Revisiting Catastrophes. Thought of the Day 134.0

The most explicit influence from mathematics in semiotics is probably René Thom’s controversial theory of catastrophes (here and here), with philosophical and semiotic support from Jean Petitot. Catastrophe theory is but one of several formalisms in the broad field of qualitative dynamics (comprising also chaos theory, complexity theory, self-organized criticality, etc.). In all these cases, the theories in question are in a certain sense phenomenological because the focus is different types of qualitative behavior of dynamic systems grasped on a purely formal level bracketing their causal determination on the deeper level. A widespread tool in these disciplines is phase space – a space defined by the variables governing the development of the system so that this development may be mapped as a trajectory through phase space, each point on the trajectory mapping one global state of the system. This space may be inhabited by different types of attractors (attracting trajectories), repellors (repelling them), attractor basins around attractors, and borders between such basins characterized by different types of topological saddles which may have a complicated topology.

Catastrophe theory has its basis in differential topology, that is, the branch of topology keeping various differential properties in a function invariant under transformation. It is, more specifically, the so-called Whitney topology whose invariants are points where the nth derivative of a function takes the value 0, graphically corresponding to minima, maxima, turning tangents, and, in higher dimensions, different complicated saddles. Catastrophe theory takes its point of departure in singularity theory whose object is the shift between types of such functions. It thus erects a distinction between an inner space – where the function varies – and an outer space of control variables charting the variation of that function including where it changes type – where, e.g. it goes from having one minimum to having two minima, via a singular case with turning tangent. The continuous variation of control parameters thus corresponds to a continuous variation within one subtype of the function, until it reaches a singular point where it discontinuously, ‘catastrophically’, changes subtype. The philosophy-of-science interpretation of this formalism now conceives the stable subtype of function as representing the stable state of a system, and the passage of the critical point as the sudden shift to a new stable state. The configuration of control parameters thus provides a sort of map of the shift between continuous development and discontinuous ‘jump’. Thom’s semiotic interpretation of this formalism entails that typical catastrophic trajectories of this kind may be interpreted as stable process types phenomenologically salient for perception and giving rise to basic verbal categories.

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One of the simpler catastrophes is the so-called cusp (a). It constitutes a meta-diagram, namely a diagram of the possible type-shifts of a simpler diagram (b), that of the equation ax4 + bx2 + cx = 0. The upper part of (a) shows the so-called fold, charting the manifold of solutions to the equation in the three dimensions a, b and c. By the projection of the fold on the a, b-plane, the pointed figure of the cusp (lower a) is obtained. The cusp now charts the type-shift of the function: Inside the cusp, the function has two minima, outside it only one minimum. Different paths through the cusp thus corresponds to different variations of the equation by the variation of the external variables a and b. One such typical path is the path indicated by the left-right arrow on all four diagrams which crosses the cusp from inside out, giving rise to a diagram of the further level (c) – depending on the interpretation of the minima as simultaneous states. Here, thus, we find diagram transformations on three different, nested levels.

The concept of transformation plays several roles in this formalism. The most spectacular one refers, of course, to the change in external control variables, determining a trajectory through phase space where the function controlled changes type. This transformation thus searches the possibility for a change of the subtypes of the function in question, that is, it plays the role of eidetic variation mapping how the function is ‘unfolded’ (the basic theorem of catastrophe theory refers to such unfolding of simple functions). Another transformation finds stable classes of such local trajectory pieces including such shifts – making possible the recognition of such types of shifts in different empirical phenomena. On the most empirical level, finally, one running of such a trajectory piece provides, in itself, a transformation of one state into another, whereby the two states are rationally interconnected. Generally, it is possible to make a given transformation the object of a higher order transformation which by abstraction may investigate aspects of the lower one’s type and conditions. Thus, the central unfolding of a function germ in Catastrophe Theory constitutes a transformation having the character of an eidetic variation making clear which possibilities lie in the function germ in question. As an abstract formalism, the higher of these transformations may determine the lower one as invariant in a series of empirical cases.

Complexity theory is a broader and more inclusive term covering the general study of the macro-behavior of composite systems, also using phase space representation. The theoretical biologist Stuart Kauffman (intro) argues that in a phase space of all possible genotypes, biological evolution must unfold in a rather small and specifically qualified sub-space characterized by many, closely located and stable states (corresponding to the possibility of a species to ‘jump’ to another and better genotype in the face of environmental change) – as opposed to phase space areas with few, very stable states (which will only be optimal in certain, very stable environments and thus fragile when exposed to change), and also opposed, on the other hand, to sub-spaces with a high plurality of only metastable states (here, the species will tend to merge into neighboring species and hence never stabilize). On the base of this argument, only a small subset of the set of virtual genotypes possesses ‘evolvability’ as this special combination between plasticity and stability. The overall argument thus goes that order in biology is not a pure product of evolution; the possibility of order must be present in certain types of organized matter before selection begins – conversely, selection requires already organized material on which to work. The identification of a species with a co-localized group of stable states in genome space thus provides a (local) invariance for the transformation taking a trajectory through space, and larger groups of neighboring stabilities – lineages – again provide invariants defined by various more or less general transformations. Species, in this view, are in a certain limited sense ‘natural kinds’ and thus naturally signifying entities. Kauffman’s speculations over genotypical phase space have a crucial bearing on a transformation concept central to biology, namely mutation. On this basis far from all virtual mutations are really possible – even apart from their degree of environmental relevance. A mutation into a stable but remotely placed species in phase space will be impossible (evolution cannot cross the distance in phase space), just like a mutation in an area with many, unstable proto-species will not allow for any stabilization of species at all and will thus fall prey to arbitrary small environment variations. Kauffman takes a spontaneous and non-formalized transformation concept (mutation) and attempts a formalization by investigating its condition of possibility as movement between stable genomes in genotype phase space. A series of constraints turn out to determine type formation on a higher level (the three different types of local geography in phase space). If the trajectory of mutations must obey the possibility of walking between stable species, then the space of possibility of trajectories is highly limited. Self-organized criticality as developed by Per Bak (How Nature Works the science of self-organized criticality) belongs to the same type of theories. Criticality is here defined as that state of a complicated system where sudden developments in all sizes spontaneously occur.

Of Phenomenology, Noumenology and Appearances. Note Quote.

Heidegger’s project in Being and Time does not itself escape completely the problematic of transcendental reflection. The idea of fundamental ontology and its foundation in Dasein, which is concerned “with being” and the analysis of Dasein, at first seemed simply to mark a new dimension within transcendental phenomenology. But under the title of a hermeneutics of facticity, Heidegger objected to Husserl’s eidetic phenomenology that a hermeneutic phenomenology must contain also the theory of facticity, which is not in itself an eidos, Husserl’s phenomenology which consistently holds to the central idea of proto-I cannot be accepted without reservation in interpretation theory in particular that this eidos belong only to the eidetic sphere of universal essences. Phenomenology should be based ontologically on the facticity of the Dasein, and this existence cannot be derived from anything else.

Nevertheless, Heidegger’s complete reversal of reflection and its redirection of it toward “Being”, i.e, the turn or kehre, still is not so much an alteration of his point of view as the indirect result of his critique of Husserl’s concept of transcendental reflection, which had not yet become fully effective in Being and Time. Gadamer, however, would incorporate Husserl’s ideal of an eidetic ontology somewhat “alongside” transcendental constitutional research. Here, the philosophical justification lies ultimately in the completion of the transcendental reduction, which can come only at a higher level of direct access of the individual to the object. Thus there is a question of how our awareness of essences remains subordinated to transcendental phenomenology, but this does not rule out the possibility of turning transcendental phenomenology into an essence-oriented mundane science.

Heidegger does not follow Husserl from eidetic to transcendental phenomenology, but stays with the interpretation of phenomena in relation to their essences. As ‘hermeneutic’, his phenomenology still proceeds from a given Dasein in order to determine the meaning of existence, but now it takes the form of a fundamental ontology. By his careful discussion of the etymology of the words “phenomenon” and “Logos” he shows that “phenomenology” must be taken as letting that which shows itself be seen from itself, and in the very way in it which shows itself from itself. The more genuinely a methodological concept is worked out and the more comprehensively it determines the principles on which a science is to be conducted, the more deeply and primordially it is rooted in terms of the things themselves; whereas if understanding is restricted to the things themselves only so far as they correspond to those judgments considered “first in themselves”, then the things themselves cannot be addressed beyond particular judgements regarding events.

The doctrine of the thing-in-itself entails the possibility of a continuous transition from one aspect of a thing to another, which alone makes possible a unified matrix of experience. Husserl’s idea of the thing-in-itself, as Gadamer introduces it, must be understood in terms of the hermeneutic progress of our knowledge. In other words, in the hermeneutical context the maxim to the thing itself signifies to the text itself. Phenomenology here means grasping the text in such a way that every interpretation about the text must be considered first as directly exhibiting the text and then as demonstrating it with regard to other texts.

Heidegger called this “descriptive phenomenology” which is fundamentally tautological. He explains that phenomenon in Greek first signifies that which looks like something, or secondly that which is semblant or a semblance (das scheinbare, der Schein). He sees both these expressions as structurally interconnected, and having nothing to do with what is called an “appearance” or mere “appearance”. Based on the ordinary conception of phenomenon, the definition of “appearance” as referring to can be regarded also as characterizing the phenomenological concern for the text in itself and for itself. Only through referring to the text in itself can we have a real phenomenology based on appearance. This theory, however, requires a broad meaning of appearance including what does the referring as well as the noumenon.

Heidegger explains that what does the referring must show itself in itself. Further, the appearance “of something” does not mean showing-itself, but that the thing itself announces itself through something which does show itself. Thus, Heidegger urges that what appears does not show itself and anything which fails to show itself can never seem. On the other hand, while appearing is never a showing-itself in the sense of phenomenon, it is preconditioned by something showing-itself (through which the thing announces itself). This showing itself is not appearing itself, but makes the appearing possible. Appearing then is an announcing-itself (das sich-melden) through something that shows itself.

Also, a phenomenon cannot be represented by the word “appearance” if it alludes to that wherein something appears without itself being an appearance. That wherein something appears means that wherein something announces itself without showing itself, in other words without being itself an “appearance” (appearance signifying the showing itself which belongs essentially to that “wherein” something announces itself). Based upon this argument, phenomena are never appearances. This, however, does not deny the fact that every appearance is dependent on phenomena.