Natural Systems Theory

by Hugh M. Lewis

http://www.lewismicropublishing.com/

 

   

Chapter Four

Systems Methodology & Modeling

 

General systems may largely be called a theory in need of a methodology, or a set of methods as well as a general set of operational instructions in the deployment and articulation of methods. If it is to be more than a theory of everything, then general systems must also become a methodology for all possible problems for all seasons. If this is asking too much from any single paradigm, perhaps this is so, but at the same time we can expect no less than a comprehensive set of applied methodologies from a purported comprehensive framework of general science.

Science based upon a general systems paradigm will not come fully of age unless and until its own distinct set of methods or methodology can be more carefully worked out and made to work in a practical manner. One of the most important aspects of developing a general systems methodology that is a method for everything is to come to terms with and deal with the problem of the anthropological relativity of our own understanding of systems, large and small. This comes to play especially I think in our identification and definition of "problem" sets with which we must deal, when we speak of applying always limited means to virtually unlimited possibilities.

To understand general systems methodologies, we must seek general purpose methodologies that are appropriate to a wide range of different kinds of systems. We must also seek to understand the general nature of the kinds of problem sets to be solved by such methodologies. The purpose of methodologies are primarily to conduct research through the solution of complex problems.

A problem may be defined as an unresolved question or condition of reality that requires a solution at some reasonable level of acceptability. A problem exists as a discrepant state of affairs between existing states or conditions and ideal or desired states and conditions, seen primarily from a human or anthropological standpoint.[1]

General systems methodologies then are concerned with the solving problem sets in a deliberate and systematic manner, being whatever it is that a person or group of people construe as being problematic based upon some calculus of ends, whether such a calculus of ends is explicit or left unstated.

Methodologies come into play as a set of possible means when the calculus of ends creates a search-solution space for the resolution of problem sets identified by these ends. This is a complex way of saying that methods attempt to systematically marry means to ends in problem resolution. We work with the understanding that, especially with complex problems, solutions, though hopefully simplifying, are unlikely to be perfect or simple.

There are some sets of methods and methodologies that seem pertinent for consideration of "general systems methodologies." I would designate two general classes of pertinent methods: the first set I would call general systems methodologies and these are a set of methods that apply generally to a broad range of systems, but which are not necessarily designative of any particular kind of system. The second set I would call special system methodologies, and they are sets of methods that are appropriate to a certain class or kind of system, but not necessarily to any other class or kind of system.

To list the set of general systems methodologies, I would include: 1. symbolic representation & strategic planning; 2. design modeling & heuristic simulation, especially involving computing and supercomputing; 3. non-linear dynamics and set-theoretic representation & manipulation; 4. inter-correlative analysis; 5. experimental prototyping of designs. I believe that for applied systems, this model automatically leads to a production or processing sequence, as well as to issues of recycling and repair/replacement of systems as well as to systems growth and regeneration. Thus I have elaborated a basic development cycle for general applied systems within which theoretically any form of applied system may be developed. 

To list sets of special systems methodologies, we need first to categorize general types of systems in some kind of logical or natural schema based upon natural stratification. In general all methodologies that are deployed in the normal sciences at each level of systems stratification are pertinent and appropriate to that level or sub-level of system, albeit usually in a fairly specialized manner. Any or all tools of the trade of any particular scientific discipline or field of inquiry are pertinent methods to be employed within the area of stratification of natural systems, eventhough some kinds of methods may be more relevant and generally deployable than others. General systems methodologies therefore encompass fully the range of analytical and investigative methods that are deployable across all fields of science.

We can generalize a methodology to a framework of applied systems of all kinds, with the recognition that all applied systems will have at least their physical, biological and human components, as well as their outcomes and consequences for the larger world. We recognize that the problem of the anthropological relativity of systems, and its influence in determining problem solving frameworks, need to be taken into account in the defining of possible search solution spaces and the realization of alternative solutions in a system. The frame of reference we adopt in defining a problem and a methodology of solution will determine the range of possibilities and thereby predetermine and constrain the outcome of the process.

Applied general systems therefore seems to entail a multi-purpose design development framework that is capable of taking a project through a series of steps in its development as part of a larger design cycle. It should be also capable of starting and maintaining multiple design-development cycles simultaneously, and upon different levels, interlinking these cycles or the components of these cycles, in a meaningful way. The design-development project cycle for any single system or kind of applied system, represents therefore a general methodology for the solution to the problem set that is related to that system or kind of system. It provides a manner of constructive application and work that allows us to investigate alternative systems and explore the possibilities for their developmental refinement and evolution as adaptive systems.

Within such a framework, specialization of systems or subsystems would follow on the heels of the development of the basic applied design-development cycle, and would represent the elaboration of such a cycle and its splitting into multiple sub-cycles. We can imagine therefore as well the higher level organization of such a framework of cycles within cycles as a single comprehensive meta-systems framework by which all projects and programs are interrelated to one another and made coordinate in their development.

Any measure of reality we may adopt may be said to be, if nothing else, discrete and therefore arbitrary. This is because our reality is anthropologically constructed in terms of symbols that are by design and of necessity discrete and arbitrary.  Measures, to be useful collectively, objectively, "inter-subjectively" must also be consistent, (i.e. standard) or else they are merely idiosyncratic constructions. We may in a sense look at our words that we speak and write as collectively shared measures of meaning, somehow pointing, however indirectly, to some form in the real world, or else some imaginary form.

Collective meaning can only be created through language and the communicative sharing of meaning, and hence we can make a claim, a very serious claim, that meaning and semantics are linguistically relative. It is the translatability of human language, largely because, no matter what the modulations of any particular pattern of speech, we share the same fundamental language (speech production/recognition) apparatus, and because all symbols, even words, are ultimately arbitrary, that we can come to mutual agreement on common forms and measures of meaning in reality. Europeans have meters and litres and Americans have yards and quarts, but because these are standardized units of measure, we can apply simple mathematical formulas to translate one into the other, and back again.

Through the sharing of measures of meaning, largely defined symbolically, human beings arrive at a collective worldview, a common, standardized frame of reference, that arbitrary design of symbols becomes thereby overlaid by convention and common agreement. All of human culture, which is largely behavioral and cognitively based in symbolically organized behavior, may be said to consist of shared conventions, whether these are explicit, in the form of meters and yards, or in the form of laws and rules, or remain implicit and indirect in our our common behavioral constraints. This in fact is an empirical, experimental, working definition of culture that allows us to take our presuppositions to the field and form conclusions about observations of behavior. It forms the basis for an empirical science of human systems and human behavior.

Conventional constraint therefore overlays arbitrary and ultimately idiosyncratic organization of symbolic reality, and comes to demarcate a common field of shared cultural meaning by which people can organize themselves on a social basis into institutional systems. Conventional constraint with underlying arbitrariness of meaning entails a built-in flexibility of our received symbolic systems that enables them to be easily carried, transmitted and transformed over space and time. At the same time, conventional constraint, ultimately arbitrary, becomes reified and naturalized as if non-arbitrary and habituated as if automatic and even reflexively instinctual. It becomes ingrained and embodied, even upon a physical level of our being, such that we are conditioned and quite comfortable with such conventional constraint, and rendered quite uncomfortable without it. Conventional constraint takes on a certain inertia and momentum in terms of its direction, rate and conservative resistance to change, and many anthropologists have confused this with issues of natural speciation and natural selection, which it is not.

Society that we are born into, raised with, and become members of, have a momentum, a mobility, and an institutional, "larger than life" presence that is greater than ourselves, and upon which we come to depend for our very survival and well being. Conventional constraint is not arbitrary. It is agreed upon, a consensus, and often also, a conflict of competing interests. It is not natural, genetic or instinctive, either. Being founded upon arbitrary principles of symbolic design, it is ultimately constructed by a process arrived at through compromise, coordination and cooperation of a group of people through time.

To reiterate, symbols mark our meaning, parsing up our phenomenal experience of the world in discrete and therefore comparable quantities or entities. In fact, we depend very much upon this symbolic process to achieve adaptive success in our life-worlds, and without it our world would be chaotic indeed. The symbols that we arrive at and are compelled to accept and use, are done so not from personal choice, but as the product of social process, group agreement, and continuous articulation and re-articulation in social contexts.

We may say our symbols, to be effective, must be achieved with consensus and agreement. They must be received in our social setting, or else they fall on deaf ears and hence are of no use beyond some psychologically solipsistic interest or need. Schizophrenics appear symbolically bound up just in this way. They are unable to use effectively the social symbolisms that are a standard coinage of the larger system, and who instead are entrapped within a private and narcissistic symbolic world of their own private construction, that is transparent from without but opaque from within.

If our terms, used to give reality an objectified sense of structure, to provide it a place in a shared symbolic universe of meaning, are our measures of reality, then the relationships we hypothesize between our terms are used to build a symbolic universe onto which we can map our conceptual systems of reality in a coordinate and understandable way. We are aided greatly in this endeavor by the fact that natural systems, for all their self-organization, tend to be naturally organized into shared patterns that fall into larger categories and groupings that allow us to label and generalize across sets of systems, and even to arrange sets of systems in relation to other systems.

Each tree in a forest may be individually unique in terms of its exact physical characteristics and measures, but fortunately our understanding of the forest is greatly aided by the fact that all the trees may belong to only a handful of groups of trees bound by homology and analogy, by common descent, shared form as a function of common adaptation, etc.

We may thus categorize and label all the trees of the forest by the several types that are found to occur there and to characterize such a forest biome. And so it seems to be with all reality. Reality is organized not only upon the level of individual systems, but in terms of sets of similar kinds of systems, either homologically (as a result of common origin) or analogically (as a result of common function). It is from the classification and understanding of these natural sets and the generalizations that are implicit to them that apply to all members of the set, that we arrive at what we refer to as natural laws that are the basis for our theories of reality.

The natural laws that apply to one set of systems upon one level of observational analysis, do not necessarily apply to other sets of systems at other levels of observational analysis. In general such laws may be said to be general statements about the periodic patterning we associate with the members of a common set, and this periodic patterning is associated with the typical or characteristic organization of the prototypical member of the set, and the emergent properties that are the consequence of this organizational patterning.

At the same time, sets of systems do not occur in nature in isolated or pre-grouped form, and it is most often the case that different sets, at different levels, overlap and interpenetrate one another in terms of shared space and time and the relationships that may occur between different but interacting members of distinct but overlapping sets. This has been the cause of much academic equivocation, especially in fields like biology and the social sciences, when the exact homological relationships between taxonomic sets, or taxons, for instance, cannot be determined in a precise or conclusive manner, or when for instance we articulate theories of natural selection based upon the speciation of populations, though in natural context we find interacting individuals of different populations with ambiguous reproductive boundaries, often with pure chance and happenstance playing a large part in selective processes.

Heterogeneous meta-systems, or systems of individually distinct and different subsystems, emerge in reality with their own characteristic properties. All eco-systems tend to be complex and heterogeneous meta-systems in this manner. The earth itself may be said to be a complex heterogeneous geophysical meta-system, composed of a variety of elements that to some extent interact with one another in regular ways. It has an iron core, and different hydrologic, plate-tectonic, and atmospheric-nutrient cycles maintain a fragile framework for the biosphere.

We symbolically group and parse up our experience of reality, and attempt to organize the totality of our phenomenal knowledge of reality, in terms of broader groupings on the basis of generalizations that we apply to all members of groups. Working with groups, instead of with individuals, is a way of simplifying otherwise complex realities and dealing upon a level of general analysis in an expanded frame of reference that leads to the formulation of worldviews and general principles about reality.

This leads to the question of alternative frames of reference for understanding the same kinds of observational phenomena. Eleventh Century Europeans saw a sun rise and set upon the earth, thinking that the earth was the center of their known universe. We see now the earth as traveling around the sun, as the earth spins daily on its axis, and even though we still refer to the rising and setting of the sun, we do so with a much clearer view of the real system than did our 11th Century counterparts.

If one is a member of a non-literate and fairly superstitious culture, then one is unlikely to view a diurnal eclipse of the sun by the moon as a natural event, and more likely to attribute it to supernatural forces at play. You would be, in terms of the logic of your own symbology, no less correct than your modern counterpart, only less realistically accurate.

Pure mathematics are examples of abstract systems in which the relational identity of all known values are founded upon the basic idea of equality. The equal sign permits us to assume that what value exists on one side is either the same or otherwise as the value occurring on the other side, and we can perform common reductive operations which demonstrate this equality in terms of reflexive identity, or that demonstrate inequality in terms of basic difference.

We can even perform manipulative operations, as long as we perform the equally on both sides of the equation sign, in order to solve the "problem" of simplifying equality. We sometimes substitute comparative signs (greater than or less than) for the equal sign, but this is usually the extent of our relational activities, but even such signs always allow for a clear dichotomous resolution of the implicit problem. The transformations we make on both sides of the sign we use are otherwise guided by the pure deductive logic that informs mathematics in terms of the axioms, laws and their corollaries that we employ.

This is the same form of positivistic, two value logic that we find with formal logical philosophy. In fact, logical philosophical positivism was derived from the logic implicit to mathematics, based as it has always been on dichotomous (true/false) values. Logical positivism or syllogistic two-value logic only works in natural language to the extent that we can clearly restrict the basic meaning of terms to dichotomous (true/false) values. Often, in such operations, conventional meaning of truth is substituted for what is presumed to be natural truth--"common sense," being nothing but the operation of conventional meaning, takes over. We do not question whether the sky is really blue, the ocean is deep or that roses are red. We simply say, modus ponens style, If all roses are red, and this flower is a rose then this flower is red. The fact that we do not normally, naturally think this way seems to have little to do with the status enjoyed by logical positivism in academia.

So how do we really think? We think symbolically, but without the necessary logical constraint of dichotomous truth value, except in very practical, common, everyday terms and applications. Our logic is less precise and more bound to the relative semantics of psychological/behavioral context, innuendo and association, whether this is conventional or arbitrary. We think rationally with a form of logic that is not constrained by two-value choices and that can move in more than one direction. We commonly employ a form of analogical association in which like is compared to like, and there is presumed similarity on the basis of proximity, co-occurrence, or pre-occurrence.

We may tend to act in dichotomous terms, and even delude ourselves that we are right in thinking in black and white truth, but we tend in fact to think in looser terms that replaces the equal sign found in abstract models of relationship with alternative signs designating similarity, one-to-one correspondence, approximation and equivalence without the necessary constraint of the law of absolute identity.

What does this entail for our general understanding of systems? In our scientific models and symbolic representations of reality, we typically employ mathematical formulations that are based upon logical positivism and that are derived from the basic relationship of identity or equality. In chemistry, the equal sign is typically changed to a reaction arrow, or set of reaction arrows in systems with equilibrium, but we are always balancing the energy/number/mass budget on both sides as if it were an equal sign. In physics, equations seem to work really well because primarily we are dealing with energy pure and simple, and we know that energy always balances--it cannot be created or destroyed. We can of course reduce everything to chemical and physical reaction terms, and hence transform all event structures in reality into nice mathematical equations, but this would indeed become quite tedious.

This is not necessarily so when we deal with macro-biological systems or social systems. We can of course apply demographics, population measures and formulas, and other statistical measures and devices to our models, and we frequently do to great benefit. But we recognize basic limitations in these approaches at this level of integration of natural phenomena.

For instance, if we have two pile of stones, seven stones to each pile, we can proceed to act and treat each pile as if they were completely identical and the same to one another, even if each stone is actually unique in terms of its exact physical characteristics. And because the stones do not act spontaneously (they are not living) and especially they do not talk back to us and behave in contradictory ways, we can treat them in our counting games as if they are in fact the same.

We may easily do the same with many living organisms, such as amoeba, dogs, trees, and even ourselves. But at some point we must come to recognize a couple of limitations to our formulations especially when it comes to living organisms, and especially thinking organisms. Even if we tend to define evolutionary processes of speciation upon a group population level, the actual selection, transmission and mutation occur effectively upon the level of the individual organism. Organisms of a common set, a common gene-exchanging population, must vary continuously upon a genetic level, otherwise they will not evolve, and they will thus lose out in the long run. Treating all organisms of a common populational set as identical therefore does not solve our basic problems of understanding the fundamental mechanics of speciation.

Beyond this, if individual organisms are enmeshed in complex webs of eco-systemic relationship with other species, then the simple classification of these organisms into their populational groupings will not get at the dynamics of meta-biotic organization and interaction that lead to certain fitness and selection regimes.

It is even more the case with human populations, complicated as these have been by culture and human civilization and all the weaknesses associated with these phenomena. There are numerous instances and times when it has been of great value to treat people in a quantitative way in statistical manipulations, but so far very few if any universal laws of human nature or human social systems have been derived in this manner, with very few exceptions.

So, the "hard" scientist used to the comfort of working with numbers and equal signs, will advocate throwing the human sciences out as "soft." This is not really coming to terms with the central problem because human systems are natural systems of their own right, at their own level. The theory of emotion is a good example to finish with. If we say that he is angry, and it is his anger that made him do it, and we then generalize that all people who do similar things do so because they are angry in the same way, we have reached a kind of hypothesis generalization based upon certain presuppositions. But in doing so we do not ask if the emotion of anger is a clear and universally shared feeling or even what it is as a feeling, or if other circumstances may co-occur to predispose a particular individual to commit a certain act, or if the sense of anger shared by all people is the same, for the same reasons, of the same quality or intensity, or may be different and even unique for different people. Upon further investigation, we may discover that in fact different people do the same sorts of things for very different sets of reasons, and the reasons are not always one and the same. There may be precedents and precursors of behavior resulting in similar consequences. Nor do we even really ask if similar kinds of acts, all lumped together, are really in fact the same acts, committed for the same sets of reasons, or perhaps different sets of acts, committed for different sets of reasons.

So, in such cases, of which there are far too many to count, do we simply throw out the problem as being somehow unscientific, or do we amend our scientific view and methodological approach to reality to be able to better account for the problem? I will only answer by stating that, in general, as we progress the hierarchy of emergent properties associated with different kinds of natural systems, we move from strictly logical, mathematical equations, to more linguistic, generally verbal generalizations in the form of basic statements based upon often imperfect classification and terminological systems. Even all our understanding of physical systems and realities cannot be completely coaxed in purely mathematical formula without reference to generalized verbal expressions. 

 

Operationalizing Systems

 

I propose a set of methodological procedures that is rooted in basic presuppositions of metasystems science and natural systems theory. I do not ascribe the same set of operations for every area of knowledge at different levels of natural stratification. Certainly the use of these procedures in dealing with human systems is fundamentally different that the use of similar or related procedures upon a biological or physical level of abstraction and analysis. I do ascribe upon a fundamental level, in terms of metasystems theory, that there are basic abstract and mathematical models that are pertinent to all classes of real or naturally occurring systems. For instance, theory of automata describe all classes of linear forms of digital computing, at least. Whether this theory, which incorporates Turing machines, is sufficient for the description of natural intelligence or naturally occurring information systems is not yet clear, and I doubt it is, at least in any unadulturated form. In this regard we must distinguish between information theory on one hand and intelligence theory on the other, and what is natural or innate, and what is artificial and preprogrammed in some arbitrary way.

Furthermore, to address theoretically in any exclusive sense the informational aspects of naturally occurring systems is to thereby ignore the energetic considerations of such systems as naturally occurring machines. A mechanistic model that is construed in a conventional, Newtonian manner is found to be insufficient to all classes or levels of systemic functioning that involves some form of energy exchange dynamics. Energy exchange dynamics in natural systems upon different levels, as well as in artificially created systems, can be demonstrated to include non-Newtonian mechanics. The conventional example of course is Heisenbergian uncertainty of quantum mechanics, but similar kinds of uncertainties exist at other levels, and in other forms. We have not yet fully modeled, for example, gravitational dynamics, and we may be quite surprised at how this form of energy exchange defies even our conventional modes of thought about quanta.

It is clear that the informational problem represented by all classes of natural and metasystems is separable analytically from the energetic considerations of such systems as real systems. These fundamental differences in natural systems theory in general reflect the mind/body duality or the material/ideal dichotomy that is typical of all western rationalist thought. In this case, both informational and energy dynamic aspects of systems can be represented in an analytic manner that is quite similar to one another--almost to the extent that they can be considered analogous or at least as two sides of the same coin. We know for instance that energy exchange without some kind of informational constraint results in random or chaotic processes. We also know that there can be no sense of informational constraint or quality within a system without some sense of energy quantities or dimensions that represent such constraint.

The real challenge of such system models is figuring out the pattern of integration that they may achieve or follow, and the principles that underlie these integrative patterns. Furthermore, this question of integration of systems leads to other questions of determining in an accurate if general manner the contextual relationships such systems have to larger systems of which they are a part, and how inter-sytems regulation occurs in natural process. Furthermore, such theoretical problems of natural integration also lead to questions about the alternative pathways any given system or set of systems may follow in their differential state-path trajectories. In other words, systems are to some extent underdetermined systems, and to the extent that they are underdetermined, no two systems will be exactly identical, nor will any two systems follow the same exact patterns of historical resolution. Finally, such questions also lead to broad and more general problems of developing a typology and taxonomy of systems in a manner that is realistically representative of the natural distribution and relational patterning of systems in a general and comparative sense.

It can be said that if all systems are by definition underdetermined, then any system will be unique in an exact sense, and will demonstrate some minimal degree of possible variability of change pattern. Even systems in nature that we hold to be fudamentally stable, such as the atomic system of the periodic table of the elements, which is held to hold true under all normal conditions on earth, must be suspect as a kind of typology that hides some degree of minimum variation of its elemental classes. It is know that isotopic configurations of elements is estimated and to some extent variable, which confers differential atomic weights and therefore estimated averages. Any particular sample of any particular element or molecule may be more or less the molecular weight that is predicted by the periodic table, with some minimal degree of isotopic variation.

We tend to assign relatively discrete mass measurements to nucleonic particles, and energy measures to electrons inhabiting what are known to be discrete orbital levels. It is possible that these measurements of mass and energy of discrete entities, which are themselves more the nature of energy-entities, may be continuously fluctuating about some normal distribution, and that they may even on occasion jump between levels. At a quantum level of measurement, we may even say that such measurements are in fact statements of a certain kind of probability, of likelihood, of finding a particular entity in a given state in a particular instance in time.

We can therefore modify even our initial statement that all systems by definition of their underdetermination will be unique in an exact sense, by say that each system will tend to be instantaneously unique and variable as a function of time--in other words systems will be unique states at any discrete instant in time, and will be variable through the longer continuum of the duration of time. This is a basic change principle:

 

1. No two systems are exactly alike in time or across space.

2. No single system is exactly like itself through time.

3. All systems are underdetermined, and hence are dynamic.

4. The only absolute about such systems is the dynamic of change.

 

If we are to get at the fundamental principle of why all systems are inherently underdetermined, we must at some level come to the problem of the relative structure of systems that is a function of their inherent complementarity. Complementarity suggests that any system may exist at any particular instant in more than one possible state with a given distribution of probability. Complementarity suggests furthermore that it is possible for the same system to exist in more than one possible state simultaneously in any particular instant, depending upon how this system is being observed. The nature of the observation affects the instantaneous state of the system, and reflects as well the basis for such distribution in the universal relativity of all systems.

We can put this another way and say that no system exists in an exactly or precisely discrete sense. All systems are inherently distributed and continuous--i.e., they are fundamentally non-discrete. Their sense of being discrete is a function of our observational constraints that we superimpose upon such systems, and are thus a result and residue of the fact of observation. To a great extent, the determination of discreteness for any system, of its exact instantaneous state, is a function of the precision of our instruments of measurement, their resolution and accuracy. It is also a function of the relative units of analysis and scale of observation that we select. It turns out that to an atom, a second, or even a femtosecond, may seem like a life-time to a human being upon a much larger scale. If a human observes a small microbe through a light microscope within the frame of a minute or two, chances are that microbe will be construed as an instantaneous event structure that has not changed during the entire period of observation. The countless numbers of biochemical transformations and processes occurring within the cell, too small to be seen even with a high-powered light-microscope, may go missed by the careful observer and therefore be discounted. In general, we see change process in such microbes within the span of generation time, be it twenty minutes or an hour, or for eukaryotic cells, within a twenty-four hour cycle. Generally, if we seek to understand processes on a molecular level within the cell, it is necessary to perform procedures leading to the death of the cell as an entity and its isolation as a momentary event structure that is arrested in time.

The complementarity of structure of all systems are due to several related properties of such systems. 1. All systems are stratified and relative to other systems upon some, and usually multiple, level levels of interaction. 2. All systems are by their basic sub-systemic structure continuous and non-discrete at multiple levels of analysis. 3. All systems are by their energetic exchange dynamics situated within a relative surrounding environment and thus are partially open within that environment. Furthermore, the surrounding external environments are by definition a part of a larger encompassing system of relations.

From an energetic standpoint, we may invoke the basic laws of thermodynamics for most mechanical systems involving energy and matter, though this may not subsume the entire class of systems or energy exchange relationships that compose such systems. Basic evidence suggests strongly that the laws of thermodynamics are covering law models that are part of a larger energy dynamic system. Thus, energy dynamics, however imperfectly understood, form fundamental mechanical constraints in the functioning of basic systems that results in inherent change and variability of all systems in time and space. These mechanical constraints can be understood in either a quantum or a classical manner with the same end results.

On a quantum level, basic phenomena can be explained that appear to violate thermodynamics upon a classical level, as for instance the phenomenon of superconducting or the tunneling of electrons through a substrate. Furthermore, these same energy dynamics appear to occur in all systems that are classifiable as real systems, no matter what the level of integrative functioning or scale upon which they occur. We may characterize biological systems in such a manner, in terms of their fundamental molecular and atomic dynamics, and we may furthermore characterize even brain-based mental systems with a similar kind of model, though the latter set of system is as yet incompletely described or understood.

It can be demonstrated though that the characterization of biological or brain-based systems by means of molecular or atomic models is inherently insufficient to the full scientific or naturalistic description of such systems, as levels of integration are complex by many orders of magnitude in such systems, leading to new sets of intrinsic properties characteristic of such systems. Systems that are integrated upon supercomplex levels can be said to exhibit both intrinsic functional properties and extrinsic state-path properties that are emergent from the integration of the system and that are, as a class, distinct from the kinds of properties of the subsystems that compose them.

In understanding the integrative stratification of systems in reality, we can make the following kinds of statements:

 

1. Functional stratification is based upon relative differentiation within systems, between subsystems, and without systems, between supersystems, which differentiation is a result of the continuous variation of such systems.

2. Functional stratification leads to increasing levels of integration that exhibit the following characteristics:

a. exponential complexity of relational patterns

b. increasing underdetermination

c. increasing alternative variation of resulting patterns

d. increasing emergent properties associated with such systems

3. We may distinguish in reality between forms of intensive stratification, or intensification, of natural process, and extensive stratification, or extensification of natural process, associated with systems.

a. All systems will exhibit some degree of both continuous intensification and extensification.

4. Such processes of intensive and extensify stratification lead to emergent forms of integration between systems at one level to create entirely new systems at another level.

5. Because such processes in nature are fundamentally underdetermined, we may say that all such processes and patterns of integration are fundamentally stochastic and unpredetermined. However unlikely such systems may be, all naturally occurring systems emerged as a result of chance distribution and occurrence without any a priori controlling force or sense of predetermination.

 

It is the case that in terms of our language of description to match in an empirical manner our level of observation and to designate our units of analysis, we are thrust upon the horns of a dilemma to the extent that we must deal not only with the physical relativity of natural systems in terms of our observational experiments, but we must deal with the anthropological relativity of our language and knowledge in terms of the designated units of analysis and description that we apply to our observations. On a naοve level, basic descriptors derived from "natural classes" in any language appear to be sufficient to the tasks of basic qualitative description. We have mathematics, the language of science, to come to our rescue especially when we are referring to basic and "average" physical processes, as for instance those entities represented by the periodic table and those energetic event structures described within the framework of classical mechanics. But even upon a micro-biological level the language of mathematics and its inherent logic begins to break down under the shear weight and complexity of the problem of natural description.

 The function that mathematical language serves upon a biological level is fundamentally different than the function it serves upon a physical level. A strong case can be made that mathematical description breaks down almost completely upon the even more complex human level of analysis, except in the form of applied statistics and rather gross and concrete numerical descriptors. But even upon the fundamental level of physical analysis and observation, resort strictly to mathematical description is inherently insufficient to the inclusive problem of descriptive explanation. Most physical properties or laws that govern systems upon these fundamental levels are defined in terms of linguistic based variables or logical syllogisms that are held to be generally if not universally applicable to all cases, and most such properties, principles or laws were derived at through empirical observation and experimentation in conjunction with deductive reasoning that is applied to the evidence at hand.

In such a context, mathematics as used in the theoretical or applied sciences takes on a basic applied function that is distinct from its abstract articulation in pure mathematical theory. In such a case, as demonstrated for instance through statistical description and manipulation, mathematics is applied to natural data sets or samples or populations of "points" which discrete point determination, as referred to previously, is inherently problematic from a linguistic and observational point of view. Dealing with natural sets of data points defined experimentally or observationally is fundamentally different from dealing with abstract sets of numbers or points defined arbitrarily or by means of logic. If we hold to our initial pressupposition that all entities and event structures are inherently underdetermined and continuous, then the application of discrete and discontinuous labels or attribution to these sets must be on some basic level fundamentally problematic.

We can often proceed, as with many covering law models, on the basic assumption that the degree of continuous variation is negligible or can be discounted and that our data sets are, for the limited purposes that they are used, sufficient in a substantive and theoretical manner. Science could not otherwise proceed in a normal manner unless we make these heuristic leaps of faith regarding the basic reliability and validity of our data sets. And even when such presuppositions become extremely suspect, especially with human systems, we even still like to invoke mathematical models and formula in a general and usually overly simplistic manner, and usually with the consequence or intention of simplifying theoretical explanation. We hold inherent complexity temporarily in check, as it might be, in order to build our model or defend or rationalize our argument. We assume the units we describe to be relatively discrete, and often ignore the relativity of our analytical indiscretion.

This problem of anthropological relativity leads us directly to the fundamental challenge in all the sciences of building reliable and empirically consistent taxonomies and typologies that allow us to systematically compare and relate different systems at different levels. The archetype of such a model is of course the periodic table of the elements in chemistry. A system of subatomic classification of fundamental particles has emerged, though its systematic definition is still incomplete. Increasingly sophisticated biological taxonomies are emerging, all fundamentally based upon a modified Linnean system that is explained in terms of an evolutionary tree model rooted in Darwinian theory. It is recognized though that upon this level, major classes and categories of biological patterning are not taken into account, and there is deep-seated desire among many biologists who feel the insufficience of their concatenated system for a new kind of "synthesis" that will integrate the many subdisciplanary focii of the overall field. A call is sometimes heard for a systematic system for classifying eco-trophic niches in ecological models, though this has not yet been accomplishe due to the enormous variability found at this level of integrative analysis. The study of human systems, at whatever level, are even less satisfactorily organized under any comprehensive framework of systematic classification, typology and taxonomy. So much is this lack of synthetic unity the case in the human sciences, that there are entire disciplines that are essentially in competition with one another over basic definitions of units of analysis and classes and nomenclature, much less the systematic relations that these descriptors imply.

 

Cross-Correlational Systems as Heuristic Models for General Scientific Description and Explanation.

 

The quest therefore in natural systems theory and metasystems science is for a generalized operational system that will permit integration and synthesis of knowledge upon a number of representative levels, and across a wide plethora of different fields.

I propose the general use of cross-correlational systems, based upon advanced number, measurement and set theory, as a sufficient heuristic model for the general description and explanation of phenomena in the sciences. These systems, in variant and modified forms, appear to have general applicability and functional utility in most if not all scientific fields of endeavor, and they lead as well to the description and explanation of real alternative systems, investigation of hypothetical systems, and the development of abstract and artificial systems as well. I do not claim that this is the only or necessarily the best set of operational procedures to be used, but I do claim its general validity and broad-based reliability.

In the delineation of cross-correlational analysis, I recognize five levels of abstraction that are involved:

 

1. Number theory deals primarily with mathematical languages, principles and problem sets. Advanced number theory attempts to work with complex numbers that are represented only or primarily as relative variables. I am concerned in relation to advanced number theory primarily with the systematic use of varables that are inherently dynamic and comosite.

2. Measurement theory is based conventionally upon descriptive and predictive statistics, but involves as well the basic issues of deriving data sets and their manipulation based upon descriptive inference. Generally the criteria of measurement is relative objectivity that is achieved by the superimposition of some conventional standard or unit of analysis that is relatively non-arbitrary, and the explicit and systematic uses of these standards in descriptive observation.

3. Set theory concerns two interrelated dimensions, the language of types and labels and the problem of the classification of things or events into some comparative framework. Set theory conventionally leads to the use of deductive and inductive inference in the construction and at least the implicit comparison of mulitple sets. Hence sets are generally constrained by the terms and rules of logic that we apply to such systems, and logical inference forms the basis by which we construct and manipulate sets in relation to taxonomic frameworks or typologies. Generally, a taxonomy will imply some kind of logical system of inference that underlies the construction of the taxonomy.

4. Relational theory is the basis of cross-correlational analysis, and concerns primarily the systematic comparison and interrelation between different or multiple sets, of the pattern of variation of the same set over time, in such a manner that we can explain processes of integration or disintegration that occur at different levels. Relational theory is concerned primarily with the scientific explanation and description of change in and between systems upon multiple levels. Therefore it is concerned with the dynamics of variation of systems, and with the ranges of alternation available to such systems over time. It is concerned as well with the problem of integration of sets into systems, and the integration of subsystems into super-systems.

5. Heuristic modeling theory concerns the use of the results derived from cross-correlational analysis to generate or construct systemic or mechanistic models of systems that permit some degree of pattern prediction and simulation under controlled circumstances. Modeling theory is primarily heuristic and experimental in orientation, but it leads secondarily to the application of model systems for solving real problem sets in a systematic and controlled manner. Heuristic modeling theory can be said to encompass most of what is received as the conventional scientific method, and it leads to the fomulation and testing of competing alternative hypothesis about the structural explanation of reality. In general, successful scientific models have not only results that are predictive, but that also can be simulative and even creative in the sense that they lead directly to the development of new and alternative kinds of systems. Models from this standpoint can be said to be theories or exemplary representations of reality in a simplified and condensed form. They can be said to be prototypical or archetypical of the full range of phenomena that they theoretically subsume. A successful model can be thought of as a correct solution for a given problems set, that, when applied under universal conditions, will lead to the same results.

6. Advanced Systematic Taxonomy depends upon the development of realistic and predictive models for the construction of larger taxonomic systems of classification based upon the principles derived from the model. A valid theoretical model should lead to at least a partial taxonomic construction--the more comprehensive the model, the more complete the taxonomic framework. The taxonomy provides the general frame of reference for the definition of the supersystem, and therefore the taxonomy comes to embody and express through its structure the theoretical model upon which it is based.

 

The basis of cross-correlational analysis is the systematic comparison of relational complexes that occur between different data sets. No degree of interdependence is necessarily presumed to exist between different sets, though there is an assumption of dependence existing between components within sets. This intradependence is not assumed to be complete, but only partial. It is also not assumed to be static but dynamic. It exists in no particular instance of an event or an entity, but is distributed throughout, unevenly and in different ways, across all possible events or entities.

The assumptions in which cross-correlational analysis are rooted include the following:

 

1. For any given system or set of systems, there are three analytical levels that must be specified: i. Subsystems composing a system; ii. The System in itself; iii. The Supersystem of which the System is a subsystem.

a. This designates the general order and suborder of systems in reality.

2. Any given System at any given level of analysis can be characterized in three ways: i. As a System in itself; ii. As a Subsystem of a surrounding super-system; 3. As a Supersystem containing subsystems.

a. Higher order systems demand analysis that is more general rationally and less precise empirically.

3. A System at any given level of analysis is subsumed by all higher Supersystems, and subsumes all lower Supersystems to which it is directly related.

a. Systems become increasingly complicated and underdetermined with the increasing order of the system. The more complex the system, the less inherently determined it will be.

4. For any given System at any given level, there will be an open class of higher and lower order systems that can be said to exist contemporaneously with that system and which can be said to be indirectly related to that system as a part of the intensive surroundings.

5. For any given System at any given level, there will be a open class of alternative systems that can be said to exist contemporaneously with that system, and which can be said to be indirectly related to that system as a part of the extensive surroundings.

6. All systems are minimially connected upon one or more analytical levels, however indirectly, hence all systems contain some minimal degree of relational similarity with other systems upon at least one level.

7. The descriptive characterization of any system is always assumed to be instantaneous and continuous, subsuming an inherent degree of variability that leads to error and uncertainty (parallax) in relation to knowledge about that system.

8. A system as a conceptual model represents in abstract form the hypothetical structures (redundant or reiterative patterns) that are observed or alleged to exist in the phenomenal pattern of experience.

9. The objective of scientific inquiry is the excoriation and explanation of such models in a manner of increasing correctness of fit between the conceptual model and the experiential patterns that it refers to and subsumes, and a corresponding decrease in the relative uncertainty or probability of error associated with that pattern.

10. All systems, at any given level, have a life-cycle trajectory and are subject to rules of random and regular change. All systems have a beginning, an indefinite intermediae period or set of periods, and an ending.

 

            Before proceeding with this digression upon operational systems and their application to general problem solving procedures at various levels in systems science, it is important to go down one or two other tangents.

 

Scientific Description as Rational Explanation

 

Scientific description is an attempt to linguistically represent the patterning of reality in a reliable and faithful manner. Such description can proceed at different levels, in alternative circumstances, and may lead to different kinds of results. As mentioned previously, description brings us to the problem of language parallax, and largely, the problem of anthropological relativity of the knowledge that such language entails.

We may say in general that the goals of scientific description are to lead to explanation in the shortest and most succinct route possible. Therefore, explanation is in a sense inherent and a part of scientific description, and should be a logical outcome of correct description. We see as well that preconceived views or models about reality can have the influence of channeling our description metaphorically in certain directions that may or may not reflect the actual patterning of reality.

Description is not necessarily to be confused with explanation. We may say that it is appropriate to separate the two problems analytically, as in a lab or field report. But we can say that description and what gets described and how is as often as not preconceived by the explanatory models we may have or want to have, and that at some point the two levels may come into dialectical conflict, in terms for instance of frame disruption, error and frame repair, or they may come into a kind of convergence, as in the case of constancy of perception that allows us to see what we want or at least think we are seeing.

The selection of descriptors and the sentential construction of a description refers to the direct perceptual response to empirical experience and observation. It connotes a studied approach to information.

Explanation refers only indirectly to the observation, or to the phenomena involved in a general sense, but refers primarily back to the description that we have formulated in relation to the observation. Observation, especially when this is constrained experimentally by systematic measurement, is itself a form of deliberate description, or at least the selective perception upon which such description is based.

Explanation carries the entire process one step further, and depends upon a deliberate "distanciation" or alienation from the source of the information, as well as upon the reliability of the descriptive information that was derived from the source. Explanation furthermore is concerned with the logic or coherence of the resulting statements concerning the prototypical patterning, or structure and its validity.

It can be seen that the primary preoccupation of description is consistency and reliability, while the primary concern of secondary explanation is coherence and validity of the models that are derived. It can be said in a reciprocal way that explanation is really a form of secondary or derivative description that takes description from a specific or methodological level of analysis to a general or theoretical level of synthesis. Again, the feedback nature of this process must be emphasized, as the development of theoretical explanation will in turn condition our initial responses and observational frameworks, and will lead to refinement of our descriptive informational background.

In a general sense, we can say that description leads us, by systematic steps based upon inference, from the particular to the universal, and from the analytical to the synthetic. It leads us from descriptive information to explanative understanding, and this continuum can be said to form a knowledge system that is defined by a certain order and kind of information upon which it is based. We develop explanatory models to organize our descriptive data sets, or information, in ways that are coherent and make sense, either from our own preconceived or arbitrary standpoint, or from a standpoint that can be said to be relatively independent of our own a priori judgement.

I have sidetracked in this essay about scientific description and explanation because, upon a fundamental level, operational systems in metasystems science occurs and works in this framework of understanding of a feed-back loop in dynamic information systems, from empirical description to rational explanation leading back to exemplifying or experimental description under rationally controlled conditions. Our knowledge is locked perpetually within such a feedback loop between our descriptions and explanations of reality, and we are always testing new frames of reference with new units of analysis to achieve some level of systemic equilibrium and sense of coordination if not control over such knowledge systems in general.

In general, it can be said that science as opposed to ideology, does not privilege any particular explanatory frame of reference that might lead to a preselection or conditioning of our descriptive units of analysis in terms that are inflexible or constrained. It tends to privilege descriptive units of analysis rooted in observational experience before it privileges explanatory frameworks, however rational or rationalized. Paradigmatically it can be demonstrated that scientific theory can frequently smuggle back into its explanation of reality ideological conceptions that may become inadvertently priviledged or in a sense a posteriori to the data, but at least in science the ultimate reference points are supposed to be the empirical observation of data that is descriptively defined in as clear and careful a manner as possible.

 

Abstract Frames of Reference and Concrete Units of Analysis

 

The basis of number theory is strictly arithmetic and mathematical. Number systems and their manipulations are considered purely theoretical and abstract. My point of departure for metasystems theory in relation to number theory is to propose a class of complex number in which a number stands as a mixed heterogenous variable that may be used differentially in a number of different kinds of systems. Each number then would be in indexical reference/inference marker, representing a complex variable, that could stand for a large number of subsets of numbers or variables, while at the same time, standing for itself, and standing as part of a larger system as well.

It may seem that this is a way of rendering mathematically systems extremely unwieldy and overcomplicated. To get at the issue, we must go to the basic meaning of what a number is and what it represents in reality beyond its own logical representation. Generally, we count things in sets. If we count a set of five pennies, we can assign the number one to each penny, and the number five as a denominator to the set as a whole, especially if we recognize a five cent piece as a whole unit of which a penny can be considered to represent a proportion of that set.

Alternatively, we can say the following:

 

1 + 1 + 1 + 1 + 1 = 5

1/5 + 1/5 + 1/5 + 1/5 + 1/5 = 5/5 = 1 nickle

We can then simplify the equation by multiplication:

1 x 5 = 5

1/5 x 5 = 5/5 = 1 nickle

 

All other manipulations from this follow, for we can subtract or divide one or more pennies from the whole to define what some number of pennies represents in relation to the entire set.

The question that I ask is what the assumptions are when we count pennies and compare a set of pennies as equivalent to a nickle. For all intents and purposes, each individual penny may and probably will not weigh exactly the same, but conceptually we treat them not only as equivalent to one another, but as mathematically identical and interchangeable within the set that can be subsumed by the name of "penny." The variation of weight and size of any particular instance of a penny is irrelevant to its estimate of value from a monetary standpoint. I do not wish to go into the symbolic dimensions of money and value, but there is a strictly logical operation performed upon the penny in which it is assigned a discrete numerical value and is classified at the same time with all equivalent pennies sharing the same value. Pennies in this case become interchangeable as numerical units, and they are used in precisely this way in the exchange of money. We could perform the same numerical operation if we count out a set of pebbles, however oddly shaped and composed, in a pile. We treat each pebble, however different, as numerically equivalent as discrete units. Anything that can be counted in this way is defined as something that is discrete as a unit, and equivalent to other similar units, no matter what the variability actually subsumed by the class.

We would say that the set of pennies or the set of pebbles (or oranges, applies, flies, etc.) are simple sets that are defined by their countability and conceptual equivalence. Any one orange would be as good as the next, no matter what their individual virtues or faults. We are essentially treating a set of real objects as if they are representatives of abstract sets, allowing thereby their mechanical manipulation in terms of abstract operations.

If we can say that simple numbers in general define simple sets, then we can say that complex numbers define complex sets. We can therefore learn what a complex number is by the kind of sets that they form. If countability is at least one of the abstract operational procedures characteristic of simple sets, then it strikes me that a complex set would be one that cannot be characterized by the procedure of counting. We cannot simply add up all the units of the set, and say that the set is (N)1 in size. There may be a number of different reasons for this non-countability of complex sets. In this regard the kind of sets I am after are those that are defined by non-discrete entities, continuous rather than discontinuous variables, unlike or nonequivalent members, open sets, non-interchangeable members, relational complexes and sets that are composed of other sets that are themselves complex.

It may well be asked, what good are sets that cannot be counted, as it would appear that from the beginning such sets are not amenable to basic arithmetic operations or manipulation. How can we determine for instance, the size of a set of air or a set of sea water if we cannot determine the number of molecules contained in our set of air and sea water? Linguistically, it makes no sense to call an area of air a set if we cannot count its fundamental units in any obvious manner. It defies the covert categories of semantic meaning that we distinguish between count and noncount values. We can count pebbles and rocks, however small, but we cannot count dirt or mud.

We can of course measure the mud out in a number of buckets, or the sea water in a number of jugs, or the air in a number of balloons, and then count the buckets, juts and balloons as countable units of mud, water and air. But this is not solving the central problem of identifying a complex set--it is rather systematically transforming a complex set into a simple one that can then be counted. This is what we do in scientific method, and this issue will be dealt with in measurement theory, but it begs the question of identifying and dealing with a complex set.

I would say that a complex set can be treated essentially as an unknown set. Its dimensions are uncertain and undescribed as is. We may not know its boundaries or its limits. We may say for instance the set of all birds in Australia, not knowing the full range of bird fauna there, the extent of any one species or the possibilies of flight by different birds from and to different surrounding land masses, or migration patterns. We do not know, for instance, the rates of death or birth of different bird populations in Australia. On the surface, "the set of all birds in Australia" is conceptually very simple, but if we try to determine or specify this set in any exact sense, we quickly run into enormous difficulty and complexity. It is the nature of complex sets, I believe, that if we try to solve them in any direct mathematical procedure in terms of their component entities, then we quickly run into an exponential increase in complexity of component variability and relationship. Take for example the following kind of set: Suppose that a set is contained of 5 variables, (x's) and each x is a composite variable of (yz) variables and each y variable is a random number between 1 and 100 and each z is yet another subset of two more variables, one of which is also a random number between 1 and 100. It can be seen that even if we eventually came down at some level to purely countable numbers, the number of operational procedures that would be required to determine the solution, or the range of possible solutions for such a complex set becomes quickly astronomical, requiring probably the assistance of a computer.

We can state in a basic way that the scientific operation is to determine a set of measurements that will simplify a complex set to a simple set that can be somehow counted and thus manipulated. Until we an perform such an operational procedure, we can say that a complex set is a kind of problem with an unknown solution or method for solution.

A complex set is a problem set of uncertain dimensionality and unknown solubility. Theoretically and methodologically, complex sets are the stuff of scientific research. Science attempts to apply systematic means to reduce complex sets without known solutions to simpler sets with known solutions.

Now that I have identified a complex set in a negative sense, we have yet to ask what it is in a positive sense. We can say that while a simple set is characterizable by countability, or what we can call the cardinality of simple numbers, we can say that a complex set is likewise characterizable by non-countable computability, or what we might refer to as the cardinality of complex numbers. A complex set is therefore characterizeable by the complex numbers that it component subsumes or represents. So, then, hedging the question a little further, what is a complex number?

Suppose for instance we have two odd assortments. The first assortment is of 10 eggs, 2 chickens, a rooster, a farmer, five flies and three ducks. The second assortment is of 10 cars, 2 trees, 25 mice and an old tire swing under one of the trees. How can we systematically compare these different kinds of sets? We can simplify the problem and count the items in each set, and say that the first set has 22 assorted items and the second has 38 assorted items. But this is, I believe, comparable to our buckets of water, in that we are lumping into the term "item" a connotation of countability and thus interchangeability and equivalence that ignores the obvious and pronounced differences between the items being counted. "Item" in this example obvious disguises more than it simplifies.

Alternatively, in this example, we can say that the first assortment has 6 subsets of equivalent items of different components, and the second set has four subsets of equivalent items of different components. In this kind of solution, we are typologizing our sets in subsets, and essentially creating a kind of matrix for each set by which to compare it to the other.

We can go the other direction and claim that a complex number is an uncertain number with an unknown solution. It may or may not have a possible solution, we just do not know. But as with the characterization of a unknown complex set, we cannot define a complex number by what it is not, rather that by what it is.

Therefore, I will venture a definition of a complex number, and say that a complex number is a polynomial variable each of which is composed of an unknown subset of other variables, which may be discrete or polynomial. At some point in this reductive analysis of our complex number, we may come to a known simple number as a constituent of the variable. In this case, we are sort of systematically chasing out what is unknown about a complex number by making it more complex than it already is, and thereby possibly factoring out as many discoverable values as simple numbers. We are factoring the problem in the hopes of obtaining a solution to it.

We can say then that a complex number, like a complex set, is an inherently undetermined and possibly undeterminable number. Any complex number remains to some extent underdetermined as a number, and any complex set remains inherently underdetermined as a set. Complex numbers are therefore capable only of partial determination through factorial analysis, and complex sets can be resolved only partially.

We may risk a generalization then, and say that a complex number is always some composite number. It is a number composed of other numbers, some of which may be known or knowable, and others of which will remain unknown. If we call the number 60 simple, then it is designated by one and only one value, however written. We could write it as 15 x 4 or as 120/2 or as 240/4 or as 10 x 6 or just as plain old 60. It would remain simple because it is reducible. But what if our complex number sixty where really the composite polynomial XY = 60, in which both X and Y could be any number in relation to the other. We end up with almost an infinite number of possibilities for X and Y if we consider not only whole numbers but fractions. If we could perchance determine one of the variables, say X, then the determination of the other variable Y could be achieved by rapid mathematial deduction. The equation XY = 60 represents therefore a kind of complex number without clear solution, while any of the other examples represents simplified numbers or equivalents of 60.

The complex number above would only increase in complexity if we split one of the variables into three, as for instance XYZ = 60 or WXYZ = 60. Then the number of possible combinations, and the required combinatorial space, jumps up exponentially.

Does a complex number exist in reality? No, not really, but then neither does a simple number which exists only as an abstraction. Just as countable objects exist that can be characterized by simple numbers in simple sets, so too do noncountable things appear to exist that can be characterized by complex numbers in complex and unknown sets. Science deals with these kinds of sets all the time, indeed most of the time. Biology is replete with examples that do not go conceptually far past the oversimplistic statement of "the set of all birds in Australia" without really being able to discretely identify this complete set. Such a set therefore represents a complex and unknown set of complex and unknown numbers that can be only possibly partially factored out and simplified. And when we really try to crunch numbers in biological systems, we quickly run into astronomical complexity and high levels of uncertainty which strongly suggests that we are playing at least conceptually with complex numbers of things we do not fully know.

The problem set "of all birds in Australia" can be said to represent a kind of conceptual solution in itself that symbolically summarizes the problem in a gross descriptive manner without solution, though this problem represents from the standpoint of a scientific solution an oversimplification of the problem. Oversimplification by conceptual definition is not always a wrong recourse, and I believe much of theoretization at the levels of biology and social science relies upon such conceptual strategies in a generalistic solution to problems where exact kinds of solutions would be impossibly complex and underdetermined.

We may say that a complex set poses a problem that entails a combinatorial explosion of possible solution space. It is interesting in this regard that only in certain computer languages, can such combinatorial explosions be handled in a logical manner that can solve for finite puzzle-type problems, however complex, in just a few lines of symbolic code.

The contrast of a simple to a complex number may seem in itself oversimplistic, or perhaps, unnecessarily complicated, but I believe it gives us a direct handle on understanding what can be considered to be fundamental designative dilemmas in normal scientific operations, and that is the determination of units of analysis among unknown variables that will permit some degree of manipulation, even systematic comparison, of these units. Indeed, scientific method is about taking complex realities and systematically simplifying them down to relatively simple solutions. And this is done by factoring out the knowns from the unknowns with the hope of eventually reducing the unknowns to a smaller and smaller subset of the knowns. There are many natural systems, of all classes and kind, that can be characterized as complex sets as I have defined this term.

A complex number can characterize a range of possible simple number solutions. We can say that any complex number will be solved by more than one possible alternate simple number, and usually by a complex combination of simple numbers.

 

Information Theory and Mechanical Systems

 

            In terms of energy transactions, all real systems that have a physical existence can be considered to be mechanical systems of some kind and order. We may deploy different "machine" models to describe different kinds of mechanical systems. The machine system can be analyzed in terms of its components, and it can be studied in terms of the patterning of interaction between components. Studying the mechanics of systems in terms of enegy transactions that involve work and relative efficiency of some kind, and some degree of entropy, invites a theoretical model of the informational correlate of the machine, as an order producing, order maintaining system that has a capacity for information and that has a certain measure or degree of noise associated with that system. To put this in short form, where we find work in systems, we find order and information about that system.

            Reductionist theory would claim that the mechanics of any system of any kind is reducible to the fundamental laws of physics governing such systems. Antireductionists, or as von Bertalanffy called, "perspectivists" would asser that such fundamental laws are insufficient to the full accounting for the behavior of the system. "…The presently existing laws of physics and chemistry may well turn out to be inadequate in the description of the living system for the same reasons that the laws of Newtonian mechanics were inadequate in dealing with the interior of the atom." (Gatlin, 1972: 16)

            According to Michael Polanyi, reductionist explanation in terms of fundamental laws of physics ignores the laws of information theory governing any information producing machine: "all objects conveying information are irreducible to the terms of physics and chemistry." Any machine is an information producing machine as well as a working machine--order required for work produces information. Any such machine cannot be understood in terms of its information processing capacity by a mere description of its hardware. Information producing machines, or real systems, are the result of higher order operational principles governing their design and function that cannot be deduced from the analysis of its hardware regardless of the accuracy and precision of its physical measurement. Any such machine is furthermore controlled primarily by its boundary conditions, and the operational principles and boundary conditions constitute a more sufficient and fundamentally relevant explanation of a machine than the systematic accounting of its hardware and mechanical operation. Higher operational principles within the hierarchy of determinations of stratified natural systems determine the boundary conditions that serve to define any information processing machine.

            Mathematical knowledge is applicable, in some as yet undefined form, to any kind of natural or real system. This application of a body of mathematical knowledge becomes, if successful, a part of the theory that is used to explain this kind of system as a general model. The description of any system mathematically becomes accurate and precise when this description becomes an explanation by laws of physics, when mathematical description of the system is "so exact in numerical terms that quantitative prediction of experimental fact inevitably follows."(Gatlin, 1973:19) Anything less than this, anything only approximate and less exact, does not constitute a covering law of the physical description of the system, but only a symbolic and hypothetical interpretation of the system.

            The information content of any possible system is defined by the number of alternative informational units or states that compose the system. Binary systems of digital computers use a bit system based upon values of 0 or 1. In any information processing system, we can denote the size of its informational content by determining the capacity of each unit and the total number of units in the system. For discontinuous or discrete data, we can ask how many bits or units of information for the total system; for continuous and indiscrete forms of data, we can ask how much information is in the total system.

            Information theory hinges on the definition of entropy we adopt. In the case of living systems, it is apparent that with the evolution of organisms, there has been a corresponding increase in the negative entropy of such organisms. In the most general sense, entropy may be defined as the degree of uniformity or sameness or redundancy in anything. Entropy comes from the theory of thermodynamic systems, and it is defined as the degree to which the energy in a closed thermodynamic system or process has ceased to be available energy. In reversible processes, entropy in systems remains the same, but in natural irreversible processes, the entropy increases. Entropy is said to be increasing for the universe as a whole. Thus, put an ice cube in a room at normal temperature and it will melt slowly. Put an ice-cube in a hot room and it will melt more rapidly. Put an ice-cube in a walk-in freezer, and it will not melt. We do not expect to see an ice-cube freeze in a warm room, for otherwise we would expect a violation of fundamental principles regarding the order of natural systems. An ice-cube melting in a warm room is an example of an irreversible process of heat gain from the environment into the ice.

            Entropy is the measure of the alternative states a system may assume, and in communication theory, it was developed as a measure of information in a system.

Any mechanical or physical system will under constant conditions approach equilibrium with its environment if the heat exchange between the system and the environment is irreversible. Equilibrium is the natural state of a system that maximizes the entropy of the system at constant energy, consistent with the constraints of the system.

            Natural processes, such as ice melting, proceed always in the direction of equilibrium, and is an irreversible physical process. Unnatural processes are impossible processes that move in the opposite direction, towards greater disequilibrium, and hence, never occur. A reversible process is an idealized natural process passing through a continuous sequence of equilibrium states, depending upon changing conditions between the system and the environment. If the temperature of the room in which the ice-cube is melting suddenly turns to freezing, then the ice cube will cease melting, and whatever water produced by melting will begin freezing once again. Work is accomplished in systems by slight changes in system state variables or boundary conditions that result in reversible processes.

            The entropy function, S, is introduced in relation to natural and reversible processes in the heat flow of systems. Lowercase q is the measure of the heat flowing into the system from its surroundings, and T is the absolute temperature of the system. Thus:

 

1. dS > q/T for a natural change and dS = q/T for a reversible change.

2. Entropy of system S is made up of the sum of all the parts of the system such that:

 

S = S1 + S2 + S3 + …..

 

            Unlike energy, entropy cannot be conserved. Increased work in a system increases the entropy of the system. The work, the conversion of energy into heat, is 100 percent efficient, and work increases the entropy of the system. Converting work into heat is an irreversible natural process, but it is impossible to reverse this process, converting heat from the environment into work in a system producing greater entropy, unless changes to the environment are made. This becomes the second law of Thermodynamics. In a cyclical system, heat can be converted to work through a system so that the system will return periodically to its initial state, but the efficiency of this process cannot be 100 percent, with a portion of the heat being lost. The lost energy results in the degradation of the original energy state. If a restoration system is used to restore lost energy to its original form, this system of restoration degrades the energy even more. Therefore, all mechanical processes occurring in the universe result in an overall increase in entropy and a corresponding degradation of energy. While the energy of the world is always conserved and therefore constant, the entropy always tends towards its maximization.

            Understanding systems upon a fundamental level of the atomic theory of matter, the increase of entropy towards its maximum value at equilibrium corresponds to the change of the system toward its most probable state, its most mixed or most random possible state, consistent with its constraints. Mixing includes configurational mixing of particles, as well as the diffusion of energy over the particles being mixed, as for instance, in the expansion of gases into one another or over a given three dimensional space. Friction spreads energy over constituent particles. Energy-spread entropy is not always compatible with configurational entropy within a system, and a compromise state of dynamic equilibrium must be obtained.

            Any substance at finite temperatures has an absolute entropy. At zero temperature, entropy vanishes from a system. Any thermodynamic state of a system at a finite temperature corresponds to many microstates of the molecular components of that system that undergo continuous rapid transitions during observation, and the entropy of the system corresponds to the logarithm of the number of available microstates. The state of the system as a whole of all the molecules is referred to as the macrostate, and entropy of the system corresponds to all possible microstates of the molecules of the macrostate of the system, and is written as W and is referred to as the thermodynamic probability of the system. At zero temperature, the thermodynamic state corresponds to a single microstate.

            Higher entropy entails higher numbers of microstates characterizing a system, and hence higher configurational variety of the system, which entails as well greater freedom of movement of elements, and greater freedom of choice, greater probability of error in the prediction of outcomes of random sampling procedures. On the other hand, greater constraint in a system results in greater reliability or fidelity and hence concomittant reduced sampling error.

            Entropy is used as a measure of information by its probability characteristics. Absence of information about a given instantaneous situation corresponds to an uncertainty (H) associated with the nature of the situation. This uncertainty is the entropy of the information about a particular state or situation of a system, such that:

 

H (p1,p2,….pn) = -εn/k=1 pk log pk

 

Where p1,p2,….pn  are the probabilities of mutually exclusive events, the logarithms are taken to an arbitrary but fixed base, and pk log pk always equals zero if pk = 0

 

            In this formula, if p1 = 1 and all other possibilities (p2,….pn) are zero, the situation is completely predictable and the entropy of the system is zero because there is no uncertainty of the state. In any other case, entropy with be a positive value and the system of a partially uncertain state. In terms of an information space, a source of information is described by its entropy H in bits per symbol. The systems relative entropy (Hr) is the ratio of the entropy of the source to the maximum rate of signaling that it can achieve with the same signals. 1 - Hr is the redundancy of the source.

            Shannon's entropy function in information theory is referred to as redundancy and is composed of two parts, D1 and D2. Any sequence of symbols has a redundancy that must be characterized by two independent numbers, one defining the amount and the other the kind of redundancy of the sequence. For any problem, the amount of increase or decrease in entropy must be determined, and the kind of entropy must also be determined.

            Information represents potential knowledge about the order or organization of a system. Information can be defined operationally like energy as the capacity to do work, as the capacity to store and transmit meaning or knowledge, not the meaning or knowledge itself. In defining information operationally, we always calculate the numerical value of its capacity, and not the qualitative value of its content. Shannon's entropy function is the measure of this capacity in information systems.

            In short, entropy measures the randomness of a system, which is determined probabilistically. The individual outcome of a random event cannot be predicted or predetermined in any other way except chance, but as a related member of a group of events that are not always identical, a random phenomenon leads to a group of outcomes that fit a natural or Gaussian curve of probabilities, allowing statistical prediction based upon likelihood. A random event is a single, particular outcome of a random phenomenon that is amenable to statistical description and prediction because its relative frequency of outcome approaches in the structure of the large and the long run a stable limiting set of values that define the probability of the random event. The random phenomena is potentially an infinite, or open-ended, series of events, and its limiting values. The limiting value of a coin toss is 50% heads or tails, but it is only by a very large number of coin tosses that the sample space of all possible tosses begins approaching this limiting value.

            A set, denoted by brackets, is a collection of things of interest, in which the identity of the individual elements or members contained in the set defines the value of the set, whatever the relative frequencies of members or their order of occurrence. A space is a set that is in principle complete, including only and every member that belongs to the set. A sample description space is the set of all possible outcomes of random phenomena, with each element being the elementary random event. Every element is assigned a number between 1 and 0 representing the probability of its event, and this is referred to as a finite probability space if the number of elements is finite.

            Independent random events are those in which the probability of the occurrence of one event does not affect the probability of the occurrence of the other event. Two random events are independent if the probability of their joint occurrence is the product of the probability of their separate occurrences, or:

 

p(ab) = p(a)p(b)

 

If a is event 1 and b is event 2.

 

            Two random events are dependent if the previous occurrence of one alters the probability of the occurrence of the consecutive event. The subsequent event is a conditional probability of the first event. For two dependent random events, the probability of their joint occurrence is the probability of the first even multipled by the conditional probability of the second event given the previous occurrence of the first. Hence:

 

p(ab) = p(a)p(b|a)

 

            Any system has a state of relative entropy, and can vary between relatively high entropy to lower entropy. A system of high entropy can be characterized qualitatively as random, disorganized, disordered, homogeneous, mixed, characterized under random sampling procedures by a high frequency of equiprobable and independent events, high configurational variety, high uncertainty of state-change or outcome, high error probability, high potential information, and a high degree of freedom of choice, if choice can be said to be involved. A system that is low entropy is said to be highly determined, structured, organized, ordered, separated, heterogenous, diverging from equiprobability (D1) and diverging from independence (D2), with restricted arrangement or configuration, high constraint, reliability of pattern, high fidelity, and much stored information.

            A system with high entropy contains high uncertainty and high probability of error in guessing the outcome any particular elementary event. Constraining and ordering a system somehow, reduces the entropy and increases the reliability of the system and reduces the probability of error.

Systems maintain equilibrium about some asymptoptically stable point within limiting constraints that keep the ratio of entropy to order in that system in a relative balance, within acceptable boundary conditions. If a system has an ordering force that arranges its elements into relationships of interdependency, then that system has lowered entropy and high determinancy.

 

A state of maximum entropy is characterized by equiprobable, independent elementary events.

A state of minimum entropy (maximum determinancy) is characterized by maximum divergence from equiprobable (D1) and independent (D2) elementary random events.

For any given macrostate, we may write:

 

S = KW

 

Where S is the entropy of the macrostate system

W is the thermodynamic probability of the system

K is an arbitrary constant

 

Entropy of one system may be additively combined with the entropy of another system, such that:

 

Sx + Sy = Sxy

 

Because the combined number of microstates of two conjoined systems is a multiplicative and not an additive function, the properties of the previous two equations are joined according to Boltzmann's definition  as:

 

S = K log W

 

The entropies for both systems may be written as:

 

Sx = K Log Wx

Sy = K Log Wy

 

These entropies, if additively recombined, become:

 

Sxy = K Log Wx + K Log Wy

 

Simplifying:

 

Sxy  = K Log Wx Wy = K Log Wxy

            In statistical thermodynamics, all microstates are equiprobable. The probability of each individual microstate of such a system becomes:

 

pi = 1/W or W = 1/pi.

 

If we substitute for W in the previous expression S = Klog W, then we have:

 

S = K log 1/ pI

 

And if the log of 1 = 0, then

 

S = - K log pi

 

This expression permits the expression of entroy in terms of probability rather than in terms of a large number such as W which is often impossible to determine. Entropy can be expressed also as a statistical average of a system, or its expectation value, which would be the sum over all possible outcomes of the probabilyt of each individual outcome multiplied by the numerical value of the individual outcome, for any numerically valued random phenomenon. This may be expressed as Shannon's formula:

 

H = -K εpI Log pi

 

In which for every arrangement of the system there is an associated a numeric value, -K log pi, which is the Boltzmann variable, and the probability of each arrangment is pi

 

Ex = εi pini

 

This formula may even be used when all microstates are not equiprobable, and serves to render the concept of entropy a part of general probability theory rather than just a function of restricted thermodynamic settings. Its value is its generality referring to the probab ilities of any elmentary events defined by any sample description space.  When K is equal to 1 and base 2 logarithms are used, the unit of entropy is the bit, the most generally used unit.

We may describe systems by two qualities--divergence from equiprobability (D1) and divergence from independence (D2).

In the first case, (D1) is the maximum value H can have in a system minus its actual value:

 

D1 = H1max - H1 = log a - H1

 

In the second case, (D2) divergence from independence, is the difference between the entropy state of the dependent event (H2D) and the entropy state of the event if it were independent (H2Ind), or:

D2 = H2Ind - H2D

 

            The sum of D1 and D2 is called the total divergence from the maximum entropy state.

 

 

 

Measurement Theory

 

A measure is a standard arbitrary unit or system of units used to determine by numerical count the dimensions or size or quantity of a system in reality. While there are many derivative measures based upon the concatenation of basic measures, like acceleration, velocity, density, volume, or gas pressure, and while there are many alternative systems of measurement, there are in fact only a few irreducible basic measures: length or distance, time or duration, mass or weight, count, temperature, direction.

Sciences depend upon measurement for establishing quantifiable and hence comparable results that can be duplicated and hence are considered objective, and the sciences have instituted standard systems of measurement to reduce the problem of conversion between competing standards.

In all scientific research methodology, there is a premium placed upon both precision and accuracy of measurement, the two values being interrelated but not the same. A large amount of research budgets is spent on acquiring instrumentation that allows for the most exact or precise measurement possible, for there is an inherent dilemma in all measurement. Even though it is critical to be as precise and exact as possible in our measurement, all measurement has a degree of residual error that creates uncertainty of measure, which is based upon the smallest unit of measure available. Any instrument of measurement is only as good as the smallest unit of measure it allows for, and any measurement that is smaller than this smallest unit creates imprecision and inaccuracy of measurement, leading to uncertainty of final values. There can be no perfect, or exactly certain measurement.

Science deals with uncertainty inherent in measurement by stating and establishing confidence limits and by statements of error assigned to any given measure. In other words, science splits the difference, and in scientific notation dealing with very large or very small numbers, it applies procedures or rules for rounding.

Measurement theory deals principally with two sets of problems and a third kind of problem interrelated the first two. First, how do we accurately measure process and distribution in reality, and, for any given kind of pattern that we might encounter, what are the best instruments of measurement that we may use. Of course, selection of the best instruments invariably hinges upon the question of what purposes we wish to put the data we collect. Generally, research resources are limited, and this imposes constraints on the kind and amount of data we can collect, so we must be selective and set priorities for research that tend to leave out many possible avenues of information for the few we prefer. Of course, we may be mistaken in this regard, and find that serendipidty and intuition in information gathering often carries the day.

The second set of problems is related to the first, and concerns the methods of analysis that we put the data through that we do manage to collect. Analysis by statistical techniques has become a standard norm in most scientific endeavors, par for the course, and it represents a second level of measurement that is derivative of and based upon the first level of actual data collection. We end up with a wide possible variety of secondary data sets (averages, Z scores, correlation coefficients, regression equations) that cannot be found anywhere in the data itself, but is implicit to the data as it was collected.

We all acknowledge that there are no 3.4 person families in America, but the may well be the average, and this average is no less real or valid (nor any more real or valid) than the raw counts upon which it was based. The second problem of measurement theory is like the first therefore in that analytical research budgets are also circumscribed by limited resources, and we must pick and choose what kinds of tests that we wish to subject our data two.

Between the first level of actual measurement and the second of data analysis, there is a third kind of problem characteristic of measurement, and that has to do largely with the results of the dichotomization of the two sets of methods such that, by the time we analyze our data, we cannot go back to the conditions of the original experiment to retrieve or reevaluate any of the information we first collected. We can conduct a repeat experiment, but the informational value of the original experiment will be mostly lost. It goes without saying that the quality of our analytical results will be directly dependent upon the quality of the data we collected, but it is probably less obvious to assert that the quality and kind of data we collect may indirectly depend upon the kind of analytical models or constructs we have created for ourselves or that lie dormant somewhere in the back of our small heads.

The third kind of problem is important to consider as well as the first two. The dichotomization of data between collection and analysis is important and most often necessary. It is in effect, like any cause effect relationship, unavoidable. Some would argue, rightfully so, that one should not mix methodological metaphors in field situations. At most it is valuable to conduct preliminary analysis of results, but full analysis must await complete samples and finished data bases.

A great deal of scientific progress has hinged critically upon the invention and development of new methods of observation and measurement. Almost any field of scientific inquiry has been made possible only by the refinement of such instrumentation that permits independent replication and non-arbitrary observation. Alvogorado's number in Chemistry has been vital to the unification of the field. The development of the microscope and optical density devices have been critical to an understanding of microbial life and its patterning. Carbon 14 dating techniques have resulted in a revolution in the paleontological and archaeological sciences, before which such fields were dominated by relativistic frameworks of chronological interpretation. Undoubtedly, a great deal of what remains unknown to us about reality is so because it remains essentially unavailable to us observationally or analytically because we have not yet devised adequate techniques or technology.

Measurement parallax begins with inherent inaccuracy of our measuring instruments, and the inherent variability of standards and inconsistency between procedures. Furthermore, there are both quantitative and qualitative degrees of freedom and innate complexity of pattern that is being thus measured. Measuring in a discrete manner 6 atoms is not equivalent to measuring discretely six ripe oranges or six successive days at the same location. Measuring complex event structures or composite phenomena is not as straightforward a proposition as weighing a gram of calcium carbonate or marking out the length of a pencil line to the nearest quarter of an inch.

Measurement parallax addresses what can be called the fallacy of measurement, which can be stated as a habitual or intentional predisposition to record and report measurements, and to think subsequently about such measurements, as if they were in fact real or reified units in and of themselves, and not just derivative and reified artifacts of our own conceptual devices. Measurement fallacy leads to the denial or ignoring of inherent variability of patterning in all natural or real phenomena and inherent error of all measurement used in analyzing and describing that phenomena.

Measurement theory becomes interesting, I believe, when it reaches a problem of having to measure in some realistic or representative way a complex set as I have defined this above, in which the total number or even types of variables may be unknown. Such a set is by definition open and incomplete in terms of the known determinants that define the set, as we cannot specify a finite limit to its size or composition without greater information about the set. In this case, the best we can do it seems is to "sample" the set as much as possible within our limited research resources. When we sample the set, we usually use some hypothesis or theory to define our sampling error or selection priorities. We look for certain kinds of patterns, probably ignoring others, without being certain in any absolute way that the patterns we choose or observe are the optimum or best possible.

Such sampling may be analytically driven by our statistical models that we will employ in their selection, as in highly developed medical research designs that target select types of population, or it may be more encounter and directively oriented, such as when an archaeologist purposefully conducts a preliminary surface survey to determine the viability of digging in a certain area.

Either way, we are never 100% clear as to the total size, limits and structure of our sample, and even presuppositions of randomness are only loosely approximated by any randomization procedures we may superimpose upon our sampling. Such problem sets tend to be context-based systems, and they are structured by the unknown variables more than by known factors.

Possibilistic statistics is rooted to advanced measurement theory in the operational problem of defining and determining what can be called complex sample sets as predeterminants of the unknown complex sets that they represent.

Possibilistic statistics is proposed as an intrinsic part of measurement theory as a means of providing a way of systematically dealing with complex sets of data that are partially factorial.

In this case, the object of possibilistic statistics is to try to determine:

 

1. The range of variation of alternative possible sets that may be represented by any given complex set, this being given as an a priori unknown. In other words, to attempt to define the possible limits of the search-solution space that is would be theoretically required to solve the problem.

2. The hypothetical "normal" distribution of the alternative possible patterning within a paradigm of a complex set in order to establish criteria of significance and for a null-hypothesis. Witin this framework, anomalies can be determined that can be rejected as non-representative of a complex set, though if such anomalies are discovered to occur they have to be given special consideration.

3. How to break down a complex problem set into a number of different subsets that may be more completely factorially determined that the whole set.

4. How to factorially determine each subset in as complete a manner as possible, part of which factorization depends upon the relational similiarity with other coterminous subsets.

5. How then to define the means by which these subsets interrelated and may be put back together to further determine the entire set.

 

In other words, possibilistic statistics is a prescribed technique of sampling large, unknown sets of undetermined complexity and size, borrowing a basic heuristic advice from computer sciences. Take big and complicated problems, and break them down analytically into small problem sets, solving each as one goes, and then interrelating the solved subsets back to the total problem. In other words, break a large problem into the smallest manageable units possible.

We must recognize that complex variability of patterning of complex sets implies an order of integration that is unknown, and complexity that defies simplistic description. In this case, presuppositions of randomness or of descriptive accuracy are possibly not even relevant to our understanding of the problem. We proceed on the assumption that all problem sets are minimally integrated and maximally variegated, and hence we seek to find both the fullest range of pattern variation and the fundamental substrate of relationship within this range.

The problem is that we have no presumed "base-line" from which to start in our differentiation of samples. The point of possibilistic statistics is sort of the continuous reiteration of clustering distances to determine best fit between multiple possible data sets. The aim and purpose of such a procedure is to define a probabilistic "base-line" from a derivative model of the problem set, from which we can then operate using more conventional probabilistic statistics. We would actually generate multiple alternative models from complex sets, each of which would then be subsequently tested for likelihood of best fit. We end up with not a single whole set, but with a fractionated ratio, of a partial set among a range of alternative sets.

In the following, what do we choose as a base-line by which to define a "normal distribution" and the limits of our distribution for any given complex sample?

 

2x/15, 10y/25, 1xy/5, 3y/10, …..

 

How can we know where to draw the line in our sampling, such that our number represents a significant proportion of the unknown whole?

First, though a complex set is open and undetermined as an unknown problem set, we may say that all complex sets are analytically finite and hence minimally determined sets. If an analytically solvable set is finite, then we can predict that in general, though we cannot know where to determine the final limits:

 

1. A larger sample set is better than a smaller one, as long as the larger one is unbiased and within scope of our sampling procedures.

2. A maximally variable sample set is better than a less variable one for a smaller than a larger system.

3. The range of countable variability within a sample set of unknown size may be partially determinable by the ratios of repeatability of different variable sets or sequences.

 

In regard to the third statement above, this has to do with the definition of noise and information in systems. We want some noise, but not too much, and some non-random order, but not too much. Any natural system is expected to have both noise and order. A noisy system will have less non-random variation of pattern, however complex, but may exibit greater simple chance non-random variations that are the result of simple stochastic probabilities. Say we flip a penny on successive trials, 0 for heads and 1 for tails, and we do this ten times, coming out with the following order:

 

0, 0, 0, 1, 0, 0, 0, 0, 1, 1

 

Without being able to repeat our experiment again, and without any other knowledge of a system of flipping pennies, we would have to assess the unknown probabilities of turning a head or a tail on each turn. Knowing nothing more about a penny, we might assume that the odds of turning a head versus turning a penny is 7/3. If we ran our experiment again, with ten flips, we might come out with a completely different ratio, of perhaps 6/4 or 2/8. Our baseline that we are searching for would of course be 5/5, but this might only be discovered after a very large number of 10 series flips. Nothing would prevent us in the long run, after 10 such series, from coming out with an average that reflected 4/6 or 6/4 rather than 5/5. Instead, if we ran one long 100 sequences series, we might find our overall average to more closely approximate 5/5, though it may still only approximate 3/7 or even 2/8. Knowning the real probabilities involved, we would know that after 100 times, the probability of turning a ratio of 5/5 is much greater than the probability of turning a ratio of only 1/9, and if we did it a thousand times, our inferable probability would be much much closer to 5/5 than any other ratio value.

This simple situation exemplifies well the requirements and types of procedures necessary for possibilistic statistics to be used. The quest of possibilistic statistics is the derivation of an estimated probability that can serve as a baseline for subsequent analysis and measurement. If we go into the entire coin-flipping affair knowing that on any independent flip our odds are always 50/50, which are good odds in the betting world, then we are likely to risk the bet that the next flip will be in our favor. Of course, most problems from a possibilistic standpoint are not so simple as this. A two by two matrix or a 3 by 3 decision tree would yield exponentially complex odds. It can be said that possibilistic statistics is a kind of decision theory, and a kind of game theory that is applied systematically to complex sets of possible outcomes.

Another problem in possibilistic statistics is defining the range and probable limits of variation in a system. For instance, if we were using a six-sided dice, not knowing how many face-numbers a dice had, how many tosses would we have to cast the dice before we could reasonable decide that the dice had a range of six possible numbers, equally distributed. Suppose for instance, that we generated dice tosses after ten trials with the following values:

 

1/x, 5/x, 1/x, 3/x, 5/x, 1/x, 1/x, 2/x, 2/x, 1/x

 

How would we analyze the results. We might conclude, even though we didn't pull up a four, that the dice had five sides. Alternatively, because we pulled up five ones, we might conclude that the dice was in fact 8 sided, with four of the non-adjacent sides having one.

Repeating our experiment over 100 tosses, we may be able to conclude, for instance, that there are indeed six sides, even if we pulled up only a handful of sixs out of a hundred. We may not know the exact relative distribution of numbers, and would increase the number of trials to 1000 before we could generate a reasonable probability of 1-6/6 odds any number 1-6 on any given toss.

In this kind of exercise, the only non-relative kind of information we may have are the partially defined real factors that are known or discovered to exist within any given sample or samples, and the inferable relationships we may derive on the basis of their cooccurrence, consequence and distribution between different samples sets.

The interest and deliberate intention of possibilistic statistics is:

 

1. Systematic non-random pattern recognition against a noisy background, presuming that:

a. non-random pattern will have inherent noise that may appear random

b. background noise will have inherent pattern that may appear non-random

c. random and non-random pattern may interact

2. from this kind of analysis, we would infer a probable likelihood of order in the patterning over multiple samples or event structures, presuming that:

a. non-random pattern will be recurrent between successive or over multiple event structures.

b. random pattern will tend to cancel itself out over the metastructure of the long run and the large.

 

It follows that possibilistic statistics is concerned centrally with the problem of stasis and change in complex problem sets, the range of variation of such sets being definable more as a function of time than of spatial distribution. We can infer that stable structures will recur over time with given rates of expectation, while non-stable structures will shift over time with given rates of expectation. We are not attempting to make predictions with possibilistic statistics, but only to state accurate expectations from our knowledge of systems from which we can then derive stateable and testable expectations within known parameters. We are attempting to narrow by focus the range of possible variation in pattern in order to more selectively make decisions regarding the "unfactored" remainder of our systems. The possibilistic baseline is the starting point for secondary probabilistic analysis utilizing more conventional statistical procedures, and not the end point.

Of course, the examples used were very simple and straight-forward to conceptualize. We quickly approach exponential complexity in the conceptualization of even slightly more variegated types of patterns. We say that in general, complex sets tend to be multiply determined, and this multiple determination of such sets is the cause of the inherent variability of pattern. Such sets are also by definition open sets, and their openness is the cause of increased random variability of the background pattern. The aim of possibilistic statistics then is to partially determine such complex sets by sample factoring of the possible determinants that may define such a set.

It is apparent that with possibilistic statistics applied to very large and complex systems, our profile of possibilities will tend to be continuously shifting with the addition of new information. As in the case of the hominid fossil record, for instance, where the evidence is few, fragmentary and far between, and the gaps of the unkown loom large on any index-horizon, each new discover tends to have significantly great impact on the understanding of the whole. This is indicative of the relative lack of knowledge relating to this fossil record, a function of its potential size and complexity.

In other words, if we have very small sample sets to infer about very large and complex real sets, then each new bit or variable of information added to our knowledge is likely to have a proportionately greater effect in restructuring our estimates of variability about the system as a whole. The next "nth" thing found in a complex system is more likely to be unlike any previous thing found than like. If this is not found to be the case, then it can be presumed that the larger system is in fact a simple and relatively stable one.

It appears that, inspite of much synchronic variation, the hominid pattern through time was quite stable and its rate of change rather slow. This lends greater credibility to the tendency for lumping versus splitting of the hominid fossil record. If new fossils are found that show significant differences from previous sample sets, it is likely that the fossil record will prove to be much more transitory and variable over time and place than is currently inferrable from the record. It does appear that there were episodic periods of sympatric speciation during certain periods of this record, with side-branches, presumably more niche-specialized, eventually coming to an end. The main line, otherwise, or trunk of the hominid family tree, appears rather stable and steady in its transition characters.

If a large sample is accumulated, with an emergent degree of order in the pattern that is recognizable, and then a completely anomalous specimen or data-point is discovered that does not fit the pattern, then the stability conferred on the entire system is not thereby jeopardized or compromised. If such anomalies are entirely unique, the possibility of a random fluke exists--if such anomalies are rare but recurrent enough, then it suggest that these occupy a special subset in an important relation to the larger set we have already accumulated, and that together these are subsets of a larger and even more complex "metaset" the nature of which has not been fully described or measured in a possibilistic manner.

Measurement of complex sets depends upon our ability to partially factor such sets into relative subsets. This type of partial measurement is relative measurement and is context dependeent. We are essentially, systematically deriving and segregating the knowns from the unknowns in any given set, while preserving the information about their relationships.

From this standpoint, the following complex set:

 

5x, 2y, 10z, 20w

 

can be said to be partially factored when we convert the known factors to fractions with a common denominator, and then apply the principle of algebraic distribution to the set as a whole. Thus, for the previous set, the following can be said to be the partially factored set:

 

1/20….(5x, 2y, 10z, 20w….)

(5/20x, 2/20y, 10/20z, 20/20w….)

(5/20, 2/20,10/20, 20/20….) + (x, y, z, w…..)

 

We can say that there would be more relative information in the first subset than in the second, and more potential variability in the second subset than in the first. This kind of set can be factored out or partially determined in more than one way, giving, for instance, the following:

 

(1/4, 1/10, 1/2, 1

/1…) + (x, y, z, w….)

(.25, .1., .5, 1.0…) + (x, y, z, w….)

 

In this case, we have expressed 1/20 as a relative cardinality factor of the entire known set, and we may predict that the value of 1/20 is important to the system as a whole, but if we discover on the very next event that the value is not within the range of 1/20 but 1/27, then we will have to redistribute the values of the entire set and we would have to readjust its relative cardinality to reflect this redistribution.

Each additional value affects the relational values of all the variables together, not necessarily because each successive event is directly dependent upon the values previous events, but because all of the events together can be said to be underdetermined by the same shared structural variables, which remain unknown and complex. In the examples above, the structural variables of our penny-flip experiment was the 50/50 odds of landing either a heads or a tails, and this could be distributed independently throughout every successive flip-event. Likewise, in our dice-toss experiment, the 1/6 odds of landing any whole number between 1 and 6, inclusively, is the shared distributional structure underlying all possible toss-events and therefore determining the structure of each event.

We can say that in a complex set, the derivative cardinality value is relative to the instantaneous event structure of the system as a whole series or distribution of sets, and this variability or stability is a relative measure of the overall variability or stability of the system as a whole.

Finally, in conclusion, we may say that possibilistic statistics has the aim of determining from a plurality of complex sets the instantaneous cardinality values relative to all the sets, and therefore the hypothetical system that these relative values define for each and every similar or related set.

The distinction between similar and related sets is an important one to make. Related sets may not appear similar, and similar sets may not in fact be related. Similar sets on the other hand may be interrelated, or indirectly related, and related sets may be similar. Possibilistic statististic can be said to have the aim of determining the relative similarity between different sets in a relatively precise manner, in the hope of stating an expectation of some direct or indirect relationship between alternative systems that defines a larger paradigmatic structure, or hypothetical model, defining such systems. In general, it can be stated that related sets will share basic underlying cardinal structures, while similar sets will only share surface patterning that are possibly shaped by external factors. In the latter case, similarity can be said to be the result of non-random patterning that is relationally spurious between the sets being related. The example of convergent evolution in natural history is sufficient as an example of similarity that is non-indicative of genetic relationship, as for instance parallel wing structures of bats, birds and pterosaurs, and genetic divergence in the genetically related structures of sea mammal flippers and mammal feet and hands. The periodicity of the elements in their specific groups with shared chemical properties is another example, I believe, of a form of similarity relationship that is a function of the same number of electrons in the outer orbitals.

In general, it should not matter whether we are dealing with genetic relationship or similiarity of sets, except that the underlying structures governing these different kinds of patterns may be fundamentally different. With genetic relationship, we expect systematic and continuous variation, or divergence of common structures. With similiarly relationship, we expect convergence of different structures due to similar underlying cardinal properties.

One of the key techniques in advanced measurement theory and possibilistic statistics is in the definition and application of an arbitrary analytical frame of reference to create comparable or differentiatable units of analysis with complex sets. As was mentioned at the start of this section, standards and instruments of measurement have greatly facilitated and made possible the advancement of science. As was mentioned previously as well, it is not always possible to determine what the appropriate frame of reference might be, given a variety of alternative possible frames to deal with. In a sense, the determination of the baseline by means of deriving the instantaneous cardinality of a factored sample representative of a more complex set is the manner proposed for developing an appropriate frame of reference for applying units of analysis in differentiation of subsequent samples. As was implied, Chemistry didn't advance very far in a numerical sense until Alvogorado count estimate reliably the number of atoms per mole of any given substance. This may be accomplished qualitatively rather than numerically, and abstractly rather than concretely, if for instance roundness were the cardinal of our complex set of all round things in Ireland. As stated below, we risk oversimplification. There may be only 10 round bowls of a certain kind in all of Ireland, but millions of round common bowls, wheels, windows, and cups. We might also discount all round balls or spheroids, which may also number several millions, if we distinguish stricktly between what is round and what is spherical.

 

Set Theory

 

Measurement theory involves the accumumlation and definition of sample sets derived from systematic observations made of patterns in reality, with the aim of deriving what can be induced as significant pattern structure from a theoretically noisy background. The result is a paradigm of limited possibilities, inclusive of possible exceptions or anomalies, by which we can conduct further experiments, and devise new means of analysis.

At some stage in this process, if it is successful, a point should be reached where there will exist multiple sample sets that require arrangement in some kind of order or frame of reference, and which may need to be partially integrated or related to one another in the definition of the kind of metasystem that the structure of the sets exemplifies.

At what point does a set of sets, or a series of sets, or some kind of set distribution, become a system, and, inferentially, a kind of "metasystem" from which we can determine the underlying predictive structures that theoretically account for our observations. We are moving by a series of steps from description and measurement analysis to metaset construction and hypohetical system development.

Set theory concerns the abstract definition of sets, the logical interrelation of sets, and the formation of metasets.

In general, we can say that a simple set is defined by its cardinality, or by the shared determinant representing each member of the entire class of members of the set. A size of a set is determined by the population of its members. In simple sets, we select some key attribute or set of attributes by which to characterize the set as a whole. In this sense, simple set theory can be found to be implicit in most typologies and taxonomies, when, for instance we can say that a Beagle is a kind of dog, or a representative of a set of dogs.

We can pick key determining traits from what can be called polythetic sets, which are sets whose membership is defined by representation of more than one kind of trait, but rather by a number of interrelated traits that may be more or less apparent to any one member of a class. A member of such a polythetic class that is defined by five key traits may in fact only possess two or three of the defining traits, but nevertheless be represented in that class. We may thus interlink different polythetic sets together, for instance, if they share members between different sets, and in complex kinds of set patterns, it is possible that such interlinkages between sets extend indefinitely or across a very wide field of systematic variation.

Set theory has important implications for both semantics, or the structure of meaning, that can be said to be symbolic, and hence culturally determined, as well as for abstract systems of mathematical quanitatization and logical relation that is rationally systematic. The definition, organization and interrelation of sets permits us to perform fundamental logical functions, and permit on a basic level a systematic unification between meaning and its correspondence within one-to-one type quantized systems. Advanced set theory therefore involves this kind of relationship with especially metasets that are derived through samples of complex sets.

 

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.

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.

 

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.

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.

 

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 other hand, 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.

 

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.

 

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.

 

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

 

Taxonomic Systems

 

Taxonomy is not context bound, it is model driven. Scientific taxonomies depend upon successful theoretical models for their construction. They do not depend upon the typological constructs upon which they may have been originally defined. Typologically defined taxonomies that lack theoretical unification, in an empirically verifiable manner, are simply ideological structures that have no scientific validity or efficacy.

Taxonomic systems are the result of successful construction and testing of hypothetical models, as complex constructs of reality. Taxonomic systems come to incorporate, and represent in basic terms, a sense of worldview to the extent that such systems can claim to be universal or at least general in application. They therefore are a statement about the conceptual organization of reality that reflects as much as possible the non-arbitrary divisions that the natural patterning of this reality takes.

Taxonomies are basically defined by implicit rules governing the order of the relations between the components of such systems. The theoretical models upon which scientific taxonomies are built define in an explicit manner the rules by which a taxonomy should be constructed. There is feedback to typological description and even observational technique and selection that creates increase in scientific knowledge upon a basic level.

 

 

Intensive systems

Extensive Systems

Hybrid Metasystems

Initial States

 

 

 

Fundamental States

 

 

 

Atomic States

 

 

 

Molecular States

 

 

 

Intermediate StateI

 

 

 

Microbiological

 

 

 

Mesobiological

 

 

 

Macrobiological

 

 

 

Intermediate StateII

 

 

 

Individual

 

 

 

Cultural

 

 

 

Social

 

 

 

IntermediateStateIII

 

 

 

Alternative Systems

 

 

 

Abstract Systems

 

 

 

Automated Systems

 

 

 

Final State Systems

 

 

 

 

Numbers and Symbols

Mathematical Mechanics & Symbolic Calculus

Terminological Systems of Functionally Complex Polynomial States

 

It has been demonstrated that a pure mathematical system describing a metasystemic model of reality is both trival and unrealistic without its hypothetical transformational applicability to any and every real system. We cannot ever prove this to be so in a nonscientific way, and any scientific proof can only be at best inductively inferred.

Nevertheless, mathematical modeling plays an important role in numerous applications in the language and operationalization of science and in our general understanding of reality. This is primarily because mathematical modeling approximates a mechanistic view of real systems and this can be deductively derived in abstract terms. Such systems are known for the internal coherence of their deductive inference structure, and this is derivative of their stable and deterministic relational patterns. If these are referentially attached to real or natural systems in a consistent manner, then they constitute the most powerful models that science has yet produced. It is critically important therefore to realistically consider and define the limited role of mathematics in its application to our understanding and elaboration of advanced systems of conceptualization if these are to have any hope of constructing and construing an alternative scientific worldview and praxis.

I offer herein only alternative deductive systems based mostly on my own limited experiences in anthropological research. They are only a point of entrance into and an alternative basis for development of analytical and synthetic operational procedures for advanced systems sciences, but they are neither the only nor the best alternative sytems that may be developed. In their construction and application, I have attempted to render them as consonant as I am capable with the theoretical primes I am most interested in understanding. It is hoped that their application to real problem sets will be as interesting as they are non-trivial.

Mathematics, as a language of scientific communication, is a limited system of signification. It achieves its power through its sense of deductive exclusion and tight terminological definition. I seek to elaborate a model of mathematics that is inherently more open and flexible as a sytem of communication, hopefully without a substantial loss in its inferential capabilities for our sciences.

At the same time, I seek to elaborate a more rigid and mathematically restrictive model of symbolic language derived from natural language models that may serve us better in the theoretical formulation and formalization of our sciences. In general, this can be achieved through precise and concise denotative definition of our symbolic primes.

Mathematical or symbolic logic is a point of departure for this alternative system, but again I see symbolic logic as being fundamentally "hung up" upon its own dilemma of identity as a dichotomized truth-value system. Symbolic language structures and mathematical signification systems are both necessary and complementary in the processes of scientific generalization, yet alone, both systems have, I believe, shortcomings that are not intrinsic to their strengths but due to their own unnecessary restriction or lack of restriction in certain basic ways.

In the applicability of natural language to the problem of truth-value, we can consider the following philosophical problem. At what point can we say, in our statements about the truth-value of a rose, that our answers go from being confirmable by some means of non-arbitrary descriptive validation, to being one of primarily prescriptive affirmation:

 

This is a rose.

This rose is red.

This rose is a flower.

This rose smells sweet.

This rose is beautiful.

This rose represents love.

 

These types of problems have mainly to do with the identification and denotation of primes, as variables or values, and their operational relations. It has as well to do with the natural flaccidity of symbolic constructs and the smuggling of tacit values into our terminological definitions and understandings of the world. The association of "truth value" to our meanings, symbols and their implicatures entails that we must understand what "truth" is in the first place and how it is attached and manipulated in our meaning systems. In otherwords, we are dealing with the problem of the language of science and general understanding, and how this constrains and enables our inquiry into nature and reality.

In terms of our natural symbolic language, we attempt to achieve a form of descriptive explanation of the underlying structures of complex phenomena, in the form of strong generalizations that have a marked degree of formalism. We approach systematically such a general theoretical model by refinement and correction of our terms and their stated and implicit relations. Such refinement occurs often by default and by lack of critical self-awareness. I believe it marks out the principle of "perfectness" in a metaphysical conception of reality that is complementary to the notion of "correctness" in our puzzle-solving efforts in science. Our theoretic generalizations, over time, become magically like Mary Poppins, "practically perfect in every way" in spite of their fundamental relativity and ultimate groundlessness of truth-value.

What is achieved by this means, I believe, is a relative degree of fit or coordination of internal frames of inference and external frames of reference about some central problematic. There appears to be little or no noise arising from the lack of coordination of these two frameworks, one abstract and ideal, the other real and natural. Perhaps Charles Darwin was the master of such argumentation when he framed his theory of evolution, but even is basic terms, like natural selection, smuggled in some undesirable if hidden connotations of value.

We cannot render a completely air-tight and unleakable generalization of the natural order based upon natural symbolic language alone, but we can get a pretty close fit that holds for most purposes.

In a natural language system, the anchor points of our truth-value are both cultural and natural experiences as these are symbolically articulated. In a sense, there can be no non-relative truth in such systems. Hence, our definitions themselves cannot obtain that molecular level of descriptive explanation that can be set without equivocation. This appears to be achievable only in the physical sciences where definitions take on precise numeric and mechanical descriptions. It appears to be partially true in the biological fields, especially as this is reducible to biochemical explanations, but it introduces greater symbolic ambiguity and parallax of meaning when it deals with naturalistic description of behavioral phenomena and events. It is especially true in the human sciences that deal with anything other than human biology.

It is partly true that in our biological and human sciences especially, we have not arrived at the degree of theoretical closure and exactitude of definition that is probably desirable. This is directly proportionate to the difficulty and degree of complexity of the phenomena being descriptively explained.

To enforce a restrictive model upon descriptive explanation, especially upon the natural sciences, is perhaps to risk loosing the artistry and power of words to animate discontinous worlds. But in itself, if it can be well done, can also be a source of artistry of our generalizations--what I will call the consistent matching of words to the ideas they represent. We cannot eliminate ambiguity completely, but we can systematically reduce it down to minimal proportions by minding our p's and q's.

In regard to metasystems, I have adopted what I construe as a mechanical model of mathematics as this is applied to the conceptual validation and demonstration of metasystems and in their inductive instantiation in terms of real systems, especially those that occur in nature. Mechanical mathematics can be thought of as an applied mathematics of systems emphasizing structural integration and functional operation. But the mechanical model of systems that I seek to employ is itself derivative from a classical and conventional conceptioning of mechanics, in a form of modeling that I call non-linear mechanics. It is therefore unconventional and leads to remarkable consequences in our understanding of systems.

The heart of a mechanical model of metasystems is the conceptioning of a machine as a relatively determined system of parts that cooperate to produce some kind of joint or coordinated effect, usually in nature an effect involving energy and motion and leading to some kind of meaningful pattern. Mechanics, I believe, provides the appropriate framework for construing metasystems as something that is scientifically interesting. One aspect of any machine is the sense of integration of its components that leads to a causal patterning of action or reaction between them. I believe that a systems theoretic approach is fit to a mechanical and mechanistic description of phenomena in a naturally mathematical way.

We can call a non-linear machine one the holistic patterning of which is not fully describable or predictable in terms of the reductionistic analysis of the cooperation of its parts. In other words, the interactions between the components of such a system are not fully determined or determinable, but only partially so, thus begetting epiphenomenal outcomes that may be variants within a range or continuum of alternative possibilities. These may in turn lead back to state changes and structural alterations within the system itself.

To the extent that the parts of a system are definable in terms of their relational identities and properties within that system, we can say that for any nonlinear system, identity of any part or element is essentially relative and also by definition "partial," within the framework the system provides itself. A theory of partial identity, or partiality, is therefore in order, which goes something like this:

 

1. Any thing is never whole to itself, but always a part-whole of something else. Thus, we have a part-whole relationship within a larger framework of possible relationships.

 

Mathematically, we may express this partial identity as:

 

A = a {ƒ (X)} + a' {ƒ (X')}

 

 where a is some presumable and significant subset of A

X is something else functionally related to subset a

And a' is the complement of the subset a, such that the union of a and a' function X equals A.

 

We may approach the problem of partial identity symbolically in terms of a delimiting system of definition, such that, we may say something like the following. Given that A represents all possible roses, then

 

A is representable by means of a particular rose (or subset of roses) of a particular kind (a) that is determinable by a transformation function X (by color, type, etc.) and all other possible alternative types of rose (and their associated functional values) and things like roses (flowers, colored things, plants, etc.).

 

Then we may say something like what follows:

 

2. In any system of abstraction, whether mathematical or symbolic, the partial realized value may stand for and represent the abstract total value of the whole as long as the operational transformations of derivation and partition is definable and the complement is assumable and sub- or superscripted.

 

3. In any system of application, we may substitute the sign of the abstract total value for the partial derivative in any occurrence of the partial, or by commutation, in any system of abstraction, we may systematically substitute any partial derivation for any abstract value, as long as the complement can be subscripted and superscripted.

 

4. In order to perform systematic substitition, we require some table of reference that allows us to clearly state the partial derivatives and direct-indirect complements of each abstract whole.

 

5. For each mathematically represented set of values, we can assign one or more relative sets of symbolic terms & their associated definitions, such that we may substitute the alternative mathematical and symbolic statements at any point in our explanation.

 

6. In all real systems, we expect that both the mathematical and symbolic forms will be used in a polynomial manner that reflects algebraic abstraction of basic terms, such that for each hypothesized abstract entity A, there is both a mathematical and a symbolic partial that cooccurs at the same time (A(Rose)). I will call this "partial duality" of our metasystems and their elements.

 

7. Finally, systematic substitution procedures are guided by frameworks and rules of inference and reference that are said to hypothetically underly and inform the metasystem in question, and in some larger sense, all metasystems.

 

For each and every metasystem in question, there are always at least two sets of governing operational rules that are applicable to that system:

 

a. A core set of universal inferential rules that relate that system and its design to a larger class of systems.

b. A derivative and relative set of inferential and referential rules that defines its pattern of variation and alternation as unique and different from other systems.

 

I will call the first (7a) unification rules and the second (7b) differentiation rules. Finally, I would say that in any given delimited metasystem, there is a third set of synergistic meta-rules that are based upon the interaction patterns of a and b above, and these will be called integration rules that apply to the metasystem as a whole. From the standpoint of set theory, integration rules can be construed as the cardinality of a system as a whole.

It is apparent in the description and explanation of any hypothetical metasystem, whether this is real or abstract, we are interested both in the systematic definition of the prime partials and of the prime rules governing the system. We can say that the partiality of any system is determined by the relatability of the parts to the whole which always includes some larger framework.

Since all systems are part-wholes of larger systems, we can say the following:

 

            1. No system is completely whole or independent.

            2. All systems are part of some larger systemic framework that is universal and infinite.

 

We cannot describe the infinite framework that embeds any particular partial metasystem, only the primary relationships that effectively determine that system as both separate and dependent upon that framework. We subsume and supersume this contextual identity through subscripting and superscripting our indirect primes.

 

We seek to outline and detail in our metasystems framework the possible ranges that state-alternation may achieve for any particular or general system we are describing. We cannot do so in an exhaustive sense, as indeed, scientific description can never be exhaustive of phenomenal reality, because it would be infinite. We substitute general explanation in a way we presume to be consistent with exhaustive description.

In fact, we prefer general explanation, over exhaustive description, because the results are more interesting and non-trivial, even if they are wrong, while exhaustive description becomes quickly tedious and does not resolve anything in the long run. At best it consumes valuable research resources. We need exhaustive description of course, as our empirical, scientific frame of reference, but we must impose generalistic limits to our scientific explanations in order that our explanations remain parsimonious and not overloaded with trivial detail.

The substitution of general explanation for exhaustive description is done in a systematic manner that should be regulated by rules of deductive and inductive inference and by terminological rules of concise description and definition. But first and foremost it needs to be externally consistent and noncontradictory to the observed or inferrable evidence. This is not to say that conceptual and symbolic systems cannot handle contradiction. Indeed, ideology is a system based upon some implicit form of tautological self-contradiction that is disguised as noncontradiction. This generally happens when the ideological constructs and their inference structures are at some level fundamentally dissociated from the external realities they purportedly represent.

Science as normal praxis and theory can tolerate a wide margin of error, indeed it thrives on error at all levels, as long as it can deal with error in a systematic way that allows it to expand and refine its knowledge system in a more realistic manner. Science often proceeds paradigmatically in spite of mounting error, so error by itself does not cause revolutions in science. They are only forms of counterevidence that eventually accumulate and build up to critical levels, and thus represent precursors, or advanced early-warning signals, that entail that science itself is as chaotic in the long run as the phenomenal patterns of nature it seeks to understand.

It seems logical to conclude that systematic inclusion of the possibility of error, and the occurrence of error, into our formulations about reality, is a good way of assuring that ideological closure will not occur in our normal scientific activity. But this is easier said than done. We, as symbolic creatures, prefer closure, even if forced, to chronic ambiguity and antinomality. We want certainty to such a degree that we are even willing to sacrifice the realism of our constructs in the name of preconceived truth.

It is the purpose of this first part, and especially of this chapter, to outline in as much detail as I can muster alternative systems of symbolic abstraction that are realistically and hypothetically applicable and appropriate to advanced metasystems.

 

Most naturally occurring systems are essentially non-linear machines. Humans have tended to conceptualize and construct real machines that are superficially and ideally linear in design, but the functioning of which usually also describes non-linear state-trajectories, especially over the long-term. In this latter regard, we must understand how such machines as finite, actual entities composed of and determined by natural proccesses, change in their composition and interrelational patterning between their components as a function of time and operation.

I believe that mathematics is the appropriate language for such metasystems, whether they are construed as linear or non-linear in design, because the relations between the parts, even indeterministic aspects of these relationships, can always be represented mathematically in terms of measurable variables and values. These are terms that are always systematic and deductively ordered in terms of logical operators occurring within a system. For such a set of conditions to hold, any such system must be finitely bounded in a discrete and deterministic way as an internally isolatable mechanism with the caveat that such bounding is never perfect but always partial.

It can be demonstrated empirically, and I believe, proven rationally, that no real system can be perfectly ordered in a "closed" sense. Hence, all real systems will in time show disintegration and decay of their normal patterns, as systems, and this is an expected aspect of any real system. Mathematically we should be able to represent this in realistic ways. The challenge and inherent problem of mathematics is that it is based on ideal models of closed systems that are therefore considered to be fundamentally unrealistic. We suffer a loss of coherence in the application of mathematics to real problem sets--it entails that we must break mathematical systems apart as systems of symbolic conceptualization, and apply them piecemeal towards the integral resolution of complex problem sets.

The mathematical system I am proposing is based upon primes that are derived ultimately from real (i.e. non-ideal) definitions of identity and relation within a metasystems model. I believe that most linear models and theories in mathematics that represent ideal systems, can be readily converted to incorporate non-linear systems in a homologous way, by means of the reidentification of the fundamental identities of the primes involved in the system as partials, derivatives and relatives. At this stage, absolute values are translated into relative values, with the sense of discrepancy or difference this involves been explicitly defined as intrinsic to the identity of the prime itself at every step of its application.

This sense of difference translates into what I believe to be a set of explicit confidence values that can be associated with defined value sets in a statistically accurate way. I will not say that it is non-arbitrary as would be expected in ideally abstract systems. I would say that the degree of aribrariness infinitely diminishes to "zero" in a non-zero reality. Some complex point in our calculations is soon reached beyond which such difference makes little difference at all. At this stage, science becomes robust both internally and externally without a sense of ideological closure or an essential loss of realism of its main lines of argument. It remains fundamentally open to error and expectable nonlinear deviation of pattern.

It can be demonstrated from this that metasystems, when regarded from a mechanistic point of view, are always isolatable and definable in general terms as such. This process gives hope for our sciences to the extent that they allow some minimal and relative degree of absolute abstraction to occur in reference to a finite system or metasystem. This is always relative to some larger system of reference and inference, but this is the best that we can do in our sciences. In otherwords, limited truth is better than untruth. By heeding and observing the limitations of our science, we can systematically violate these limits in interesting ways.

 

The fundamental question becomes therefore how do we delimit truth-value in our conceptual formulations and abstract contructions of reality. We need to do so in an empirically consistent way and yet remains logically coherent in a rational manner. We know of the fundamental trade-off between description and explanation. We know that parsimony of explanation cannot be served by infinitely extending our linguistic constructs and by exhaustively describing the minutia of reality. We know also that usually parsimony of internally elegant conceptual models cannot be achieved without some fundamental leap of faith beyond which we tend to sweep contradictory evidence or patterns of variation under the carpet as just so much clutter and confusion.

What I am proposing is a built-in system of allocational trade-offs between opting for empirical consistency and rational coherence in our model building. This system is built into the very language of scientific description and explanation itself in several ways.

It proposes relatively tight denotative primes when it comes to our descriptive language, even involving, of course, quantitative measures. These are abstractly representable as non-quantitative variables that define the system or the parts of the system in question. These primes, if need be, as variables of our metasystem, are expandable in either a qualitative or quantitative manner (preferably in both ways at the same time). These primes should be relatively restrictive, especially and even in very complex and derivative real systems where the identification of such primes usually remains ambiguous and without clear points of reference.

Thus, a great deal of effort must go into the concise definition and refinement of the primes at every point. This constitutes the basis for what I would call Scientific Philology and this represents a companion project that I will attempt to undertake consequently to this work. Of course, explicit elaboration of denotative primes entails and demands a clear and concise framework of reference/inference within which its definitions can be constructed. This of course describes a metasystem of phenomenological epistemology and metaphysics. The definitions themselves are usually constructed from looser models of natural symbolic language, and this invites a substrate of a groundless ground of meaning in our knowledge systems into which a great deal of essential arbitrary values can be imported surreptitiously or unintentionally. Elaboration of a systematic framework of reference/inference is thus a complementary part of such a work in scientific philology.

At the same time, it proposes a relatively  unrestricted identification and application of the primary operational relations that articulate within any system. Classical scientific methods were based upon mathematical and logical models that implied, among other things, a kind of strict causality of implicature and truth-value. This has been clearly mechanistic in a linear and deterministic sense. It entailed, among other things, a blanket application of an additive construction of systems in which there was a clear-cut boundary demarcating parts, sets and samples from one another.  In other words, it imposed a kind of abstract sense of discontinuity upon systems that were in reality relatively continuous, and it did so in a manner as to hide the arbitrary nature of this superimposition.

In embracing the inherent complexity of nonlinear systems, we must sacrifice the language of description based on finite unidirectional causes, what might be called a "chemical reaction" view of natural relations. This is not any great sacrifice, I believe, as the search for causes has often led us on wild goose chases in our theoretical constructs, to the implicit foreclosure upon construal of systems functioning as such.

Natural relations appear to be not so much deterministic, as they are interdependent, and not so much causal, as they are correlational. If such opening of our models confers upon them a basic sense of directionlessness, the absolute directness of time and change comes to our rescue, and also the notion that most patterning in sytems is cyclical rather than linear. If we confuse cyclical process as linear time-ordred cause and effect, we restrict our understanding of such natural processes that distorts the real relations that occur.

Particularly appropriate in the adjustment of our langauge, is the search for ultimate causes and prime movers in complex, multi-determined systems. It can be said that usually there are no clearcut prime movers that can be said to account for systemic patterning, except if these are destructive in their consequences. Most systems can handle some threshold of change without disintegration of the system being the net consequence. Even in the cases of catatrophic events, prime movers can be construed more as the catalysts precipitating systemic crises, rather than as the efficient cause of such events themselves.

I therefore propose in the spaces of this work to undertake a revision of this system of mathematical abstraction and mechanical modeling of reality as much as is possible. I do so with the purpose of making explicit the ways and points at which arbitrariness enters into the application of abstract systems to real systems.

There is an important proviso in this. Abstract mathematical systems are, in the purest sense possible, absolutely non-relative constructions. This is the basis of their sublime power and irreducible truth-value. But as such ideal systems of abstraction, they are essentially, unmodified, unrealistic systems that cannot exist in pure form in reality. In this regard, I propose that there is a fundamental dichotomy between a priori and noumenal systems of abstraction, which pure mathematics represents, and essentially a posteriori and phenomenal systems of realization that are represented by applied mathematical systems. What I propose herein is essentially an applied system, but one that hopefully transcends this dichotomy in important ways. I attempt to do so by means of demonstrating as explicitly as possible the transformational operators necessary to the application of pure mathematical constructs to real systems. Hopefully in this regard we can retain a limited sense of the abstract truth value inherent to such ideal systems, without sacrificing at the same time the applicability and descriptive consistency to real world problem sets.

Perhaps this is somewhat of a compromise approach, a bastard of science that will prove to be an infertile oddity and hybrid.  But even if it is only a freak of an abstract system, it may open the door to something better beyond that we do not yet understand or know.

 

Natural language finds its sense of order in the symbolic-relational structure that the human brain creates within a larger cultural system. It gains its power by indirect contextual reference to abstract meaning as well as lived experience.  The power of language is realized in its capacity for reification, for making seem real what is in fact imaginary.

This sense of order is minimally constrained internally in terms of its semantic value by loosely implicit principles of non-contradiction, or what we can call the dialectical contrast of opposites, and analogical association. For the most part it relies upon its external reference coordinate system to achieve its degree of realissimum. In essence, one thing cannot mean its opposite at the same time. This is imaginable and possible in the symbolic universe, especially in mythology, but it is not structurally desirable as it creates dissonance within the meaning system it embraces. Otherwise, almost anything is relatable to anything else, and the actual deterministic patterns of relationship are included only by progressive degrees of direct relationship. Thus, in the symbolic structure of natural thought and language, almost anything can stand for anything else, except the opposite of that thing. Technically, a thing can come to embrace and stand for its antithesis, as long as it is marked in an acceptable manner that allows it to do so within a larger system of symbolization. This is the power and potency of natural human symbolization, especially as this is articulated and expressed by natural human language. It is the power to resolve contradition and ameliorate "marginal" realities that contradict our knowledge. This is the basis of the natural symbology underlying most human ideological systems.

Mathematical language is a subsystem of the more general form of symbolic system. Its main difference is that mathematical language is internally constrained in ways that normal language is not. Thus, mathematical language achieves a degree of extreme internal coherence that is often lacking in natural language. It pays a price for this in not being fully or sufficiently functional as a natural symbolic system. It lacks the power that natural language can achieve in its description of reality and in its ability to resolve contradition. But it gains a power of internal coherence of structure that is much greater, and finds a broad range of applicability in precise, formulaic and scientific descriptions of physical reality, especially in mechanical systems.

Mathematics is not even a true symbolic system--it is a system reduced to one of signification that does not depend upon communicative efficacy. It lacks the duality of patterning found in natural language, but it achieves thereby the consistency of exact correspondence between terms. It is true that the functional design of natural language, that of making sense of and promoting adaptation to the real world, demands an inherent flexibility and external reference orientation of its linguistic structure that precludes the possibility of setting up such a restrictive tautological system.

There are several clear implications of this. A mathematical model of structural linguistics is not sufficient to a full description of natural human language--it is at best a limited heuristic device applicable mostly to grammar. Furthermore, to arbitrarily restrict natural language by the superimposition of rules of relation and definition, is to curtail and cut short its symbolic capacity. This is not the most desirable thing to do if we depend upon the full power of our language to describe reality at any level.

Between symbolic natural language and mathematical language that is essentially a system of signification lacking many of the design features of true language, there is a trade-off. Mathematics works well, especially in mechanical and physical descriptions of reality where measures predominate and in the abstract generalization of closed models or universal relations that are essentially mechanistic in nature.

Natural language remains the preferred, indeed, necessary, mode of communication when it is important to try to encapsulate and describe complex realities that resist denotative analysis in every way. Of course, this trade-off is never very clear-cut. Science requires both the language of natural description and rational explanation, as much as it needs mathematical formulas for achieving theoretical validation. This is true at almost every level, and from a scientific standpoint, natural language and the language of mathematics are not mutually exclusive in theory building, but are mutually complementary to one another.

Of course, attempts have been made to try to constrain natural language and semantic systems in ways similar to mathematical systems. Mathematical or symbolic logic is perhaps the best and most productive example of this kind of deliberate deductive constraint. Ideological systems that are fundamentally closed and symbolically restrictive usually impose some restrictive constraints upon the language process as this is employed in ideological articulation, though at some level or other non-logical leaps of faith and unquestioned presuppositions are smuggled into the system of rationalization. The consequence is that if a religion teaches us that two plus two equals five or six, we are liable to believe this even if it represents an internal contradiction of formal logic.

Such systems are fundamentally "closed" systems of rationalization that do not permit a testing of its truth propositions on any level, either logically or empirically. Even mathematics is an inherently open system in this regard. Because it is based on deductive logic alone, it does not require faith for its apprehension or extension in the world. Accepting mathematically that 2 plus 2 equals 4 is correct from a logical standpoint, and so does not require any other form of conviction or symbolic legitimation. It does not require that we agnostically abnegate or publically confirm our faith in God or the Devil or in any other form of belief. It only confirms our own confidence in our objective knowledge.

From this standpoint, objective knowledge has always two facets--internal and external. Not all knowledge has these two facets simultaneously. Subjective knowledge, feelings, intuitions, and dreams, do not need necessarily a set of external reference points or an internally air-tight system of deductive inference. Belief systems have two facets, but the external facet of belief is conditional upon social sanctioning of the system, and not upon the validation of phenomenal experience. The internal facet of belief is conditional not upon the application of deductive logic, but actually upon the suspension of logic or else the employment of "symbology" that is relatively unconstrained and at least from one standpoint would be considered illogical.

It is clear that mathematical language is at its best, though not exclusively so, in its internal coherence. It is clear though that mathematics can be used to reinforce an empirical description of reality at every point. There is no sense in abandoning what is best about both language systems in the extension of these systems to advanced systems science, especially just to offer some mixed system which is specious at best and at worst trivial and spurious. On the other hand, we should also recognize the intrinsic limitations of design and applicability of both systems, and try through our advanced systems science to overcome as much as possible such limitations.

I propose that we need to try to work towards a broader paradigm of the limitations and strengths of language in the sciences, according to something as what follows:

 

Scientific Language Systems

Natural Symbolic--Restrictive/explanatory

Natural Symbolic--

Inclusive/descriptive

Math--Restrictive/deductive

1. Pure mathematics

2. Symbolic--Applied Math

Math--Inclusive/inductive

3. Mathematical Logic

4. Symbolic Language

 

We normally have at our disposal mostly systems of types 1 and  4 above. Limited systems have been developed in type 3 and also some type 2 systems can be found, particularly in the application of math to especially complex derivative systems. These hybrid type systems are at best ambitious and at worst over-extended and clumsy, bogging down in their own top-heavy structures.

It is difficult at this point to tell where descriptive statistics, as a form of mathematical language, would be applied, but it is a form that is important to our integration of our scientific languages. Statistics includes types 2 and 3 respectively, and is a good starting point in the elaboration of a procedural language appropriate for science, but it is not itself without important limitations.

I propose that we need to try to work systematically to achieve a tight interfunctional integration of all four types of language systems. We must as well work to elaborate a more realistic and abstractly integrated system for each of the types if we are to achieve the degree of functional comprehensivity that we hope in our advanced systems sciences. In the course of the first part I work towards development of such a broader language base for our sciences through the development of the ideas and operational systems of these types in an abstract sense. In the second part, I propose to work towards the extension and procedural application of our language as operational systems.

The basis of symbolic mathematics that I propose herein is to be able to extend a mathematical model to the description of complex derivative systems without the necessary overloading of variables and functions that usually characterizes such constructions. Elegance can be preserved and consistency conserved if we are careful and precise with our definitions and formulations. We must be careful in this regard to hit with our scientific hammer the proverbial nail squarely on the head, and not on our own thumbnails.

 

The basis of symbolic mathematics is first to provide a concise formulaic description of the core operational procedures of advanced systems science. It then provides a means for its systematic extension to the development of hypothetical and working heuristic models relating to any possible system. It should be powerful enough to accurately and hypothetically describe any actual system in a minimally sufficient way such that its complex event structures can be comprehended in a realistic manner, and its epiphenomenal outcomes made known such that this knowledge relates the system to all other systems. The core operational procedures are derived ultimately from a mathematical model of the scientific knowledge of mechanical systems. They are therefore considered to constitute a purely abstract system that is based upon strict hypothetical-deductive rules of logic and measure, and which is nonetheless hypothetically and experimentally applicable to all and every system in a scientific manner.

We must ask to begin with "exactly what is mathematics" and how is it used and useful in our sciences? The root of mathematics, manthanein, originally meant: "to learn, what is learned, or learnable knowledge." Mathematics is formally defined as "a group of sciences (including arthmetic, geometry, algebra, calculus, etc.) dealing with quantities, magnitudes and forms, and their relationships, attributes, etc., by the use of numbers and symbols." (Webster's Unabridged, 1979)

I will offer a minimal definition of mathematics as the system of relating quantitative measures of some standard kind in an internally coherent way that always results in some kind balanced equation. Implicit to such a definition is the notion of "measure" as a definable quantity upon some standard, arbitrary interval-event scale and that has some kind of numeric value that can be at least theoretically assigned to it. This has important implications in its relationship to science, which is operationally and theoretically based upon the principle of measurability and therefore the systematic relatability of constructs and phenomena to one another.

As will be demonstrated in the course of this text, defining and superimposing interval scale measures has important theoretical implications for our knowledge, especially as this relates to our advanced systems sciences. It allows us, among other things, to generalize and extend our range of knowledge from a finite and semi-ordered set of phenomena to increasingly larger realms of possible phenomena. It allows us then the capability of testing our theoretical knowledge by the application of the same measurement devices to other hypothetically relatable sets of phenomena.

Mathematics is not science, at least not in a natural or applied sense, though scientific methodologies are almost always based upon some form of mathematics applied to the knowledge contexts of that science. If mathematics is scientific in and of itself, it is so only in an ideal sense as a science of abstraction. Attempts are made to philosophically validate mathematical ideas and knowledge derived from natural sets and relations. Mathematics is a field of knowledge inquiry unto itself that makes applied scientific method possible. If mathematics is a science, it is purely a science of abstract ideas and relations, forms and systems that exist only hypothetically in an ideational space. In a pure sense, mathematics does not deal with empirical phenomena or objects in the external, material world, at least not directly. It deals with ideational constructs purely that are considered noumenal, a priori and totally abstract. It imagines therefore the most perfect of possible worlds, whether this world is assumed to be completely determined or completely random in its foundation.

The internal sense of validity of mathematics is considered mostly unquestionable and as being fundamentally independent of the cultural conditions or constraint which normally occurs with symbolic knowledge. Proofs for theorems in mathematics are derived purely by logical deduction, and strict classical logic based upon the principle of exclusive identity is the basis for mathematical coherence and validation. Hence it is in its purest form universal to human knowledge, and often the conception of universal structures in human patterning is construed within a mathematical form or model, as for instance, structural linguistics. We hypothesize the psychic unity of humankind largely on the basis for people of all cultures to be able to understand and employ the same mathematical concepts and constructions in the mechanical ordering of their experiences. Thus, mathematical languages and constructs form a foundation for an objective but non-empirical basis of science. It permits the possibility, occassionally realized, of deriving valid scientific theories by deductive reasoning alone, without initial or final resort to empirical tests.

We can say that mathematically speaking, a mechanical view of the world that deals with relations, strengths and potentially observable, hence measurable, values, however indirectly, is inherently non-symbolic. Therefore a purely mechanical view of reality is inherently non-arbitrary except in some minimal sense of the conventional standards of our measurement or design of our experiment or operational methodology. While the latter set of considerations is non-trivial for the metaphysical status of science in reality, it can be temporarily overlooked in consideration of the neutral and amoral application of a mechanical viewpoint or worldview that is free of cultural constraint. Mechanical technology has readily crossed cultural boundaries, such that we can find Moslems, Hindus, Buddhists, Catholics, Jews and Agnostics all driving the same Mercedes-Benz cars in the world, all with equal moral indifference about the internal working order of the car they are driving in. A nonevaluative, a-symbolic mechanical perspective on reality extends directly from an immediate and unconstructed phenomenological experience of reality. We know this to be true in the fundamental knowledge structures of our brain and how we construe reality. We cannot afford to process reality in its original and natural form in any other way, as we would soon be overwhelmed and overloaded with sensory iputs. Thus, in spite of preconceptual frames, a mechanical view of the world informs our first selective cut of reality in an experiential sense.

As a purely abstract system of comprehension, mathematics is yet its own system of knowledge. It is a primary objective of this chapter therefore not only to understand the general application of mathematics to advanced systems sciences, but also to understand mathematics generally as itself a naturally occurring "possibilistic" system that is purely abstract in character. In other words, as a pure and independent knowledge system, it informs our understanding of natural order in basic and important ways. Indeed, it informs our understanding of order itself in critical ways, as somehow systemic and nonrandom. The rational order we are capable of in our mathematical constructs, with such great precision, reflects ultimately the general patterning of systemic order itself as this occurs at all levels of phenomenal event patterning in nature.

Of course, natural phenomenal patterning is always a chaotically, complexly "mixed" and heterogeneous system of relations. This makes inherently problematic the application of mathematical models to natural systems.

In mathematics, we can conceive of a pure, ideal sense of order, and this is contrasted with an implicit notion of absolute disorder or ideal randomness, just as the principle of exclusive, absolute identity can be contrasted with its dialectical complement of absolute non-identity. In a similar manner, so too can positive be contrasted with negative and affirmation with negation. And if we look about us in the natural world, we see symmetrical complementarity of structure at very basic levels of the ordering of natural patterning.

Mathematics is not subservient to science, and science is not absolutely bound to mathematics. Mathematicians do not need to be concerned with science, and some scientists ply their trade without much concern with mathematics. But from both a theoretical and methodological perspective, mathematics is the operational language of science in the deepest sense possible, and therefore it critically informs the structure of our scientific knowledge at almost every level of its articulation. Symbolic mathematics is the primary form of communication of science, by which science operates and achieves transmission and progress in its functional application and theoretical validation in the world.

There is more than a little epistemological & metaphysical relativity about this. Just as language not only facilitates and makes possible thoughts, but also creates new thoughts, so too does the language of mathematics not only specify and define the concepts and constructs of science, but it in turn often creates these new ideas and operations for science.

We can say therefore, from the standpoint of the inherent anthropological relativity of knowledge, that scientific knowledge is fundamentally relative to the mathematical frame of reference that it becomes defined within.

Another way of construing this is to state that if we are to get at the foundation principle of Reality in a scientific and systematic way, then we must be able to do so in mathematical terms. Any system evinces some kind of structure that should be representable in a mathematical form. If a system cannot be represented mathematically, then it is not a true system scientifically, but only a fictive one, or at best a hypothetical system lacking in any precise structural coherence. In other words, we do not understand it well enough yet, and our theoretical constructs can only be partially correct.

Again, most systems occurring in nature are phenomenologically observed as inherently "mixed" and heterogeneous systems. Many systems are in fact complex epiphenomenally derivative systems of more basic, but still complex patterns underlying them on another level of analysis. Our observations of phenomena are therefore always inherently "contaminated" with noise and ambiguity. We seek to understand pattern in a complex field of apparent disorder, which pattern is always construed against a background of disorder. Hence our ability to represent this underlying sense of implicit order in natural systems is often fundamentally compromised, not only by the noise, but by the inherent complexity of the epiphenomenal patterning of the system itself which fundamentally defies attempts at abstract and simplifying mathematical formulations.

All mathematics is symbolic in a strict sense, and this points up the applicability of mathematics, as a single informational system that is broad and powerful in scope, to the understanding of systems whether in abstract and ideal or actual and real forms. I employ the term symbolic mathematics to refer to the special case of the intentional application of mathematics to advanced systems science. It encompasses and embodies in its most basic constructs of identity the inherent duality of mathematics as at one time an idealized construct that can be symbolically represented as an abstract and exclusive entity. It can be represented at the same time as a set of actual, measurable realities that underlie and are represented by that identity. This inherent duality of knowledge patterning in mathematical formulations can be put to good use in the operational integration of systems sciences at its various levels, particularly upon complexly derivative levels where the language of description tends to resist even accurate definition, much less quantifiable denotation. In this regard it borrows something from symbolic logic, or what is known as "mathematical logic" though it does so in a sense that is more flexible and realitistcally adaptable to alternative operational constructs.

It will be stated at the outset that there is a general progression in a common continuum of knowledge as it moves from more basic to more derivative constructs in its application to empirical realities. Scientific knowledge varies along a continuum between what can be called the strictly mathematical and measurable to the loosely denotational and fundamentally immeasurable.

 

 

 


 

 

 

 

We can clearly mark out upon such a continuum where the human sciences sit versus the biological and physical sciences. The concern of this model is to point out the unification of perspective that is possible by means of mutually constraining both mathematical and symbolic language forms by means of one another, to constitute its own operational system. Thus, however quantative we may become in our numerical measurements, we maintain some minimal attachment to symbolic constructs, such that we never forget the ultimately arbitrary and anthropological relativity of even our measurements, and these are always attached in turn to some foundation in empirical phenomena. Similarly, on the other end, no matter how loosely symbolic we may become in our ideas and terminologies, some residuum of mathematical precision and measurability must be preserved in our conceptual formulations and operations.

And it is in formulation, or in the construction and testing of formulas, that we can find the necessary operational unification for our advanced systems science. Formulaic thinking underlies the structure of mathematical inquiry. Mathematical systems of conception are based upon formulas, which are defined as symbolic strings that are strictly subject only to specific general rules of composition. Formulas in mathematics are almost always equations, or at least potential equations or transformations. The same formulaic structure of inquiry is applicable to the physical sciences as much as it is applicable, albeit in less precise forms, to biological and human scientific inquiry.

Formulaic thinking is based upon deductive reasoning within an explicit and well defined system.

Thus, I propose such a deductive system for our metasystems.

The same standards and style of formulaic thinking is applicable in advanced systems science as an implicit structure of operational inquiry at all levels of informational complexity. We reach a level where the symbolic entities and constructs we are dealing with, as complex variables, become inherently non-numeric in structure, though on some level they can be hypothesized to be reducible to numerically definable entities or measures. We have no choice but to proceed in such a manner.

 

We can say that a scientific worldview is inherently a systematic view of the world, and that a systematic view of the world that is based upon the hypothetical design of working systems is a fundamentally mechanical or mechanistic view of the world. Even the abstract ideas of pure mathematics itself can be said, as ideational as they are, to be structurally and fundamentally mechanistic in character. The original definition of mechanics was the study of behavior of systems under the action of forces. Statics dealt with systems that were motionless or else motion was considered irrelevant to the description of the system. Statics dealt primarily with equilibrium or stable states of "rest." Kinematics has been a special subdivision of classical mechanics that is concerned principally with the study of motion itself without concern for explaining the causes of motion in a system. Dynamics dealt with systemic motions that were the result of forces operating upon or with a system and that entailed some form of state changes or alternation. The extension of a mechanistic view of science to naturally occurring systems is fundamental to the operational design and organization of advanced systems sciences.

We can distinguish between classical Newtonian Mechanics, fluid continuum mechanics or classical field theory, and quantum mechanics. We can distinguish large order or large-scale systems, and small, microscopic scale systems. Statistical mechanics is applied to dealing with systems that entail large sample sizes.

The notion of relativity is inherent to a mechanistic view of reality, and it informs our understanding of systems upon all levels. The definition of mechanism, as "an assembly of movable parts having one part fixed with respect to a frame of reference, and designed to produce a specific effect," embodies the notion of classical relativity. We can say that two similar but independent systems within the same frame of reference will produce similar effects. Our scientific methodologies are based upon this principle. Generally, as working systems, in a broad and most general sense, a mechanism is defined a constituent, self-organized system of parts that mechanically directs and transforms motions and energies. This is true if we are describing the system of the total universe, or we are describing the system of life occurring on earth, or the system of human symbolization popping in and out of the human brain. Natural informational patterning is the result of this mechanical sense of order and direction, and leads to an understanding of the implicit structure and natural laws underlying any mechanically definable system.

Thus, in our scientific and mechanistic view of the world, we often employ many analogies, whether derived from abstract mathematical models or actual mechanical systems. These are often simplified representations of the more complex systems we are attempting to describe, and such analogies, or "exemplars" are important to the theory building, testing and comprehension of science. The rootedness of mechanical models in mathematical relations makes this kind of model building and heuristic problem-solving possible in the first place.

Classical mechanics dealt with the description of the states and positions of material objects in space under the action of forces as a function of time. This was conventionally construed in a non-relativistic framework, though it always implied a more general form of relativism. We know that in natural patterning, few systems are purely linear in the sense represented by classical mechanical models, but we can also understand that such linear models are subsets of more complex, larger, nth-scale non-linear systems. The models of classical mechanics were based on mathematical description and utilized symbolic logic to derive a precise explanation for any observable system of classical motion. It defined the basis for the derivation of subsequent fields of physics. Many mathematical formula that were derived purely by internal logic, and which, by themselves, appeared to have no direct foundation in empirical reality, were found to be subsequently useful in the elaboration of non-classical physical theories of reality. Often they became applicable as working mathematical analogies that described in precise ways the functional patterns and attributes of physical systems.

The point of departure for understanding the role of symbolic mathematics in advanced systems sciences is therefore to make the following kind of statement. Reality (the Reality Principle) is inherently problematic, whether we want to solve it or not. If we choose to construe reality as fundamentally unproblematic, then we are living in a world with intelligence but without using our intelligence. Since intelligence is functional in a problem-solving manner, it is impossible to live in a world without applying our sense of intelligence to somehow solve its problems. Even the abnegation of responsibility to define and solve problems in reality is a kind of minimally intelligent solution. The inherent aspect of our anthropological relativity to all our knowledge is the problematic nature of our reality, especially in any shared or collective sense.

How does science solve problems systematically in reality, and in an objective manner? It adopts standards of measure that are ultimately numerical in character. Only by such a means can it achieve an objective frame of reference that is external to the subject knower, or a collection of subject knowers, in a non-arbitrary manner.

Mathematics is a powerful system of constrained symbolic signification that can be said to be truly internally non-relative to itself, though it is applied relativistically to external contexts in reality in the descriptive explanation of mechanical systems. Hence pure mathematics is noumenally independent in reality, and is based only and exclusively upon its own achieved internal coherence for its validation. This is derived, I believe, from the natural, internal countability of discrete things in reality. That so much that is so basic to our reality and our sense of reality, can be demonstrated in rather pure and basic mathematical terms, demonstrates the degree to which naturally self-organizing systems follow and must obey in their mechanical design fundamental mathematical precepts.

Pure mathematics is almost entirely based upon principles of constrained internal coherence that are inviolable. Applied mathematics, upon which science has successfully constructed its operational methodologies, has been based not directly upon internal consistency of its mathematically constructs, but on their generalizability and consistency with external experience. In the scientific use of mathematics, internal coherence is usually always implicit to the use of these formulas, but their efficacy is based upon their external consistency to empirically measureable realities and to their appropriateness in leading to successful teleological applications and predictions.

The foundation of mathematics I believe to be the presupposition of absolute identity, such that something at any one time and place can only be itself, and not something else. This is also basic to classical mechanical identity of things in physical reality. This is not to say that we cannot have composite entities that are more than one thing at one time. But in an absolute sense, we can at least say something like the following: one equals one, and not two or any other value. Classical two-value truth logic derives its strength from this same presupposition when it is applied to qualitative or non-quantitative values. Hence we can say the following: blue is blue and not red or any other color. We can say that in a fictive world, blue can be red and one can equal two, but in reality this kind of statement violates something fundamental about our basic sense of identity, and thus must be rejected as inherently false or fictive.

Derivative from this principle of identity in mechanical reality, are the basic arithmetic computational formulas in mathematics that are built mechanically upon the principle of addition. One plus one equals two (and not three or some other number). Logically, we say that blue and yellow make green (and not red or some other color). All other computational operations, subtraction, multiplication and division, are elaborated extensions of our ability to make one and one always equal to two.

Up to this point, standard logic and fundamental mathematics are closely tied, but beyond this level they diverge and go their separate paths. Logic, dealing with semantic meaning that is inherently qualitative, hence subjective, quickly breaks down in the face of the inherently symbolic values of natural language and discourse. Mathematics, dealing with ratiocinative values that are inherently and fundamentally quantitative, hence non-subjective, leaps to the next level of algebraic abstraction involving basic principles defined by substitution, distribution, and association, as well as to geometric analysis of basic forms and shapes. From here it leads ultimately to extremely complex and sophisticated permutations and elaborations in analytical geometry, trigonometry, calculus, non-euclidean geometries, probability and statistics. Mathematics has been highly successful, so successful in fact, that we could not have had science without it.

The beginning of understanding the role of symbolic mathematics in the operationalization of advanced systems science is to get at the fundamental philosophical aspects of mathematics and how this relates to reality, and especially to our scientific understanding of reality. In a sense, it can be said that all mathematics is fundamentally symbolic in at least a restricted sense that attaches value to some coordinate sign system. Mathematics would not survive as a successful system of rationcination if we loosened its standards to embrace the symbolic aspects of natural human language, for instance. It would be reduced to a trivial system of notation that oversimplifies reality.

If we go back to Kuhn's critique of science, we can understand that what sets science apart from other forms of knowledge is its "puzzle solving" character. Science identifies and defines problems that have, at least in theory, some definite single solution that is correct to that problem. They are thus like puzzles and less like the dilemmas of meaning and value that we encounter in literature and literary critique. The measure of success and progress of any scientific endeavor is the extent to which it is capable of solving complex puzzles that scientists come to ask methodologically about reality.

Thus, if we are to posit an alternative variety of symbolic mathematics as somehow non-trivial and operationally useful to the functional integration of advanced systems science, then we must define it in a clear and concise way. This precise definition allows it to identify the problems encountered in our understanding of reality in such a way as to be "puzzle-posing" and hence "puzzle-solving." If we cannot accomplish this in some minimal way, then we should stop before we start.

I believe that in one limited and limiting sense symbolic mathematics selectively and potentially encompasses all the areas of mathematics, both pure and applied. It organizes all the areas of mathematics in terms of the comprehensive functional integration of systemic problem solving. The use of mathematics as a procedural language for advanced systems science is not spurious or superfluous. It has been designed with the idea of permitting computational and programmatic integration across all the mathematical fields, and in terms of its possible applicability to any system in whatever area or field it is identified within. It is necessary to the structure of this approach in order to render it procedurally systematic. If terms and events cannot be expressed clearly in mathematical language with measurable and assignable values, then it is likely that we both do not understand the systems in question sufficiently enough, and that we are therefore also unable to "operate" upon the system whether experimentally or through alternative application.

It requires therefore an encompassing grasp and command of mathematical theories, formulas and principles. It is not my intention in the course of this work to elaborate all of mathematics, which would be a voluminous and lifetime affair. It is safe to say that as long as we understand the basic principles involved, we can put our skeptical trust in the capacity of computers to do a great deal of mathematical processing for us. This is not only a time saving issue, but an issue of fostering a system that is of greater efficiency both in terms of work and in terms of its informational capacity.

 

The point of departure of symbolic mathematics is to attach all possible measures, hence all potential numeric values, to some symbolic system of non-quantitative denotation within a standard relativistic framework that reflects ultimately the relativistic foundation of our reality and our knowledge of reality. In a sense, algebra already does this to some extent, as a direct extension of basic arithmetic equations to embrace non-discrete variables.

Underlying this is the fundamental principle of unity of identity, such that on a basic level there is no difference between qualitative and quantitative, but they are inherently alternative aspects of the same physical identity. Hence, if we are going to identify something occurring in reality as distinct in some qualitative sense, we must also isolate that "thing" as somehow distinct in some quantitative sense as well. Hence, we do not talk about blueness in a qualitative way only unless we can offer up some mathematically quantitative description of blueness, as being somehow a range of light on the electromagnetic continuum. We can say one blue thing, and also one green thing. We can add the two things together, as things, but not as two blue-green things. We can say, one blue thing plus one green thing equals two things that are blue and green respectively.

When we apply mathematical formulas to real world descriptions, we are always assuming some state of ideal equivalence of discrete value between objects that is not necessarily or exactly so. This is especially problematic in statistical descriptions of large populations of things. Reducing complex sets to simple count numbers often conflates and disguises a great deal of intrinsic/extrinsic variability between things. We treat a classroom of forty men as all essentially equivalent in our experiment, both qualitatively and quantitatively, for the purposes of solving our basic problem. We cannot proceed otherwise in reality without superficially overcomplicating things to an inordinate and disagreeable level.

Thus, in order to generalize between events or entities in reality, we must assume some minimal degree of finite equivalence and discreteness occuring between these events or entities. The elaboration of empirical reality otherwise leads to infinite differentiation and particularization between separate events and entities.

In probability theory, that is applicable especially to physics, we adopt standard terms that describe elementary entities, outcomes or events as fundamentally isolatable and indivisible constructs, or units, that we call sample points. Compound events or entities are usually described in terms of set theory, and defined in terms of our conceptual experimental model in relation to some possible problem set. They are called a set of sample points that is united and differentiated on the basis of their relative identity to the theoretical constructs, and are referred to as the sample space. Every compound event or state is represented by an aggregate of sample points that are regarded as relatively equivalent or synonymous in terms of the set theory we are employing. This is the basis for our scientific generalization and operational procedures.

Now it can be seen that in reality any set of events or entities that are hypothetically equivalent are actually differentiable on some level as realistically non-identical. To presume otherwise is to violate a basic principle of physics that says that the same thing cannot be in two different places at the same time. Thus on some level, scientific generalization commits itself to a basic fallacy of spurious equivalence of identity between discrete state-events. This is especially true in the more derivative sciences of biology and anthropology, but it even happens regularly at the basic levels of physics. Indeed, it is at the level of physics that we have unusual properties operating based on Bose-Einsteinian statistics. We need this fallacy for the sake of preserving parsimony in our theoretical generalizations--for the most part this works well enough if we do not become too picky with our data points.

Thus, scientific generalization normally depends upon the categorical conflation of data points in the sampling of reality and in the generalization of its concepts and their relations. To proceed otherwise is to quickly overwhelm our procedures with a great deal of spurious and nonessential complexity.

But to completely ignore what can be considered as spurious to our constructs and to conflate inherently complex realities as simplex samples is to commit ourselves to a basic kind of error in our sampling procedures. Indeed, it is in sampling error, usually the result of inherent variability of our sample points, that we usually and unexpectedly learn something interesting about the inherent structure of reality as this is different from our constructs. This is especially true in non-deterministic and stochastic sampling errors that arise from what can be considered in our constructs as "random" error.

Underlying this kind of presumption of sampling error is the implicit presupposition underlying the presumed equivalence of sample points. Because they are equivalent, they are considered to be interchangeable with one another. Hence they are considered to occur fundamentally independent of one another, and thus they occur in what are considered to be essentially randomized sets. In reality, this underlying assumption of perfect randomization of equivalent sample points is rarely realized in reality. It is safely presumed on basic levels--otherwise most that passes for statistical evidence would fall through the screens of biased sampling procedures. The presumption of ideal randomization of a sample set is attached to the idea of a perfect descriptor for an entity in reality, and the conflation of variation within the sample. Indeed, scientific learning and progress largely arises as the result of the violation of these presumptions in our data sets. It is in the deviations of patterns from the ideal parameters of the sample that results in the ability to detect non-random deterministic relations underlying the sample. Thus, it forces, at some point, a revision of theory to take this non-random pattern of determination into account.

If we can generalize from a random collection of relatively discrete data points to a sample of a set of such points, we can also generalize from a set of samples to a larger abstracted sample that is a compound aggregation of the sample sets themselves. We can even generalize from very large sets of numbers to a very large sample set that is, at least in theory, infinite and unboundable. This is theoretically accomplished in probability theory by limiting procedures that define intervals as aggregate point sets instead of points as the limit of an infinite sequence of contracting intervals. The probability of relative zero is assignable to each individual point.

The law of large numbers is derivable from from this kind of limiting procedure, which states that the relative frequency of alternates tend toward their natural expected frequency as the sample size "n" tends towards infinity, and this aggregate event has a probability outcome of relative one. This is a basic situation in measurement. If we begin with a set of basic events, or intervals, that we attribute probabilities to, by simple and natural limiting procedures, probabilities can be assigned to any broader class of events by applying set theoretic operations to intervals (union). To each event-interval there corresponds an associated probability that is greater than or equal to zero. The total probability for the larger class of events is merely the summation of the probabilities of the aggregated intervals, or the measure of the Borel field of the interval set.

Thus:

 

P{A} = ΣP{Ai} = 1

 

Where A = the union of the mutually exclusive event intervals A1, A2, A3.....

 

We can see that in our sampling procedures it technically may make no intrinsic difference to have large samples or small samples if we can always assume perfect randomization of data sets. But in the real world a larger set of data points tends to minimize the adverse effects of non-random patterns of variation not accounted for by the theory (limiting procedure such that P for any particular point equals relative zero). At the same time optimization of the positive affects of random exclusive event-intervals, such that the probability of expected frequency patterns of the total set equals relative 1, can be accomplished by the presupposition of continuity and the extension of the addition rule from finitely to infinitely many summands.

These are important considerations that effect both the realism of our constructs and the ability to generalize based upon our samples. It is easy to see that how we define our conceptual constructs directly determine how we identify our data points and how we limit and constrain our samples as event-intervals and sets. That this is so frequently overlooked in the design and evaluation of statistical projects in our "sciences," particularly in our social sciences, is simply amazing. It points to the degree to which any pure or applied science, lacking in either world vision or operational efficacy, becomes the servant of political controlling structures.

Therefore, I have made the point of departure for symbolic mathematics as a procedural language and set of operations for advanced systems sciences the central issue of the presumed and differentiable realism of particularistic data points as complex compound event-intervals upon whatever level we define our sampl. This is intrinsic to the definition of our sample points at whatever level of generalization we choose to operate at, or in whatever area of application or level of derivative phenomenal distribution. Thus, I impose a uniform set of terminological and relational variables as intrinsic/extrinsic derivates and alternatives operating implicitly on each level of analysis and synthesis that we define our samples upon.

The intrinsic disparity between the idealized data set represented by our conceptual definition of our sample as a randomized set of exclusive event-intervals, and the realized instantiation of the actual data points representative of and by the sample, is made in every expression structurally explicit and intrinsic to the definition in the first place in a systematic way. If we choose at any level to expand the formula through systematic differentiation and substitution, a process I call "functional object embedding" and then we have built into the design of the procedures a means for doing so. On the other hand, if we wish to replace differentiated chains of values with a sample set that is ideally defined by a single variable, we still carry subscripted with that variable the possibility for its elaboration.

The point of departure for symbolic mathematics from other forms of mathematics is the realization of a model of a mathematical system of transcription that is symbolically defined by and defining of complex polynomial states. These polynomial states are implicitly embedded in the definition of the key variables at whatever level of sample generalization we are operating upon. These states at least purportedly represent the hypothetical underlying event structures relating to any particular system or set of hypothetically related systems and that would be normally conflated or systematically excluded in our simplifying procedures.

I believe this is accomplished within the following kind of framework:

 

            1. Assignment of absolute values of zero and unity (absolute 1) as the relative limits to any system.

            2. Representation of all hypothetical systems within the same hypervolumetic space called the unification space.

            3. In such a system, all discretely occurring values are transformed into ratio values by means of systematic procedures in which they are transformed into "ideal numbers."

            4. Representation of all states as complex polynomial variables that are always differentiable into a composite of at least three non-absolute derivatives:

            a. A numerical value based on some scale of measure or set of measures or scales.

            b. A instantiated variable that may itself be a complex polynomial

            c. A derivative that represents the difference between the idealized state-variable, and its instantiated values and variables.

            5. Representation of all relations as complex events differentiable into alternative sets of determination.

 

Important to this procedural system at the same time is the discontinous determination of key variables and their associated discrete or expected values in any system set, and the capability of contextually relating this set to supersystems or subsystems to which it is hypothetically related. In other words, we require some grander sense of a universal inference-reference coordinate system that is defined at least on a general conceptual level in terms of ideal, discontinous variables with relatively concise functional explanations.

Another way of looking at this is to say that the normal procedures of an applied symbolic mathematics cannot really occur outside of appropriate theoretical contexts that define the conceptual parameters of its operation in a hypothetico-deductive and empirical-inductive sense. We can advance a relatively "pure" model of such procedures in an abstract way, but it has little value unless and only until we can apply it to real and generalizable problem sets.

To this end, our analysis of systems must be intentionally contextualized within an abstract frame of reference that is general and metalogically constitutes advanced systems science as a whole, and operationally the unified framework of a mechanical-systemic approach to all phenomena. We can relate all phenomena as a part of some system to which the appropriate units of analysis, or interval measures, are definable by nature of its positioning within the overall framework.

All systems are part of a larger, total universal system that is most basic and derivative of systems. Furthermore, we can specify scientifically a rather precise order or stratification of systems in natural classes or categories depending upon their level of derivation.  I attempt to set up this kind of generalistic frame of reference in the second and third parts of this works, with an eye to showing the operational systematization occurring for all levels and in any area. In the last part I return to issues of functional integration in advanced systems science, especially as this deals with issues of applied and artificially constructed systems, demonstrating how the basic operational procedures can be used in alternative ways.

In a more fundamental way, we may say that the language of mathematics, especially in its purer forms, is equipped only to deal with ideal states, and that it achieves its systematic coherence only when it can assume some degree of equivalence or correspondence with ideal states. Applied mathematics must deal with the issue of the translation of the ideal procedures and coherence of math to the description of real sets of events. This works well enough for physics especially, which is usually based on a fairly mechanistic set of relationships between fairly quantifiable forms of data. It also works well in engineering that deals with some form of mechanics that is derivable from physics, but this kind of mathematical language tends to break down and become spurious when we deal with complex derivative phenomenal patterns in biology and in the social sciences.

We bring advanced systems of statistics to aid us in the extension of mathematics to these levels of phenomenal complexity, and take great care in the definition of our data types and their implications for our procedures. But even these are usually inadequate to cope with the intrinsic scope of complexity embodied in such systems, especially when we wish to deal with issues that are synthetically significant and not analytically over-reductionistic.

Symbolic mathematics has been designed therefore from the point of view of allowing us to more realistic model complex realities without the risks of over-simplification that are rooted in presuppositions of ideal mathematical descriptions. If it is done well, it should permit us to systematically generalize from data points and sample sizes large and small in a manner that achieves simplification while retaining a sense of empirical realism. Ultimately, this should lead to more accurate statements of expectability of frequency distributions and prediction of deterministic outcomes of non-random event structures.

 

I presuppose first a common hypervolumetric space within which any and all hypothetical event structures occur, and we can model these event structures mathematically within this space. I call this the space of total unification. All terms, variables and values within the framework of symbolic mathematics are set to occur within this single space. The entire space encompasses what I call total unity, and reflects the principle of unification and the Reality principle. Unity is depicted as absolute 1, and disunity is depicted as absolute zero.

In this event space, we may conceptualize it in n-order dimensions. Each dimension would have a total unity value of one plus its complement of negative one. Any form of possible event may be represented as occurring in this possibilistic space, and any kind of mathematical procedure can be represented within this space once necessary transformational operations have been performed.

I call this space the unification space, and it constitutes the basis for the procedural unification of all mathematical constructs with the framework of advanced metasystems. It is a space that is inherently differentiable. Its boundaries or limits can never be overpassed, hence functions can never be completely linear except in narrow intemediate ranges of its limits.

The space is reversible, such that unification at absolute 1 can be represented by the origin of the x-y axii, or else the origin can be used to represent absolute zero. It is useful to construe the space as reversible, because, I believe, it represents a fundamental complementarity of order and disorder. Ordered systems can be considered to be represented in the reversed direction, such that disorder occurs at the limits of the system. Ordered systems are seen from the "inside out" in the nonreversed view.

The D axis represents the temporal dynamic dimension of the system. It can be represented in 2nd or 3rd dimensional systems as reiterated diagrams that represent transformation. We can superimpose these transformations within the same space, especially if we are to consider it as presentable within the space of a computer screen.

We can arbitrarily represent the D dimension as either occurring cyclically in the spinning of the system in a clockwise direction, or as a straight line that suggests the temporal reiteration of the time-arrow. It's only constraint is that it is always unidirectional or only clockwise in orientation. We can specify a negative D dimension that would be represented by an arrow in the opposite direction or a counter-clockwise turn of the knob. This is a sense is most closely approximated by our imagination of history.

In this unification space, there is no need usually to represent Nth dimensions. I have set them to potentially rotate in a counter-clockwise direction, in order to fundamentally segregate them from the temporal dynamic.

We can imagine the entire universe flowing in a backward direction in some fundamental way, even though it appears to be moving forward temporally, or else moving or changing in some way that we do not comprehend or immediately apprehend. Nth-order dimensions exist only as hypothetical or possible dimensions, and suggest the cooccurrence of multiple realities. The actual existence of such realities is at this stage only conjectural.

This is not exactly the same notion as the contemporaneous existence of parallel universes. Such universes could be construed to exist within the same meta-temporal dimension in fact. This is analogous to the synchronous existence of two independent people, who nonetheless occupy the same temporal frame. Each additional dimension represents some strange form of reiteration of the lower dimensions, as a unified system. We cannot say what these dimensions might be.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


It can be clearly seen that in just this depiction of unification space, we have represented a great percentage of what appears to be most basic about any system. This suggests that functional integration of any system always occur at least in terms of such potential unification space.

The presupposition of this kind of space as the basis for all mathematical modeling brings up an important relationship between mathematics and graphic representation, or what I would call geometrical modeling. Any mathematically ordered system should be describable as an orthogonal translation in some form of geometricized space. This entails that if all science is mathematically expressible, it should also be geometrically describable.

Minimally speaking, though we may loose a great deal of information in the translation, we can depict any 3 dimensional system as a 2 dimensional topographical transformation. Within the context of this work, all diagrammatic representations are essentially 2-dimensional. Two dimensionality of a single construct is the minimal integrational requirement for any system. Less than this and we deal only with straight lines which are construed as fundamentally unrealistic and functionally useless to our system. Thus any system should be minimally representable in terms of plane geometry, though most systems can be projected and translated into terms of spherical geometry. Ideally, though, it is intended to be used to represent functional descriptions of curvilinear relationships that are based upon the application of analytical geometry.

Though we may represent Euclidean systems by this space, and it is itself essentially Euclidean, the basic requirements of all values within this space are that none can equal absolute 1 or 0. This sets the space to be essentially Non-euclidean in design. No line of any kind may actually pass the perimeter or boundary of the system. The boundary of the system is representable as either a perfect circle (3-Dimensional sphere) or as a square (or cube) that is either contained within the circle (or sphere) at its vertices or that contains it at its midpoints. The implications as to whether the square contains the circle or the circle contains the square is important I believe to our ability to represent with certainty any system, especially infinite or else infinitesimal systems. I will speculate at this point that the former condition represents the outer limit of uncertainty and the latter condition represents the inner limit of certainty, and the perimeter of the circle itself  represents the midpoint of no return or vanishing point at which certainty and uncertainty become essentially equal.

My first presupposition in the construction of this procedural system is to state that:

 

All possibly occurring values are presentable within this Borel unification field. Any scale or type of measurement may be defined in terms of this space.

 

Another way of looking at this is to state that whether we are dealing with a hypothetical space of some expected probabilities or frequency distribution patterns, or with an actual space of realized phenomenal event patterns, we are always also dealing with a finite sample that is somehow and in some way a part of a larger system of relations. It is in the largest sense infinite, and to some unknown extent prestructures and influences the system we are dealing with, real or ideal.

 

In order to relate this hypothetical unification space to mathematics in general and to our operational procedures in advanced systems sciences, it is necessary to define some standard terms of notation relevant to this system. X, Y, Z, D and Nth have already been utilized as reference terms naming the principal axii and dimensions of our unification space. Lower case x, y, z, d and nth will be used to represent any discrete instantaneous ratio values that are attacheable to any variable in a system.

I will represent absolute zero in the system as the stand O, with lower case "o" representing the concept and derivative value I call relative "o," which can be defined as:

 

O = o/O

 

I will use A to represent the value of absolute unity, or 1, in the system, and similarly, lower case "a" to represent relative achieved or instantanous unity within any given system.

I have reserved U to represent uncertainty and "u" relative uncertainty. I use S to represent some hypothetical original Start state or initial state, and "s" some actualized or infered beginning state. F is used to represent some hypothetical end state, or final state, and "f" is some actualized end state. S and F can also be used under subscripted conditions to represent "success" or "failure." P is used as a standard probability value associated with any possible event, and "p" the actual estimated probability of that event.

I have selected the variables J and H to represent, arbitrarily, any given global variable. J would be the primary variable, and H would be the third derivative associated with J. J would be a variable that is partially dependent upon H in its derivation. M stands for any numeric or measurable or parameter value that may be associated with either H or J in their derivation. Lower case h, j and m all represent instantaneous actualized derivative values of these systems.

A second presupposition to impose on this operational system is to state that:

 

Any discrete or nondiscrete variable or term is in fact always a trichotomous term that contains at least three intrinsic derivatives.

 

I presuppose in this the notion that for any given hypothetical system, we can define at least one state that is approximately discrete and that is at least partially determinable upon some "numerical scale" of measurement. Thus, any variable represents a complex polynomial that has mixed numeric-symbolic values. Symbolic values are nothing but labels, and in computers, also addresses for storage. The presupposition is that these entities can be mixed in a systematic way without the symbolic variable having to be ultimately determined numerically or parametrically, but can be relativistically determined in a discontinuous and non-parametric way by the principle of relational self-identity. All variables or terms always have at least some derivative numeric and non-numeric value, as well as some residual value that makes up the difference between the derivative and the ideal value.

 

1. Any term encompasses some value/variable and can be expressed as some systemic derivative.

 

Hence, for any given variable J, we can have at least the following variametic breakdown:

 

J = M(j) + H

 

Where N subsumes some complex derivative numerical value or weight assignable to (j) which is some particular instance or delimited set representing J and H is some other complex polynomial construct representing the differential between J and its actualized derivative M(j), hence:

 

H = J - M(j)

 

 

And 1 = (M(j) + H)/ J = (J - M(j))/H

 

If we hypothesize that X is also a similar complex polynomial, we get:

 

H = Mh(h) + Hm

 

Where Hm is some derivative nth value of the difference between H and M(h).

And Mh is some other numerical weight or value associated with the derivative of H.

 

This set of equations is meant to demonstrate only the complex algebraic and polynomial structure of symbolic mathematics that combines numeric and symbolic components in the same model. We can imagine that each variable and value is complexly determined by some other set of variables that are themselves complexly determined, and so on ad infinitum. It can be clearly seen that this kind of formula is applicable directly to the modeling of our operational systems developed in the Introduction, if we consider the J variable in the original formula to be some hypothetical state, and the M(j) + H to be the polynomal expansion or differentiation of this state in some subsequent or alternate state or in some theoretical construct of that state.

The original complex derivative polynomial M(j)  can be thought of from an artificial intelligence language standpoint as representing a basic CAR/CDR relation where the address points to some numeric value stored there. We can thus talk about intrinsic polynomial expansion such that M will be able to be designated by some set of subsets each with their own (j) values. H always stands for some complex set of relative residuals that are attached to the system by virtue of its relation to the hypothesized ideal system.

The point of symbolic mathematics is to emphasize that any discrete state or value is always representable as a complex derivative. There are no absolute values in this system, only values that are relative to the derivative functions. Thus symbolic mathematics is ultimately, as I conceive it, an entirely relative system. In this system, there is absolute Zero but it exists always as an ultimate end-state state that cannot be reached. Hence Zero is expressed by the same kind of equation as above in the following form:

 

O = M(o)  + H

 

where H = O - M(o)

 

Another presupposition of our operational system is to state that in any and every given system:

 

3. Any relation subsumes a range of varying relational determinations and can be expressed as some systemic alternative or set of systematic alternatives.

 

A relational value between points or sets always assumes a parenthetic embedding of these points or sets in some relatively differentiated way.

In mathematics, formulas normally circumscribe symbolic strings that are ordered systematically by means of statable and precisely ordered logical relations. These are considered to be "rules of composition" that order the symbols, usually in a manner that expresses an equation or else a transformation. Formulas are considered applicable to defined sets of points that are part of a population of possible points in reality. A point in this sense can be considered a particularized or particularistic event-interval or entity-interval that has some kind of relatively discontinuous quality tha t is considered elementary and fundamental within the general or standard frame of reference being employed. It implies among other things, a kind of "instantaneity" or instantaneousness of its phenomenal occurrence.

The test of a formula, for its generality, is that it is hypothetically applicable or relevant to any particular instance or point event of any class that the formula defines. Thus, all the points of the set should be, at least in theory, susceptible to the uniform application of the same formula or set of formulas that are contingent upon that definition of a set. In a sense, the formula therefore defines a hypothetical or ideal set of relatable and relatively equivalent points that is generalized on some level, and in the larger sense, is held to be universal if the validity of the formula is claimed to be universal.

It occurs in reality that exact equivalence cannot always be presumed for members of a common set, and that the formulaic operations, or "functions" that apply to the members of such a set apply in an exactly equal or undifferentiable manner to all members of the set. In the most ideal view of science, we would have a minimum paradigm of universal laws that underlie and explain all phenomena, and by deduction result in all other general and covering laws that are valid within the system. Science has not yet obtained that point of comprehensive integration or theoretical unification, and it will never reach the point where it will proffer unequivocally and with uncritical doubt or unquestionable certainty a paradigm of a few universal laws of reality underlying all sciences. But this does not mean that Science cannot or should not, at least in theoretical construction, progress toward such a goal. Neither does it mean that there is no place for differentiation of multiple scientific applications in reality, or that these themselves cannot be brought under a common umbrella of functional integration.

In this system, we have already the expression of the four basic arithmetic operations of addition/subtraction and multiplication/division. Addition and subtraction implies a system that is a composite of subsystems that are relatable in complex ways. Among other things, these relational signs imply an essential equivalence between members of a common set or sets. Thus addition and subtraction subsume, I believe, a variety of possible interactions between subsystems. The signs themselves, (+) or (-) would themselves take on alternative relational significances (conjunction, disjunction).

From a set theoretic standpoint, we can talk about union and intersection of sets, which implies conjunction and disjunction respectively. We can also talk about the multiplication of sets if we consider sets to stand for matrix structures.

In the foregoing basic equation, we may also express what can be called relative dependence/independence. We can say in the original form of the equation, that Z is a term that is relatively dependent upon X that is itself relatively independent in a complementary way, such that if we return to our third equation above:

 

1 = (M(j) + H)/ J = (J - M(j))/H

 

Then we get:

 

1 - H/J = N(j)/J

 

and

 

1 - M(j)/J = H/J

 

or

 

1 - M(J)/H = J/H

 

and

 

1 - J/H = M(j)/H

 

The arithmetic functions of multiplication and division express relational values of integration & distribution. Any implicit multiplication sign subsumes and implicit matrix in the formula, such that in the first equation above:

 

J = M(j) + H

 

The M x (j) would represent the dimensions of a martix subsumed by J and of which H is a differential derivative. This implicit matrix describes a range of alternative derivative values-variables that are encompassed internally by J, plus the range of other alternative derivative values subsumed by H that would itself be some matrix. Thus in the equation above M (j) comprises a size dimension of the intrinsic matrix that implicit to J subtract H.

I propose a set of transformational operations to be performed for all numerical values. I will call these relational numbers. Essentially, any discrete numerical value x will be derived as 1/x

 

In specifying a terminological basis for our metasystemic understanding, I believe it is necessary to answer the following basic questions:

 

What is a thing (or an entity, a part, an element, a component, an entity, a point, a state, an interval)?

What is a limit (or a boundary, or constraint)?

What is a relation (or an operator, a dependency, a function)?

What is a set (or a sample, a collection, a matrix, a group)?

What is a string (or a formula, a series, a vectorial)?

What is a system (or a machine, a mechanism)?

What is a framework (or a context)?

What is a size (or a dimension, a magnitude) ?

What is a space?

 

Science cannot descriptively account for all phenomena that occur in reality. Scientific knowledge can only represent a selective subset of the total reservoir of possible knowledge of reality, and yet that subset should at least in theory lead to and be able to account for all possible knowledge of reality. In the allocational tradeoffs between rational coherence of our explanation and empirical consistency of our observational descriptions, some middle ground has to be marked out. We can speak of the selective procedures that lead to the systematic simplification of scientific knowledge that represents a generalized substitution of phenomenological knowledge of reality. We seek this form of simplification in both our mathematical and linguistic-symbolic constructs.

It can be demonstrated that scientific praxis is based upon the superimposition of selective constraint upon our observations and our conclusions derived from our observations. This constraint is progressive in the sense that it leads to greater and greater resolution of the problems inherent to a scientific worldview--i.e., the systematic excoriation and explanation of the structural relations implicit to and deterministically accounting for the observed phenomenal patterns of nature.

If we could not selectively limit our knowledge base in rational and interesting ways, we could not have a science. Ultimately, we would like our scientific theories to be expressible in rather elegant and simple formulas or grand equations that can be expressed in abstract mathematical terms, or else in as few words as possible. But if we cannot achieve such elegance, especially in our depiction of inherently complex non-linear systems, which all naturally occurring systems can be demonstrated to be, our science is thereby not fundamentally weakened or rendered imperfect.

 

Symbolic calculus begins at the other end of the continuum of mathematical mechanics. The paradox of the comparision of abstract mathematical systems and natural language symbol systems is that mathematics enables us to express infinitudes and the notion of continuous variation with quite clear terms. Natural symbolism that is based on the positing of discontinous entities as if concrete makes the conceptioning of infinitudes and continuities between things seem inherently paradoxical and problematic. The obverse of this conditionality of our knowledge, which I take to be a form of linguistic relativity of different systems of discription, is that in some vague sense the detailed and accurate description of finite realities in mathematical terms becomes quickly overcomplicated. At the same time, natural language that is constrained by a sense of realism is very robust in this task, and in the task of articulating and describing inherently complex but dicontinuous systems.

I have proposed a kind of symbolic calculus as the complement of a mathematical mechanics. I would propose symbolic calculus as a kind of systematic integration of infinite and continuous change states in reality in terms of differential integration of discrete states that are defined symbolically in natural categorical terms. It is like narrative description that fosters the illusion of a motion-picture projector. If mathematical mechanics contributes uncertainty values and weights to our basic formulas, then symbolic calculus is intended to coordinate and make consistent the use of symbolic terms and definitional meanings in the articulation and elaboration of such formulas.

 

Systems Modeling

 

I propose metasystems theory as the basis for the integration of sciences upon a new level of articulation, or for the elucidation of what I would call meta-science, which would comprise the methodologies and knowledge stock of metasystems theory. The basis for metasystems theory and meta-science rests upon the inference that all things in reality are interconnected, however remotely, upon one level or another, and this interconnection between things is the basis for the integration of reality. It is the regular and recurrent nature of these interconnections, as well as the variant processes of change that occur within such interactions, that constitutes the basis of knowledge and metasystems science. The disparate nature of knowledge in different scientific domains has tended to occlude what can be considered an interdisciplinary approach to natural and real world problem sets in reality, much of which by nature demands input from a variety of different disciplines and perspectives. What is occluded I believe is not only a coherent and comprehensive worldview that can be called scientific, but also, and more important, a general operational approach to the understanding of reality that rests upon such comprehensiveness of perspective. If reality is an undichotomized whole, if real systems that occur within it happen in a naturally integrated manner, then it stands to reason that the knowledge systems we derive from and bring to bear upon this reality might be also similarly integrated and reflect this holism and comprehensiveness of perspective.

Science has proceeded upon foundations that have been empirically and methodologically strong, but theoretically and conceptually weak. It has been weakened in part by the lack of an overarching worldview that can be considered to be scientific. This central and general weakness pervades all fields of science, more or less. It is not so much the case that human beings are creatures with limited conceptual abilities, so much as it is the symbolic form and function that human conceptuality takes, and the inherent constraints placed upon conceptual systems by the fact of their symbolization. Symbolization involves more than metaphorical encapsulization or linguistic expression. It also entails a level of organic embodiment of the symbolisms such that they seem real. Such concretization of symbolizations tends to obscure the facticity of their abstract character and origin, the result of which are the perpetuation of certain kinds of informal fallacies of reason and undue and unself-critical attachment to received points of view. This creates the foundation, as Kuhn remarked, for making scientific though paradigmatic and for its constructive reification.

It comes to me as a paradox perhaps, that it is often the case that scholarship in the humanities and affiliated social sciences tends to achieve a much stronger conceptual foundation and prowess than in the sciences, though the former disciplines by their nature lack a strong empirical or methodological orientation that is comparable to the sciences.

It is the case as well that conceptual systems and the languages that encode these in the sciences tend toward a strong mathematical model that constrain conceptual abstraction in certain ways that lacks the flexibility that symbolization and a concern with a looser system logodaedaly permits.

The strength of conceptual development rests in several parameters:

 

1. A strong and detailed knowledge of facts and realities.

2. A critical and reflexive approach to all such knowledge.

3. The capacity to construct alternative systems to fit realities.

4. The critical development of such systems and their reality testing.

 

This approach is not fundamentally different from a general form of scientific method that incorporates heuristic problem solving and hypothetico-deductive experimentation. Indeed it is not, except that it tends, I believe, to be looser and more powerful on the abstract end of things than are the received realities of scientific theoretization.

My concern in the development of a general metasystems approach for the sciences is two-fold at least. First it is my desire to offer to the general sciences a means for developing conceptual systems that are at once stronger and more flexible both because they are less prone to the ideological and paradigmatic conundrums of their own facticity as constructions, and because they offer a more powerful means of conceptual construction than that afforded by a strict reliance upon mathematical description. Secondly, it is to provide for general science an actual set of conceptual constructions that stand as a set of alternative constructs for further development of ideas surrounding central issues in the sciences.

The Greek philosopher's realized a form of conceptual development that was far stronger and more powerful than any other period of human history. They used largely a critical approach to naturalistic observation, combined with a rigorous logic tied to language and a notion of "truth" that permitted them to construct models of their world that were far in advance of their actual technological state. We find in Leonardo da Vinci and in Albert Einstein a similar conceptual prowess of mind, and in Charles Darwin a realization of this prowess for the biological sciences.

I believe that it is Einstein's analogy of attempting to figure out the mechanisms of a watch by the external examination of a pocket-fob that provides us the clue to the understanding of a natural systems theoretic approach. In this, the role of both inductive inference in the face of empirical uncertainty, and hypothetico-deductivism in the midst of rational uncertainty, are critically important as a way for logically deriving and evaluation different kinds of conclusions.

Often it seems that ideas and theories surrounding reality are set in the stone of social consciousness, with a sense of commitment and investment into them that is all too humanly real. Conceptual systems are nothing but framing devices that can be applied for best fit to anything we want to use them for. They can be concocted and constructed for almost any context or situation that we wish to deal with. They permit insight, as beyond the face of the pocket-fob, and they permit understanding of hidden realities beyond the face that leads to a form of vision with the mind.

 

The methodological/operational basis of systems theory and method are the development of coherent representational models, in a variety of forms, that serve to accurately represent structural patterns, properties and principles of real systems. It is through the construction, development and refinement of representational models that we gain greater understanding of the structural patterns of systems of all kinds, and it is these models that are eventually applied in the development of new systems or in the progressive control of change and moderation of established systems.

All models are primarily conceptual and symbolic constructs in our minds, that are worked in some form in reality. The basis of all art and artistic creativity in human systems is in the development of representative models of reality, in some media or set of media, that are tied to conceptual models and frameworks of understanding or seeing the world.

Modeling and heuristic representation of real or ideal systems in the form of models provides an exploratory and experimental platform of the development of alternative systems by means that are relatively economical and efficacious in terms of cost of resources input into the creation of such systems, and the potential heuristic outcomes and benefits coming from such systems. Construction, prototyping and testing of models is a standard practice in most engineering efforts, and is always a precursor to the actual development of a real system.

Supercomputing has permitted a level of authentic virtual representation of extremely complex systems in a manner that is true and reliable, and has itself constituted a major technological advancement for the sciences, especially in those areas dealing with intrinsically complex data-sets and systems, like meteorology or ecology.

We may recognize certain design principles that might be appropriate to the construction and development of systems-based models relevant to our further understanding of real or ideal systems. We must distinguish in this regard between what can be referred to as general design principles that are appropriate across and for all kinds of systems, and what might be referred to as "system specific" or particular design principles that are appropriate of only a given kind or particular system to which we are referring.

Clearly it is the case that scientific domains have largely emerged around a distinctive body of knowledge and technical/technological methods used to access and augment this knowledge. We cannot conceive of the field of microbiology without a microscope, and we would be hard pressed to articulate a meaningful astronomy without access to even a rudimentary telescope. We must learn to recognize and appreciate the unique differences and specialized assets relevant to each field and domain of scientific research, and to consider these as a part of a larger collection and body of tools available to extend our knowledge of reality in systematic ways.

It is equally clear as well that principles and theoretical models that are appropriate for one area of knowledge or domain of scientific research, do not necessarily translate very well into any other areas or domain of scientific endeavor. The models that apply upon physical levels of stratification in natural systems are completely different than the systems-based models that apply upon biological or human systems levels.

All systems that we can think about are essentially knowledge systems that are symbolically constructed. The natural systems they represent are in and of themselves inert and incapable of self-reference or a sense of identity in the world. They are by themselves without the human intelligence component systems in which trees fall silently in a forest without notice and in which stars collide and burst on a regular, semi-random basis without further mention of the deed. We say that natural systems are implicit to the patterns in terms of the redundancy and stochastic structures that these patterns reveal to the human observer, or rather in terms of the information they yield upon observation. And no observation is or can be conducted in a completely naοve apperceptive sense without the automatic and built-in filtering processes that are the result of our conscious awareness and the conceptual models and understanding that we bring to our organization of experience and to our making sense of our awareness of the world. This is to be aware of the world, of the experience of reality, in terms fundamentally different in kind than that of a dog or a rat or a bird or a fish. It is to be not only consciously self-aware in the world, but reflexively so. It is to be aware not only of the world but of one's own awareness in that world, moment by moment, breath by breath. And we may say even when we are wide awake we are never fully or completely aware or conscious of our world, but we always perceive it, and conceive it, in a partial and partly distorted form. But however imperfect and incomplete, this kind of human awareness is enough to effect a kind of transcendence of existential context, of biological imperative, that I would call symbolic.

Thus all systems as knowledge systems are symbolic in organization and reflect the human being as both knower and articulator of knowledge in the world as well as the general life-situation of that human being. We like to call them rational but they are in fact as much rationalized and rationalizing as they are actually logical or factual about the world. They represent symbolic models we have of the world, or of parts of the world, and these models are built from parts and pieces we define and the relationships that we decide to interconnect the many pieces with.

It is our dilemma as human beings that we have no choice but to see the world in this way, with our symbolic models, in a manner that gives order to our relations and apprehensions about the world. These models are hardly static affairs, but are continuously changing and developing depending upon the changes in the relationships and patterns of response we maintain and are capable of carrying on with the world. Even if we attempt to deliberately suspend the influence of these models, they remain unconsciously embedded, not only in the subconscious background of our own brains, implicitly prestructuring how and even what our experiences with the world are, but they are also similarly embedded in the field of social relationships and the sense of order we bring into the world and shape the world by. Even if we could rid ourselves of our own preconceptions and biases in this regard, it proves virtually impossible to rid other people of theirs, especially if they are not even cognizant, much less willing, of a need to do so. And so when it eventually comes to pass that we must interact with such people, as life always constrains us to do, we are forced to reshape and yield our own models, however independently achieved, in order to do so.

In this way we must see all systems, as general, abstract theoretical systems, as knowledge systems that are representational and explanative in function, and as ultimately constrained by the symbolic-cognitive relativity of the human subject as central knower and articulator of these systems. This I call the anthropological relativity of all knowledge systems, and hence of all systems we are capable of knowing in however an objective, scientific manner.

 

First, Second, Third and Nth order Systems & Relational Theory

 

We refer to systems complexity in a relative sense of the position and level at which they occur in a larger metasystems framework--relative to encompassing systems these systems become subsystems, and they in turn become supersystems for the subsystems components that are encompassed within the boundaries of their definition. As we proceed from one level to the next, either ascending or descending in the hierarchy, it is clear that the order of complexity that we encompass in our metasystems framework increases exponentially. We cannot describe this exponential increase in clear and uncertain numerical terms. We cannot assume there to be a doubling, trebling or quadupling of complexity, thus we must leave the exponent as well as the main term as variables. We can write an expression for this exponential increase of complexity of order in a system in the following manner:

 

((X (x))y)z

 

We have to have a way of handling the terms, and we know from the logarithms that exponents are added together or multiplied. We can address any system in the following manner. For any given level, there is at least one higher order of generality or abstraction which should represent an order of magnitude of simplification. We would address this kind of model indicating ascending superordination and descending subordination in the following manner:

 

c(b(a)X (x)y)z

 

For the same level, there is always also one lower order of increasing differential specification which should represent a corresponding order of magnitude of complication. We can say as a rule in general metasystems that generalization implies specification, and simplification implies complication. As a consequence, we may identify 3, 5, 7 or even 9 or 11 orders of magnitude to comprehensive metasystems, and we find that expert knowledge sometimes attains these levels, at least descending if not always in the ascending comprehension of systems. We would thus identify a 3 level stratified system as a first order system, a five level stratified system as a second order system, a sevel level stratified system as a third order system, and so on.

We must understand that the variable terms themselves would represent what could be called complex non-linear instantaneous state-values. In other words, the central term X would denote in most natural systems not a single value or variable, but a set or matrix of multiple values or variables that would be related by some function. At the same time, it is assumed that the exponential values are related to the central variable in terms of some functional set of derivatives or integrals. We would state that the ascending terms would represent integrals of the system, and the descending terms would represent derivatives of the term--derivatives and integrals being defined in an instantaneous manner. In the application to a calculus of space-time dynamics, this model represents simultaneous systems that co-occur independently upon the same levels of stratification in accordance to the cosmological principle.

            I have coined the term relational theory in reference primarily to the understanding of the structure of human symbolic systems in order to get a handle on the structural aspects of naturally occurring metasystems. In relational systems it can be said that there are no apriori primes or starting values, but each term is definable in reference to some set of other terms within the system. There are thus no anchor points by which to ground the system or upon which to build the system. I believe metasystems as these naturally occur in reality represent such relational structures. We assign to these relational structures properties and values that are associated with a given level of specificity/generality in such a system, but we cannot designate in a non-arbitrary manner the upper or lower limits of such a system. I would state anthropologically, from the standpoint to the anthropology of knowledge and anthropological relativity, that this central paradox of reality is as much an artifact or consequence of our own knowledge or way of understanding reality, as it is anything intrinsic to reality itself. We are referring to a set of limiting conditions at which epistemological and metaphysical considerations converge. We do not say that this patterning is intrinsic to the order or patterning of reality in an of itself. We only infer this sense of order from our own knowledge frameworks and filters. Reality in and of itself, divorced from the experience of human knowledge, is none-self-aware. It can be said to contain information in an implicit and theoretical sense in its patterning and organizational structures that it assumes, but this patterning is stochastic and ultimately blind.

            The paradox in a physical sense though is that physical reality appears to reflect and embody this kind of relational patterning, and all physical aspects of reality can be said to constitute a grand relational meta-structure within which there are no fixed or predetermined coordinate reference systems. In other words, we must contend not only with the paradox of anthropological relativity of knowledge systems about reality, but we must contend as well with extrinsic limits to this knowledge in terms of the physical relativity of our systems of understanding and our capacity to observe naturally occurring systems without influencing these systems by means of our observation.

 

Equilibrium & Super-systems

 

It may be said that naturally occurring systems that exhibit redundant and consistent properties upon an organismic level attain a certain relative equilibrium of structure that permits us to refer to it as a system that is at least partially closed and partially self-determining. This equilibrium exists as a kind of dynamic balance that is maintained through self-organizational patterning with the frameworks in which the system exists in the first place.

Equilibrium can be said to be complex, dynamic and inherently underdetermined. In nature it is almost always non-linear in its patterning, and hence its equilibrium is used to account for its state-path trajectory, or developmental patterning, within a larger metasystemic context.

In short form we refer to a system of natural patterning as a "system" because it exhibits a relative structure that we associate with a set of properties that we refer to as emergent or synthetic to the system. When we analyze such a system, we break it down into its definitional or componential primes, which we treat as if given and non-relative, the emergent properties of the higher order suddenly disappear and we attempt to determine the network and transition structures that occur between the component parts without the benefit of a holistic integration of the system in terms of its transcendent properties. This represents a basic dilemma of scientific theory and explanation between analytical reductionism of the system into its component parts, and synthetic generalization of the interaction of the component parts in relation to the system as a whole integrity.

There are certainly properties that are evident upon one level that are not fully accountable for by the terms and relations of the underlying levels. Thus analytical explanation falls frequently short of its intended aim of full comprehension when it is done without the aid of synthetic theoretical hypothetization about the system as a whole and its metasystemic provenience in a larger scheme of things. This constitutes what I refer to as the scientific dialectic that is continuously switching back and forth between analytical explanation on the one hand and synthetic generalization on the other.

 

Emergent properties associated with metasystems are the consequence of the operation of the metasystem upon a transcendent level of integration. These properties depend greatly on the fidelity of order of the underlying system upon which the emergent properties are based. Emergent properties really can be seen only as the sensible qualities that are available to our knowledge at some level, by which we understand systems and their composition in the first place. Emergent properties can be seen as dependent upon the integrity of the underlying system, and these properties are those primarily that we attribute to such systems. Emergent properties define systems in a stratified sense and entail that a system is integrated to its surroundings in relation to other parallel systems, and form together what can be called a supersystem.

Nature is thus organized at multiple levels of integration, each level exhibiting its own independent sets of properties, and yet each based upon the systems resting beneath it. The stratification of nature was not achieved in an instant, and represents probably the result of a series of highly unlikely events, which can be described as an occurrence of change within a situational context. That this stratification exists is undeniable, and yet there is what can be considered to be a central dogma of this stratification, and this is that all systems tend toward increasing size and scale of complexity in their integration. This integration is achieved in a basically physical and mechanical model, at all levels. The organization of emergent properties at different levels, or their stratification and ranking between levels, is a derivative consequence of this physical integration of natural systems. It follows that the basis of scientific explanation is always physical, and this this explanation will grow increasingly general as we move from the physical to the higher emergent orders of natural systems. The degree of complexity of such systems can be seen to expand exponentially as well, such that we can consider the following kind of model:

 

…….(V3(W2(X1(Y00 )z)z) z)z)z……

 

where X is the starting point (zeroth entity), superscript z is the relative power or exponent of increased complexity, subscripts represent successive orders of levels, and …..VWX represent in creasing emergent properties associated with the subsystems.

 

All scientific explanation begins in and leads back to the explanation of the physical processes that underlie and account for the basic emergent properties that are associated with any given level of integration of reality. Secondarily, scientific explanation is concerned with the problem of the derivative or resultant systems that emerge or are developed as a result of the interactions of physical process in some kind of order.

            All naturally occurring systems exhibit emergent properties upon discrete levels of stratification, and there is no natural system that is not so endowed and that is fully comprehensible in a completely constitutive manner. The emergent properties of all natural systems are an indication of the fundamental relativities of such systems, both physically and anthropologically in the sense of our knowledge and understanding, and even observation of such systems.

Natural systems theory breaks down and stratifies reality in this manner into natural and logically ordered sets occurring upon different levels of superordination-subordination. In fact natural systems stratify in terms of a spectrum ranging from purely physical phenomena on one extreme to purely symbolic and metaphysical phenomena upon the other extreme, with biological systems ranging somewhere between these two extremes. We can range along this spectrum from one end to the other and notice discontinuity only in terms of the emergent properties that are associated with a particular level of the spectrum. If we sought a purely analytic approach, we would find for instance that this emergent discontinuity of systems breaks down and systems appear more or less continuously reducible in terms of components and components of components and so on ad infinitum.

We can say that the most comprehensive natural system is the physical system, and of the physical systems the most comprehensive is probably the fundamental unified field system that encompasses the total universe as a metastate and possibly multi-state system. At the same time, when it comes to the emergent properties of energy, of various forms of elemental matter, and of organic molecules, cells and biological systems, each of these is a sub-set of the larger and more basic system in which it rests. We arrive at human systems, which relate ultimately to other possible intelligent systems in the universe, at the other end of the extreme as a form of natural system that is capable of automaton self-awareness, or consciousness, and to some extent a measure of self-determination tat is relatively non-stochastic or non-random.

 

Abstract States & Natural Orders

The Systematics of Identity, Property, Relation & Inferential Structures

 

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.

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.

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 alternative 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.

 

 



[1] How a problem will be understood, or even what problem occurs, will be largely a condition of the frames of reference adopted by the problem solver. What may seem problematic about reality for one person may not be so problematic for another individual or group of people. We may therefore distinguish also between primary or direct problem sets that deal with immediate, instantaneous conditions of reality, and secondary or indirect or derivative problem sets that are the consequence of the differential or parallax of perception of primary problems or other secondary problems. I would also distinguish what I would refer to as "tertiary" problem sets that are distinguishable as "pseudo" problems or false problems that arise as the result of error of processing or recording, the transmission of misinformation, or erroneous apprehension of either direct or indirect problem sets.

 


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: 08/25/09