by Alan Sokal
(Deleuze and Guattari 1994, p. 122, italics in the original)
Here Deleuze and Guattari recycle, with a few additional inventions (infinite speeds, chaotic virtual), old ideas of Deleuze’s that originally appeared in the book Michel Foucault judged ‘among the greatest of the great’, Difference and Repetition. At two places in this book, Deleuze discusses classical problems in the conceptual foundations of differential and integral calculus. Since the birth of this branch of mathematics in the seventeenth century through the works of Newton and Leibniz, cogent objections were raised against the use of ‘infinitesimal’ quantities such as dx and dy.205 These problems were solved by the work of d’Alembert around 1760 and Cauchy around 1820, who introduced the rigourous notion of limit – a concept that has been taught in all calculus textbooks since the middle of the nineteenth century.206 Nevertheless, Deleuze launches into a long and confused meditation on these problems, from which we shall quote just a few characteristic excerpts:207
Must we say that vice-diction208 does not go as far as contradiction, on the grounds that it concerns only properties? In reality, the expression ‘infinitely small difference’ does indeed indicate that the difference vanishes so far as intuition is concerned. Once it finds its concept, however, it is rather intuition itself which disappears in favour of the differential relation, as is shown by saying that dx is minimal in relation to x, as dy is in relation to y, but that dy/dx is the internal qualitative relation, expressing the universal of a function independently of its particular numerical values.209 However, if this relation has no numerical determinations, it does have degrees of variation corresponding to diverse forms and equations. These degrees are themselves like the relations of the universal, and the differential relations, in this sense, are caught up in a process of reciprocal determination which translates the interdependence of the variable coefficients. But once again, reciprocal determination expresses only the first aspect of a veritable principle of reason; the second aspect is complete determination. For each degree or relation, regarded as the universal of a given function, determines the existence and distribution of distinctive points on the corresponding curve. We must take great care here not to confuse ‘complete’ with ‘completed’. The difference is that, for the equation of a curve, for example, the differential relation refers only to straight lines determined by the nature of the curve. It is already a complete determination of the object, yet it expresses only a part of the entire object, namely the part regarded as ‘derived’ (the other part, which is expressed by the so-called primitive function, can be found only by integration, which is not simply the inverse of differentiation.210 Similarly, it is integration which defines the nature of the previously determined distinctive points). That is why an object can be completely determined – ens omni modo determinatum – without, for all that, possessing the integrity which alone constitutes its actual existence. Under the double aspect of reciprocal determination and complete determination, however, it appears already as if the limit coincides with the power itself. The limit is defined by convergence. The numerical values of a function find their limit in the differential relation; the differential relations find their limit in the degrees of variation; and at each degree the distinctive points are the limits of series which are analytically continued one into the other. Not only is the differential relation the pure element of potentiality, but the limit is the power of the continuous [puissance du continu],211 as continuity is the power of these limits themselves.
(Deleuze 1994, pp. 46–7, italics in the original)
Just as we oppose difference in itself to negativity, so we oppose dx to not-A, the symbol of difference [Differenzphilosophie] to that of contradiction. It is true that contradiction seeks its Idea on the side of the greatest difference, whereas the differential risks falling into the abyss of the infinitely small. This, however, is not the way to formulate the problem: it is a mistake to tie the value of the symbol dx to the existence of infinitesimals; but it is also a mistake to refuse it any ontological or gnoseological value in the name of a refusal of the latter. ... The principle of a general differential philosophy must be the object of a rigourous exposition, and must in no way depend upon the infinitely small.212 The symbol dx appears as simultaneously undetermined, determinable and determination. Three principles which together form a sufficient reason correspond to these three aspects: a principle of determinability corresponds to the undetermined as such (dx, dy); a principle of reciprocal determination corresponds to the really determinable (dy/dx); a principle of complete determination corresponds to the effectively determined (values of dy/dx). In short, dx is the Idea – the Platonic, Leibnizian or Kantian Idea, the ‘problem’ and its being.
(Deleuze 1994, pp. 170–1, italics in the original)
[T]he differential relation presents a third element, that of pure potentiality. Power is the form of reciprocal determination according to which variable magnitudes are taken to be functions of one another. In consequence, calculus considers only those magnitudes where at least one is of a power superior to another.213 No doubt the first act of the calculus consists in a ‘depotentialisation’ of the equation (for example, instead of 2ax – x2 = y2 we have dy/dx = (a – x)/y). However, the analogue may be found in the two preceding figures where the disappearance of the quantum and the quantitas was the condition for the appearance of the element of quantitability, and disqualification the condition for the appearance of the element of qualitability. This time, following Lagrange’s presentation, the depotentialisation conditions pure potentiality by allowing an evolution of the function of a variable in a series constituted by the powers of i (undetermined quantity) and the coefficients of these powers (new functions of x), in such a way that the evolution function of that variable be comparable to that of the others. The pure element of potentiality appears in the first coefficient or the first derivative, the other derivatives and consequently all the terms of the series resulting from the repetition of the same operations. The whole problem, however, lies precisely in determining this first coefficient which is itself independent of i.214
(Deleuze 1994, pp. 174–5, italics in the original)
There is thus another part of the object which is determined by actualisation. Mathematicians ask: What is this other part represented by the so-called primitive function? In this sense, integration is by no means the inverse of differentiation215 but, rather, forms an original process of differenciation. Whereas differentiation determines the virtual content of the Idea as problem, differenciation expresses the actualisation of this virtual and the constitution of solutions (by local integrations). Differenciation is like the second part of difference, and in order to designate the integrity or the integrality of the object we require the complex notion of different/ciation.
(Deleuze 1994, p. 209, italics in the original)
These texts contain a handful of intelligible sentences – sometimes banal, sometimes erroneous – and we have commented on some of them in the footnotes. For the rest, we leave it to the reader to judge. The bottom line is: What is the point of all these mystifications about mathematical objects that have been well understood for over 150 years?
Let us look briefly at the other book ‘among the greatest of the great’, The Logic of Sense, where one finds the following striking passage:
In the first place, singularities-events correspond to heterogeneous series which are organized into a system which is neither stable nor unstable, but rather ‘metastable,’ endowed with a potential energy wherein the differences between series are distributed. (Potential energy is the energy of the pure event, whereas forms of actualization correspond to the realization of the event.) In the second place, singularities possess a process of auto-unification, always mobile and displaced to the extent that a paradoxical element traverses the series and makes them resonate, enveloping the corresponding singular points in a single aleatory point and all the emissions, all dice throws, in a single cast. In the third place,
singularities or potentials haunt the surface. Everything happens at the surface in a crystal which develops only on the edges. Undoubtedly, an organism is not developed in the same manner. An organism does not cease to contract in an interior space and to expand in an exterior space – to assimilate and to externalize. But membranes are no less important, for they carry potentials and regenerate polarities. They place internal and external spaces into contact, without regard to distance. The internal and the external, depth and height, have biological value only through this topological surface of contact. Thus, even biologically, it is necessary to understand that ‘the deepest is the skin.’ The skin has at its disposal a vital and properly superficial potential energy. And just as events do not occupy the surface but rather frequent it, superficial energy is not localized on the surface, but is rather bound to its formation and reformation.
(Deleuze 1990, pp. 103–4, italics in the original)
Once again, this paragraph – which prefigures the style of Deleuze’s later work written in collaboration with Guattari – is stuffed with technical terms;216 but, apart from the banal observation that a cell communicates with the outside world through its membrane, it is devoid of both logic and sense.
To conclude, let us quote a brief excerpt from the book Chaosmosis, written by Guattari alone. This passage contains the most brilliant mélange of scientific, pseudo-scientific and philosophical jargon that we have ever encountered; only a genius could have written it.
We can clearly see that there is no bi-univocal correspondence between linear signifying links or archi-writing, depending on the author, and this multireferential, multidimensional machinic catalysis. The symmetry of scale, the transversality, the pathic non-discursive character of their expansion: all these dimensions remove us from the logic of the excluded middle and reinforce us in our dismissal of the ontological binarism we criticised previously. A machinic assemblage, through its diverse components, extracts its consistency by crossing ontological thresholds, non-linear thresholds of irreversibility, ontological and phylogenetic thresholds, creative thresholds of heterogenesis and autopoiesis. The notion of scale needs to be expanded to consider fractal symmetries in ontological terms. What fractal machines traverse are substantial scales. They traverse them in engendering them. But, and this should be noted, the existential ordinates that they ‘invent’ were always already there. How can this paradox be sustained? It’s because everything becomes possible (including the recessive smoothing of time, evoked by René Thom) the moment one allows the assemblage to escape from energetico-spatio-temporal coordinates. And, here again, we need to rediscover a manner of being of Being – before, after, here and everywhere else – without being, however, identical to itself; a processual, polyphonic Being singularisable by infinitely complexifiable textures, according to the infinite speeds which animate its virtual compositions.
The ontological relativity advocated here is inseparable from an enunciative relativity. Knowledge of a Universe (in an astrophysical or axiological sense) is only possible through the mediation of autopoietic machines. A zone of self-belonging needs to exist somewhere for the coming into cognitive existence of any being or any modality of being. Outside of this machine/Universe coupling, beings only have the pure status of a virtual entity. And it is the same for their enunciative coordinates. The biosphere and mecanosphere, coupled on this planet, focus a point of view of space, time and energy. They trace an angle of the constitution of our galaxy. Outside of this particularised point of view, the rest of the Universe exists (in the sense that we understand existence here-below) only through the virtual existence of other autopoietic machines at the heart of other bio-mecanospheres scattered throughout the cosmos. The relativity of points of view of space, time and energy do not, for all that, absorb the real into the dream. The category of Time dissolves into cosmological reflections on the Big Bang even as the category of irreversibility is affirmed. Residual objectivity is what resists scanning by the infinite variation of points of view constitutable upon it. Imagine an autopoietic entity whose particles are constructed from galaxies. Or, conversely, a cognitivity constituted on the scale of quarks. A different panorama, another ontological consistency. The mecanosphere draws out and actualises configurations which exist amongst an infinity of others in fields of virtuality. Existential machines are at the same level as being in its intrinsic multiplicity. They are not mediated by transcendent signifiers and subsumed by a univocal ontological foundation. They are to themselves their own material of semiotic expression. Existence, as a process of deterritorialisation, is a specific inter-machinic operation which superimposes itself on the promotion of singularised existential intensities. And, I repeat, there is no generalised syntax for these deterritorialisations. Existence is not dialectical, not representable. It is hardly livable!
(Guattari 1995, pp. 50–2)
Should the reader entertain any further doubts about the ubiquity of pseudo-scientific language in Deleuze and Guattari’s work, he or she is invited to consult, in addition to the references given in the footnotes, pages 20–4, 32, 36–42, 50, 117–33, 135–42, 151–62, 197, 202–7 and 214–17 of What is Philosophy?,217 and pages 32–3, 142–3, 211–12, 251–2, 293–5, 361–5, 369–74, 389–90, 461, 469–73 and 482–90 of A Thousand Plateaus. These lists are by no means exhaustive. Besides, the article of Guattari (1988) on tensor calculus applied to psychology is a real gem.218
10
PAUL VIRILIO
Architect and urban planner, former director of the École Spéciale d’Architecture, Paul Virilio poses questions about speed and space starting from the experience of war. For him, the mastery of time refers to power. With an astonishing erudition, which mixes space-distances and time-distances, this researcher opens up an important field of philosophical questions that he calls ‘dromocracy’ (from the Greek dromos: speed).219
(Le Monde, 1984b, p. 195)
The writings of Paul Virilio revolve principally around the themes of technology, communication and speed. They contain a plethora of references to physics, particularly the theory of relativity. Though Virilio’s sentences are slightly more meaningful than those of Deleuze–Guattari, what is presented as ‘science’ is a mixture of monumental confusions and wild fantasies. Furthermore, his analogies between physics and social questions are the most arbitrary imaginable, when he does not simply become intoxicated with his own words. We confess our sympathy with many of Virilio’s political and social views; but the cause is not, alas, helped by his pseudo-physics.
Let us start with a minor example of the astonishing erudition vaunted by Le Monde:
Recent MEGALOPOLITAN hyperconcentration (Mexico City, Tokyo...) being itself the result of the increased speed of economic exchanges, it seems necessary to reconsider the importance of the notions of ACCELERATION and DECELERATION (what physicists call positive and negative velocities [vitesses positive et négative selon les physiciens]) ...
(Virilio 1995, p. 24, capitals in the original220)
Here Virilio mixes up velocity (vitesse) and acceleration, the two basic concepts of kinematics (the description of motion), which are introduced and carefully distinguished at the beginning of every introductory physics course.221 Perhaps this confusion isn’t worth stressing; but for a purported specialist in the philosophy of speed, it is nonetheless a bit surprising.
Drawing inspiration from the theory of relativity, Virilio continues:
How can we fully take in such a situation without enlisting the aid of a new type of interval, THE INTERVAL OF THE LIGHT KIND (neutral sign)? The relativistic innovation of this third interval is actually in itself a sort of unremarked cultural revelation.
If the interval of TIME (positive sign) and the interval of SPACE (negative sign) have laid out the geography and history of the world through the geometric design of agrarian areas (fragmentation into plots of land) and urban areas (the cadastral system), the organization of calendars and the measurement of time (clocks) have als
o presided over a vast chronopolitical regulation of human societies. The very recent emergence of an interval of the third type thus signals a sudden qualitative leap, a profound mutation in the relationship between man and his surroundings.
TIME (duration) and SPACE (extension) are now inconceivable without LIGHT (limit-speed), the cosmological constant of the SPEED OF LIGHT …
(Virilio 1995, p. 25; Virilio 1997, pp. 12–13; capitals in the original)
It is true that, in the special theory of relativity, one introduces ‘space-like’, ‘time-like’ and ‘light-like’ intervals whose ‘invariant lengths’ are respectively positive, negative and zero (according to the usual convention). However, these are intervals in space-time, which do not coincide with what we habitually call ‘space’ and ‘time’.222 Above all, they have nothing to do with the ‘geography and history of the world’ or the ‘chronopolitical regulation of human societies’. The ‘very recent emergence of an interval of the third type’ is nothing but a pedantic allusion to modern telecommunications. In this passage, Virilio shows perfectly how to package a banal observation in sophisticated terminology.