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1984

Catastrophe Theory

Examples of Integrated, Multi-set Concept Schemes (Annex 3)

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See other Examples of Integrated, Multi-set Concept Schemes. The concept scheme described here is discussed in the paper on: Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation (1984). This was prepared for a sub-project meeting of the Forms of Presentation group of the Goals, Processes and Indicators of Development (GPID) project of the United Nations University (UNU). The annexes were published in Patterns of Conceptual Integration. Brussels, UIA, 1984, pp. 161-204

Summary

This theory is believed by many to have considerable significance for a wide range of domains, whether the concern of the natural or the social sciences. The following points are extracted from:

0.1 "One of the central problems studied by mankind is the problem of the succession of form. Whatever is the ultimate nature of reality (assuming that this expression has meaning), it is indisputable that our universe is not chaos. We perceive beings, objects, things to which we give names. These beings or things are forms or structures endowed with a degree of stability; they take up some part of space and last for some period of time. Moreover, although a given object can exist in many different guises, we never fail to recognize it; this recognition of the same object in the infinite multiplicity of its manifestations is, in itself, a problem (the classical philosophical problem of concept) which, it seems to me, the Gestalt psychologists alone have posed in a geometric framework accessible to scientific investigation.... If the change of forms were to take place at all times and places according to a single well-defined pattern, the problem would be much easier....The fact that we have to consider more refined explanations...to predict the change of phenomena shows that the determinism of the change of forms is not rigorous, and that the same local situation can give birth to apparently different outcomes under the influence of unknown or unobservable factors." (p. 1-2]

0.2 "We may then ask whether we cannot, by a refinement of our geometric intuition, furnish our scientific investigation with a stock of ideas and procedures subtle enough to give satisfactory qualitative representations to partial phenomena. It is necessary to emphasize one point: we can now present qualitative results in a rigorous way, thanks to recent progress in topology and differential analysis, for we know how to define a form and can determine whether two functions have or have not the same form or topological type. We therefore endeavor in the program outlined here to free our intuition from three-dimensional experience and to use much more general, richer, dynamical concepts, which will in fact be independent of the configuration spaces."(p. 6)

0.3 "...when we speak of "information" we should use the word "form". The scalar measures of information (e.g. energy and entropy in thermodynamics) should be geometrically interpreted as the topological complexity of a form." (p. 127)

0.4 "It is sometimes said that all information is a message, that is to say, a finite sequence of letters taken from an alphabet, but this is only one of the possible aspects of information....To conclude, it must not be thought that a linear structure is necessary for storing or transmitting information (or, more precisely, significance); it is possible that a language, a semantic model, consisting of topological forms could have considerable advantages, from the point of view of deduction, over the linear language that we use, although this idea is unfamiliar to us." (p. 144-5)

0.5 "This methodology goes against the present dominant philosophy that the first step in revealing nature must be the analysis of the system and its ultimate constitutents. We must reject this primitive and almost cannibalistic delusion about knowledge, that an understanding of something requires first that we dismantle it, like a child who pulls a watch to pieces and spreads out the wheels in order to understand the mechanism....Our method of attributing a formal geometrical structure to a living being, to explain its stability, may be thought of as a kind of geometrical vitalism; it provides a global structure controlling the local details..." (p. 159)

n 6 "Summary: 1. Every object, or physical form, can be represented as an attractor C of a dynamical system on a space n of internal variables.

0.2. Such an object is stable, and so can be recognized, only when the corresp- onding attractor is structurally stable.

0.3. All creation or destruction of forms, or morphogenesis, can be described by the disappearance of the attractors representing the Initial forms, and their replacement by capture by the attractors representing the final forms. This process, called catastrophe, can be described on a space P of external variables

0.4. Every structurally stable morphological process is described by a structure ally stable catastrophe, or a system of structurally stable catastrophes,on P.

0.5. Every natural process decomposes into structurally stable islands, the chreods. The set of chreods and the multidimensional syntax controlling their positions constitute the semantic model.

0.6. When the chreod C is considered as a word of this multidimensional language, the meaning (signification) of this word is precisely that of the global topology of the associated attractor Cor attractors) and of the catastrophes that it (or they) undergo. In particular, the signification of a given attractor is defined fay the geometry of its domain of existence on P and the topology of the regulation catastrophes bounding that domain." [p. 320-1)

0.7 "It may seem difficult to accept the idea that a sequence of stable transform ations of our space-time could be directed or programmed by an ongoing centre consisting of an algebraic structure outside space-time itself. The important point here, as always, is to regard it as a language designed to aid the intuition of the global coordination of the partialsystem controlling these transformations." (p. 119)

0.8 "Catastrophe theory is a controversial new way of thinking about change -- change in a course of events, change in an object's shape, change in a system's behaviour, change in ideas themselves. Its name suggests disaster, and indeed the theory can be applied to literal catastrophes such as the collapse of a bridge or the downfall of an empire. But it also deals with changes as quiet as the dancing of sunlight on the bottom of a pool of water and as subtle as the transition from waking to sleep." (Woodcock, p. 9)

0.9 "Greek geometry was essentially timeless: any triangle or circle in the real world was considered an imperfect, changeable 'shadow' of ideal, eternal, mathematical form. Thom uses differential topology to start from the opposite premise: that changes of form tin processes as well as objects) are real, and that the aim of science is to grasp what he calls the universe's 'ceaseless creation, evolution and destruction of forms.'....Thom's goal is to describe the origin of forms, which he calls morphogenesis, borrowing a word used by the Greeks and also by modern biologists. To do so, he has created a mathematical language -- catastrophe theory -- built on the assumption of structural stability and stressing qualitative rather than quantitative regularity. He believes that it is general enough to fit the snowflake, the butterfly, and the processes that shape them, as well as the more complex and highly organized processes by which the words 'snowflake' and 'butterfly' enter our minds and speech." (Woodcock, pp 15-18)

1.1 "When Laplace was working on celestial mechanics at the beginning of the eighteenth century, he developed a convenient mathematical shortcut to represent the action of gravitational force. This was the potential, a concept that summed up all the forces acting on an object in a single quantity....It has become customary since then to view many systems as governed by the tendency to seek a minimum of potential energy, although the energy may be of many different kinds....Biology, social science and ordinary language also use the concept of potential....The concept of potential is closely linked to that of equilibrium." (Woodcock, pp. 43-44)

1.2 n-dimensional space-time manifold

2.1 "If each of the attractors involved in a catastrophe enters into competition in the neighbourhood of the given point only with one or several other attractors, this is called a conflict catastrophe....When an attractor enters into competition with itself...it is called a bifurcation catastrophe." (p. 47)

3.1 "The rule of three states: A macromolecular structure can occur in three different states.

The normal evolution of a structure is from stage 1 via stage 2 to stage 3, but the transitions 1 to 2 and 2 to 3 are often reversible, for example, in mitochondria and nuclear membranes." (p. 273)

3.2 "The umbilic catastrophe graphs (hyperbolic, elliptic and parabolic) are respectively five-, five- and six-dimensional. Instead of one behaviour axis [see 4.3 below), they have two, so that a catastrophic transition must be imagined not as a point jumping along a straight line...but as a line jumping across a plane. Obviously, these three types of umbilic catastrophe are 'elementary' only in a technical sense. Their geometry is very rich...." (Woodcock, pp. 64-5)

3.3 When a system is governed by 3 control factors, whether it has one or two behavior axes, there are 5 elementary catastrophes

4.1 "This type of phenomenon, which we call generalized catastrophe, occursfrequently in nature. The topological appearance can be varied, and we can give here an overall qualitiative classification...Formally a generalized catastrophe is characterized by the destruction of a symmetry or homogeneity...

  1. Lump catastrophes...
  2. Bubble catastrophes...
  3. Laminar and filament catastrophes...
  4. Catastrophes with a spatial parameter..." (p. 103-4)

4.2 4 "simple" archetypal interaction morphologies [in a set of 16) whose graphs, linear in form, are labelled as follows:

4.3 When a system is governed by 4 control factors and one behaviour axis, there are 4 elementary catastrophes. If the number of behaviour axes is increased to two, there are then 7 elementary catastrophes. (Woodcock, p 53)

5.1 When a system is governed by 3 control factors, whether it has one or two behaviour axes, there are 5 possible elementary catastrophes. (Woodcock, p 53)

5.2 "Since 1965 the theorem has been extended to describe systems with five control factors, thus adding another four catastrophes, even more complex than the original seven. For catastrophes with more than five control factors, there is an infinite number of singularities without unique unfoldings." (Woodcock. P 65)

7.1 "In situations involving three dimensions of space and one of time, the number of elementary catastrophes is seven. These are:

7.2 "In any system governed by a potential, and in which the system's behaviour is determined by no more than four different factors, only seven qualitatively different types of discontinuity are possible. In other words, while there are an infinite number of ways for such a system to change continuously [staying at or near equilibrium), there are only seven structurally stable ways for it to change discontinuously [passing through non-equilibrium states') . Other ways are conceivable, but unstable; they are unlikely to happen more than once...To put it very simply, in a wide range of situations -- physical, biological, even psychological -- where experience tells us that 'something's got to give' (i.e. there is a potential and a possible discontinuity), the classification theorem indicates that there are only seven fundamentally different ways it could happen." (Woodcock, pp. 52-3)

7.3 "There is a subtle paradox here: each model summarizes the appearance and disappearance of stability, but it does so in a stable way. This is possible, as Thom discovered, because the equilibrium points for general classes of equations can be represented as unfoldings of topological singularities, and because for each of the seven simplest singularities, there is only one stable unfolding: others are possible, but the 'collapse' into the stable form at the slightest disturbance." [Woodcock, pp. 49-52)

7.4 "...Thom reached a remarkable conclusion in 1965: that for a very wide range of processes, only seven stable unfoldings, the seven 'elementary catastrophes', are possible. The unfoldings are called catastrophes because each of them has regions where a dynamic system can jump suddenly from one state to another, although the factors controlling the process change continuously. Each of the seven catastrophes represents a pattern of behaviour determined only by the number of control factors, not by their nature or by the interior mechanisms that connect them to the system's behaviour. Therefore the elementary catastrophes can be models for a wide variety of processes, even those in which we know little about the quantitative laws involved. This is an extraordinary idea: how is it possible that two processes can have features in common even when they are on different physical scales, operate under different quantitative laws and are affected by different sets of causes ?" (Woodcock, p 34)

11.1 "Since 1965 the theorem has been extended to describe systems with five control factors, thus adding another four catastrophes, even more complex than the original seven. For catastrophes with more than five control factors, there is an infinite number of singularities without unique unfoldings. When this occurs, it is no longer possible to distinguish among the possible catastrophe surfaces." (Woodcock, p 65)

12.1 12 "complex" archetypal interaction morphologies (in a set of 16)whose graphs are labelled as follows:

- capturing - sending - crossing

- "almost" - fastening - giving

- rejecting - failing - taking

- stirring - emitting - cutting (p. 307)

16.1 "In fact every linguistically described process contains privileged domains of space-time bounded by catastrophe hypersurfaces; these domains are the actants of the process, the beings or objects whose interactions are described by the text....At each moment...we contract each actant to a point, and when two actants interact this implies that their domains come into contact in a region of catastrophe points which we also contract to a point of intersection of the lines of the two contiguous actants. In this way we associate a graph with every spatiotemporal process. I then propose that the...interaction subgraph...belongs to one of...sixteen archetypal morphologies...,(made up of a "simple" linear group of 4 and a more "complex" group of 12)." (p. 311 and 307; see 4.2 and 12.1 above)

16.2 The parabolic umbilic may be represented on a plane surface by a cyclic sequence of sixteen sections, of which several depict conditions characterized by other elementary catastrophes. The cycle includes eight "principal changes of topological type" [p. 85-6)

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