Numbers and Symbols

Mathematical Mechanics & Symbolic Calculus

Terminological Systems of Functionally Complex Polynomial States

by Hugh M. Lewis

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

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

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

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

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

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

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

This is a rose.

This rose is red.

This rose is a flower.

This rose smells sweet.

This rose is beautiful.

This rose represents love.

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

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

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

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

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

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

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

*****

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

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

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

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

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

Mathematically, we may express this partial identity as:

A = a {ƒ (X)} + a' {ƒ (X')}

where a is some presumable and significant subset of A

X is something else functionally related to subset a

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

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

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

Then we may say something like what follows:

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

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

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

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

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

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

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

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

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

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

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

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

1. No system is completely whole or independent.

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

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

*****

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

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

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

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

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

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

*****

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

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

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

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

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

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

*****

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

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

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

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

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

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

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

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

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

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

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

*****

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

*****

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thus:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

O = o/O

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

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

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

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

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

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

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

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

J = M(j) + H

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

H = J - M(j)

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

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

H = M_h(h) + H_m

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

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

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

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

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

O = M(o) + H

where H = O - M(o)

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

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

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

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

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

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

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

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

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

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

Then we get:

J = M(j) + H

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

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

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

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

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

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

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

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

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

What is a framework (or a context)?

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

What is a space?

Deductive-Inductive systems

Non-linear mechanics.

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

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

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

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

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

Blanket Copyright, Hugh M. Lewis, © 2005. Use of this text governed by fair use policy--permission to make copies of this text is granted for purposes of research and non-profit instruction only.

Last Updated: 04/19/05