The theme of the subject divided between knowledge and truth is raised throughout the seminar. Lacan responds to the alternative between the mathematical model and metaphor by stating that "topology is not a metaphor, but a rigorous montage with the objet a." Thus the use of four mathemes: the disk with a hole, the Moebius strip, the torus and the Klein bottle. "The hole of the lack of the objet a would be located at the intersection of the fields of truth and knowledge": such is the conribution of psychoanalysis. It can therefore question science as to the truth whose contingency is missed or forgotten; the same happens with religion. Lacan both splits and unites his audience in two categories: "those who use my word for analytic purposes," and "those who prove that it can be followed in all its coherence and rigor, that it fits in a structure valid even outside its present practice." He also distinguishes between the analyst who at the moment of knowledge is divided (and he knows it), and the status of the subject-supposed-to-know (the subject of science) who restores the prestige of méconnaissance by thinking that he is uniting knowledge and subject.
Lacan goes to the Graph of Desire and relates them to his topology. The objet a is situated on four sides:
1. the demand of the Other (objet a is feces)
2. the demand on the part of the Other (objet a is the breast)
3. desire on the part of the Other (objet a is the gaze)
4. desire of the Other (objet a is the voice)
In this perspective he gives an account of his lectures in the United States, organized by Roman Jakobson, notably "Of Structure as an Inmixing of an Otherness Prerequisite to Any Subject Whatever," at Johns Hopkins University. Michel Foucault talks about Velasquez's Las Meninas. His address allows Lacan to conjure his theory of the painting as "a trap for the gaze," a gaze in which what falls is objet a. The little girl is the slit in the perspective and the vanishing point, the hidden center of the painting, and "in this gap, béance where there is nothing to see, it is impossible to recognize the structure of the objets a: underneath the Infanta's dresses, 'it looks at me,' while the eye is made not to see..." Georges Bataille's Histoire de l'oeil is quoted as a text that establishes a connection among all the objets a in their rapport to the feminine sexual organ. Therefore, the phallus is the sign that occupies the place of this gap, the impossible or untenable real. This entails a reshaping of the unconscious around language and the gaze (excluded by Freud). Lacan goes back to to the Freudian dimension of desire and of the subject whose foundation is castration. The vagina, the feminine sexual organ, becomes the objet a, which fascinates and leads to ruin unless there is the screen of the phallus, even under the form of - Φ. In the end, the penis, as a manifestation that is seen, hardly hides the presence of an objet a that would be an enigmatic - a.
The gaze, it should be noted, is not found on the side of the subject, but on that of the object. "It marks the point in the object (the picture) from which the viewing subject is already gazed at" (Slavoj Zizek). The gaze is a spot in the picture, which does not warrant the presence of the subject and by blurring its visibility, introduces a split in the rapport between the object and the subject: the latter cannot see the picture at the point from which it is gazing at him. Zizek brings out Psycho, where Norman Bates' house is rendered uncanny because Hitchcok's viewpoint switches from the house coming closer (as seen by the approaching woman) to the same woman coming closer (as seen from the house), giving the anxious impression that the house is gazing at her.

Lacan stresses the importance of the signifying structure in fantasy. He takes as his starting point the matheme $, which is the logical articulation of fantasy. The matheme was already introduced in Les formations de l'inconscient, in the graphs of desire, and was later developed in 1960 in "The subversion of the subject and the dialectic of desire in the Freudian unconscious" (Écrits: A Selection) as the first topology of the subject.
$ represents the division of the subject barred by the signifier that constitutes him. The sign <> enunciates the relation either of inclusion/implication, or of exclusion between the two terms. It's a binary system where the verb as such disappears to leave room for the algebraic sign of a pure relation. Definitions of objet a will vary over the years; to understand it here, one should go back to the part object of La relation d'objet et les structures freudiennes, and then address its analysis in L'angoisse and L'objet de la psychanalyse. In 1960, however, Lacan mentions the fascination of the fantasy in which "the subject becomes the cut that makes shine in its inexpressible oscillation".
The objet a would be the primal object, forever lost, the remainder or the product, which cannot be assimilated because it is real, of the cut operated by the primal signifier engendering the subject when it repeats itself in absolute difference (L'identification). "If a is the frame of the subject, this frame falls at the level of the most fundamental act of life, the act in which the subject as such is engendered, i.e the repetition of the signifier." This is the symbolic paternal mark or the phallic mark since there is no signifier of sexual difference: "The phallus alone is the sex-unity." The objet a creates a hole constantly filled, in the partial frives, by the different objets a, the breast, feces, the penis, the gaze or the voice, objects that are in themselves caught in imaginary substitutions. To understand fantasy, one should try to determine the logical status of objet a, which can only be accomplished by way of a topology dealing with gemetrical figures. Is objet a situated on the side of the drive or of desire of which it is the cause? Is it born out of the separation from the placenta as a part of the body proper or from the division from oneself from the signifier, the cost that the speaking being has to pay to become a subject? Is there really an alternative? Lacan talks of a surface where "desire and reality" are "the right and the wrong sides"; however, the passage from one side to the other is unnoticeable, as if there were only one side, because "the relation of texture does not entail any break." Might the fantasy allow oneself to go from the drive to desire and from desire to the drive, to link them or to disjoint them?
Lacan oscillates between exaltation and bouts of anxiety: "The logic of fantasy is the most fundamental principle of any logic that deals with formalizing defiles," and at the same time defers his presentation of "alienation in terms logical calculation" because its formulation is not yet ready. The reason might have been that "truth is related to desire," which "creates difficulties for handling it like logicians do." His aim is to define "a logic that is not a logic, an entirely new logic that I have not named yet, for it needs to be instituted first." Using the character of Diotima from The Symposium, he mentions academic Penia (the lack) before psychoanalytic Poros (male resource) and wonders "up to what point, between the two, he could let the obscurity go."
The seminar shifts its course toward the search for a logic of the subject around the Cartesian cogito, then toward "the sexual act," questioning "the impossible subjectivization of sex," and of jouissance.
The multiple transformations of Descartes' cogito ergo sum (either I think or I am"; "either I don't think or I am not"; "I am where I don't think," or "I think where I am not") end with a play of words: Cogito ergo es. The Latin es (you are) marks the fundamental dependency on the Other and raises the problem of the passage from objet a to the Other or from the Other to objet a. Applied to desire, "I desire you" means "I implicate you in my fundamental fantasy" as objet a. Applied to love, "You are not, therefore I am not"; "You are nothing but what I am"; "You are the nothing that I am." Now, in German, Es is the id, defined as the "non-I," the impersonal id, is it the reservoir of drives? Is it the cauldron (with a hole in it) of Freud's witches? Or is it an aggregate of signifiers?
Lacan elaborates on the notion of "unbeing," désêtre, which would become the mark of the end of analysis. He elaborates on puns: the unconscious desire is "pure desire," dés-être like dés-espoir, despair, is an irpas, from the Latin ire, to go and the negation pas, not, which is an impassé, something that had not gone through, linked to the desire of the Other, but also an impasse, a dead end, due to repetition. The interpretation does not entail any solution, issue, to the interpreted desire because there is no solution, issue, to the unconscious desire that "will always remain a désirpas (desirenot)." For lacan, is this knowledge of the truth of the unconscious desire really the solution, issue to be offered in analysis, the solution to the unfulfilled desire" of hysteria, to the "prevented desire" of phobia, or to the "impossible desire" of obsession? By itself, the objet a upholds "the truth of alienation"; to discover this truth is to discover that "there is no universe of discourse" because something real (something impossible because not symbolizable) eludes it.
"The big secret of psychoanalysis is that there is no sexual act," all there is is sexuality, a very different thing. The act has a specific definition: it provokes a signifying doubling that allows for an insertion of the subject in a chain in which he inscribes himself. Or it raises the institution of the signifier. But there is no signifier of sexual difference and none of the feminine sex either. Between man and woman "there is this third object, objet a, whose always sliding function of substitution makes it impossible to keep them opposed in whatever eternal essence. It is impossible then to posit a subject inscribing himself as sexed in the act of conjunction to the subject of what is called the opposite sex." At the symbolic level, "there is no sexual rapport": there is merely (a + 1) and (a - 1), where a term marks the difference as a plus or a a minus. The phallus is "the sex-unity": the 1 symbolizes the incommensurable.
Lacan uses Marx's analyses of use value and exchange value, and starts his theory of "man-he" (l'homme-il). The "man-he" is also the man-standard and the man-stallion (l'homme-étalon), as well as the bull and the poor bearer of the symbol of sex, doomed to symbolic castration. He does not know how to live since there is no Other to guarantee him, not even if he were God, marked as he is by castration. The only safeguard is the construction of a protective society based on masculine homosexuality. The Father of Freud's primal horde, because he supposedly jouis all women, sees "his jouissance killed." Then, if the almighty phallus circulates, it is due to women. "Woman represents the phallus as an exchange value among men; and, if the power of the penis bears the mark of castration, it is because fictitiously she becomes what is enjoyed, ce dont on jouit, and circulates as an object of jouissance: she is the locus of transference of this jouissance value" represented by the phallus. Through her identification to the use value embodied in the phallus, woman transforms herself into an object-good. Yet, she does not lack resouirces, such as masquerade, to act as "man-she," l'homme-elle. "She is inexpugnable as a woman precisely outside the system of the sexual act," or "she has a different use of her own jouissance outside this ideology." Thus, Lacan establishes "the radical heterogeneity" of the jouissance of the two sexes whose rapport could only be problematic. He begins here a reflection that will lead him to Encore, the twentieth seminar.

Since La logique du fantasme, where he states that there is not "sexual act," Lacan questions the difference between the act, l'acte and a mere action, agir. To make love would be an action, un agir, and to get married an act, un acte, because there is a commitment and a recognition, which entail repetition and the inscription in the Other. The signifier will appear soon: the absence of contradiction between Saint John's "In the beginning was the Word," and Goethe's "In the beginning was the action." Lacan then asserts "the irreducibility of the sexual act to any truthful relation." Since love is itself purely narcissistic, a social pact is what remains of a possible rapport between the sexes.
As to the different types of acts in psychoanalysis, there is the founding act: before, the effects of the unconscious existed, but nobody knew that they existed. There is the entrance into analysis and the fact of becoming an analyst, which are decisions and commitments. On the side of the analysand, there are slips and failures, which lead Lacan to give an Éloge de la connerie, Praise of Folly. In analysis it is almost impossible to answer simply to the injunction "render unto truth the things that are truth's and unto folly the things that are folly's," because the two overlap and then one finds "the folly of truth even more often than the truth of folly." The passage à l'acte and the "acting out" are activities that, although they fill a distressing hole, reproduce the past instead of remembering it in words. On the side of the analyst, "outside the manipulation of transference, there is no psychoanalytic act." In order for the analysand to move to the function of analyst, the latter - while pretending to be the upholder of the subject-supposed-to-know - must accept being "reduced to the function of cause of a process in which the subject-supposed-to-know is undone." Moreover, in the end the analyst must accept to be "nothing more than a waste of the operation represented by the objet a," which will produce an effect of truth. The position of the analyst is untenable, and this is why he opposes "the most violent misconstruction, méconnaissance, as to the analytic act itself." Besides, the analysand who experiences désêtre discovers, when becoming an analyst, that he is forced to restore for another the subject-supposed-to-know. The transmission would thus be compleed, very different from the passe itself. The psychoanalytic act, a "setting into act of the subject" and a "setting into act of the unconscious," is like a tragedy where the hero falls in the end as a piece of trash.
"In the beginning of psychoanalysis is transference," without any intersubjectivity, because between the two partners the subject-supposed-to-know acts as a third, as "the pivot from where everything that goes on in transference is articulated." This pivot is the signifier introduced in the discourse instituted by it, a formation as though detached from the analysand, which has nothing to do with the analyst's person. It is "a chain of letters that leads the not-known to frame knowledge," which concerns desire. The Graph of Desire still guides the analysis but an identity is asserted between the matheme of the subject-supposed-to-know and the agalma of Plato's The Symposium, which presents "the pure angle of the subject as the free rapport to the signifier, a signifier from which both the desire of knowledge and the desire of the Other are isolated."
Lacan wants to establish, as to the passage from the analysand to the analyst, "an eaquation whose constant is the agalma" (this term being a sort of compromise between objet a and the phallus). Once "the desire that, in its functionning, uphelds the analysand has been resolved, the analysand no longer wants to remove the possibility of such desire, the remainder which, insofar as it determines his division, makes him fall from his fantasy and destitutes him as subject." Lacan interprets the depressive position often noticed as the end of the analysis in terms of désêtre and "subjective destitution. "The subject sees its assurance sink, a self-assurance that comes from the fantasy in which everybody's opening onto the real is constituted." The subject realizes that the grasp of desire is nothing other than that of a désêtre. "In this désêtre what is unveiled is the nonessential nature of the subject-supposed-to-know; the analyst-to-be is dedicated to the agalma of the essence of desire, even if it means that the analyst-to-be has to be reduced to an ordinary signifier, since the subject is the signifier of the pure signifying relation." Does going through the fantasy, then, mean going toward the drive or toward a confrontation with the signifier? Thus Lacan answers: "The being of desire meets the being of knowledge to be reborn from their knot in a strip formed by the only side on which only one lack is inscribed, that which upholds the agalma." The agalma becomes the signifier of the bar that is put on the Other (A); the gap of (- Φ) opens in the Other; and the (a) falls from the Other.
Slavoj Zizek argues that "here we find the inescapable deadlock that defines the position of the loved one: the other sees something in me and wants something from me, but I cannot give him what I do not possess - or as Lacan puts it, there is no rapport between what the loved one possesses and what the loving one lacks. The only way for the loved one to escape this deadlock is to stretch out his hand toward the loving one and to return love, that is to exchange, in a metaphorical gesture, his status as the loved one for the status of the loving one. This reversal designates the point of subjectivization: the object of love changes into the subject the moment it answers the call of love. And it is only by way of this reversal that a genuine love emerges: I am truly in love not when I am simply fascinated by the agalma in the other, but when I experience the other, the object of love, as frail and lost, as lacking 'it', and my love none the less survives this loss."

Lacan takes a stand in the crisis of the university that follows May 1968: "If psychoanalysis cannot be articulated as a knowledge and taught as such, it has no place in Academia, where it is only a matter of knowledge." He rejects nonconceptualization: structure is the real. Dealing with the passage from objet a to the Other and from the Other to objet a, Lacan analyzes and combines Pascal, Marx and the logic of the link between l, the unbroken line, the trait unitaire of L'identification and a as follows:

To that, he adds questions on feminine jouissance (is it the place of the Other or of the Thing?), on the nullibiquité, non-ubiquitousness, of the phallus that testifies that jouissance is real but cannot be symbolized, on the Phallus as a symbol that is lacking or outside system, and the repetition of the Graphs of Desire.
Marx invented surplus-value, plus-value, and he, Lacan, invented the objet a. He asserts that he is going to construct the plus-de-jouir so as to isolate the objet a, he will do so by homology with surplus-value. In the matheme of fantasy, $<>a, "the being of a is the plus-de-jouir, surplus-jouissance.* At the level of the enunciation, perversion reveals "surplus-jouissance in its bare form." The rapport between surplus-jouissance and surplus-value is the function of the objet a. The perverse has given to God his true plenitude by giving a back to the Other. Hence, a is in A (the small other is in the big Other); however, a makes a hole in A. Jouissance is excluded, the Other is the place where it is known, a is the effect of fall that results from it. So, after going from a to A, one must go from A to a.
"I mainly talk about a dead God, maybe in order to better free myself from my relation to a dead Freud." Yet, in Le Pari, Pascal raises the question of the existence of God. The only true question is that of the subject: Does I exist? Do I exist? "The nothing that life is," which is at stake for Pascal, is the surplus-jouissance. The assumption of the loss creates the gap, béance, between the body and its jouissance: such is the effect of the objet a, the lost object, in the field of the Other. For Pascal, the central point is "the infinite nothing"; the only salvation is grace, for God's mercy is bigger than His justice. Grace allows proximity to the desire of the Other in its various forms: "I ask myself what you want," then "I ask you what you want," which leads to "Thy Will be Done!" However, this sentence is uttered to a faceless Other. God's will, for not being our will, comes to lack; then, for lack of God, we are left with the Father as dead, the Father as a name (the pivot of discourse) and as the rapport of jouissance to castration. "The Name-of-the-Father is a rift that remains wide open in my discourse, it is only known through an act of faith: there is no Incarnation in the place of the Other."
Slavoj Zizek aptly describes surplus-jouissance (lacanian ink 15) as follows:
"So in the case of the caffeine-free diet Coke, we drink the Nothingness itself, the pure semblance of a property that is effectively merely an envelope of a void. This example makes palpable the inherent link between three notions: that of the Marxist surplus-value, that of the Lacanian objet a as surplus-jouissance, and the paradox of the superego, perceived long ago by Freud: the more you drink Coke, the more you are thirsty; the more profit you have , the more you want; the more you obey the supergo command, the more you are guilty. In all three cases, the logic of balanced exchange is perturbed in favor of an excessive logic of "the more you give, the more you owe (or the consumerist version "the more you buy, the more you have to spend"), of the paradox which is the very opposite of the paradox of love where, as Juliet put in her immortal words to Romeo, 'the more I give, the more I have'.
The key to this perturbation is the surplus-jouissance, the objet a which exists (or rather insists) in a kind of curved space in which, the more you approach it, the more it eludes your grasp (or, the more you possess it, the greater the lack).
Perhaps, sexual difference enters here in an unexpected way: the reason why the supergo is stronger in man is that it is man, not woman, who is intensely related to this excess of the surplus-jouissance over the pacifying functionning of the symbolic Law. In terms of the paternal function, the opposition between the pacifying symbolic Law and the excessive supergo injunction is the one between the Name-of-the-Father (the paternal symbolic authority) and the "primordial father," allowed to enjoy all women. This rapist "primordial father" is a male (obsessional), not feminine (hysterical), fantasy: it is man who is able to endure his integration into the symbolic order only when this integration is sustained by some hidden reference to the fantasy of the unbridled excessive jouissance embodied in the unconditional supergo injunction to enjoy, jouir to go to the extreme, to trangress and force constantly the limit. It is man in whom the integration into the symbolic order is sustained by the superego exception."
* Bruce Fink notes that the translation of plus-de-jouir rendered in Television (New York: Norton, 1989) as "over-coming" is deficient. Since plus-de-jouir is based on plus-value (Marx's surplus value), it means a surplus, extra or supplemental jouissance: the plus should be understood in the sense of Encore, More. He stresses that "the more sensual sense of being 'overcome' with or 'overwhelmed' by pleasure is related to the Other jouissance.

Lacan identifies four viables types of social bond which regulate intersubjective relations. Articulations of the symbolic network, the Four Discourses get structured throughout dramatic reflection: plus-de-jouir and jouisance; the master and the slave; Marx; knowledge, truth and jouissance; the Father of Totem and Taboo who is all love - or all jouissance - and whose murder generates the love of the Dead Father, a father to whom Lacan opposes both the Father presiding over the first idealization - the one deserving love - and the Father who enters the discourse of the Master and is thereby castrated ab initio. For Lacan "the death of the father is the key to supreme jouissance, later identified with the mother as aim of incest." Psychoanalyis "is not constructed on the proposition 'to sleep with the mother' but on the death of the father as primal jouissance." The real father is not the father of biological reality, be he who upholds "the Real as impossible." From the Oedipus complex Lacan only saves the paternal metaphor and the Name-of-the-Father which "is positioned where knowledge acts as truth. Psychoanalysis consolidates the law."
The novelty in this seminar is the return of the hysteric, with Dora and la Belle Bouche erre - the Beautiful Mouth wanders - an allusion to the dream of the beautiful butcher's wife analyzed by Freud and carried on in "The direction of the treatment and the principles of power" (Écrits: A Selection). Three questions: the rapport between jouissance and the desire for unfulfilled desire; the hysteric who makes man - fait l'homme or the Master - she constructs him as "a man prompted by the desire to know"; a new conception of the cure as a "hystericizaton of dicourse," which the analyst introduces at the structural level. This leaves untouched hysteria as attributed to woman - the only discourse where sexual difference comes openly into play. Castration is "the deprivation of woman," insofar as "she would fulfill herself in the smallest signifier." Woman is absent from the field of the signifier.
As to the mathemes "a fundamental starting relation" functions as a postulate:

Discourse of the Master:

Discourse of the University:

Discourse of the Hysteric:

Discourse of the Analyst:

In any case, this algebra is concerned with the positions which are fixed:

"Radiophonie" (Autres écrits) is an interview recorded while L'envers... is taking place. In it Lacan declares that if "language is the condition of the unconscious, the unconscious is the condition of linguistics." Freud anticipates Saussure and the Prague Circle when he sticks to the patient's words, jokes, slips of the tongue, and brings to light the importance of condensation and displacement in the production of dreams. The unconscious is the fact "that the subject is not the one who knows what he says. Whoever articulates the unconscious says that it is either that or nothing." Linguistics has no hold on the unconscious since it leaves as a blank that which produces effects on the unconscious, the objet a, the focus of the analytic act - of any act. "Only the discourse that defines itself in terms given by psychoanalysis manifests the subject as other, whereas science, by making the subject a master, conceals him, so the desire that gives way to him bars the subject for me without remedy." There is only one myth in Lacan's discourse: the Freudian Oedipus complex. "In psychoanalysis, as well as in the unconscious, man knows nothing of woman, and woman nothing of man. The phallus epitomizes the point in myth where the sexual becomes the passion of the signifier." There is, however, no algebraic formula for the unconscious discourse: "...the unconscious is only the metaphorical term designating the knowledge only sustained when presented as impossible, so that it can conform by being real - real discourse."