Intellectual Impostures

by Alan Sokal

  HARRY WOOLF: May I ask if this fundamental arithmetic and this topology are not in themselves a myth or merely at best an analogy for an explanation of the life of the mind?

  JACQUES LACAN: Analogy to what? ‘S’ designates something which can be written exactly as this S. And I have said that the ‘S’ which designates the subject is instrument, matter, to symbolize a loss. A loss that you experience as a subject (and myself also). In other words, this gap between one thing which has marked meanings and this other thing which is my actual discourse that I try to put in the place where you are, you as not another subject but as people that are able to understand me. Where is the analogon? Either this loss exists or it doesn’t exist. If it exists it is only possible to designate the loss by a system of symbols. In any case, the loss does not exist before this symbolization indicates its place. It is not an analogy. It is really in some part of the realities, this sort of torus. This torus really exists and it is exactly the structure of the neurotic. It is not an analogon; it is not even an abstraction, because an abstraction is some sort of diminution of reality, and I think it is reality itself.

  (Lacan 1970, pp. 195–6)

  Here again, Lacan gives no argument to support his peremptory assertion that the torus ‘is exactly the structure of the neurotic’ (whatever this means). Moreover, when asked explicitly whether it is simply an analogy, he denies it.

  As the years passed, Lacan became increasingly fond of topology. A text from 1972 begins by playing on the etymology of the word (Greek topos, place + logos, word):

  In this space of jouissance, to take something that is bounded, closed [borné, fermé] constitutes a locus [lieu], and to speak of it constitutes a topology.

  (Lacan 1975a, p. 14; Lacan 1998, p. 9; seminar originally held in 197219)

  In this sentence, Lacan has used four technical terms from mathematical analysis (space, bounded, closed, topology) but without paying attention to their meaning; the sentence is meaningless from a mathematical point of view. Furthermore – and most importantly – Lacan never explains the relevance of these mathematical concepts for psychoanalysis. Even if the concept of ‘jouissance’ had a clear and precise meaning, Lacan provides no reason whatsoever to think that jouissance can be considered a ‘space’ in the technical sense of this word in topology. Nevertheless, he continues:

  In a text soon to be published that is at the cutting edge of my discourse last year, I believe I demonstrate the strict equivalence between topology and structure.20 If we take that as our guide, what distinguishes anonymity from what we talk about as jouissance – namely, what is regulated by law – is a geometry. A geometry implies the heterogeneity of locus, namely that there is a locus of the Other.21 Regarding this locus of the Other, of one sex as Other, as absolute Other, what does the most recent development in topology allow us to posit?

  I will posit here the term ‘compactness.’22 Nothing is more compact than a fault [faille], assuming that the intersection of everything that is enclosed therein is accepted as existing over an infinite number of sets, the result being that the intersection implies this infinite number. That is the very definition of compactness.

  (Lacan 1975a, p. 14; Lacan 1998, p. 9)

  Not at all: although Lacan uses quite a few key words from the mathematical theory of compactness (see note 22), he mixes them up arbitrarily and without the slightest regard for their meaning. His ‘definition’ of compactness is not just false: it is gibberish. Moreover, this ‘most recent development in topology’ goes back to 1900–1930.

  He continues as follows:

  The intersection I am talking about is the same one I put forward earlier as being that which covers or poses an obstacle to the supposed sexual relationship.

  Only ‘supposed,’ since I state that analytic discourse is premised solely on the statement that there is no such thing, that it is impossible to found [poser] a sexual relationship. Therein lies analytic discourse’s step forward and it is thereby that it determines the real status of all the other discourses.

  Named here is the point that covers the impossibility of the sexual relationship as such. Jouissance, qua sexual, is phallic – in other words, it is not related to the Other as such.

  Let us follow here the complement of the hypothesis of compactness.

  A formulation is given to us by the topology I qualified as the most recent that takes as its point of departure a logic constructed on the investigation of numbers and that leads to the institution of a locus, which is not that of a homogeneous space. Let us take the same bounded23, closed, supposedly instituted space – the equivalent of what I earlier posited as an intersection extending to infinity. If we assume it to be covered with open sets, in other words, sets that exclude their own limits – the limit is that which is defined as greater than one point and less than another, but in no case equal either to the point of departure or the point of arrival, to sketch it for you quickly24 – it can be shown that it is equivalent to say that the set of these open spaces always allows of a subcovering of open spaces, constituting a finity [finitude], namely, that the series of elements constitutes a finite series.

  You may note that I did not say that they are countable. And yet that is what the term ‘finite’ implies. In the end, we count them one by one. But before we can count them, we must find an order in them and we cannot immediately assume that that order is findable.25

  What is implied, in any case, by the demonstrable finity of the open spaces that can cover the space that is bounded26 and closed in the case of sexual jouissance? What is implied is that the said spaces can be taken one by one [un par un] – and since I am talking about the other pole, let us put this in the feminine – une par une.

  That is the case in the space of sexual jouissance, which thereby proves to be compact.

  (Lacan 1975a, pp. 14–15; Lacan 1998, pp. 9–10)

  This passage illustrates perfectly two ‘faults’ in Lacan’s discourse. Everything is based – at best – on analogies between topology and psychoanalysis that are unsupported by any argument. But, in fact, even the mathematical statements are devoid of meaning.

  In the mid-1970’s, Lacan’s topological preoccupations shifted towards knot theory: see, for example, Lacan (1975a, pp. 107–23; 1998, pp. 122–36) and especially Lacan (1975b–e). For a detailed history of his obsessions with topology, see Roudinesco (1997, chapter 28). Lacan’s disciples have given full accounts of his topologie psychanalytique: see, for example, Granon-Lafont (1985, 1990), Vappereau (1985, 1995), Nasio (1987, 1992), Darmon (1990) and Leupin (1991).

  Imaginary Numbers

  Lacan’s predilection for mathematics is by no means marginal in his work. Already in the 1950s, his writings were full of graphs, formulas and ‘algorithms’. Let us quote, by way of illustration, this excerpt from a seminar held in 1959:

  If you’ll permit me to use one of those formulas which come to me as I write my notes, human life could be defined as a calculus in which zero was irrational. This formula is just an image, a mathematical metaphor. When I say ‘irrational,’ I’m referring not to some unfathomable emotional state but precisely to what is called an imaginary number. The square root of minus one doesn’t correspond to anything that is subject to our intuition, anything real – in the mathematical sense of the term – and yet, it must be conserved, along with its full function.

  (Lacan 1977a, pp. 28–9, seminar held originally in 1959)

  In this quote, Lacan confuses irrational numbers with imaginary numbers, while claiming to be ‘precise’. They have nothing to do with one another.27 Let us emphasize that the mathematical meanings of the words ‘irrational’ and ‘imaginary’ are quite distinct from their ordinary or philosophical meanings. To be sure, Lacan speaks here prudently of a metaphor, though it is hard to see what theoretical role this metaphor (human life as a ‘calculus in which zero was irrational’) could fulfill. Nevertheless, a year later, he further developed the psychoanalytic role of imaginary numbers:


  Personally, I will begin with what is articulated in the sigla S(Ø) by being first of all a signifier ...

  And since the battery of signifiers, as such, is by that very fact complete, this signifier can only be a line [trait] that is drawn from its circle without being able to be counted part of it. It can be symbolized by the inherence of a (–1) in the whole set of signifiers.

  As such it is inexpressible, but its operation is not inexpressible, for it is that which is produced whenever a proper noun is spoken. Its statement equals its signification.

  Thus, by calculating that signification according to the algebraic method used here, namely:

  with S =(–1), produces: s = √ –1 .

  (Lacan 1977b, pp. 316–7, seminar originally held in 1960)

  Here Lacan can only be pulling the reader’s leg. Even if his ‘algebra’ had a meaning, the ‘signifier’, ‘signified’ and ‘statement’ that appear within it are obviously not numbers, and his horizontal bar (an arbitrarily chosen symbol) does not denote the division of two numbers. Therefore, his ‘calculations’ are pure fantasies.28 Nevertheless, two pages later, Lacan returns to the same theme:

  No doubt Claude Lévi-Strauss, in his commentary on Mauss, wished to recognize in it the effect of a zero symbol. But it seems to me that what we are dealing with here is rather the signifier of the lack of this zero symbol. That is why, at the risk of incurring a certain amount of opprobrium, I have indicated to what point I have pushed the distortion of the mathematical algorithm in my use of it: the symbol √ –1, which is still written as ‘i’ in the theory of complex numbers, is obviously justified only because it makes no claim to any automatism in its later use.

  ...

  Thus the erectile organ comes to symbolize the place of jouissance, not in itself, or even in the form of an image, but as a part lacking in the desired image: that is why it is equivalent to the √ –1 of the signification produced above, of the jouissance that it restores by the coefficient of its statement to the function of lack of signifier (–1).

  (Lacan 1977b, pp. 318–20)

  It is, we confess, distressing to see our erectile organ equated to √ –1. This reminds us of Woody Allen, who, in Sleeper, objects to the reprogramming of his brain: ‘You can’t touch my brain, it’s my second-favourite organ!’

  Mathematical Logic

  In some of his texts, Lacan does less violence to mathematics. For example, in the quote below, he mentions two fundamental problems in the philosophy of mathematics: the nature of mathematical objects, in particular of the natural numbers (1, 2, 3, ...), and the validity of reasoning by ‘mathematical induction’ (if a property is true for the number 1 and if one can show that its truth for the number n implies its truth for the number n+1, then one can deduce that the property is true for all natural numbers).

  After fifteen years I have taught my pupils to count at most up to five which is difficult (four is easier) and they have understood that much. But for tonight permit me to stay at two. Of course, what we are dealing with here is the question of the integer, and the question of integers is not a simple one as I think many people here know. It is only necessary to have, for instance, a certain number of sets and a one-to-one correspondence. It is true for example that there are exactly as many people sitting in this room as there are seats. But it is necessary to have a collection composed of integers to constitute an integer, or what is called a natural number. It is, of course, in part natural but only in the sense that we do not understand why it exists. Counting is not an empirical fact and it is impossible to deduce the act of counting from empirical data alone. Hume tried but Frege demonstrated perfectly the ineptitude of the attempt. The real difficulty lies in the fact that every integer is in itself a unit. If I take two as a unit, things are very enjoyable, men and women for instance – love plus unity! But after a while it is finished, after these two there is nobody, perhaps a child, but that is another level and to generate three is another affair. When you try to read the theories of mathematicians regarding numbers you find the formula ‘n plus 1’ (n+1) as the basis of all the theories.

  (Lacan 1970, pp. 190–1)

  So far, this is not too bad: those who already know the subject can recognize the vague allusions to classic debates (Hume/Frege, mathematical induction) and separate them from some rather questionable statements (for example, what does it mean to say ‘The real difficulty lies in the fact that every integer is in itself a unit’?). But from here on, Lacan’s reasoning becomes increasingly obscure:

  It is this question of the ‘one more’ that is the key to the genesis of numbers and instead of this unifying unity that constitutes two in the first case I propose that you consider the real numerical genesis of two.

  It is necessary that this two constitute the first integer which is not yet born as a number before the two appears. You have made this possible because the two is here to grant existence to the first one: put two in the place of one and consequently in the place of the two you see three appear. What we have here is something which I can call the mark. You already have something which is marked or something which is not marked. It is with the first mark that we have the status of the thing. It is exactly in this fashion that Frege explains the genesis of the number; the class which is characterized by no elements is the first class; you have one at the place of zero and afterward it is easy to understand how the place of one becomes the second place which makes place for two, three, and so on.29

  (Lacan 1970, p. 191, italics in the original)

  And it is at this moment of obscurity that Lacan introduces, without explanation, the alleged link with psychoanalysis:

  The question of the two is for us the question of the subject, and here we reach a fact of psychoanalytical experience in as much as the two does not complete the one to make two, but must repeat the one to permit the one to exist. This first repetition is the only one necessary to explain the genesis of the number, and only one repetition is necessary to constitute the status of the subject. The unconscious subject is something that tends to repeat itself, but only one such repetition is necessary to constitute it. However, let us look more precisely at what is necessary for the second to repeat the first in order that we may have a repetition. This question cannot be answered too quickly. If you answer too quickly, you will answer that it is necessary that they are the same. In this case the principle of the two would be that of twins – and why not triplets or quintuplets? In my day we used to teach children that they must not add, for instance, microphones with dictionaries; but this is absolutely absurd, because we would not have addition if we were not able to add microphones with dictionaries or as Lewis Carroll says, cabbages with kings. The sameness is not in things but in the mark which makes it possible to add things with no consideration as to their differences. The mark has the effect of rubbing out the difference, and this is the key to what happens to the subject, the unconscious subject in the repetition; because you know that this subject repeats something peculiarly significant, the subject is here, for instance, in this obscure thing that we call in some cases trauma, or exquisite pleasure.

  (Lacan 1970, pp. 191–2, italics in the original)

  Thereafter, Lacan tries to link mathematical logic and linguistics:

  I have only considered the beginning of the series of the integers, because it is an intermediary point between language and reality. Language is constituted by the same sort of unitary traits that I have used to explain the one and the one more. But this trait in language is not identical with the unitary trait, since in language we have a collection of differential traits. In other words, we can say that language is constituted by a set of signifiers – for example, ba, ta, pa, etc., etc. – a set which is finite. Each signifier is able to support the same process with regard to the subject, and it is very probable that the process of the integers is only a special case of this relation between signifiers. The definition of this collection of signifiers is that they constitute what I call the
Other. The difference afforded by the existence of language is that each signifier (contrary to the unitary trait of the integer number) is, in most cases, not identical with itself – precisely because we have a collection of signifiers, and in this collection one signifier may or may not designate itself. This is well known and is the principle of Russell’s paradox. If you take the set of all elements which are not members of themselves,

  x ∉ x

  the set that you constitute with such elements leads you to a paradox which, as you know, leads to a contradiction.30 In simple terms, this only means that in a universe of discourse nothing contains everything,31 and here you find again the gap that constitutes the subject. The subject is the introduction of a loss in reality, yet nothing can introduce that, since by status reality is as full as possible. The notion of a loss is the effect afforded by the instance of the trait which is what, with the intervention of the letter you determine, places – say a1 a2 a3 – and the places are spaces, for a lack.

  (Lacan 1970, p. 193)

  First, from the moment that Lacan claims to speak ‘in simple terms’, everything becomes obscure. Second – and most important – no argument is given to link these paradoxes belonging to the foundations of mathematics with ‘the gap that constitutes the subject’ in psychoanalysis. Might Lacan be trying to impress his audience with a superficial erudition?

  Overall, this text illustrates perfectly the second and third abuses on our list: Lacan shows off, to non-experts, his knowledge in mathematical logic; but his account is neither original nor pedagogical from a mathematical point of view, and the link with psychoanalysis is not supported by any argument.32

  In other texts, even the supposedly ‘mathematical’ content is meaningless. For example, in an article written in 1972, Lacan states his famous maxim – ‘there is no sexual relation’ – and translates this obvious truth in his famous ‘formulae of sexuation’33: