Chapter XIX

Operational Systems

I propose a set of methodological procedures that is rooted in basic presuppositions of metasystems science and natural systems theory. I do not ascribe the same set of operations for every area of knowledge at different levels of natural stratification. Certainly the use of these procedures in dealing with human systems is fundamentally different that the use of similar or related procedures upon a biological or physical level of abstraction and analysis. I do ascribe upon a fundamental level, in terms of metasystems theory, that there are basic abstract and mathematical models that are pertinent to all classes of real or naturally occurring systems. For instance, theory of automata describe all classes of linear forms of digital computing, at least. Whether this theory, which incorporates Turing machines, is sufficient for the description of natural intelligence or naturally occurring information systems is not yet clear, and I doubt it is, at least in any unadulturated form. In this regard we must distinguish between information theory on one hand and intelligence theory on the other, and what is natural or innate, and what is artificial and preprogrammed in some arbitrary way. Furthermore, to address theoretically in any exclusive sense the informational aspects of naturally occurring systems is to thereby ignore the energetic considerations of such systems as naturally occurring machines. A mechanistic model that is construed in a conventional, Newtonian manner is found to be insufficient to all classes or levels of systemic functioning that involves some form of energy exchange dynamics--energy exchange dynamics in natural systems upon different levels, as well as in artificially created systems, can be demonstrated to include non-Newtonian mechanics. The conventional example of course is Heisenbergian uncertainty of quantum mechanics, but similar kinds of uncertainties exist at other levels, and in other forms. We have not yet fully modeled, for example, gravitational dynamics, and we may be quite surprised at how this form of energy exchange defies even our conventional modes of thought about quanta.

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

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

It can be said that if all systems are by definition underdetermined, then any system will be unique in an exact sense, and will demonstrate some minimal degree of possible variability of change pattern. Even systems in nature that we hold to be fudamentally stable, such as the atomic system of the periodic table of the elements, which is held to hold true under all normal conditions on earth, must be suspect as a kind of typology that hides some degree of minimum variation of its elemental classes. It is know that isotopic configurations of elements is estimated and to some extent variable, which confers differential atomic weights and therefore estimated averages. Any particular sample of any particular element or molecule may be more or less the molecular weight that is predicted by the periodic table, with some minimal degree of isotopic variation. We tend to assign relatively discrete mass measurements to nucleonic particles, and energy measures to electrons inhabiting what are known to be discrete orbital levels. It is possible that these measurements of mass and energy of discrete entities, which are themselves more the nature of energy-entities, may be continuously fluctuating about some normal distribution, and that they may even on occasion jump between levels. At a quantum level of measurement, we may even say that such measurements are in fact statements of a certain kind of probability, of likelihood, of finding a particular entity in a given state in a particular instance in time. We can therefore modify even our initial statement that all systems by definition of their underdetermination will be unique in an exact sense, by say that each system will tend to be instantaneously unique and variable as a function of time--in other words systems will be unique states at any discrete instant in time, and will be variable through the longer continuum of the duration of time. This is a basic change principle.

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

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

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

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

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

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

From an energetic standpoint, we may invoke the basic laws of thermodynamics for most mechanical systems involving energy and matter, though this may not subsume the entire class of systems or energy exchange relationships that compose such systems. Basic evidence suggests strongly that the laws of thermodynamics are covering law models that are part of a larger energy dynamic system. Thus, energy dynamics, however imperfectly understood, form fundamental mechanical constraints in the functioning of basic systems that results in inherent change and variability of all systems in time and space. These mechanical constraints can be understood in either a quantum or a classical manner with the same end results. On a quantum level, basic phenomena can be explained that appear to violate thermodynamics upon a classical level, as for instance the phenomenon of superconducting or the tunneling of electrons through a substrate. Furthermore, these same energy dynamics appear to occur in all systems that are classifiable as real systems, no matter what the level of integrative functioning or scale upon which they occur. We may characterize biological systems in such a manner, in terms of their fundamental molecular and atomic dynamics, and we may furthermore characterize even brain-based mental systems with a similar kind of model, though the latter set of system is as yet incompletely described or understood.

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

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

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

a. exponential complexity of relational patterns

b. increasing underdetermination

c. increasing alternative variation of resulting patterns

d. increasing emergent properties associated with such systems

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

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

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

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

It is the case that in terms of our language of description to match in an empirical manner our level of observation and to designate our units of analysis, we are thrust upon the horns of a dilemma to the extent that we must deal not only with the physical relativity of natural systems in terms of our observational experiments, but we must deal with the anthropological relativity of our language and knowledge in terms of the designated units of analysis and description that we apply to our observations. On a naïve level, basic descriptors derived from "natural classes" in any language appear to be sufficient to the tasks of basic qualitative description. We have mathematics, the language of science, to come to our rescue especially when we are referring to basic and "average" physical processes, as for instance those entities represented by the periodic table and those energetic event structures described within the framework of classical mechanics. But even upon a micro-biological level the language of mathematics and its inherent logic begins to break down under the shear weight and complexity of the problem of natural description. The function that mathematical language serves upon a biological level is fundamentally different than the function it serves upon a physical level. A strong case can be made that mathematical description breaks down almost completely upon the even more complex human level of analysis, except in the form of applied statistics and rather gross and concrete numerical descriptors. But even upon the fundamental level of physical analysis and observation, resort strictly to mathematical description is inherently insufficient to the inclusive problem of descriptive explanation. Most physical properties or laws that govern systems upon these fundamental levels are defined in terms of linguistic based variables or logical syllogisms that are held to be generally if not universally applicable to all cases, and most such properties, principles or laws were derived at through empirical observation and experimentation in conjunction with deductive reasoning that is applied to the evidence at hand. In such a context, mathematics as used in the theoretical or applied sciences takes on a basic applied function that is distinct from its abstract articulation in pure mathematical theory. In such a case, as demonstrated for instance through statistical description and manipulation, mathematics is applied to natural data sets or samples or populations of "points" which discrete point determination, as referred to previously, is inherently problematic from a linguistic and observational point of view. Dealing with natural sets of data points defined experimentally or observationally is fundamentally different from dealing with abstract sets of numbers or points defined arbitrarily or by means of logic. If we hold to our initial pressupposition that all entities and event structures are inherently underdetermined and continuous, then the application of discrete and discontinuous labels or attribution to these sets must be on some basic level fundamentally problematic.

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

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

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

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

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

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

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

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

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

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

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

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

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

1. For any given system or set of systems, there are three analytical levels that must be specified: 1. Subsystems composing a system; 2. The System in itself; 3. The Supersystem of which the System is a subsystem.

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

2. Any given System at any given level of analysis can be characterized in three ways: 1. As a System in itself; 2. As a Subsystem of a surrounding super-system; 3. As a Supersystem containing subsystems.

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

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

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

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

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

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

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

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

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

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

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

Scientific description is an attempt to linguistically represent the patterning of reality in a reliable and faithful manner. Such description can proceed at different levels, in alternative circumstances, and may lead to different kinds of results. As mentioned previously, description brings us to the problem of language parallax, and largely, the problem of anthropological relativity of the knowledge that such language entails. We may say in general that the goals of scientific description are to lead to explanation in the shortest and most succinct route possible. Therefore, explanation is in a sense inherent and a part of scientific description, and should be a logical outcome of correct description. We see as well that preconceived views or models about reality can have the influence of channeling our description metaphorically in certain directions that may or may not reflect the actual patterning of reality.

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

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

Explanation refers only indirectly to the observation, or to the phenomena involved in a general sense, but refers primarily back to the description that we have formulated in relation to the observation. Observation, especially when this is constrained experimentally by systematic measurement, is itself a form of deliberate description, or at least the selective perception upon which such description is based. Explanation carries the entire process one step further, and depends upon a deliberate "distanciation" or alienation from the source of the information, as well as upon the reliability of the descriptive information that was derived from the source. Explanation furthermore is concerned with the logic or coherence of the resulting statements concerning the prototypical patterning, or structure and its validity. It can be seen that the primary preoccupation of description is consistency and reliability, while the primary concern of secondary explanation is coherence and validity of the models that are derived. It can be said in a reciprocal way that explanation is really a form of secondary or derivative description that takes description from a specific or methodological level of analysis to a general or theoretical level of synthesis. Again, the feedback nature of this process must be emphasized, as the development of theoretical explanation will in turn condition our initial responses and observational frameworks, and will lead to refinement of our descriptive informational background.

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

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

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

The basis of number theory is strictly arithmetic and mathematical. Number systems and their manipulations are considered purely theoretical and abstract. My point of departure for metasystems theory in relation to number theory is to propose a class of complex number in which a number stands as a mixed heterogenous variable that may be used differentially in a number of different kinds of systems. Each number then would represent a complex variable, that could stand for a large number of subsets of numbers or variables, while at the same time, standing for itself, and standing as part of a larger system as well. It may seem that this is a way of rendering mathematically systems extremely unwieldy and overcomplicated. To get at the issue, we must go to the basic meaning of what a number is and what it represents in reality beyond its own logical representation. Generally, we count things in sets. If we count a set of five pennies, we can assign the number one to each penny, and the number five as a denominator to the set as a whole, especially if we recognize a five cent piece as a whole unit of which a penny can be considered to represent a proportion of that set. Alternatively, we can say the following:

1 + 1 + 1 + 1 + 1 = 5

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

We can then simplify the equation by multiplication:

1 x 5 = 5

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

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

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

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

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

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

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

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

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

A complex set is a problem set of uncertain dimensionality and unknown solubility.

Theoretically and methodologically, complex sets are the stuff of scientific research. Science attempts to apply systematic means to reduce complex sets without known solutions to simpler sets with known solutions.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The second set of problems is related to the first, and concerns the methods of analysis that we put the data that we do manage to collect. Analysis by statistical techniques has become a standard norm in most scientific endeavors, par for the course, and it represents a second level of measurement that is derivative of and based upon the first level of actual data collection. We end up with a wide possible variety of secondary data sets (averages, Z scores, correlation coefficients, regression equations) that cannot be found anywhere in the data itself, but is implicit to the data as it was collected. We all acknowledge that there are no 3.4 person families in America, but the may well be the average, and this average is no less real or valid (nor any more real or valid) than the raw counts upon which it was based. The second problem of measurement theory is like the first therefore in that analytical research budgets are also circumscribed by limited resources, and we must pick and choose what kinds of tests that we wish to subject our data two.

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

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

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

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

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

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

Such sampling may be analytically driven by our analytical models that we will employ in their selection, as in highly developed medical research designs that target select types of population, or it may be more encounter and discover oriented, such as when an archaeologist conducts a preliminary surface survey to determine the viability of digging in a certain area. Either way, we are never 100% clear as to the total size, limits and structure of our sample, and even presuppositions of randomness are only loosely approximated by any randomization procedures we may superimpose upon our sampling. Such problem sets tend to be context-based systems, and they are structured by the unknown variables more than by known factors.

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

Possibilistic statistics is proposed as an intrinsic part of measurement theory as a means of providing a way of systematically dealing with complex sets of data that are partially factorial. In this case, the object of possibilistic statistics is to try to determine:

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

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

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

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

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

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

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

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

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

2x/15, 10y/25, 1xy/5, 3y/10, …..

How can we know where to draw the line in our sampling, such that our number represents a significant proportion of the unknown whole. First, though a complex set is open and undetermined as an unknown problem set, we may say that all complex sets are analytically finite and hence minimally determined sets. If an analytically solvable set is finite, then we can predict that in general, though we cannot know where to determine the final limits:

1. A larger sample set is better than a smaller one, as long as the larger one is unbiased.

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

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

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

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

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

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

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

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

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

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

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

The interest and deliberate intention of possibilistic statistics is:

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

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

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

c. random and non-random pattern may interact

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

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

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

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

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

It is apparent that with possibilistic statistics applied to very large and complex systems, our profile of possibilities will tend to be continuously shifting with the addition of new information. As in the case of the hominid fossil record, for instance, where the evidence is few, fragmentary and far between, and the gaps of the unkown loom large on any index-horizon, each new discover tends to have significantly great impact on the understanding of the whole. This is indicative of the relative lack of knowledge relating to this fossil record, a function of its potential size and complexity. In other words, if we have very small sample sets to infer about very large and complex real sets, then each new bit or variable of information added to our knowledge is likely to have a proportionately greater effect in restructuring our estimates of variability about the system as a whole. The next "nth" thing found in a complex system is more likely to be unlike any previous thing found than like. If this is not found to be the case, then it can be presumed that the larger system is in fact a simple and relatively stable one. It appears that, inspite of much synchronic variation, the hominid pattern through time was quite stable and its rate of change rather slow. This lends greater credibility to the tendency for lumping versus splitting of the hominid fossil record. If new fossils are found that show significant differences from previous sample sets, it is likely that the fossil record will prove to be much more transitory and variable over time and place than is currently inferrable from the record. It does appear that there were episodic periods of sympatric speciation during certain periods of this record, with side-branches, presumably more niche-specialized, eventually coming to an end. The main line, otherwise, or trunk of the hominid family tree, appears rather stable and steady in its transition characters.

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

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

From this standpoint, the following complex set:

5x, 2y, 10z, 20w

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

1/20….(5x, 2y, 10z, 20w….)

(5/20x, 2/20y, 10/20z, 20/20w….)

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

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

(1/4, 1/10, 1/2, 1/1…) + (x, y, z, w….)

(.25, .1., .5, 1.0…) + (x, y, z, w….)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Uncertain inference and inductive reasoning Unknown sets. Descriptive Typology

Complex sets, Heisenberg-Einstein Sets

Transforming sets into matrices, topological translation of sets, unknown sets and partial translations

Hypotheticals and model construction

Description

Distribution

Transmission

Transition

Dimensionless variables

Dimensional variables

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

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

Taxonomies are basically defined by implicit rules governing the order of the relations between the components of such systems. The theoretical models upon which scientific taxonomies are built define in an explicit manner the rules by which a taxonomy should be constructed.

There is feedback to typological description and even observational technique and selection that creates increase in scientific knowledge upon a basic level.

Descriptive typology versus an explanatory taxonomy