Chapter XXI

In attempting to get at abstract systems and general theories of natural systems that have universal validity, it is important to resolve a basic consideration having to do with the inherent complexity of naturally occurring systems. Multi-variable nonlinear differential equations are insufficient to express the entire problem set at even rudimentary levels of naturally occurring phenomena. Such equations remain beyond proof or even solution, but it remains possible, within a paradigm of possibilities that are defined by basic parameter variables, to create a generative system of differential equations that are interdependent upon one another in terms of their input and conditional values and variables. The consequence and possibility is to be able to produce a parsimonious mathematical model of complexly developing systems without having to resort to working out solution sets for every differential equation that is encountered. Such equations would be derivative and based upon other equations, which in turn would be based upon and derived from yet other equations, which eventually would resolve to single variable, soluble equations that operate within a possible range of discrete or continuous input values. A complex natural system would then be expected to be solved in terms of an interrelated set of complex differential equations that would be framed within a paradigm of possibilities, a matrix that is composed of simpler sets of equations, and so on. The solution we would seek would be in terms of a simulation of the pattern based upon the set of governing equations that we define for that system. The degree of achieved detail and representative accuracy would be a measure of the degree of closeness of fit between the real system and the artificially contrived one. It becomes possible to express a complex theory of systems accurately in terms of a single general differential equation that can be unpacked by its systematic qualification of variables as derivatives of nested differential equations within a matrix hierarchy. 

It would be necessary as well to build into such equations the uncertainty factors that would provide for the under-determination of structure that all naturals systems exhibit.

It has been shown that metaphysically and naturally, chaos underlies and is more basic to order, and all real systems tend, in the structure of the long run, to return to a state of greater disorder. It is worthwhile therefore to take into more careful consideration the problematic that the notion of chaos implies for our understanding of advanced systems.

In an abstract sense, absolute systems, as for instance, mathematical systems are only possible if they have an implicit and antithetical counter-reference to absolute disorder and chaos. We can say that they achieve their coherence by the absolute determination of their values and relations, leaving no room for uncertainty. Thus, in such a world, uncertainty is excluded to a domain of the implicit. Underlying this is a sense that uncertainty and disorder are inherently and ideally disordered. Consider trying to generate a list of random numbers off the top of your head--can you be sure the list of numbers you generate are completely random. If you think about them, and attempt to rearrange them so that they appear to exhibit less patterning, might you not be imposing some sense of order upon them?

Much of probability theory can only be construed from the standpoint of a hypothetical "null space" that is defined by total randomness and randomization, which is itself an ideal state that is never attained in nature or real systems. Indeed, the entire structure of mathematics as we know it could not exist without the central notion of zero as a common point of reference. Without the notion of zero that implies nothingness and hence disorder, we could not have equations or perform many operations that are common to mathematics.

To try to treat disorder in a systematic manner, to deal with it in terms that are complementary and integral to systems theory, is to try to put a handle upon a significant aspect of reality that influences every real system that exists. The outcome of chaos theory is that even high complex systems can be based upon relatively simple operations, and relatively simple formulas can generate highly complex and unusual outcomes. But not all disorder is capable of being patterned--in any system, there should always be some residual sense of true disorder that cannot be accounted for by any means.

I hope to demonstrate thereby that there can be found order in disorder, and we can superimpose a sense of system upon a sense of disorder itself. We do it not out of some strange pathological compulsion to minimize uncertainty and chaos. We do it rather out of necessity in our theorems. If we don't, then there remains a residual possibility that, in failing to deal adequately with the tasks at hand, these issues will somehow creep into our formulas and undermine our ability to functionally extend our theories to real systems.

In attempting to do this, I am not so interested in stochastic theory and probability, as I am interested in a system of possibilistics that must underlie any kind of stochastic estimation. Before we can judge the odds, we must know the playing field we are dealing with in a manner that allows us to make such choices. However uncertain, our knowledge must somehow move from remaining remote and unknown to being proximate and at least inferable.

If we wish to derive some kind of sample, whether representative or randomly, then we must at first understand the possible sample spaces or regions that are available to being sampled, and that are defined as those that are interesting by the criteria of our theory and its operationalization. But often we cannot know beforehand the possible sampling spaces that might be important to our operational procedures. Much that might be of value to us in possible sampling domains must remain unknown--this is part of the reason we sample in the first place. It represents a kind of exploration of unknown areas. Similarly, if we seek some solution to a problem, we are at first confronted with a potentially infinite number of possible choices and alternatives. We must pick and choose a pathway based upon some series of choices that will lead to a successful solution to the problem. Often, we cannot know not only the correct choices, but even the possible range of choices to begin with. In complex problems, we may construct initially complex search tree structures, but none of the possible outcomes may necessarily lead to the correct solution.

The question of possibilistics therefore leads directly into the problems of operationalization of procedures, an issue that will be undertaken in the next part. It deals especially with the heuristics of problem solving, and many issues broached in this chapter relating to initial problem definition and identification will be taken up further in the second part. At this point, all I wish to do is to elaborate a form of continuous and nondiscrete statistics, or variable statistics, that can be used to conceptualize alternative possibilities for any given problem. In this regard, the problem is especially the issue of the constructive representation and application of alternative metasystems models to real working models in any number of different areas. This in itself creates a large space of possible alternatives that should be considered part of the issue.

If we start with a simple 10 by 10 matrix, such that any slot within the matrix may be filled with a number 0-9, and our job is to create all possible combinations or permutations of strings occuring in rows or columns of the matrix, we quickly find that we have an overwhelmingly complex number of possibilities, something of the order of 2 x 10¹⁰ . What appears at first as a rather simple square matrix of a very manageable size, quickly zooms to astronomical complexity when we begin searching for its solution. If we built a computer program to generate by recursion or reiteration all these possible strings, it would require a very long running time, and would be liable to consume the working memory resources of the whole computer.

As one computer science teacher told me, consider trying to realistically represent and explain the orbits of all the lunar bodies of the solar system about their planets, as these spin about the sun, and then try to fit this into a larger pattern of motion of the sun within the galactic system it occurs in. Though the motions are elegantly described by mathematical equations and are sublime observed through telescopes in the night sky, actually plotting these complex astronomical ballet moves is virtually impossible.

We could impose rules upon our matrix problem to narrow its search space. For instance, we could specify all strings that are only of a certain length, or that have a certain initial order, say 999. Doing so would limit the total space of possibilities considerably.

It is perhaps part of the project of possible statistics, or possibilistics, to be able to get an idea of the inherent complexity represented by any problem set, without having to reach a complete solution to the problem. In other words, if a simple problem proves to have an astronomically complex solution set, then it is better to represent the problem as some kind of recursion function than as a complete solution set of alternative sample points. This is clearly the case in most of the sciences, even on very basic levels. We always prefer the correct formulas to the actual solutions to any particular problem.

It is often the case that unintentionally, rather sophisticated and straightforward statistical problems require strict randomization criteria that prove almost impossible to meet, especially with large sample sets. It is the epitomy of wisdom in such cases to systematically restrict the problem set down to some narrow range of possbility within the larger spectrum in order to achieve more control and accuracy of the results. It is a case in statistics that the law of large numbers does not necessarily apply if you have a genuinely or relatively random sample--you can have the largest sets possible but it would mean nothing if they were not randomly selected.

Many theories, especially in statistics, rest upon a presupposition of purely random samples. Often, this is taken for granted, or fudged, when in fact it is truly difficult if not completely impossible to create a truly random sample, especially with people. Determinisms creep into our database in many different ways, often without our understanding. But this in itself is not necessarily a bad thing. Some kinds of surveys that can generate deep knowledge and understanding are not necessarily contradicted by the presence of bias in samples. It is possible that even with great bias, samples remain true to life and representative of the reality they purport to explain.

Probability theory has been well worked, as many people have purported to depend upon it. But possibility theory remains something of an unsolved mystery, and therefore is something more worthy of understanding for its own sake. I would call possibilistics a form of statistics that comes before description. Perhaps it can be called observational statistics. It does not necessarily presume randomness in the research design. Rather it presumes only a natural self-organization of pattern irrespective of our own observational biases we may introduce into the sample organization.

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