by Alan Sokal
Everything can be held to develop itself around what I set forth about the logical correlation of two formulas that, to be inscribed mathematically ∀x·Φx, and , can be stated as:34
the first, for all x, Φx is satisfied, which can be translated by a T denoting truth value. This, translated into the analytic discourse of which it is the practice to make sense, ‘means’ that every subject as such – that being what is at stake in this discourse – inscribes itself in the phallic function in order to ward off the absence of the sexual relation (the practice of making sense is exactly to refer to this ab-sense);
the second, there is by exception the case, familiar in mathematics (the argument x=0 in the exponential function 1/x), the case where there exists an x for which Φx, the function, is not satisfied, i.e. does not function, is in fact excluded.
It is precisely from there that I conjugate the all of the universal, more modified than one imagines in the forall of the quantor, to the there exists one with which the quantic pairs it off, its difference being patent with what is implied by the proposition that Aristotle calls particular. I conjugate them of what the there exists one in question, to make a limit on the forall, is what affirms or confirms it (what a proverb already objects to Aristotle’s contradictory).
...
That I state the existence of a subject to posit it of a saying no to the propositional function Φx, implies that it inscribes itself of a quantor of which this function finds itself cut off from the fact that it has at this point no value that one can denote truth value, which means no error either, the false only to understand falsus as fallen, which I already emphasized.
In classical logic, to think of it, the false is not seen only as being of truth the reverse, it designates truth as well.
It is thus correct to write as I do: .
...
That the subject here proposes itself to be called woman depends on two modes. Here they are:
Their inscription is not used in mathematics.35 To deny, as the bar put above the quantor indicates, to deny that there exists one is not done, much less that the forall should notforall itself.
It is there, however, that the meaning of the saying delivers itself, of that which, conjugating the nyania that noises the sexes in company, it makes up for the fact that, between them, the relation isn’t.
Which is to be understood not in the sense that, to reduce our quantors to their reading according to Aristotle, would set the notexistone equal to the noneis of its negative universal, would make the µήπάντες come back, the notall (that he was nevertheless able to formulate), to testify to the existence of a subject to say no to the phallic function, that to suppose it of the contrariety said of two particulars.
This is not the meaning of the saying, which inscribes itself of these quantors.
It is: that in order to introduce itself as a half to say about women, the subject determines itself from the fact that, since there does not exist a suspension of the phallic function, everything can here be said of it, even if it comes from the without-reason. But it is an out-of-universe whole, which is read without a hitch from the second quantor as notall.
The subject in the half where it determines itself from the denied quantors, it is that nothing existing could put a limit on the function, that could not assure itself of anything whatsoever about a universe. So, to ground themselves of this half, ‘they’ (female) are not notalls, with the consequence and for the same reason, that none of them is all either.
(Lacan 1973, pp. 14–15, 22)
Tout peut être maintenu à se développer autour de ce que j’avance de la corrélation logique de deux formules qui, à s’inscrire mathématiquement ∀x·Φx, et , s’énoncent:
la première, pour tout x, Φx est satisfait, ce qui peut se traduire d’un V notant valeur de vérité. Ceci, traduit dans le discours analytique dont c’est la pratique de faire sens, ‘veut dire’ que tout sujet en tant que tel, puisque c’est là l’enjeu de ce discours, s’inscrit dans la fonction phallique pour parer à l’absence du rapport sexuel (la pratique de faire sens, c’est justement de se reférer à cet ab-sens);
la seconde, il y a par exception le cas, familieren mathématique (l’argument x=0 dans la fonction exponentielle 1/x), le cas où il existe un x pour lequel Φx, la fonction, n’est pas satisfaite, c’est-à-dire ne fonctionnant pas, est exclue de fait.
C’est précisément d’où je conjugue le tous de l’universelle, plus modifié qu’on ne s’imagine dans le pourtout du quanteur, à l’il existe un que le quantique lui apparie, sa différence étant patente avec ce qu’implique la proposition qu’Aristote dit particulière. Je les conjugue de ce que l’il existe un en question, à faire limite au pourtout, est ce qui l’affirme ou le confirme (ce qu’un proverbe objecte déjà au contradictoire d’Aristote).
...
Que j’énonce l’existence d’un sujet à la poser d’un dire que non à la fonction propositionnelle Φx, implique qu’elle s’inscrive d’un quanteur dont cette fonction se trouve coupée de ce qu’elle n’ait en ce point aucune valeur qu’on puisse noter de vérité, ce qui veut dire d’erreur pas plus, le faux seulement à entendre falsus comme du chu, ce où j’ai déjà mis l’accent.
En logique classique, qu’on y pense, le faux ne s’aperçoit pas qu’à être de la vérité l’envers, il la désigne aussi bien.
Il est donc juste d’écrire comme je le fais:
...
De deux modes dépend que le sujet ici se propose d’être dit femme. Les voici:
Leur inscription n’est pas d’usage en mathématique. Nier, comme la barre mise au-dessus du quanteur le marque, nier qu’existe un ne se fait pas, et moins encore que pourtout se pourpastoute.
C’est là pourtant que se livre le sens du dire, de ce que, s’y conjuguant le nyania qui bruit des sexes en compagnie, il supplée à ce qu’entre eux, de rapport nyait pas.
Ce qui est à prendre non pas dans le sens qui, de réduire nos quanteurs à leur lecture selon Aristote, égalerait le nexistun au nulnest de son universelle négative, ferait revenir le µή πάντες, le pastout (qu’il a pourtant su formuler), à témoigner de l’existence d’un sujet à dire que non à la fonction phallique, ce à le supposer de la contrariété dite de deux particulières.
Ce n’est pas là le sens du dire, qui s’inscrit de ces quanteurs.
Il est: que pour s’introduire comme moitié à dire des femmes, le sujet se détermine de ce que, n’existant pas de suspens à la fonction phallique, tout puisse ici s’en dire, même à provenir du sans raison. Mais c’est un tout d’hors univers, lequel se lit tout de go du second quanteur comme pastout.
Le sujet dans la moitié où il se détermine des quanteurs niés, c’est de ce que rien d’existant ne fasse limite de la fonction, que ne saurait s’en assurer quoi que ce soit d’un univers. Ainsi à se fonder de cette moitié, ‘elles’ ne sont pastoutes, avec pour suite et du même fait, qu’aucune non plus n’est toute.
(Lacan 1973, pp. 14–15, 22)
Among the other examples of sophisticated terminology thrown at the reader, let us note in Lacan (1971): union (in mathematical logic) (p. 206) and Stokes’ theorem (a particularly shameless case) (p. 213). In Lacan (1975c): gravitation (‘unconscious of the particle’!) (p. 100). In Lacan (1988): theory of the unified field (p. 239). And in Lacan (1998): Bourbaki (pp. 28, 47), quark (p. 36), Copernicus and Kepler (pp. 41–43), inertia, mv2/2, mathematical formalization (p. 130).
Conclusion
What should we make of Lacan’s mathematics? Commentators disagree about Lacan’s intentions: to what extent was he aiming to ‘mathematize’ psychoanalysis? We are unable to give any definitive answer to this question – which, in any case, does not matter much, since Lacan’s ‘mathematics’ are so bizarre that they cannot play a fruitful role in any serious psychological analysis.
To be sure, Lacan does have a vague idea of the mathematics he invokes (but not much more). It is not from him that a student will learn what
a natural number or a compact set is, but his statements, when they are understandable, are not always false. On the other hand, he excels (if we may use this word) at the second type of abuse listed in our introduction: his analogies between psychoanalysis and mathematics are the most arbitrary imaginable, and he gives absolutely no empirical or conceptual justification for them (neither here nor elsewhere in his work). Finally, as for showing off a superficial erudition and manipulating meaningless sentences, the texts quoted above surely speak for themselves.
The most striking aspect of Lacan and his disciples is probably their attitude towards science, and the extreme privilege they accord to ‘theory’ (in actual fact, to formalism and wordplay) at the expense of observations and experiments. After all, psychoanalysis, assuming that it has a scientific basis, is a rather young science. Before launching into vast theoretical generalizations, it might be prudent to check the empirical adequacy of at least some of its propositions. But, in Lacan’s writings, one finds mainly quotations and analyses of texts and concepts.
Lacan’s defenders (as well as those of the other authors discussed here) tend to respond to these criticisms by resorting to a strategy that we shall call ‘neither/nor’: these writings should be evaluated neither as science, nor as philosophy, nor as poetry, nor ... One is then faced with what could be called a ‘secular mysticism’: mysticism because the discourse aims at producing mental effects that are not purely aesthetic, but without addressing itself to reason; secular because the cultural references (Kant, Hegel, Marx, Freud, mathematics, contemporary literature ...) have nothing to do with traditional religions and are attractive to the modern reader. Furthermore, Lacan’s writings became, over time, increasingly cryptic – a characteristic common to many sacred texts – by combining plays on words with fractured syntax; and they served as a basis for the reverent exegesis undertaken by his disciples. One may then wonder whether we are not, after all, dealing with a new religion.
3
JULIA KRISTEVA
Julia Kristeva changes the order of things; she always destroys the latest preconception, the one we thought we could be comforted by, the one of which we could be proud: what she displaces is the already-said, that is to say, the insistence of the signified, that is to say, silliness; what she subverts is the authority of monologic science and of filiation. Her work is entirely new and precise ...
(Roland Barthes, 1970, p. 19, concerning Kristeva’s Séméiotiké:
Researches for a Semioanalysis)
The works of Julia Kristeva touch on many areas, from literary criticism to psychoanalysis to political philosophy. We shall analyse here some excerpts from her early work on linguistics and semiotics. These texts, which date from the late 1960s to the mid-1970s, cannot properly be called poststructuralist; they belong, rather, to the worst excesses of structuralism. Kristeva’s declared goal is to construct a formal theory of poetic language. This goal is, however, ambiguous because, on the one hand, she asserts that poetic language is ‘a formal system whose theorization can be based on [mathematical] set theory’, and on the other hand, she says in a footnote that this is ‘only metaphorical’.
Metaphor or not, this enterprise faces a serious problem: What relation, if any, does poetic language have with mathematical set theory? Kristeva doesn’t really say. She invokes technical notions concerning infinite sets, whose relevance to poetic language is difficult to fathom, especially since no argument is given. Moreover, her presentation of the mathematics contains some gross errors, for example with regard to Gödel’s theorem. Let us emphasize that Kristeva has long since abandoned this approach; nevertheless, it is too typical of the type of work we are criticizing for us to pass it over in silence.
The excerpts below come mainly from Kristeva’s celebrated book Séméiotiké: Researches for a Semioanalysis (1969).36 One of her interpreters describes this work as follows:
What is most striking about Kristeva’s work ... is the competence with which it is presented, the intense singlemindedness with which it is pursued, and finally, its intricate rigour. No resources are spared: existing theories of logic are invoked and, at one point, quantum mechanics ...
(Lechte 1990, p. 109)
Let us therefore examine some examples of this competence and rigour:
... science is a logical endeavor based on the Greek (Indo-European) sentence that is constructed as subject-predicate and that proceeds by identification, determination, causality.37 Modern logic from Frege and Peano through Lukasiewicz, Ackermann or Church, which moves in the dimensions 0–1, and even Boole’s logic which, starting from set theory, gives formalizations that are more isomorphic to the functioning of language, are inoperative in the sphere of poetic language where 1 is not a limit.
It is therefore impossible to formalize poetic language using the existing logical (scientific) procedures without denaturing it. A literary semiotics has to be made starting from a poetic logic, in which the concept of power of the continuum38 would encompass the interval from 0 to 2, a continuum where 0 denotes and 1 is implicitly transgressed.
(Kristeva 1969, pp. 150–1, italics in the original)39
In this excerpt, Kristeva makes one correct assertion and two mistakes. The correct assertion is that poetic sentences cannot, in general, be evaluated as true or false. Now, in mathematical logic, the symbols 0 and 1 are used to denote ‘false’ and ‘true’, respectively; it is in this sense that Boole’s logic uses the set {0,1}. Kristeva’s allusion to mathematical logic is thus correct, though it adds nothing to the initial observation. But in the second paragraph, she seems to confuse the set {0,1}, which is composed of the two elements 0 and 1, with the interval [0,1], which contains all the real numbers between 0 and 1. The latter set, unlike the former, is an infinite set, which, moreover, has the power of the continuum (see note 38). Besides, Kristeva lays great stress on the fact that she has a set (the interval from 0 to 2) that ‘transgresses’ 1, but from the point of view she purports to adopt – that of the cardinality (or power) of sets – there is no difference between the interval [0,1] and the interval [0,2]: both have the power of the continuum.
In the subsequent text, these two errors become even more manifest:
In this ‘power of the continuum’ from zero to the specifically poetic double, one notices that the linguistic, psychic and social ‘prohibition’ [interdit] is 1 (God, the law, the definition), and that the only linguistic practice that ‘escapes’ from this prohibition is poetic discourse. It is no accident that the inadequacies of Aristotelian logic in its application to language were pointed out, on the one hand, by the Chinese philosopher Chang Tung-sun who came from another linguistic realm (that of ideograms) where the Yin-Yang ‘dialogue’ is deployed in place of God, and on the other hand, by Bakhtin who attempted to go beyond the Formalists by a dynamic theorization carried out in a revolutionary society. For him, narrative discourse, which he assimilates to epic discourse, is a prohibition, a ‘monologism’, a subordination of the code to 1, to God. Consequently, the epic is religious and theological, and any ‘realist’ narration obeying the 0–1 logic is dogmatic. The realist novel that Bakhtin calls monologic (Tolstoy) tends to evolve in that space. Realist description, the definition of a ‘personality type’ [caractère], the creation of a ‘character’ [personnage], the development of a ‘subject’: all these descriptive elements of narrative belong to the interval 0–1 and thus are monologic. The only discourse in which the 0–2 poetic logic is fully realized would be that of the carnival: it transgresses the rules of the linguistic code, as well as that of social morality, by adopting a dream-like logic.
... A new approach to poetic texts can be sketched starting from this term [dialogism] that literary semiotics can adopt. The logic implied by ‘dialogism’ is simultaneously: ... 3) A logic of the ‘transfinite’,40 a concept that we borrow from Cantor, which introduces, starting from the ‘power of the continuum’ of poetic language (0–2), a second formative principle, namely: a poetic sequen
ce is ‘next-larger’ (not causally deduced) to all the preceding sequences of the Aristotelian series (scientific, monologic, narrative). Then, the ambivalent space of the novel presents itself as ordered by two formative principles: the monologic (each successive sequence is determined by the preceding one) and the dialogic (transfinite sequences that are next-larger to the preceding causal sequence).
[Footnote: Let us emphasize that the introduction of notions from set theory in an analysis of poetic language is only metaphorical: it is possible because an analogy can be established between the relations Aristotelian logic/poetic logic on the one hand, and denumerable/infinite on the other.]
(Kristeva 1969, pp. 151–3, italics in the original)
At the end of this passage, Kristeva concedes that her ‘theory’ is only a metaphor. But even at that level, she provides no justification: far from having established an analogy between ‘Aristotelian logic/poetic logic’ and ‘denumerable/infinite’, she has merely invoked the names of these latter concepts, without giving the slightest explanation of their meaning or, above all, their relevance (even metaphorical) for ‘poetic logic’. For what it’s worth, the theory of transfinite numbers has nothing to do with causal deduction.
Later on in the text, Kristeva returns to mathematical logic:
For us poetic language is not a code encompassing the others, but a class A that has the same power as the function φ(x1 ... xn) of the infinity of the linguistic code (see the existence theorem, cf. p. 189), and all the ‘other languages’ (the ‘usual’ language, the ‘meta-languages’, etc.) are quotients of A over more restricted extents [étendues] (limited by the rules of the subject-predicate construction, for example, as being at the basis of formal logic), and disguising, because of this limitation, the morphology of the function φ(x1 ... xn).
Poetic language (which we shall henceforth denote by the initials pl) contains the code of linear logic. Moreover, we can find in it all the combinatoric figures that algebra has formalized in a system of artificial signs and that are not externalized at the level of the manifestation of the usual language. ...