Catastrophe at Warwick

A catastrophe is only violent in its uncalled for appearing, and its coming is all around us in the smallest things.

This year’s Critical Legal Conference takes place at Warwick under the title Catastrophe.  This is not without reason, for it sees the notion of catastrophe returning to one of its theoretical homes.  It was Warwick thinkers that coined the term ‘catastrophe theory’ for an assemblage of physico-mathematical techniques that explore ruptures, or jumps in behaviour as diverse as stretched elastic bands, ships at sea, human creativity, and stock market crashes.  Led by Christopher Zeeman, from the mid-1960s a team which also included Ian Stewart and Tim Poston married practical applicability of the theory to its popularisation, and may be considered as paving the way to full-blown chaos theory by the 1980s.  This trajectory led catastrophe theory to rejoin its metaphysical origins in its partial deployment in Gilles Deleuze’s The Fold: Leibniz and the Baroque ([1988], 1994), completing an arc that had its origin with French mathematician René Thom.

A dark history

Apparently, catastrophe theory did not begin catastrophically; René Thom notes (1980) that it is difficult to discern just when the pre-existing corpus of mathematical work on bifurcations in physical phase dynamics developed into the ‘new’ work of Thom on the one hand, and Warwick’s Christopher Zeeman on the other.  Yet in his explanation of the theory’s aims, Thom is careful to indicate that its objects – complicated systems – are marked by their secret histories; histories unobservable either by some barrier (the ‘black box’ system) or due to sheer uncountability (which demand statistical methods of approximation).  The purpose of catastrophe theory is to project a light into the depths in search of these moments of change.

Contextual determinations certainly seem evident in the Warwick approach to catastrophe.  While Zeeman tended to underline his linkages to Thom’s Structural Stability and Morphogenesis, which had been circulating since the mid-60s but was only formally published in 1972, Thom preferred to underline the bifurcation of their approaches.  Whereas Zeeman emphasised the concrete applications of the theory in physical systems, and thereby enlarged its scope to all kinds of material phenomena, Thom’s predilection was for metaphysics and the speculative aspects of the theory.  This is most evident in the way in which the Warwick academics cited Thom as the source for the nomenclature ‘catastrophe’, whereas Thom rejected this.  Thom averred that he had only used the phrase ‘points of catastrophe’ within Structural Stability and Morphogenesis as a part of a wider investigation of the development of biological systems.  It had been Zeeman, he claimed, who had seized on the word and made of it a new point of departure for further research.

No doubt this passing back and forth of the dubious honour of naming was linked to the violence which the word expresses.  Many of the initial physical phenomena on which the theory proved tractable were quite mundane – problems of tipping points in statics – and the name ‘catastrophe’ seemed to be overselling an undue radicalism to a post-68 student body.  It also led to a degree of hostility from some academics, such as VI Arnol’d, no doubt fired by a perceived whiff of pseudo-science.  Nevertheless, even Arnol’d had to admit that Zeeman had an incontrovertible basis for the theory: he was describing physical phenomena; the man could actually build experimental machines which made the theory flesh.

The Zeeman Catastrophe Machine

For both Zeeman and Thom the common object for catastrophe theory was the system, defined as a domain of all connected states.  Connected is used here in the topological sense that a mapping of the domain is constant and preserves the structure of the domain.  Thom gives the negative example of a table setting: the knife and a cup are part of the table setting but they are not connected.  I might have a domain {table setting} and I could map from this to the range {knife, cup}, but to do so makes clear a fundamental ‘break’ in {table setting} – some of the points of {table setting} are mapped to {knife} then we make a jump and all the remaining points are mapped to {cup}.  A positive intuitive example is the cup itself.  Notice that the cup has a loop (its handle).  Topologically this is the principal defining characteristic.  We can deform {cup} to any other shape, such as an American doughnut, but if we retain that loop we have preserved the cup’s structure.  The ability to perform this mapping without jumps tells us all the points of {cup} are connected.

Let’s consider the case of a physical system.  An object might be described by various parameters, such as space-time but also heat, pressure, energy and other qualitative but measurable grades.  This provides an abstraction from the extensive position of our object; rather we consider it located in a ‘phase space’ in which a given position designates not location in space but its state(s) according to measures of our choosing.  Thus, if an axis is potential energy, then moving positively along the axis is an increase in energy, though not necessarily movement in space.

So we can imagine our object as the set of all its possible connected states in this abstract ‘space’ = its system, and in particular we can study our object’s smooth changes of state and so mark a line in this phase space of its ‘life’ according to the application of various inputs e.g. we increase the pressure as input and the output shows an increase of heat.  A continuous line is drawn linking input = heat to output = temperature.  What had drawn the catastrophe theorist’s attention was that occasionally these continuous lines tended to ‘jump’, for example when pressure is so increased in a liquid system that it ceases to be a liquid at all, but ‘boils’ and becomes gas.

Both Thom and Zeeman were interested in such cases, and it was Thom’s ingenuity to propose a specific examination and categorisation of the geometry of phase spaces in which these jumps occurred.  Imagine a two dimensional phase space, but rather than a steady increasing line linking the variables of the two axes (a ‘normal’ system), the line is ‘S’-shaped though both extremities of the S continue respectively left and rightwards indefinitely.  Now imagine that we increase the x value along the bottom of the S.  The value curves upwards gently, with the y value increasing accordingly, but then we reach a critical point: the lower bow of the S.  Here our x value wants to continue rightwards but the graph ‘S’ wants us to curve back leftwards first; impossible because to do this would require the x value to be able to assume three values simultaneously for each tier of the S.

There is only one way out: catastrophe.  Thom theorised that in such a case the system state leaps (saut) from the lower curve of the S to that part of the upper extremity directly above it, and then was able to continue rightwards indefinitely.  This jump was experienced phenomenologically by a rupture in the system, be it a tipping over, a collapse, a snapping, a boiling, a flash or a crash.  With his metaphysical leanings Thom no doubt had Leibniz , who claimed nature never proceeded by jumps1, in mind when he proposed precisely this doctrine.  Notice, however, that in one sense the states of the system are still connected.  They are connected as part of the ‘S’ shape.  When Thom speaks of a jump it is a leap locally, but considered globally the two states of the jump are connected overall.

Over three weeks in the Warwick of 1969 Christopher Zeeman developed a machine which would illustrate this jumping behaviour.  The Zeeman Catastrophe Machine (pictured) can indeed be built by anyone with access to good catapult rubber (the stuff which is thick and has a square section when cut across).  The machine consists of an oblong board to which is attached a disc that is permitted to rotate.  At one end the elastic is fixed, and then it is fixed again to a point on the circumference of the disc.  The elastic is extended beyond the disc in the opposite direction and attached to a stylus which is capable of drawing on paper affixed to the board.  The elastic is always held taut and the disc is gradually rotated.

What one discovers is that the stylus will draw continuous lines at many points on the paper, but near the centre there is a space which may be described as a concave diamond shape, in which the stylus will not tolerate to rest.  The stylus will approach the space quite gradually, but as soon as it passes the boundary the whole apparatus ‘snaps’ the stylus across the space to the opposite boundary, whence it may once more pursue a gradual and smooth journey.  It is as if the space of the concave rhomboid does not exist for the stylus, or, better, the opposing boundaries of the rhomboid are closer than they appear, much like the lower curve and upper extremity of our phase space S.

If such a simple machine could be the source of great mathematical interest, would other phenomena could be assessed for catastrophic tendencies?  Zeeman’s subsequent papers include an analysis of stock markets (1976), his argument being that the interaction between liquidity and capital reserve preferences proceeded smoothly up to a point, at which the market system snapped from boom to bust.  He illustrated the idealised phase space as a system-fold whose cross-section manifested the characteristic ‘S’ that permitted jumps between levels.  He also believed that psychological experiences of creativity, and crowd phenomena could be subjected to catastrophe theoretic analysis.  Yet his critics accused him of overreach.  He had established that so-called catastrophes occurred in mathematically describable ways in very simple physical phenomena, but little experimental evidence existed of the theory’s utility for social phenomena.

The Fold

An attractive feature of the Zeeman Catastrophe Machine is that, via the stylus, it projects the machine’s phase space onto the 2-dimensional paper.  It is by means of this complex projective geometry that Thom had collected and categorised various types of catastrophic point which he named: the fold, the cusp, the butterfly, the swallowtail, and the elliptic, parabolic and hyperbolic umbilic.  What, though, is the physical status of these projections?  Are they simple indicia of an underlying physical event, much like the movement of tree branches ‘shows’ us the action of wind?

The analogy however is misleading.  The bending branch is immediately apparent and appears smoothly; the bough that breaks comes as a surprise even if experience suggests the result is possible.  The catastrophes described are phenomenally immediate; the jumps are before us as integral parts of our perceptions.  If every individual is their perspective, as Leibniz held, then the catastrophe has the potential to be reintegrated into our life-world as part of a new phenomenalism.  The philosophical model of existential projection (e.g. Bergson’s image of the cone of memory whose apex is the present) is reinterpreted to regard the leaps that appear to occur on the plane of the collective present as projections of folds that are contracted up in memory.  We are invited to project from the leap to the kind of catastrophe which instantiates it, and to now regard the ‘plane’ of existence as mobile, undulating, subject to singular points.  Thom describes a state-point as a ball moving around a phase space, but then asks us to take into account that the phase space moves itself like the sea.  From the ball’s perspective it rises to the cusp of a wave, rides it, then falls in a moment, all as if it was traversing a flat plane of certain, continuous and smooth development.

It was this notion in part which emerges openly in Deleuze’s work of the 1980’s on Leibniz and the notion of the fold.  It seems likely that Deleuze was drawn to Thom’s work because of its theoretical applications in biological morphogenesis (a strand of thought which was explored with particular skill at Warwick by Keith Ansell-Pearson).  Be this as it may, I would agree with Simon Duffy that aspects of projective geometry are foremost in The Fold.  To be brief, Deleuze develops the notion that the mathematical point is an idealisation of something integral in the metaphysical point that we find in the plane in which we exist. This thing is the real relation (to use Scholastic terminology), which is to say that the metaphysical point is not ‘one’, but rather at least two contracted together with maximal intensity.  What the point represents is the relation of their difference as such, and Deleuze believes that this insight informs the theory of infinitesimals in Leibniz’s calculus.  Here the infinitesimal point is not a unity, but the least possible change between two related variables.  Indeed, it is the particularity of this relation, not the variables, that the infinitesimal expresses.

Deleuze then posits the present as just such a ‘least possible’ – a contraction of the virtual differences of kind into the actuality of their real relationship as a point: the subject.  And indeed the whole world is just such a contraction of interrelationships, the whole projected plane expressing compossibility or the integral relationships capable of existing simultaneously as the present.

The engagement with catastrophe theory provided a further avenue for elaboration of this projective theory.  What coexists is now capable of expressing a ‘folded up’ virtual whence come leaps and bounds; the appearance of the event.  By shining a light through the present we can project its contours into an opposing conic, one whose apex is the present and which recedes from us.  In this imagined conic we can ‘see’ projected the folds of the virtual.  By projecting through the present into the future we discover those strange incorporeal animals that, like obscured puppets, act out the pre-individual singularities nested in the virtual’s incessant swell.  Indeed, the shadows of the cave are not without purpose in consciousness’ journey to freedom.

A catastrophe then is only violent in its uncalled for appearing, and its coming is all around us in the smallest things.  Always sensitive to the depth of language, perhaps René Thom had this in mind when he wrote of points catastrophiques.  For originally katastrophē referred to the rupture between the drama of the play and the preparation of the theatre audience for their return to the street outside, to perceive once again their own context with keener eyes.

Stephen Connelly is the author of Spinoza, Right and Absolute Freedom, and is currently writing a book about Leibniz and law.  He is Assistant Professor at the University of Warwick and is co-organising CLC2017.

Further reading

—Keith Ansell-Pearson, Germinal Life: The Difference and Repetition of Deleuze (NY: Continuum, 1999)
—Gilles Deleuze, The Fold: Leibniz and the Baroque (London: Continuum, 1994) [I have to say though that the translation is not great, and the rendering of the figures betrays a miscomprehension of the mathematics.] —Tim Poston & Ian Stewart, Catastrophe Theory (NY: Dover, 2012)
—René Thom, Structural Stability and Morphogenesis, trans. W. A. Benjamin, (Westview Press, 1994 [1972])
—René Thom, Modèles mathématiques de la morphogenèse (Paris: C. Bourgeois, 1980)
—Christopher Zeeman, ‘Catastrophe Theory’ (April 1976) Scientific American 65-70, 75-83

Show 1 footnote

  1. This is however a common partial quotation.  Leibniz actually said that nature never proceeds by jumps, but that nevertheless it does ‘incline’.
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