II. - Patterns, analogy and isomorphism.
III. - Auto organisation Theories
V. - Auto referential patterns
-4A- Theoretical frame
The term of "catastrophe" means a sudden change of behaviour at the time of very gradual and continuous changes of external parameters, but also a general type of systems where such changes produces.
The theory describes the morphology of discontinuities in living (and not living) processes, to qualitatively foreseeing the evolution of complex systems (systems which number of control parameters is immense or infinite), and to formalise change of state processes in physic sciences or in human sciences. It leads to the recognition of new characteristics of dynamic systems and of regulation processes (living beings) : structural stability and change properties.
Its theoretical statement supposes a system which state is described by N internal variable, and by P external parameters. This system will be endowed of a potential F, (function of the internal variable) governing the system so :
- to all value imposed to external parameters, system state characterised by internal variable itself is on a local minimum of the potential F function.
- Minima of potential F defines the local steady systems so called "strange attractors(*)" of the system.
(*) points or spaces in the space of control to which the system tends asymptotically.
-4B- Generalised chaos theory.
The animate or inanimate metabolic forms present a character of circular periodicity. Their evolution is submitted to recurrence ; the trajectory of the dynamic fiber of system coming back to the neighbourhood of his starting point and cutting up the initial space an infinite number of time ; phenomenon that characterises an "attractor" which dimension is superior to 1.
The dynamic of the "attractor" of a metabolic form, insofar as it possesses its own periods, is susceptible of entering in resonance with some similar metabolic forms. It is thus "bearing significance" (René Thom, 1972). A generalised chaos intervenes by destruction of the recurrence of this fiber dynamic, caused by a perturbation that exceeds regulation possibilities of the system, that is going to fix the system in a characteristic form of fractal curve ( so called Julia curves, or Mandelbrot sets) of different shapes (filamentous, tubular, laminar).
Samples : creation of bubbles, of curds, or arborescent shapes.
Some are catabolic :
- the death, the psychosis.
Other are anabolic :
- Genealogical arborescent shapes ;
- ontogenesis (Cf. epigenetic theory.)
- the learning of speaking process
(every word becoming object of a kind of imprinting and circumscribed familiarisation).
Strictly speaking, only the living dynamic is subject of such anabolic arborescent shapes, but artificial intelligence tries to simulate such genealogical arborescent processes, with systems also able to learn thus the ANELLA system [ Associated Network with capacity of Emergent Logic and Training ] ; intelligent protosystem issued from Cognitive Sciences researches of the British Telecom and CNRS (88-89), that required a new mathematical object created by Gisèle de Meur ; the P-DUAL of a graph.
Examples:
- genealogical arborescent of computer viruses capable to adapt and modify by themself ;
- the very classic "game of the life" simulating a cellular clones;
- games with capacity of training, such as electronic games named Tamagochi ;
-4C- Elementary chaos theory (R. Thom 1972, E. C. Zeeman 1980).
For some values of the external parameters, several minima of potential could coexist: there are then two or several attractors in conflict. An elementary disaster included such a structure: the lighter disruption of the external parameters will conduct the system to jumping brutally in of new steady states.
- "An elementary catastroph, as could say René Thom, corresponds to the apparition to a certain t moment, of a discontinuity of the morphological appearance in the development of a trajectory, of a space, discerned like continuous before and after the catastroph."
The seven elementary catastrophs are:
1. The FOLD; 2. The GATHER ; 3. The DOVETAIL; 4. The BUTTERFLY; 5, 6, and 7. THE UMBILICUS
The description of these morphologies was given back possible to the following of the demonstration of theorems of differential [topology] (Theory of universal spreading).
-4D - Typology of the Elementary catastrophs
1. The stratum of FOLD: It corresponds to the archetypes of birth, of death, of the beginning and of the end, etc.
2. The GATHER : It visualises two steady regimes in conflict ; a bimodal behavior : laugh/ tears; fear/ agressivity.
3. The DOVETAIL: For Jean Petitot (1983), it corresponds to the morphology of A. J. Greimas semiotic square, joining contradictory and complementary metaterms. By ex: must make/ do not have to make/ must do not make/ do not have to make.
4. The stratum of BUTTERFLY: It makes to intervene two situations where bimodal conflicts are embedded one each other. For. ex. : communication source/ message/ receptor.
5. HYPERBOLIC UMBILICI : Morphology of breakink like a wave on a beach ;
6. ELLIPTIC UMBILICI: sword, tip, intrusion, etc.
7. PARABOLIC UMBILICI: Collision of a tip with a surface, of jaws closing again.
In human sciences an interaction is a dynamic sequence of exchanges of several messages between two people at least. It exists a specificity of the interaction, not reducible to the personality of actors. The concept of interaction appears at the same time in a psychodynamic perspective and in the general theory of systems. A domestic interaction describes the way members of a family are connected each others, communicate, and play each one their role. This interaction, or acted relation, fantasised, dreamed is working also in the therapeutic or educational relation, in so far as the observer (therapist, teacher, ethnologist) is a part of the system. - Behind Gregory Bateson (ethnologist), one interaction can be said complementary, symmetrical, reciprocal, parallel, divergent, or convergent (shismogenesis), depending on whether the observer is inside or outside to the system, and the way he modulates his intervening in a real situation. According to the cases the approach will insist on fantasy, oniric, or transferential aspects of the interaction ; or rather on communicational, behavioural, ethologic, or symbolic aspects.
-5B- Intricate Hierarchies & strange loops
- The intricate hierarchies are two or several hierarchies coupled each others, as the coupling is constituted of "strange loops"; loops in which the lower level of a hierarchy is retroacting on the superior level of an other, while the superior level of the second is retroacting on the lower level of the first : _ Is H1 and H2 two hierarchies, A and B the subsystems coupled belonging to H1 and H2; one has then A under B in H1, and B under A in H2. If he one considers the system in his totality, something interior of the system leaves of it in order to act on the system. Such profiles appear in non oriented topologic spaces no oriented, such as the "ribbon of Moebius" (also called wheel of Lao-Tzu) or the "bottle of Klein". For Douglas Hostadter (1980), an intricate hierarchy stands on an "auto modifying software level" and an "unchanging material level," because it exists in any auto-organised system a "protected" level that cannot be reached by other levels rules, whatever could be the entanglement of their interactions. For the author of "Gödel, Escher, Bach," a strange loop is definite as the return to the starting point at the time of a coming down or of a rise in a hierarchical system.
- such loops exist in the biologic systems, some musical or pictorial works...
In the interactions between the individual, domestic and social system, some strange loops appear also, creating therefore some intricate hierarchies. For J.L. The Moigne, "A strange loop is the levelling of an observed level and of a meta level of observing.
-5C - Multi-level hierarchies
Hierarchies with mutiple levels are one of the fundamental properties of general complex systems. They were particularly studied by Mr. D. Mesarovic, D. Macko, Y. Takahara (1970). These authors propose a distinction between several types of interconnected hierarchical levels :
_ Level of description and of abstraction ; Multi strata hierarchies
_ Level of the complexity of the decision ; Multi layer hierarchies
_ Level of the organisation ; Multi grade hierarchies
Contrarily to the B. Russel's theory (logic types theory), the mathematic theory of categories" (Andrée C. Ehresmann and Jean-Paul Vanbremeersch, 1986) permitted to modelise the junction between these different levels. It ensues a general theory of the co-ordination between these three fundamental systemic levels : modelisation, intervening and organisation. Human mind, domestic system, micro and macro-social organisations gets a hierarchized internal structure, and are mutually hierarchized ; - Some entities (individual personalities, groups domestic, social structures) are borderline assuring the transition from a hierarchical level to an other ; so the domestic system, intermediate between the individual and collectivity. The nature of domestic interactions would determine for an individual his future position in the social hierarchies. However ethology allows to don't reduce hierarchies to power, as noted L. Dumont about macro-social level.
Answer to Bertrand Russell's theory (Logical Types theory), the incompleteness theorems demonstrated by Kurt Gödel (1931-1936) excludes definitely the possibility of a mathematical formalisation auto consistent and complete, in the case it should be sufficiently powerful.
Its demonstration uses a coding that makes reference to himself, an autorefering system. It conducts to statements that affirm the impossibility of them own validation inside their own axiomatic setting.
Gödel's statement says "I am not demonstrable": it is true because one can demonstrate that it is not demonstrable, and yet it is no demonstrable because true...
-6A- Formulation of the first theorem of incompleteness
1." No formal theory of the arithmetic could be at a time finalizing presentable, consistent and complete."
[No one numbers theory doesn't can at a time include a finite number of axioms, obey to the law of no internal contradiction and provide a real interpretation some symbols, axioms and theorems who gives back these last true; conferring to it a statute of a theoretical pattern ;
All the theoretical statements are so-called decidable if either one or other of the statement or his negation is a theorem.]
- 6B- Formulation of second theorem of incompleteness
2." If a T theory is a number theory with a finite number of axioms, the formalisations (formal statements translating an assertion) of the assertion "T is consistent" cannot be deducted in T."
Corollary : No one finite and consistent theory can pretend covering all the assertions and in particular assertions concerning itself.
-6C- Poetic epilogue
For the quantic physicist Basarab Nicolescu (link toward his biographic page and poetic extracts on the Site of Sylvaine Arabo),
..."Our representation of the Nature in terms of level of Reality leads to a gödelian structure of our knowledge."(1)
The internal contradiction of macrophysic level (duality of the nature of the light ; at once material and non material ) is resolved to an other level (2) of reality ; (quantum of energy), the microphysic level that follows no more classic logic rules but rules of the third Included: . "If he one stay to one level of Reality, either the description is contradictory or it implies indecidable propositions."
For B. Nicolescu ; "...The introduction of Third included to a second level of Reality resolves the contradiction to the first level but it opens the contradiction to this second level..."
... "It is crucial to observe that the new logic of the Third included is, like the classic logic, no contradictory."
Following with insistence on the "discontinue nature of Reality" he pursues, refering to C.G;Jung :
. "The unification of the contrary to a superior level is not a rational business nor further a question of will, but a psychic process of development who expresses in some symbols." (3)
. "The hermetic thought. affirms that more our language is ambiguous and versatile, more it uses symbols and of metaphors, better it is able to name an Unity where achieves the coincidence of the contrary."
Basarab Nicolescu is a brightness defender of such intuition as a writer in his " Théorêmes poétiques (poetic Theorems), compilation of French poems published by Le Rocher Ed, 1994.
(link toward an extract of its poems on the Sylvaine Arabo Web site).
(1) article "Jung et la science. Histoire et perspectives d'un malentendu", Basarab Nicolescu - Paris
Cahiers jungiens de psychanalyse, N°80, automne 94, pp 73-80 ("Jung and the science. History and perspectives of a misunderstanding")
(2) Level of Reality: set of invariant systems.
(3)"Commentary on the Mystery of the golden Flower" (link to geomance's bibliography)
