The concept of metasystems implies set theory, as well as a number of other related theories that deal with the organization of elements and relations between elements. Exactly how set theory and other related mathematical theories might be implicated in the understanding of metasystems theory is dealt with in this chapter. We can say that a metasystem implies one or more transformable sets. We can consider that each instantaneous state transition that we measure or mark off for a system constitutes a subset of the total set comprised by the metasystem.
Any metasystem in theory has a start state, or beginning, and an end state, or terminus. In actuality, it can be demonstrated that in natural systems, there is rarely a clear-cut line that marks a beginning and an end of a system. It is more a question of descriptive short-hand and the need to impose a sense of discontinuous boundary upon systems that are otherwise continuous and in their essence unending.
We impose some qualitative definitional shorthand of life upon an organism. We say that a human being has a beginning at the moment of conception and an end at the moment of its final expiration. If we look more closely, though, we can see that conception was preceded by the life-forms and processes of the parents, and represents this essential continuity of process in life. Even in death, neither can we mark the exact moment of final expiration, in which the system quits all at once, nor can we say that the system, in return to nature, does not reenter some larger event-cycle of nature, which it clearly and always does. But from the standpoint of talking about that distinct, individual entity as a living person, we must mark the boundary as such. During that period, it constitutes its own unique system that is independent of the systems that come before, or after or that encircle it in every way. We mark this uniqueness by our discontinous superimposition of definitional boundaries.
We can hypothesize that, for any metasystem, there is some original start state, AS, and some final end state, AF. There is an indeterminate range of intermediate states that are describable by the state-transformations from the start-state to the endstate according to some complex non-linear transformational function, such that we may write:
AS → ƒ (AS → AF ) ? AF
The endstate may be the direct transformation of the start-state, but the indirect by-product of a whole series of transformations of intermediate states. The arrow implies one thing--it is change over time. It is irreversible, and unequal.
It appears that in our model of metasystems, the notion of equality or the equal sign is an ideal and absolute that connotes a static system. It thus cannot connote change of systems as these occur in reality. The closest I believe is to impose an equivalence sign, such as ≈, denoting that one hypothetical entity is approximately the same, or remains relatively equal or unchanged in relationship, to another entity or to itself, in time.
We can say that all metasystems are time-ordered systems. Equality is reserved in our denotations for the specification or ideal identification of entities and their partial values for when we impose substitution upon systems that allow their relational embedding and differentiation in abstract terms of other systems.
We can say that time's arrow in our formulas are arrows of change and difference, and implie therefore an additive or substractive comparison of values, or state-differential. We can say that they represent definite "intervals" of transition between "states."
In metasystems, we refer to "states" as complex entities. We can infer a kind of "state-theory" that perhaps shares many aspects of set theory and order theory in mathematics. A state is a kind of subset of a metasystem. Metasystem models must therefore elaborate state-theory as somehow relevant to its abstract representation of real systems. We can describe for any metasystem a hypothetical metastate that is the series of all subsets of the state. Series implies a form of union that occurs in both time and space, what I will call state-integration.
A state is a sequentially ordered subset. It is also, necessarily, a spatially ordered subset. Any real system that exists in time, must occupy some kind of discontinuous space. As such, its position is always at least implicitly definable within a larger "meta-matrix" of alternative states. We can refer to pre-states, post-states, super-states, sub-states, and alter-states which we can designate in a disjunctive way as either (right-hand) or (left-hand) states. Each state would have some direct or indirect relational function with our "center-state." The minimal construct for a metasystem model can be seen to be an orthogonal projection of a four-dimensional reality onto a three-dimensional spatial representation.
Each state is a unique subset of a "metastate." I will also impose what I refer to as the hypothetical "zero-state" which can be considered to be a non-state. A zero-state can be defined as any state plus its complement, less its metastate. The complement of a state is therefore all alternative states, and this complement defines the matrix structure and implicit reference-inference framework of the state.
Each center-state constitutes its own point of origin in the larger metastate framework, and also simultaneously an extension of another, infinite numbers of origin points in alternative states.
We may characterize a state as an instantaneous or momentary set of interrelated points, in a momentary sense, or as some extended set of such subsets that constitutes a discrete interval of a larger meta-state or metasystem. Ultimately, all states are continuous and therefore our superimposition of interval measures or discrete momentary "snap shots" are possible only in an abstract sense.
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The problems we encounter in the abstract representation of real systems is precisely the kinds of problems encountered in the recording of living realities by means of movie-cameras and still-frame photography. We can with any meta-system only adopt one point of view at one time by which we configure the metasystem. This is so because we cannot adopt the point of view of the center of origin for any metastate or state within a metasystem. Any metasystem presents to us the possibility of an infinite number of alternative points of view, and there is no single correct set or number of points of view that is best or exclusive to a valid representation of the system.
We are rescued in this daunting form of relativism when we consider that every and any point of view is approximately equivalent to any other point of view. There is no single best or worst point of view, though some may be relatively better than others, especially in terms of what functions they are serving.
This has a great deal to do with our knowledge and descriptive explanation of complex systems. My wife was perusing an old medical anatomy book of my father's medical school days in San Francisco. The detail of the book was amazing. It presented numerous points of view of the body in different angles and at different levels, some highly schematized and others highly realistic, including actual photographs. To a great extent, what points of view were included was primarily determined by the purposes that it was intended to serve in the larger structure of the text itself.
The human body, as a metasystem of nature, presents to us the possibility of an infinite number of viewpoints that focus on an infinite number of center-states as an alternative and equivalent point of reference. We cannot say that one overall viewpoint was best, or that there was any single-point of view that is without value.
We really have no way of proceeding otherwise in our abstract representations and descriptive explanations of systems, other than the elaboration of alternative center-states from a variety of "angles" and different points of view, depending upon our functional purposes to which they are put.
This digression about our relative knowledge, what I will call the representational state-relativity of metasystems, is important to our consideration of state-theory. I believe it demonstrates clearly that we cannot adopt any point of view that does not serve some extrinsic functional purpose that is not inherent to the metasystem under inquiry. Not only can we never describe any metasystem in its entirety, in a complete or exhaustive manner, but we can never describe any metasystem in a completely non-arbitrary or a priori way. Our understanding of any metasystem remains always tied to the functional framework within which we ourselves are embedded. It serves us well in our descriptive explanation of metasystems to always remember and mention at least in passing some sense of functional rationality underlying our description. This is clear in anthropological fieldwork, but it is not so readily apparent in the telescopic observation of distant stellar systems.
It is clear that in our scientific explanations, we seek to impose a set of standards and explicit limitations upon our descriptions such that we are able to abstractly represent any metastate or metasystem in a minimally sufficient manner. This cannot be done by means of exhaustive elaboration of alternate states. We seek a metastate of metastates, a description of the order and relation that underlies the metasystem in its entirety. We hypothesize the existence of some underlying sense of order of relations that governs a system, of which any particular state-description is but one imperfect and partial representation.
Scientific theoretization and generalization is a form of systematic simplification of metastates used to functionally explain metasystems as these occur in reality. Any operational procedure we may apply to our descriptive explanation of states and systems must lead to a simplification rather than an elaboration of a system. If we seek to elaborate some point of view in detail, it is in the interest of applying this particular description to alternative state-descriptions, as an example. Simplification rules are based upon the notion of relative equivalence and substitutability of states and metastates such that we may derive refined abstract models that are representative of most alternative states occurring for any given system.
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A state can be said to be an abstractly objective relational set that is the partial instantiation of a system. A metastate can be said to be some hypothetical description or theory of a state or set of states composing some metasystem. A metastate is always partial to the whole metasystem.
A state has been described as a relational subset of a system. Its identity and composition as a state, its sense of integrity, is defined relationally by the transformational functions that are pertinent to that state. This is usually always supercomplex and multiply connected at several levels of analysis. We must identify what we can call the principal or prime relational cardinality of any state as the minimal set of relational determinants that can be used to relate and describe the most number of point values for a given state. The degree of integration of this set of functional determinants can be said to be the extent to which they can be successfully unified within a single transformational equation. If it requires two or more separate sets of transformational formulas to describe a system, we can say that the state is heterogeneously underdetermined by that number of degrees.
Since any nonlinear state can always be said to be only partially or imperfectly determined, then we can hypothesize that for any real system, there are always at least two or more basic sets of equations that determine the values of that state. The primary function can be said to be the set of those deterministic relational functions that determine the most number of values of the system. The complement function can be considered to be that residual set of nondeterministic relational functions that determine the remainder of the values of the system.
While we can speak of positive functions that determine the ordering of a system, it is difficult to imagine what can be called negative complementary functions that "determine" the relative disorder of a system. That disorder may be somehow represented in an ordered manner, or that chaos may be somehow determined in a functional manner, seems self-contradictory and presents something of a paradox in our understanding of reality. We can say that just as there can be no perfectly ordered states or systems, there also cannot be perfectly disordered states or systems. Hence, we can describe some kind of improper integral function for any state of relative disorder that hypothetically characterises any real state or system.
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The total set of a metasystem would in theory be the total number of instantanous state transitions between the time of origination to the time of ultimate disintegration of the system as a system. In fact, the total set of such a metasystem would be a continuous set of alternative state-vectors, each of which would constitute a subset of the total. To view the system synchronously at an instantaneous point in time would be to view the subsets of the system in a way that is distinct from that if we viewed each of the state-vectors of the system from the point of their initiation to their respective terminus. We could plot this on a matrix in which the horizontal axis represents the temporal vector of the system, and the vertical axis represents the spatial vector or distribution of points. It can be seen that from one instantaneous interval to the next, that the distribution of points of the set would not necessarily be the same.
To understand set theory in terms of our metasystems model, it is therefore necessary to construe sets as dynamic entities. Dynamic set theory would derive from a nonlinear topography, and would lead to continuous intercorrelational matrices. There are, I believe, many implications in this model, and it demonstrates as well the character of applying basic mathematical theories to the model of a metasystem.
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Set theory deals with the abstract organization of collections, or sets, of entities. All systems are composed of or compose sets or collections of things that are identifiable in some abstract sense. Because of the paradox that any set or collection of things must necessarily be both a subset of some larger collection and also a set containing subsets of smaller systems, we must be careful in our specification and identification of things that determine their order and relation to other things.
All pattern in nature, if it is recognizable as such, exhibits some sense of "order" that is symbolically resonant with our understanding of reality. Often we observe pattern in complex phenomenological events and see no pattern or sense of order whatsoever. We construe only what appears to us to be somehow random or at best some subliminal sense of pattern that we do not notice and construe as only something of the background.
Our ability to recognize pattern in natural phenomena, or in the larger sense, in our phenomenological experience of reality, is directly contingent upon the preconceptions and gestalt frameworks of symbolic attention and understanding that we bring to bear upon such experience. We will not see in natural order what does not accord with our prior knowledge structures, and which, also paradoxically and somewhat systematically, in turn derives from our previous experiences.
In a sense, as we peer through a telescope or through a microscope, or we just peer out a window onto the outside world, we embrace the whole of the structural patterning of nature, indeed, the basics of all reality, in a single instant. This would be true if we understood clearly what we were looking at and what to look for in the patterning of what we observe. Ascetically, we could develop the whole of a very successful natural science just based alone upon our ability to look out of a single window onto the natural world, at least in theory. Technically, we could claim this to be hypothetically true, because everything is connected somehow to everything else, and thus the infinite set of all things is indirectly inferrable from any finite set of small things it contains. The only requirement, again, is that we knew what to see or how to see what we were observing.
But our history of science and current sense of scientific worldview did not arrive full blown in a single vision from some window, nor did it come overnight in a single passage of the moon. It was built slowly with with many stops and starts, over a long period of the accumulation of experience and observation by many different people from many different points of view. It arrived to where it is today only after a long struggle with alternative arguments and different points of view. It marched with falsehood and folly as much as it cavorted with truth and wisdom. And except perhaps for Kepler and Galileo, few scientists have also been saints.
But it has clearly arrived at the doorstep of the 3rd Millenium with a self-conscious awareness of its own resolving and inferential capabilities. In the structure of a subatomic particle it is viewing the entire universe, and in the structure of the nucleus of a cell it is viewing all of life, and in the structure of a simple book or poem, it is viewing the structure of all human reality. This is its power and its sublime elegance, that in all the confusion and apparent chaos of our reality, as infinite and open-ended as this is, there reigns a supreme and supremely simple sense of order. And, except for the admonishments of Einstein, if we have science, we almost do not need God any longer. Of course, I say "almost" in an agnostic rather than an atheistic manner. I will not go so far as Kierkegaard, Marx or McCluhan to claim that "God is dead."
It is the effort of this third chapter of this first part, to attempt to reconcile our limited understanding of set theory, especially as this underlies much of what we do in mathematics and in the scientific organization of knowledge, with our equally limited understanding of patterns of natural order, simple and basic or complex and elaborated, especially as these are encountered apperceptively and apprehended immediately in our phenomenological experience, unconstrained by the preconceptions and points of view we bring to every event.
Hopefully, in the process of this reconciliation between abstract theory and concrete experience, we can transcend the limitations of both forms of knowledge, to arrive at a transcendent sense of order that is both synthetically holistic and analytically systematic.
It is quite clear to me that if we are to move forward with our metasystems models based on mathematical symbolisms and symbolic mathematics in nontrivial ways, then we must achieve such reconciliation.
Implicit to the preceding argument is the sense that the application of set-theory to our apprehension of phenomenogical order is greatly conditioned by the sense of order we bring to such experience. If we dichotomize our abstract systems of meaningful identification, hence of definition and accounting, from our experiential systems of meaning and pattern recognition, then we are sundering what is in fact a unity of experience and our sense of reality. Reality is necessarily dichotomized only if we make it so, and only if we emphasize difference over unity of experience. If we construe this process of pattern recognition and conceptual construction as interdependent, as part of a knowledge system itself involving dynamic feeback, then we are able to step beyond the boundaries implied by such a dichotomization between the real and the ideal.
But it is also quite true that not all patterning we construe in nature, especially upon very basic levels of apperception and response, are necessarily "preconditioned" by our own preconceived constructions. Many reponse patterns are direct and rooted in our nature, and I am sure as well that there are basic universal patterns of perception that we are born with and that forms a substrate, however unconscious, to our meaning systems. But at the same time, it is in the selection and interpretation of experience, beyond mere fright reactions, natural curiosity and inchoate feelings we bring to our experiences, that we find the work, necessarily, of our cultural and conceptual constructions.
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Implied in this kind of understanding is of course the basis for an argument about the validity of a gestalt approach to scientific phenomenology and theoretical construction. Consideration of formal set theory and its applicability to real systems, and consideration of the limits and facets of our sense of order in natural phenomenal patterning, upon which our inference abilities and our sciences are based, is the beginning move toward a systematic excoriation of abtract metasystems.
Nature seems to organize things in one way, and abstractly ideal entities are organized in some related, but not exactly equal way. The fundamental disparity between our abstract systems and systems of realization are essentially measurable or determinable in terms of the basic identities or thingness of groups or collections and the relationships between things and groups. Thus, set theory is really a theory about grouping and groupability, or the ability to sort and arrange things into groups. It is in a sense foundational to our ability to organize reality in some coherent way that makes sense, whether abstractly or realistically.
A great deal of abstract set theory is implicit to most of mathematics. I will construe what is technically known as a mathematical series as an implicit and special kind of set. In deed, it seems to be the case that our ability to deal with things at all in any general sense is based on our ability to group and form sets and to relate sets and things of sets to one another. It furthermore provides us with the means of relating our abstract notions and ideas, or rather our generalizations, with naturally occurring sets of things that are alledged to be representative of our generalizations. A generalization can be construed as being at least an implicit set, or an explicit statement about an implicit set, that is made explicit through systematic definition. Systematic definition would proceed through both the application of a mathematical mechanics to the description of real systems, and by means of an elaborated symbolic calculus that serves to integrate the sense of reality in a gestalt framework pertinent to such a system, as a hypothetical metasystem.
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Technically, set theory refers to the mathematical study and description of collections and sets. In a larger sense, in terms of logic and semantics, it deals with taxonomy and the systematic organization of knowledge based upon relational properties, similarities and differences. Thus, it is very important to science on a number of levels. It is easy to find the role of taxonomic organization of knowledge in many different areas, for instance, in biology. Evolution or the engine of natural selection would make no sense and demonstrate no apparent order or dynamic outside of an understanding of natural taxonomic systems. Indeed, a natural taxonomy as framed by Carolus Linnaeus preceded and had to come before the development of a realistic theory of natural evolution. Also, we cannot understand natural history in any deep sense if we do not have the common reference-inference framework that our natural taxonomic system provides. Of course, the natural taxonomy of biological life is imperfect and many arguments still rage about what group is related to what. But we couldn't have developed biological sciences, especially not in any comprehensive sense, without such a taxonomic system being constructed in the first place. And once such a taxonomic tree was consistently, and mostly correctly constructed, the theory of evolution was implicit to its structure and sense of order. The relational similarities and divergence of species could only be explained by some mechanism of change as applied to such a system of classification.
Taxonomic classification is implicit to all our knowledge, especially as this is organized scientifically and systematically to serve functional purposes in our world. Evidence indicates that children are creating their own taxonomic classifications of their life-world long before they begin learning to apply the rules of language to it or act within it in any meaningful way.
Set theory underlies in an ideal and abstract sense all our systems of classification and taxonomy. A set is a collection of any kind of objects that may be denoted by a variable, say Z. Set Z may be formed by identifying a property (P) that is possessed by certain elements of a given set X. Z would be the set of elements of X with the property P.
That p is an element with property P of X is designated by the following:
p ε X
Therefore:
Z = {p/ p ε X and p has property P}
The Z set can be said to be characterized by determinative properties that characterize its membership. But those properties are also characteristic of the Z set as a whole irrespective of what its elements are in any exact sense. In set theory, a basic property assigned to all sets in a hypothetical sense, what can be called a meta-set, is the property of cardinality.
Cardinal in its root means "cardo" or hinge, and rfers to that on which something turns or depends. In reference to the property of cardinality in set theory, it refers to the basic sense of chief, or principal or primary or fundamental properties that are definitive of a set, or upon which, the definition or collection of a set depends. Dependency that is implicit to the term also implies the notion of a functional and determinant relationship that defines the set as such. A cardinal number is one that is in answer to the question "how many." Thus, a cardinal is a member of a set. More technically and mathematically, cardinality has a more exact denotative definition of one-to-one correspondence as this is construed as a system of positive integers or absolute numbers. This has important applications and implications in the extension of set theory to advanced systems analysis.
Technically, two sets are said to have the same cardinal written C(A) = C(B), if there is a presumable one-to-one correspondence between the elements of A and the elements of B. In other words, both sets are relative to the same cardinal number system by virtue of their one-to-one correspondence. The two sets are said to be matched along the cardinal property of C, which is the shared or common determinant or denominator of both sets.
In finite sets this implies the notion of equal sized sets such that we can say A has the same number of elements as set B. It implies in a loose symbolic form an exact quaternary analogy between sets A and B. Two symbolic sets can be said to be analogically cardinal if for each symbolic element of set A there is a corresponding analog in set B.
For infinite sets the application of cardinality yields interesting consequences. If A equals the set of integers and B the set of odd integers, then the function ƒ(n) = 2n - 1 represents the cardinality of C(A) = C(B). This can be interpreted that an infinite set A may have the same cardinal (functionally defined) as its subset B. The cardinality of an infinite set A and its subset B suggests the polynomial expandability of infinite sets. This paradox has interesting implications, for instance, in its application to the understanding of the physical structure of the total universe, if this is presumed to be an infinite system.
The notion of subset is intrinsic to this paradox. A subset of a set is one in which each element of subset A is also an element of set B. Hence, a subset may be smaller than a set, whether finite or infinite, or any set may be a subset of itself. This allows us, among other things, to subordinate or rank or order properties that are determinative of the same set.
Another way of forming a set Z is to assume that Z is the set of all subsets of a given set X, such that it can be show that:
C(X) < C(Z)
On the otherhand, the collection of all sets cannot be regarded as a set. If a collection X were called a set, and Z denoted the set of all subsets of X, then the impossible ordered relation would exist:
C(X) < C(Z)
If an infinite set cannot be put into a one-to-one correspondence with positive integers, then the set is referred to as uncountable. Any statement of functional cardinality of such a set is referred to as the continuum hypothesis and has as yet been unproven and remains unprovable in conventional set theory. It remains one of the unsolved puzzles of pure mathematics. It is stated thus:
If X is an uncountable subset of the reals R, is C(X) equal to C(R)?
This broaches one of the basic dilemmas of improper integration of real, infinite sets. It is a dilemma underlying the application of ideal and abstract systems to real systems.
Cardinality of sets are said to be comparable if one-to-one correspondence is said to exist between the elements of set A and the elements of some subset of B, such that:
C(A) ≤ C(B)
Any two sets are said to be comparable if:
C(A) ≤ C(B) (and)/or C(B) ≤ C(A)
The cardinality of any two sets is comparable if each set is less than or equal to the other, such that if:
C(A) ≤ C(B) and C(B) ≤ C(A)
Then: C(A) = C(B)
Cardinality is established by means of setting up one-to-one correspondence between two sets by means of ordering the sets. An order relation is designated by the sign < if the following three conditions are satisfied for a set X:
1. If x1, x2, are two elements of X, either x1 < x2 or x1 > x2 . In this case, any two elements in set X are relatable.
2. If x1 is not less than x1 . In this case, no element is less than itself.
3. 1. If x1 < x2, and x2 < x3 , then x1 < x3 . In this case, the relations between the elements is transitive.
Ordering implies a countable series of elements, or a sequence that is rankable. An odering of a set is called a well ordering if it satisfies a fourth condition:
4. Each non-null subset Y of X has a first element. In this case, there is an element y0 of Y such that if y' is another elementof Y, y0 < y'.
Well ordering of sets invites theorems about sets that are considered strange and counterintuitive, and that are frequently used as "pathological" counterexamples for various kinds of conjectures. Positive integers are naturally well ordered, but neither the integers nor the reals is a well ordering. A well ordering for real numbers cannot be written, but it can be proven that there is one.
Sets may be related to one another in operations of addition, subtraction, multiplication and mapping. The sum or union of sets A and B is given by the following:
(A + B ) or (A U B) is the set of all elements in either A or B; that is:
A + B = {p/ (p ε A or p ε B) }
The intersection, product or common part of sets A and B are given by (A · B, AB, A ∩ B) and is the set of all elements of both A and B, such that:
AB = {p/ (p ε A and p ε B) }
If A and B share no common elements, then they do not intersect and their intersection is written as:
AB = 0
The difference between A and B is written A - B and consists of the collection of elements of A that do not also belong to B, or:
A - B = {p/ (p ε A and p ε/ B) }
If A is a subset of B, then the difference between A and B is zero. Some boolean algebraic relations follow from these considerations:
A + B = B + A
A ∙ (B + C) = A ∙ B + A ∙ C
X - (A + B) = (X - A) ∙ (X - B)
X - A ∙ B = (X - A) + (X- B)
Boolean algebra underlies a theory of relations and closely relates set theory to probability and computer circuit design. It describes combinations of the subsets of a given set I of elements, taking the intersection of S ∩ T or the union S U T of two such subsets S and T of I, and the complement S' of any one such subset S of I. Thus, we can write the following:
S ∩ S = S
S ∩ T = T ∩ S
S ∩ (T ∩ V) = (S ∩ T) ∩ V
S U S = S
S U T = T U S
S U (T U V) = (S U T) U V
S ∩ (T U V) = (S ∩ T) U (S ∩ V)
S U (T ∩ V) = (S U T) ∩ (S U V)
If an empty set is denoted by 0, and I is the set of all elements under consideration, then:
0 ∩ S = 0
I U S = I
0 U S = S
I ∩ S = S
S ∩ S' = 0
S U S' = I
From these fundamental laws, other algebraic laws can be deduced. If the logical connectives and, or or not are substituted for union, intersection and null set, respectively, then the same laws hold. Deductive propositions and assertions also hold when these laws are combined by the same connectives.
Set X may be transformed into set Y by means of a transformation function that assigns a point of Y to each point of X. At this point, sets are representable as matrices. The point assigned to X under a tranformation function ƒ is called the image of x and is denoted ƒ (x). The set of all points x sent into a particular point y of Y is called the inverse of y and dnoted by ƒ-1(y).
The transformation ƒ(x) = x2 takes each real point x into its square. Geometry provides many examples of transformations. Generally, transformations change the size and shape of an object. From set transformations, topology can be studied.
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It can be said that each system comprises some hypothetical matrix structure, and the diagrammatic representation of such a system can be derived from the compounded matrix that the system represents, and it can lead to a construction of the implicit structural matrix embodied by the system. It can be said that such matrices tend to be compound, integrated, multi-factorial, and open. They frequently subsume other matrix structures, and are part of a larger multiple matrixes.
The matrix structure that is comprised by any hypothetical system emphasizes the relational functions occuring between points at whatever level of analysis we are upon. I will hypothesize that, just as there is a single unified space within which to represent all systems in uniform and comparable ways, this space embodies and expresses an implicit matrix structure that is implicit and that can be used differentially and alternatively for the expression of any system.
Just as we can minimally represent most systems in a two-dimensional plane geometricized translation, we can minimally represent most systems by a hypothetical discrimination table of M x (j) rows and columns.
The most minimal representation we can make is a simple chi-square table that represents the values geometricized over the x or y axis:
|
(X, Y) |
X + |
X- |
|
Y+ |
+X +Y |
-X +Y |
|
Y- |
+X -Y |
-X -Y |
The chi-square type table above is quite common in scientific theory, and is the maximally congruent between idealized and non-parametric values. On the other hand, it tends to represent the most simplifed form possible and therefore disguises the most variability occuring in any system.
Thus, most systems, it is worthwhile to elaborate tables systematically by elaborating the diminsional characteristics embedded by each idealized variable. This is done by the backward chaining extrapolation of the functions underlying each data point on some ordered scale of measurement. In general, I've adopted and assume for most instances a cardinal scale of measurement that is sufficient for both parametric and non-parametric sets of values. It must be seen that the actual frequency distribution represented by an actual system may be composed of multiple alternative matrixes that would result in the same distributional pattern.
All possible matrices, which may be infinite, represents the total possibilistic space or potential sample area that the actual distribution would represent. In such a system, each instantiated point or event interval is always represented by some translated and interdependent point upon both the x and the y axis. Each point is represented by a complement pair (x, y) that is projected from the x, y axis. Each point would therefore be represented by some complex equational relation with at least x and y values that would always be expressed as ratios greater than O and less than 1. The total size of the elaborated matrix would be determined by the total number of points or the sample size. The R-C dimensions of the data points would always be equal and the matrix would always be squared.
The actual data points themselves may have been derived by another set of dimensions that can be labeled qualitatively and that may not be squared. M in the formula above is usually a complex set of parametric values that represents both the number of data points and the main ideal dimensions of the actual matrix. Setting these values to the x and y axis, respectively, embodies that the values of these composite variables represented by M are minimally differentiable on the basis of some standard equation or set of equations applicable to all members of the set. These values may be mapped in common space along the same x and y scales. The actual dimensional characteristics may be lost in the translation of the sample to the x-y coordinate system, and these cannot be recovered from the table except by labeling the individual data points with their dimensional headings.
Understanding the matrix structure of any complex equation is critical because it determines a great deal that can be done with the equation. Spreadsheet functions and databases that integrate multiple matrixes in feedback control structures are derivable from these. By extension, it allows us, among other things, to build and functionally organize computing functions that enable us in turn to more dynamically model a system in virtual space.
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Matrix theory is extremely important then to the operational definition of symbolic mathematics as the basis of advanced systems science. Matrix structures can be hypothesized to occur at every level that we can analyze. A matrix is in a sense a translation of any unification space of a common set of points definable within a Cartesian coordinate system to a common framework of a discrimination table. Such a table allows us to systematically compare and relate values along critical dimensions of differentiation that are implicit to the structural relations that define the identity of the points.
The point of departure here is to hypothesize that total reality as expressed by the Reality principle, can be represented as a single complex, composite matrix structure of infinite size and complexity. Any subset of Reality, at any level, can be represented as a component matrix of the unified matrix structure, and each specifiable sample of points in reality, can also be represented as a constituent and derivative matrix of the unified matrix structure. All occurring or representable matrices are therefore partial matrices of the unfied matrix structure.
It is a central design of symbolic mathematics in advanced systems sciences that all forms of data that are measureable upon some scale, are representable within the framework of some kind of matrix that is defined by the units of measurement. This entails that we may build matrices representative of all systems at all levels of naturally occurring phenomena. Furthermore, if we hypothesize that all systems are in fact composite systems of more basic systems, then we can see all matrices as being composed of, and in part determined by, the underlying sub-matrices that compose the data points upon which the matrix is based. This presupposes that reality is composite, because it is constituent, and that therefore our analysis of reality is composite. It also presupposes that we may construct larger and derivative sets systematically from more basic and smaller sets.
We may build our unifying matrix structure empirically from the ground up, or we may build it hypothetically from the abstract top down. Ultimately, in our operational procedures, we must attempt to do both at the same time, hopefully meeting somewhere in the middle.
Matrix theory is conventionally rooted in a linear conception of reality. Matrices only really become interesting to advanced systems sciences when the nonlinear control aspects of their functional operators are taken into account, and when the derivative structure of embedded functions underlying matrix stratification and integration is taken into account. At this stage of their developmental application, computation devices must be relied upon to generate the solutions for such complex structures.
For definitional purposes, a matrix can be said to be any rectangular array of numbers or elements with m rows and n columns, such that any matrix A has a size of m by n, and is representable in the compact form when the size is given as:
A = (aij)
Where a is the element in the i-th row and the j-th column and aij is known at the typical element of A where i takes on the values of 1, 2, 3...m and j takes on the values of 1, 2, 3,...n
This describes a table of A that can be depicted as follows:
|
A |
n = 1 |
n = 2 |
n= ....n |
n |
|
m = 1 |
a (1,1) |
a (2,1) |
a (....n,1) |
a (n,1) |
|
m = 2 |
a (1,2) |
a (2,2) |
a (....n,2) |
a (n,2) |
|
m =....m |
a (1,....m) |
a (2,....m) |
a (....n,....m) |
a (n,....m) |
|
m |
a (1,m) |
a (2,m) |
a (....n,m) |
a (n,m) |
Conventional matrices are useful computational devices with a number of useful applications in diverse fields of applied mathematics. They are used in mathematics especially in the study of linear systems of algebraic equations and linear differential equations. In such structures, the rows are usually used to represent string formulas that are aligned in parallel fashion and are of equal size.
If m = n, then A is called a square matrix of order n. If m = 1, then A is called a row matrix and if n = 1, then A is called a column matrix. The elements aij of A, for which each i = j, are known as the principal diagonal elements. A diagonal matrix is one where aij = 0 if i ≠ j. A scalar matrix is a square diagonal matrix with equal diagonal elements. An identity matrix is a sclar matrix in which the common diagonal element is the number 1. An n by n identity matrix is denoted In.
Matrices are regarded as generalized numbers, and they can be combined in certain definite ways. The matrix operations of addition, subtraction and multiplication are defined in terms of these same operations for the elements, and they satisfy some, but not all, the rules of ordinary algebra.
Two matrices A = (aij) and B = (bij) are equal if they have the same size m by n and (aij) = (bij) for all i, j. Two matrices of the same size can be added by adding the elements of the corrsponding positions of each matrix together, such that A + B above equals C = (cij) and meets the criteria stated above for equal matrices. Matrix addition is therefore associative and commutative, such that (A + B) + C = A + (B + C) and A + B = B + A.
A null matrix is a matrix with zero in every position and is denoted as 0. A + 0 = 0 + A = A. The matrix -A = (-aij) is the negative of matrix A and it follows that A + -A = 0. Subtraction of m by n matrices is defined by B - A = B + (-A) = (bij - aij)
A matrix B is said to be conformable with matrix A if B has size n by q and A has size m by n, such that B has the same number of rows as A has columns. The product of AB is defined only if B is conformable with A, such that the product matrix C = AB is an m by q matrix and the element in the i, j position of C is obtained by multiplying the n elements in the ith row of A into the ne elements in the jth column of b, term by term, and adding these products.
If two matrices are square and of the same size, then the product of both matrices is commutative. Matrix multiplication is commutative such that If A is m by n, and B is n by q, and C is q by r, then (AB)C = A(BC) and both are m by r matrices. If A, B and C are of the proper sizes for the operations to be defined, then A (B + C) = AB + AC and (A+B)C = AC + BC. If A is m by n, then for identity matrices of the proper sizes, A In = ImA = A I may happen for matrices that AB ≠ BA and AB = 0 if A ≠ 0 and B ≠ 0.
The product of a matrix A and a number a is called a scalar product and is obtained by multiplying every element of A by a. The transpose of an m by n matrix A is an n by m matrix B in which the n column of A is the n row of B and the m row of A is the m column of B, for every element of B and A. If the transpose of A is denoted A', and B is conformable to A in every respect, then (AB)' = A'B'. A matrix is symmetric if A = A' and it is always square. A square n by n matrix is nonsingular if the determinant of A is not zero. Otherwise, A is singular.
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This kind of relative mathematics becomes more useful when we consider that H stands for some hypothetical state of an implicit system that is represented by the relations of the matrix M x (j) and H. When we do this, we can see that the original equation represents a cyclical feedback pattern that fits our original conception of the operational model. At this point, we must entertain a nonlinear form of matrix calculus, in which matrices consist of elements that are functions of one or more independent variables.
The original state matrix that defines the principle elements and determinants of the system, become articulated n number of times, such that each subsequent state matrix is of the same size as the original matrix. Though a matrix represents a set of parallel linear equations, multiple reiterated matrix structures represent a non-linear function such that the results obtained in the first transformation are outputs which are feedback to the values of the original matrix, resulting in an intermediate nth state matrix that begins the reiterative cycle over again.
We will also assume what I will call the "almost closed" system where we assume that for each system is almost completely represented by a number of continous/discontinuous states with a definite start state and an eventual definite end state.
If we go back to our principle of unification and to the reality principle, we can state that in the total sense, absolute A stands for the total unity or total system of reality in some ultimate sense. All other systems are derivative subsystems of A and are fit together in some complex composite way to constitute absolute A as a total system. I will state that in the total system, A will equal 1 or the principle of total unity. But like absolute zero, total unity cannot be achieved, but will always be expressed as relative unity, such that:
U = M(u) + H
where H = U - M(u)
and M = Z
This same sort of equation can be used for any system, or any subsystem that is a derivative of the system. In the differential expansion of our system to encompass subsystems, we must always retain the original and intermediate values in the successive embedding of the formulas, such that the original values will always be embedded in N as a derivative.
If we wish to capture the cyclical reiteration of a system we can begin by assuming some initial start state that can be represented above as Zs. We will speculate that eventually some end state represented by Z0 will be reached through an (n) number of intermediate states represented by Zn such that:
Z 0 =Zs - [Zn - Z(n-1)]
The interval limits intrinsic to a system define its contraining our boundary limiting factors. The size of a system is defined by the degree
The dimensions of the system:
Size, Polarity, Parity, Periodicity, Limits, Inputs, Outputs, Duration, Variance
Blanket Copyright, Hugh M. Lewis, © 2005. Use of this text governed by fair use policy--permission to make copies of this text is granted for purposes of research and non-profit instruction only.
Last Updated: 03/08/05