Piaget, DeLanda, and Deleuze

In Deleuze’s philosophy, the concept of structure is not important. In one of the places where he discusses the concept, Deleuze criticizes a structural conception of the calculus proposed by French mathematicians for being a ‘static’ structuralism, and calls for a dynamic concept of structure that would be capable of explaining evolution – but there is no mention of Piaget’s genetic structuralism or of the man himself. As this hint at the static nature of structuralism indicates, it is a central concern for Deleuze, too, to do away with all ideas about structures or, in Deleuze’s case, with ideas about ‘timeless forms’ or ‘essences’ that emanate from some Platonic heaven to give shape to the world of real things. Deleuze finds that such ‘essentialism’ pervades our normal perception and ways of thinking, and much empirical science, too, because – as a result of a certain necessity, and for philosophically good reasons – things are thought of as belonging to categories and sub-categories which are defined in terms of invariant properties or, again, essences. Space and time too, are ascribed essential properties when space is thought of as a measurable three-dimensional continuum in which all things are contained, and in which changes take place as time trickles along at an even pace from past to future.

Deleuze designates this state of affairs as ‘the actual’, and he wants to show that although the actual is indeed a part of reality, it is also a kind of illusion that conceals or ‘covers up’ another realm of reality, which he calls ‘the virtual’. In order to grasp the virtual, we should think of everything as ‘individuals’, each of which has its own history. The word ‘individual’ is used here in an extended sense. In biology, for instance, species and genera as well as organisms are defined as individuals. On another spatio-temporal scale, ecological environments in which species are embedded and evolve are individuals, and at the opposite end of the scale each molecule and atom is an individual. In every case individuals are meshed together in fuzzy aggregates without distinct borders, and are in an incessant state of ‘becoming’ in a sense that escapes our perception of actual processes and states of affairs.

Apart from the common aim of doing away with Platonic structures and essences, at some crucial points Deleuze’s conception of the virtual and the actual as aspects of the real invites comparison with Piaget’s description of the psychogenetic process. This comparison is facilitated by an interpretation or reconstruction of Deleuze’s ontology, which Mario DeLanda presents in Intensive Science and Virtual Philosophy. The book addresses physicists and biologists and is not concerned with psychogenesis, but the basic Deleuzian concepts are explained in terms of Klein’s architecture of geometries and in this way link up with Piaget. DeLanda points out that Deleuze himself never referred to the theory of groups, although the distinction between metric and non-metric geometries is essential to his philosophy. This is all the more noticeable since Deleuze draws on almost all other branches of mathematics – number theory, the theory of sets; catastrophe theory, the theory of fractals and other branches of topology and in particular calculus and differential geometry. DeLanda compensates for this by including differential geometry in Klein’s hierarchy, putting it on a level between continuous transformation and perspective transformation (I don’t know whether this is legitimate; according to Morris Kline, to whom DeLanda refers several times, differential geometry does not feature in Klein’s scheme, although Klein did speak of the group of transformations that leave the expression for ds2 invariant). On the other hand, DeLanda does not mention the scattering group, the most abstract level at the top of the hierarchy, which Klein did not know of – it was only recognized in more recent times – and I believe this actually strengthens DeLanda’s argument.

In the place where DeLanda introduces Klein’s architecture of geometries, he says that metaphorically the hierarchy ‘topological – differential – projective – affine – Euclidean’ may be seen as an abstract scenario for the birth of real space. As if the metric space which we inhabit and which physicists study and measure was born from a non-metric topological continuum as the latter differentiated and acquired structure. The hierarchy of geometries is a metaphor, DeLanda says, in the sense that to a mathematician the relations it describes are purely logical; the problem is to give the cascade of transformations an ontological dimension by relating the geometrical transformations to real processes – a task which Piaget, although he was not concerned with ontology, took upon himself within his specific field of research.

To explain how Deleuze constructs the ontological dimension, DeLanda introduces the concepts of ‘intensity’ and ‘extensity’. Extensive properties are defined as those that are intrinsically divisible, including metric magnitudes: for instance, a volume of 2 m3 is divisible onto two volumes of 1 m3. Obviously, this does not apply to the space of topology: in this space, ‘length’ is not invariant, hence a line segment is not divisible into segments of ‘equal length’; and moving on to scattering, the very notion of ‘length’ evaporates. In these terms, then, the space described by scattering is the most intensive one, while the metric space at the bottom level is the most extensive. DeLanda says that by the same token, one should think of an intensive, topologically non-metric time out of which extensive metric time is born; indeed one should think of all quantities and qualities of our actual perception as emerging through downward cascades from intensive states. The upper, least differentiated level, DeLanda goes on, corresponds to what Deleuze designates as the virtual, while the bottom level corresponds to the actual.

In these terms, the Deleuzian virtual is all but indistinguishable from the primordial continuum of Piaget’s psychogenesis, out of which space and time, the ego, mechanical and religious ideas, causality, etc. are differentiated as they evolve along intertwined lines that lead to the world of our normal experience – which then covers over the original condition (although, as Piaget points out, the cover is sometimes lifted when advanced ideas pick up threads from the deep unconscious past). But in that case there would be a problem with defining the equivalent of the Deleuzian actual. The actual space has the structure of metric geometry, and as an example of spatial actualization DeLanda mentions embryogenesis, where the organism’s metric properties emerge through transformations and differentiations of the topologically fluid fertilized egg. The psychogenetic equivalent of this is illustrated by the example of pictorial representations presented in plate 2 in the picture gallery: the child’s earliest drawings show precisely how metric space emerges in a child’s space of imagination from a non-metric topological continuum as the latter is differentiated and acquires structure. In this process complexity increases in terms of the operations that the child has to perform in order to abstract straight line, square, triangle, circle, out of the squiggle. But then a new story begins, which is about ‘counter-actualization’, as it were; that is, ‘upward’ movements within Klein’s hierarchy – as affine transformations at first, after which projective transformations are actualized. In this process complexity increases in the sense that the metric structure arrived at in the first place is comprised by the affine structure, and the affine by the projective.
I would like to elaborate on this by jumping to one of the places where DeLanda discusses evolution. He says that the idea that evolutionary processes possess an inherent drive toward increased complexity reintroduces teleology – another kind of essentialism – into Darwinism. In this connection he mentions a mechanism in biological evolution called neoteny, which shows that novelty need not be the effect of terminal addition of new features; on the contrary it can be the result of a loss of certain old features. Humans, for example, may be regarded as juvenilized chimpanzees – that is, primates from which a developmental stage (adulthood) has been eliminated. More generally, the loss of features may provide an escape route from morphologies that have become too rigid and specialized, allowing organisms to explore new developmental pathways. In these terms, psychogenesis and cultural evolution could be described as a process where the juvenilized chimpanzee first consolidates the metric space which the chimpanzee, too, inhabits, at a new level, that of imagination; and then escapes from this too rigid and specialized space along a route that leads through a sequence of spaces of increased flexibility.

So far, DeLanda’s use of Klein’s geometrical scheme has conveyed the image of a virtual fluid state, leaking cascades of transformations that end up in the rigid actual state, but obviously this is misleading: ontologically, DeLanda says, the relation between the virtual and the actual cannot be thought of as one-way traffic, since in that case the virtual would hardly be distinguishable from a Platonic structure. To construct the concept of the virtual, it is necessary to show how the actual could work back on the virtual through ‘quasi-causal operators’, which are in the nature of ‘affects’ rather than effects. This amounts to ‘counter-actualization’ in an ontological sense that presumably differs from the one hinted at above. DeLanda also points out that communication between the virtual and the actual cannot be thought of as taking place in directional, metric time, nor as a sequence of cause and effect, since metric time and causality are properties of the actual that exist only potentially or as intensities in the realm of the virtual. The concept of the virtual implies a time that is not only ‘topologically’ non-metric, but extends towards the past as well as towards the future from a present of infinite duration, in a manner that allows for the present to work back on the past, and the future on the present, and thus also with reversed causality. Within this conceptual frame, the truly new – new ideas, new biological organisms, new societal forms – cannot result from chains of transformation within the actual, or be explained in terms of causes and effects; newness must be thought of as the actualization of reconfigurations within the virtual, even when these reconfigurations are affected by changes within the realm of the actual.
It is from the standpoint of this ontology, the complexity of which has only been intimated here, that Deleuze refutes evolutionism. From our standpoint the interesting question is whether this refutation is confirmed when it comes to real or actual historical processes and states of affairs. Returning to DeLanda’s example, in terms of genetic structuralism neoteny is a fine example of the way structure grows out of structure in a process that at bottom yields increased complexity by generating a new developmental level. The problem that makes discussions of evolution difficult is that Deleuze rejects the notion of epistemological and developmental ‘levels’, which is essential to Piaget. Instead, Deleuze introduces the concept of ‘strata’, which are intermingled or folded into one another and shot through by escape routes or ‘lines of flight’. At one point Deleuze says that among strata there is no fixed order, and one stratum can serve directly as a substratum for another without the intermediaries that one would expect from the standpoint of stages or degrees. Or the apparent order can be reversed, with cultural or technical phenomena providing a fertile soil, a good ‘soup’ for the development of insects, bacteria, germs or even particles. The industrial age defined as the age of insects...
As mentioned earlier, Deleuze’s arguments draw heavily on calculus, which DeLanda includes in Klein’s hierarchy. The point is to show that individuals in the Deleuzian extended sense of the word can be structured or determined in different ways, and to different degrees; that a diversity of such partially determined things can be determined as a whole in different ways; and that such wholes can in turn be un-done and assembled in yet other ways. Even when this is true, there could still be a real, as distinct from a merely actual, overall tendency towards increased complexity, and close reading of scattered and often obscure pages in Deleuze’s works where development is discussed or hinted at shows that in crucial cases Deleuze does think of stepwise, cumulative processes where each new step transforms and enriches the elements assembled in the preceding steps. As regards the evolution of cultural space, I believe that what Deleuze in fact shows is that the course of history traced in The Architecture of Space – The Space of Architecture could equally well be traced in terms of the Deleuzian concepts of space (see section 5. Deleuze and Space: The smooth and the striated).


The book ‘The Architecture of Space - The space of Architecture’ is a historical survey of space and architecture covering the theories of Deleuze, Piaget and others.
Author of the book is the Danish architect and lecturer Lars Marcussen.

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