International Encyclopedia of Systems and Cybernetics

The mathematician Hermann WEYL wrote in 1928: "Occidental mathematics has in the past centuries broken away from the Greek view and followed a course which seems to have originated in India and which has been transmitted, with additions, to us by the Arabs: in it the concept of number appears as logically prior to the concepts of geometry. The result of this has been that we have applied this systematically developed number concept to all branches, irrespective of whether it is most appropriate for the particular application. But the present trend in mathematics is clearly in the direction of a return to the Greek standpoint…" (Quoted by W.H. HUGGINS – 1963, p.132).

This view proved to be prophetic.

W.H. HUGGINS observes that "traditional mathematics does not provide us with adequate denotations of the following kind of entities: signals, patterns, operators and, generally, scaled sets of physical observables in space" (p.133).

To the "science of magnitudes", as A. BOGDANOV described mathematics (1980, p.46), a complementary science of static and dynamic connexions is now growing swiftly. It includes graphs, groups, topology, sets (common or fuzzy), catastrophes, fractals, chaos, criticality, and possibly more.

G. KLIR acknowledges this: "Mathematically derived systems knowledge is the subject of various mathematical systems theories, each applicable to some class of systems. It consists of theorems regarding issues such as controlability, stability, state equivalence, information transmisssion, decomposition, homomorphism, self-organization, self-reproduction, and many others" (1991, p.101).

From a systemic viewpoint, G. KLIR observes: "Mathematical theories are most frequently developed for assumptions that are interesting or convenient from the mathematical point of view. As a consequence, they produce methods that cover rather small and scattered parts of the whole spectrum of systems problems… Hence, the primacy of problems in systems methodology is in sharp contrast with the primacy of methods in applied mathematics" (1993, p.39).

As a result, only parts of classical mathematics are well adapted to systemics. However, various new developments of mathematics appeared during the 20th. century, independently or not from G.S.T. and Cybernetics, as well as some new theories, not purely mathematical, but with a strong mathematical basement, which are obviously useful for systems sciences and methodology.

In a sense, systemics adopts or even spawns a new type of qualitative mathematical theories.

Hereafter, a tentatively classified list of these theories.

1. Older mathematical and logical theories and concepts well adapted to systemics: (partly based on L.von BERTALANFFY's "Outline of a G.S.Theory") (1962, p.134-165).

- Theory of categories (N. HARTMANN)

- Differential and integral calculus

- Isomorphic laws: the exponential law; the logistic law (VERHULST); the law of allometric growth (W. PARETO).

- Mathematical theory of populations (V. VOLTERRA – A. LOTKA)

- Binary logic (BODLE)

- Theory of sets

- Theory of graphs

- Theory of nets

- Theory of cyclical phenomena

- Stability theory (LIAPOUNOV)

2. Mathematics of cybernetics

- Theory of communication (C. SHANNON and W. WEAVER)

- Theory of feedback, oscillations and control (N. WIENER)

- Theory of information (C SHANNON, L. BRILLOUIN, D. MacKAY and others)

3. G.S. generated mathematics

- Mathematical theory of systems (von BERTALANFFY)

- Systems dynamics (J. FORRESTER)

- Theory of hierarchical multi-level systems (M.D. MESAROVIC, D. MACKO & Y. TAKAHARA)

- Arithmetic relators (groupe SYSTEMA France)

- Reconstructability theory (G. KLIR)

4. Independently created mathematical theories important for systemics

- Topology

- Catastrophes theory (R. THOM -E.C. ZEEMAN)

- Fuzzy sets theory (L. ZADEH)

- Fractals (B. MANDELBROT)

- Theory of games (J.von NEUMANN)

- Theory of algorithms

- Theory of recursivity (G. SPENCER BROWN)

- Deterministic chaos (LORENZ, SMALE, RUELLE, etc.)

5. Systemic mathematical theories originated in some specific sciences

- Theory of dissipative structures and bifurcations in non-equilibrium systems (I. PRIGOGINE)

- Synergetics (H. HAKEN)

- Theory of hypercycles (M. EIGEN & P. SCHUSTER)

G. KLIR emphasizes "the current trend of generalizing mathematical theories…" and gives the following examples:

- " from quantitative theories to qualitative theories,

- from functions to relations;

- from graphs to hypergraphs,

- from ordinary geometry (Euclidean as well as non-Euclidean) to fractal geometry,

- from ordinary automata to dynamic cellular automata;

- from linear theories to nonlinear theories,

- from regularities to singularities (catastrophe theory);

- from precise analysis to internal analysis;

- from classical measure theory to fuzzy measure theory,

- from classical set theory and logic to fuzzy set theory and logic;

- from regular languages to developmental languages,

- from intolerance to inconsistencies of all kinds to the logic of inconsistencies;

- from simple objective criteria optimization to multiple objective criteria optimization" (1993, p.52-3).

6. Other models and formulations

- attractors of distinct types games of life (J. CONWAY) (in fact: cellular automata)

- Markov Chains and Matrixes

- Markovian systems

- Matrixes

- Networks

- Percolation

- Power laws

- Renormalization

- Scaling laws

(these last three closely related)

- Solitons

- Tilings

(In most cases, see corresponding article)