Catastrophe Theory

Nik Boyd


In his seminal work Structural Stability and Morphogenesis, René Thom applies topology and catastrophe theory to a broad variety of domains in order to develop the outline of a general theory of models, with a special emphasis on qualitative models. His explorations cover: form and structural stability; catastrophes and morphogenetic fields; general morphology and semantic models; the dynamic of forms: their mechanics, complexity, information, and significance; biology, embryology, and the dynamics of living beings and cells; and a final chapter on thought and language. Thom's speculations regarding the application of qualitative models to linguistics and semantics provide many interesting insights into the natural origins of language and the processes of language usage. This page contains a number of excerpts from his work.


Archetypal Morphologies

In section 13.4.B of Structural Stability and Morphogenesis, René Thom develops a theory for the spatial origin of syntactical structures. He proposes that we can associate a graph with every spatiotemporal process. Thom suggests that the interaction subgraph of every process belongs to one of sixteen archetypal morphologies.

This theory of the spatial origin of syntactic structures accounts for many facts, for example, the restriction to four actants in an elementary phrase and most of the cases in a language with declension: the nominative, for the subject; the accusative, for the object; the dative, for verbs having the gift morphology; the instrumental, for verbs having the excision morphology of cutting, or ablative. The only classical case that cannot be interpreted by this tableau is the genetive, which is an operation of semantic destruction, dislocating a concept into its regulating subconcepts in a kind of inverse embryology.


Semantic Models

In section 13.8 of Structural Stability and Morphogenesis, René Thom proposes the following definitions and their implications.

  1. Every object, or physical form, can be represented as an attractor of a dynamical system on a space of internal variables.

  2. Such an object is stable, and so can be recognized, only when the corresponding attractor is structurally stable.

  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 of external variables.

  4. Every structurally stable morphological process is described by a structurally stable catastrophe, or a system of structurally stable catastrophes, on the space of external variables.

  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.

  6. When the chreod 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 (or attractors) and of the catastrophes that it (or they) undergo. In particular, the signification of a given attractor is defined by the geometry of its domain of existence on the space of external variables and the topology of the regulation catastrophes bounding that domain.

One result of this is that the signification of a form (chreod) manifests itself only by the catastrophes that create or destroy it. This gives the axiom dear to the formal linguists: that the meaning of a word is nothing more than the use of the word; this is also the axiom of the "bootstrap" physicists, according to whom a particle is completely defined by the set of interactions in which it participates.


Conceptual Stability

In Appendix 2 of Structural Stability and Morphogenesis, René Thom indicates that

We have seen that concepts have a regulation figure, a logos, analogous to that of living beings. We might regard a grammatical category (in the traditional sense) as a kind of abstract logos, purified to the point that only the rules of combination and interaction between such categories can be formalized. From this point of view, we say that a grammatical category C is semantically denser than a category C' if the regulation of the concept of C involves mechanisms intervening in the regulation of C'. For example, take the name of an animate being, say a cat: this cat must make use of a spectrum of physiological activities -- playing, purring, and the like. Similarly, each substantive has a spectrum of verbs describing the activities necessary for the stability and the manifestation of the meaning of the concept. Since the verb is indespensible for the stability of the substantive, it is less dense than the noun. The adjective shares in the stable character of the noun, but it is defined on a space of qualities, deeper than space-time, that support the verb.

When a category C is denser than C', there is, in general, a canonical transformation from C to C'. The inverse transformation, however, is generally not possible.

These rules lead to the following order, in decreasing semantic density, for the traditional grammatical categories: noun-adjective-verb-adverb-affixes and various grammatical auxiliaries.


René Thom. Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Benjamin-Cummings Publishing, Reading, Massachusetts, 1975.