Excessive Subjective Transversalities. Thought of the Day 33.0

In other words, object and subject, in their mutual difference and reciprocal trajectories, emerge and re-emerge together, from transformation. The everything that has already happened is emergence, registered after its fact in a subject-object relation. Where there was classically and in modernity an external opposition between object and subject, there is now a double distinction internal to the transformation. 1) After-the-fact: subject-object is to emergence as stoppage is to process. 2) In-fact: “objective” and “subjective” are inseparable, as matter of transformation to manner of transformation… (Brian Massumi Deleuze Guattari and Philosophy of Expression)


Massumi makes the case, after Simondon and Deleuze and Guattari, for a dynamic process of subjectivity in which subject and object are other but their relation is transformative to their terms. That relation is emergence. In Felix Guattari’s last book, Chaosmosis, he outlines the production of subjectivity as transversal. He states that subjectivity is

the ensemble of conditions which render possible the emergence of individual and/or collective instances as self-referential existential Territories, adjacent, or in a delimiting relation, to an alterity that is itself subjective.

This is the subject in excess (Simondon; Deleuze), overpowering the transcendental. The subject as constituted by all the forces that simultaneously impinge upon it; are in relation to it. Similarly, Simondon characterises this subjectivity as the transindividual, which refers to

a relation to others, which is not determined by a constituted subject position, but by pre-individuated potentials only experienced as affect (Adrian Mackenzie-Transductions_ bodies and machines at speed).

Equating this proposition to technologically enabled relations exerts a strong attraction on the experience of felt presence and interaction in distributed networks. Simondon’s principle of individuation, an ontogenetic process similar to Deleuze’s morphogenetic process, is committed to the guiding principle

of the conservation of being through becoming. This conservation is effected by means of the exchanges made between structure and process… (Simondon).

Or think of this as structure and organisation, which is autopoietic process; the virtual organisation of the affective interval. These leanings best situate ideas circulating through collectives and their multiple individuations. These approaches reflect one of Bergson’s lasting contributions to philosophical practice: his anti-dialectical methodology that debunks duality and the synthesised composite for a differentiated multiplicity that is also a unified (yet heterogeneous) continuity of duration. Multiplicities replace the transcendental concept of essences.

Catastrophe Revisited. Note Quote.

Transversality and structural stability are the topics of Thom’s important transversality and isotopy theorems; the first one says that transversality is a stable property, the second one that transverse crossings are themselves stable. These theorems can be extended to families of functions: If f: Rn x Rr–>R is equivalent to any family f + p: Rn x Rr–>R, where p is a sufficiently small family Rn x Rr–> R, then f is structurally stable. There may be individual functions with degenerate critical points in such a family, but these exceptions from the rule are in a sense “checked” by the other family members. Such families can be obtained e.g. by parametrizing the original function with one or several extra variables. Thom’s classification theorem, comes in at this level.

So, in a given state function, catastrophe theory separates between two kinds of functions: one “Morse” piece, containing the nondegenerate critical points, and one piece, where the (parametrized) family contains at least one degenerate critical point. The second piece has two sets of variables; the state variables (denoted x, y…) responsible for the critical points, and the control variables or parameters (denoted a, b, c…), capable of stabilizing a degenerate critical point or steering away from it to nondegenerate members of the same function family. Each control parameter can control the degenerate point only in one direction; the more degenerate a singular point is (the number of independent directions equal to the corank), the more control parameters will be needed. The number of control parameters needed to stabilize a degenerate point (“the universal unfolding of the singularity” with the same dimension as the number of control parameters) is called the codimension of the system. With these considerations in mind, keeping close to surveyable, four-dimensional spacetime, Thom defined an “elementary catastrophe theory” with seven elementary catastrophes, where the number of state variables is one or two: x, y, and the number of control parameters, equal to the codimension, at most four: a, b, c, d. (With five parameters there will be eleven catastrophes). The tool used here is the above mentioned classification theorem, which lists all possible organizing centers (quadratic, cubic forms etc.) in which there are stable unfoldings (by means of control parameters acting on state variables). 

Two elementary catastrophes: fold and cusp

1. In the first place the classification theorem points out the simple potential function y = x3 as a candidate for study. It has a degenerate critical point at {0, 0} and is always declining (with minus sign), needing an addition from the outside in order to grow locally. All possible perturbations of this function are essentially of type x3 + x or type x3 – x (more generally x3 + ax); which means that the critical point (y, x = 0) is of codimension one. Fig. below shows the universal unfolding of the organizing centre y = x3, the fold:


This catastrophe, says Thom, can be interpreted as “the start of something” or “the end of something”, in other words as a “limit”, temporal or spatial. In this particular case (and only in this case) the complete graph in internal (x) and external space (y) with the control parameter a running from positive to negative values can be shown in a three-dimensional graph (Fig. below); it is evident why this catastrophe is called “fold”:


One point should be stressed already at this stage, it will be repeated again later on. In “Topological models…”, Thom remarks on the “informational content” of the degenerate critical point: 

This notion of universal unfolding plays a central role in our biological models. To some extent, it replaces the vague and misused term of ‘information’, so frequently found in the writings of geneticists and molecular biologists. The ‘information’ symbolized by the degenerate singularity V(x) is ‘transcribed’, ‘decoded’ or ‘unfolded’ into the morphology appearing in the space of external variables which span the universal unfolding family of the singularity V(x). 

2. Next let us as organizing centre pick the second potential function pointed out by the classification theorem: y = x4. It has a unique minimum (0, 0), but it is not generic , since nearby potentials can be of a different qualitative type, e.g. they can have two minima. But the two-parameter function x4 + ax2 + bx is generic and contains all possible unfoldings of y = x4. The graph of this function, with four variables: y, x, a, b, can not be shown, the display must be restricted to three dimensions. The obvious way out is to study the derivative f'(x) = 4x3 + 2ax + b for y = 0 and in the proximity of x = 0. It turns out, that this derivative has the qualities of the fold, shown in the Fig. below; the catastrophes are like Chinese boxes, one contained within the next of the hierarchy. 


Finally we look for the position of the degenerate critical points projected on (a,b)-space, this projection has given the catastrophe its name: the “cusp” (Fig. below). (An arrowhead or a spearhead is a cusp). The edges of the cusp, the bifurcation set, point out the catastrophe zone, above the area between these limits the potential has two Morse minima and one maximum, outside the cusp limits there is one single Morse minimum. With the given configuration (the parameter a perpendicular to the axis of the cusp) a is called the normal factor – since x will increase continuously with a if b < 0, while b is called the splitting factor because the fold surface is split into two sheets if b > 0. If the control axes are instead located on either side of the cusp (A = b + a and B = b – a) A and B are called conflicting factors; A tries to push the result to the upper sheet (attractor), B to the lower sheet of the fold. (Here is an “inlet” for truly external factors; it is well-known how e.g. shadow or excessive light affects the morphogenetic process of plants. 


Thom states: the cusp is a pleat, a fault, its temporal interpretation is “to separate, to unite, to capture, to generate, to change”. Countless attempts to model bimodal distributions are connected with the cusp, it is the most used (and maybe the most misused) of the elementary catastrophes. 

Zeeman has treated stock exchange and currency behaviour from one and the same model, namely what he terms the cusp catastrophe with a slow feedback. Here the rate of change of indexes (or currencies) is considered as dependent variable, while different buying patterns (“fundamental” /N in fig. below and “chartist” /S in fig. below) serve as normal and splitting parameters. Zeeman argues: the response time of X to changes in N and S is much faster than the feedback of X on N and S, so the flow lines will be almost vertical everywhere. If we fix N and S, X will seek a stable equilibrium position, an attractor surface (or: two attractor surfaces, separated by a repellor sheet and “connected” by catastrophes; one sheet is inflation/bull market, one sheet deflation/bear market, one catastrophe collapse of market or currency. Note that the second catastrophe is absent with the given flow direction. This is important, it tells us that the whole pattern can be manipulated, “adapted” by means of feedbacks/flow directions). Close to the attractor surface, N and S become increasingly important; there will be two horizontal components, representing the (slow) feedback effects of N and S on X. The whole sheet (the fold) is given by the equation X3 – (S – So)X – N = 0, the edge of the cusp by 3X2 + So = S, which gives the equation 4(S – So)3 = 27 N2 for the bifurcation curve. 


Figure: “Cusp with a slow feedback”, according to Zeeman (1977). X, the state variable, measures the rate of change of an index, N = normal parameter, S = splitting parameter, the catastrophic behaviour begins at So. On the back part of the upper sheet, N is assumed constant and dS/dt positive, on the fore part dN/dT is assumed to be negative and dS/dt positive; this gives the flow direction of the feedback. On the fore part of the lower sheet both dN/dt and dS/dt are assumed to be negative, on the back part dN/dt is assumed to be positive and dS/dt still negative, this gives the flow direction of feedback on this sheet. The cusp projection on the {N,S}-plane is shaded grey, the visible part of the repellor sheet black. (The reductionist character of these models must always be kept in mind; here two obvious key parameters are considered, while others of a weaker or more ephemeral kind – e.g. interest levels – are ignored.)