Meta-science is the philosophy of science framed meta-logically from a scientific point of view, in terms that can be said to be denotatively scientific. Science as a human knowledge system has its own history and sense of development. Meta-systems science is a formal and systematic approach to scientific knowledge systems of various kinds, achieved through independent verification and logical non-contradiction. It can be said in general that any ideological system, to achieve symbolic unity and closure in reality, will encompass and embody certain kinds of logical contradictions that are the result of certain kinds of pragmatic fallacies in the application of scientific knowledge. For a science to claim the metaphysical status of being truly non-ideological, it must pass certain tests of empirical or observational realism, communicative efficacy and logical non-contradiction. This is most often easier said than done, because symbolic ideological implications enter into even the basic terms and terminologies that we use to define a scientific worldview, at any level. This forms the ultimate limit to science, the limitation of the anthropological relativity of our scientific knowledge. This limit underlies other kinds of critical limitations in science, for instance certain physical relativities in the observation limits of our knowledge. Though we may not be able to ultimately transcend these kinds of basic constraints in scientific knowledge, we can develop devices or means for achieving indirect observational and ideological parallax for the objectification of these limits as such, and for ascertaining in some probabilistic manner the possible realities beyond.

This work Metasystems deals with a level of reality that involves interconnections between knowledge domains on one hand, and real or hypothetical connections between real systems on the other hand. It is evident that real systems are interconnected almost upon every level of their articulation, and that we only need to scratch the surface of any one particular system to find nested each within another and so forth ad infinitum. As a result metasystems science is an attempt to deal in a coherent and systematic manner with the inherent complexities of both real systems in natural contexts, as well as with our knowledge and understanding of these systems, and also, as a result, the systems that are created as a result of our knowledge and interaction. The central thrust of this work is to seek a common ground and operational approach to the study of any and every possible system, in order to comprehend that system in its larger metasystemic context, in terms that do justice and faithfully represent the real complexities involved in its description and understanding. Metasystems approaches and possibly answers the problem of very large and very complex systems, or super-systems, but it does not claim to be complete or sufficient to the challenge of such representation.

There is a critical sense in the consideration of metasystems from a scientific and hypothetical perspective that variables and values occur as statistical results that do not in fact exist in any real form. The classic example is the family with 2.5 children. There is in fact no "average" anything, much less a family with two and a half children, but as a composite value, statistically defined to represent large classes or numbers of entities, or sets, such composite statistics may in fact be more accurate and more real than the physical entities they purport to describe. We do not throw away reliable statistical measures because they do not directly describe anything in reality. We consider them to be valid in a directly non-empirical manner, though they were ultimately derived, at least in principle, from empirical measures or parameters. As such statistical measures are part of a pure, abstract conceptual realm that, in the form of Plato and Aristotle, are no less real than the objects to which they get attached.

This issue is dealt with in the first part of the work, but it is important to emphasize the role and place of mathematical and conceptual abstraction in the heuristics of scientific operations and methods. Science only as empirical description is vacuous of meaning and cannot elicit the general rule patterning that governs natural behavior of systems. Just because the rules cannot be directly observed, except only their effects, does not mean that we throw the rules or the rule book away in favor of direct experience only. This is what distinguishes a "system" and by inference, a "metasystem" from the random patterning and otherwise chaotic ordering of natural phenomena that are merely observed but not understood.

It is the central effort of metasystems theory to raise this process of scientific abstraction and conceptualization to a new level or order of complexity and reliability, in order that we may treat systems that may in any direct form seem unrelated. Thus, the notion of causality becomes equally complex as well, as we go beyond the search for mechanical causes for ultimate or original reasons for the patterning of systems. Metasystems is more than about mere statics or mechanics. It is about dynamics and concerns therefore centrally the problem of change and the challenges of understanding change in a coherent manner. Many questions therefore involve those of ultimate cause and origins, and final ends and consequences.

The point of departure for metasystems theory is the observation that all systems in the universe, however remote or distant from one another, are interconnected at multiple levels with every other system, however indirect. As a result, all systems are hypothetically interrelated on an abstract, theoretical or "metasystemic" level. At the same time, all systems appear to be unique, underdetermined, complex and chaotic in their systemic patterning and epiphenomenal order.

We cannot conceive of any real system that is perfectly isolated from the universe--if we could we would have a kind of perpetual motion machine that fundamentally violated the basic laws of thermodynamics. It would violate these basic laws because in its perfect isolation such laws would not apply. It is interesting that in this regard the closest we might come to such a system is the observation of superconduction at or near the value of absolute zero. We may in fact define near perfect perpetual motion machines commonly in the universe. The point is that whatever system we adopt or consider, we must understand that this system will be critically linked, at multiple levels, to other systems, super-systems and subsystems, and that in the grandest sense, everything scientific is literally connected with everything else, but always in a directly underdetermined and partial way. It is this sense of being underdetermined that allows for the dynamics and systematics of change to occur at all.

Metasystems therefore takes the challenge of studying systems both as unique instances of larger classes or samples or populations of similar or different kinds of systems, and at the same time of understanding the interconnections between any particular system and the larger metasystemic framework that serves to define and regulate that system. Metasystems science is therefore as much about delimiting and defining systemic context as backgrounds to systems, as it is about understanding the internal functional structures of such any such system in an ideal sense of being a mechanical isolate. It follows that different classes and kinds of systems and subsystems will interrelate to other similar or different systems in certain ways, and these forms of interrelationships between systems will serve to define and further characterize systems at all levels.

Metasystems offers a new level of scientific comprehension and comprehensiveness with the promise of at least partially stepping beyond the paradigmatic boundaries of disciplinary institutions and of the anthropological relativities of our own knowledge. As such it offers a worldview, or view of the world, that is essentially coherent and intelligible from a scientific standpoint. Such a worldview is essentially mechanistic and dynamic at the same time. It is systematic, holistic and analytic. It offers us for once the possibility of entertaining a view of the world that is undichotomized by methodological priorities or by partial philosophical points of view or hidden ideological commitments regarding what we know or how we know it.

From the standpoint of metasystems, alternative theories are seen as competing possible paradigms, and are regarded heuristically in terms of their implications and productivity on a theoretical and operational level for the respective fields that they encompass. They are also considered secondarily from the standpoint of alternative fields of inquiry, for cross-over and feedback that leads to greater productivity. This interdisciplinary function of metasystems science reflects both the comprehensiveness and integrity of natural systems that are upon some partial level connected to every other system.

In this regard, no theory is to be preferred over any other, except perhaps on grounds of internal coherence, external non-contradiction, and plausibility of hypothetical inference structures such theories generate. Productivity is also an important measure of success, and productivity is defined as proof in the final pudding. Another way of seeing this is to realize that from a hermeneutic and critical standpoint, no cosmological theory, however well conventionally received, is to be regarded as privileged or so sacrosanct that it cannot be brought into question or that alternative competing models cannot be heuristically considered and proposed instead. This constitutes the basis for the interdisciplinary nature of natural systems theory. Knowledge does not exist in privileged domains, so much as it exists upon an evolving noetic landscape of competing ideas and relations.

Natural systems theory deals with the sciences of naturally occurring phenomena that are complexly interconnected. It broaches several levels of the natural stratification of physical phenomena in different sets and subsets. The differentiation of these sets and their subsets is largely based upon the application of different generalizable rules of relation and operation that determine the state-path trajectories and functioning of these systems at their respective levels. Underlying these tacit rule systems are even more intricate mechanical interconnections or devices that also stratify upon different levels of natural integration.

Rules that occur in natural systems are implicit to the a priori patterning and expectation or predictability of the patterning of natural systems in terms of their behavior and design. Such tacit rules of function and operation can be said to be equivalent to the informational value of these systems as they occur and change naturally. We observe such systems, either in natural or experimental conditions, and from our observations we determine the rules that apply to the observed system. This constitutes the basis for all science, as well as for our realistic understanding of the world, both in an everyday sense and in a more general or collective way. The rules we state explicitly govern the statics, mechanics and dynamics of all naturally occurring systems that we have studied. We test the rules in our trials and applications, and we redefine our rules to fit the new information we have learned about such systems.

The systemic relations occurring between parts of systems are often similar regardless of the level or kind of system we are referring to. Natural systems of all kinds and at all levels share certain affinities of relation and interaction with one another. Of course, it is difficult to compare human systems with the behavior of subatomic systems in any but the most superficial manner. We find the relationship between such systems in understanding that there is a natural order in the stratification of nature. We can say that all human systems, by definition, are composed of atomic systems, but not all atomic systems are necessarily composed into human systems. Human systems therefore represent a very small sub-class of a much larger field of atomic systems, albeit a very special set of subsystems with very special derivative properties. It is easier to find behavioral relationships between baboon troops and wolf-packs, for instance, than it is to find similar kinds of analogies between ant colonies and birch forests. We can say that in a vast matrix of naturally occurring systems, systems that are more closely related to another in both history and in shared affinities, will have more in common than those systems that are more distantly or indirectly related. While this may be too obvious to warrant further consideration, it is important not to overlook the implications for understanding the structure of this kind of matrix of relationships that affects the life-cycles and outcomes and indeed the matrix itself.

First, it appears that natural systems, even at very fundamental levels, are emergent systems and possess an inherent characteristic of self-integration, regeneration and synthetic manifestation. If we seek fundamental answers about the origins of the universe for instance, we must conclude that either the universe originally derived from something out of nothing, a process which, given our conventional understanding, would appear to defy basic principles of thermodynamics, or else, it came out of something else. And if we conclude that the second alternative is more acceptable to science, then we still need to explain the origin of that something else from which our universe sprang.

All systems are therefore stratified within a larger metasystemic matrix. It is true that in nature, more composite and complex systems are composed of more basic but often no less complex sub-systems. The entire universe can be said to constitute a vast super system possibly characterized by infinite complexity. It comprises all components and subsystems of all levels, and encompasses the totality of nature.

A theory of the systemic universe is in essence a theory of metasystems, or, in other words, of the scientific metasystem that orders all naturally occurring relationships.

Another characteristic, or set of characteristics shared by all naturally occurring systems, is that they all involve some form of energy exchange process, almost always including some measure of heat exchange. Thus all systems are both dynamic and mechanical in the classic sense of these terms. There is no system that is of interest to scientific inquiry that does not involve in some form or other the exchange or transfer of energy.

While most naturally occurring systems are thermodynamic in their energy processes, it is also true that there are classes of natural systems that appear to follow as well relatively non-thermodynamic pathways of energy exchange.

This leads to an understanding of a second important characteristic of all systems. It can be put this way--to the extent that the energy exchange processes of a system serve to delimit and describe a system in terms of its statics and dynamics, the system can be said to contain information that is intrinsic to its patterning of behavior and organization of its relations. All naturally occurring systems can be said to be both energetic and informational in terms of their organization and behavior.

It is the relationship between information and energy transfer in naturally occurring systems that is of greatest interest. A non-systems can be said to be totally chaotic and disorganized. It contains no meaningful order or pattern, hence no information, and its energy relations can be characterized as completely entropic. When a system decays or breaks down at the end of its life cycle, it can be said to enter into such a relative state at which point it is no longer recognizable as an organized system.

The relationship between energy exchange and information is so close that terms used to describe one kind of process are often used to describe the other process as well.

All systems therefore exist in a background context that is definable by means of a relatively random and disorganized pattern, which can be referred to as chaos. This sense of background chaos can be said to comprise a universal energy sink or informational reservoir by contrast and relation to which we come to understand systems. We cannot fully think about systems in natural contexts, in terms of their life-cycles and their operational environment, without reference to this notion of background entropy. Background entropy at least indirectly relates all systems to one another, by means of providing a common ground or reference.

Universal entropy or chaos is defined itself by one characteristic, and that is its state of non-isotropic zero-equilibrium. This equilibrium balances and defines the limits of all naturally occurring systems, and determines that all such systems must return eventually to a state of total chaos.

Total chaos is related to a condition of a complete lack of information, or rather of absolute or essential meaninglessness. Another way of saying this is that stable and self-maintaining energy systems are informational because they are anti-entropy systems. They resist in some way the natural tendency towards dissolution to complete chaos of relation and entropy transfer of energy. To the extent that such systems are informational, they are meaningful and amenable to scientific inquiry.

This universal chaos determines another important property of all naturally occurring systems that is related to the fact of their life-cycles. All natural systems are bound by parametric constraints ultimately determined by this background relationship--these constraints ultimately determine the state-path trajectory taken by any and every system.

1. There can be no non-natural systems that are not constrained in some basic manner by the fundamental rules of universal entropy.

2. Artificial systems are a derivative sub-class of natural systems, hence remain subject to the same basic rules of universal entropy.

At the same time, all systems exist in some form of meta-relation with other systems at multiple levels. These meta-relations may be more or less organized into meta-systemic patterns themselves, or they may resemble relatively chaotic or non-isotropic patterns of occurrence.

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Nature is the primary object of concern for science, and scientific activity has its starting point and final objective in the excoriation and formal explanation of the natural laws underlying all naturally occurring phenomena at any level that may be discernable or at least inferable. Thus the close and careful observation of physical phenomena at one level of integration of nature may lead to an intimate and profound understanding of similar or related kinds of phenomena at other levels.

1. All natural systems have an inherent life-cycle and state-path trajectory-- particular systems rise and fall with the changing tides of time, and no system remains permanent for all time.

2. All natural systems maintain a boundary relation between internalized states and the external environment.

3. All natural systems are composed of subsystems, and are part of larger super-systems.

4. Most natural systems exist in a paradigm of possible alternative systems or of alternative possible states.

a. Usually, there is more than one set of co-occurring systems of the same kind or population. Systems are usually not unique except in their discrete and composite physical characteristics.

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The point of departure for natural systems theory and metasystems science is the observation, first made by Heraclitus, that all naturally occurring systems must change, and change is a continuous and intrinsic aspect of all such systems. The problem of change creates a dilemma in terms of theoretical description, and especially in terms of systems science, as in a conventional way systems are seen as being synchronous and structurally unchanging. Most of our terminology and vocabulary to describe systems carries with it the connotation of static and unchanging realities. When we seek the scientific essence of most systems, the immutable laws governing systems, we seek the eternal verities, the absolute noumenal truths that are held to be perfect and unchangeable. As far as we now, most, if not all naturally occurring systems that we deal with obey the basic laws of thermodynamics, and remain relatively imperfect processes that are forever modulating in some chaotic manner. We run into an even greater sense of dilemma when we come to the realization, for instance, that even our basic laws of science may be somehow changing in ways we scarcely understand.

It seems a basic presupposition, that gains some support in physical systems theory, that all naturally occurring systems, upon all levels, are subject to continuous change at some rate. This casts a relativistic shadow of ultimate uncertainty over all our knowledge and science, but the classical quest for classical and immutable laws governing the order of natural relations does not seem to hold forever and for all things that science must consider. We end up instead of grand and comprehensive theories, with partial "covering law" models that hold exceptionless for a certain range or level of phenomena, but which must be replace on other levels by other theories or models.

We are led in such a manner to ask some ultimate kinds of questions about physical reality--such as whether the universe is infinite or not, and whether there is some ultimate beginning or end to this reality, and how did it all come into being anyway. And we are faced only with the understanding that we will not be able to ultimately answer these kinds of questions, but also with the imperative that we must answer them to try to make sense of our world. And it follows therefore that the kinds of answers we provide to such fundamental questions, end up having tremendous implications in terms of our scientific worldview and our operational approaches in science, upon very basic levels.

Metasystems science does not throw up its hands to concede that it cannot be bothered by such questions, as their answers creep into our formulations and view of the world however implicitly, however indirectly. We invariably end up trying to answer these kinds of questions in terms of our theoretical constructions whether we intend to or not. The point of departure for metasystems science is to make explicit what otherwise would remain implicit and surrepetitious in this regard--by explicitly evoking the terms of the basic arguments and dialectics, we gain control over the theory construction processes that we otherwise relinquish in the name of disinterested inquiry and neutral scientific method.

Metasystems science proceeds from several interrelated presuppositions regarding the natural ontological status of knowledge in reality. In general, it can be said that the structure and pattern of our formal and functional scientific knowledge comes to reflect the patterning of the empirical phenomena it represents in certain critical ways. This is a first metalogical precept of metasystems theory.

1. All natural systems are a priori to our understanding or knowledge of them, but such systems are only made known through our knowledge. Without our ability to observe and record and remember such event patterning in nature, we would have no coherent knowledge of them, and they would therefore exist in reality as implicit only to the phenomenal structures that underlie such understanding.

2. Natural systems are in essence possible abstract and general structures that remain implicit to the informational patterning of natural phenomena at all levels--the role and goal of scientific theoretization is the excoriation and empirical substantiation of these underlying patterns of order in nature, at whatever level they are found to occur. Such systems are in essence knowledge structures that are a product of our own reasoning, experience and imagination--they are not the actual patterning in and of itself, though the illusion of symbolic reification may make them substitutable for such patterning. In general, such patterns are demonstrated through relational similarities and event anomalies that can be defined in terms of formal or heuristic rules and their exceptions.

3. All natural systems are:

interconnected upon multiple levels,

complexly underdetermined

non-isolatable

Therefore, all naturally based knowledge systems are also:

Interconnected upon multiple levels,

Complexly underdetermined

And non-isolatable.

There is no empirical pattern in the universe that is completely isolatable from the natural context of its occurrence. Similarly, there is no possible theory about any such pattern that is also completely independent or isolatable from its primary source of reference, or from other forms of knowledge about other patterns that are interconnected.

There occurs a single set of exceptions to the above stated generalization, and this reflects the status of a certain class of abstract knowledge that is logically coherent and which has no primary reference to naturally occurring systems. This class of knowledge primarily constitutes the language of science, mathematics, in its various forms. It is because mathematical knowledge in its logical coherence and abstraction stands completely apart from the things it is used to represent, that it can function effectively as an objective basis for the communication and articulation of scientific knowledge. Related to this kind of knowledge are forms of possible knowledge that relates to questions of the ultimate and the ideal. Because some kinds of logical relations are abstractly necessary and unavoidable, this same system sets up the paradox of being able to imagine other kinds of logically correct and ideal systems that are in a strict sense non-mathematical in form. This provides the motivational basis and inquisitive force to scientific research--because we can imagine ideal systems.

Metasystems science allows us to step objectively outside of the normal dialectics of science, between holism and analytical reductionism, and to see in a metalogical way a system that is both superorganic and synergistic and also individualized, particularistic and unique in its instantaneous event structures. There is no sense of giving preference to one way of looking at a problem over the other, opposed viewpoint. Instead, it seeks a comprehension that allows such contraposed points of view to be reconciled and put together as facets of the same system. The possibility of metasystems is realized when we come to understand that such dichotomies reflect the limitations of our knowledge and not of the systems we seek to understand in nature--in other words there is no necessary basis for a false dialectic between contrapuntal perspectives if both perspectives are simultaneously true at the same time. Then it becomes important to reconcile these perspectives and to try to understand the system for what it is, beyond the limits of our knowledge.

The point of departure for metasystems science is in the objectification of Goedel's Theorem, which states ultimately that there is no escape from the kind of paradox represented by such a statement "This is false." This kind of statement introduces us to the liar's dilemma, that the man from Crete said all Cretans are liars, and it points up a very basic and specific design feature inherent to human communication and language--that is the duality of patterning that results in the possibility for prevarication. When we consider that a conventional, or standard, Popperian view of science rests upon presuppositions of falsifiablity and falsification, we recognize the critical importance of Goedel's Theorem. We may never be able to absolutely prove the truth of any relation drawn empirically from observation, though any such statement can be easily disproven by the demonstration of exceptions. A science to be empirically based must be fundamentally inductive in the derivation of such generalizations as "All swans are white." The basis for all relativity, scientific and anthropological, is in the introduction of the exception for any rule that we may formulate derived ultimately from external reference points in reality. Again, the only form of knowledge not subject to such prevarication is abstract logical knowledge that is internally referential--but that is the ultimate difference between artificial intelligence and natural intelligence as we know it. The former can only refer back to itself in a closed system, while the latter must always refer beyond itself to the larger field of relations from which it is derived.

In a sense, scientific theory and truth only emerges as rules are gradually developed to which no counter exceptions have arisen--so far few such theories have proven to be completely or totally unexceptionable. There is in this sense an operative heuristic principle that the larger the subset successfully subsumed by the theory or generalization, the more mileage it gets, the "truer" and more valid it is. We have really no choice but to proceed in such a manner.

The only other way out of such a conundrum of our knowledge is to see that any referential system that has as its locus a range of phenomenal patterning beyond itself is ultimately a language system that has certain inherent limits and features of its own design that constrain it in certain ways. Scientific relativity becomes as much a linguistic and anthropological constraint of our knowledge systems, or rather in our ability to know in any objective sense, as it is intrinsic to the physical phenomena itself. Objectivity is in this sense ultimately a question of communicability of knowledge as it is a question of the realistic representation of external pattern or the truthful abstraction of any sense of underlying order.

When we recognize that Goedel's theorem is ultimately a language problem upon which any external or natural logical system is based, then we can see that only by strict external reference of such a system can we determine its truth or sense of non-contradiction. "This" as a simple subject-noun is of an indeterminate reference, and can refer to more than one thing--if we specify in more certain terms exactly what "this" refers to, especially as something that is available to independent observation, then we are in a better position to assess and resolve its sense of paradox.

It remains the case that if our knowledge is forever imperfect to the task of understanding the underlying sense of order to the patterning of natural phenomena, then it is even more true that our language employed for the representation of such knowledge is even less adequate. It does not serve our purposes to so restrict our terminological framework to precise denotative terms that we can describe nothing that is real or complex--the strength of human language is its inherent weakness, and this is a part of its paradoxical relativity as well. The symbolic power of language to describe, imagine and intuitively fill in the gaps of understanding is as much a fundamental instrumentality of science as it is of worldview. The critical linguistic difference between science and worldview is, I believe, that in science language should at least in theory always have some empirical point of departure and return--a common reference framework that is rooted in basic ways to phenomenal experience. Ideological language ultimately must point only inwardly to its own sense of truth, and to its own intra-linguistic reference points. Any external reference to such a closed system is ultimately secondary and peripheral to its main sense of legitimization or validation.

Another way of looking at the meta-scientific solution to the problem posed by Goedel's theorem, is to understand that the primary reference of any such statement is and must always be some form of empirically or observationally verifiable experience--preferably that can be verified through some non-arbitrary system of standard measurement and reference. The alternative solution can only be ideological and hence non-scientific because unfalsifiable. In this latter case, primary reference extends to the internal logic of a system of rationalization used to justify a determination of the statement in the first place.

The point of departure for meta-systems science can be seen therefore to extend from a certain kind of methodological solution to the class of dilemma inherent to the linguistic and symbolic structure of knowledge, as is represented by the paradoxicalness or antinomality of a statement that can be inherent contradictory--that can be either or both true and false simultaneously. It is to be found that only by the superimposition of some arbitrary but standard system of conventional measurement, can we hope to overcome such a dilemma in a manner that is sufficient for a scientific worldview. Measurement theory remains largely neglected and taken for granted--but its influence in the procedures of scientific verification are as important for the social and psychological sciences as they are for the biological and physical sciences. It is beyond the purview of this outline to elaborate more fully a meta-scientific theory of measurement, except to note its central relevance to the operational definition of meta-systems science.

Measurement involves the superimposition of some arbitrary standard of reference that has certain characteristics:

A zero reference point

A standard or set scale of incremental measure

Scalability

A substitutability of scales or standards by conversion.

Associated dimensionless or dimensional properties that exist independently of the exact measure or value of the scale.

Predefined units of analysis readily exist in the physical sciences but are not very obvious in the social sciences. Atoms have fairly discrete properties that make them predefined as units of analysis, as do molecules, cells and organisms. The closest approximations to units of analysis in social sciences are individuals in a gross sense, within a population dynamic framework. Beyond this, we can designate various forms of social units, or alternatively, "cultures" as discrete historical entities that have some kind of relative or areal boundary separating them from other groupings at various levels. But beyond a superficial definitional sense, the agreement usually ends. In spite of a century of cultural anthropology, there remains as yet no pat or standard definition of its principle object of inquiry, culture. Self is also something that can be defined in a multi-faceted manner and for which there remains little if any agreement.

This kind of measurement refers mostly to a parametric standard, and does not necessarily include a non-parametric standard, though it is possible to extend measurement theory and practice to non-parametric systems if certain kinds of assumptions are made and certain limits acknowledged. Non-parametric values have no precise or equal scale to determine the value of the thing being measured, but it implies that all things so measured are of an equal or more or less equivalent value, even though such value is not determinable in any precise way. Equivalence of scale remains implicit only in non-parametic measures, and there is no necessary zero-reference point by which to arrange such values relative to one another.

There is another related set of dimensions important to measurement theory, and this involves the question of the human dimension of measurement, and the possibility especially for error of measurement and for inexact estimation. Sources of possible error in any measurement system are multiple and overlapping, and this invites the dilemma of Zeno's arrow to the problem of measuring static what is in fact dynamic, and measuring as finite and discrete what is ultimately, or on some other level, infinite and indiscrete. This does not even broach issues of error in recording or translating such measurement, or theoretical issues of its interpretation and framing in a larger system of reference and knowledge. An arbitrary system of measurement is just that--it is arbitrary to an agreed point reference point or shared point of departure. Such a system is a true system only in as much as they are derivatives of mathematical systems in an abstract sense, and are constructions of such systems in an applied sense. Nevertheless, the statement can be made that all such scales or measurements are ultimately relative to the person doing the measurement and to the implicit social contract of agreement that makes a metric system legitimate, for instance, over a pound system, etc.

Sources of error in measurement systems are perhaps more obvious in human sciences or observational biological systems when there exist few if any non-arbitrary points of departure for such systems, nor any standard means for ascertaining discrete values in complex behavioral patterns, etc. In general, nonparametric measures are more applicable and useful in the human sciences than they are in the physical sciences, when even what is being observed may be so complex that it is subject to multiple interpretations without any standard frames of reference.

Measurement theory brings us directly to the issues in metasystems science of the paradigmatic complexity of sciences as these articulate as informational patterns upon different scales and levels of analysis. Progress in scientific theory depends upon the ability to create a common ground of uncontestable agreement in our knowledge structures based upon objective measurement. Scientists far removed in time and place can return with the same instruments and derive the same accuracy and reliability of measurement of a discrete event structure as their predecessors. Paradigmatic agreement may be easier to achieve or at least more obvious in the physical sciences, where entire paradigmatic systems emerge with a great deal of predictive efficacy, than in the social sciences where there can be said to be very little paradigmatic agreement, even upon a definitional scale.

Measurement theory brings us to the question of statistics, number and set theory and the use of these in the description of complex natural based systems. It goes almost without saying that there are different kinds of statistics. Statistics are really nothing more than a systematic means of description of relational patterns of complex phenomena, and depend upon fundamental units of analysis and appropriate measures for these units. Descriptive statistics is readily elaborated to predictive statistics and to even a form of prescriptive statistics that rests upon game theory and probability theory. Statistics also implies a complementary system of knowledge, what can be referred to as "dynamics" and it is at least implicit that most statistical description has as its goal generalization of dynamic patterns underlying statistically defined relationships. Dynamics involves change in stable systems; statistics involves the description of such systems as stable structures. We may also speculate on the role of synergistics of systems, which incorporates both an understanding of the organismic patterns of structure represented by systems, and by the emergent patterns and properties of such systems that are a function of their operation and that are approached holistically as if the systems were themselves somehow discrete entities. In general, I would claim that these relationships involve one form or other of operational and general systematics, and the patterns of relationship that are predictive and that can be said to be mechanical. Mechanical systems can be said to be systems that exist naturally or artificially (human made) in reality, and that exhibit certain types of intrinsic/extrinsic properties--primarily of energy exchange and informational capacity. No organized system of energy exchange is without informational capacity, and no informational system that has external reference can not be about some form of energy exchange. Such energy exchanges, upon one level or another, are always reducible to scalable measures. It is the case that there are informational systems that have no direct representation in terms of energistics, unless we consider the bioenergetics of the functioning of the human brain that produces such systems in the first place.

The operational basis of metsystems science rests upon the operational elaboration of advanced set theory, or what I refer to as the description of set theory, via means of the articulation of alternative possiblist statistics. A meta-set can be said to be a simple collection of complex entities.

All sets are subsets of some larger hypothetical set, and all sets are composed of some collection of subsets. It appears that in reality, there is no end to the infinite regress of sets. In general, the level of set that we work upon depends upon the units of analysis and frames of reference we impose upon our data. We may look at the same set of events, say a couple kissing.-a phsysicist might see a collection and demonstration of subatomic forces, a chemist the articulation of a bunch of molecules, a biologist.

All sets have a complex multi-level identity--they are or can be complex representations of simple things composing it. All sets are simultaneously subsets of some larger meta-set, paradigmatically alternative sets of some class simultaneous sets, and a meta-set of some system of composite subsets.

From an operational point of view we must adopt a more restrictive definition of sets in general. We can say that they are a collection of things that share some sense of affinity, or what is known as cardinality making those things common members of a shared set. Sets are composite entities. The type of set can be designated by the nature of the cardinality principle or relations governing the shared identity of its constituents. A set may be a grouping that is intrinsically organized in a more or less complex manner, or that are relatively independent of one another except in some indirect way. All sets have a certain size in the sense that they are bounded entities.

In general, the cardinality of a set is defined by the determinants that characterize the members of a set--in this sense any set is a collection or sample of related objects or points that may be said to be relatively determined or undetermined. Most sets in nature can be said to be partially determined, and the determinants that define the cardinality of the set can be said to consist of implicit rules of ordering defining the identity of the members and the possible relations occurring between members.

A meta-set can be said to be a superset of one or more sets, either defined through time or across space or both, in which the cardinality defining the membership and relational order of the parts of the set is either variable or alternate. A meta-set is inherently dynamic, changing in its order and composition, and this sense of dynamic change in the constitution and behavior of a set can be topographically mapped in hyper-volumetric or complex multidimensional space.

Sets can be combined and related to one another, and set theory largely involves the possible logical relations between sets and leads into other forms of mathematical theory. In a sense, statistical samples and derivative data points represent virtual sets that can be superimposed upon different forms of data, rendering this data in terms of sample points manipulated to various statistical techniques of description and analysis.

A matrix can be said to be a special kind of meta-set that is defined in a reiterative or recursive manner, and that has a fixed set of constraints defining the breadth and size of the set, and also the nature of the relations between the members of the set. In general, a matrix can be said to be a fully determined set of a special type, and I believe matrices are actually rare in nature with a few noteworthy exceptions, and are demonstrated more through the reiterative or cyclical articulation of natural systems over time than by distribution of systems in space. An exception to this might be said to be the reiterative structure of DNA in genetic encoding. Crystallic structures of atoms and molecules are said to have matrix type structures, but these matrices constitute gross geometric descriptions of the distribution of molecules within a lattice framework, rather than a fully determined matrix that functions as a system.

It is beyond the scope of this introduction to elaborate in more detail the mathematical aspects and permutations of set theory as this may apply to meta-systems analysis. It is important to state though that statistical systems constitute the principle means of expression of meta-systems analysis in terms of set theory, and the main aspect of general statistical description and analysis are in terms of what can be called "possiblist statistics." All statistics constitute a form of heuristic representation, analysis and modeling that describe what can best be described as hypothetical meta-sets that are purported representative of natural systems and that exhibit patterns that are relatively non-random in distribution or reiteration. We do not need to invoke more exotic forms of statistics, as for instance Einstein-Bose Statistics, to explain the possiblist aspects of statistical analysis and description of meta-sets.

It is true that statistical measures can generate profiles and descriptive variables of meta-sets that only indirectly represent reality, or that do have any direct representation in reality. Such systems are nonetheless considered to be relatively valid within a certain range of statistical probability, and to the extent that they can be said to be valid, they exist as a meta-system independent of the actual sample points from which they were derived.

Set theory is important to meta-systems in that it provides an operational and methodological handle to the description of complex systems, and it provides a means for framing and analyzing possible relationships occurring at different levels of complex system. In general, it comes as a paradox that as long as systems can be accurately and reliably represented, the larger and more complex the system, the better and more realistic the possible representation that may be forthcoming for it.

Another point of departure in the elaboration of meta-systems analysis is in the definition and conception of what I refer to as alternative number theory as complex variables. A particular data point is a set may represent a particular range of values, or, even a number of different ranges of alternative values. The identity of any particular number within such a system is therefore inherently multidimensional and complex. Furthermore, any point may be a variable that is defined by both dimensional and dimensionless parameters, and may in fact constitute a subset of possible points, each of which in turn is represented by another complex alternative variable. Each alternative variable can therefore be said to be completely relative to the variables that in turn define it within a relational complex. Discrete values may be associated with such variables at different points and times.

In general, a meta-system can be said to be an integral of a meta-set that is articulated within a larger superset framework or context, and which determine or govern the relations occurring in the state-path behavior of a meta-set. A meta-system is primarily concerned with the problem of integration of sets or meta-sets over time and space, in a manner that there arises emergent or synergistic properties of the system as a whole that cannot necessarily be predicted by the analysis of its parts or constituent subsets. A meta-system comes to have a particular dynamic state-path behavior that usually fluctuates over time. A meta-system can be thought of to be composed therefore of one or more meta-sets in interrelation that are changing through time or across space in significant ways.

Consideration of meta-systems analysis invariable brings up the issue of contextual analysis of the superset or the framework for the articulation of the meta-set. The question of context is largely a question of identifying the significant figure ground relations that serve to define the identity of a meta-set in contrast to and identity with a shared background of other relations. In general, the question of appropriateness or relevance of contextual information is important to an understanding of a meta-set in its natural context. In a world in which everything is at least indirectly connected to everything else, it is easy to see how too much of everything else can serve as so much noise in the understanding of background relations affecting the dynamics and behavior of a meta-system.

There is also a critical sense that inherently underdetermined meta-systems can form a complex set of alternative pathways, or alternative event spaces, in the unfolding of their possible state path behavior--these alternative pathways are nowhere ever fully predictable, though we can expect certain probabilistic pathways forthcoming from them. Such systems in their unfolding are said to be chaotic.

The relationship of matrix theory to set theory comes when we understand that much of this apparent chaos in complex systems can be organized within a framework of possiblist event spaces that may be defined statistically as a discrimination table through which all possible pathways of a state-path trajectory of a system may be defined and identified.

It is possible to construct such matrix discrimination structures by means of inter-correlational analysis of sample sets in which at least some indirect relation can be hypothesized, even if no direct causal relation can be demonstrated. Inter-correlational analysis provides a means for mapping complex sets in multidimensional space, in a manner that the relationships become sample points in a relational structure. The sense of the reality of the original data point is lost completely in such representation, and only relational structures are represented in hyper-volumetric space. Such inter-correlational analysis is based upon a principle of cardinal numbers as alternative data points, derivative as partial determinants of complex systems. Such a method allows us a means of comparing complex relational systems in a shared or common meta-space.

From discrimination tables and inter-correlational representations and matrices of partially determined alternative variables, it is possible to apply a set of arbitrary decision rules, or what are more formally defined as heuristics for the selection of alternative pathways and for the determination of relational rules that can be said to be implicit to and underlie the complex organization of a system.

From this, rule-based systems may be devised that provide a more formal description of the complex state-path behavior of meta-systems, and from these systems we may test for the accuracy of simulated models to our observational measures of real systems.

Governing these kinds of systems can be said to be a form of hypothesis generalization that isolates key operational rules or relationships that govern the articulation of a system and that regulate and partially determine its outcomes and dynamics. Paradigmatics can be said to be a form of dialectical counterpoint and argumentation of alternative competing rule sets or systems, often defined conceptually or symbolically by key operating metaphors, that govern meta-systems in a general if not in a universal way. Paradigmatics lead to a competition of competing ideas and theories, and eventually to a sense of progressive understanding of the structural nature of systems in a parsimonious manner.

Heuristics, in modeling, in learning and in exploration of alternative possibilities, can be said to play an important role in this process overall. Heuristics has rarely been formally defined--it consists of modeling and modeling theory, and the representation of real systems in abstract terms. It can involve game theory and simulation, as well as in scenario forecasting.

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I have undertaken to write about natural systems within the framework of an emerging scientific perspective that is rooted to the advanced understanding of information systems, knowledge, and intelligence as this has rapidly developed with the information revolution. It is difficult, indeed ultimately impossible, to separate the human dimensions of knowledge and intelligence from the natural order and patterning discovered at all levels and in every area of natural phenomena.

The challenge of meta-systems theory in essence is the problem of overcoming human prejudice, especially that is implicit and tacit to the background of the world as humans have constructed it and to seek to understand and control it through their sciences. Behind the prejudice that tends to set restrictive paradigmatic boundaries around knowledge domains, are domains of knowledge that are as often as not implicitly arbitrary in determining what is essential and focal in importance, and what is beyond the normal boundaries of knowledge.

It is found repeatedly that realistic understanding of natural problem sets at all levels tends in the larger frame of reference demanded by meta-systems theory to cross-cut and intersect conventional knowledge domains. Thus meta-systems theory is about the interdisciplinary integration and coordination of disciplines and knowledge across a broad field of application. It is furthermore a question of symbolic integration of this knowledge upon levels of comprehensiveness that are unprecedented. This integration is symbolic in a fundamental sense of being representational of the physical or possible realities that stand behind it. It is symbolic too in the sense that all human understanding, no matter how analytical or seemingly rational, is ultimately symbolic in structure, and from an anthropological point of view, this symbolic structure of all information and knowledge of our world, what is referred to as anthropological relativity, constrains us in everything we do and in every possible way we may know.

I have adopted a system's theoretic approach to the comprehension of all natural patterning because this approach is the most comprehensive possible and the best suited to the analysis and modeling of natural patterning of phenomena at all levels. It combines both an information theoretic and cybernetic approach with a mechanistic and ergonomic or energy exchange perspective. Aspects of information theory are parallel to theory of energy efficiency or thermodynamics.

On the other side of the coin, I do not want to thereby discount the unique, the individual and the sublime aspects of the particular complexity of nature at all levels, especially of human nature and culture and of biological systems in general. Particularism of description and analysis complements at every level the generalism of labeling and synthesis of systems.

In hindsight, having been primarily an anthropologist interested in human knowledge patterning, I've come to a belated understanding of the role of human prejudice and intelligence plays in the real world, and in the ways that it can constrain our actions, both individually and collectively. If we are to move ahead with our sciences in a significant manner, then we must have the courage and willingness to think in new directions, and to assume new frameworks of understanding that seemingly violate the unspoken sanctions of our received conventions and paradigms.

The stratigraphy of knowledge and natural information is complex in an infinite sense. The layers are manifold and convolute at many different points and in many ways. The conventional academic solution to this problem is through bureaucratic-administrative organization of increasingly specialized sub-fields and problem sets. The price paid overall for this hyper-specialization and "hyper-comparmentalization" of knowledge is that there is yielding of overall control or outcomes or sense of responsibility for knowledge either in general or in specific frameworks. There is a loss, in other words, of a coherent worldview that would give each of us, as citizens and members of humankind, a greater measure of control and influence over either our own relation to such knowledge or to the total framework of such knowledge.

I find this particularly so in my own professional field of Anthropology, constrained and scientifically defunct as it has become by a blind and almost mindless political culture of correctness. The dilemma is that no matter what the social structures of professional reinforcement, opportunity and social legitimization, the mind remains essentially free and the field, from a purely theoretical point of view as a possible science at least, essentially neutral and uncontaminated by the boils that fester beneath the social skin. Great Academic names will rise and fall, come and go, with the ebb and flow of the social tides of power, funding and status, but the science of anthropology will remain in the end as a yet unanswered set of questions about human reality, however adorned.

Thus meta-systems science is largely about pulling back the veil of socially contrived illusion that interferes with a more objective and realistic view of the world. Science is no less prone to such illusion than religion, except that it is saved fundamentally by the implicit unanswered questions it asks of the unknown, and by a rigorous commitment to an objective empiricism that "seeing is believing." Thus, as social praxis, science can be no less ideological than religion or other forms of ideology promulgated in the name of truth, except for its fundamental attachments to an open and objective approach to comprehending reality.

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If I had to summarize a central point of departure for a book entitled "Meta-systems Science," I would claim that all good science does not begin or end in a classroom. It begins with basic observation, experience, common sense, sound reason, a high regard for the natural patterning found in reality, and the ability to ask fundamental questions to which we know no clear answers.

Meta-systems science, put succinctly, is about model building and playing with different kinds of models that relate one way or another to different aspects of reality. The kinds of models built are primarily conceptual and hypothetical constructs, designed to investigate and explore a search-solution space for a wide range of problems. It has emerged that there are relatively systematic, or should I say it, meta-systematic means for proceeding in such a process in the identification of the important from the trivial and the central from the peripheral. A great deal of model building must proceed and accompany any venture we make into scientific research and development--to fail to do so is to mistake the trivial for the significant and to charge down pathways that have no final destination though they may seem to be leading somewhere.

Meta-systems science is also, basically, about asking questions about what is unknown, and for having a fundamental curiosity for the profound questions that define the unknown in such a way that makes it available to our imagination and conjecture. Some will dismiss a meta-systems approach as conjectural only and therefore not rigorous. I can only counter such a criticism that one's approach to and in meta-systems science, and respect for it, is a relative function of one's attitude and worldview in general, of one's openness and ability to deal with uncertainties and the unknown in ways that seem to make some sense.

Models of complex phenomena depend for their resolution on several types of procedures--the identification of key variables or focal questions; the partitioning and reintegration of the problem into natural subsets; the ability to ask and seek comprehensive frames of reference within which subsystems can be effectively integrated and explained as a course of logic. Some call this last procedure a hypothetico-deductive approach, and indeed conceptual model building is a heuristic procedure that is largely so. But I believe meta-systems science is more than just hypothetical deductive theory building, which seems to be a fairly formal definition for a fairly informal kind of process in scientific theorization. It involves the construction of alternative frames of reference in a heuristic manner. I would say that meta-systems science involves the construction of alternative models, and a kind of noetic competition and equilibrium being established between these models. It is a clear case that concepts and conceptual models exist in a kind of dynamic landscape in which some succeed and others fail. A dialectic between alternative models can lead to a resolution and even more, an identification of the central issues involved in the dialectic. It is also the ability to step beyond the dialectic, while keeping one foot in the dance, so to speak, in order that an objective reference can be given to the entire frame of the dialectic. The dialectic is transcended, and this is an important function in considering any meta-system.

Many of the key questions that are broached in meta-systems theory concern the creation of hypothetical "meta-spaces" for the alternative construction of theories that reflect the following:

1. Natural origins of all systems, assuming natural systems had some kind of beginning that can be described and explained in terms available to scientific observation and experimentation.

2. Natural dynamics of all systems, assuming that natural systems all have certain complex state-path trajectories that leads to continuous kinds of changes of such systems, that can be measured and to some extent expected or predicted by our scientific models.

3. Natural mechanical structures of all systems, that explains the relational distribution and functioning of observable phenomena in terms of general-specific rules that are empirically and inductively consistent over very large, virtually infinite sets.

4. Natural static descriptions of systems of all kinds, as complex sets and meta-sets constituted by inherently complex entities that are themselves variable and composite in constituency. Here self-consistency competes with constituency in description if not in full and complete explanation--any explanation must adopt analytically a constituent approach, any description must frame holistically a self-consistent view.

5. Natural systems all are defined theoretically by generalizable rules that govern in a partial manner the pattern of such systems. Such determinants are partial always in natural systems because such systems are thermodynamic and therefore entropy. In other words, natural systems contain significant information about their underlying sense of order, and it is the objective of meta-systems science and theory to elucidate and clarify this information--making explicit what is otherwise implicit to the patterning of nature.

6. Natural stratification and contextual interrelation of all systems that occur in a common framework of physical reality--how do systems interrelate with other systems and how do systems become embedded within other systems in complex ways and yet with a measure of partial determination.

We must live with a grand sense that all natural systems, at whatever level, are not fully determined systems. This means that we cannot always predict the outcomes of complex event structures based upon our models, and that our models, no matter how precisely designed and constructed, can never fully predict the patterning of natural epiphenomena except in some probabilistic framework of well defined expectancies. To put it more concisely and mathematically, all natural systems are inherently complex and chaotic, following pathways determined by non-linear control mechanisms.

Related to this is the critique that such an approach tends not to be as systematic as, for instance, chemistry. But there is no field of scientific inquiry that does not falter at the edge of complexity and chaos, in the confrontation of unknown structures and unanswered questions that make our methods appear weak and inadequate. Systematicity in this regard is for those whose squamous worlds are well ordered places with minimal tolerance for error or disorder. To counter such a criticism, I would only say that in fact Meta-systems science becomes quite complex and there is a fundamental systematicity of heuristic structure in meta-systems science that applies to all levels and areas of application. This systematicity is defined explicitly in a mathematical way, understanding that pure mathematics is a unique form of abstract knowledge with no necessary external evidentiary proof, and that applied mathematics must sacrifice this sense of eternal, a priori truth-value for goodness of fit and inductive mileage in real world systems.

In this regard, the systematicity is expressed in several ways: in terms of the systematic conceptual approaches to abstract model building and testing, rooted in an anthropological theory of human knowledge; in terms of analytical techniques for understanding complex systems; and in terms of systematic approaches to integrated applications. It is difficult to describe in short-hand the operational rationale for these procedures. It is always the case that the proof is in the pudding, and the understanding how things work always follows knowing that they work.

The systematicity, briefly described, appears to be the following kind of structure. Variables, as complex composite entities, are defined as parts of a general set or sets derivable from naturalistic observations. These variables can be considered to be members of a sample population or a kind of systematic grouping. Rules of relationship guide both the relations between members and sets, and the transitions that occur within each set at each point. These rules form matrix structures which are more constrained dimensionally and determined than the sets or samples they are built upon, and describe in a repetitive manner the functioning of the set as a meta-set either through time or across space, or both. From this we can derive a set of nonlinear control functions that, given initial inputs into the meta-set and matrix design, determine at least in a partial (probabilistic or stochastic) manner the outcomes given certain constraining factors. At another level, these functions are in turn controlled by multi-factorial periodic harmonic oscillatory mechanisms that serve to maintain the stability of the meta-system as a whole. These oscillatory mechanisms may be built into the design of the system itself, occur as the product or part of sum key subsystem, or be found to exist externally from the system itself in an independent manner in the super-systemic context in which such as system occurs. The description of the stability of the whole then becomes encapsulated into a set of output variables at another level, and feedback can lead to the initial input stage of the cycle to repeat, or the system can be stepped up to a higher level of integration that incorporates the lower level as a subsystem. This is a highly structured approach that has applicability to a very broad range of natural systems applications, in varying form, as well as to certain systems relating to alternative intelligence. The stages described in this complex meta-system serve as functions with determinants that define the character and behavior of the system as an entity that gains expression in some manner in the world.

Advanced approaches to labeling and number theory, set theory, possibilist statistics and mathematical matrix design are also a part of this research design.

This brief outline requires greater elaboration that is beyond the immediate scope of this text, but it is important to suggest that this kind of patterning can be found to occur naturally in many complex systems in physics, biology and in the theoretical description of human systems. It arose originally in the anthropological sciences in the need to describe in an operationally sufficient manner the complexity of such systems. It arose in conjunction to certain artificial intelligence and multidimensional representation strategies applied to human systems, but it has found a wider basis for application at many different levels scientifically. It also describes a framework for a general model of abstract intelligence by which we can understand the application and design of alternative systems that are informatively complex and sophisticated in terms of their real world application.

To contrast an analytic or reductionist approach versus a synthetic approach or holistic approach to natural systems theory is something of a false dichotomy. In our knowledge, we live with the paradox that the properties of all naturally occurring systems are simultaneously both the sum of their parts and emergent as more than the sum of their parts. Such dichotomies are more a residuum of the limits of our conceptual abilities or systems than they are intrinsic to the natural patterning itself. Systems as larger entities often describe state-path trajectories in nature that are fundamentally independent of the component parts that compose these systems. At the same time, if we seek to analyze and understand the internal relations and interactions of such systems at any given moment, or over a prolonged period of time, we are faced with the conundrum of dealing only with many individual elements that are often themselves analytically reducible to smaller and smaller components.

The boundaries recognized in natural systems theory are thus directly not the boundaries of conventional knowledge domains---biology versus chemistry, psychology versus sociology, or physics versus philosophy. The boundaries that exist are those naturally occurring differences that stratify natural phenomena at different levels and areas of functioning and articulation, particularly in terms of relative and absolute size, complexity and duration. That they are reflected in the functional organization of our knowledge domains is after the fact of their natural stratification and before the possibility of their reintegration as meta-systems.

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It is important to reiterate the important and unique role that mathematics plays in meta-systems science. Mathematics is the language of science, and science cannot achieve the degree of objectivity or logical non-contradiction necessary except by means of the strict adherence to and application of rather formal mathematical rules and relations. Pure mathematics has no external reference beyond the rules of its own logical relations and identity. As a language it is a sign system that, unlike natural human language, permits no internal contradiction. It is beyond the purview to speculate about the a priori nature of abstract mathematical systems, except to remark that they exist as ideal possibilities that appear to be structurally immanent in all naturally occurring systems. Another way of saying this is that they are the consequence of the fact of the systematic ordering of nature in the first place--without such ordering science as we know it would not be possible. The first rule of a natural science is therefore to state that abstract order preexists in the natural patterning and chaos, before our observation or understanding of it. By virtue of our own intelligence and capacity for knowledge and understanding, we bring to reality the possibility of abstractly representing these ordered relationship underlying all classes of natural phenomena in a way that is mathematically correct and accurate--we bring with our models the possibility, indeed the realization, of alternative ideal systems, mathematically defined and represented, that have no clear manifestation or reference in physical reality.

On the other hand, we must advance an empiricist rejoinder that applied mathematics entails the application of mathematical formulas and models to natural problems sets such that the goodness of fit between the two is always imperfect and in need of improvement. If abstract mathematically systems are internally perfect, applied mathematical systems are externally imperfect. Translation of pure to applied systems, and the induction of purer from applied systems, is a challenge altogether different from the deduction of a logical proof from a set of formal presuppositions. The former is a practical problem of everyday science, the latter largely a pedantic issue of metaphysical philosophers and theoretical mathematicians.

I believe it to be a source of infinite sublime wonder and awe that this is repeatedly demonstrated in nature--that we should have evolved a unique ability to imagine such possibilities, and by their imagination, to make them a part of our reality. Once we have defined the rules of science, they become as much a part of our world as if they were set in concrete or stone. Whether we wish it or not, we can realistically see the world in no other way than that defined by science. Science permanently alters our view of the world. It then becomes our choice how we make use of this understanding.