01. What does Modern Topological Theory do?: (Poincare) It permits one a. to model relationships whose complexity surpassed the limits of algebraic expression and b. to study not only the translational changes within the system but the qualitative transformations that the system itself undergoes. Topology describes transformational events (deformations) that introduce real discontinuities into the evolution of the system itself. Those events are determined by the singularities of the flow space and not the quantitative substrate.
02. What do we mean by singularities?: a designated point where a merely quantitative or linear development suddenly results in the appearance of a ‘quality’ (i.e. rainbow) All such forms that owe their existence to singularities.
03. What is form?: Forms constitute nothing absolute but rather structurally stable moments within a system’s evolution; yet their emergence (their genesis) derives from the crossing of a quantitative threshold that is, paradoxically, a moment of structural instability (…) All forms are irruptions of a discontinuity, not on the system but in it or of it (…) All forms of the universe are produced as by-products or maps of particular evolutionary segments of one or another dynamical system. Forms are not fixed things but continuous metastable events (…) Forms are always new and unpredictable unfoldings shaped by their adventures in time.
04. Why are forms susceptible to change?: This is possible because forms are dissipative systems, thus open, dynamic systems where information is flowing out if it and into it as well. This energy comes from other systems contiguous to it and operating within it or upon it: that is, at entirely different scales of action.
05. What keeps a system dynamic?: It is the continual feeding and siphoning of energy or information to and from a system -simultaneously in continuous transformation locally and in dynamic equilibrium globally.
06. What does the flow of energy do to the system?: A. Information from outside the system will pass to the inside. The outside of the system becomes depleted, the inside’s flows are perturbed and same effects are now produced now in reverse. B. Information from certain levels in the system is transported to other levels, with dramatic results as some parts or strata can not withhold the minimum change and thus precipitate a catastrophe forcing the system to find a new equilibrium.
07. What is catastrophe theory?: Every event or form enfolds with it a multiplicity of forces and is the result of not one, but many different causes. Form is the result of strife and conflict. Catastrophe theory is interested in the effects of forces applied on a dynamical system from outside, forces that it then becomes the task of the system to neutralize, absorb, or resolve. A catastrophe can occur only in the region of a singularity. The catastrophe set is a three dimensional, unpredictable and open ended irruption of a 2d surface and thus an example of virtual form.
08. What is a virtual form?: It is a real ‘fold’ in real n-dimensional space that can give rise to indeterminate morphogenetic events in the n+1 space.
09. What is an epigenetic landscape?:it is invented to depict the relation between phenomenal forms and the morphogenetic fields in which they arise. It is an undulating topographical surface in phase space whose multiplicity of valleys corresponds to the possible trajectories (shapes) of any body evolving on it. Seen from above any form is susceptible to to any exogenous forces. Seen from below, it features a complex network of interaction underlying it.
Kwinter, S., Boccioni, U., 1992, ‘Landscapes of change: Boccioni’s Stati d’ animo as a General Theory of Models’, in Assemblage No. 19, MIT Press, pp. 50-65