by Alan Sokal
What comes next is even more surprising:
Listen to the physicist speaking about the logic of particles: ‘A representation is defined by a complete set of commuting observables.’ [G. Cohen Tannoudji and M. Spiro, La matière-espace-temps, Paris, Fayard, 1986.] There is no better description of the macroscopic logic of the REAL-TIME technologies of this sudden ‘teletopical commutation’ that completes and perfects what until now had been the fundamentally ‘topical’ nature of the City of Man.
(Virilio 1995, p. 26, capitals in the original223)
The sentence ‘A representation is defined by a complete set of commuting observables’ is a rather common technical expression in quantum mechanics (not in relativity). It has nothing to do with ‘real time’ or with any ‘macroscopic logic’ (quite the contrary, it refers to microphysics), much less with ‘teletopical commutation’ or the ‘City of Man’. But above all, in order to understand the precise meaning of this sentence, one needs to have studied physics and mathematics seriously for several years. We find it incredible that Virilio could consciously copy a sentence that he manifestly does not understand, add to it a completely arbitrary comment, and still be taken seriously by editors, commentators and readers.224, 225
Virilio’s works are overflowing with this pseudo-scientific verbiage.226 Here is another example:
What happens to the transparence of air, of water, of glass – one could say, of the ‘real space’ of things surrounding us – when the interface of ‘real time’ takes over from the classic ‘interval,’ and when distance suddenly gives way to the power of transmission and instantaneous reception? ... Transparence is no longer composed of light rays (solar or electric) but instead of elemental particles (electrons and photons) that are transmitted at the speed of light.
(Virilio 1989, p. 129; Virilio 1990, p. 107; italics in the original)
For what it’s worth, electrons, unlike photons, have a non-zero mass and thus cannot move at the speed of light, precisely because of the theory of relativity of which Virilio seems so fond.
In what follows, Virilio continues to throw around scientific terminology, supplemented by his own inventions (teletopology, chronoscopy):
This displacement of the direct transparence of a material is due primarily ... to the effective use of undulatory optics alongside classic geometric optics. In the same way that alongside Euclidean geometry we find a non-Euclidean, topological geometry, the passive optics of the geometry of camera lenses and telescopes is accompanied by the active optics of the teletopology of optoelectric waves.
... Traditional chronology – future, present, past – has been succeeded by CHRONOSCOPY – underexposed, exposed, overexposed. The interval of the TIME genre (the positive sign) and the interval of the SPACE genre (the negative sign, with the same name as the inscription surface of film) are inscribed only by LIGHT, that interval of the third genre in which the zero sign means absolute speed.
The exposure time of the photographic plate is therefore merely time’s (space-time’s) exposure of its photosensitive material to the light of speed, which is to say, finally, to the frequency of photon-carrying waves.
(Virilio 1989, p. 129; Virilio 1990, pp. 108–9, 115; italics and capitals in the original)
This mélange of optics, geometry, relativity and photography needs no comment.
Let us round off our reading of Virilio’s writings on speed with this little marvel:
Remember that the dromospheric space, space-speed, is physically described by what is called the ‘logistic equation,’ the result of the product of the mass displaced by the speed of its displacement, MxV.
(Virilio 1984, p. 176; Virilio 1991, p. 136227)
The logistic equation is a differential equation studied in population biology (among other fields); it is written dx/dt=λx(1–x) and was introduced by the mathematician Verhulst (1838). It has nothing to do with M × V. In Newtonian mechanics, M × V is called ‘momentum’; in relativistic mechanics, M × V does not arise at all. The dromospheric space is a Virilian invention.
Of course, no work in this genre would be complete without an allusion to Gödel’s theorem:
This drifting of figures and geometric figuring, this irruption of dimensions and transcendental mathematics, leads us to the promised surrealist peaks of scientific theory, peaks that culminate in Gödel’s theorem: the existential proof, a method that mathematically proves the existence of an object without producing that object.
(Virilio 1991, p. 66)
In reality, existential proofs are much older than Gödel’s work; and the proof of his theorem is, by contrast, completely constructive: it exhibits a proposition that is neither provable nor falsifiable in the system under consideration (provided the system is consistent).228
And, to top it all off:
When depth of time replaces depths of sensible space; when the commutation of interface supplants the delimitation of surfaces; when transparence re-establishes appearances; then we begin to wonder whether that which we insist on calling space isn’t actually light, a subliminary, para-optical light of which sunlight is only one phase or reflection. This light occurs in a duration measured in instantaneous time exposure rather than the historical and chronological passage of time. The time of this instant without duration is ‘exposure time’, be it over- or underexposure. Its photographic and cinematographic technologies already predicted the existence and the time of a continuum stripped of all physical dimensions, in which the quantum of energetic action and the punctum of cinematic observation have suddenly become the last vestiges of a vanished morphological reality. Transferred into the eternal present of a relativity whose topological and teleological thickness and depth belong to this final measuring instrument, this speed of light possesses one direction, which is both its size and dimension and which propagates itself at the same speed in all radial directions that measure the universe.
(Virilio 1984, p. 77; Virilio 1991, pp. 63–4; italics in the original)
This paragraph – which in the French original is a single 193-word sentence, whose ‘poetry’ is unfortunately not fully captured by the translation – is the most perfect example of diarrhoea of the pen that we have ever encountered. And as far as we can see, it means precisely nothing.
11
GÖDEL’S THEOREM AND SET THEORY: SOME EXAMPLES OF ABUSE
Ever since Gödel showed that there does not exist a proof of the consistency of Peano’s arithmetic that is formalizable within this theory (1931), political scientists had the means for understanding why it was necessary to mummify Lenin and display him to the ‘accidental’ comrades in a mausoleum, at the Centre of the National Community.
(Régis Debray, Le Scribe, 1980, p. 70)
In applying Gödel’s theorem to questions of the closed and the open, as they relate to sociology then, with one gesture Régis Debray recapitulates and concludes the history and the work of the previous 200 years.
(Michel Serres, A History of Scientific Thought, 1995, p. 452)
Gödel’s theorem is an inexhaustible source of intellectual abuses: we have already encountered examples in Kristeva and Virilio, and a whole book could easily be written on the subject. In this chapter we shall give some rather extraordinary examples in which Gödel’s theorem and other concepts taken from the foundations of mathematics are extrapolated in a totally arbitrary way to the social and political domain.
The social critic Régis Debray devotes one chapter of his theoretical work, Critique of Political Reason, to explain that ‘collective madness finds its ultimate foundation in a logical axiom that is itself without foundation: incompleteness’.229 This ‘axiom’ (also called ‘thesis’ or ‘theorem’) is introduced in a rather bombastic fashion:
The ‘secret’ of our collective miseries, of the a priori condition of any political history, past, present or future, may be stated in a few simple, even childish words. If we bear in mind that surplus labour and the unconscious can both be defined in a single
sentence (and that, in the physical sciences, the equation for general relativity can be stated in three letters), there is no danger of confusing simplicity with over-simplification. The secret takes the form of a logical law, an extension of Gödel’s theorem: there can be no organized system without closure and no system can be closed by elements internal to that system alone.
(Debray 1983, pp. 169–70, italics in the original)
Let us pass over the allusion to general relativity. What is more serious is the invocation of Gödel’s theorem, which concerns the properties of certain formal systems in mathematical logic, to explain the ‘secret of our collective miseries’. There is quite simply no logical relationship between this theorem and questions of sociology.230
Nevertheless, the conclusions that Debray draws from his ‘extension of Gödel’s theorem’ are rather spectacular. For example:
Just as it would be a biological contradiction for an individual to give birth to himself (integral cloning as biological aporia?), the government of a collective by itself – verbi gratia ‘of the people by the people’ – is a logically contradictory operation (‘generalized workers’ control’ as political aporia).
(Debray 1983, p. 177)
And likewise:
It is quite natural that there should be something irrational about groups, for if there were not, there would be no groups. It is positive that there should be something mystical about them, because a demystified society would be a pulverized society.
(Debray 1983, p. 176)
According to Debray, therefore, neither a government of the people by the people nor a demystified society are possible, and this apparently for strictly logical reasons.
But if the argument were valid, one might as well use it to prove the existence of God, as suggested by the following statement:
Incompleteness stipulates that a set cannot, by definition, be a substance in the Spinozist sense: something which exists in itself and is conceived by itself. It requires a cause (to engender it) and it is not its own cause.
(Debray 1983, p. 177)
Nevertheless, Debray denies the existence of God (p. 176), without explaining why it would not be an equally ‘logical’ consequence of his ‘theorem’.
The bottom line is that Debray never explains what role Gödel’s theorem is supposed to play in his argument. If he wants to employ it directly in reasoning about social organization, then he is simply wrong. If, on the other hand, Gödel’s theorem is intended to serve merely as an analogy, then it could be suggestive but certainly not demonstrative. To support his sociological and historical theses, he would have to supply arguments dealing with humans and their social behavior, not mathematical logic.
Gödel’s theorem will still be true in ten thousand years or a million years, but no one can say what human society will look like so far in the future. The invocation of this theorem thus gives the appearance of an ‘eternal’ quality to theses that are, at best, valid in a given context and at a given time. Indeed, the allusion to the ‘biological contradiction’ supposedly inherent in ‘integral cloning’ looks nowadays a bit out of date – which shows that one must be careful when ‘applying’ Gödel’s theorem.
Since this idea of Debray does not seem terribly impressive, we were quite surprised to see it elevated to a ‘Gödel–Debray principle’ by the renowned philosopher Michel Serres,231 who explains that
Régis Debray applies or discovers as applicable to social groups the incompleteness theorem valid for formal systems, and shows that societies can only organize themselves on the express condition that they are founded on something other than themselves, outside their own definition or border. They cannot be sufficient in themselves. He calls this foundation religious. With Gödel he completes the work of Bergson, whose Les Deux Sources de la morale et de la religion [The Two Sources of Morality and Religion] differentiated between closed and open societies. No, he says, internal coherence is guaranteed by the external: the group only closes if it is open. Saints, geniuses, heroes, paragons and all sorts of champions do not break institutions but make them possible.
(Serres 1995, pp. 449–50)
He continues:
Since Bergson, the most notable historians have copied from the Two Sources ... Far from copying a model, as they do, Régis Debray solves a problem. Where historians describe the crossing or transgression of social or conceptual limits, without understanding them, because they have borrowed their model ready-made from Bergson, which Bergson constructed on the basis of Carnot and thermodynamics, Régis Debray has constructed his own, and has therefore grasped a new model, based on Gödel and on logical systems.
This decisive contribution from Gödel and Debray frees us from ancient models and from their repetition.
(Serres 1995, p. 450)
Serres goes on to apply the ‘Gödel–Debray principle’ to the history of science232, where it is as irrelevant as it is in politics.
Our last example is reminiscent of Sokal’s parody, where he plays on the word ‘choice’ to forge an absurd link between the axiom of choice in mathematical set theory233 and the movement for abortion rights. He goes so far as to invoke Cohen’s theorem, which shows that the axiom of choice and the continuum hypothesis234 are independent (in the technical meaning of this word in logic) of the other axioms of set theory, to claim that conventional set theory is insufficient for a ‘liberatory’ mathematics. Here again, one finds a completely arbitrary leap from the foundations of mathematics to political considerations.
Since this passage is one of the most openly ridiculous in the parody, we were rather surprised to find similar ideas put forward in utter seriousness – or so it appears – by the philosopher Alain Badiou (in texts that we emphasize are rather old). In Theory of the Subject (1982), Badiou happily throws together politics, Lacanian psychoanalysis and mathematical set theory. The following excerpt from the chapter entitled ‘The logic of excess’ gives an idea of the book’s flavour. After a brief discussion on the situation of immigrant workers, Badiou refers to the continuum hypothesis and continues:
What is at stake here is nothing less than the fusion of algebra (ordered succession of cardinals) and topology (excess of the partitive over the elementary). The truth of the continuum hypothesis would make law [ferait loi] of the fact that the excess in the multiple has no other assignment than the occupation of the empty place, than the existence of the nonexistent proper of the initial multiple. There would be this maintained filiation of coherence, that what exceeds internally the whole does not go beyond naming the limit point of this whole.
But the continuum hypothesis is not provable.
Mathematical triumph of politics over trade-union realism.235
(Badiou 1982, pp. 282–3)
One cannot help but wonder whether a few paragraphs were inadvertently omitted between the last two sentences of this quotation; but no such luck, the jump between mathematics and politics is as abrupt as it appears.236
12
EPILOGUE
In this last chapter, we shall address some general questions – historical, sociological and political – that arise naturally from a reading of the texts quoted in this book. We shall limit ourselves to explaining our point of view, without justifying it in detail. It goes without saying that we claim no special competence in history, sociology or politics; and what we have to say must, in any case, be understood as conjectures rather than as the final word. If we do not simply remain silent on these questions, it is principally to avoid having ideas attributed to us against our will (as has already been done) and to show that our position on many issues is quite moderate.
Over the past two decades, much ink has been spilled about postmodernism, an intellectual current that is supposed to have replaced modern rationalist thought.237 However, the term ‘postmodernism’ covers an ill-defined galaxy of ideas – ranging from art and architecture to the social sciences and philosophy – and we have no wish to discuss most of these areas.238 Our focus
is limited to certain intellectual aspects of postmodernism that have had an impact on the humanities and the social sciences: a fascination with obscure discourses; an epistemic relativism linked to a generalized scepticism toward modern science; an excessive interest in subjective beliefs independently of their truth or falsity; and an emphasis on discourse and language as opposed to the facts to which those discourses refer (or, worse, the rejection of the very idea that facts exist or that one may refer to them).
Let us start by recognizing that many ‘postmodern’ ideas, expressed in a moderate form, provide a needed correction to naive modernism (belief in indefinite and continuous progress, scientism, cultural Eurocentrism, etc.). What we are criticizing is the radical version of postmodernism, as well as a number of mental confusions that are found in the more moderate versions of postmodernism and that are in some sense inherited from the radical one.239
We shall begin by considering the tensions that have always existed between the ‘two cultures’ but that seem to have worsened during the last few years, as well as the conditions for a fruitful dialogue between the humanities and social sciences and the natural sciences. We shall then analyse some of the intellectual and political sources of postmodernism. Finally, we shall discuss the negative aspects of postmodernism for both culture and politics.
For a real dialogue between the ‘two cultures’
Interdisciplinarity seems to be the order of the day. Though some people worry that the dilution of specialization may lead to a decline in the standards of intellectual rigour, the insights that one field of thought can bring to another cannot be ignored. By no means do we wish to inhibit interaction between the mathematico-physical sciences and the human sciences; rather, our aim is to emphasize some preconditions we see as necessary for a real dialogue.
Over the past few years, it has become fashionable to talk about a so-called ‘science war’.240 But this phrase is quite unfortunate. Who is waging war, and against whom?