Review of Intellectual Impostures

by Henry Krips

[To appear in Metascience 9.3]

Sokal, Alan, and Jean Bricmont.
Intellectual Impostures: Postmodern Philosophers' Abuse of Science.
London: Profile Books, 1998.


In a series of recent publications, Alan Sokal has launched a series of stinging attacks against contemporary cultural studies. In Intellectual Impostures, for example, written together with Jean Bricmont, the authors (hereafter S&B) criticise the way in which French poststructuralist critics, such as Julia Kristeva, Jacques Lacan and Gilles Deleuze, have abused the scientific terminology to which, Sokal claims, they exhibit slavish adherence. Many authors, such as Andrew Ross and Stanley Aronowitz, have taken up the cudgels against S&B. But their replies often miss the mark either by arguing at too abstract a level against S&B's project as a whole or employing strategies and principles drawn from the self same body of texts which S&B criticize. In this article, by contrast, I reply to specific criticisms S&B direct against Lacan's use of topological concepts. By showing that S&B miss a perfectly reasonable, mathematically acceptable way of reading Lacan's appropriation of topology, I intend to raise more general doubts about S&B's erudition in connection with the works they criticise.

S&B's Double Standards

S&B claim that Lacan's use of scientific and specifically mathematical terminology does violence to its original meaning, that Lacan "use[s] scientific (or pseudo-scientific) terminology without bothering much about what the words actually mean (p. 4). S&B qualify this criticism by allowing that Lacan uses scientific terminology metaphorically, on the basis of structural analogies: "Everything is based...upon analogies between topology and psychoanalysis" (23, see too p.9). However, unlike scientists who appropriate technical vocabulary on the basis of analogies, Lacan offers "not the slightest conceptual or empirical justification" for such appropriations (4, and see too p.23): "If a biologist wanted to apply, in her research, elementary notions of mathematical topology...she would be asked to give some explanation. A vague analogy would not be taken very seriously by her colleagues".(4) S&B's criticism may be put in the form of a dilemma: either Lacan intends the technical terms he uses to be understood literally, in which case they are meaningless, or he uses them metaphorically, in which case their usage is unjustified.

S&B also claim that Lacan's adaptations of scientific concepts fail to have any coherent meaning in their own right. Thus Lacan not only does violence to the original meanings of the scientific terms he uses but also, having lifted them from their original context, fails to provide with any new coherent meanings: "Lacan uses quite a few words from the mathematical theory of compactness...he mixes them up arbitrarily and without the slightest regard for their meaning. His 'definition' of compactness is not just false: it is gibberish" (21). Harsh words, which we shall turn back upon S&B's own writings.

I address S&B's criticisms by first considering how in general terms metaphoric extensions are justified in science. To this end, I consider an example drawn from an elementary text book on optics by R.W. Ditchburn. Ditchburn concedes that the notions of "wave" and "particle" employed in modern optics violate the original meaning of these terms as developed for "things like waves on water, or moving particles" (Ditchburn 3). Nevertheless, he says, "we often find it convenient to 'translate' part of this theory [the mathematical theory of light] into....a wave picture of light" (2). And, he continues, because "most people think more readily in terms of words than in terms of equations....we...use the analogies as far as possible" (3). In short, the use of metaphors or analogies in optics, and by implication in science more generally, is justified by their heuristic or mnemonic value, their role in helping readers visualize and think about the processes described by the mathematical equations. In making these points, Ditchburn draws an analogy with the situation in meterology, where notions like "cold front" and "depression", themselves already metaphors, provide "a convenient way of summarizing certain aspects of the observations" charted on the isobaric charts (2). As he puts it, these notions help the meteorologist "think quickly and clearly about the meteorological situation" (2).

S&B require an additional justification for the metaphoric extension of concepts. He requires that such extensions must be "tested empirically" (p. 9). Lacan, he claims, fails to satisfy this requirement: "his analogies between psychoanalysis and mathematics are the most arbitrary imaginable, and he gives absolutely no empirical or conceptual justification for them".(34) But S&B's requirement is too strict. Many of the metaphoric extensions of terms in science fail to contribute new empirical content to the theory in which they are introduced. For example, the introduction of the terms "depression" and "cold front" does nothing for the empirical content of isobaric charts. Instead, as Ditchburn concedes, these terms have a purely heuristic value. Similar comments apply to the metaphoric usage of the terms "particle" and "wave" in optics, which add nothing to the theory's empirical content. In short, it seems that S&B areguilty of double standards: criticizing Lacan's theory for failing to live up to standards which science itself fails to meet.

Lacanian Topology

So far my reply to S&B has been at an abstract level. I now shift to a more concrete engagement with Lacan's text, and consider his metaphorical extension of topolgical terminology, in order to show that (a) his understanding of the terminology does not display what S&B call a "superficial erudition" and that (b) his analogical extensions do not produce "meaningless gibberish". On the contrary, I argue, by missing the creative use to which Lacan puts topological concepts, it is S&B who misunderstand and displays a superficial acquaintance with Lacan. Consider the following two sentences from Lacan:

Let us take the same bounded, 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 to the point of arrival or departure, to sketch it for you quickly -- it can be shown that it is equivalent to say that the sets of these openspaces always allow a subcovering of open spaces, constituting a finity [finitude], namely, that the series of elements constitutes a finite series.(22)
This extract attracts stern criticism from S&B: "In this sentence, Lacan gives an incorrect definition of open set and a meaningless definition of limit" (22). But this, S&B remark (in footnote 24 to p.22) "is a minor point compared to the overall confusion of the discourse". In short, Sokal find in Lacan nothing but confusion, nonsense, and elementary mistakes.

On closer inspection, however, Lacan's remarks are far from "meaningless", "incorrect" or "confused". Topological spaces are defined by mathematicians as spaces in which there exists sets of subspaces which are closed under the operations of union and intersection. That is, if we take the union or intersection of any two subspaces belonging to one such set then the result also belongs to the set. Even when the union or intersection of subspaces is "extended to infinity", to use Lacan's words, the result belongs to the set. Any such subspace is called an "open set". Metric spaces are special sorts of topological spaces for which there exists a distance function (a "metric") measuring the distance between pairs of points. The open sets in a metric space are defined (and here I use the terminology Lacan employs in the above extract) as subspaces which "exclud[e] their own limits". (Equivalently, a set is defined as "closed", that is as "not open", if it "contains all its limit points" - Patterson 30). Lacan also introduces the concept of "boundedness": a subspace in a metric space is said to be "bounded" if any two of its points are separated by a finite distance. A standard theorem in topology tells us that:

If X is a closed bounded subspace of Euclidian n-space, then every open covering {U} of X contains a finite open covering; that is, from the open sets {U} which cover X, there can be selected a finite number which still cover X.(59)
This theorem, proved originally by Borel, maintains that for any space embedded in a Euclidian n-space, any set of open subspaces which cover the whole space contains a finite subset of subspaces which also cover the whole space, or, in Lacan's terms: "the sets of these open [sub]spaces always allow a subcovering of open [sub]spaces, constituting a finity [finitude], namely, that the series of elements constitutes a finite series"(22). Thus, far from being meaningless, the above extract from Lacan contains a succinct statement of one of the standard theorems of topology, Borel's theorem. Borel's theorem can be restated even more economically by introducing a new concept, that of "compactness", which is introduced in such a way that it applies to topological not merely metric spaces: "We define a topological space T to be compact if any open covering of T contains a finite open covering" (59). Borel's theorem may then be restated succinctly as saying that "closed, bounded subspaces of Euclidian spaces are compact" (59).

What use does Lacan make of Borel's theorem? How does he extend it metaphorically from the domain of topology to psychoanlysis? His argument on this point is sketchy and far from clear. (Had S&B restricted their criticisms of Lacan to accusations of unclarity or incompleteness, his case would have been stronger, albeit less dramatic.) According to Lacan (who follows Freud on this point) the space of sexual relations is constituted in relation to a forbidden, and in that sense "excluded" or "limiting", object, namely the Mother. Thus, in a metaphorical sense, the space is closed (fermé), that is, it incorporates its own limit point, namely the (M)Other. Indeed, Lacan goes further, and claims that the space of sexual relations - also referred to as the space of Jouissance J - is bounded (borné), that is, any two points in it are physically accessible to each other in the sense of being a finite distance apart: "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" (19). (Unfortunately Lacan never explains the metrical concept which underlies the concept of boundedness relevant to the space J. This is a flaw in his presentation, but more by way of an incompleteness than an incoherence of the sort S&B envisage.) But if a space is closed and bounded (as well as embedded in an n-dimensional Euclidian space) then, according to Borel's theorem, it is also compact. This in turn, means that even though the space as a whole is closed, there exists a finite open cover, that is, the space can be covered by a finite number of open subspaces. If we follow Lacan's suggestion to transfer this theorem metaphorically to the space of sexual relations, then we are licensed to infer the following conclusion. Despite its own closure (that is, despite containing its own limit point in the shape of the Mother) if the space J of sexual relations is covered by a set of open subspaces then "in the end, we can count them one by one [un par un]" (p. 22). What could Lacan mean by this enigmatic claim? What metaphoric meaning, if any, does it assume in the context of Lacan's sustained metaphor that the space J is topological, and more specifically "closed", "bounded", and implicitly "embedded in an Euclidian n-space"?

Here we must turn to Lacan's theory of the constitution of the subject, in particular his theory of the mark - the unary signifier, the single stroke, the one (the un) (Lacan 1981, 141-142 and 256-257). His point is that in order for a thing to appear as something which can be counted in its own right, as a single, separate entity, there must already be two entities: the thing which is counted but also the "mark" by which it is counted, and hence too, a position from which it is counted - the position that in the Freudian architectonic corresponds to the ego ideal. Thus, if the subject is to emerge as an un, a subject distinct from others, then, according to Lacan, it must incorporate a mark, which interestingly may appear as a cut or stroke, such as the notch which the primitive hunter makes into the bone in order to count the number of times he gets his target (Lacan 1981, 256). As Lacan makes the point:

The subject himself is marked off by a single stroke, and first he marks himself as a tatoo, the first of the signifiers. When this signifier, this one, is established - the reckoning is one one, at the level of the reckoning. It is at this level, not of the one, but of the one one, at the level of the reckoning [that is, at the level of the one who counts] that the subject has to situate himself as such. In this respect, the two ones are already distinguished. Thus is marked the first split that makes the subject as such distinguish himself from the sign in relation to which, at first, he has been able to constitute himself as subject.(141)
Lacan's point, then, is that for subjects to emerge they must be "countable" in the sense of marked by what may appear literally as a cut (a cut which may, however, take on a more metaphorical identity as a splitting of the subject, a point which I cannot discuss here). Now think of the set of distinct sexually related subjects as associated with some set S of disjoint open subspaces in the space of sexual relations. Assume additionally that all sexual relations are between elements of such subspaces. (Informing the latter assumptions is a relational conception of the subject, that is, a view that subjects are constituted as termini of a system of "intersubjective relations", together with the Freudian view that subjects are ontogenetically sexual in nature.) It follows, then, that the space J is "countably covered" in the sense that it can be covered by subspaces belonging to S, subspaces which "in the end" (that is after a suitable selection) are countable in the sense that "we can count them one by one". And this property of having a countable cover is, of course, exactly the form of property which Borel's theorem establishes for closed bounded spaces (embedded in a Euclidian n-space).

Of course, the sense in which the space of sexual relations may be said to be "closed" and "bounded", or its subspaces "countable" differs from, indeed is a metaphoric extension, of the sense in which a metrical space is closed and bounded and finitely covered. But that does not affect the point Lacan makes here, which is that Borel's theorem comprises a way of organizing thoughts about a basic connection between, on the one hand, the "closure" of the field of sexual relations, meaning the fact that sexual relations are constituted in relation to an excluded (or "limiting") Other, and, on the other, the "countability" of human subjects, that is the fact that the constitution of subjectivity necessarily involves a cut or mark.


In sum, S&B have, it seems, failed to understand the substance or depth of Lacan's analogies with the field of mathematical topology, specifically he has failed to spot Lacan's references to Borel's theorem as they apply metaphorically to the space of sexual relations. S&B's failures might be forgivable were it not for his turning them into evidence for the "superficial erudition", indeed, "meaninglessness" of Lacan's work ("gibberish", he calls it), a superficiality for which Sokal suggest the most venial of motives: "Might [Lacan's] goal be to pass off as profound a rather banal philosophical or sociological observation, by dressing it up in fancy scientific jargon?" (p. 9).

The unfairness of S&B's criticisms tempts one to turn them back against the critic. One might speculate unflatteringly that S&B's motive for claiming familiarity with post-structuralist is "to pass off as profound" his own "rather banal philosophical or sociological observation, by dressing it up in fancy scientific jargon". Here S&B's own words in repudiation of Lacan return to haunt him. Other speculations concerning S&B's motive for attacking Lacan are equally tempting. For example, a remark by S&B upon Lacan's analogy between the turgid penis and the imaginary number -1, tempts one to invoke an unconscious motive of penis envy: "It is", S&B confess, "distressing to see our erectile organ equated to -1" (25). As fascinating as such speculations may be, I fear that they take us well beyond the bounds which reason, academic courtesy and common decency permit.


Ditchburn, R.W. 1959. Light. London: Blackie.

Lacan, Jacques. 1981. The Four Fundamental Concepts of Psychoanalysis. Trans. Jacques-Alain Miller. New York: Norton.

Patterson, E.M. 1969. Topology. New York: Oliver and Boyd.

Henry Krips
Departement of Communication
University of Pittsburgh
1117 Cathedral of Learning
Pittsburgh, PA 15260