The Seminar of Jacques Lacan, Book XXV: The Moment to Conclude

by Jacques Lacan

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Synopsis

Lacan's Seminar 25, delivered under the title "Le moment de conclure" (1977–1978), stages a sustained attempt to ground psychoanalytic theory — and ultimately the whole of clinical practice — in a rigorous material topology rather than in intuitive or linguistic metaphor alone. Opening with the provocation that psychoanalysis is not a science but an irrefutable practice of equivocation, Lacan gradually displaces the linguistic register that dominated earlier seminars in favour of a geometry of threads, fabrics, tori, and Borromean knots. The central argument is that the three registers — Real, Symbolic, Imaginary (RSI) — are not merely conceptual categories but topological objects whose mode of knotting materially determines what psychoanalytic experience is: the triadic Borromean structure, realised through a closed threefold chain, is posited as the only adequate formalisation of the clinic's triad. A substantial portion of the seminar is devoted, in collaboration with mathematician Pierre Soury, to working through the specific topological operations on the torus — reversal by holing versus reversal by cutting, longitudinal versus transversal sections, the construction of the Möbius strip and its triple variant — in order to demonstrate that the structural gap between the Imaginary and the Real is inscribed in the very fabric of these surfaces, not added to them by interpretation. Fantasy is identified with the torus within the Borromean structure, the end of analysis is reframed as recognising one's captivity to the sinthome, and the passé is tentatively reassigned to writing rather than speech, since writing alone stands a chance of approaching the Real. The seminar thus functions simultaneously as a technical topological workshop and as Lacan's final, most austere reflection on what it means to conclude — analytically, institutionally, and theoretically.

Distinctive contribution

Seminar 25 is the most rigorously technical of Lacan's late topological seminars and the one that most systematically attempts to substitute material geometry for linguistic metaphor as the operative language of psychoanalytic theory. Where earlier seminars (notably Seminars XX and XXII) introduced topology as a supplement to or illustration of concepts already established linguistically, Seminar 25 treats topological operations — toric reversal, the distinction between holing and cutting, Borromean knotting, the construction of the threefold Möbius strip — as themselves constitutive of psychoanalytic argument. The claim is not that topology illustrates the RSI triad but that the triad has no adequate formulation except in topological terms: the Borromean knot is not a metaphor for the clinic's structure, it is that structure. This produces an extraordinary density of mathematical-topological content, including extended exchanges with Pierre Soury and Jean-Claude Terrasson that bring professional mathematical reasoning directly into the seminar room.

A second distinctive contribution is the seminar's sustained meditation on writing as the privileged mode of access to the Real, as against speech. The analyst's intervention is characterised not as interpretation in a hermeneutic sense but as a cut — an act of writing that equivocates orthographically, producing an effect that the analysand's speech cannot intentionally achieve. This leads to a remarkable re-reading of the passé as a procedure better conducted in writing than in speech, and to a conception of psychoanalysis as a form of poetry rooted in fantasy, where science, history, and analytic discourse all share the same fantastical substrate. The convergence of topology, writing, and the Real — against any purely linguistic or dialectical account of the analytic process — is the seminar's most original and least widely assimilated theoretical contribution.

Main themes

• Topology as the operative language of psychoanalytic structure rather than mere illustration
• The Borromean knot as formalisation of the RSI triad in clinical practice
• Toric reversal and the structural distinction between holing and cutting
• Writing versus speech: writing as the privileged approach to the Real
• Fantasy as torus within the Borromean structure; the sinthome as what captivates the subject
• The non-existence of the sexual relationship as foundation of psychoanalysis
• Equivocation as the defining practice of analytic intervention
• The Möbius strip (single and triple) as model of the subject's one-sided topology
• Lalangue and the materiality of language beneath the signifier
• The end of analysis: concluding as recognising captivity rather than achieving liberation

Chapter outline

• Seminar I: Wednesday 15 November 1977 — p.2-10
• 13.12.77 — Toric Reversal and the Borromean Knot (Session II) — p.11-15
• Seminar 3: Wednesday 20 December 1977 — Writing, Metaphor, and the Analyst's Cut — p.16-23
• Two Lines of Numbers — Writing, the Real, and the Passé (Session continuation) — p.24-34
• Soury's Presentation — Arithmetic of Chains and the Borromean Threefold as Generator — p.35-42
• Seminar 6: Wednesday 14 February 1978 — Construction of the Threefold Borromean Knot from a Double Loop — p.45-49
• Seminar 7: Wednesday 21 February 1978 — Mirror-Image and Essential Difference — p.50-55
• Seminar 8: Wednesday 14 March 1978 — Holing versus Cutting: Two Modes of Toric Reversal — p.56-70
• Seminar 9: Tuesday 21 March 1978 — Toricity, Holing, and the Toric Mirror — p.76-84
• Seminar 10: Tuesday 11 April 1978 — The Sexual Non-Relation, Lalangue, and the Triple Möbius Strip — p.85-88
• Seminar 11: Tuesday 18 April 1978 — Möbius Strip, Half-Twists, and the Threefold Knot — p.89-98
• Seminar 12: Tuesday 9 May 1978 — Cut, Strip, Knot, and the Imaginary-Real Gap — p.99-102

Chapter summaries

Seminar I: Wednesday 15 November 1977 (p.2-10)

Lacan opens by announcing the seminar's title — 'Le moment de conclure' — and immediately establishes the epistemological status of psychoanalysis: it is not a science, not because it fails scientific criteria but precisely because, as Popper showed, it is irrefutable. Yet this irrefutability is not a defect; it is the condition of a practice of equivocation, of 'chit-chat' (bavardage) that spatters meaning rather than containing it in propositions. The opening gambit is thus to identify the analytic practice with speech at its most materially unstable — not the proposition but the word, and within the word, the equivocation that turns toward sex. The sexual relationship is formally declared an empty set: the signifier has hollowed out whatever biological complementarity might have existed, leaving only the fantasy of relation.

From this epistemological clearing, Lacan transitions to topology. Euclidean geometry is characterised as itself a fantasy — 'the straight line is manifestly a fantasy' — and topology is offered as what restores 'weaving' (tissage) to thought, replacing the fantasy of the line with the idea of neighbourhood, consistency, and body. The word, Lacan argues, makes the thing (fait la chose), but this is immediately re-written equivocally as 'fêle achose' — it splits the thing. Writing and equivocation are thereby introduced as the tools through which analysis works, and the analyst is positioned as a 'rhetor' who rectifies by equivocating. This first session thus lays out the seminar's entire programme: the inadequation of the Symbolic to the Real, the necessity of topology, and the primacy of writing over speech.

Key concepts: Borromean Knot, Topology, Real, Symbolic, Equivocation, Lalangue Notable examples: Karl Popper on irrefutability; Euclidean geometry as fantasy

13.12.77 — Toric Reversal and the Borromean Knot (Session II) (p.11-15)

This session, presented in a draft format, is the seminar's first extended foray into technical topology. Lacan works through a series of figures in which one torus of a Borromean knot is reversed, demonstrating that the result depends critically on which torus is privileged and in what direction the cut on the reversed torus is made. A longitudinal (concentric) cut dissolves the Borromean knot; a transversal (perpendicular) cut does not. This is not a merely formal observation: 'the privilege that is at stake is not something univocal.' The asymmetric sensitivity of the Borromean structure to where and how a rupture occurs is taken to model the differential consequences of structural breaks in the clinical field.

Lacan also proposes a 'six-fold Borromean structure' — a more complex arrangement of coupled tori — and raises the question of whether reversing one element of this arrangement produces the same results as in the simpler threefold case. The session thus establishes what will be a recurrent concern throughout the seminar: topology is not a neutral diagram of pre-established concepts but a site of genuine discovery, where Lacan acknowledges uncertainty and poses open questions. The dialogue with Soury is already operative, with Lacan registering objections to Soury's constructions while affirming the mathematical rigour they bring.

Key concepts: Borromean Knot, Toric Reversal, Topology, Real, Gap Notable examples: Six-fold Borromean structure; Concentric vs. perpendicular cuts on a reversed torus

Seminar 3: Wednesday 20 December 1977 — Writing, Metaphor, and the Analyst's Cut (p.16-23)

Lacan opens by distinguishing saying (dire) from speaking (parler): the analyser speaks and produces poetry; the analyst 'slices' (tranche) — makes a cut that carries the characteristics of writing rather than speech. Analytic intervention is not interpretation in a hermeneutic sense but an orthographic equivocation, a different writing that makes ring out something other than what is consciously intended. Neither what the analyser says nor what the analyst says is anything other than writing — a formulation that radically displaces the talking-cure's self-understanding. The gap between what one intends and what one says is precisely where the unconscious operates.

The session then asks how Lacan 'slipped' from the Borromean knot to imagining it composed of tori, and from there to toric reversal. This biographical-intellectual reflection serves to introduce the concept of metaphor as itself material: 'the stuff of metaphor is that which in thought constitutes matter.' The body, represented in fantasy, is Cartesian extension imagined — a gap-filling that the analyst fills with sexuality's colour. The question of knowledge is also raised here: knowledge as what guides, as instinct, as 'l'appensée' (thought-support). Fantasy is identified with the torus within the Borromean structure, and three coupled pairs are mapped — drive/inhibition, pleasure principle/unconscious, Real/fantasy — onto a 'six-fold torus.' The end of analysis is quietly redefined as recognising what one is captive of: the sinthome.

Key concepts: Fantasy, Borromean Knot, Topology, The Act, Metaphor, Drive Notable examples: The World of Mathematics (four-volume work); Descartes on extension

Two Lines of Numbers — Writing, the Real, and the Passé (Session continuation) (p.24-34)

This section advances Lacan's argument about the relationship between writing and the Real. The Real, he asserts, 'does not cease to be written' — it is only through writing as artifice that the Real appears at all. This is not a claim about linguistic representation but about the ontological status of writing: the Real is there through my way of writing, through the forcing that writing produces. Speech, by contrast, is linked to truth — and truth is precisely what cannot be said entirely. The passé is raised in this context: Lacan suggests that the passé might better be conducted in writing than in speech, since writing has a better chance of approaching the Real, though he is pessimistic ('these writings will not be read').

The unconscious is then characterised as containing writing — dreams, slips, jokes are all defined by the readable, by what is subsequently interpreted. The transference is reaffirmed as the subject-supposed-to-know, but now rewritten as the 'supposed-to-know-how-to-read-otherwise' (autrement), and 'otherwise' is equated with S(Ø) — the signifier of the barred Other — designating not an alternative reading but a lack, a different kind of lacking. The torus is introduced as the model of the living body and of sexuality, because unlike the sphere it introduces structural asymmetry between inside/outside and hole/consistency. The arithmetic of zero and one (hole/consistency) provides an analogue for chain topology: the Real is grounded in writing, and zero as hole and one as consistency are the arithmetic correlates of topological structure.

Key concepts: Real, Topology, Unconscious, Subject Supposed to Know, Transference, Letter Notable examples: Freud, The Interpretation of Dreams; S(Ø) as signifier of the barred Other

Soury's Presentation — Arithmetic of Chains and the Borromean Threefold as Generator (p.35-42)

This session records Pierre Soury's extended presentation on the systematisation of Borromean chains. Soury opens by acknowledging that systematisation depends on writing — that speech cannot adequately carry systematic content. He then develops an arithmetic analogy: within chain operations, the threefold Borromean chain plays the role of 'one' (the generative/exemplary element), while the twofold chain plays the role of 'zero' (a degenerate, neutral element). Citing Milnor's 'Link Groups,' Soury demonstrates that any Borromean chain of any number of elements can be obtained starting from the threefold chain, establishing it as the arithmetic generator of all chain structures.

Lacan intervenes to identify a conceptual gap in Soury's framework: the distinction between 'interlacing' and 'interlocking' remains unmastered, the categories bleed into each other. This intervention is characteristic of the seminar's methodology — Lacan uses collaborators' rigour to expose structural lacunae that carry theoretical weight. The degenerate case (twofold chain, zero) takes on special importance precisely within a systematic framework: like the zero in arithmetic, it only matters once one is preoccupied with systematisation. The implicit claim is that psychoanalytic theory needs exactly this kind of systematic formalism — not to reduce clinical reality to mathematics but to discover what structural relations the clinic depends on.

Key concepts: Borromean Knot, Topology, Real, Gap, Signifier Notable examples: Milnor, Link Groups; Arithmetic analogy: threefold chain as one, twofold chain as zero

Seminar 6: Wednesday 14 February 1978 — Construction of the Threefold Borromean Knot from a Double Loop (p.45-49)

Lacan begins from a practical construction problem: given a figure presented as a double loop, can one produce a threefold Borromean knot? He affirms that one can, but only on condition of making the structure circular — the Borromean chain of more than three elements does not hold unless closed into a ring. The circularity here is structural, not merely topological convenience: the ring that ends the chain is the same ring that inaugurates it, a figure of self-coincidence that the threefold case alone satisfies without additional artifice.

The explicit motivation for introducing the Borromean knot is then stated with unusual directness: 'I introduced it because it seemed to me that it had something to do with the clinic. I mean that the trio of Imaginary, Symbolic and Real seem to me to have a sense.' The threefold knotting — where each pair of rings is free but the triple is bound — directly mirrors the clinic's triad. The session demonstrates the construction through multiple figures, working through the drawing difficulties with characteristic candour ('I am extremely awkward in these drawings'). The circularity of the threefold Borromean knot is thus both a topological result and a model for the RSI triad's self-sustaining, mutually conditioning structure.

Key concepts: Borromean Knot, Topology, Imaginary, Symbolic, Real, Unconscious Notable examples: Double-loop starting configuration for threefold Borromean knot; Circularity as structural necessity

Seminar 7: Wednesday 21 February 1978 — Mirror-Image and Essential Difference (p.50-55)

This session turns on a dispute about whether two topological figures — one described as the mirror-image of the other — are identical or essentially different. Lacan maintains, against the view that the figures are 'the same turned over like a pancake,' that 'a figure placed in a mirror is not identical to the figure, to the original figure.' The mirror produces an essential difference, an inversion that is not trivially undone. Soury intervenes to enumerate the multiple kinds of inversions at stake (mirror-image inversion, wicker-work inversion, above/below exchange, front/back stitch exchange), showing that there is not 'just one inversion' but a family of them — and that they do not all commute.

This topological dispute does substantial theoretical work. The non-identity of a figure with its mirror image translates directly into the non-coincidence of the subject with its mirror representation — the Imaginary register's founding misrecognition. But here the claim is more rigorous: it is not that the subject 'identifies' with an image and thereby misrecognises itself, but that the topological operation of mirroring is itself an asymmetric, non-trivial transformation that produces a structurally distinct object. The session thus grounds the mirror stage's essential insight not in phenomenology but in topology.

Key concepts: Imaginary, Topology, Borromean Knot, Real, Gap Notable examples: Mirror-image inversion vs. wicker-work inversion (Soury's taxonomy)

Seminar 8: Wednesday 14 March 1978 — Holing versus Cutting: Two Modes of Toric Reversal (p.56-70)

Lacan announces that he has transformed Borromean chains 'not into tori but into toric fabrics' — it is now tori that carry the rings of string, meaning the topology of surfaces rather than of closed curves is at stake. He introduces, via Soury's work, a fundamental distinction between two modes of toric reversal: reversal by holing (introducing a hole from outside, through which one reverses the torus so that its core becomes its axis) and reversal by cutting (which also substitutes core and axis, but through a cut that additionally dissociates the coupling between the two faces and the inside/outside couple). At first Lacan is inclined to treat the two cases as homogenous, but Soury's insistence on their difference impresses him: 'I have great confidence in Soury.'

Soury then demonstrates the difference in detail using a colour-coded system (blue/red for core/axis, green/yellow for the two faces). Reversal by holing preserves the link between the two faces and inside/outside; reversal by cutting dissociates this link. This distinction carries theoretical weight for how topological transformations model psychoanalytic concepts: the coupling between inside/outside and the two faces of a surface is precisely what is at stake in the distinction between the subject's two faces, between demand and desire, or between the surface of the body and its inner lining. Cutting is strictly more powerful than holing — it contains holing as a special case while enabling additional reversals — and this asymmetry is not reducible to a difference of degree.

Key concepts: Topology, Borromean Knot, Real, Imaginary, Toric Reversal, Möbius Strip Notable examples: Core/axis vs. face/face colour-coded torus diagrams (Soury)

Seminar 9: Tuesday 21 March 1978 — Toricity, Holing, and the Toric Mirror (p.76-84)

Lacan opens by articulating a worry about 'the relationship between what can be called toricity and holing' — Soury has suggested there is no intrinsic relationship between the two, while Lacan suspects he may be operating with a confused concept of the torus. The session works through what holing enables: by introducing a hole, one can insert a hand into the torus and reverse it, converting the inside into the outside. But there is a second, more radical possibility: by pushing the entirety of the torus through the hole, one obtains a reversal effect that is topologically different from simple inversion.

Lacan introduces the concept of a 'toric mirror': what appears on the inside of the torus, when inverted, is that surface's mirror-image. This mirror-image is not a coincidence but a structural feature — defined by the fact that it is inside the torus rather than outside. The inversion of above/below that accompanies this toric mirroring 'complicates the affair,' introducing an asymmetry that is not merely apparent but structural. The session thus continues the seminar's broader argument that topological operations — reversal, holing, cutting — are not neutral transformations but produce qualitatively different structural relations, each of which corresponds to a distinct configuration in the clinic.

Key concepts: Topology, Real, Imaginary, Borromean Knot, Gap, Möbius Strip Notable examples: Toric knitting as holing demonstration (Soury); Toric mirror as defined by inside/outside inversion

Seminar 10: Tuesday 11 April 1978 — The Sexual Non-Relation, Lalangue, and the Triple Möbius Strip (p.85-88)

Opening from 'there is no sexual relationship,' Lacan now specifies the structure of this non-relation more precisely: sexual relationship exists only between neighbouring generations (parents/children), and this is precisely what the incest prohibition wards off. Knowledge is always related to 'l'asexe(ualité)' — asexuality — meaning that knowing how to deal with sexuality is always a kind of knowing-in-relation-to-a-prohibition. Lacan recalls that he once symbolised sexuality with a Möbius strip, and announces that he wants now to 'correct' this strip by tripling it.

The triple Möbius strip is introduced: unlike the simple Möbius strip (which has one face, front coinciding with back once), the triple strip has the property that front and back appear on the same face twice, making it 'bilateral' in a paradoxical sense. What is lost in the abstraction from fabric to formal strip is precisely the stuff, the woven material — which is why Lacan strives for 'a geometry of fabric, of thread, of stitching.' Analysis as a social fact is founded on lalangue — written in a single word to capture its pre-syntactic, material character — and free association is interrogated: does dreaming about a dream count as free association, or is Freud's own interpretation of dreams another instance of dreaming? This leads to a citation of Jean-Claude Milner's 'From syntax to interpretation,' which is taken as a significant but insufficient step toward the problem of how interpretation can be grounded without becoming yet another fantasy.

Key concepts: Möbius Strip, Lalangue, Real, Imaginary, Oedipus Complex, Unconscious Notable examples: Triple Möbius strip construction; Jean-Claude Milner, 'From syntax to interpretation'; Freud, The Interpretation of Dreams (critiqued)

Seminar 11: Tuesday 18 April 1978 — Möbius Strip, Half-Twists, and the Threefold Knot (p.89-98)

Lacan demonstrates that cutting a triple Möbius strip down the middle produces a threefold knot — a result he finds 'quite striking.' The Möbius strip is reprised in its canonical form (the image from the cover of Scilicet) and distinguished from the triple-twist variant. Jean-Claude Terrasson is introduced as a collaborator who contributes the precise vocabulary of 'half-twist' versus 'complete twist,' enabling a systematic comparison of Möbius strips with different numbers of half-twists.

The session then works through the relationship between the Möbius strip with a single complete twist and the torus: a single complete twist, drawn as a loop that passes behind and in front of the torus, is equivalent to what Terrasson calls a complete twist, and this equivalence is demonstrated through the torus's own geometry. Terrasson takes the floor to address the problem of paving the plane regularly with flattened Möbius strips — a combinatorial-geometric question that functions as preparatory groundwork for the broader question of whether a Borromean knot can be constructed from a threefold knot. The session thus operates as a technical workshop in which topology functions not as illustration but as the operative language for investigating structural relations.

Key concepts: Möbius Strip, Borromean Knot, Topology, Real, Letter Notable examples: Scilicet cover image of the Möbius strip; Terrasson on half-twists and plane-paving

Seminar 12: Tuesday 9 May 1978 — Cut, Strip, Knot, and the Imaginary-Real Gap (p.99-102)

Lacan returns to the result that a cut on a torus, redoubled into a strip, realises the threefold knot. The argument is careful: a cut alone is not enough to make a knot — there must be 'stuff,' a tube, a material substrate. The triple Möbius strip cannot lie flat on a torus; therefore cutting the tube directly produces something unexpected (a fourfold-folded figure) rather than the threefold knot. The threefold knot only appears when one cuts the triple Möbius strip down its middle, and this is what 'excuses' Lacan's earlier 'absurdity' of saying it was impossible to establish a knot on a torus.

The session concludes with a theoretical reflection: the requirement to 'imagine' how things behave implies the use of the Imaginary. Fabric is 'particularly linked to imagination' to the point that the support of fabric is what Lacan calls the Imaginary. But the structural gap between the Imaginary and the Real — the gap that constitutes inhibition — is what the triadic RSI structure inscribes. The topology of Möbius strips, tori, and Borromean knots is thus not merely a formal apparatus but the material articulation of what distinguishes representation from object, Imaginary from Real. Lacan closes by affirming that this triadic structure is intrinsic to psychoanalysis and to the distinction it makes — and must make — between representation and the object it can never fully capture.

Key concepts: Borromean Knot, Topology, Imaginary, Real, Symbolic, Gap Notable examples: Triple Möbius strip cut to produce threefold knot; Fabric as support of the Imaginary

Main interlocutors

• Pierre Soury
• Jean-Claude Terrasson
• Jean-Claude Milner
• Sigmund Freud, The Interpretation of Dreams
• Sigmund Freud, Beyond the Pleasure Principle
• Sigmund Freud, Jokes and their Relation to the Unconscious
• Milnor, Link Groups
• Cantor, Set Theory
• Descartes, René — Meditations / Principles
• Althusser, Louis
• Karl Popper
• Marx, Karl
• Jacques Lacan, Seminar XX
• Jacques Lacan, Seminar XXII
• Jacques Lacan, Seminar XXIV
• Jacques Lacan, Écrits
• Jacques Lacan, Seminar VII

Position in the corpus

Seminar 25 sits at the far end of Lacan's topological-Borromean period, the culmination of a trajectory that runs through Seminars XIX–XXIV. It presupposes familiarity with Seminar XX (Encore, 1972–73), where the non-existence of the sexual relationship and lalangue were first systematically developed, and with Seminar XXII (RSI, 1974–75), where the Borromean knot was first deployed as the structural model of the RSI triad. Readers should also have some acquaintance with Seminar XXIII (Le sinthome, 1975–76) and Seminar XXIV (1976–77), where the sinthome and the end of analysis were last elaborated before this final seminar. Without this preparation, the topological arguments of Seminar 25 — which presuppose rather than introduce the RSI framework — will be difficult to follow. The seminar also benefits from being read alongside Lacan's collaborator literature: Soury's mathematical contributions and the broader tradition of Lacanian topology (e.g., work by Pierre Skriabine, or Joël Dor's more accessible topological accounts) help contextualise the mathematical content.\n\nWithin the wider corpus, Seminar 25 occupies a singular position as the most technically uncompromising of the late seminars and the one most resistant to translation into discursive commentary. It has no close parallel in the Žižekian tradition (which tends to bypass the mathematical topology in favour of the dialectical-Hegelian register) and is more closely related to the work of mathematically literate Lacanians such as Alain Badiou (whose set-theoretic approach shares the commitment to formalisation) or the French clinical topology school. Readers coming from the linguistic Lacan of Seminar XI or the Écrits will find Seminar 25 a significant reorientation: the talking cure has here become, provisionally and provocatively, a writing cure.

Canonical concepts deployed

• Borromean Knot
• Topology
• Real
• Symbolic
• Imaginary
• Möbius Strip
• Toric Reversal
• Lalangue
• Letter
• Fantasy
• Gap
• Unconscious
• Subject Supposed to Know
• Transference
• Das Ding
• Drive
• Demand
• Desire
• Metaphor
• Surplus-jouissance