Mathematical Meta-systems--Ideal & Applied

by Hugh M. Lewis

The question of mathematical systems theory brings up issues of non-linear & dynamic systems & control theory. It is beyond the scope of this brief article to address this in any sufficient manner. I only want to address within this introduction to mathematical meta-systems some basic developments that I have pursued in the past and have a continuing interest in developing in the future.

Mathematical systems are pure symbolic systems of representation that are restricted and constrained by pure logical relationship and logical processes of deductive reasoning. Mathematics is referred to as the language of science (and engineering, in applied forms) and basically all computational systems constructed by human beings are mathematical systems in either an analogical or digital form. Artificial intelligence breaks down primarily at the point at which purely mathematical computational models are deployed to represent symbolic systems that are not necessarily mathematical in design, and artificial intelligence solutions have succeeded best in those restrictive problem sets, like chess playing and super-computing, in which complex solutions are derivative of purely mathematical & logical relationships. This leads to a rather interesting question of whether natural biological intelligence systems may be fully characterized by mathematical models & designs, as for instance, neural networks.

Formally speaking, mathematics in the pure theoretical sense represent the only known form of "ideal system" of knowledge that is purely abstract and that has no necessary forms of representation in reality, though it appears that upon a physical level all real systems and all relations of real systems are characterizable in a consistent mathematical manner. In other words, in terms of mathematical theory we may devise mathematical relationships and structures that have no necessary representation in any real form, and these structures may be claimed to be formally "true" in a strict logical sense, though they admit of no realized representation beyond the symbolic formula used to verify and express these structures. Ideal systems are the only form of knowledge system which may in the strictest sense be claimed to exist objectively independent of any necessary empirical verification of such systems, though many phenomenological & objective empirical verifications may be made of such systems, and though the first mathematical systems were in fact derived from empirical observation and logical deduction from real systems.

Upon a physical level of organization, all real systems are able to be characterized in terms of mathematical relationships, expressible in terms of one or more mathematical formulas, and to the extent that such systems are measurable in terms ultimately of time, space & count, are amenable therefore to parametric statistical description in terms of continuous, nonlinear & linear relationships between multiple variables. 

My pursuit of mathematical systems has led me to basic philosophical inquiries in mathematics and logic, and the problems of abstractive characterization of real systems by means of ideal systems represented by mathematics. It has led to basic philosophical investigations of number theory, theory of estimation, error & approximation, measurement theory, set theory, matrix theory, and to basic questions in statistical description of populations and points or entities of populations.

The challenge in the construction of such mathematical systems frameworks is in terms of the parsing of continuous variables in discontinuous terms, and manipulation of discrete, qualitatively defined variables in terms that are of a parametric and continuous character. It is also a challenge of handling relative systems in a non-relative manner, as it can be said unequivocally that ideal mathematical systems are the only non-relative form of knowledge we have, and it is for this kind of reason that some people speculate that the only manner that we may be able to communicate with alien life forms that are intelligent will be through the language of mathematics. Finally, it seems to be a challenge of deriving an ought from an is, or of working in a non-arbitrary manner from a descriptive system of denotation to a prescriptive system of connotation. Most philosophers would probably maintain that this last challenge is an impossibility, that there are no non-relative, non-arbitrary systems of value attribution in the human sense that are ultimately not the product of human choice, volition and preference.

When we characterize and describe entities in the real world as a system of some kind, we must take care in the terms we use to summarize, define and elaborate such a system as a whole or in part in terms of the relationships it may maintain within its meta-systemic context. If we seek to describe such entities as members of a larger set or population, we must take care in our description and designation of such an entity on the basis of count, presence or absence. In short, we should seek to understand that monothetic descriptors might disguise polythetic sets of traits and interacting variables with a very high degree of relational complexity occurring within such a system.

General Systems Essays, Vol. I