INTRODUCTION

In systems thinking we can name several basic kinds of abstract system, which is a general class of system that exists only or primarily in a noumenal or noetic manner, as an idea, rather than having any necessary phenomenal demonstration. We can identify, for instance, ideal or ideological systems, mathematical systems, hypothetical or theoretical systems, and imaginary or possible systems, and pure knowledge or conceptual systems, all of which are in a sense primarily or purely systems of conceptual abstraction, whether based or derived from empirical or objective reality or not.

Ideological systems are a class of abstract system that has important implications in all areas of human knowledge and endeavor. All scientific theories have a degree of ideology and ideological bias about them. This bias tends to appear beyond the horizon of our empirical knowledge, in the quest to answer fundamental, or perhaps, monumental problems beyond the true scope of the data. But ideological systems have a degree of arbitrariness about them that is of little interest or value to a general systems perspective except in as much as they constitute a kind of human system, a noetic system, that serves frequently to guide and influence human behavior. The kind of abstract system we are more concerned with is that kind of rational system that seems primarily guided by logic, and that has truth value either through internal coherence or external consistency, that seems to stand independently of our faith or commitment to sets of symbolic ideals.

The principle concern of science is in the theoretical comprehension and controllable application of knowledge to reality. It is a grand paradox that the foundation of science is rooted in a purely abstract system of mathematical language and logic that owes nothing of itself in any fundamental sense to the external world of phenomenal experience. At the heart of this paradox is something fundamental and quite interesting about our sense and sensibility of reality. There is a sense of absolute truth that rests at the base of our scientific knowledge and that drives this knowledge forward in its quest for both a greater sense of realism and a grander sense of rational understanding. It ultimately stems from the fact that on some most basic of all levels, we, with all our fanciful ideas and ideals, share the same realities, the same processes, the same structures, the same sense of order, as the physical universe and all possible realities that surround us. Ultimately, there is a convergence of what is real and what is true about reality, such that they become one and the same thing in terms of our systems of abstraction, as long as we remain faithful both to a sense of rational order and a sense of empirical consistency.

The beginning of advanced systems science is in the conceptualization of the abstract system that lies at the foundation of human knowledge and informs it upon every level and in all areas of its articulation. This abstract system is rooted in an alternative conceptualization of mathematics in both its pure and applied senses. We must ask basic questions of reality and of mathematics as something that is unusual in reality and the answers we give to these questions provide us a foundation for the development of our abstract conceptual systems.

The beginning of advanced systems science is therefore not a question we must ask of physical reality per se, as this is a naturally occurring system. It is rather what we must ask of total Reality, or of our sense of reality, as this is encompassed by our relative sphere of knowedge. It embraces what can be called "meta-physical" reality in that it includes both physical reality, as this is normally construed by natural science and our sense of perception, and our knowledge of reality that to some extent transcends physical reality. It comes to include at the same time, by its systematic extension, our understanding of physical reality in a larger set of knowledge systems.

Reality is encompassed by our knowledge, and includes all things known or knowable, however indirectly. It is only bounded by the unknowable. We cannot directly know what is unknowable. Whereas the work Natural Systems was rooted in a fundamental presupposition about nothingness in the world, this work is rooted in a similar, and in some ways, homologous presupposition about the ontological status of the unknown.

The unknown is a boundary around our knowledge, but as a boundary condition, it is itself an important part of our knowledge system. This follows from the first presupposition of the work in Natural Systems, that nothing cannot be known, and therefore there cannot be nothing because all things are known or are by definition knowable. Thus, whatever remains unknown remains part of a larger implicit system of unrealized knowledge that is, by its attachment to our knowledge of reality, an intrinsic and important part of that system. It is not unknown in any absolute sense of being unknowable, but only relatively unknown from the always limiting perspective that our own knowledge always presents to us and imposes upon our sense of reality. The philosophical paradox is that we can never be certain of what we do not know, and this sets the limits to our science in a way that nothing else can do.

Our sciences then become a matter of always attempting to test the limits of what we know by making forays, however blind and ill fated, into what we do not know. We begin these forays with only intuitive and vague questions. We try to do this more systematically by imposing arbitrary constraints upon our methods and our procedures, in order that we can approach the problem of the unknown by measurable degrees that we can easily deal with. Going slowly to the finish line is always better than racing headlong into oblivion. Our sense of reality has to be testable as a fundamental limiting condition of our scientific validation procedures. Scientific method can afford no leaps of faith, though scientific theory must make such leaps. Knowledge that cannot be tested in some way by means of our independent experience is knowledge that cannot ultimately be validated, and hence cannot be clearly segregated from what we do not and cannot know.

This brings up a grand paradox about our sense of reality and our place in reality. Our reality, as a sphere of knowledge bound by the unknown, always encompasses the physical reality that is its principal object of reference. But this sphere of knowledge can be said, at the same time, to be encompassed by this physical reality upon which the ability to know has been based. This is the grand paradox posed by the anthropological relativity of our knowledge, at which we are ourselves always and forever at the center of our sense of reality. It leads to another kind of relativity of knowledge, that I call metaphysical relativity of our ability to know, that says that while the unknown is always a diminishing domain on the horizon of our knowledge, our knowledge is always encompassed within this larger horizon and a subset of what remains unknown.[1]

The presumption underlying our definition of physical systems, at whatever level of our analysis and reference, is that this sense of reality is inherently mechanical in both abstract form and in its phenomenal expression. Science as a description of reality is therefore a fundamentally mechanistic view of the world, and this is particularly true if we add the constraint of referring to phenomenally occurring systems. Systems viewed as machines, either analytically or synthetically, and the machine model of systems is deployed both ways, remains a fundamental mechanistic view of reality.[2]

The questions we must ask is about our knowledge itself, assuming that this always embraces the principle of total reality defined by our comprehensive but always limited sphere of knowledge. It is about our ability, and inability, to know, and the structural patterning that knowledge itself takes independently of the naturally occurring systems that it becomes attached to.[3]

To a great extent, mathematics is articulated in a manner that does not ask or answer implicit philosophical questions about its own knowledge base or its ontological and metaphysical status as a knowledge system in reality. Mathematical signs are assigned and operations are performed on a deductive basis without making any deeper inquiry into the implications of the constructs involved. Inquiry in mathematics is usually not in terms of the epistemological foundations of its truth value, but in its ability to solve basic problems that are intrinsic to the knowledge itself, based on certain a priori precepts, such as deriving a formula for calculating all prime factors. But philosophers have involved themselves from the beginning in basic mathematical questions, and they have frequently made important contributions to mathematical knowledge by means of their inquiry into its foundations in reality and knowledge. This is just as scientists have both used mathematics and contributed themselves to the expansion of mathematics by means of their creative and constructive applications in theory and experiment.

The concern in the first part of this work is in the consideration of the fundamental relationships between the three fields of abstract inquiry into reality:

1. Philosophy, which concerns especially the status of our knowledge in reality,

2. Mathematics, which occurs as an independent and abstract knowledge system in reality, and which relies to some extent upon philosophical constructs,

3. Scientific systems of knowledge, which depends to a great extent upon validation and description procedures derived from mathematics and indirectly from philosophical conceptual systems, but which depends for its validation not on the internal criteria used in mathematical systems, but upon empirical substantiation.

It is not difficult to see how, in the history of all these areas of inquiry, the three fields are bound up with one another in very basic and important ways, such that progress in one area usually leads to advances in the other two areas as well. By means of progress achieved in any one of the areas, our sense of reality becomes somehow expanded by our enlarged horizon of knowledge. We embrace a wider region of the previously unknown, but we create a wider boundary of the still unknown, which surrounds us like a wall of unasked questions begging to be answered.

As an abstract system, we are therefore attempting to describe a model of the following kind stated in terms of the fundamental questions that are asked in each area:

I will call this the fundamental conceptual framework of reality, and for the interests of our advanced systems science, and I will make the following kinds of initial generalizations:

1. All reality is a unified system that is based upon abstract information that is intrinsic and implicit to the patterning of order in systems.

2. Abstract knowledge, in its purest form, is absolute (i.e., it is nonrelative to the subject knower, hence it is a priori to our realization of knowledge, which is always relative, and it will remain valid whether we exist to realize it or not).

3. As such, Abstract knowledge, in its purest form, does not change. It can be said to be perfect.

4. Our ability to know reality in any abstract (non-concrete, or conceptual) sense, is rooted in our ability to apprehend abstract information in either pure or applied senses, however imperfectly, through logically derived inference.

5. Abstract knowledge does not exist itself physically in reality, though physical manifestations in reality exemplify and represent abstract information.

6. Abstract reality exists only abstractly. "Order" is only inherent to order itself.

7. We can only know abstract reality indirectly by means of its physical or conceptual demonstration in our knowledge, whether this is pure or applied in terms of some mechanical system.

8. The validation of abstract knowledge is completely internal to itself in its own system of ordered relations, what I call an inference framework, and does not depend upon its empirical manifestion or external reference. Only science, which deals with naturally occurring phenomena, requires ultimately an empirical frame of inductive reference by which to achieve progress.

9. Abstract knowledge that is valid is objective and nonsubjective, though it can only be subjectively apprehended as such.

10. We can only know and validate abstract knowledge by means of hypothetical deductive inference. Our methods of deduction have been arbitrarily and unnecessarily constrained by overrestrictive versions of logical standards of truth.

11. Abstract truths, if they are valid, can never be contradicted in reality. Hence, empirical counterevidence is a means of falsification for untrue abstractions.

12. Though abstract reality is absolute and a priori, our ability to know it is always relative to our system of knowledge itself. Hence, the paradox of our knowledge is that it is absolutely relative to itself, and what is known absolutely is always relative to our knowledge of it.

These are somewhat radical and also probably unprovable assertions to make, especially considering the commitment of advanced systems science to an empirical and scientific view of the world. But they are made as a necessary leap of faith that is intended to give us some kind of foundation for our ability to know reality in some fundamental way, regardless of the relativities that seem to so inform reality at every level. It serves as a final anchor point and central fulcrum to the articulation of our advanced systems science in a broader sense. We plant the foundation of this field of inquiry in a framework that is ultimately unassailable except perhaps by means of blind faith and conviction alone. It is necessary to do this, I believe, if our advanced systems science is to have any nontrivial consequence in our world and in our reality.[4]

We can rest assured that there is absolute truth in the world, that two plus two always equals four, and not any other number, but we can never move past our fundamental sense of uncertainty about what we are observing and know in reality. We can say that our knowledge systems as abstract systems of conceptualization inform our ability to know and what we select to know about empirical reality, and that our phenomenal experience of empirical reality necessarily feeds back to test and challenge our fundamental presuppositions and propositions about our reality. This kind of critical feedback and epistemological dialectic between what we know ideally and what is known in reality and how it is known is a crucial part of the dialectic that informs our sciences.

We can say that though abstract systems are by tautological self-definition closed and ultimately independent in their validation to our experience of reality, they always inform and shape our experience of reality in basic ways, and are always paradoxically subject the verification of our sensory experience. In other words, they are no use to us whatsoever, at least as science, and not as simply closed ideology, if we do not have some kind of external reference-coordinate system within which to frame such abstract understanding and knowledge. If our abstract system cannot be used to validate our external knowledge of reality, it is either useless or false as a scientific system of knowledge.

An abstract system, to be functionally useful and true, must have some frame of reference and inference, whether this is external or internal to the system, by which its conceptual constructs can be validated or rendered invalid. This essentially relativizes all systems in some basic way, at least in terms of the criteria of essential anthropological and metaphysical relativity. In other words, systems of abstraction must somehow be plausible systems if they are to be believable, and if they are non-relative in some absolute sense, they must yet retain some means of validation that is independent from their conceptual construction but upon which their design and credibility may be ultimately based. If follows that if we are to devise an abstract system for advanced systems science, this must entail as well in its fundamental design a framework of coordinate reference and inference by which its constructs can be validated in some reasonable way.

It also follows that if we are to understand what is mathematical knowledge, we must also seek to understand forms of knowledge that are essentially non-mathematical by way of fundamental contrast and complementarity between different kinds of knowledge systems. It seems to me that the fundamental difference between these kinds of systems are that mathematical systems are constrained systems of signification, where as non-mathematical systems, particularly metaphysical systems of truth, are not constrained in the same basic ways, and occur as naturally defined systems of symbolization. Pure mathematical systems do not use nature or experience as its fundamental frame of reference. Its frame of reference is a completely abstract system of relations that occurs in terms of deductive logic and some non-arbitrary scale or scales of measurement. Natural symbolic systems ultimately depend upon some real frame of reference for their achieved realism, though they may adopt some arbitrary and relatively abstracted frame of reference as well, and though such frames of reference are still not constrained in the same way as are mathematical systems. They remain arbitrary in a fundamental sense and represent ideological or mythological systems that remain essentially non-mathematical though they may take on the guise of mathematical systematicity.

There are hypothetically abstract ideals or values that are essentially non-mathematical and yet which may be claimed by some to be absolute and by definition universal. Some candidates to this are "right" and "love" and "beauty"--all these ultimately moral valuations are not incontestable in the same way that mathematical systems can claim to be. We can find no metaethical definitions of right or love that are not somehow culturally or historically relative. "Truth" itself is perhaps the ultimate such ideal. Philosophy in a classical sense is based upon the abstract excoriation of "truth" which is often held to be noumenally a priori to its empirical manifestions. Absolute or non-relative truth remains a questionable issue.

We must understand that math is a language system with exact and precise denotation of its terms. We cannot have quantitatively exact equalities of operation or value when we deal with qualitatively abstract systems that are rooted in basic non-numerical symbolic language in which implication and connotation cannot be completely removed from the process of naming and definition. Even in the case of logical positivism and logical empiricism, we are hard pressed to prove our initial presuppositions.

Mathematical systems that are correct in their logic are inarguably true, and this is a form of mathematical correctness that embodies a kind of non-relative truth. We can have non-relative mathematical truth in the sense of "correctness" which is ultimately non-arbitrary. I believe, if we attach ideal or abstract systems of generalization to natural systems then we can also derive another kind of non-relative truth that is essentially scientific and "applied" abstraction in nature. This kind of applied truth also has the sense of "correctness" in that it involves puzzle-solutions to problems that have finite and definite kinds of solutions closely delimited and restricted by careful measurement and observation. Mathematics often enters into this kind of limited truth formulation, but always as an applied rather than a pure system.

I will speculate as well upon the hypothetical possibility of a third form of non-relative truth. I believe, lacking any clearer terms, I will call it metaphysical truth that is inherent to the structure of knowledge and our knowledgeability itself as somehow something a priori and absolute. I believe that knowledge as a general system of abstraction of reality has its own system of order that must be founded upon some sense of absolute truth that serves as its fundamental frame of reference and inference. This is neither empirically derived from the observation of natural phenomena nor is it necessarily derived by the same logical and constructive means by which mathematical truths are made known and gain expression.[5]

I would see it rather as some sense of an ideally "perfect" system that is always abstractly and at least implicitly contrasted with real and by definition always "imperfect" systems. The systematics derived from such a truth-system would be the measurement of the discrepancy between ideal and real states exhibited between perfect and imperfect systems, and would implicitly entail as well therefore a precise definition of what a sense of "perfection" would be. In a scientific and mechanistic sense, we have only different kinds of perpetual motion machines as representing somehow perfect systems (i.e., systems of perfect efficiency without the condition of entropy). We know that such systems are impossible, but we also know that they exist as abstract and ideal conceptions that help us to think about such troubling realities like relative efficiency and entropy.

Its would go something like this: Given such and such variables and initial values, what is the best possible system that can be derived for such limiting factors. These factors do not have to be necessarily quantitatively defined. They can include qualitative definitions, as long as we are clear and concise about our definitions in a manner that leaves little room for doubt or "play."

I have undertaken therefore to get to the core of an abstract system that underlies all systems. Hence it is in this sense meta-systemic in being both a system in the most abstract sense possible and about all systems in the widest possible way or largest applied sense of the term. I would say that it is both mathematical and non-mathematical at the same time, and thus we need to enlarge our sense of understanding the implications of what mathematical and non-mathematical knowledge systems, to the extent that they influence our meta-systemic comprehension of any and all systems. Therefore, I believe that in order to outline and detail the essential abstract features of a meta-ystem, we must approach it from all angles at the same time, as both a mathematical, a scientific, and a philosophical or metaphysical system of abstraction.[6]

If comprehensiveness is the measure of the universality of the system, then systems of abstraction stand as the most comprehensive, hence universal, at least in potential, of all kinds of systems we can think of. But what are they really? Philosophies East and West, Ideologies of any collective mind, mythologies, human realities, worldviews. Of all the possible alternative systems of symbolic abstraction we might consider, perhaps pure mathematics stand as the truest and best example of a universal abstract system. And if mathematics is held to be the language of science, we must caution with a mindful proviso that it is a language of some of the sciences, but not all. Even those knowledge systems like chemistry and physics that are endowed with absolute equations, these equations reflect principles, processes and properties occurring in nature, relative to the systems they occur within. But these mathematical systems of abstraction are all symbolic systems, of our own imagination and construction. They share a common language, a symbolic language with the capacity to express representative ideas; notions that stand for realities that are only imaginary, remote, not connected to the realia of lived experience.

I would argue that abstract systems are a special case of applied human system that are derivative of human symbolic capacities. Many would want to debate this assertion. If we look for the reasons for the origins of the idea of the triangle, or for human language, we can venture many "just so" stories. Some have effectively argued that abstract ideas like the triangle, nearly perfect in form, arose from abstraction from everyday experience in dealing with things shaped or modeled in a triangular form. Others have argued just as effectively that the triangle was possibly an "a priori" archetype, a noumenal idea without any necessary phenomenal form, itself an embodiment of the principle of perfection.

This dilemma of identifying and classifying the objective provenience of abstract systems underlies a central sense of dilemma in all theorization about systems. Theories, however imperfect and human-made, embody logical relations and rational arguments that appeal to reason, not emotion. Systems at all levels appear so wonderfully organized and regulated, that there must be some prior sense of order or design upon which they are based. And in this we face perhaps the ultimate dilemma of the anthropological relativity of our own situation in reality, for though our logic and thought systems are a product of our symbolic awareness, our minds are a product of the organization of our brains, an evolutionary product of nature, with a biological seat in cellular organization and function.

Truly then it can be said that the source of our ability to reason in a logical manner derives from the logical sense of order by which our brains developed in the first place. To claim that symbolic awareness is an emergent property of the system created in the complexity of the integration of our brain does not go far enough I believe to unravel this sense of paradox. I say this because we cannot step fully outside of our own heads, either our own brains or our sense of mind and abstraction that we bring to the experience of reality.

Our own intelligence that is rooted in the wonderful neural organization of our small brains, constitutes a kind of logical framework upon which abstraction can be based. I would argue though that while logical relations are possible in the neural pathways that lead to human consciousness and the seat of reason, they are not obligatory, especially not without linguistic coda with associated relational values by which to precipitate these relations in some reified form.

This may be the critical difference, especially in what separates a literate mind in a cultured context from an illiterate mind in semi-feral context. The mind, it seems, does not function bereft of a symbolic metasystems context, socially derived through communication, primarily linguistic, but symbolic nonetheless in form and function, by which to organize its thoughts and by which to order relations between things in some manner that can be then called "objective" in the key sense of it being inter-subjectively shared and available to people independent of their own ideation.

If it seems to us that the most universal systems are de facto abstract systems, this is both true and utterly wrong at the same time. We have really a kind of Goeddelian dilemma working upon us that something like the Cretan liar, in being proved true, proves itself false at the same moment. We are forever in quest of the most universal system of abstraction by which to frame all relations and things in reality. It is like our Holy Grail for which we must endlessless quest.

The correspondence of relationships, found in all comparison, in all identification, in all relationships, may be the basis for a very common system of abstraction, shared by any and every particular symbolic system that we've constructed. But how we go about drawing out and upon this system varies tremendously and leads to all kinds of different results. It is not enough either to merely say that such systems are "symbolic" and leave it at that without defining concisely how they are symbolic, how they function in a symbolic manner, and how this is important to systems of abstraction.

A fairly accurate and probably universal anthropological model of just this kind of process of symbolization exists, but might not address the crux of the issue, which really is to find a single system of truth, that is good for all kinds of possible situations and all alterantive manners of truth as they occur in reality or might possibly occur. I do not claim by this an upper case "Truth" in the Platonic sense of rational idealism, neither do I claim the anthropological absence of many different lower-case "truths" all competing with one another in an equal and relative playing field.[7]

Anyone acquainted with biology knows that mathematics is not always the most suitable language for descriptive explanation of biological patterns, even if many models are eventually reduced to mathematical or at least statistical (applied mathematical) terms. As an anthropologist who accepts a definition of anthropology as a clear science, one can hold even less to a strict mathematical language by which to define human behavior and cultural phenomena. This does not mean that the theories, insights and understanding that the anthropologist gains from research, largely naturalistic research in cultural settings, is any less "scientific" or less scientifically significant or interesting than that knowledge gleaned in a particle accelerator laboratory. It only means that we usually must frame our science with the luxury of clear logical and mathematical relationships occurring.[8]

The problem we seem to have in general is that we never seem to know what is possible or not until we get there. We know it is probably impossible to chop a persons head off and then to successful re-attach it so that the person will continue living. "Scientists" in Russia claimed to have reattached severed dogs heads which survived but a very short time. There are some kinds of things I think just won't happen, like reviving the dead, or using a transporter system to beam people from one place to another, or cryogenically suspending a body and then reviving it a long time later. We may never reach Warp 9, but it is fun to imagine if we could. We call statistics, a kind of applied system of abstraction, to our rescue to help determine what is likely from unlikely, but it is very difficult to clearly tell what is possible from what is ultimately impossible.[9]

I like to think dealing generally and abstractly with philosophy, mathematics and possibility has some grander and more significant connection. This would be the place to spell that connection out. Science was born of philosophy, and mathematics was a product mainly of philosophical inquiry. But science grew up, and went its own way, and a divorce occurred and no longer needed the archaisms of the old school that referred to scientists as natural philosophers, born in scholastic ignorance.

The two Academic cultures, the hard Sciences and the soft-headed Humanities, emerged, and once sundered, never the twain shall meet again. But in this separation, something was lost on both parts, something vital to the whole enterprise of knowledge and learning, especially when this concerned basic problems of truth and reality, which, in my big but insignificant book at least, are one and the same. Scientists usually make lousy philosophers, and philosophers probably make poor scientists, at least in the unquestioning manner that they are trained today. The most that stands between them is a 101 introduction to scientific method--glossed over by the scientists, hacked to death by Popperian and Lakatosian philosophers. Everyone in all the sciences, from Grade Four up to Post-post Doctoral specialization, preaches the mantra of the scientific method, but very few bother to question or deeply understand its fully implications. Scientists would be loathe to admit the softer side, for instance, of mathematics, or the inherent uncertainties underlying many of their taken for granted truths and theories. Philosophers would be quick to jump to the defense of their pet assertions about the paradigmatic structure of knowledge or the constructive or structural aspects of ideas, without worrying about their linkages to facts or experienced realities.

Primarily, I am interested in a philosophy of systems, and mainly, scientific systems, but this can just as easily be reversed, as I am searching for a systematic science of philosophy as well as a systematic philosophy of science. I am in fact looking for a common ground in all three areas, and then I want to extend this, if possible, to improved comprehension that the role that mathematical modeling may play in the symbolic representation of ideas, and, even more, in the application of symbolic systems to the descriptive explanation of reality.[10]

I have sought to define an abstract system of specialized and general mathematics as this is shaped within the systems framework. I have posited the central notion of the metasystem as an abstract hypothetical construct that can be used to model alternative systems in theory. Each instance of a hypothetical metasystem can be said to represent a generalized and integrated theory about the system, hence each such alternative theory would have a minimal number of component dimensions of description that would be considered sufficient to that system. These dimensions are derived from the application of alternative mathematical models and theories to the system. The description of the system would be deductively derivative from the theoretic construction of the metasystem. In theory, the correct hypothetical system would yield predictable values as compared to the patterning of the actual systems upon which its description is based or to which it is referred. In this case, it must be seen that two or more hypothetical constructions of the same metasystem might yield correct values for any test case. Competition between alternative constructions should, in time and through testing, yield the correct model, or what we might say, the model that best fits, parsimoniously speaking.

This approach serves to integrate various and diverse theoretical aspects of mathematics under a common operational umbrella, in a unified systems framework. It entails also some new and interesting permutations of mathematical theories in a number of areas. The unification of various theoretical constructs within a metasystems model entails that the metasystems as a model would be part of a larger theoretical paradigm about such metasystems. Such a paradigm would in time, through continuous refinement by the inclusion of new metasystem constructs, lead to a universal comprehension of advanced systems theory that would embrace in theory most possible systems that are known, and many that are yet unknown. Such a paradigm would as well come to encompass a growing set of alternative theoretical constructs that would also become improved in their generality and refinement of applicability.

At the same time, the concept of the metasystem provides a theoretical and metaphysical framework for the definition and symbolic description of alternative real systems that is basically non-mathematical or semi-mathematical in character. It allows us to develop a common parlance about which to describe alternative kinds of systems in different frameworks, and by which to coordinate the understanding of different kinds of systems within a common theoretical framework.

In this regard, I propose a form of mathematics that applies to the abstract description and analysis of systems at any level or in any form of expression. Such an approach is also, I believe, to some extent meta-mathematical in the sense that it is a mathematical system of systems, and hence its constructs and concepts embody what is basic about all systems that it seeks to describe, including mathematical systems themselves.

The point of departure for this approach is to posit the existence of an ideal abstract system that has the minimal structural features underlying any and every possible system of a general kind. I call this a general thereotical model of the meta-system. The study of the possible state permutations that such an abstract mathematical meta-system can undertake I have called "structural metasystematics."

Underlying this approach is an implicit argument that mathematical systems of systems are in some as yet unspecified way an ideal abstract system of conceptualization that meets all the requirements of pure mathematical systems. This underlying system is not just mathematically abstract, but also metaphysically universal and scientifically general.

It follows from the fact that systems are universally observable to underlie any complex pattern of order in reality, and that all systems are at least hypothetically interconnected in some minimally defining sense, however indirectly. It also follows from this that we can hypothesize that such a meta-mathematical system of systems should provide the basis for the integration of mathematical systems, as well as for the systematic and structural-functional integration of other abstract symbolic systems that are essentially non-mathematical.

We must ask therefore, from a mathematical point of view, what is a system in the abstract sense of the term, and how can we define and describe it using mathematically appropriate formulas?

The mechanical model of a machine is important to this abstract conceptioning of metasystems. A machine by definition is an integrated system of component parts that co-operate to achieve some form of energy exchange or transfer in expectable or predictable ways. Machines generally operate over a period of time, or in an interval duration, during which they exhibit certain state transitions that characterize the system functionally and operationally.

The classical conceptioning of a machine is of course a fairly linear model occurring in some kind of classical Euclidean space-time. Though a classical mechanical model is useful as a starting point to the basic understanding of all machines, it is inherently incomplete when we are dealing with complex second order machine systems that are essentially non-linear in their characteristic design.

If we take a simple machine, for instance a screw, we can understand that the thread of a screw imparts special properties to the screw that makes it a machine that is unlike a nail or some other kind of fastener. The screw can be said to have a specially designed characteristic, or trait, like its thread, that imparts a special set of properties to it, like twist. A screw functions in a special and particular way that characterizes it as a typical simple machine. A spring is another similar simple machine. A helical coil spring has certain elastic properties that allow it to be compressed or stretched, and that permit a form of resistance resulting in predictable functions.

In order to get at the nature of a simple machine, we must ask what components or characteristics of this machine impart to it properties that separate it functionally from other types of devices, for instance the minimum features of contrast between a screw and a straight nail, or between a spring and a simple twisted string. We can say that even a rope that has twisted fibers bundled in some complex way about a central axis, or else woven in a sophisticated way, represents a similar kind of complex machine, as might be a nail that has a shaft that's twisted for superior fastening power.

I will go back to Archimede's screw and speculate that all classical machines have at least one axis about which a twist or turning characteristic is defined. This axis may be orthogonally projected as a central pivot point or fulcrum about which a lever or rocker mechanism operates. The turning action, however it is defined, imparts a fundamental non-linear periodicity of a machine by which the machine gains its characteristic properties. We can see that a bow and arrow is not obviously a machine, unless we understand that the arrow is launched from a central fulcrum of a shaft that has springlike qualities.

Generally, we can say that all systems, as some kind of machine, exhibit a defining set of determinants and dimensions that are functionally interdependent, and that are structurally and uniquely characteristic of that system, or kind or type or class of system. These determinants and dimensions in their operational organization, impart structural order, hence "information" to the system. Furthermore, in first order classical systems, these dimensions and determinants tend to be discrete and measurable on some linear scale. In second order non-standard systems, these dimensions tend to be continuous and nonlinear, hence difficult or impossible to measure in some finite and regular interval scale.

I will also claim that any system exhibits an integral design that is characteristic and unique to that system. We can refer to a form of structural relativity that the function and organization is characteristic to a particular system or kind of system, and hence any system has its own internal frame of reference to which its states and values are relative.

Any system also functions as a kind of device, a device being defined as some kind of machine or mechanism that functions in some particular way. We can refer to functional relativity of any system as being the characteristic patterns of behavior of a system depending upon its internal organizatio as a device.

Finally, all systems maintain what can be called a developmental pattern or cycle that defines the system as a series of state-stages over time. All systems exhibit an inherent dynamic pattern of change or fluctuation of states.

I would say that all systems by definition have some finite set of characteristic values that are part of the initial start state or entrance state into the system. We can call these input values, our what I might coin as "in-go" values. Each system therefore has some predefined initial start state determined by this input set. Systems therefore also have some final stop state, at which point their final "output" values or "outcomes" can be stated. Inputs and outputs can occur periodically, randomly or continuously within a system, and serve to connect the system semi-determinisitically with other systems within a meta-systemic framework. Inputs and outputs, or in-go's and outcomes, can be said to be the value determinants of the system that are independent (initial) or derivative (final) factors that relate the system to a larger meta-systemic context containing other systems.

Defining the positions where inputs and outputs are expected to occur within any given system gives us a picture of the points of articulation of the system within a larger framework of alternative systems. Any other system is an alternative system to the one in question that we are directly observing. All epi-phenomenal patterning found in nature are basically the outputs of some systemic patterning underlying the structural ordering of this phenomena in some quasi-determined manner. They are frequently the outcomes of the interdependent operation of multiple systems simultaneously. These same outputs either expire and have no further influence within any system, and hence are truly epiphenomenal, or in some way become subsequent inputs back into the system on some other level or another.

How we organize and relate our output-inputs and how we express these as discrete values and how they are changed as a result of their functioning within a system, has great importance to our understanding of systems. For naturally occurring systems, and for all real systems, we have no other way of knowing their structure except through the patterning and organization of relationship between these sets of values and how they change over time relative to one another. The structure can be said to be always implicit to these patterns of phenomena, and our systems of generalization are means by which to render the structural patterning as explicit as possible.

It is clear that a meta-systemic comprehension of reality attempts to derive an abstract and generally valid conceptualization of the sense of structure that is at least implicit to any and all systems. All systems can be said to be internally ordered and functionally integrated in some minimal sense that serves to define these and set them apart as systems. A meta-systemic system of generalization about all systems must seek to apprehend and concisely explain what is considered to be universal and most basic to any system.

1. Mathematically, we can identify at least one central function that serves as an internal point of reference about which a system is articulated in a directive manner. In this sense, a system in a minimal sense can be said to be internally relative to some central reference point that defines the main directional property of the system, and which constrains the system such that no other kinds of motions or interactions are permitted. This central "fulcrum" can be referred to as the determining control factor, or the main controlling determinant of the system. Such controlling determinants of any system are effectively scalar determinants.

2. Any system is also, in the classical sense of a machine, a singular entity and therefore has finite properties. All classical systems therefore are finite and bounded in some way that sets the constraining factors of the system, as well as the size of the system. We can call this the limitsor limiting factors of the system.

3. Any conventional system, as a finite machine, can also be said to have some sense of size about it that is in some way measurable or determinable.

4. Any system can also be said to exhibit a static order and shape about it, an internal sense of organization that is unique to that type of system.

5. Systems, as conventional machines, can be ranked as well on the order of their extrinsic scale and their intrinsic complexity.

6. All machines, as a result of their design constraints, perform some kind of action or function in a non-static sense.

7. We know some other characteristics based upon experience at least. We can say that a real machine is one that exists in reality, whereas an ideal machine is one that is only abstract. All real machines approximate imperfectly ideal machines, and approach the ideal machine by diminishing degrees. Thus for any given kind of machine of some kind of design, there is some hypothetical ideal that can be defined for that machine of which each example is but an imperfect approximation. In a metaphysical sense at least, we can speculate upon a hypothetical perfect machine of a given type, however specified.

8. We can specify at least one state or a set of states for any given system, that are unique to that system or type of system, however complex. The states exhibited by a system are unique to that system and functionally defining of that system.

9. All real systems therefore exist in some finite condition that can be said to be instantaneous and in which it exhibits some state-trajectory that defines its characteristic direction of transition or state-change. All ideal systems can be said to exist in some definable condition that is hypothetically instantaneous.

10. The same or similar systems that recur over time or across space simultaneously can be said to form sets of multiple systems. Real sets of systems can be said to be imperfectly equivalent such that they comprise a normal range of variation. Ideal sets of multiple systems can be said to be perfectly equivalent to one another and hence exibihit no normal variation.

If ideal sets of multiple systems are variants of a single system, then they comprise multiple possible sets of alternate systems, and these can be hypothetically said to exhibit an idealized normal range of variation. These sets are real sets that occur in some relatively non-abstract state. The degree of generalizability of such systems is extendable almost infinitely based upon the relative particularity of any and every included subsystem or member system.

Natural sets of systems can furthermore be defined as real sets of systems that have no non-relative ideal standard and that comprise a range of variation that is both normal and abnormal. They are underdetermined systems in which some form of nonlinear and random variation normally occurs as part of the system.

For metasystems from an abstract perspective, we can speculate on the preceding pattern of order. We can speculate that in an abstract sense, all systems are generalizable. But systems vary considerably to the extent to which the can be generalized in a relative or non-relative way. Perfectly abstract systems, for instance, mathematics itself is a non-relative system of abstract generalization. Naturally occurring systems are scientifically generalizable systems, but they are essentially relative systems that lack any ideally defining archetypes. We may thus define a continuum of relativity along which any abstract system may be placed. We may also say that in the application of mathematics to the description of naturally occurring system, mathematical descriptions must take increasingly relativistic frames of reference into account.

From this, we can speculate that mathematically, scientifically and metaphysically, a meta-system is a system of sets of system, or generalized hypothetical systems that can be derived from either real or idealized instances of systems. Thus we are capable of moving from an abstract description of a single simple system to a more sophisticated and elegant general description of alternate, complex and multiple systems as they occur either realistically or hypothetically. We can represent hypothetical meta-systems statistically and mathematically in ways that are precise and valid. But we depend at least upon the principle of duration and state alternation or variation of systems, or what might be called the intrinsic multiplicity and variability of systems, in order to derive a meta-systemic understanding. In doing so, we must deposite key systems variables that serve as state or condition descriptors and boundary parameters to the operational functioning and integrational equilibrium of systems.

Naturally occurring systems therefore represent self-organizing machines that are usually quite complex and chaotic and hence rarely ever closely approximate to some idealized system. Such natural systems incorporate complex relations and chaos into their design patterns and state transition pathways to such an extent that we cannot discount the chaotic epiphenomena, the variability and noise, as extrinsic to the system. The natural system itself comes to incorporate entropy and disorder as being on some levels intrinsic to its sense of patterning and order. Thus, I believe that in natural systems, we cannot hypothesize some ideal state in exactly the same way that we can for, say, artificial systems like a stop-watch built on some idealized design in which we normally try to minimize or systematically exclude through precision all kinds of non-deterministic influence. A perfect machine would be an ideal self-winding pocket-fob, that after its first winding, would never need rewinding and would keep perfect time forever, without running down or needing readjustment.

We cannot say in some nonrelative way what an ideal tiger should look like in the same way we might hypothesize a perfect equilateral triangle or a perfect screw with such and such dimensions. Perhaps if we were to judge tiger breeds, we could arbitarily impose some conventional standards upon tiger-traits. We do this with all kinds of domestic breeds. What we have are only numerous evolutionary instances of various tigers, more-or-less similar and yet all different to varying degrees. By natural standards, the tiger that survives to propagate successfully best approximates some ideal standard, but this is itself a purely stochastic mechanism based mostly upon chance.

We can hypothesize that at any one time, there is an instantaneous if complex range of "tigerness" that is represented by the total range of variation of all extant occurring tigers. This is about the closest we can come to understanding tigerness, unless we are to impose some kind of statistical measures of populational parameters upon our definition of tigerness as an abstract system.

It appears therefore that in our accounting of metasystems mathematically, scientifically and metaphysically, we must also account for the problem of relativity of knowledge in some systematic way. I identify a kind of "metaphysical relativity" of general and abstract systems as informing out knowledgability about abstract systems, or our ability to know them in some ideal and perfect way. This forms an inherent horizon of abstraction in our knowledge, and is complementary to the forms of physical and anthropological relativity that I have identified previously. I have already referred to forms of structural and functional relativity of systems that is intrinsic to systems. Systems as they occur in reality tend to be unique, and the defining characteristics of systems tend to be specific to an internal frame of reference within the system. In other words, all systems define to some minimal extent their own independent states, and their states are largely dependent upon its internal conditions.

Ideally abstract systems, as for instance those evident in pure mathematics, are said to be non-relative in the sense that their standards for validation are wholly internal, being an internalized frame of reference that requires not other external reference point. At the other end of the continuum, naturally occurring systems, which are the applied problematics of science, are said to be completely relevant to the external frame of reference in which they occur. They cannot be validated by internal inference alone. Abstraction in science permits a degree of generalization, but this generalization is always subject to measures of its consistency with its external frame of reference. We can say in another sense, that in purely abstract mathematical systems, validity is intrinsically self-evident, hence always explicit to the generalization itself, while in natural scientific systems of generalization, validation is always implicit to the phenomenal patterning from which it derived and to which it refers.

A related consideration of this aspect of the metaphysical relativity of abstract systems is the degree to which generalized systems can be said to be determined. In general, a non-relative system can be said to be a completely self-determining system, hence a total deterministic one. On the otherhand, natural systems that are said to be relative, can be said to be relatively undetermined systems, their structure being at least partially determined by the contexts of their occurrence.

It is evident above that natural scientific systems of generalization and pure mathematical systems of abstraction are mutually exclusive sets. They can be said only to possibly intersect at their boundary, but they form sets of "truths" that are in a fundamental sense different from one another. This is not to say that mathematics cannot be applied to science,as it regularly is, or that science cannot contribute to pure mathematical knowledge, which it often has done.

The question of metaphysical relativity of metasystems brings up the question of the problem of unification of metasystems, or the possibility of representing all systems in a universally comprehensive sense as a single integrated system. While it is evident and more useful to approach systems from the standpoint of multiple subsystems, it is also necessary to at least consider the challenge of conceptualizing a grandly unified system of all systems.

The problem of metaphysical relativity also brings up a related issue that leads us back to basic aspects of knowledge. Relativity of knowledge always implies an inherent amount of uncertainty about the validity of what we know, and it therefore also implies a sense of unknown. The unknown surrounds what we know with a wall of reflexive uncertainty. We can say that what we know absolutely, like two plus two equals four, we know with sufficient certainty. We cannot say this in a non-relative sense regarding our generalizations about science. We can form universal generalizations that we know with relatively high degress of certainty, but there is always a residuum of critical doubt like a background shadow cast by gigantic cloud over our scientific knowledge.

This is not a bad thing as it drives scientists forward in quest of ever greater certitude about their knowledge. But it does entail that scientific knowledge is always bound by a greater domain of the unknown. The unknown always sets a limit to our knowledge. This same domain of the unknown does not necessarily exist as such in the realm of pure mathematical abstraction. Of course, there are probably many mathematical systems that have not yet been worked out or described, that remain basically unknown to us at this time. But what we do know about mathematics cannot be fundamentally changed or weakened by what we don't know in the same way that the discovery of new evidence in nature or of a new scientific theory might undermine or revise in a revolutionary way what we do know about the natural universe.

Based on what we have said so far, it can be seen that metasystems as a science and as a basis for our advanced systems applications, is itself primarily about abstract generalization and the metaphysical limitations that govern our ability to generalize about what we know. We can say that all generalizations require some frame of reference or inference by which they can be validated as part of some larger system of relations. This frame of reference determines the relativity of the system of generalization.

It is a virtue of systems that they are in some limited sense always, by definition as systems, internally relative to themselves. Another way of saying this is that systems of generalization are never completely relative or nonrelative, but are metaphysically complementary in this regard. Even strongly relativistic systems exhibit yet some minimal degree of non-relativity about them, from the standpoint at least of their internal structural coherence. What we do generalize and know about them obtains only within an ideal and absolute set of conditions, however partial this may really be in real or natural systems.

Hope for science remains in the relatively non-relative basis for knowledge that can be found in limited truth models. We can be minimally and sufficiently confident that our tiger is representative as a generalizable case of all tigers in some minimal structural sense. We call this inductive empiricism, and our confidence grows on the basis of the law of large numbers. This allows us the possibility of inductive generalization that permits us to take a statstical profile of an instantaneous population as defining of the relatively general ideal type. This is perhaps the best we can do with our sciences. Even our hypothetico-deductive models in science, as for instance the theory of Evolution, rests ultimately upon an inductive and empirical basis. An "average" everyday tiger is minimally sufficient at least from a systems perspective for the understanding of most if not all tigers. In such a system of generalization, we can say that all systems are internally nonrelative to themselves. Therefore they cannot be self-contradictory as non-systems. They are discrete entities in some minimal sense of relative independence.

If we observe 999 tigers and they are all striped, then we can conclude with reasonable confidence that "all tigers are striped" even if we discover just beyond the range of our observations that the thousandth swan is in fact white and without stripes. Actually, to be more accurate, we always say that "most all swans are white" and this kind of statement remains true even if we discover a black swan.

It is in the intermediate range of real systems, which are usually artificially constructed systems, where a sense of non-relative truth-value emerges as independent from the instantiatio of such systems. In this sense, our anthropological relativity becomes turned inside-out, and at the center of our knowledge base is a presupposition of ultimately non-relative absolute knowledge of truth. We use our applied math to real and naturally occurring systems in effect to undo the results of the intrinsic anthropological relativity of our knowlege, to gain a relatively restricted sense of internal coherence. In the intermediate range of real systems, and in the validation of all systems of generalization, it is expected that internal coherence and external consistency become matched up and non-contradictory to one another.

The complementary relativeness of systems of generalization is directly tied to the degree of determination that a hypothetical system exhibits. A fully determined system is a non-relative system. It is also an invariant or constant system. A minimally determined system is one in which random variants are the norm, and not the exception, and hence the system is inherently chaotic in its structural patterning. This idea of relative determinancy sets the "degree" of coherence that a system may manifest. It also sets the complementary degree to which random factors of change that are essentially uncontrollable are inherent to the system. We can make the following kinds of generalizations about systems of generalization:

1. The complementary relativity of any system is measured by the degree of relative determination inherent to the system that is a function of the degree to which random factors control the system in an antithetical manner.

2. Relatively underdetermined systems tend to be metaphysically relative systems which also entails that they are inherently more problematic in their generalizability.

4. Applied mathematical systems can only be partially sufficient to the description of natural systems.

5. Our certainty values and sense of confidence in our generalizations is directly proportionate to the intrinsic determinancy of any system. If a system demonstrates a high degree of complexity and chaos, then our confidence limits for our generalization must be high because our sense of certainty will always remain low.

6. We can say that fully determined systems are systems that are completely known and knowable, and hence are fully predictable in their functional patterning. Relatively undetermined systems are those that remain in some residual sense unknown and unknowable, hence uncertain, and hence at best are only expectable in their functional outcomes.

7. We can also say that we can only completely know systems that are fully determinable. Fully determinable systems are ones that have a completely internalized frame of inference that defines the system as such.

8. The degree of relative determination of any system will be reflected in the distribution and composition of its input-output variables and values. In general, a determined system will have constant and stable values, while relatively undetermined systems will have values that are dynamic and unstable because they are at least partially random. In a fully determined system, input-output values are completely knowable and hence predictable. In a relatively undetermined system, similar values remain partially unknwn and hence at best expectable.

So far, we have begged the central problematic of rendering a concise analytical description of a purely abstract meta-system, one that can be said to be internally coherent and hypothetically independent of any real system that represents it. It is evident that the quest of a grand unified theory of all science that is rooted upon a single set of mathematical formulas is probably misguided and impossible. The differential formulaic application of mathematical equations to the abstract description of naturally occurring phenomena upon all levels of analysis and synthesis determines that problem sets in the sciences have to be approached separately, in different scientific axioms in terms of different idioms, and each kind of system on its own level of natural integration and control, and each in its own descriptive merit and particularistic characteristics.

A metasystems approach does have a integrating effect in allowing us to see the inherent design of all naturally occurring systems as being in the most general sense possible as "systems" (albeit of different kinds) that incorporate both order and disorder into themselves.

5. They express relative complementarity of deterministic/indeterministic elements and relations.

Discussion of the inherent metaphysical relativity of abstract systems of generalization, as this is especially dependent upon the complementary degree of determinancy and indeterminancy any system exhibits, brings us back to a fundamental issue about science in dealing with the nature of order and chaos as this is manifest in the natural phenomenal patterning of reality. We cannot say that order and chaos are clearly separable on any level, such that extended deterministic functions can rapidly lead to apparently chaotic patterning, and fundamentally random variables can beget ossified structures that appear overly determined. Another way of saying this is the following:

1. There cannot be in naturally occurring systems deterministic relations that are not minimally chaotic.

2. There cannot be in naturally occurring systems absolutely chaotic relations that are not minimally deterministic.

Take an explosion, for instance. We would use an explosion as a classical example of maximum entropy, in which a system of energy is suddenly, in a brief moment, transformed into a non-system. But on another level, even an explosion can be seen to be a minimally ordered process of expectable transformations. Explosions are themselves systemic events, and as such even an explosion is never 100% efficient.

These principles can be seen to be consonant with what we know of naturally occurring thermodynamic systems, and we can conclude that any metasystem is an abstract model of some mechanism that is relatively determined. We can say that chaos and order are both intrinsic and extrinsic to any system, and from this we can propose the following kind of paradigm:

In any system, order and chaos impose a kind of fuzzy relationship to one another, that is its own form of relativity, where we cannot say clearly where order leaves off and chaos begins.

I would propose a minimal differentiation formula characterizing any system A, such that:

And where (1- y) equals the extrinsic random factors impinging on the same system

ƒ (y) and ƒ (x) are functional interdependencies of system A within a larger framework, such that the functions of y and of x are unique to the system A or the type of system A and the original values of y and x stand for initial input and state values of system A.

Such a kind of system might describe the minimal differentiation needed to unite any potential metasystem.

Such a metasystemic model can only be understood from the standpoint of its structural reiteration or recursion as a dynamic system, such that we may posit some start state A and some end state A' and any number of intermediate states that unite and characterize the transformation [F(A)] of the system A ↔ A'. This implies a basic formulaic framework for our understanding of systems and dynamics, according perhaps to the following kind of paradigm:

Condition 0:A totally undetermined system is one in which A ≠ A' under any condition and in any sense of relation. This embodies a basic principle for all real systems, and for any ideal system, and that is the principle of non-contradiction. A thing cannot be itself and something else at the same time, in some absolute sense.

Condition 0a: A relatively undetermined system is one in which A ≠ A' in some approximate and imperfect sense.

Condition 1: We can state that any abstract ideal system is abolutely ordered and determined if for any variable relation or transformation occurring, the results are always the same, such that: A = A' under any and every condition, no matter what the intervening processes or variables.

Condition 1a: A relatively determined system is on in which A ± A' in some approximate sense under most conditions in the large. We can call such a system generally determined.

Condition 2: We can say that for any general real system, there is some composite linear equation such that each alternate transformational state A" is the linear equivalent of A → A'.

Condition 2a: A relatively real system is one in which A ≈ A' in some approximately equivalent sense under most conditions.

Condition 3: We can say that any general natural system is generally determined if for any occurring state A there is some composite nonlinear formula such that there is an infinite number of each subsequent A' or any alternate A" in at least a statistically approximate sense upon some normal curve of possibilities, such that A ∞ A'. We can say that A is infinitely representable by A' in some general sense.

Condition 3a: A relatively natural system is one in which A ∞ A' in some normal stochastic sense for most cases.

If this kind of paradigm or order and chaos in metasystems resembles the paradigm of the laws of universal dynamics, it might be perhaps more than a little bit serendipitous. We can posit a general metaphysical homology between real systems as these are generalistically known and natural systems as these are scientifically known, and this kind of paradigm is the metaphysical equivalent of the laws of universal dynamics in real knowledge systems.

We may state that these conditions governing meta-systems are themselves ordered such that condition 3 can be composed of relative conditions 0, 1, and 2. Condition 2 that describes what I call real systems can be composed of relative conditions 0, and 1, but condition 0 even in a relative sense cannot be composed of condition 3, 2, or 1 and condition 2 or 1 cannot be composed of condition 3. We might put it another way and say that the paradigm of metasystems is stratified in a inclusive/exclusive sense such that lower order conditions can underlie higher order conditions, but not vice-versa. This may be modeled thusly:

We can say that any realization of a lower order condition in a higher order system is always relativized to that order of system. In other words, the lower order system would be subject to the control of the boundary constraints determined by the higher order system.

It is apparent in this kind of metasystemic model that we can say that in a fundamental and absolute sense, disorder underlies all order, and disorder is always more basic to a system than order. We can say as well that order is always derivative or disorder in a relative way. We can say that in any naturally occurring system, there is a universal tendency in the large or in the structure of the long run, to return to a state of relative disorder. Only "non-real," hypothetically absolute, systems of condition 1 can be said to not return to some state of relative disorder, because such systems cannot change. They would represent the minimal differentiation from disorder that is possible. Of course, such systems are in a sense non-existent because they have no direct phenomenal instantiation. We only can know them in an indirect and relative way by abstract representation.

It follows from these considerations of general models of metasystems that the description of the system proceeds from a consideration of its order-to-chaos spectrum, or the manner in which it becomes realized in the continuum of order-to-chaos, as part of a larger system of similar relationships. Another way of seeing this is to say that in meta-systemic models we are attempting to desribe the systematic statics and dynamics of a mechanistic system, and both statics and dynamics are only comprehensible from within the framework of the order-to-chaos continuum relative to the mechanical frame of reference governing the system. We may imagine dynamic conditions that lead to fairly stable or "relatively static" states, perhaps like cyclones, and we may imagine relatively simple statics that result in fairly complex dynamics, like general weather patterns on earth.

Static-dynamic descriptions integrate the understanding of order-to-chaos in terms of the functional parameters and limits of any system. We may speculate on some basic sets of relative operational relations, or "transformations" that are definable by a static-dynamic description. I will offer the following alternative possibilities:

Unification: the general realization of any system within the meta-systemic framework of reality. A totally unified system would by definition be a minimally or maximally ordered system.

Integration: the functional consolidation or structural organization of a set of parts into a holistic framework. Integration can be considered to be a relativized instantiation of unification.

Differentiation: the order-to-chaos continuum determines that in the largest unified structure, most systems eventually seek a state of unification in the minimal sense. Differentiation is the result of systems to seek alternative pathways, as a result of the chaos factor in their articulation and reiteration. Differentiation can be considered to be the consequence of integration operating against a background of long-term systemic unification.

Elaboration/Extension: Differentiation proceeds on the basis of continuous-discontinuous elaboration of parts, which can be considered to be the variation of pattern leading to systemic differentiation. These describe I believe basic processes of growth and development of systems by which order-to-chaos may become enlarged.

Alternation: Alternation can be defined as the minimal defining differences between two otherwise similar systems, or two possible states of a generally integrated system. In other words, systems may develop differentially along different pathways, which lead to variation and alternative instantiations of the same system.

Construction/Composition: Systems in their transformation may become constructively unified with other systems in a synthetic and metasystemic sense leading to relative integration/unification of the system. Constructive processes determine the part-whole relationships of static-dynamic systems.

Reiteration/Recursion: Technically, reiteration and recursion describe different programming or control facets of systems, one being relatively continuous and the other by contrast discontinous is its state-trajectories. But both describe a basic transformational characteristic of systems that describes their duration or durability, or systemic durability as a function of the temporal dimension. Transformation implies change, and change isa temporally ordered process. Thus change is coherent, and we cannot imagine a change process that is in a fundamental sense discontinuous and incoherent.

Systemic integration implies some degree of deterministic stability that implies a relatively static state. We know these as event intervals defined systematically by the conditions and terms of the system it occurs within. Reiteration describes the set of changes in a gradational or step-wise fasion, and describes a periodicity of fluctuation patterns such that periods of stasis are interrupted by intervening periods of relative disorder. Recursion implies the continuousness of change processes within any system, such that change is alledged to be a constant variable of any system as a function of time. If we imagine a film as a long string or series of frames, then we can imagine reiteration as the stepwise progression from one frame to the next. Realistically speaking, the synergism of a film projection system is only achieved if we run the film in a recursive manner, rather than reiteratively, such that the boundaries of discontinuity of the basic state system is phenomenologically eliminated.

Nihilation/Destruction: A final descriptor of metasystemic transformations is, I believe, the process of nihilation or destruction of any system, which can be defined as the point beyond which any system must be returned to a larger and more fundamental pool of possible systems in a state that, from the standpoint of the system itself, is fundamental disordered or "nonsystemic."

Destruction processes are those in which patterns of chaos appear stronger and statistically more prevalent than patterns of order, and lead to the disintegration of the system as a system. We can say that there occurs a reunification of the elements of the system on another level of relational order.

From the standpoint of metasystemic dialectics, it can be seen that order to chaos are thetic and antithetic elements constraining any pattern of unfolding, and leading to a larger synthetic pattern of relationship in a larger context of relations. At the point of ultimate destruction of the system, we can say that a system has reached a point of no return, beyond which chaotic processes are no longer antithetical, but basically non-thetical to the system. In otherwords, random change process become essentially non-systemic in character, and non-systemic relations intruded to greater and greater degree within a system that is disintegrating.

It is apparent that if we can look for chaos in order, we can also seek order in chaos, as long as we can refer to some kind of metasystemic framework in which to construe such patterning. In a totally chaotic framework, or even in a relatively chaotic framework, we can refer to chaos without apparent metasystemic order.

It is apparent that reiteration and recursion can be put upon a continuum of continuity to relative discontinuity of transition. We might contrapose this with the previous order-to-chaos continuum to imagine a space as follows:

A classical conception of science was implicitly one of linear systems. If we knew the intial start variables for any system, and we understood the exact linear laws that described the dynamics of the system, then we could predict with relative certainty the long term state trajectories of any system. It became apparent in the last thirty years especially, with the convergence of Chaos theory from diverse fields of inquiry into naturally occurring systems, that almost all natural systems are non-linear and therefore in a basic sense do not fulfill the implicit conditions of a conventional scientific worldview that was, by the way, somewhat Euclidean, Aristotelian, Cartesian and Newtonian. Einstein upset the apple cart of science in a fundamental sense, but scientific worldview has been slow to catch up with the implications of nonlinear systems in nature.

Another way of seeing this, is to say that we cannot necessarily predict the outcomes of any natural system in any exact or fundamental sense, especially when we are talking about large structures or structures of the long-run, but we can derive alternative sets of expectations of possible state-transition pathways or trajectories of any such system, given any set of primes. For any given interval period or area, we can even assign probabilistic or stochastic confidence limits to these sets of expectations, such that we can say that, perhaps there is a 99% chance of a major earthquake occurring in Southern California within one hundred years.

We do not call such a statement a prediction in an exact or precise sense. We do not give an exact date or time, and there is always the residual likelihood that our statement may prove wrong. We call such a statement a declaration of expectancy or a definitive expectation within certain confidence limits. We would like to be able to push our confidence limits in our "inexact" natural sciences to be able to make a statement like "In a week's time there is a 90% chance of sunshine" but most of us know that the weather has not yet been that systemtically modeled.

Behind this, is a theory of confidence or of relative certainty, that affects decision theory. There is also a theory of error that is also important to understand in a basic sense. These will be taken up in the second part on operationalization of systems. I would only say that in a revised synthesis of scientific worldview, we cannot any longer afford to assume a basic predictability of patterning occurring at any level of phenomenological instantiation, even upon those levels of classical mechanics that were considered absolute and inviolable for all cases. It follows also that in the methodological operationalization of our sciences, the goals of falsification, prediction and control are perhaps unrealistic in the larger sense at least. The philosophical implications for science will be taken up in another work.

Control, predictability and falsifiability are in a more realistic view of science only relative conditions that we somewhat arbitarily impose upon our phenomenal observations. Even our observations themselves are somehow "controlled" by definitions of measurement and selection, which in turn are often predetermined by the inherent design of the experiment itself. Error may be introduced at any level in such an operational system. Many purportedly objective statistical studies frequently ignore or forget to declare this in their final statements.

We can say that whatever can be controlled can also be uncontrolled, and what is fundamentally controllable, is also paradoxically fundamentally uncontrollable. This is not apparent in our basic world of molecular interactions, but it is more obvious at the margins of our phenomenological event horizons.

It appears that scientific worldview suffers inherently a basic set of paradoxs, or what can be called a fundamental antinomial uncertainty, about its own ability to know, or knowledgeability about reality. We may choose to ignore this issue and be complacent about its implications for our sense of the world, but we cannot do so without at some level risking a sense of realism about our fundamental relationships with and in reality.

Control is a conceptual construct inherent to systems theory. Our ability to introduce controls systematically in the experimental manipulation of phenomena is the basis for falsification and for our scientific method. But control implies an absolutistic quality of being able to manipulate the variables and outcomes involved in any transition system. We can do this in approximate ways in chemistry. I would replace the term "constraint" for the term control in a nonconventional view of science. Constraint suggests a form of relative control operating in any system. We can see that even our observations themselves may be a form of constraint imposed on the system and thus affecting its outcomes in basic ways. Talking about constraint instead of control leads to a theory based on limits and limiting factors that may in some sense predetermine a system and its consequences in a less than precise fashion.

Repeatability is a basic experimental design concept underlying the notion of control. The ability to replicate experimental conditions to derive the same sets of consequences over repeated independent trials, is held to be the methodological basis of conventional science. It is the foundation of conventional control, which allows us to predict systematically the variation of outcomes as we modify the conditions in a controlled manner. Indeed, scientific progress is usually achieved not through replicability of controls, but by the ability to take into account the apparently random and unpredicted and unexpected exceptions to the rules of our generalization, leading to a revision or modification of our conceptual constructs. In other words, it is in the chance encounter with unexpected outcomes that nevertheless demonstrate an unknown sense of pattern and order that we come to more realistically and more generally revise our scientific systems of generalization.

The experimental system of control that we superimpose upon our observations of phenomena, relating to a theory of measurement, which itself relates to other theories of sets, series and samples, will be dealt with in greater detail in subsequent chapters. It is important to emphasize here that such theory is a conventional construct that is not wholly realistic to actual scientific progress. Its revision leads us to the construction of a systematic frame of reference within which to construe, define and operationalize the concept of metasystems.

Control from a systems framework takes on entirely different but not necessarily unrelated meanings. Control refers to the boundary parameters and operating variables that constrain the behavior of a system, and that determine its pattern of response or reaction to changing external variables or conditions. What we recognize usually as the integrative properties of systems are the epiphenomena of the control of the system, and control is therefore the direct consequence of the relative integration a system achieves at any state or instant. Control becomes in systems thinking something that is a function of the system itself, to be observed and explained.

Thinking about the relationship between order and chaos in metasystems leads to some interesting conclusions. Systems science provides a heuristic and general framework for understanding the complex relationships between order and chaos in all naturally occurring phenomena, and in elucidating the structural mechanisms underlying such patterning. A system can be seen therefore to embody both chaos and order in its general description and modeling, and a metasystemic approach is, I believe, the appropriate framework for construing advanced systems science in both analysis and synthesis.

To begin with, we can speculate that any system always ranges along a continuum of order-to-chaos, but that there is no real system that is ever completely ordered or disordered:

The idea of a perfectly determined system or a perfectly undetermined system (a perfect non-system) is unrealistic and impossible in the real world, though it can be modeled hypothetically in abstract space. These boundary conditions of our metasystems bring our metasystems into alignment with our theoretical understanding of naturally occurring systems, such that naturally occurring systems always obey the rules of thermodynamics and whatever other operational principles that may apply.[11]

Another interesting outcome of thinking about the relationship between order and chaos in metasystems theory is to conclude that there can be no completely closed system, or a real system that is isolated and totally independent of other systems. Another way of saying this is according to the following paradigm:

1. All real systems are interconnected to compose a single heterogeneous & stratified metasytem that constitutes total Reality.

2. Any particular system we can describe is always connected to other systems within the framework of the metasystem, sharing attributes of the larger system.

3. Any particular system in a real sense is always a part-whole description of a set of relationships within a larger metasystemic framework.

4. There can be no such thing as a perfectly isolated system. All naturally occurring systems are interconnected functionally in some minimal sense. We can call this the reality continuum that defines the functional interrelationships of the metasystem.

From this, we can reintroduce the principle of anthropological relativity of our knowledge as underlying the sense of fundamental metaphysical relativity of systems. We can see that even absolutely abstract systems, as these occur in mathematics for instance, are still embedded in human knowledge organization as a system and as part of a larger system of understanding. One derivative of this would be to say that our measurements can never be perfect, and that our understanding of natural phenomena is always somehow minimally conditioned by the frameworks of our understanding and our inherent positionality as ourselves a complex heterogeneous system.

It follows as well that to search for a single "ideal" formula for the perfect or universal metasystem, as some kind of grand unified theory, is perhaps unrealistic and maybe even undesirable. It is perhaps more useful to develop a frame of reference that serves to integrate different problem sets relating to alternative systems with different fields of inquiry and application. The beginning of advanced systems science is about providing such a standard frame of reference for systematically construing our reality.

A meta-system is about a universal frame of reference for reality that serves to comprehend and contextualize all systems, and about the systematic differences and interrelationships that hypothetically must occur in any real or particular system as itself a part-whole example of a meta-system.

Gestalt theory enters into this understanding of metasystemic frames of reference, as it embodies to some extent the basic principles occurring in human pattern recognition, and that are possible in the abstract organization of knowledge of all kinds. Gestalt theory gives us a handle for getting outside of the internal dynamics and systematics of any particular system, while still maintaining a handle upon such dynamics. It follows from the notion that any "figure" (implicitly, any systems of relations) must always be configured and contextualized within a field (implicitly, any metasystemic framework) by which that figure gains contrast and value. Gestalt recognition is the organization of consciousness, of pattern-recognition, based upon perception of figure-ground relationships, and by extension, underlying symbolic and conceptual organization of human knowledge.

To go one step further, there in a sense of infinite regress in the contextualization of frames of reference-inference about systems. For any set of relations or specific framework we may specify, it is always itself a figure-frame relationship of a larger system of relations, and we may in turn reiterate the entire construct ad infinitum. We may also see that any figure or part-whole system is itself a framework for an entire plethora of subsystems, and so on to infinitesimal levels. We can see that this sense of reality is corroborated by our understanding of natural phenomena in the physical universe quite well and at many levels.

The application of gestalt theory to metasystems and to their functional operationalization may be more than fortuitous in more than one way. Not only does it figure centrally at the basis of human symbolic knowledge, and as the basis for heuristic problem solving techniques, but it also figures theoretically and generalistically in our comprehension of abstract and real systems. Gestalt theory permits both the analytical excoriation of structure underlying phenomenal event patterns, but also the synthetic reconvergence of pattern based upon our structural analysis of underlying systems. Gestalt theory and method provides the operational and functional foundation for the integration of advanced systems sciences in terms of nested metasystemic frameworks of understanding.

If we go back to some basic principles of gestalt theory, we can find some fundamental statements regarding the relationship of order to disorder. We can say that any figure achieves its coherence and unity by the degree of achieved contrast with its background. It becomes a figure of order in a background of disorder, or in a negative sense, a hole of disorder in a field of order. In other words, if a system is to a great degree similar to its background framework, then, as a separable system, it is more problematic than one that has a high degree of dissimilarity with its background.

It follows that no description of a system can be offered without also first describing as well the expected and existing framework in which it is embedded. Any structural description of a system can never be completely internal, but must be both internal and external, relating the structure of its relationship to its naturally occurring context.

We can ultimately say too, that this description must always include as well the point of view of the observer in the final verdict, no matter how remote or indirect. This always introduces a residuum of arbitariness into our observations.

A gestalt framework proceeds from the understanding that in our observations of natural phenomena, we bring to it the conceptual constructs of our sciences, and without such basic conceptual constructs, which are ultimately symbolic constructs, then our observations would not be meaningful. We see rocks in the ground as separate entities because we have some conceptual notion of what a "rock" should be like. This is inherent gestalt in nature, as it allows us to configure a rock from a background of similar and different entities. It can alsobe said to be phenonemological in the sense that each rock is but an instance of a prototypical form of a rock that underlies our symbolic construct. The more instances we have of the more kinds of rocks we encounter in our experience, the better our construct of a rock becomes.

It can be seen that conceptual constructs therefore order our phenomenological experience of reality in meaningful ways, allowing us to selectively focus on some kinds of information as important, and to interpret this information as somehow important to our systems of conceptualization. Thus, gestalt systems permit us to do a continuous reality testing to see if new experiences are congruent with our preconceived notions and expectations of experience, or what we can refer to as the metaphysical constancy of our worldview.

We build constructs from our experience, by means of inferring relationships between phenomena that are not otherwise obvious, based on what we can refer to as implicit patterning of the phenomenal points. This is inherently a gestalt pattern-recognition process. A great deal of scientific generalization is indeed built up indirectly from remote inferences based upon properties that can be consistently associated with discontinuous and independent observations.

Consideration of a gestalt framework as the basis for a functional operational methodology to advanced systems science invites a number of fairly radical considerations. Phenomenology as a philosophy has been attacked by both rationalists and empiricists as somehow fundamentally problematic. I believe that gestalt theory implies a theory of the phenomenology of experience. This is not to embrace whole heartedly and without reservation phenomenological philosophy. We must understand that even phenomenological experience can be fundamentally non-Euclidean in structure and pattern.

A phenomenological accounting of reality accords with a metasystemic model in a number of ways. Reality can be said by our definition to constitute an ordered stream of consciousness. As such, it is constrained in certain basic ways--for instance, it always flows forward in time, in one direction. It is through the instantaneous expression of phenomena that we come to know pattern and structure in reality, and it is through the periodic recurrence of such pattern that we come to understanding the sense of structure in a larger framework through extended reference and inference.

A gestalt theoretic approach to metasystems can be understood possibly on the basis of the following levels of systematically ordered description and explanation:

To understand gestalt phenomenology, I believe we must invoke a model of the French Art Gallery. Imagine an oil painting of a subject hung on a wall. We, you and I, were invited to attend the premier showing of the painting, in order that we may publish a review of the picture the next morning, and we arrive to find it hanging in a fancy frame on a wall. We stand back at arm's length to view the picture. We study the picture for some minutes to try to get a feeling for its essential design and its meaning. We can say that the theme of the picture is the central part of the system. On the face of the central figure, we notice a curious detail of color and stroke, and so we edge as close as possible to see what the medium and method of execution the artist employed, and there in the corner we find the artist's signature.

But as we study it, we notice the geometrical flow of the pattern of the picture, and the hidden spaces of the background, and the multipoint perspective that is atmospherically reinforced. We realize that the picture is a large picture and we are too close to see the whole thing in a systematic way, so we stand back from our original position several feet. At our new distance, we better see the picture in relation to the frame it is in, and we understand that the person who chose the frame for the painting did a good job, as the frame seems to "fit" with the overall motif and feeling of the picture.

And then we notice that the color of the wall, a neutral sherbert color, also complements well the main values and tonalities of the painting, and we realize that juxtaposed next to it is another very dissimilar painting by the same artist, smaller in size, and on the other side is a similar kind of painting by another artist, also of smaller size. As we take in our view, we can't but help notice the ugly wall socket in the corner of the wall, and we look up to the ceiling to find a filtered skylight with indirect sunlight filtering into the room from above, and giving a nice carmine glow to the room. And then we notice the flow, the molding, the oak boards nicely varnished, and the rug that sits in the middle and is quite fancy.

We then notice a small statue in the corner which also appears to complement the subject of the picture. And while we are noticeing these things, we are interrupted by another spectator who makes some comment about the picture, and this comment redirects our thoughts and brings our attention to a detail of the painting that we hadn't noticed before. And then we grow a little tired and wish to take a break, and we notice a small bench that is strategically place along one wall, from which we can take in a view of the "whole" affair of the museum, and we notice a plain-clothe's guard watching us and we notice a museum staff person going through a door next to the room.

The point of this digression is to emphasize the essential contextuality of our gestalt understanding of the "painting" as a phenomenological event, or even more appropriately, as an encounter. We cannot say that the painting was viewed in isolation from anything surrounding it. We agreed that the colors of the room and other paintings and objects of art complemented the picture well, but that the wall plug stood out in an insulting way from the corner of the room.

When we go to write our review of the event, how much of the context should we include in our recounting of our experiences? Can we say that the sight of the wall plug did not affect our final judgement of the piece, or that the interruption, if it hadn't occurred, might have changed the outcome of how we write our final review? If we focused exclusively on the internal design details of the painting itself, as if it existed in a vacuum of isolation from everything around it, would we be giving our readers a fair sense of what we actually encountered? On theother hand, if we pay too much attention to all the surrounding context, we might as well talk about the war that was happening on the other side of the globe at the time, if we can indirectly tie this to the painting somehow.

It is my estimation that reality always presents us, especially as scientists, with this kind of inherent gestalt problematic of the configuring the central figure to the background context of its actual phenomenal instantiation. We cannot say that the structure of our understanding of the painting was not influenced by the external context of its presentation--that our point of view might have been considerably altered had the museum staff made a different set of choices about its frame, and its placement in the gallery. Perhaps if the sky outside were overcast instead of the sun shinning, the hues of diffuse light cast upon the room might have created a different effect and feeling to the chromaticities of the painting. This kind of effect cannot be discounted completely, especially if a critical part of the painting were its chromatic effects.

We can say that conventional science generally tends to leave out the contextual details in an exclusive focus upon the central subject as if it existed in some kind of hypothetically neutral space or abstract vacuum in reality. A gestalt approach implying a phenomenology of experience would by definition be nonexclusive of the context, but one sense of control to such an approach would be the progressive delimitation of its contexts of phenomenological presentation, on the basis of the degrees of directness/indirectness of its context. We are then construing events containing things, rather than things in and of themselves. And the event, not the thing, tends to be integrative of the entire gestalt experience of the figure and ground.

Gestalt design I take to be a second step in the process of description, and it is the first step towards what we can call a general explanation. Gestalt design can be construed as the holistic experience of the entire structure of the event, with the painting as the central point of reference and articulation about which the event unfolded. Gestalt design can give a treatment of the internal design of the painting itself, or even of the central figure of the painting, exclusive to the receding contexts of its presentation. But the contexts of presentation and the internal design of the central subject of the painting somehow appear to interact with one another. The frame fits and complements the central subject well, and the subject appears to be looking out past the boundary of the frame, even into the center of the room, and points in some other imaginary direction. Gestalt design construes the synthetic and holistic sense of integration that the painting achieves, as an event, or, more precisely, as a thing that is the center of an event.

From Gestalt design, we proceed to a third step in our process of explanation, that of gestalt analysis, at which we seek detailed refinement and resolution to our understanding of the overall design. We may take greater notice of the details. We may search for correspondences with the artist's other paintings, and we may search for clues in the artist's life that relate his experiences in the painting to our own. In gestalt analysis, we seek to understand the structure of the invent, and the holism of the design, in terms of its parts, that are construed as a part of the structure of the whole.

In the fourth and final step of gestalt theoretization, I propose that we attempt a resynthesis of our experience of the event in some manner that allows the event to become cognitively consistent with our other experiences and understandings of the world, particularly of the art world, and particularly the world of the artist and the painting itself. I would call this a symbolic resynthesis of our understanding, as this is what it in fact is, but this seems antithetical to what good science should be about. I would claim though that when we seek scientific generalizations relating to phenomena that we experience, we are doing precisely this, albeit in a manner that may be more constrained in terms of its framework of reference and inference. This last step may come in our actual writing of the review itself, or even subsequently after we read what we've written and that's been published in the paper.

To get at the internal analytical structure of a gestalt event, I would proposing the following lists of attributes or parts of a meta-system:

1. Prime variables, those elementary factors that are self-constituent to the system.

2. Key variables, or constraining factors that determine the patterning of the event.

3. Relational variables that are incidental or coincidental to the patterning of the event.

4. Phenomenal variables (or points) that are the instantaneous factors that we can observe.

I would go further to suggest that in any gestalt framework of a meta-system, we can ask the following kinds of questions of a system as a whole:

What is the center-point or central region of the system that defines its most stable center of balance?

What are the apparent principal axii of the system and what is the apparent range of its variation of pattern?

What is the background and essential context of the system, and how does this appear to affect the pattern evident in the system?

This last question ties this approach to gestalt synthesis of the whole in reference to its presentational or phenomenological context, back to the complementary dialectic of gestalt analysis of the whole in terms of the parts and explanation of the parts in terms of the whole.

Before leaving off with this mostly metaphysical discussion of meta-systems, I wish to go back to the minimal differentiation formulas presented midway in this digression. They were presented as follows for any hypothetical metasystem "A":

And where (1- y) equals the extrinsic random factors impinging on the same system

In consideration of the conclusions drawn from our understanding of metasystems, we can possibly modify the basic formula to be in keeping with what we know of systems. The function of x and y that determines its values at any particular instantaneous state can be said to be fundamental factors determined by time as a scalar value. Thus:

where c and c' represent change constants for a given period of time t and d and d' represent change dynamics for the same given period of time t and z equals some other unknown set of determinantive scalar values underlying the entire system.

We can also say that for each set of articulations, begining with some initial set of point values, we can derive a transformational relationship between start state A and end state A' such that:

We can see that z would affect x and y differentially in its instantiation at any particular moment in the system. From this we derive a conclusion that is very basic about the patterning of phenomena in the systems. In general, any differential set of phenomena that shows variation is indicative of some underlying scalar determinant that contrains the system in a basic way.

A metasystem is not just a model of any possible system. By itself, however abstractly expressed or mathematically formalized, such a model is rather trivial. In being about all systems or any system, it tells us almost nothing interesting about any given system in particular or even any possible system in general. A metasystem is not just a model, but it is more importantly, a standard frame of reference and inference by which all models and all instances of systems can be organize and compared and contrasted in a systematic way. A metasystem framework should provide therefore a taxonomic system of classification of systems based upon their organizational and operational principles and parameters.

In this kind of metasystemic framework, it is apparent that inference becomes equal in importance to reference. We must be able to infer relationships of constraint and change in the unfolding phenomenal patterning of events. Inference relates what we know and can observe directly to our models and structures of understanding in systematic ways. If we can observe pattern of similar variation, we can infer underlying determination. If we can observe increasing instability within or around a system, we can infer underlying orders of chaos. The issue of inference brings us back to structure or ordered relationship, which implies some form of deterministic variable operating.

In general, we might say that whatever is implicit to the patterning of phenomena, can be inferred, and what can be inferred can be done so in reference to similar and dissimilar sets or points. Reference "points to" things as somehow discontinuous or focal in importance. It is a method of marking out the figure from the background, distinguishing it from its field. Inference implies reference, and reference implies inferability. Any matrix or phenomenal event pattern or set or sample of event intervals can constitute reference points. Indeed, phenomenological experience is itself a vast system that is the first and last frame of reference for our understanding of reality. Not only have we built up conceptual constructs from experience in reality, but we have in turn applied these same conceptual constructs to theordering of our experience of reality.[12]

Advanced systems are fundamentally alternative abstract models and conceptual constructs rooted in mathematical theory and application, and serving primarily the heuristic function of facilitating our understanding and science about complex real systems. This heuristic function does extend systematically to general problem solving and experimental applications of alternative designs in reality, but the main purpose of this work is not such extension, but the explicit and general elaboration of the abstract representations that stand behind such possible applicability.

While I advance a grand prototypical model of a "metasystem" I see this as mostly a philosophical problem that serves as a central touchstone to the integration of advanced systems. It deals with a central philosophical problematic that underlies all our knowledge of reality, and that is the problem of the integration of reality. I can take the easy rode, suggested by the pressupposition of anthropological relativity of knowledge, and just say that human reality is symbolically integrated. But this sidesteps the central issues of truth and possibility, and presumes too much in the first place. From the standpoint of advanced systems sciences, this kind of answer is insufficient and unrevealing of the true complexity of the problem that brings to bear almost the whole breadth and scope of our philosophies, our mathematical knowledge systems and our sciences.

At this stage, solving the problem of the integration of reality require solving the implicit problem of the integration of our knowledge systems at a highly abstract, philosophical and theoretical level. In essence, I would say that both kinds of integration are two sides of the same central problem posed to us by our sense of reality in the first place. The two cannot be fundamentally alienated or dichotomized in our consideration of alternative central solutions.

The hypostatization of metasystems is therefore as much a paradigm as it is a central problem, based primarily upon a non-linear mechanical view of reality, as it is a functional model or general theory of integration of reality. I stake no claims as to its metaphysical validity or ontological status in regard to any particular real system, however large in scope or particular and finite in detail.

I seek to employ a diverse range of mathematical theories and methods to tackle an equally diverse range of problems that occur upon different levels of our knowledge and experience of reality. Within this range, the central problem of the integration of reality is approached and broached in many ways, but nowhere adequately resolved.[13]

If we are to make great strides in our understanding of the complexity and order presented to us at almost every point in our shared experience of reality, then we must strive to achieve a higher level of integration of our knowledge in systematic and, hopefully, nontrivial ways. This being said, it is easier said than done.

Advanced systems science has been approached so far with the purpose of outlining a core set of theoretical structures that are hypothetically applicable to all identifiable and self-contained systems, what I call a metasystem, whether these are naturally occurring or artificially constructed and maintained systems. The potential value of such an abstracted model of advanced systems science is in its heuristic and creative application to the identification, understanding and construction of new systems in reality and to the resolution of complex problems created by actual systems. I believe that to effectively realize such a metasystem would be to allow us to better control both our sciences and the functional purposes to which science is generally or specifically put in the world. It serves as a touchstone for the theoretical, operational and functional integration of science at all levels and in all areas, whether this is pure or applied.

In the previous companion work Natural Systems I dealt mainly theoretically and constructively with the problem of natural systems at these primarily have been found to occur in some larger phenomenological sense, or what I would call the problem of Nature or Natural reality. In this work, I deal with a complementary set of problems that I would define as the general problem of Reality, and of how we come to know reality. This is a larger and more basic issue compared with the problem of nature. I call it in general the Reality Problem, and I propose herein the "Reality Principle" as the foundation for advanced systems science. The reality problem subsumes a larger set of problems that I call real problems, and these encompass as a subset the class of natural problems that normal science typically deals with. The implication of this is that science as a general knowledge system is tied and a part of a larger sytem of understanding reality. If we are to get at a clearer, more comprehensive, and more systematic science, then we must first get at this background system of understanding and knowledge in a more explicit manner.

Science, as this is conventionally construed, is in the birth throes of yet another revolution. This is a more general knowledge revolution than most people realize. It is an entire frameshift of our entire worldview. It encompasses the so-called Information revolution at its heart, but this information revolution is itself but one aspectual manifestation of this more general background movement. Scientific knowledge is rapidly transforming our world and our realities in ways we do not yet fully comprehend. We cannot know the long-term consequences of these kinds of tranformations, and they are as noetic and behavioral as they are tehnological and informational.

Scientific knowledge has as its central problematic the understanding of nature and natural systems, but science as a knowledge system is actually much larger in scope than this. It comes to embrace a broader range of applied fields that derive largely artificial and constructed systems that do not normally occur in nature. It then reflexively rebounds on itself at some point to attempt to understand itself as a natural system. If we are to understand and develop science as an advanced system of abstraction, inquiry and application to reality in the broadest senses of the term, then we must be willing to step beyond the normal boundaries of conventional scientific praxis. This is true especially as this has been rooted to a highly successful past, and be daring enough to explore new possibilities and new manifestations of its development and implication in our world.

The approach I take in advanced systems science is to see that for any given set of related phenomena, there is at least one possible hypothetical explanatory paradigm that works on two levels, generalistically, and applied or particularistically. These lead to construction and development of alternative working models of the system in question. The interaction between the two levels provides dialectical feedback enough to drive alternative model building in the area of concern. Underlying any coherent paradigm is in theory at least one or a set of related working models that defines the paradigmatic exemplars upon which substantiation of the theory rests. These models are generally defined in relation to some central problem upon which the parent paradigm is based.

The relationship of the paradigm to the model is that of the relationship between parent and child. Multiple models may share a single paradigm, but rarely will multiple paradigms share the same set of models unless they are competing for possession of the same space. Paradigms exist in mutually exclusive space to one another, and hence there is usually an intrinsic competition upon this level of theoretical integration & unification. Models may or may not be exclusive and hence competitive to one another. Models, I believe, exist at an intermediate level of theoretical construction and serve the purpose of mediating the very general with the phenomenal instantiation or particular events themselves. They are largely symbolic constructs that mediate our experience of reality in self-consistent ways. Models are approximate constructions rooted in our language of descriptive explanation, and lead to what can be called prototypical or archetypical exemplars that govern the articulation of paradigms. Paradigms in general define the theoretical and practical limits to systemic construction and knowledge--they are articulated and organized by means of models, and provide the ground plan and foundation for normal operationalization of scientific procedures.

While paradigms function in a manner that is comprehensive and all encompassing, tending to integrate diverse ranges of phenomena at multiple levels, models tend to be analytically particularizing and synthetically generalizaing in allowing us to recognize instances of problem sets and hence variations of phenomenal patterns. They permit is to extend the range and embrace of our understanding by systematic incorporation of new phenomena and by recognition and excoriation of new problem sets. They are an intrinsic part of the normal dialectic of scientific language and praxis by which integration of knowledge and progress in the sciences is achieved.

This dichotomization between paradigms and models may be more apparent than real, and more arbitrary than necessary, in our understanding of advanced systems science. A model may be a paradigm under some cases, especially if a paradigm is represented by a single model. A paradigm may be a model, or set of models, if it lacks a central or cohesive theoretical framework by which to interrelate the models in a more comprehensive manner. Thus paradigms can be represented by some class of models even if they lack clear or concise theoretical definition or outline. The paradigm would be said to be implicit to the class of models that are used to represent it.

But I believe that the distinction between a paradigm and a model in our theory building is important and critical in the development of our scientific understanding of systems. They do appear to be two clear and distinctive sets of entities in our theoretical constructions of science. Models can be said to represent prototypical exemplars in demonstration of a paradigmatic explanation. Paradigms can be set to define the theoretical boundaries and permutations of a hypothetical system to which models are related and by which they are themselves explained as exemplars.

It can be said that advanced systems sciences are fundamentally and paradigmatically dialectical in their articulation and pattern of development. This is an important concept that has had an unfortunate history of misunderstanding and misappropriation. In a sense, all systems can be said to be dialectical in that there is intrinsic and extrinsic feedback both between the system and other systems, and between the system and our understanding of the system. The dialectical framework can be represented by the following kind of table:

This kind of table suggest that in our theoretical construction and reality testing processes there is in fact a complex set of relationships occurring that inform the discourse and development of models and ideas in different areas of science. I became first aware of this patterning of dialectic in my attempt to understand the paradigmatic aspects of the Anthropological sciences about a decade ago. I believe it has as much relevance of any area of scientific inquiry as it has had in the history of Anthropology or in its various sub-disciplines.

The point of emphasizing this dialectical structure of scientific inquiry is to point up its counter-paradigmatic possibilities. In whatever system of ideas we may posit, the dialectical approach allows us to step beyond the constraints of any such system, and to adopt critically and hermeneutically the antithetical viewpoints of alternative multiple systems. Thus, in the structure of our inquiry, we are able to fundamentally transcend the paradigmatic constraints of any system, and to embrace a more comprehensive synthetic understanding that allows us to understand the limited value of alternative systems.

Properly speaking, a dialectical approach that is itself non-paradigmatic, is inherently meta-paradigmatic in that it permits us to realistically weigh and compare alternative systems of understanding, while simultaneously to escape the ideological conundrums of adopting any one viewpoint exclusively.

I believe dialectical structure defines the structure of inquiry and systems as theoretical constructs, and is intrinsic to our systems sciences. Whatever point we may adopt or construe, there is always at least one contraposed counterpoint that serves to relativize our understanding, and therefore forces us to reconcile alternative realities.

The process of synthesis in dialectical development of idea systems allows our sciences to achieve the degree of transcendent integration of our understanding. It allows us to synthetically transcend analysis and to reconstruct our sense of reality in a holistic and synergistic sense. This is vital to the successful achievement of such inquiry.

It is apparent that in the sciences, as these have been formally and conventionally construed, there is a strong emphasis on analysis and often a devaluation of the role of synthesis in theoretical and operational construction. Overemphasis upon analysis to the exclusion of the role of synthesis in model building leads to a strong and pervasive form of reductionism that is quite common in research. Often times, holistic and synthetic approachs are regarded as beyond the purview of science, as essentially ascientific in character, especially when such approaches deal with mixed type problems. In general, applied fields that demands some degree of cross-disciplinary integration are construed as "impure" type sciences, or even as non-scientific approaches in engineering design. Sometimes, those who prefer to practice what they preach, often develop a negative antipathy or reaction against theory and the place of "pure" scientific activity, even to the detriment of their possible outcomes.

One of the main rationalizations of advanced systems science is the notion that this dichotomization between analytic and synthetic approaches to science can be transcended and reintegrated through the systematic achievement of both theoretical comprehensivity and functional integration of the fields of inquiry. Thus, those who do holistic and applied sciences, often regarded as impure, do not have to be segregated as non-scientists, from those who presume to pontificate on scientific theory and conduct pure research without having to be concerned with the problematics that science leads naturally to. I believe that a studied and built-in dialectical transcendence can accomplish this set of goals.

A systems approach inherently transcends and embraces the analytic-synthetic dichotomy that divides conventional science into contraposed camps of activity. Being "systems" they are both simultaneously analytically reducible as nothing but the sum of the parts and synthetically holistic and synergistic as something more than that mere sum. It can be seen that most systems actually operate or function at multiple levels simultaneously, and thus come to define a complex set of contextual relations at these different levels. To construe the functioning of a system at only one level to the neglect of its epiphenomenal patterning at other levels is to short-change the role of systems science as nothing but a hand-maiden of conventional scientific activity that is paradigmatically and somewhat dogmatically committed to narrow disciplinary interests.

Certain basic principles of constraint seem to inform all systems. These are kinds of fundamental design features regarding all systems, as "knowledge" systems or as theoretical constructs about some set that phenomenally occurs in reality. These can be identified as the following:

Anthropological Relativity: We as subject knowers are at the center of knowledge of all systems. In general, it can be said that we can only know a system in a partial sense, and we can only infer the entire system from this partial knowledge. Our knowlege of any system is therefore by definition always incomplete and imperfect. Anthropological relativity conditions our knowledge at all levels. It is true that we are bound within our own human event horizon, which would be a solipsistic shell of objectivity if we did not have some sense of inter-subjective communication by which to test the objectivity of our constructs. But even then, often our standards of measurement themselves are bound within the same event horizon

There is more than a little bit of "Heisenbergian" uncertainty involved in all our knowledge, as it is quite true that there is a fundamental complementarity of our knowledge structures. The means we use to relate to and understand and measure our reality, affect the reality that is being measured in fundamental ways. Thus for each construct, there is always a hypothetical alternative construct available that might prove more accurate.

Anthropological relativity of knowledge does not spell disaster for our scientific understanding. In fact, it spells greater realism and objectivity for it, because it allows us self-reflexive recognition of the limitations of our own knowledge. Surely, we will never achieve absolute knowledge of reality, but we can push the relativistic limits of what we know infinitely towards the goal of an absolute model. The paradox of this is that the natural phenomenal event patterning of the universe itself is fundamentally relativisitic itself, what I will call universal physical relativity. Hence, even if we could develop a sense of objective parallax about our shared reality, especially in relation to some alternate alien form of intelligence, then we would still have the physical event horizon of our observable universe to impede our awareness of the whole of reality.

I will hypothesize therefore two sets of event horizons that operate in the theoretical-phenomenal construction of our sciences:

I would hypothesize as well an intermediate event horizon that is constituted by the biological foundation of our human intelligence. But that is an issue that I do not believe to be central to this argument except to point out that non-human biological forms of phenomenal experience and intelligence may broaden the range of possible observation and inference possible, if we can seek to comprehend these.

Universality: All delimited systems are subsets of a larger infinite system, hence they are embedded and contextually defined within larger systems. The total system is what I will call "Total Reality" and it subsumes as its objective foundation in physical reality the total physical universe.

The presumption of the principle universality is extremely important to our understanding of advanced systems sciences for several reasons. First, it governs what can be called the "Reality Principle" upon which advanced systems sciences is based. This Reality Principle is a first principle of advanced systems science and will be further elaborated in the next section.

Several points about the principle of universality are important to explicate here:

First, whatever the basis and limits of our knowledge about any given system or possible system, there is always some larger realm of possible knowledge relating to that system that remains unknown but not unknowable. This defines the basis for the extra-physical relativity of our knowledge and our sciences.

Second, all systems are at least indirectly interconnected in a multiplex network of relationships that is infinitely complex. The ultimate aim of our advanced systems sciences is ultimately the objective understanding of the total reality.

Third, to the extent that we have objective understanding of physical reality in terms of systems, this objective knowledge is congruent in at least some minimal way with the overall patterning of Reality.

In other words, objective, scientific understanding of reality is universally congruent, such that the same principles governing the function of the system under a specific set of circumstances can be predicted to be the same for any other systems of identical characteristics in some other time or place.

Structure: All systems exhibit some minimal sense of order that is defined by deterministic relations within and between systems that have predictable or expectable consequences. Structure of a system is defined and measurable by its degree of achieved integration.

Structure is, systemically speaking, the minimal deterministic ordering of relationships necessary to the self-definition of any system. Our sciences attempt to get at this minimal definition of structure governing any particular system or set of systems. It describes the central problematic of our systems sciences.

As part of the Reality Principle mentioned above, we can state that all of reality is minimally ordered or structured in some complex way that we probably do not yet fully understand, nor can ever completely comprehend.

Dynamics: All systems change or are subject to change. Systemic change is dynamic in the sense that it can follow alternative transition pathways, such that the system may enter alternative states in its historical development.

In general, it can be said, that change is the basis for both order and disorder in reality, such that systemic variation of phenomenal pattern leads to increasing differentiation and relational complexity of reality, which in turn results in increasing degrees of chaos.

It can be clearly stated, that, except for change, we could not comprehend reality fully or even partially, and, in deed, we could not have reality as we know it. All systems are in fact temporally organized systems that are defined by change processes through time. The diachronic patterning of systems is the basis of their phenomenal event-structures.

The explanation of change is paradigmatically vital to the objective imperative of advanced systems sciences in understanding the minimal structural patterning of reality at all and any level of its phenomenal expression. In deed, paradigms in the larger sense are defined by the ability to explain and predict changes in event systems.

We can state clearly that in any system, there is always some minimal sense of order that is evinced through expectable events occurring over time. Observation and systematic measure of these events constitutes the empirical basis for our science.

Entropy: Entropy is extremely important to the understanding of systems, because it guarantees that all systems, being part of larger systems, are noisy and imperfect in their function. They exhibit in the patterning of the long run the tendency towards decay and total annihilation or dissolution or disintegration as systems.

We can say that entropy is the consequence of change in reality, and the basis for the fundamental relativities underlying our limits of understanding reality at any and all levels. It guarantees that no systems in reality are perfect or absolute, and that therefore neither can our scientific comprehension of systems be perfect or absolute as well.

We can say that entropy is the ultimate expression of constraint in Reality, such that, in the structure of the long run and in the large, all systems are "absolutely" entropic in some ultimate and final sense. We can use it to explain death of biological organisms and extinction of species. We can use it to explain the fundamental dynamics of the universe itself.

We can say also that in change process, entropy is always contraposed to some minimal sense of order within systems. It constitutes therefore an indirect measure of structural order within any system.

Whatever systems we encounter in our explorations, they will at whatever level they happen upon exhibit these basic characteristics. There are three other operating constraints in our understanding of systems:

1. We may say that our systems science, whatever its form or subject area, is always based upon a limited fund of knowledge, from which we must infer the larger system as it occurs in reality. We ever only understand a part of the larger system, and we must use the patterning that we find in the observable area, to infer the patterning in the areas of the system that we cannot directly observe. To go one step further, we normally would impose some degree of statistical measure of non-random occurrence of pattern in our determinations. Statistical description and determination of probability plays an important role in our systematic knowledge of reality.

2. We can therefore hypothesize a kind of pie chart of our systems understanding as follows:

This pie-chart of our advanced systems sciences is derivative of the previous diagram of the relativities of our event horizons.

3. For any given system, a similar kind of pie of knowledge distribution chart can be developed, that can stand for the larger, generalized system, or for any particular instance of a generalizable system or for the total system at the same time.

It can be seen that when we begin defining the discrete values and discontinous boundaries of a particular or general system, we begin to isolate and distinguish that system from all other possible systems. The chart would come to take on distinguishing characteristics that would represent its development as a separate and unique system occurring in reality. All systems, or any possible system, or some generalizable "hypersystem" can be similarly represented in this way by modifications to the same basic knowledge pie.

Systems science is concerned both with the analytical differentiation of systems within systems, and with the synthetic unification of all systems within a single abstracted total system that can be represented by the same chart. Thus each time we achieve the relative comprehension of some subsystem, we improve our understanding, however slightly, of the total possible system.

I will call this the principle of holothetic unification of systems that underlies their identification as such, or as "systems" in the larger sense, and that underlies our comprehension of systems science as a purposive and ordered kind of inquiry.

Each system in its design and function therefore contains partial information that is relevant to all other systems, as well as to the total system of which all are a subset. It follows that if we can understand any particular subsystem well enough, we may be able to infer in a sufficient manner the total system, without having to comprehend other or most other systems.

There would be a caveat to this, and that is that our understanding of the total would be critically limited by the particularistic constraints upon which we based our comprehension. Thus it is in the systematic inclusion and extension of our understanding to multiply occurring systems, especially upon different levels of integration, that we achieve a more comprehensive, and hence more realistic, comprehension of reality.

With improved abstracted understanding of the total system, we can increasingly apply this knowledge to other systems in order to facilitate our understanding of these subsystems as well. Hence information contained in each subsystem provides information about all other subsystems also.

Our systems science is therefore "meta-systemic" in the sense that it comprises a system of systems, in a form that resembles the structural patterning and design principles of all systems. We can hypothesize therefore a kind of second-order feedback cycle operating in our advanced systems sciences that permits us to alter our understanding and even our observability and applicability of our knowledge, based upon what we learn about alternative systems. Our understanding of a system becomes a part of the system.

The more we learn about some systems, the more we understand about all systems, and the easier it is to discover information not previously known. Furthermore, this feedback cycle describes a cyclical growth process that defines a node that is asymptoptically stable in the large and in the structure of the long run. Our systems science is therefore intrinsically progressive in the development of its knowledge, and, because the total system is regarded as infinite and unlimited, there is no intrinsic or apparent extrinsic limit to this pattern of knowledge development in advanced systems sciences beyond the phenomenal event horizons themselves.

Discovery and exploration of systems, whether these are naturally occurring or artificially constructed, does not have to be based on blind serendipidity in our quest of new knowledge and understanding. Advanced systems science is based on the presupposition that, because all systems share similar design features and are interrelated on some level or in some way, no matter how indirectly or remotely, a metasystemic system of all systems can itself be discovered that facilitates and systematizes to some extent the processes of exploration and scientific discovery.

The heuristic and functional operationalization of advanced systems science is therefore justified on the basis of its ability to be applied to all areas of knowledge and understanding in an equivalent and productive way. There is one important caveat in this claim. It is that in any particular area of application, advanced systems science and the models derivative from its operation must be consonant and consistent with the informational patternings of the phenomena that it seeks to describe or replicate. These particularistic systems, at whatever level, will tend to be isolating, particularizing and unique to the system they occur within, and thus our systems science in its application will always tend to be incomplete and imperfect.

In the previous section I mentioned in passing what I call the Reality Principle and I consider this reality principle to be extremely important to the comprehension of advanced systems sciences. It therefore needs to be more thoroughly elaborated before we can begin to construct science more comprehensively and systematically. The Reality Principle can be called the first and most basic positive principle underlying advanced systems sciences. It goes something like this:

1. There is a single, minimally integrated reality that encompasses all things possible, known or knowable.

a. In an objective sense, it is a shared or intersubjective reality, such that we, as its principal subject knowers, are a part of its reality. The limits of our shared reality is the physical relativity of the phenomenal event horizon.

b. It embraces all aspects of reality, even subjective realities, such that ultimately our subjective experience is at the center of reality. The limits of our subjective reality is the anthropological relativity of the phenomenal event horizon.

The subjective and objective are fundamentally two sides of the same coin of reality. Reality is the entire coin of our experience. We cannot escape reality, even in our flights of unrealistic fancy or in our ideological falsehoods.

2. Reality is infinite and unbounded except in relativistic ways. Therefore all finite systems are partial and incomplete parts of the whole system of reality.

3. Reality is ultimately chaotic and entropic in the structure of the long run and in the largest sense possible, because it is always subject to the law of universal change.

4. There cannot be no reality, or anything that is not a part of reality somehow. Hence,

a. All things are somehow explanable in terms of reality, even noise and entropy.

1. The total system is infinite, and composed of an infinite number of subsystems of infinite dimensions.

b. Any observable or knowable system is a partial instance of a general class of similar kinds of systems within a larger framework of systemic determination.

2. All systems are working systems--that is, they perform some kind of function that is based upon energy and information exchange.

a. Working systems can be defined as systems that maintain order in either one or both of two ways:

1. Maintaining stasis within the system and of the system independently and separately within some larger context.

2. Creating regular change or state-transition within the system and between systems.

b. Limits in a system describe boundary conditions or constraints that serve to maintain the internal stasis of the system as a self-consistent and self-defining entity.

4. All systems are imperfectly bound. Hence, they are always minimally "open" systems which openness is a measure of their relative entropy, among other possible things.

a. Limits that bound and demarcate systems generally fluctuate and change over time, leading to the developmental state-alteration of the system in the structure of the long run.

5. All systems are bound within larger systemic contexts and in turn come to encompass sub-systems. Hence,

b. Each distinct instance of a system has its own relatively unique context, hence all systems tend to be historically particularistic in their phenomenal event patterns.

c. A system as a theoretical and descriptive construct can represent a general class of independently instantiated systems as a unified or differentiated set of systems in the large.

6. Ultimately, in the total system, all systems are interconnected as subsystems to all other systems however indirectly and remotely.

a. Contexts are all encompassing in a non-exclusive sense. For any given system, we may define a contextual hierarchy of relations that are pertinent to that particular system such that relations can be construed on the following levels:

b. Contexts define both exogenous and endogenous sources of change that are both stabilizing and chaotic, such that we can describe the following quadratic paradigm for all changes:

7. Systems define cyclical patterns of deterministic relations, or control or regulatory structures, occurring within and without the systemic context upon multiple levels, and these patterns of interrelationship generally preclude the possibility of defining ultimate causes or strictly prime mover or unicausal arguments.

a. There is a regularity and periodicity of recurrent interactions that define relationships in all systems and that provide a sense of long-term asymptoptic stability to such systems.

b. There is an inherent source of uncertainty and undetermined variation of pattern of relation in all systems that provide a sense of instability to systems leading to alternative state-transition pathways. No system is perfectly determined.

8. Our understanding of systems, whether these are pure or applied, natural or artificial, is always imperfect and incomplete.

a. Generally, systematic approaches to understanding systems are both analytic and synthetic at the same time. We may seek to analyze a system in terms of its parts and their functional interrelationships, or we may seek to synthetically comprehend the synergistic functioning of the system as an integral whole.

b. Our understanding of all systems is always fundamentally anthropologically relative to our own anthropocentric and subjective horizon of knowledge.

c. Criteria of scientific objectivity are based upon inter-subjective trans-substantiation of knowledge, and upon presumed relative independence of the event phenomena to our phenomenal experience as determined by their measurability upon some objective scale or set of scales.

9. All systems demonstrate a minimal coherence that is describable in terms of mathematical equations and symbolic definitions.

a. All systems have a minimal and discrete set of core values or variables that define and govern the behavior of the system.

b. These core values and variables are intrinsically continuous and non-discontinuous, but they are unique to the system and hence are definable in mathematical or symbolic terms.

10. All systems can be said to manifest some developmental pattern as self-definitional systems, such that they exhibit some kind of developmental life-cycle that can be described in concise general terms.

a. The developmental cycles of systems are complex and tend toward chaos, such that in the long run they lead to asymptoptic instability and systemic disintegration.

b. Systemic cycles frequently recur with some expectable or predictable measure of periodicity.

Other features serve to distinguish systems as such self-definitional entities. Our definitions of systems are of course tautological to the need to define a system as such. All things are systems and are simultaneously part of other systems, and systems abound and interpenetrate one another in complex and multiplex ways. All things, at whatever level, therefore share the basic design aspects common to any and every system. We have not exhausted the list in this regard. It is important to understand how such explication of systems design features permits us some operational handle upon our scientific and theoretical constructs pertaining to systems in general and in particular instances.

Advanced systems sciences face a common set of structural challenges. These can be seen as the challenges of complexity, contextuality, comprehensivity, causality and chaos:

1. Complexity: The challenge of complexity in systems approaches is the difficulty of resolving the information explosion when analysis is carried progressively and gradationally to lower and lower levels, or to encompass broader and broader systems.

2. Contextuality: The challenge of contextuality is to recognize that all systems are embedded at whatever level within other larger systems, and in turn encompass other sub-systems. At each point in each system, there are a larger set or framework of relationships that structure and give systemic significance to that particularistic set. In the problem of contextuality, we must ask, how much context is sufficient theoretically and operationally.

3. Comprehensiveness: The challenge of comprehensivity resides in the issue that all systems are interconnected with every other system. We cannot conceive of a perfectly isolated system. Systems are also embedded at multiple levels within other systems, giving rise to the problem of multiplexity and the interpenetration or multiple integration of different systems. Our approachs on some level must embrace and resolve the challenge of comprehensivity.

4. Causality: Causality is inherently a challenge to systems approaches because the classical conception of causality implies, among other things, linear relationships between variables, and logical relationships between variables and undirectional relationships between variables. Such relationships are rarely found in systems analysis. Causality becomes multiply deterministic and underdeterministic in many systems and usually yields to complex descriptions of systemic feedback cycles and developmental state-trajectories or pathways.

5. Complementarity: Complementarity is the obverse of causality, and concerns the interrelationships between component variables of systems and their patterning interaction that can be described as complementary in the general service of the system as a whole. In other words, it describes a challenge of the interdependencies of the components of a system by which a system achieves holistic integration.

6. Uncertainty: The problem of chaos stems from the issue of inherent randomness of under-determined relations, and from variation of patterning found within systems, and from indirect causal chains that impact upon systems at many points. The challenge of chaos is the uncertainty it produces within our comprehension and modeling of systems.

1. The assumption of total systems: All systems are a part of a total system, which can be comprehended scientifically as such, and its subsystems thus contextualized within its framework.

2. The assumption of critical feedback cycles: Sub-systems gain identity in the larger system by means of feedback mechanisms that are inherent to the functional integration of the system as somehow holistically separate from its component parts or their behavior.

3. The assumption of solutions: We can develop what constitute simplifying solutions to complex systems, by identifying the main or key variables involved in the feedback relationships established within any system.

4. The assumptions of models: We can develop on the basis of our simplifying solutions alternative models of the system that allow us to manipulate the variables of the system in a controlled manner.

5. The assumptons of dynamics: All systems exist in the long run in some form of asymptoptic stability or instability. External relations and internal variations that affect the system result in state changes within the system that can lead to its structural modification, its disintegration or its development. Systems thus follow alternative state trajectories, and possible pathways such a system might follow is the basis for defining the paradigm of the system as a hypothetical and theoretical construct.

What I propose is a meta-systemic science of systems, that in itself comes to comprise a system of all systems. It is based on the study of all systems in a systematic and controlled fashion, and leads to the experimental application of artificial systems, as models, to a variety of specific problem sets.

The key design characteristics of all systems seem to be something like the following:

a. All systems can be mechanically described in terms of their functional interrelationships.

a-1. The functional relationships between parts condition the descriptive analysis of the parts.

b-1. This synergism is conditioned by and is part of some larger system or set of systems.

c. Each part forms its own synergistic unity that is conditioned by and subsumes some other sets of systems.

d. All systems define some complex form of functional equilibrium or stability of integration.

f. This pattern of fluctuation in the long term leads to state-transition changes that affect the parts and the system as a whole.

g. The patterns of fluctuation of any system obey some theoretical set of limits that define the range of normal function, within which equilibrium can be maintained, and beyond which equilibrium must yield to entropy.

b. Information can be defined as the functional parameters of relationship between the parts of a system.

4. All systems can be understood and described in terms of their information, in a manner that is scientifically useful such that:

b. We can use the models to test propositions regarding the nature of the system.

e. We can use validated models for our theoretical construction and dialectical revision of alternative paradigms.

Natural systems theory is organized upon three strata of natural information function. At all three levels there appears to be a core of related systems-theoretic principles that are applied in fundamentally different but similar ways, based upon a machine model operating within alternative mechanical frameworks. The same core appears to me to underlie and account for almost all levels of programming in artificial intelligence research. I have come to the conclusion that in all working systems, there may be a core set of theoretical principles that operate in the determination of such systems.

As working systems, involving measureable energy transactions and also informational order amenable to logical processes of abstraction, these are systems that are amenable to mathematically based analysis and logical modeling whatever the level of our understanding. It entails that we may reduce our units of analysis to discrete or continuous variables that enter into structural relationships. This can be applied with equal effect to human systems as to physical systems. I believe this describes a systems paradigm. My central hypothesis has been to delineate and explicate this core working system, that underlies all systems, and then to functionally extend it to alternative systems development.

The problem of theoretical integration of our advanced systems science underlies the requirement that our solutions are in some sense sufficiently simplifying, or at least, satisfy the problem of complexity. We cannot afford to account for all phenomena in a particularistic and detailed sense, hence our models and paradigms must get a lot of mileage before they can be accepted as such. A single theoretical construct must be at least potentially capable of construing a great deal of information, however indirectly.

Mathematical and symbolic explanation serves the purpose of theoretical integration of systems sciences in a number of important ways. They operate hypothetically at all levels of advanced systems theory, and serve the purpose of model building in systematic terms that allows us to heuristically extend and test the models.

Because systems are about the relations between things, relations can be described as variables themselves that relate the things to the system. Thus the identity of the component parts is critically defined by their relational identity within the "structural" relations of the system they function within. Mathematical explanation is important to the definition of the relational values and variables of such systems, such that they describe the nature of the system that is to some extent independent of the intrinsic values of the component parts as isolatable entities. In theory at least, the same forms of mathematical description can be applied to relational values across of broad range of different kinds of systems.

Mathematical description and explanation is a constrained form of symbolic definition of variables and values, but symbolic definitions of terms and their relations encompasses a broader range of semantic and theoretic values than can be accounted for by mathematical formula alone. Thus, in many areas of more complex systems that are derivative of more basic systems, especially in human and social systems, the language of description is frequently insufficient to the task of simplification that is required in these areas. Symbolic language itself, as a broader base than the more constraining mathematical description, often falls short of the task of sufficient explanatory power in these super-complex derivative systems.

Mathematical integration depends upon our capability to express our primes as symbolic variables to which a range or alternative series of values, a set or space, may be attached. These values are held to be numerically definable upon some scale of measurement, hence they are inherently quantifiable in at least some parametric or non-parametric form. We can presuppose a broader class of qualitative non-parametic or discontinuous values that can be described in a kind of quasi-mathematical or abstracted form of equations. This is sometimes found in highly systematized social science applications and descriptions, but their explanatory power or generalizability is frequently left wanting.

I have proposed to develop a body of mathematical models as heuristic devices that can serve in model building and theoretical construction work in all areas. These can be used as tools in advanced forms of analysis and constructive synthesis in systems science, and serve as its foundation. These models can be used in the construction of various programming functions that serve the purpose of hypothetical modeling of systems in various state-transition trajectories. These synthetic functions can be used in the analysis and heuristic testing of applied systems to various contextual frameworks, with limiting contextual conditions being built-in or defined within the framework of the function itself as systematic inputs.

"Science," as it is published in textbooks and journals, can be distinguished by several defining features. First, there tends to be the use of mathematical formulas for the concise, non-rhetorical description of basic laws or principles. Second, such literature appears to be logically coherent and empirically as consistent as possible, as permitted by the explanatory model of the thematic point of view of the paper itself. Third, and not least importantly, is the use of diagrams, graphs and illustrations of central points or complex realities. In this last sense, often a single picture can be worth ten thousand scientific words. Fourth, generally speaking, most scientific texts are defined and codifed in terms of some specialized jargon that is shared by a community of like-minded scholars and that embodies a set of shared exemplars that informs that particular paradigm.

And so it must become with advanced systems science that we develop a language, both mathematically and jargonistically, as well as a logical system of reference and inference, and a system of shared exemplars appropriate to its constructs and operationalization. We must also liberally but carefully employ diagrams, tables and illustrations as an intrinsic part of our dialectic. I believe this trend in scientific literature reflects also the folk notion that "seeing is believing." Legitimacy and credence appears to be much stronger based if and when we can reinforce our arguments with at least two dimensional visuals constructs.

I propose that there is a special mathematical system, jargon, logical-symbolic system, system of shared exemplars and working models, and pattern of illustration that in particular informs and helps to define advanced systems science in its basic core areas as well as in its derivative and applied regions. The mathematical system I have proposed to develop I have termed symbolic mathematics. The logical system I developed I propose to call relational symbolic logic. Shared exemplars and working models in a comprehensive sense describe a natural set of models that exemplify important theoretical design features of systems. Diagrammatic illustration of advanced systems science and its related and applied areas have already been developed and will continue to be so developed.

It can be said in general that advanced systems science involves a distinctive style of mathematical modeling, computational programming design, alternative design analysis and the unconventional use of knowledge and heuristic designs or devices in the contruction, operationalization and functional integration of its fields.

The symbolic mathematical language I propose for the development of these working heuristic models is based upon an understanding of an applied conception of math as symbolic language in the possibilistic and probabilistic description of statistical reality. Math is based upon rigorous, faultless numerical and relational logic. It is a restricted variety of symbolic language in terms of string theory that occurs more naturally and more loosely in human language. A strong and perhaps irrefutable case can be made for noumenal archetypes of a priori concepts like zero and the triangle, that inform mathematical theory in basic ways.

The mathematical language I propose must therefore range upon a continuum that embraces both a more restricted and rigorous set of constraints on one extreme, and one that embraces on the other extreme a minimally ordered set of constraints necessary to the symbolic reasoning on the other extreme. In general, the distinction between predictability and expectability of results is the kind of differential that this continuum expresses.

In general, I would state that the goal is the development of a single, cohesive, middle range system of terminology and constructs that embraces both dialectical extremes in an effective manner. Therefore, the operational and terminological system I propose as being appropriate for advanced systems science consists of an intermediate range of mixed and hybrid forms.

The kind of symbolic math I am concerned with is a form of math that is applied to real world problem sets at all levels, and that therefore requires additional criteria of external consistency and in turn looser criteria of strict internal coherence. Hence, this math must be capable of introducing and dealing with uncertainty variables and values at all levels of its articulation as uncertainty is manifest in the naturally occurring world at all levels.

Furthermore, this math must be capable of dealing with the inherent complexity represented in all systems, as well as with the contextual interconnectedness of all systems upon multiple levels. This means that while we may strive for elegant and simple equations to understand complex realities, these elegant equations cannot be straight-forwardly derived from direct experience. They tend to subsume enormous complex formulations that define and lead to the values expressed in such formulas.

My artificial intelligence teacher taught me in my struggle to compete with young computer wizards, that complex problems have to be approached by breaking them down into simpler units, then followed by subsequent synthetic recomposition into larger functions. I would say, in addition, based on further experience, that the initial analytical attack to any complex problem or problem set must be based upon some implicit hypothesis or meta-logical conception or definition of the problem set as a hypothetical possibility or set of possibilities in imaginary hyperspace.

Such possibilistic hyperspace is in essence an unknown landscape, or what can be called a search-solution space, that has astronomical, almost infinite size or magnitude when it is seen in a random and chaotic way such that any solution has equal probability of being the correct solution. It is an "exploded" space in its minimally ordered form. The unknown space represented by an unsolved problem set can be said to be in its original form minimally ordered, and thus maximally entropic and chaotic. It becomes the functional goal of our systematic understanding, best exemplified in terms of our mathematical languages, to be capable of resolving this problem by the reduction of systematic complexity to manageable proportions. Ideally, the most parsimonious solution is the best, most optimal and most correct solution to any delimited problem set, though we can never ultimately know if there might not be some better solution available.

We can make the following kinds of statements regarding such uncertainty and randomness in our problem-solving capacities:

1. Systemic integration produces information and yields energy for directed purposes, hence solutions to systemic problems result in working models.

2. Disordered and chaotic anti-systems consume energy without purpose and yield energy entropically. Informationally such systems are noisy and also inflexible and maladaptive.

3. The anti-entropic maintenance of inefficient systems consumes energy in an entropic way.

4. Optimizing solutions tend to minimize the inefficiency of systems and maximize their integrated functioning in realistic ways.

5. No system can be absolutely ordered or disordered, but remain relativistically both ordered and disordered on some level.

6. All systems tend to fluctuate about some stable state-trajectory, which will itself fluctuate. (i.e., all systems change in time)

For complex problems, there may be more than one optimal or best solution, or a number of competing alternative solutions that are polythetic by comparison, each solution involving trade-offs in one dimension or trait compared to the others. There may be only one ideal "correct" solution for such a complex problem that would be the most optimal with the least trade-offs. In other words, this would be the most efficient possible solution among a large number of alternative solutions. It is evident that continuous streamlining in technological development, as for instance in the history of the automobile industry or the aviation industry, leads us eventually, over many generations of new models and designs, to increasingly optimal solutions.

Computers and computer programs generally cut through such vast spaces in a very systematic and extremely rapid way. They can do this kind of linear or even multi-linear processing far faster than can people, who remain relatively slow to calculate all the variables if left to their own neural devices. But human beings have a certain advantages and an innate facility for thinking divergently that computers lack. This allows people to occassionally jump to higher orders of organization in such space, and hence to solve problems that computers are fundamentally incapable of solving, at least not without a tremendous volume of linear or parallel processing beforehand.

b. Identify and define the prime or key variables and ranges of values articulated within the delimited system.

c. Identify and define in some measured way the relational variables, both endogenous and exogenous, that lead to state-transitions within delimited systems.

d. Define optimal solutions and progressive integration of definitions leading to a more exact ideal end-state.

e. Define alternative pathways and solutions that systems may take in their development.

f. Embed the delimited system in its proper context in the total metasystemic framework, and relate it differentially to all other systems.

It is the last function that is of greatest importance in understanding the significance of relying upon mathematical description for advanced systems sciences. It should at least in theory provide a common objective language by which to achieve the necessary comprehensiveness, unification and integration of systems sciences in all areas and at all levels. For mathematical description to be capable of achieving this primary function, it must be adapted and demonstrate adaptability to a universal range of alternative phenomena. It must, in other words, be transformed and rendered applicable in a flexible way that preserves both its internal coherence and maximizes its external consistency with the world. This is of course a systemic challenge to mathematical language and modeling, as there will always be some sense of trade-off between rational coherence and empirical consistency in our modeling.

Two sets of things come to our rescue in overcoming this dilemma in the development of a suitable language of description. First, progressively integrated and complex mathematical functions can be handled efficiently by means of computational devices through systematic and exhaustive analytical parsing and processing. Secondly, human intellect, which always remains at the center of our understanding, is capable of synthetically jumping to final solution states that encompass entire orders of magnitude of complexity.

Thus, in the first pass of the delineation of our mathematical language, we must embrace the central dialectic between these two alternative analytic and synthetic procedures. In the first pass, we must be capable of delimiting and clearly identifying the problem set in the first place, in terms of its key components and synergistic functioning as a system. Problem identification is itself problematic, and has its own dilemmas. We may say something like the following:

a. Any hypothetical problem set is itself part of a larger problem set and there is some infinite or total problem set of which all others are some subset. In a sense, the entire issue of human comprehension of reality is a total problem set, of which all else is some derivative subset. I will call this the total reality problem.

b. The ultimate goal of advanced systems science is optimal comprehension of the total reality problem.

c. Specific solutions to particular problem sets tend to be relative and unique to that problem set.

d. General solutions to non-particular or encompassing problem sets tend to be generally applicable to some larger hypothetical range of problem sets.

e. General solutions are derived from the correlation of specific solutions (empirical inductionism), and in turn deduce to alternative particular solutions (hypothetical deductivism).

This ultimately defines the central goal and purpose of advanced systems science, and that is the progressive global identification and resolution of the total reality problem.

The mathematical language of description I propose for advanced systems science I will call applied symbolic mathematics and inferrable mathematical symbolism upon which it is based. As an applied form, it resembles engineering on many levels, and we can call the form of engineering forthcoming from it as heuristic and experimental symbolic engineering. Intrinsic to this language are several built-in capacities:

1. Symbolic mathematical definition of all significant terms and relations minimally applicable to a delimited problem set.

4. Progressive integral refinement of linear to nonlinear problem sets through programmable functional procedures at all stratified levels.

6. Statistical-intercorrelational sampling & measurement procedures, including heuristic uncertainty procedures.

10. Differential & rank-ordered feedback between the last and all the other levels.

In the construction of our symbolic mathematical language certain pressuppositions appear to me to be in order. These are something like the following:

1. Total Unification: Unity may be represented by the integer 1 or by the variable U, which implies a perfect state for a delimited finite, or total infinite system. We can say that in an ideal sense, 1 or U may represent either a particularistic, finitely delimited instance of a system, or it may represent the total possible system that encompasses all systems. We can subscript 1 depending on its implicit representational value. We can substitute 1 by some hypothetical variable that stands for some value or system that is a part of the total system.

In all instances and in all forms, 1 describes and encompasses both the initial hypothetical start-state and the final end-state. 1 therefore represents some inherent variable of unification, and it will be represented alternatively as 1_U or U₁

2. Perfect Equation: In any total or delimited finite system, the actual system is represented by a composite set of variables that occur on the left-hand side in the initial start-state and final end-state or stop-state, and always equals unity on the right hand side.

2a. The corollary of reversible identity. If equatability exists on both sides, then they are interchangeble and of isomorphic identity.

Only the largest sense, with perfect unification, can we achieve perfect equations.

3. Systematic Realization : imperfect Actualization or actual Imperfection: Whatever the hypothetically ideal equation, there will be some imperfect imbalance in the equation between left and right side, which will require resolution in some systematic manner. Realized expression on the left-hand side is always somehow unequal to the idealized state unity on the right hand side. The relational expression between them becomes in a sense an implicit ratio.

4. Abstract Representation or transcendant abstraction: Variables may be infered from and represent real values. Central to the success of advanced systems science is the presupposition that our language of description is sufficient and accurate to the problem of representing the reality in a minimally consistent way.

5. Improper Integration: The final end-state on the right-side of the equation of unity is always unbounded and infinite, and thus cannot be properly represented in its equal form by the expansion and development of terms on the left-side of the equation.

5. Stochastic Differentiation or Chaotic Transformation: Left-sided values are transformable in an infinite number of ways that are partially random.

6. Relative Uncertainty: There is always some measure of relative uncertainty that is expressed on the left-side of the equation and that is itself always a complex and inherently undetermined set.

6a. The principle of contextual indeterminancy. Contextual relations between systems tend to be underdetermined to the degree that they are indirectly removed or remote from the central system in question.

7. Absolute Limitation: Certain absolute limits exist for any system which define constants that can be approached by elements or instantaneous states of a system, but which can never be obtained in an absolute way.

8. Stratified Determination: Systems stratify based upon deterministic relationships that are more basic or derivative. Basic systems underlie derivative systems such that derivative systems are synergistically greater than the basic systems that compose them. Stratified determinations allow derivation of more complex formulas from simpler sets, and, differentiation of basic general formulas into more complex sets that they represent.

9. Stadial Substitution: One set of formulas may take the place of another set of formulas at each state-stage of the development of a mathematical description. This is accomplished through systematic substitution of equivalent terms within strings.

10. Gradational Alternation: Formulas may be fundamentally changed or transformed on a structural level based on several alternative principles, such that the resulting sets are not equivalent to the preceding sets.

11. Dialectical Correlation: Alternative sets or matrix structures may be contrasted or contraposed or compared within one another in systematic ways to render some measure of difference operating or occuring between them.

12. Programmable Distribution: Functional operations and complex sets may be distributed through programming structures.

13. Infinite Chaining: Open-ended and infinite strings may be produced on one or the other side of an equation. Also, an infinite number of permutations may be developed on either side of an equation.

14. Control Regulation: Key defining or regulating variables are factored into formulas that express limiting or boundary maintaining factors associated with the variables.

15. Nonlinear Exchangeability: Values may be exchanged between left and right sides of relational sets to create non-linear patterns of state-alternation.

Advanced systems sience accomplishes functional integration of its areas of involvement by means of progressive application and refinement of general problem models to integrated problem sets in reality. Thus, through functional integration that is inherently problem oriented, advanced systems science naturally crosses conventional disciplinary boundaries, and provides the framework for a truly cross-disciplinary approach to real world and hypothetical problem solving.

At the heart of the core system is a working model of advanced systems analysis and progressive application of working models. In doing the work on natural systems theory, I have come to the understanding of the similarity of principle and design underlying all naturally occurring systems, and I expect, to be found in most artificial systems as well. I believe this core system constitutes a theoretical paradigm governing all systems, hence its explication should be useful in the application of systems sciences at all levels.

The basis of advanced systems science is its comprehensiveness of approach and its functional integration as a working system. Underlying the comprehensiveness is what appears to be a basic set of universal patterns of structural relations that can be mathematically modeled in approximate and general form. The basis of this is, I believe, that a systems approach is about systematic interrelations between things however we must define these things. On some level or another, these things involve some level of energy exchange or tranfer. Often these energy transfer systems are "multiplex."

The identity and character of the things involved in these relational systems is to some extent determined by these systems. This allows us to map things at different levels as variables with continuous and even dissimilar values.

In general, systemic pattern or order has certain design characteristics about it. First, it is generally a working system of relationships, or the result of some kind of working system. Therefore, it follows patterns that are fundamentally thermodynamic. We expect several things from thermodynamic systems. First, they rely upon some form of energy entrapment or control mechanisms. Secondly, they tend to be variable in patterning, and tend towards entropy or decay. Always, there is some kind of similarity or correspondence of pattern, both spatially and temporally, occurring by relative degrees, and recurring over time, either on a continuous (recursive) or discontinuous (reiterative) basis. As a system, also, we would expect that it would be by definition bounded in some way, and that it has a finite number of discrete variables or values that compose and account for the patterning we observe. Also, to go a step further, we would expect systems to be historically developmental or evolutionary in the structure of the long run, so we should expect that the system will advance through a series of patterned state-alterations.

Operationalization allows us to construct, use and revise models in testing environments that can be regulated and controlled. Below is an outline of potential application procedures within this system:

1. Theoretical-Methodological Dialectics: The dialectics of Analysis/Synthesis arise from the multi-level interrelatedness and interconnectedness of all naturally occurring systems such that the y are everywhere and always, at the same time, systems within systems. Once we specify a certain level we must recognize that there are both synergistic patterns and analytical part-whole patterns related to that level simultaneously.

2. Correlational Pattern Analysis/Synthesis: Consideration of this kind of analysis proceeds from the observation that strict causality is historically chaotic, but only epiphenomenal to systemic organization of patterning in the universe. In other words, things are caused to happen, like hens laying eggs and eggs hatching young hens, because they are part of a larger framework of deterministic relationships. To get at the deterministic structure of this larger framework, one must step beyond the logic of direct causal explanation as inherent insufficient language.

3. Transformational Analysis/Synthesis is the complement of correlational pattern analysis, in that it entails a study of the effects and causes of pattern changes within a system, which can be called state alternation.

4. Contextual Analysis/Synthesis: Any thing that happens, or any inferrable system of relations, always occurs and is configured against a background field of contextual relationships, and of contexts within contexts, that requires some degree of specification and delimitation in our definition. The challenge and dilemma of contextual analysis is the specification of how much context is enough, necessary and sufficient, for an explanation of a particular pattern, process or phenomenal event.

5. Relational Structure Analysis/Synthesis: This suggests the operational establishment of experimental control structures & artificial research conditions & environments for the testing of theories in various levels, as well as the use of various forms of heuristic designs for modeling & prototyping of alternative design configurations.

Advanced systems theory is metalogical in the sense that it provides general knowledge structures about theory itself, in patterns homologous to the structure of theory at any level of natural information patterning. We can say that basic theoretical designs found at all levels of natural informational pattern are homologous in design principles. This is so not only because they arise one from another in a larger "hiearchy of correlations," because natural information patterning in reality appears everywhere to follow certain basic principles of non-linear self-organizational design.

This suggests a noumenal design science that is independent of any given instantiation of its patterning, as well as possibilities for its elaboration and application to artificial and alternative systems of informational construction.

Ideally, these five areas should represent a single tool kit, or rather a single set of related procedures, however mixed or complex they may be. This functional meta-model for operational systems goes something like this:

In general, within any coherent system we attempt to delimit, we can describe a circle that represents a closed feedback loop or regulatory control structure. Any such circle will not be completely closed, but will most likely have inputs intruding from it at any one of several critical junctures.

Such loops tend to be complex in the sense that they are multifactorially composed of many variables, each of which may represent its own set of loops. The entire loop itself may be part of one or more larger control structures.

At this point we employ a variant of what I call symbolic mathematics to describe the main variables and their values involved in the system, as well as to detail the interrelationships known to occur and account for the system.

Within a larger framework, we adopt relational logic to make decisions regarding the implicit rule patterns and determinations that drive the system.

In such a system, we can usually identify clearly some degree of polarity and parity, which describes a harmonic oscillatory cycle of the system, which polarity is defined by constraints or key limiting factors in the system. There may be more than one pole of oscillation within any system, in fact there may be n-poles of oscillation simultaneously cooccurring, either synchronously with one another or randomly.

It becomes apparent that the determination of the axii relevant to the system is critically important to the integration and definition of the system as such. Such axii determine the fundamental characteristics of the system, and its state-transition pathways. They describe the principle moments possible in a system, and the constraining factors that define these moments. We invoke our relational logic to determine and define the axii. It is clear that changes are most likely to occur as the result of amplitude variation along one or more axii of control. The result is that the pathway or directional axis of a system can be altered as the result of these modifications.

The polarity of a local semi-closed system describes in a local sense a pendulatory state-transition pathway in which the extreme nodal states are usually unequal or assymmetrical to one another. We can describe complex axial patterns, and we can describe as well a pendulatory cycle that represents a second order spin characteristic if we can see the central directional axis as rotating in a parallel manner about itself.

This kind of movement or moment within a system is a complex secondary pattern within the system that cannot only be accounted for by the main controlling directional axis. If it contains its own feedback cycle, then it creates complex patterns in the main system. This describes a clock pattern of cyclically recurring periodicities.

In its directional iteration, it describes the sinoidal wave patterns of a string that is plucked. Thus, such a system is said to travel in a pathway of state-transition from one point to the next within the oscillatory framework it has established for itself internally and in relation to its context. This describes a basic "chain of pearls" structure of state-reiteration. We can also describe more complex patterns of spirals and helices by such means.

Each system has some hypothetical initial start state, and the values realized at this initial stage determine in part the possible degrees of freedom and alternative pathways that the system may eventually follow.

Each system has some number of alternative state-trajectory pathways that are intermediate and which describe the hypervolumetric space represented by the system. In terms of the main directional axis, which may in fact be a bundle of composite axii in complex interrelation, we can describe various alternative possible tree or dendritic structures.

Such a branching-tree structure is commonly known to occur in nature. We can infer that states tend towards increasing differentiation and complexity, and hence towards greater potential order/chaos in the overall system. Systems can converge, but convergent systems are a sign of the emergence of new systems or the operation of some larger system. We also have the dilemma of Zeno's arrow, such that between any original state and subsequent state, we may hypothesis an infinite number of intermediate possible states. To reconstruct systemic trees from an historical perspective of hindsight confronts this problem directly. Clear transition points are often very difficult if not impossible to determine. This follows from the integrated multi-factorial aspect of all systems.

This tree structure can define either possible state transition pathways or actual state transition pathways. The directional axis that defines such a structure do not have to be linear, but may actually be either curvilinear or non-linear in structure. We can thus imagine very complex patterns that describe what can be called strange and chaotic chain reactions.

Each system has also some eventual ultimate end state, at which point the decay of the cycle is such that the following state is so assymmetrical and dissimilar to the preceding state, that it is no longer identifiable as a system or as the same kind of system.

It is apparent in the description and systematic comparison of states, whether they are presumed to be causally or deterministically related or only correlated, that we can invoke a model of statistics. This allows us to handle such comparisons such that we can identify and map the trajectories in a hypervolumetic space.

The length of time, (length of the event-string structure) of the system is its relative measure of its asymptoptic stability in the long run. The degree of fluctuation of the system about the central line of its development is the measure of the intrinsic variability or variance of the system that is a measure of its stability in the large. Systems are frequently characterized by critical transition events or episodes which result in a fundamental change in their structural patterning, such that we can say that the directional trajectory of the main line of its development is shifted or turned to some new direction.

We seek to describe for any system its expected and predictable values, its size, its variability, its complexity, the length or period of its cycles, its constraints, etc. We seek to define also its various state-stages in its developmental pattern over time, and to identify its possible pathways, its probable pathways and its history and any critical determining factors that by chance may have impacted the system. At this stage, we invoke a model of alternative intelligence programming to construct functions that can deal with this level of description for the system.

To a great extent, historical patterning of systems is a patterning of unintended consequences. We invoke experimental heuristic systems to explore alternative possibilities and to factor random or uncertain determining factors into the system. Our heuristic variables and designs subsume and thus encapsulate uncertainty in our system, and allow us to put a handle upon such values in the testing of our models.

We yield therefore a hypothetical model for any given delimited system as well as some defined set of conditions by which to test and revise our model for its degree of relative fit to reality.

We can within such a framework of description develop sets of mathematical and logical formula that describe each part of such a system. We can also develop algorithms and functions that can develop such a system in a complex way that remains as representationally as realistic and accurate as possible.

Thus, the operational paradigm that we seek consists of a core abstracted set of formula that are capable of describing such structural systems in basic ways, and that are also capable of being adapted to the abstract representation of actual systems occurring in reality.

We can define correspondence as a measure of the degree, direction and manner that two different systems or sets of points respond in a similar ways to similar kinds of stimuli. The similarity of two systems or sets is not just analogical in character, nor is it directly causal, rather they can be said to be of a complementary nature. Complementariness of relationship, or the degree of interdependence, is the basis for correspondence between systems. We can further distinguish between general or theoretical, natural, logical or hypothetical or mechanical or experimental correspondence depending upon the relational terms that we identify, the analytical dimensions that these terms frame, and the standards of comparative measurement that we adopt as evidence for these dimensional frames of reference.

Correspondence is a general term that I have adopted for the systematic identification of relationships and relational patterngs that cooccur and recur in complex systems and chaotic super sets. Correspondence is related statistically to the correlation coefficient, or a general measure of how to comparable sets "move together" or are similar or dissimilar, but it moves beyond correlation and intercorrelational analysis in permitting us to specify deterministic directions and unequal equilibria in relational patterns that we can observe, and thereby to predict in a general way, or to state an "expectation" of likelihood of behavior given such and such conditional constraints. We may say that two similar sets of relationships that occur in parallel fashion with two different and independent sets or systems, correspond with one another, even upon different analytical levels of integrative articulation, because they coexist within the same relativisitic framework in which similar kinds of mechanical principles are said to be operating. Thus, we understand that two different cultures correspond to one another, even if the patterning of either cultures is very different, if both these cultures articulate in certain ways consistent relationships that correspond. Similarly, we can state the same kind of correspondence for two different kinds of living systems that operate in similar ways but in different ecological systems. Thus we observe in the natural biotic world many examples of parallelism and convergence between completely independent systems, in large part because the framework under which both sets of systems are operating correspond to one another in basic ways. Correspondence theory and method becomes therefore a systematic approach to handling very large systems in a manner that permits us to speak in simplified terms that realistically represent the meta-system of which the real systems that we observe are construed as examples or demonstrations.

Correspondence between complex systems or super sets is not the kind of one-to-one correspondence that is generally associated with this term. What can be called complex correspondence is a measure of the degree to which the profiles and landscape of systems resemble one another, especially along certain dimensions of analysis. Two very different kinds of systems may exhibit near perfect correspondence along a single or limited set of dimensions that are not otherwise obvious to observation unless analysis along those dimensions are systematically made. We may say that the correspondence of two systems is not direct correspondence, which implies a kind of analogy of relationship, but are abductively mediated as being similar results of similar kinds of eidetic structures operating upon different sets of data points.

Complex correspondence can be said therefore to be indirect correspondence, and to be implicit to a system. Evidence for such correspondence may be obvious to the naked eye or presented to raw experience, but this evidence will appear incomplete and only partial in nature. Complex correspondence is therefore also never, or at least, very rarely, perfect correspondence.

Another way of looking at the problem of correspondence is to ask ourselves to what degree do two systems or things correspond to one another, versus some other or alternate system by which both might be compared. In a sense, it gives us a way of comparing systematically apples and oranges.

The basis for correspondence theory is the presupposition that all things and all systems in nature and in reality are indirectly related to one another. Independence of systems becomes relative to the frameworks that we specify for such systems, and only coincidental or contemporaneous systems that coccur can be said to be relatively independent. For such a hypothesis of independence to be made, we must determine that the two systems are spatially separated in such a manner that, at least upon the level of analysis we identify, no significant interactions are taking place between the two systems. If we were comparing the size, shape and weight of two large objects of mass, we might speculate that such systems may be relatively independent of one another in terms of their chemical composition, but we would be hard pressed to prove that the two systems were not interacting with one another in a gravitational sense. If we are examining two different archaeological sites that are widely space in time and place, and yet we find remarkable patterns of correspondence between them, we would be hard pressed to say whether or not the sites were somehow "related" in a chronological way or whether they just exhibit a kind of independent correspondence within similar kinds of frameworks.

Relationships in corresponding systems can be said to be structured in similar ways.

The basis of determining correspondence between two unrelated systems is what can be called the identification of complementarity that relates the two systems to one another. Complementarity of design or relational patterning between two independent systems can be said to be the measure of the degree of holistic integration achieved by these two systems.

Determining systematic correspondence between two or more complex systems or supersets was an impossibility before the advent of digital and analog computing, and with the increasing power and efficiency of mechanical computing, the capacity to establish in a systematic manner correspondence between different kinds of systems has increased many fold.

The kind of correspondence that we are speaking about is in a basic sense quantitative correspondence, and this is by definition non-qualitative. Qualitative correspondence between systems can only be achieved in broadly or loosely descriptive terms, or alternatively in theoretical and rational terms, depending upon the similarities of property or framework of two different systems that we are comparing. Qualitative correspondence works best upon a very basic level of description and upon a very general level of theoretical abstraction, but we cannot depend upon this form of correspondence alone without the evidentiary support of quantitative forms of correspondence. Usually, with quantitative correspondence of complex systems, numerous variables along multiple dimensions are systematically compared for relationship and degree of similarity between the two systems. We can specify that two sets of complex variables will show a significant correlational pattern of similarity or contrast, if the two sets of variables correspond in a complementary manner to one another. In other words, both sets of variables are governed by similar principles or general frames of reference.

Cross-correlational analysis represents a species of quantitative analysis linked to a specific form of data. Cross-correlational analysis involves tabular data arranged in arrays of rows and columns. This type of data is common in social scientific and psycho-cultural research, and should be considered to be a natural form of knowledge representation of information, especially of certain categories and classes of data. This type of data is referred to as "paradigmatic" in the sense that it is theoretically unified and underlying this paradigmatic presumption is the implication that the data contained within the table is somehow inter-linked in a theoretic unity to some partial and imperfect degree.

Cross-correlational analysis therefore involves the systematic, quantitative elucidation of what may be referred to as underlying correlational structures that are implicit to the organization and patterning of data in such matrix tables, and that are held to be possibly "meta-paradigmatic" in the sense that this structural patterning may be repeatable across alternative matrices and may encompass and serve to explain a number of different alternative data-tables drawn from the same host population and their implicitly related paradigms.

Cross-correlational analysis involves descriptive statistics, but is itself not strictly statistical in the sense of the basic presuppositions of randomness underlying statistical testing does not underlie cross-correlational analysis in the same way. Cross-correlational analysis picks up where conventional statistics leaves off, with the presupposition of underlying structure influencing the patterning of the data distribution. Cross-correlational analysis is therefore not a means of testing hypothesis in a statistical manner, but for exploring the data in alternative ways that render hypothesis formulation more available and explicit in the first place.

The underlying correlational structures of data tables are always (a) implicit and always (b) hypothetical. Thus they are never available for direct determination, but must always be inferred in a probabilistic manner. Because they are always hypothetical, they are subject to the conditional constraints of statistical description and statements of probability. Hence they are always relative and nonabsolute in a mathematical sense, though the mathematical procedures that are involved in their analysis are true and correct.

The type of solution that cross-correlational analysis represents for complex informational systems is therefore only partial and semi-deterministic. It does not represent a total or net solution to the problem of underlying order in the sense of resolving a Von Neuman type bottleneck by reducing the "information explosion" algorythmically induced by a search for a simplifying solution to complexity problems. But by being linked to tabular data that represents supposedly real relations upon a theoretical level, cross-correlational analysis offers the possibility of partially determining significant underlying structures that are empirically based, and upon which discrimination tables and other forms of inference-drawing frameworks can be subsequently constructed. These processes may prove to be heuristically very useful to the construction of expert systems and other forms of A.I. programming such as genetic algorthyms and neural networks.

My interest in cross-correlational analysis arose out of a need to deal effectively with a tremendous amount of diverse kinds of reponse data from multiple, partially overlapping samples, many of which in a strict conventional sense did not meet the criteria of randomness, and yet that showed without a doubt a realistic patterning of relationship and determination.

Cross-correlational arose out of a need to coordinate and establish systematic patterns of difference and similarity between complex distributions of data--in ways that might otherwise be inobvious, counter-intuitive or not directly ascertainable through conventional forms of analysis and representation. Cross-correlational analysis arose in need of a basic descriptive system for complex data sets derived from realistic conditions of elicitation that rendered the presuppositions of statistics impossible--a form of analytical description or descriptive representation which precedes and in part preconditions explanatory inference in the framework of such inherent complexity of data.

Cross-correlational analysis provides a context for the integration of quantitative and qualitative forms of information, and for the inductive construction, operationalization and validation of theory and topical domains that are otherwise nonquantitative and only qualitatively represented. To the extent to which such a form of analysis is ultimately tethered to a specific set or range of data sets that are empirically rooted and measurable, such analysis can be claimed to be empirically consistent and representative of actual, if hidden, relationships of patterning in the data.

The possibility of cross-correlational analysis rests upon the following premises:

1. A strong correlation may represent a functional relationship, but a weak correlation represents a probable lack of a direct functional relationship.

1 a. A stronger correlation is more likely to represent such a relationship than a weaker one.

1 b. In a complex and large correlational matrix, even low correlational values may be significant indicators of a functional relationship.

2. Correlational sets of a matrix may be grouped and described statistically in a meaningful way--they form a curve the characteristics of which (mode, median, mean, etc.) may be evaluated.

3. A correlational matrix can be reorganized in different ways, resulting in different descriptions of the resulting tables--these values can be compared (i.e. rows versus columns) and evaluated for their relative fitness and theoretical import.

4. The rows and columns of a correlational matrix represent hypothetical dimensions of analysis that can be topically characterized.

5. In a complex and large correlational matrix, even low correlational values may be significant indicators of a functional relationship.

6. Statistical description in Correlational Analysis assumes a special data type, that I refer to as "cardinal" form of data, that allows itself to be construed alternately as either parametric or non-parametric, or discrete or continuous, depending on the circumstances of its definition. This data alternate type is, furthermore, a relative and non-absolute form of data.

A correlational matrix forms an uneven landscape of values that can be topographically mapped in detal. It is the pattern of this landscape that yields insight into the functional organization of the data. This patterning may be carried over into second and subsequent order cross-correlational matrices.

Underlying cross-correlational analysis are numerous suggestions of structure and functional relationships between alternate sets of data. The central hypothesis always to be tested in cross-correlational analysis is whether or not there might be some form of predictive "structure" underlying correlational patterns within different matrices, and if so, then how might these implicit correlational structures be further elucidated and evaluated. Evidence for such correlational structures suggests that not only will strong correlations occur within a matrix, but these correlations will co-occur and even reoccur, and will tend to cluster in meaningful and repeatable ways in numerous alternate data sets.

At this point, cross-correlational analysis resembles cluster analysis, and a form of cluster analysis as representation is a spin-off of correlational analysis. But cross-correlational analysis extends beyond mere clustering of the data to make systematic assertions and to draw inferences about the underlying structure that yields a dynamic and partially predictive model--one that can be tested and deliberately manipulated in subsequent experiments.

The close relationship between correlation and linear regression suggests that linear relationships may underlie correlations in significant ways. There is a tendency in large correlational matrices for data sets or scores to form "natural" clusters or statistically uneven distributions which strongly imply such underlying linear structure. It is the search for, representation and analysis of and the inferences drawn from such distributions and clustering of data into natural sets that drives cross-correlational analysis as a fruitful form of research.

Cross-correlational analysis constitutes an "hypothetical-inductive" form of exploratory, empirical research, that is bolstered at points by deductive hypothesis testing of derived inferences. The theoretical implications of cross-correlational analysis will be explored more thoroughly in the final chapter.

The standard Pearson product moment correlation coefficient contains a great deal of information about a distributed set of data. That correlation matrices are grouped by column-row headings entails a higher level patterning between coefficients, which represent an astounding level of complexity. The comparison between similar or different matrices, or of parts of a matrix with one another, makes possible a level of analysis that is intuitively interesting, and yet which is difficult to represent graphically in any simple, straight-forward way, or that is also difficult to demonstrate mathematically or statistically.

A correlation may be calculated for any equal-sized set of points, whether an actual relationship exists between the points or not. Correlations may be indicative of only spurious or quite superficial relationships of otherwise quite different sets of points. Apples and oranges may have high correlations on measures of roundness and diameter. Thus, unrelated sets of points may by chance happen to have a high correlation, and very closely related sets of points may by chance have a very low and apparently insignificant correlation.

Thus, strong correlations of apparently related data sets may actually belie the indirect influence of a third or more unknown set of variables, or else may just be a random ocurrence between two data distributions between which there is little or no other functional relationship at all. On the other hand, correlation coefficients that are seemingly low and insignificant, may actually belie an important functional relationship between two sets of data points.

Two main considerations drive the use of correlation coefficients in the analysis of data--that sets of data are united theoretically or hypothetically in a non-random way, and random sets of data will tend, on average, towards low correlation. When sets of data are linked theoretically, then even minor correlations may have significance. When especially large, naturally occuring sets of data tend toward unusually high correlation, it is probably indicative of some kind of structural, non-random relationship between these points.

The second problem is that though a high correlation may be indicative of a functional, deterministic relationship, the precise nature of this relationship is neither obvious nor directly available. Thus a great deal of correlational analysis has to do with the systematic elucidation of the underlying functional structure of a given correlational matrix, if one can be said to probably exist.

A correlational matrix represents a set of relative values of interrelationship between points that are hypothetically united under related topical domains. Any correlational matrix contains a great deal of information--more than the superficial landscape of its topography--that is available to analysis. Correlational matrices are therefore under-utilized for their potential analytical value. That correlational matrices are hypothetically united under supposedly related domains subsumed by the row and column headings entails that there is a hypothetical and implicit internal structure to the table which allows for a sophisticated and systematic reading of the data, and also that allows for the possibility of the systematic comparison and interrelation of different matrices that can be held to be theoretically or paradigmatically united.

It is important to construe correlational analysis in the kind of framework in which it arose, and that is the systematic comparison and description of psyco-cultural data that was gathered and organized on the presupposition of "cultural consensus"--that cultural patterning is shared by culture bearers. This sharing is somehow fundamental to the understanding of this patterning and its daily reiteration and reinforcement, and that sharing is empirically available for analysis and is partially indicative of the influence of culture. Now sharing may be incidental, may belie a lot of diversity within a cultural orientation, and may be the by-product of rather complex reasons. Cross-correlational analysis appears to be especially suitable for the analytical description of these patterns, and for the hypothetical elucidation of the underlying "structures" which may be held theoretically to account for such patterns.

Elucidating the patterning of complex psycho-cultural data represents the basis of cross-correlational analysis, but as a technique it can be systematically extended to embrace a more realistic description of naturally occuring "cosmographical" systems, and we can see that it may represent a fundamental aspect of information theory. To the extent that "culture" is ordered as a naturally occuring system, we can see that the cross-correlational elucidation of its patterning is but the beginning of many possible applications for this form of analysis in human social life.

A correlational matrix contains a field of information about the interrelationships between the "things" labeled in the column/row headings. It is a central thesis of this study that correlations represented in such matrices may be treated for heuristic purposes like any distributed set of data points--at an ordinal level as relative expressions of "distance" between the things they associate, such that the normal sets of descriptive/predictive statistics that are applied to any other data points, may be applied to this correlational data in a similar manner.

Thus we may compute averages, z-scores, or linear regression from these points--at the same time we can employ chi-square analysis and perform non-parametric correlational analysis upon the table or related tables. These statistical computations are virtually identical to those done on normal sorts of data, except that they come to have special heuristic importance--they allow us to make statements about the relationships between the "things" rather than about the things themselves. Thus what it is that cross-correlational statistics is describing is fundamentally different from what formal descriptive statistics derived--they are measures made on relational measures, rather than on "points" or isolatable "entities" actually occuring in reality.

Furthermore, a correlational table represents a "closed sample" with known data-points and specific degrees of freedom. The cross-correlational statements we can make in this regard remains the same no matter what the size of the original sample, or the size of the host population that was sampled. The correlational matrix always has the same basic primary structure regardless of the dimensional size of the matrix or arrangement of data within it. Thus each matrix contains a set of hypothetically related values that can be arranged within the same dimensional space, and the structural patterning of which can therefore be compared. This aspect of correlational matrixes and cross-correlational analysis is fixed and in a sense "absolute." It therefore provides a common anchor point by which very different correlational structures can be systematically compared and evaluated.

We can therefore calculate the p-value of the cross-correlational description of the data according to original sample or population sizes. If known, it is a value that is to some extent the measure of significance or independence of the derivative cross-correlational structures in relation to the orginal data from which they were derived.

If several correlational matrices were computed from different samples drawn from the same host population, we would then be able to evaluate the degree to which the sampling and correlational matrices resembles the actual relational characteristics of the host population by the extent to which the cross-correlational patterns are significantly different or similar to one another.

Correlational matrices contain information about hypothetical "correlational structures" which are held to be inherent to the actual patterning of the host population or original data set. Correlational objects are not objects in the conventional sense but relational representations of the dimensional space within which all the related correlational values are theoretically contained. They are statements of hypothetical spaces in which functional relations are held to occur with a predictable frequency pattern. Correlational structures therefore are fundamentally spatial or "areal" structures, rather than being point values. Such dimensional space has signficance in the sense that tightly defined areas represent "peaks" of a complex landscape. These peaks sit on the surface of the table like ice-bergs float on the surface of the ocean, disguising and yet indicating a mass of hidden information below the surface.

Correlational values thus contain information about the probable relation between any two or more sets of points within this space, the likelihood that these points are coincidental, simultaneous, or functionally interdependent. Thus movement within the correlational space represents a continuous projection and modulation of the structure through time, and problems of temporal patterning can be represented through the use and analysis of cross-correlational analysis--revealing linear relationships between alternative or related data structures.

A correlational table (an inverted matrix) then comes to also comprise a special kind of discrimination table and hypothetical search space--we are interested in deriving inference trees from such tables based upon the relative saliencies and "peaks" of probability values within the structure. This allows us to speak of precedence of changing structural patterns of the underlying structure, a pattern of change that is to some extent predictable and partially determined by the structure.

No single correlational matrix contains enough information by which to completely analyze the underlying structure from which it is drawn. This is why cross-correlational analysis proceeds with the descriptive and comparative analysis of two or more matrices that are theoretically related. The process of cross-correlation allows us to elucidate more information about the underlying correlational structure than would otherwise be possible from the analysis of only one matrix.

Not every correlation of a matrix is necessarily significant or represents part of the underlying structure. There will occur within any matrix a clustering of correlational values and it is this natural clustering that is most indicative of the underlying pattern within the matrix--two clusters of values may have high positive values within each, and yet have a high negative correlation between the two clusters themselves. Any such clustering will tend to encompass only a few of the total possible number of values represented within such a matrix, though there may be indirect correlational clusters that interconnect these with other value sets.

In general, a correlational matrix that exhibits some meaningful ordering or clustering, is indicative of an underlying correlational structure, such that rearrangement or alteration of values will produce predictable patterns of clustering in the resulting matrix. If such an underlying structure indeed exists within a matrix, then this structure will tend to be consistent and to manifest itself in varying but stable forms in different matrices or arrangements drawn from data of the same original host population.

It is possible that humans think, at least in part and upon a very basic level, in cross-correlational patterns by which we derive estimates of similarity/difference of successive or alternate structures of meaning. We may therefore refer to cross-correlational "networks" and chaining activity of the human mind, which may be something more specific or more than metaphorical compared to "analogical chaining." This suggests that our minds may actually be hard-wired at some basic level to automatically perform a kind of complex analysis that is involved in cross-correlation, and that symbolic templates may exist, constructed and precipitated from experience and more or less available within our memory, which in effect constitute cross-correlational transform operators. Such structural patterning of the brain may in fact represent mental "devices" by which we organize and analyze our realities at a basic level of pattern recognition, intuitive preunderstanding and rational inference production. Because such processing would be mostly done automatically, we are rarely reflexively aware of exercising this mode of thinking.

The logic which guides this mode of thinking may also be different from the more conventional understanding of logic--it may utilize a more flexible and informal form of abductive reasoning structure which permits modus ponens type fallacies of deriving an antecedent from a consequent, if such derivations are rooted inductively in the experience of past occurrences or in conventional preunderstandings or "common knowledge" or "common sense."

This type of reasoning is more befitting correlational structures in which simultaneity of co-occurence is always present, and, being a less restrictive nonstandard form, it therefore permits more flexibility in the inferential interpretation of the correlational structure than would otherwise be possible.

The nature of the data in cross-correlational analysis takes a particular form--it consists of what I refer to as X number parallel data sets which are aligned such that for each data in any given set, there is at least a one-to-one correspondence with a subset of data points from all the other matched sets. Such data sets are common in psycho-cultural research where individuals or subsamples are being compared along a suite of the same sets of traits, or where multiple dimensions occur, each represented by a range of similar data. This permits the analysis of correlational data in both depth-wise "parallel" fashion, and breadth-wise or "cross-wise" fashion, a form of dual analysis that opens the possibility of cross-correlational analysis in several different directions.

Secondly, the data, being thus constrained, while hindering our ability to use data that does not meet the requirements (i.e. unequal sized data sets), at the same times allows us a degree of control over the data that we would otherwise not have. All correlational data tables of whatever dimension X are X squared in size, and this permits us to conduct cross-correlational analysis upon the matrix.

Cross-correlational analysis represents an experimental and exploratory form of data analysis based upon the manipulation, comparison and inter-correlation of different correlational matrices that are derived from the correlation coefficients which measure the degree to which equal sized sets of variables move together, regardless of their relative magnitude. In short, cross-correlational analysis takes two or more correlation matrices of equal dimensionality, and rearranges the data in order to construct a correlational matrix based upon the aligned first order matrices. Alternatively, it takes different, equal sized sections of the same correlational matrix, and compares these by a second order correlation matrix.

Sometimes third and even fourth order matrices can be constructed out of previous intercorrelation matrices, especially if such analysis is strongly motivated by theory and the occurrence of a significant patterning of the correlation.

The first and second order correlation matrix can be reconstructed in different ways, or analyzed in alternative ways, enabling different kinds of intercorrelational matrices to be derived. For instance, rows and columns of a first order matrix, if equal in number, may be compared in a subsequent table based upon their intercorrelation--they may be rank-ordered and compared in this manner in an alternative form of correlational matrix.

Techniques of cross-correlation lead to different ways of representing data and the relationships between sets of data that enable us to draw inferences systematically and to statistically evaluate certain kinds of inferences about such data, as if the data were nominal, ordinal or rank order in type. It enables us to see patterns of interrelationship between sets of data that would not otherwise be apparent, and to infer from such patterns the likelihood and degree to which a functional relationship may cohere between the data points.

In a sense, the information contained within an intercorrelational matrix is fundamentally different from that of the usual correlational matrix, as it demonstrates the degree to which each part is related to and representative of the whole set of variables--information about the total set of relationships is therefore evenly distributed between all the different points contained within such a matrix. This information itself can be rank-ordered or arranged in meaningful ways that allow us to make inferences about the data. It is this underlying distribution of relational information that ulimately permits structural analysis of compared matrices to proceed--where gross unevenness occurs or recurs in the redistribution of values, especially in repeated alternate instances, where none would otherwise be expected, then such unevenness points to a semi-deterministic patterning between the original dimensions being compared.

Any set of points may be correlated with any other set of points, whether an actual relationship exists between such points or not. The only stipulation of different sets of data in their correlation is that they be of the same magnitude in terms of equal number of degrees of freedom. Though anything may be correlated with anything else, it is also the case that when and if a functional relationship exists between two or more sets of data, then definite patterns of correlation will be apparent and will tend to be statistically significant, and will take on certain characteristic patterns that will not be evident among unrelated and basically random sets of data. Thus, techniques of intercorrelation can be systematically exploited for the discovery of hidden functional relationships between different kinds of data, and for providing insight into the causal direction and theoretical significance of such relationships.

Correlation and intercorrelation matrices can also be converted into discrimination tables or histograms, or used in different forms of data analysis and representation, such as multidimensional scaling, cluster analysis or factor analysis.

Cross-correlational analysis can be time-consuming and therefore costly in research resources. The number of possible intercorrelational matrices that can be constructed and derived grows exponentially with the size and number of sets of data. It can yield a great deal of information at a level of complexity that is frequently difficult to decipher or interpret theoretically, or which may belie a fundamental spuriousness or triviality of real relationship. To some extent these draw backs can be offset by the introduction of systematic means of cross-correlational search and analysis, especially by means of computer, and by the close coordination of such analysis with theoretical interpretation and inference derivation, and, finally, by the construction of an integrated data-base or computer program that can take over and exhaustively conduct such search analysis on raw data.

But the reward of conducting such analysis and search is not to be measured in terms of its cost or complexity, but in terms of its productivity of theoretical insights, inferences and in providing a systematic means by which to construct a sophisticated rule-based inference engine derived from the patterning of the data. These advantages far outweigh the disadvantages. It is simply wrong to discard cross-correlational analysis on the basis of its fundamental lack of statistical determinancy that is all to often presumed away in other forms of data analysis, as it is a method that is far too natural and productive of insight.

[1] The point of departure for this work in advanced systems science, and what fundamentally separates it from the previous work in natural systems, is that it begins with the problem of the self-reflexive role of knowledge in the construction of our sense of reality. This is ultimately a separate problem than the question of physical reality itself, in which the question of our ability to know is assumed away and held in control except perhaps on the horizons of our knowledge of the physical universe.

[2] It can be demonstrated that a mechanistic view of the world is an inherently relativistic one, and therefore it can be extended systematically to embrace exotic phenomena that are not conventionally a part of mechanical explanations, especially not in any classical sense. Because it is relativistic, we can also correctly say that all naturally occurring systems are fundamentally non-linear and underdetermined control systems, by innate design.

[3] In the previous work, we took for granted the question of the objectivity of our knowledge, and thus planting it as a universal reference point, we assumed that the physical objective reality was all encompassing and embraced even our ability to know itself. In this work, we do not take this question for granted, but we plant instead as our universal frame of reference the inherent problematic of our ability to know reality, especially our physical reality, in fundamental ways. It is presumed therefore that what always lies behind this problematic framework is the solution of reality itself. We have shifted our fundamental coordinates and reference points, but the central dilemma of the relativity of our knowledge remains regardless of our starting point or frame of reference.

[4] The danger of this is to fall into the complacency of accepting a classical conception of a non-relativistic world in which knowledge has some final sense of certainty that is attachable to it. We know this not to be true, or we've learned that this is never the case in reality. Hence, there is always some sense of fundamental discrepancy between what we know as a conceptual system that is at least internally coherent, and what we experience in reality as a phenomenal system that is at least minimally consistent with our knowledge

[5] I would call it a form of rational truth, but it is not clear to me exactly what "rational" might mean in this case. Neither would I call it "logical" truth as well. I do not know if a moral system that is descriptive/prescriptive would be forthcoming from its elaboration or not. I would not prematurely say so, or otherwise, though any kind of moral formulation, no matter how metaethical, must be always critically suspect of some kind of relativity of values.

In a sense, such truth cannot be scientific, as science cannot ask and answer fundamentally non-scientific questions. It is essentially a form of metaphysical truth, thus it transcends the physical parameters of scientific systems. It is neither a purely abstract system in the same sense that mathematics is. I will not go so far as to call it a form of spiritual truth either, though this may be related to it, and may point towards a fourth kind of truth system lying somewhere beyond.

[6] I have attempted to do this in a synthetic and integrated manner that attempts to account for all types of abstract systems in equal measure. I have intended to do it in a way that remains faithful to the objectives of advanced systems science as a nontrivial system of knowledge and inquiry in the world. To do justice to this problem set would require perhaps an entire series of related works, but it is beyond the scope of this primer to attempt such detailed excoriation of all the implications entailed by such problems. I soon exhaust my own limits of tolerance of the obtuse and esoteric.

[7] What I do hold strongly to is what I refer to as an objective standard of scientific truth as the basis for all knowledge, and as a cornerstone for all systems theory which makes a claim to being scientific. I am not interested in quibbling on the head of a pin, or in exploring the infinite meanings of a grain of sand. This does not mean either that I do not have a strong streak of humanism or the humanities coarsing madly and somewhat illogically through my veins.

[8] Before the hard scientists of the physical persuasions should be so overly proud of their accomplishments in physics and chemistry, I would like to remind them that in their theorization they are not completely free of the conundrums and paradoxes that their symbolic language, or their paradigmatic communities, are prone to. In theoretical construction, physicists can be as muddle-headed, if not more muddle-headed, than the soft-shelled, namby-pamby brained social scientists. Indeed, because clear-headedness is in such demand and has such a premium precisely where the data is least clear as such and precisely where relationships are least certain and most equivocal, I would venture the claim that one-on-one a good anthropologist worth her salt tends to be less muddle headed, not more, than his hard-science counterpart, who too often takes his facts, his ideas, and his truths, for granted.

[9] My aim in this first part is not to determine a universal system of abstraction, as I do not think one exists and I think it is silly to presume anything we can construct can accomplish this. My aim is first to elaborate what I would refer to as a systems-based philosophy of science, one that is capable of standing sufficiently for all scientific knowledge and brings all scientific knowledge under a common symbolic umbrella. The next concern is to explore what I take to be relatively unchartered areas of mathematical systems, as systems of abstraction and as "abstractions" of systems. Finally, my aim is to elaborate a framework for what I prefer to call "possible systems" which, I guess, would be a framework for knowledge systems in general.

[10] Furthermore, and perhaps most importantly, I want to comprehend how alternative logical relations cohere in symbolic systems, gain expression through the language and sense of mathematics, and can lead to the construction of knowledge systems and the organization, manipulation and transmission of information, particularly as this is advanced by theory of Automata and the question of Intelligence, to which I will hopefully return in a later chapter, in a later part, of this text. Perhaps this is already overly reductionist, as one scientist's model may be another philosopher's muddle, but it is the best I can come up with at this short notice.

[11] A perfectly ordered or determined system would be some kind of perpetual motion machine. A perfectly undetermined system would be a total energy reservoir or perfect vacuum that reached absolute zero. It would be the equivalent of saying that there is nothing. I believe these ideas may be interdependent to one another, such that our sense of order, whether absolute or relative, is founded implicitly upon some sense of disorder, and vice versa.

[12] Thus a standard frame of reference should permit us to make systematic inferences relating to the functional structure and structural functioning of a system. I believe that gestalt theory applied to metasystems may provide such a standard frame of reference.

[13] . I deal only with those issues that I consider most important to our sense of reality and worldview, and that cannot be excluded if we are to entertain any illusions about being comprehensive. Though I make a stake in comprehensiveness of perspective, I make no claims about being complete or exhaustive in this regard or in any other way. It is a beginning for other work, hopefully basic, ground breaking and foundational in some respects, but it does not stand in place of work that remains to be accomplished in the future.

Meta-Systems	Intrinsic	Extrinsic
Chaotic
Deterministic

	Exogenous/extrinsic	Endogenous/intrinsic
Stabilizing	Stabilizing Exogenous	Stabilizing Endogenous
Chaotic	Chaotic Exogenous	Chaotic Endogenous

	Philosophical Precepts/ Theoretical Constructs	Computational Modeling/ Heuristic Design	Historical Analysis/ Synthesis	Experimental Testing/ Emprical Research	Engineering Design Development/Production Cycling
Advanced systems
Physical systems
Biological systems
Human systems
Artificial systems

Metasystems	Abstract	Real	Natural
Linear Simple
Linear Complex
Nonlinear