by Alan Sokal
171 For a deeper but still non-technical discussion, see Ruelle (1991).
172 At least to a very high degree of approximation.
173 Note that this does not rule out, a priori, the possibility of statistically predicting the future climate, such as the average and fluctuations in temperature and rainfall for Britain during the decade 2050–60. Modelling the global climate is a difficult and controversial scientific problem, but is extremely important for the future of the human race.
174 In technical terms: in the first case the imprecision increases linearly or polynomially with time, and in the second case exponentially.
175 It is important to add one qualification: for some chaotic systems, the fixed amount that one gains when doubling the precision in the initial measurements can be very long, which means that in practice these systems can be predictable much longer than most non-chaotic systems. For example, recent research has shown that the orbits of some planets have a chaotic behaviour, but the ‘fixed amount’ is here of the order of several million years.
176 Kellert (1993) gives a clear introduction to chaos theory and a sober examination of its philosophical implications, although we do not agree with all of his conclusions.
177 Strange attractors, Lyapunov exponents, etc.
178 ‘Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it – an intelligence sufficiently vast to submit these data to analysis – it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes’ (Laplace 1902 [1825], p. 4).
179 The purpose of quoting these remarks is, of course, to clarify the distinction between determinism and predictability, not to prove that determinism is true. Indeed, Maxwell himself was apparently not a determinist.
180 This verbal formulation actually confuses the problem of linearity with the very different problem of causality. In a linear equation, it is the set of all the variables that obeys a relation of proportionality. There is no need to specify which variables represent the ‘effect’ and which the ‘cause’; and indeed, in many instances (for example, in systems with feedback) such a distinction is meaningless.
181 Often called total order.
182 [For the experts:] Here ‘natural’ means ‘compatible with the field structure’, in the sense that a, b>0 implies ab>0, and a>b implies a+c>b+c.
183 Let us note in passing that it is false to assert that intuition plays no role in ‘traditional’ science. Quite the contrary: since scientific theories are creations of the human mind and are almost never ‘written’ in the experimental data, intuition plays an essential role in the creative process of invention of theories. Nevertheless, intuition cannot play an explicit role in the reasoning leading to the verification (or falsification) of these theories, since this process must remain independent of the subjectivity of individual scientists.
184 For example:
‘These [scientific] practices were rooted in a binary logic of hermetic subjects and objects and a linear, teleological rationality ... Linearity and teleology are being supplanted by chaos models of non-linearity and an emphasis on historical contingency.’
(Lather 1991, pp. 104-105)
‘As opposed to more linear (historical and psychoanalytic as well as scientific) determinisms that tend to exclude them as anomalies outside the generally linear course of things, certain older determinisms incorporated chaos, incessant turbulence, sheer chance, in dynamic interactions cognate to modern chaos theory...’
(Hawkins 1995, p. 49)
‘Unlike teleological linear systems, chaotic models resist closure, breaking off instead into endless “recursive symmetries.” This lack of closure privileges uncertainty. A single theory or “meaning” disseminates into infinite possibilities ... What we once considered to be enclosed by linear logic begins to open up to a surprising series of new forms and possibilities.’
(Rosenberg 1992, p. 210)
Let us emphasize that we are not criticizing these authors for employing the word ‘linear’ in their own sense: mathematics has no monopoly on the word. What we are criticizing is some postmodernists’ tendency to confuse their sense of the word with the mathematical one, and to draw connections with chaos theory that are not supported by any valid argument. Dahan-Dalmedico (1997) seems to miss this point.
185 For example, Harriett Hawkins refers to the ‘linear equations describing the regular, and therefore predictable movements of planets and comets’ (Hawkins 1995, p. 31), and Steven Best alludes to ‘the linear equations used in Newtonian and even quantum mechanics’ (Best 1991, p. 225); they commit the first mistake but not the second. Conversely, Robert Markley claims that ‘Quantum physics, hadron bootstrap theory, complex number theory [!], and chaos theory share the basic assumption that reality cannot be described in linear terms, that nonlinear – and unsolvable – equations are the only means possible to describe a complex, chaotic, and non-deterministic reality.’ (Markley 1992, p. 264) This sentence deserves some sort of prize for squeezing the maximal number of confusions into the minimal number of words. See pp. 246–7 below for a brief discussion.
186 See Ruelle (1994) for a more detailed discussion.
187 For thoughtful critiques of applications of chaos theory in literature, see, for example, Matheson and Kirchhoff (1997) and van Peer (1998).
188 We do not deny that if one understood these systems better – enough to be able to write down equations that describe them at least approximately – the mathematical theory of chaos might provide interesting information. But sociology and history are, at present, far from having reached this stage of development (and perhaps will always remain so).
* * *
* * *
189 What is a non-Euclidean space? In Euclidean plane geometry – the geometry studied in high school – for each straight line L and each point p not on L, there exists one and only one straight line parallel to L (i.e. not intersecting L) that passes through p. By contrast, in non-Euclidean geometries, there can be either an infinite number of parallel lines or else none at all. These geometries go back to the works of Bolyai, Lobachevskii and Riemann in the nineteenth century, and they were applied by Einstein in his general theory of relativity (1915). For a good introduction to non-Euclidean geometries (but without their military applications), see Greenberg (1980) or Davis (1993).
190 See our discussion (p. 133 above) concerning abuses of the word ‘linear’.
191 To illustrate this concept, consider a collection of billiard balls moving on a table according to Newton’s laws (without friction and with elastic collisions), and make a film of this motion. Now run this film backwards: the reversed motion will also obey the laws of Newtonian mechanics. This fact is summarized by saying that the laws of Newtonian mechanics are invariant with respect to time inversion. In fact, all the known laws of physics, except those of the ‘weak interactions’ between subatomic particles, satisfy this property of invariance.
192 The experiments of Benveniste’s group on the biological effects of highly diluted solutions, which seemed to provide a scientific basis for homeopathy, were rapidly discredited after being hastily published in the scientific journal Nature (Davenas et al. 1988). See Maddox et al. (1988); and, for a more detailed discussion, see Broch (1992). More recently, Baudrillard has opined that the memory of water is ‘the ultimate stage of the transfiguration of the world into pure information’ and that ‘this virtualization of effects is wholly in line with the most recent science’ (Baudrillard 1997, p. 94).
193 Not at all! When zero is an attractor, it is what one calls a ‘fixed point’; these attractors (as well as others known as ‘limit-cycles’) have been known since the nineteenth century, and the expression ‘strange attractor’ was introduced specifically to refer to attractors of a different sort. See,
for example, Ruelle (1991).
194 Examples of the latter are variable refraction hyperspace and fractal scissiparity.
195 Gross and Levitt (1994, p. 80).
196 For other examples, see the references to chaos theory (Baudrillard 1990, pp. 154–5), to the Big Bang (Baudrillard 1994, pp. 115–16), and to quantum mechanics (Baudrillard 1996, pp. 14, 53–5). This last book is permeated with scientific and pseudo-scientific allusions.
197 For a more detailed critique of Baudrillard’s ideas, see Norris (1992).
* * *
* * *
198 Gödel: Deleuze and Guattari (1994, pp. 121, 137–9). Transfinite cardinals: Deleuze and Guattari (1994, pp. 120–1). Riemannian geometry: Deleuze and Guattari (1987, pp. 32, 373, 482–6, 556n); Deleuze and Guattari (1994, pp. 124, 161, 217). Quantum mechanics: Deleuze and Guattari (1994, pp. 129–30). These references are far from being exhaustive.
199 Indeed, Deleuze and Guattari, in a footnote, refer the reader to a book by Prigogine and Stengers, where one finds the following picturesque description of quantum field theory:
The quantum vacuum is the opposite of nothingness: far from being passive or inert, it potentially contains all possible particles. Unceasingly, these particles emerge out of the vacuum, only to disappear immediately. (Prigogine and Stengers 1988, p. 162)
A little later, Prigogine and Stengers discuss some theories on the origin of the universe that involve an instability of the quantum vacuum (in general relativity), and they add:
This description is reminiscent of the crystallization of a supercooled liquid (a liquid that has been cooled below its freezing temperature). In such a liquid, small crystalline kernels form, but they then dissolve without consequences. For such a kernel to unleash the process leading to the crystallization of the entire liquid, it has to reach a critical size that depends, in this case too, on a highly nonlinear cooperative mechanism called ‘nucleation’. (Prigogine and Stengers 1988, pp. 162–3)
The conception of ‘chaos’ used by Deleuze and Guattari is thus a verbal mélange of a description of quantum field theory with a description of a supercooled liquid. These two branches of physics have no direct relation to chaos theory in its usual sense (namely, the theory of nonlinear dynamical systems).
200 Deleuze and Guattari (1994), p. 156 and note 14, and especially p. 206 and note 7.
201 For example: infinite, speed, particle, function, catalysis, particle accelerator, expansion, galaxy, limit, variable, abscissa, universal constant, contraction.
202 For example, the statement ‘the speed of light ... where lengths contract to zero and clocks stop’ is not false, but it may lead to confusion. In order to understand it correctly, one must already have a good knowledge of relativity theory.
203 For an amusing exegesis of the above passages, in the same vein as the original, see Alliez (1993, chapter II).
204 This sentence repeats a confusion of Hegel (1989 [1812], pp. 251–3, 277–8), who considered that fractions such as y2/x were fundamentally different from fractions like a/b. As noted by the philosopher J.T. Desanti: ‘Such propositions could not but surprise a “mathematical mind”, who would be led to regard them as absurd.’ (Desanti 1975, p. 43)
205 Which appear in the derivative dy/dx and in the integral ∫f(x)dx.
206 For a historical account, see, for example, Boyer (1959 [1949], pp. 247–50, 267–77).
207 Further comments on calculus can be found in Deleuze (1994, pp. 43, 170–8, 182–3, 201, 209–11, 244, 264, 280–1). For additional lucubrations on mathematical concepts, mixing banalities with nonsense, see Deleuze (1994, pp. 179–81, 202, 232–4, 237–8); and on physics, see Deleuze (1994, pp. 117, 222–6, 228–9, 240, 318n).
208 The previous paragraph contains the following definition: ‘This procedure of the infinitely small, which maintains the distinction between essences (to the extent that one plays the role of inessential to the other), is quite different to contradiction. We should therefore give it a special name, that of “vice-diction”.’ (Deleuze 1994, p. 46)
209 This is, at best, a very complicated way of saying that the traditional notation dy/dx denotes an object – the derivative of the function y(x) – which is not, however, the quotient of two quantities dy and dx.
210 In the calculus of functions of a single variable, integration is indeed the inverse of differentiation, up to an additive constant (at least for sufficiently smooth functions). The situation is more complicated for functions of several variables. Conceivably Deleuze is referring to this latter case, but if so, he is doing it in a very confused fashion.
211 The correct translation of the mathematical term ‘puissance du continu’ is ‘power of the continuum’. See note 38 above for a brief explanation of this concept.
Deleuze notwithstanding, ‘limit’ and ‘power of the continuum’ are two completely distinct concepts. It is true that the idea of ‘limit’ is related to the idea of ‘real number’, and that the set of real numbers has the power of the continuum. But Deleuze’s formulation is, at best, exceedingly confused.
212 Quite true; and as far a mathematics is concerned, such a rigorous exposition has existed for more than 150 years. One wonders why a philosopher would choose to ignore it.
213 This sentence repeats the confusion, going back to Hegel, mentioned in note 204 above.
214 This is an extremely pedantic way to introduce Taylor series, and we doubt that this passage could be understood by anyone who did not already know the subject. Furthermore, Deleuze (as well as Hegel) bases himself on an archaic definition of ‘function’ (namely, by its Taylor series) that goes back to Lagrange (circa 1770) but which has been superseded ever since the work of Cauchy (1821). See, for example, Boyer (1959 [1949], pp. 251–3, 267–77).
215 See note 210 above.
216 For example: singularity, stable, unstable, metastable, potential energy, singular point, crystal, membrane, polarity, topological surface. A defender of Deleuze might contend that he is using these words here only in a metaphorical or philosophical sense. But in the next paragraph, Deleuze discusses ‘singular points’ using mathematical terms taken from the theory of differential equations (cols, nœuds, foyers, centres) and continues by quoting, in a footnote, a passage of a book on differential equations that uses words like ‘singularity’ and ‘singular point’ in their technical mathematical sense. See also Deleuze (1990, pp. 50, 54, 339–40n). Deleuze is, of course, welcome to use these words in more than one sense if he likes, but in that case he should distinguish between the two (or more) senses and provide an argument explaining the relation between them.
217 This book is, in fact, densely filled with mathematical, scientific and pseudo-scientific terminology, used most of the time in a completely arbitrary way.
218 For examples of academic articles that elaborate Deleuze and Guattari’s pseudo-science, see Rosenberg (1993), Canning (1994) and the recent academic conference devoted to ‘DeleuzeGuattari and Matter’ (University of Warwick 1997).
* * *
* * *
219 As Revel (1997) has noted, dromos does not mean ‘speed’, but rather ‘course, race, running’; the Greek word for ‘speed’ is tachos. Probably the error is Le Monde’s, because Virilio (1997, p. 22) gives the correct definition.
220 Translation ours. See note 221 below for a critique of the published English translations (Virilio 1993, p. 5 and 1997, p. 12)
221 Acceleration is the rate of change of velocity. This confusion is systematic in Virilio’s work: see, for example, Virilio (1997, pp. 31, 32, 43, 142). One of Virilio’s translators (Virilio 1993, p. 5) made things worse by rendering vitesse as ‘speed’ rather than ‘velocity’. In English physics usage, ‘speed’ designates the length of the velocity vector and thus can never be negative. The other translator (Virilio 1997, p. 12) tried to improve matters by inserting the words ‘vector quantities’ (which do not appear in the French original) before ‘positive or negative velocities’; but this interpol
ation, while correct, leaves untouched the fundamental confusion between velocity and acceleration.
222 The book by Taylor and Wheeler (1966) gives a beautiful introduction of the notion of space-time interval.
223 Translation ours. See note 223 below for a critique of the published English translations (Virilio 1993, p. 6 and 1997, p. 13).
224 Virilio’s English translators – who can hardly be expected to possess a technical knowledge of physics – have likewise made a hash of this sentence. One rendered it as ‘A representation is defined by a sum of observables that are flickering back and forth’ (Virilio 1993, p. 6), while the other came up with ‘A display is defined by a complete set of observables that commutate’ (Virilio 1997, p. 13).
225 Here is how a book containing this essay of Virilio’s was lauded in an American journal of literary studies:
Re-thinking Technologies constitutes a significant contribution to the analysis of techno-cultures today. It will definitely contradict those who still think that post-modernity is merely a fashionable term or an empty fad. The nagging opinion that cultural and critical theory is ‘too abstract,’ hopelessly removed from reality, devoid of ethical values and above all incompatible with erudition, systematic thinking, intellectual rigor and creative criticism, will simply be pulverized. ... This collection assembles some of the most recent and fresh work by leading culture critics and theoreticians of the arts and sciences, such as Paul Virilio, Félix Guattari, ...
(Gabon 1994, pp. 119-120, emphasis added)