by Alan Sokal
The conclusion we can draw from this research (and much more not mentioned here) is that the continuous differentiable function167 is losing its preeminence as a paradigm of knowledge and prediction. Postmodern science – by concerning itself with such things as undecidables, the limits of precise control, conflicts characterized by incomplete information, ‘fracta,’ catastrophes, and pragmatic paradoxes – is theorizing its own evolution as discontinuous, catastrophic, nonrectifiable,168 and paradoxical. It is changing the meaning of the word knowledge, while expressing how such a change can take place. It is producing not the known, but the unknown. And it suggests a model of legitimation that has nothing to do with maximized performance, but has as its basis difference understood as paralogy.
(Lyotard 1984, p. 60)
Since this paragraph is frequently quoted, let us examine it closely.169 Lyotard has here thrown together at least six distinct branches of mathematics and physics, which are conceptually quite distant from one another. Moreover, he has confused the introduction of nondifferentiable (or even discontinuous) functions in scientific models with a so-called ‘discontinuous’ or ‘paradoxical’ evolution of science itself. The theories cited by Lyotard of course produce new knowledge, but they do so without changing the meaning of the word.170 A fortiori, what they produce is known, not unknown (except in the trivial sense that new discoveries open up new problems). Finally, the ‘model of legitimation’ remains the comparison of theories with observations and experiments, not ‘difference understood as paralogy’ (whatever this may mean).
Let us now turn our attention to chaos theory.171 We shall address three sorts of confusions: those concerning the theory’s philosophical implications, those arising from the metaphorical use of the words ‘linear’ and ‘nonlinear’, and those associated with hasty applications and extrapolations.
What is chaos theory about? There are many physical phenomena governed172 by deterministic laws, and therefore predictable in principle, which are nevertheless unpredictable in practice because of their ‘sensitivity to initial conditions’. This means that two systems obeying the same laws may, at some moment in time, be in very similar (but not identical) states and yet, after a brief lapse of time, find themselves in very different states. This phenomenon is expressed figuratively by saying that a butterfly flapping its wings today in Madagascar could provoke a hurricane three weeks from now in Florida. Of course, the butterfly by itself doesn’t do much. But if one compares the two systems constituted by the Earth’s atmosphere with and without the flap of the butterfly’s wings, the result three weeks from now may be very different (a hurricane or not). One practical consequence of this is that we do not expect to be able to predict the weather more than a few weeks ahead.173 Indeed, one would have to take into account such a vast quantity of data, and with such a precision, that even the largest conceivable computers could not begin to cope.
To be more precise, let us consider a system whose initial state is imperfectly known (as is always the case in practice). It is obvious that this imprecision in the initial data will be reflected in the quality of the predictions we are able to make about the system’s future state. In general, the predictions will become more inexact as time goes on. But the manner in which the imprecision increases differs from one system to another: in some systems it will increase slowly, in others very quickly.174
To explain this, let us imagine that we want to reach a certain specified precision in our final predictions, and let us ask ourselves how long our predictions will remain sufficiently accurate. Let us suppose, moreover, that a technical improvement has allowed us to reduce by half the imprecision of our knowledge of the initial state. For the first type of system (where the imprecision increases slowly), the technical improvement will permit us to double the length of time during which we can predict the state of the system with the desired precision. But for the second type of system (where the imprecision increases quickly), it will allow us to increase our ‘window of predictability’ by only a fixed amount: for example, by one additional hour or one additional week (how much depends on the circumstances). Simplifying somewhat, we shall call systems of the first kind non-chaotic and systems of the second kind chaotic (or ‘sensitive to initial conditions’). Chaotic systems are therefore characterized by the fact that their predictability is sharply limited, because even a spectacular improvement in the precision of the initial data (for example, by a factor of 1,000) leads only to a rather mediocre increase in the duration over which the predictions remain valid.175
It is perhaps not surprising that a very complex system, such as the Earth’s atmosphere, is difficult to predict. What is more surprising is that a system describable by a small number of variables and obeying simple deterministic equations – for example, a pair of pendulums attached together – may nevertheless exhibit very complex behavior and an extreme sensitivity to initial conditions.
However, one should avoid jumping to hasty philosophical conclusions.176 For example, it is frequently asserted that chaos theory has shown the limits of science. But many systems in Nature are non-chaotic; and even when studying chaotic systems, scientists do not find themselves at a dead end, or at a barrier which says ‘forbidden to go further’. Chaos theory opens up a vast area for future research and draws attention to many new objects of study.177 Besides, thoughtful scientists have always known that they cannot hope to predict or compute everything. It is perhaps unpleasant to learn that a specific object of interest (such as the weather in three weeks’ time) escapes our ability to predict it, but this does not halt the development of science. For example, physicists in the nineteenth century knew perfectly well that is impossible in practice to know the positions of all the molecules of a gas. This spurred them to develop the methods of statistical physics, which have led to an understanding of many properties of systems (such as gases) that are composed of a large number of molecules. Similar statistical methods are employed nowadays to study chaotic phenomena. And, most importantly, the aim of science is not only to predict, but also to understand.
A second confusion concerns Laplace and determinism. Let us emphasize that in this long-standing debate it has always been essential to distinguish between determinism and predictability. Determinism depends on what Nature does (independently of us), while predictability depends in part on Nature and in part on us. To see this, let us imagine a perfectly predictable phenomenon – a clock, for example – which is, however, situated in an inaccessible place (say, the top of a mountain). The motion of the clock is unpredictable, for us, because we have no way of knowing its initial state. But it would be ridiculous to say that the clock’s motion ceases to be deterministic. Or to take another example, consider a pendulum: When there is no external force, its motion is deterministic and non-chaotic. When one applies a periodic force, its motion may become chaotic and thus much more difficult to predict; but does it cease to be deterministic?
Laplace’s work is often misunderstood. When he introduced the concept of universal determinism,178 he immediately added that we shall ‘always remain infinitely removed’ from this imaginary ‘intelligence’ and its ideal knowledge of the ‘respective situation of the beings who compose’ the natural world, that is, in modern language, of the precise initial conditions of all the particles. He distinguished clearly between what Nature does and the knowledge we have of it. Moreover, he stated this principle at the beginning of an essay on probability theory. But, what is probability theory for Laplace? Nothing but a method that allows us to reason in situations of partial ignorance. The meaning of Laplace’s text is completely misrepresented if one imagines that he hoped to arrive someday at a perfect knowledge and a universal predictability, for the aim of his essay was precisely to explain how to proceed in the absence of such a perfect knowledge – as one does, for example, in statistical physics.
Over the past three decades, remarkable progress has been made in the mathematical theory of chaos, but the idea that some physical syst
ems may exhibit a sensitivity to initial conditions is not new. Here is what James Clerk Maxwell said in 1877, after stating the principle of determinism (‘the same causes will always produce the same effects’):
There is another maxim which must not be confounded with [this one], which asserts ‘That like causes produce like effects.’
This is only true when small variations in the initial circumstances produce only small variations in the final state of the system. In a great many physical phenomena this condition is satisfied; but there are other cases in which a small initial variation may produce a very great change in the final state of the system, as when the displacement of the ‘points’ causes a railway train to run into another instead of keeping its proper course.
(Maxwell 1952 [1877], pp. 13–14)179
And, with regard to meteorological predictions, Henri Poincaré in 1909 was remarkably modern:
Why have meteorologists such difficulty in predicting the weather with any certainty? Why is it that showers and even storms seem to come by chance, so that many people think it quite natural to pray for rain or fine weather, though they would consider it ridiculous to ask for an eclipse by prayer? We see that great disturbances are generally produced in regions where the atmosphere is in unstable equilibrium. The meteorologists see very well that the equilibrium is unstable, that a cyclone will be formed somewhere, but exactly where they are not in a position to say; a tenth of a degree more or less at any given point, and the cyclone will burst here and not there, and extend its ravages over districts it would otherwise have spared. If they had been aware of this tenth of a degree, they could have known it beforehand, but the observations were neither sufficiently comprehensive nor sufficiently precise, and that is the reason why it all seems due to the intervention of chance.
(Poincaré 1914 [1909], pp. 68–9)
Let us turn now to the confusions arising from misuse of the words ‘linear’ and ‘nonlinear’. Let us first point out that, in mathematics, the word ‘linear’ has two distinct meanings, which it is important not to confuse. On the one hand, one may speak of a linear function (or equation): for example, the functions f(x) = 2x and f(x) = –17x are linear, while the functions f(x) = x2 and f(x) = sin x are nonlinear. In terms of mathematical modelling, a linear equation describes a situation in which (simplifying somewhat) ‘the effect is proportional to the cause’.180 On the other hand, one may speak of a linear order:181 this means that the elements of a set are ordered in such a way that, for each pair of elements a and b, one has either a < b, a = b, or a > b. For instance, there exists a natural linear order on the set of real numbers, while there is no natural such order on the complex numbers.182 Now, postmodernist authors (principally in the English-speaking world) have added a third meaning to the word – vaguely related to the second, but often confused by them with the first – in speaking of linear thought. No exact definition is given, but the general meaning is clear enough: it is the logical and rationalist thought of the Enlightenment and of so-called ‘classical’ science (often accused of an extreme reductionism and numericism). In opposition to this old-fashioned way of thinking, they advocate a postmodern ‘nonlinear thought’. The precise content of the latter is not clearly explained either, but it is, apparently, a methodology that goes beyond reason by insisting on intuition and subjective perception.183 And it is frequently claimed that so-called postmodern science – and particularly chaos theory – justifies and supports this new ‘nonlinear thought’. But this assertion rests simply on a confusion between the three meanings of the word ‘linear’.184
Because of these abuses, one often finds postmodernist authors who see chaos theory as a revolution against Newtonian mechanics – the latter being labelled ‘linear’ – or who cite quantum mechanics as an example of a nonlinear theory.185 In actual fact, Newton’s ‘linear thought’ uses equations that are perfectly nonlinear; this is why many examples in chaos theory come from Newtonian mechanics, so that the study of chaos represents in fact a renaissance of Newtonian mechanics as a subject for cutting-edge research. Likewise, quantum mechanics is often cited as the quintessential example of a ‘postmodern science’, but the fundamental equation of quantum mechanics – Schrödinger’s equation – is absolutely linear.
Furthermore, the relationship between linearity, chaos and an equation’s explicit solvability is often misunderstood. Nonlinear equations are generally more difficult to solve than linear equations, but not always: there exist very difficult linear problems and very simple nonlinear ones. For example, Newton’s equations for the two-body Kepler problem (the Sun and one planet) are nonlinear and yet explicitly solvable. Besides, for chaos to occur, it is necessary that the equation be nonlinear and (here we simplify somewhat) not explicitly solvable, but these two conditions are by no means sufficient – whether they occur separately or together – to produce chaos. Contrary to what people often think, a nonlinear system is not necessarily chaotic.
The difficulties and confusions multiply when one attempts to apply the mathematical theory of chaos to concrete situations in physics, biology or the social sciences.186 To do this in a sensible way, one must first have some idea of the relevant variables and of the type of evolution they obey. Unfortunately, it is often difficult to find a mathematical model that is sufficiently simple to be analyzable and yet adequately describes the objects being considered. These problems arise, in fact, whenever one tries to apply a mathematical theory to reality.
Some purported ‘applications’ of chaos theory – for example, to business management or literary analysis – border on the absurd.187 And, to make things worse, chaos theory – which is well-developed mathematically – is often confused with the still-emerging theories of complexity and self-organization.
Another major confusion is caused by mixing the mathematical theory of chaos with the popular wisdom that small causes can have large effects: ‘if Cleopatra’s nose had been shorter’, or the story of the missing nail that led to the collapse of an empire. One constantly hears claims of chaos theory being ‘applied’ to history or society. But human societies are complicated systems involving a vast number of variables, for which one is unable (at least at present) to write down any sensible equations. To speak of chaos for these systems does not take us much further than the intuition already contained in the popular wisdom.188
Yet another abuse arises from confusing (intentionally or not) the numerous distinct meanings of the highly evocative word ‘chaos’: its technical meaning in the mathematical theory of nonlinear dynamics – where it is roughly (though not exactly) synonymous with ‘sensitive dependence on initial conditions’ – and its wider senses in sociology, politics, history and theology, where it is frequently taken as a synonym for disorder. As we shall see, Baudrillard and Deleuze–Guattari are especially shameless in exploiting (or falling into) these verbal confusions.
8
JEAN BAUDRILLARD
Jean Baudrillard’s sociological work challenges and provokes all current theories. With derision, but also with extreme precision, he unknots the constituted social descriptions with quiet confidence and a sense of humour.
(Le Monde, 1984b, p. 95, italics added)
The sociologist and philosopher Jean Baudrillard is well-known for his reflections on the problems of reality, appearance and illusion. In this chapter we want to draw attention to a less-noted aspect of Baudrillard’s work, namely his frequent use of scientific and pseudo-scientific terminology.
In some cases, Baudrillard’s invocation of scientific concepts is clearly metaphorical. For example, he wrote about the Gulf War as follows:
What is most extraordinary is that the two hypotheses, the apocalypse of real time and pure war along with the triumph of the virtual over the real, are realised at the same time, in the same space-time, each in implacable pursuit of the other. It is a sign that the space of the event has become a hyperspace with multiple refractivity, and that the space of war has become definitively non-Euclidea
n.
(Baudrillard 1995, p. 50, italics in the original)
There seems to be a tradition of using technical mathematical notions out of context. With Lacan, it was tori and imaginary numbers; with Kristeva, infinite sets; and here we have non-Euclidean spaces.189 But what could this metaphor mean? Indeed, what would a Euclidean space of war look like? Let us note in passing that the concept of ‘hyperspace with multiple refractivity’ [hyperespace à réfraction multiple] does not exist in either mathematics or physics; it is a Baudrillardian invention.
Baudrillard’s writings are full of similar metaphors drawn from mathematics and physics, for example:
In the Euclidean space of history, the shortest path between two points is the straight line, the line of Progress and Democracy. But this is only true of the linear space of the Enlightenment.190 In our non-Euclidean fin de siècle space, a baleful curvature unfailingly deflects all trajectories. This is doubtless linked to the sphericity of time (visible on the horizon of the end of the century, just as the earth’s sphericity is visible on the horizon at the end of the day) or the subtle distortion of the gravitational field. ...
By this retroversion of history to infinity, this hyperbolic curvature, the century itself is escaping its end.
(Baudrillard 1994, pp. 10–11)
It is to this perhaps that we owe this ‘fun physics’ effect: the impression that events, collective or individual, have been bundled into a memory hole. This blackout is due, no doubt, to this movement of reversal, this parabolic curvature of historical space.
(Baudrillard 1994, p. 20)
But not all of Baudrillard’s physics is metaphorical. In his more philosophical texts, Baudrillard apparently takes physics – or his version of it – literally, as in his essay ‘The fatal, or, reversible imminence’, devoted to the theme of chance: