measured. Furthermore, one does
not have any means of estimating just how frequently superimposing deviations occur.
To be sure,
Riemann may be less interested at this point in the idea of measurement than he is
interested in trying to determine the structural character of the fundamental unit of
space- namely, curvature. However, as suggested previously, Riemann has not really
established that the fundamental unit of space is that of curvature.
What he has
established is that one can use the idea of curvature as a fundamental unit of relational
measurement and, thereby, produce heuristic results. Such results allow one to model
various facets of the magnitude of quantity to which one is making identifying reference
through the measurement process. In other words, Riemann has found a means of
operationalizing the concept of quantity as a function of curvature, but he has not
necessarily fathomed the fundamental structural character of the magnitude of quantity per
se.
Curvature in
Riemann's sense is a relational concept that exists among a set of points or values and
does not necessarily reflect the fundamental structural character of a unit of space. As a
result, once again, a distinction has arisen between the structural character of
methodology and the structural character of the ontology such methodology is attempting to
engage as a means of helping the individual to orient himself/herself with respect to some
aspect of experience.
According to
Riemann: " About any point, the metric [measurable] relations are exactly the same as
about any other point.". In other words, the same fundamental units are
involved in the construction of lines, surfaces and spaces, irrespective of whether those
lines, surfaces and spaces are linear or curved. Each of these geometric structures is
determined by, and a function of, the property of curvature.
Riemann
claimed to demonstrate that when one analyzes the magnitude of flatness in terms of its
most fundamental aspects or units (namely, points), one discovers that these fundamental
units are but a special case of the property making up the fundamental units of curved
geometric structures. In effect, the fundamental linear units making up the structure of
straight lines, flat surfaces and rectilinear spaces give expression to the property of
zero curvature.
For Riemann,
geometry, of whatever sort, was constructed from fundamental or elementary units of
curvature, and curvature was a manifestation of the character of the relationship among a
set of points or values. These relationships could assume a positive, negative or zero
value, and, taken collectively, they represented a spectrum of infinite
curvature with respect to which any possible geometric figure could be subsumed as a
simple or complex function of such curvature.
Riemann's
position is not anti-Euclidean. Riemann is attempting to show that geometry does not begin
and end with the Euclidean methodology.
Moreover, he
is attempting to show there are limits to what Euclidean methodology can be fruitfully and
accurately applied. Euclidean geometry works quite well in the context of simple and
uncomplicated spaces, planes and dimensions. However, Euclidean geometry is incapable of
handling geometry involving infinitely small regions.
Moreover,
the structural character of the Euclidean plane is such that one could never show that
parallel lines are capable of crossing. The reason for this is because the Euclidean plane
is constructed from units displaying zero curvature. However, in those geometric planes
constructed from units of non-zero curvature, one is able to show there are cases in which
the appropriate kind of curvature will permit parallel lines to cross at some point.
In general
terms, Riemann holds that the shortest distance between two points is the path showing
least curvature among all the paths which might be drawn between those two points. In the
case of Euclidean geometry, the shortest path is the one displaying zero curvature. This
is expressed as a straight line.
Riemann held
a dynamic understanding of what Pietsch refers to as the idea of "active zero".
This is the zero between +1 and -1, not the zero of nothingness.
It is a
relational concept forming part of a continuum with other values. It is not an absolute
emptiness. Active zero is a relational but neutral presence.
As such,
zero space identifies that part of the infinite spectrum of continuous curvature which
lies between positive and negative curvature and which serves as a connecting link between
positive and negative curvature. Zero space geometry encompasses those aspects of the
infinite continuum of curvature involving units of construction displaying zero curvature,
and this is the realm with which Euclidean geometry deals.
In summary,
there are at least three basic principles characterizing Riemann's position:
(a)
Geometric coordinates are a function of the elements of curvature and not vice versa.
(b) point
(a) follows from Riemann's discovery that the relationship of points in the neighborhood
of any given point is the same as the relationships of points in the neighborhood
surrounding any other point. This means the geometric properties describing a given
coordinate system will actually be a transform of the properties describing some other
coordinate system. This is the case since underlying both coordinate systems will be a
common structural bond in the form of the basic unit of curvature.
(c) the
property of least curvature constitutes the structural theme that is at the heart of the
transform operation linking one coordinate system with any other coordinate system.
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