During the final phase of his work extending from roughly 1964 to the end of his life, Lacan came to focus increasingly on the role of the Real in the triad composing the Symbolic, Imaginary and Real. This entailed understanding the formations of the unconscious– roughly symptoms –as attempts to recreate a harmony with the Real. As Lacan puts it,

Whenever we speak of cause… there is always something anti-conceptual, something indefinite. The phases of the moon are the cause of tides– we know this from experience, we know that the word cause is correctly used here. Or again, miasmas are the cause of fever– that doesn’t mean anything either, there is a hole, and something that oscillates in the interval. In short, there is a cause only in something that doesn’t work. Well! It is at this point that I am trying to make you see by approximation that the Freudian unconscious is situated at that point, where, between cause and that which it affects, there is always something wrong. The important thing is not that the unconscious determines neurosis– of that one Freud can quite happily, like Pontius Pilot, wash his hands… For what the unconscious does is to show us the gap through which neurosis recreates a harmony with a real– a real that may well not be determined” (Lacan, The Four Fundamental Concepts of Psycho-Analysis, 22).

The product of this attempt to re-create a harmony is of course the symptom. A symptom can be anything from the dramatic compulsion to repeatedly wash one’s hands to a simple slip of the tongue or dream. What is important is that the symptom is a response to a gap, lack, or absence which is characteristic of the Real.

Lacan gives two key formulations in characterizing the specific difference of the Real: On the one hand Lacan claims that the Real is that which always returns to its place. In the middle Lacan something qualifies as real if it has this quality of always returning to its place. Here, then, we might think of the movement of the planets. We can see how this characterization of the Real evolves over the course of his thought insofar as the symptom comes to increasingly be conceived as that which always returns to its place. In fact, we might even think of that final moment of analysis which involves identification with the symptom, as consisting in the eternal return of the symptom. While it is certainly true that the movement of the symptom produces an endless variety of symptomatic formations, the lack or absence around which these formations occur is always the same. Analysis thus consists in the mapping of this lack in its sheer nonsensical being (the movement from symptoms imbued with meaning and the sinthome as a pure process). This mapping of the real was what was at stake in Lacan’s discussions of zeros and ones in seminar II and the Seminar on the Purloined Letter. Part of traversing the fantasy consists in coming to stand before this fundamental void borne of castration covered over by fantasy.

On the other hand, Lacan characterizes the Real as what is impossible. It is with this formulation of the Real that we truly enter Lacan’s mature thought. Here the claim that the Real is the impossible should not be equated with idiotic common sense platitudes to the effect that man will never fly or pigs and donkeys cannot mate. As Lacan remarks in Seminar XX, impossibility is not to be understood as the opposite of possibility. Moreover, we ought not understand impossibility as being defined in terms of what people or a given culture believes is possible or impossible. Rather, the sort of impossibility Lacan has in mind are formal impossibilities like the sort that arise in logic or mathematics. Most often these formal impossibilities have to do with sets that do not include themselves like the set of all sets that do not include themselves. Such entities generate irresolvable paradoxes. Thus there is a special relationship between paradox and impossibility as it pertains to the Lacanian Real. The graphs of sexuation, along with the stances of hysteria (“am I a man or a woman”) and obsession (“am I alive or am I dead”) can be seen as variations of these set theoretical paradoxes.

Thus, for instance, the problem with the set of all sets that do not include themselves is that if the set of all sets that do not include themselves includes itself, then it is simultaneously a part of itself and its own whole. On the other hand, if the set of all sets does not include itself as a member of itself, then it would appear that it is not, in fact, the set of all sets that do not include themselves. The set of all sets us therefore a paradoxical notion. We encounter a very concrete example of this in the paradox of the Barber of Seville. If the Barber of Seville cuts everyone’s hair but those who cut there own hair, who cuts the Barber’s hair? According to this proposition the Barber cannot cut his own hair because he cuts everyone’s hair except those who cut their own hair. Yet someone else cannot cut the Barber’s hair because he cuts everyone’s hair except those who cut their own hair. The symbolic thus generates paradoxes, these paradoxes express formal impossibilities, and these formal impossibilities are what characterize the Real. Moreover, these impossibilities are intriguing in that they always return to their place. They always occur in the same place and thus mark a certain invariance in the symbolic which otherwise does not exist (cf. Seminar XX where Lacan remarks that language does not exist. We’ll see why in a moment). Lacan will define three fundamental fantasies revolving around these paradoxes: the non-existence of the sexual relation, the origin of our being as subjects, and the non-existence of Woman.

Although Lacan would not explicitly formulate this position until the sixties, he had already been moving in this direction with respect to his understanding of neurosis as early as the mid fifties. For Lacan a neurosis isn’t a pathological deviation from normality, but is rather a specific structure organizing a specific sort of symptom. In Seminar III, The Psychoses, Lacan articulates these structures in terms of specific questions. Thus, in the case of obsession, the structure is expressed by the question of whether I am alive or dead. Am I alive or am I dead? This question is only intelligible when thought in terms of how the obsessional relates to the master or the mythological primal father as articulated in the masculine side of the graph of sexuation. By contrast, hysteria is expressed in terms of the question of whether I am a man or a woman, which only becomes intelligible in terms of how feminine desire is organized in terms of identification with the desire of the Other. Now, what might not be evident at first glance is that both of these questions are expressions of specific impossibilities or the Real.

If we refer obsession to the masculine pole of the graphs of sexuation presented in Seminar XX, then we see that the paradox characterizing obsession has to do with the nature of castration and jouissance. According to Lacan, the claim that there is no being that is not submitted to the phallic function implies that there exists at least one being that is not subordinated to the phallic function. The male side of the graphs of sexuation can therefore be understood as involving the dialectic between the master and the slave. Recalling that obsession is closely related to the discourse of the master, when the obsessional asks whether he is alive or dead, we can translate this as a question as to whether all jouissance is ceded to the master (that mythological being that isn’t subordinated to the phallic function) or whether he, the slave, maintains some jouissance for himself. The question of whether I am alive or dead is thus a question about whether or not I must sacrifice all my jouissance, and is premised on the fantasy that I might eventually obtain total or complete jouissance by besting the master. What we have here is a formal impossibility characterizing the logic of the phallic function or castration. When I enter the symbolic order I am forced to sacrifice my jouissance (by observing the laws of kinship exchange), yet I still imagine (unconsciously) that somewhere there is some being that makes no such sacrifice. It is this that leads Lacan to characterize obsession as the desire for an impossible desire.

The case is similar with respect to hysteria, for in asking whether I am a man or a woman the formal impossibility encountered is the non-existence of The Woman. La Femme n’existe pas. I will not get into the details of Lacan’s claim that The Woman does not exist or that there is no unified category of Womanness capable of including all women. Most fundamentally Lacan’s claim is premised on the claim that there is no signifier for Woman or that the big Other is incomplete, thereby entailing that L’Autre n’existe pas. In this connection, Lacan, in seminar XX, equates Woman as such with the Other. In a moment we’ll see why the big Other cannot be said to exist. For the moment it’s important to note that this formal impossibility gives rise to the hysterical desire for an unsatisfied desire. For in his search to discover what Woman is, the hysteric’s desire is referred to the desire of he who desires Woman so as to discover what Woman is. In short, the unsatisfied desire desired by the hysteric isn’t simply her own unsatisfied desire, but the unsatisfied desire of an Other through which she might learn what Woman is and thereby become the semblance of Woman as such. It is in this regard that Lacan was led to claim that “The Woman is the symptom of man” (Seminar 22: RSI). For insofar as The Woman does not exist– meaning their is no consistent category of women capable of including all women –any woman (note the case) is but a semblance of Woman arrived at through identification with an-Other’s desire. The symptoms encountered in the treatment of hysterics and obsessionals are thus products of the specific form that these formal impossibilities take. Identifying with your symptom entails identifying with the specific formal impossibility or Real governing your subject position.

Beginning in 1964-65, with Seminar XII, Crucial Problems for Psychoanalysis, Lacan came to focus more seriously and intensively on the formal structure of these impossibilities characterizing the Real. Here what we have is a sort of Goedelian psychoanalysis, or a psychoanalysis founded on the formal impossibility of producing a closed system. Lacan expressed this impossibility with the aphorisms that “there is no universe of discourse” and “there is no metalanguage”. Although these two claims are closely related, they both express slightly different points. Thus when Lacan says “there is no universe of discourse” what he is essentially claiming is that discourse, language, can never form a closed totality, unity, or whole. This refers to Cantor’s paradox or the impossibility of forming a set of all sets insofar as the subsets of such a set would always be greater than the initial set by the power-set axiom. It can also be taken to refer to Russell’s paradox, insofar as the signifier has the feature of not belonging to itself and thus cannot form a totality.

Language is always constitutively incomplete. This isn’t simply a contingent accident such that we could finally rectify it by adding one more (encore!) signifier, but is an essential feature of any system or the mark of systematicity as such. From a psychoanalytic perspective this logic is seen most clearly in Totem and Taboo and Group Psychology and the Analysis of Ego. The consistency of the social system is only made through the subtraction or addition of a particular element, such that this element has the paradoxical status of simultaneously being a part of the system and outside the system. By contrast, when Lacan claims that there is no metalanguage, he is essentially claiming that there is no point of view one can adopt on language that would allow one to survey the whole from the outside. We are always already inside of language such that we are dialecticized by language. Thus we have two formal impossibilities or Reals characterizing the being of the symbolic.

Why, then, is there no metalanguage or is it impossible to take a point of view on language that would survey the totality from outside of language. There is, of course, the mathematico-logico demonstration that there is no whole of any formal system whatsoever. Consequently, if there is no whole of any formal system whatsoever, there can be no whole to survey. However, if there is no metalanguage, this is also by virtue of the fact that language is diacritical such that every “element” of language takes on its identity by virtue of its difference to the other elements. Insofar as each element only takes on its identity with respect to the other elements, no element is every simply present, but is already dispersed or “contaminated” by the other elements.

In Seminar XIV, The Logic of Fantasy, Lacan expressed this point with the aphorism that “the signifier cannot signify itself” which is equivalent to the matheme S1 —> S2. Every signifier only produces effects of meaning through its relation to other signifiers. Insofar as every signifier only produces effects of meaning through its differential relation to other signifiers, it follows that any attempt to formulate a metalanguage, to give a description of language from the “outside”, is already differentially included in language and thus a part of the very thing it seeks to describe. Thus we encounter another formal impossibility or Real, characterizing the impossibility of ever arriving at simple identity with oneself. As many post-structuralist thinkers have observed, identity is always already contaminated by difference by virtue of the diacritical play of language. This is just another way of saying S1/$ in the discourse of the master.

Zizek gives a terrific example of this principle in his magnum opus, For They Know Not What They Do: Enjoyment as a Political Factor. As Zizek remarks in the context of a discussion of Hegel’s distinction between boundary and limit,

National identification is an exemplary case of how an external border is reflected into an internal limit. Of course, the first step towards the identity of the nation is defined through differences from other nations, via an external border: if I identify myself as an Englishman, I distinguish myself from the French, Germans, Scots, Irish, and so on. However, in the next stage, the question is raised of who among the English are ‘the real English’, the paradigm of Englishness; who are the Englishmen who correspond in full to the notion of English…However, the final answer is of course that nobody is fully English, that every empirical Englishman contains something ‘non-English’– Englishness thus becomes an ‘internal limit’, an unattainable point which prevents empirical Englishmen from achieving full identity-with-themselves. (110)

Zizek’s point is that insofar as a nation is defined by a boundary, it’s identity can only be established in its difference to other nations. We can readily observe this phenomenon at work in personal identity as well; for as Lacan shows in the second cell of the graph of desire, my identity is only arrived at differentially in relation to others.

What we have here is thus the Real of identity or the way in which identity, properly speaking is impossible. Neither a nation nor a person is able to ever arrive at identity with itself insofar as it is differentially structured with respect to other nations and identities. Thus when Zizek claims that social antagonisms are always structured around an impossible Real, one way of understanding him would be to point to this formal impossibility of achieving identity. This impossible Real is not without consequences; for as a traumatic impossibility it turns the accomplishment of identity into an insistent demand. Despite the fact that identity is formally impossible insofar as it is always-already contaminated by difference, identity or respite from the play of diacritics is nonetheless demanded. Just as the Real of castration produces desire in the subject, the Real of impossible identity produces a sort of collective desire or fantasy. Identity must be accomplished even if impossible. In this respect, identity is not established through a totalization of the system in question, but is instead produced by having some contingent entity stand for the totality of entities. For instance, some particular type of Englishman– perhaps the working man –comes to stand for all Englishmen. This addition to the system is simultaneously a part of the system and outside it, and functions in such a way as to grant the system a semblance of identity with itself. It is notable that the unconscious functions in exactly this way. The function of the symptom is in fact that crazy addition that allows the otherwise untotalizable unconscious to hang together as a consistent whole. The symptom is always a +1 that stands in the place of the absence lying at the center of the unconscious structured like a language. It is therefore a S(-A-) or a signifier of the barred Other. Yet in functioning in this manner it simultaneously reveals and conceals the fact that the Other is barred. In this respect, the symptom recreates harmony with what would otherwise be infinite deferral. This is why the symptom can also be understood as a metaphor. By contrast, the operation of addition by which an untotalizeable system takes on the semblance of totality is itself subject to the diacritical movement which effaces identity and is therefore in danger of collapsing. For this reason the addition of one element is never enough. In addition to this +1 there must also be a subtraction (-1) which accounts for the failure of this totalization in advance. It is here that the logic of contamination emerges in connection to those fantasies of collective wholeness. For in every semblance of totality there is always a contamination or cries of a virus corrupting the identity of the system. This contamination is a strict corollary of the crazy identity established through the addition of that one extra signifier and functions to account for the failure of this signifier or the manner in which this signifier itself is effaced by the diacritical play of differences. The subtracted signifier or contaminant is always the immigrant, the ethnic other, women, liberals, etc. It is for this reason that those discourses most characterized by the call for identity (nationalistic discourses, individualistic discourses premised on the ego, etc.) are always most characterized by discussions of their Others or those supposed invaders contaminating the identity of the discourse. In fact, what the discourse encounters in these Others is its own disguised Real or the manner in which it always already differs from itself. In short, these Others are the objets a that the identity has had to sacrifice in order to constitute itself in the semblance of totality. For that which is repressed always returns.