Math Reflections and Resonances - Part 1
Relation and functions
Suppose one
has two sets of objects and, then, joins them through the operation of intersection.
Suppose, further, one establishes some sort of correspondence between the objects of these
two sets. Such a correspondence can be described in terms of a relation F between
the elements of the intersection of the two sets. One of the two sets is stipulated to be
the domain of definition or D(F), and the remaining set is said to be the range
or R(F).
The
essential feature of a function concerns the nature of the correspondence through
which one set of objects is assigned to objects of another set. Thus, the character of the
assignment or correspondence is at the heart of the concept of a function.
A function
is a special kind of relation F which assigns or maps elements from the domain into
elements of the range. More specifically, if there is one and only one element (b) of the
range, R(F), which corresponds to each element (a) of the domain, D(F), then the relation
(F) is termed single-valued and the relation is called a function. Moreover, the range
element (b) which is assigned to the domain element (a) is referred to as the image
of (a).
The
foregoing can be summed up in the following way. A function is a set of ordered pairs
(a,b) whose first element is a member of the domain of definition D(F) and whose second
element is a member of the range R(F).
A mapping of
A into B requires that every element of A be an original element. A mapping of A onto B
requires that every element of B be an image of an element of A. The element
(b) which is assigned to an element (a) by the function f is symbolized as: 'f(a)'. This
correspondence relation is written either as: ab = f(a), or as b f(a). The element (a) is
referred to as the argument, and the element (b) is called the function value f(a)
at the point a.
Virtually
all operations in mathematics can be construed in terms of a correspondence which gives
expression to a rule of linkage that specifies how to map each and every object or element
of some abstract space S to a corresponding image object or element in some other abstract
space O. Although O may be distinct from S, this is not necessarily the case.
The
foregoing sort of correspondence is known as a mapping of S into O, and the rule of
linkage or correspondence is referred to as an operator. One can write the
correspondence in the following form: y = A(x), where A stands for a given operator, and x
stands for the element of a given space which is being operated on and linked to y by
means of A.
In short, in
order to be able to characterize a function, one must be able to give: (i) the domain of
definition; (ii) the range, and, finally, (iii) the assignment or mapping procedure which
relates the elements of the domain to the elements of the range.
In the
hermeneutical context, the domain of definition consists of the various products of the
hermeneutical operator. The range is expressed in terms of the spectrum of
phenomenological structures which arise in the experiential field. The mapping between
domain and range is the latticework of understanding that attempts to account for why the
components of the range have the structural character they do.
Vector spaces: mathematical and hermeneutical
The idea of
a vector space can be described in terms of a set of elements which can be combined
through different operations (usually addition and multiplication) in such a way that the
results of these operations will not be an element falling outside the structural
parameters of the set in question. Linear algebra is considered to be concerned with the
theory of vector spaces.
Vector is
the term given to any element of a vector space. Scalars refer to the numbers which
are used to multiply vectors. A set of scalars can consist of rational, real, or complex
numbers, as well as such structures as fields.
There are
eight basic rules which define the general properties of a vector space:
(1)
associative law of addition - (a + b) + c = a + (b + c);
(2)
commutative law of addition - a + b = b + a;
(3)
existence of zero - there exists an element '0' such that a + 0 = a for all values of
‘a’ in the vector space;
(4)
existence of inverses - for each value of 'a', there is an element, -a, such that a + (-a)
= 0;
(5)
associative law of multiplication - n(ma) = (nm)a, where n and m are scalar in character;
(6) unital
law - 1 a = a;
(7) first
distributive law - n(a + b) = na + nb, where n is a scalar quantity;
(8) second
distributive law - (n + m)a = na + ma, where n and m are scalars.
Whenever the
elements of a set can be combined through the operations of addition, as well as can be
multiplied by scalars, without either: (a) exceeding the structural boundaries of that set
of elements, or (b) violating any of the above eight rules, then, that set is said to
constitute a vector space. The set of real numbers, the set of complex numbers, as well as
the sets of all integrated and differentiable functions form vector spaces.
In the
hermeneutical context, rules stipulate realms of constraints and degrees of freedom. The
interaction of these various sets of constraints and degrees of freedom leads to a
dialectic that spins the woof and warp of the latticework which gives expression to the
structural character of the sort of 'space' or context being described.
A
hermeneutical or phenomenological space may be held together by a set of "rules"
or principles or themes in a way that is comparable, in the sense of an analog, to the way
in which a vector space is held together by the aforementioned eight rules. Part of the
task of hermeneutics is to try to specify, to whatever extent is possible in a given set
of circumstances, what these analog rules and/or principles are.
A shift in
vector space occurs whenever there is a point E (known as the 'end-point') which
can be associated with each original point O (known as the 'initial point' such
that: (1) a directed or oriented line segments connecting any given point O with its image
E (a directed line segment of this sort is called a 'representative of the vector')
will be parallel with a new directed line segment that connects a point On to
its image En; and, (2) the new directed line segment is of the same length as
the original directed line segment. The distance between the initial point 'O' and the
end-point E is called the norm or modulus.
A transition
of directed line segments satisfying these two conditions is referred to as a
'translation' or 'vector'. As such, a vector can be considered to be a shift in
3-dimensional space.
The
hermeneutical/phenomenological counterpart for the aspect of parallelism among vectors may
be rooted in the structural character of the analog relationship between, or among,
different latticework's. More specifically, as indicated above, a directed line segment
must be parallel with, and the same length as, another directed line segment in order for
both line segments to be considered to be representatives of the same vector. Similarly,
in order for an oriented or directed hermeneutical latticework to be considered to be an
analog for some other oriented or directed latticework, there must be a constancy to the
structural character of the "distance" between the latticeworks, together with a
constancy to the structural character of the "length" of the respective
latticeworks.
When
translated into the hermeneutical or the phenomenological context, the ideas of 'distance'
and 'length' become qualitative in character rather than quantitative. As such, 'distance'
and 'length' both will be construed in terms of the hermeneutical operator's dialectical
interaction with a given set of structures and concomitant phase relationships. Therefore,
the hermeneutical counter-part for 'length' might be construed in terms of congruence, and
the hermeneutical counterpart for 'distance' might be construed in terms of inferential
functions.
In order for
two latticeworks to be analogs for one another, the two must be:
(a)
expressed through different mediums;
(b) capable
of preserving a set of phase relationships within a latticework which is congruent with
the phase relationships of its counterpart image latticework in the character of their
respective inferential structures and dialectical interactions;
(c) the
phase relationships which are preserved will be continuous, rather than discrete, (or vice
versa) by virtue of the manner in which they are linked together by the orderedness of the
dialectic of dimensionality that makes a latticework of such structural character
possible.
All
representatives of the same vector will be parallel as well as have the same length.
Similarly, all representatives of the same oriented analog structure will display the
hermeneutical counterparts to being parallel as well as having the same length. Thus, they
will show congruence through preserving the inferential structure of phase relationships
in such a way as to be reflective of their analog images.
Two or more
vectors can be added together by taking the sequence of representatives which are to be
combined and, then, proceeding so that the end-point of the first representative in the
sequence serves as the initial point of the next representative in the sequence to be
added. One should continue on in this fashion until all the representatives in the
sequence have been exhausted. The sum of this combination of representatives is given
expression by the line segment extending from the initial point of the first
representation to the end-point of the final representative in the sequence.
The zero
vector (also known as the null vector) refers to that vector or translation
that leaves unchanged any given point 'O' to which it is applied. In effect, the zero
vector is a translation which "shifts" a point 'O' onto itself. This vector has
zero length and has no direction or orientation.
One might
want to treat reflexive consciousness as a sort of operation which, when added to a given
aspect of the experiential field, is capable of (but may not always realize its
capability) leaving everything as it was or is. It is something like a dynamic zero or
null vector which allows one to examine a given aspect of the experiential field from a
variety of different viewpoints without altering the structural character of the
latticework being examined. The idea of reflexive consciousness as a sort of dynamic null
vector is somewhat reminiscent of the way Paul Pietsch talks about Riemann's idea of zero
curvature being an active rather than a passive zero.
The inverse
of a vector is obtained by interchanging the initial point and the end-point of a vector.
This is done in such a way that although the length remains the same, the direction or
orientation of the vector has been reversed.
Suppose, in
the hermeneutical context, one takes some point in the spectrum of a given focus as the
initial point. Suppose, further, that one takes some point in the spectrum of horizon as
the end-point. If the starting orientation of the 'line segment' joining these two points
goes from focus to horizon, then, one could say that the inverse of this representative or
vector is a line connecting the same two points but with an orientation extending from
horizon to focus.
For example,
the hermeneutical vector extending from horizon to focus could be considered as an
expression of the dialectic which occurs when memory comes to bear on some on-going focal
point of the experiential field or the phenomenology of the experiential field. On the
other hand, the vector extending from focus to horizon could be construed as an expression
of the dialectic which occurs during the process of learning, when focus links up with
some aspect of the horizon in order to place a new piece of data in an appropriate
neighborhood or latticework category.
In both
cases a shaping or modulation process is involved. However, the difference concerns the
direction or orientation of the shaping process and whether it is focus or horizon
(learning or memory respectively) which is the primary instigator and/or influence during
the shaping process. This would make learning and memory inverses of one another in the
context of hermeneutical/phenomenological "vector spaces".
In an
orthogonal or Cartesian system, the unit vectors that have a positive direction in the x-,
y-, and z-axis are referred to as basis vectors of the coordinate system. Moreover,
if one considers any given vector 'v' along any of the three axes x-, y or z-, and if 'r',
's' and 't' are the positive unit vectors for, respectively, the x-, y- and z-axes, then,:
vx = vx times r, vy = vy x s, and vz
= vzx t, where vx, vy and vz are all real
numbers and are known as the coordinates of 'v' for the coordinate system of which
they are a part.
A system of
vectors - v1, v2, ..., vn, is said to be linearly
independent, if and only if, each and every vector of this system can be expressed
either: (a) as a unique (i.e., it can be expressed in just one way) linear combination of
v1, v2, ..., vn; or, (b) not at all. The basis vectors of
a coordinate system are considered to be linearly independent. Furthermore, every vector
in that coordinate system is dependent on them, since any given vector v = vxr
+ vys + vzt.
Therefore,
"a basis of a vector space V is a system B of vectors of V such that every vector in
V can be represented in exactly one way as a linear combination of vectors of B."
Not only is the basis of a vector space a linearly independent system of vectors of
V, but every vector in that vector space is linearly dependent on that basis.
If the basis
for a vector space is finite, then, the vector space is referred to as finite-dimensional.
On the other hand, if the basis for a vector space is infinite, then, the vector space is
called infinite-dimensional.
In the case
of a finite-dimensional vector space, any two bases in that vector space will have the
same number of elements, and this number of elements constitutes the dimension of the
vector space V. Thus, in the case of V3, the dimension is 3 since
there are three elements: 'r', 's' and 't' which form the basis of this vector space.
Therefore, irrespective of the actual structural character of a given basis is in V3,
there will always be three elements which make up that basis.
No matter
how one constructs the structural character of the unit vector of a given vector space,
the dimension of that vector space always will tell one how many elements go into making
up the basis of that particular vector space. In other words, if the basis of
a vector space is changed (i.e., if the character of what constitutes a unit vector is
re-defined), then, the coordinates of a vector v will change as a result of the change in
the character of the basis (after all, vectors are linearly dependent on the basis), but
the number of coordinates will still be the same as the dimension of the vector space.
Consequently, for any given basis in a n-dimensional vector space, one will be able to
associate a unique n-tuple with each vector of that vector space.
In view of
the foregoing, if one has two vectors 'm' and 'n' which are given by their coordinates in
terms of the same basis, then, these vectors can be added together by means of their
coordinates. Furthermore, any given vector which is given by its coordinates can be
multiplied by a scalar.
One might
want to think in terms of treating experiential intensity as a
hermeneutical/phenomenological counterpart to the notion of a scalar. As such, it would
serve as a quantitative structure that does not add a new orientation or direction to an
already existing vector.
Intensity
emphasizes the themes of constraints and degrees of freedom which already exist in a given
vectoral latticework. It accomplishes this process of emphasis through a sort of density
distribution function in which different ratios of constraints and degrees of freedom of a
given structure are highlighted in different circumstances.
However,
beyond certain boundary limits, intensity may take on vectoral properties and transcend
the purely quantitative properties of a scalar structure. For example, this may be the
case in relation to extremes of pain or pleasure.
In the
context of hermeneutics and phenomenology, the idea of a basis is like a point-structure
that serves as the fundamental building block out of which vectors are constructed in a
given hermeneutical vector space. This may be where characterization becomes so important
since it often tends to set the tone (in the form of values, attitudes, beliefs,
impressions, biases and so on) for what the structural character of the basis will be for
a given hermeneutical or phenomenological space. In other words, characterization often
establishes the initial set of constraints and degrees of freedom that constitute the
hermeneutical unit vectors which will be used in generating the phase relationships that
give expression to latticeworks in a given hermeneutical or phenomenological space.
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