top study-index prior next ------------------------------------------------------------------------ ABCX SUMMARY 1 ------------------------------------------------------------------------ Cognitive dichotomies require DIALECTIC MODELS as support of describing the reality and of dealing with it. As we have seen, dialectic models are relative and polar. Nevertheless, most of current scientific, or applied models are still logical, i.e. absolute and single-term based. Dialectic models can be found mainly in physics and mathematics. We shall briefly present a few of them. Infinitesimal calculus. ----------------------- The proper dichotomy of the calculus is TOPOLOGY with its terms CONTINUOUS and DISCRETE. The calculus both possible orientations and two correspoding branches: 1.orientation discrete --> continuous: differential calculus. 2.orientation continuous --> discrete: integral calculus. Its dialectic character manifests itself immediately through: 1.absence of a common name for both orientations, 2.the uneasiness that each sincere student experiences at his first contact with the calculus. He finds himself expulsed from the apparent paradise of clarity and stability. Things which used to "be something" now "tend towards something", they are no more "equal", but "differ by less than an arbitrary epsilon". Even graduated mathematicians may have problems with basic concepts of the calculus. It is not unusual for instance to hear discussions about the nature of the differential: dy = f'(x) dx, and about it being a finite or infinitesimal increment. Now, this question is ill-formulated, the nature of dy being that of a function. This example should not be interpreted as an attempt to criticize the calculus, or, indirectly, any other similar model. We wish to point out with its help that dialectic requires specific thinking habits and specific types of intuition, which may at first disconcert people trained and experienced exclusively in logical type of thinking. The general relativity and the unified theory. ---------------------------------------------- The proper dichotomy is here again the TOPOLOGY. The general relativity is an attempt to approch it with the orientation continuous - discrete. Its great merits are too well known and too commonly admitted to be discussed here. However, from Einstein's point of view, the general relativity has been only a part of his life's task. Indeed, he dedicated most of his life to develop a unified theory and finished by being discouraged as to its feasibility. Having placed his preference at the continuum side of the barricade, he has most fairly admitted that: ABCX SUMMARY 2 1.the field theory cannot explain the atomic- nor the quantum structure of the (observable (our comment)) reality, due to lack of pertinent mathematical methods. 2.one can forward good arguments for the reality not being at all representable by a continuous field, but by a pure algebraic theory. However, nobody knows what could such a theory be based upon. We see in this statements one of the best arguments in favour of the dialectic principle of alternative otientation, according to which a unified theory is a limit of the series of successive approximations based upon alternative orientations. The general relativity appears to be one of such approximations based upon the orientation continuous - discrete. Quantum theory of field. ------------------------ As in the case of the general relativity, the proper dichotomy is here the TOPOLOGY. Unlike the general relativity, however, the quantum theory of field approches it with the orientation discrete - continuous. It is the most recent trial aiming at a unified theory. It is in developpement phase, which makes it difficult to establish a balance. Nevertheless, it is a perfect example of the dialectic approach and a confirmation of our belief that this approach determines currently the trend of scientific investigation, which seems to advance with progressive steps following the dialectic principle of alternative orientation. Synergetic models. ------------------ Under the name "synergetic models" we understand dialectic models, whose proper dichotomy is SYNERGY with its terms CHAOS and ORDER. We have seen in ABCAX that the dialectic character of the apparent contradiction of chaos and order has been only recently understood. This understanding made it possible to formulate and approach rigorously this apparent contradiction and to conceive such theories as the MORPHOGENESIS of R. Thom and SYNERTGETICS of H.Haken. In the part B of the present study we shall illustrate synergetic models with help of simple examples and we shall discuss their implications for epistemology and informatics.