chattopadhyay.indd S: Journal of the Circle for Lacanian Ideology Critique S13: 119-137 A r k a C h a t t o p a d h y a y L o g I C A L S p A C e I n L A C A n : F r o m p o e ’ S L e t t e r t o V A L d e m A r ’ S B o d y I. Spaces: Imaginary-Symbolic-Real t his article attempts to build on the somewhat underdeveloped theme of spatial dynamics in Jacques Lacan’s “Seminar on ‘The purloined Letter’” in order to make a larger argument about the logical approach to real space in his later teachings.1 The purpose of this intervention is to promul- gate a notion of ‘logical space’ in Lacan, analogous to his construction of ‘logical time’ that underwrites the reading of edgar poe’s story, “The purloined Letter.” By evoking another poe tale also admired by Lacan, namely, “The Facts in the Case of m. Valdemar,” I wish to show how we can situate ‘logical space’ as a corporeal aspect of the real that responds to the category of ‘logical time.’ developing Freud’s characterization of the unconscious as timeless, Lacan suggests that unconscious time is not chronological but logical. In his 1945 essay on the subject, “Logical time,” Lacan presents an inter-subjective modulation of temporality in which the subject encounters the tension between hesitation and haste in the logic of the unconscious. The three scansions of logical time, mapped through the problem of the prisoners’ dilemma are: “The Instant of the glance,” “The time for Comprehending” and “The moment of Concluding.” As Lacan concludes, the subject anticipates a certainty in the logical form of an assertion, expressing different modulations of time. But what about space in all this? In other words, is there a logic of space in Lacan’s thought that connects back to logical time and its complex moments, suspending anticipation between hesitation and haste? This question is of interest to this article. Let me begin by approaching this logical space through the Lacanian topology of the Symbolic, the Imaginary and the real as three registers of this spatial logic. to clarify, when I relate logic to the three Lacanian orders, I refer to the operative principle or principles by which the Imaginary, the Symbolic and the real orders may work. Having said that, there is a special status of logic for the real order. As the later Lacan repeatedly observes, the real can only be approached through logic. Arka Chattopadhyay: Logical Space in Lacan S13: 120 He engages with Frege’s mathematical logic to arrive at a notion of ‘+1’ or the ‘one- multiple’ that has real written all over it. This ‘+1’ creates an impasse in the system by adding itself to the number series as the signifier of inexistence or the originary zero, counted as one. In a similar way, Lacan’s use of Aristotle’s propositional logic produces a real logic of the ‘not-all’. to echo the definition of the real from Seminar XX, it is “an impasse of formalization” (1998, 93). to put it in another way, the real is the impossible of mathematical and logical formalizations. In Seminars XIX and XX, Lacan relates the real to a logic of writing. moving through Aristotle’s modal categories of necessity, possibility and contingency, he inscribes a fourth modality, i.e., the real as the impossible through a logic of double-negation: “what doesn’t stop not being written” (1998, 59). moreover, the Borromean triad itself is established as a real knot in Seminars XXI to XXIII. So, there is quite evidently a logic of the real. However, what interests me in this article is a spatial dynamic of this logic of the real, or in other words, what I am calling ‘real space’ in Lacan. In what follows, I will treat Lacan’s late work with the geometry of warped surfaces in the Borromean knot as a topo-logical extension of his interest in the mathematical and the logical. The Borromean topology provides Lacan with a final paradigm of logical inscription to approach the unconscious topography (the spatiality of the unconscious as a Freudian legacy). He writes the knotting of the three orders of the psyche (real-Symbolic-Imaginary) in the topological form of the Brunnian link/ chain, i.e. the Borromean knot. This topology offers a logic for the real writing that happens in the unconscious. Stated differently, the Borromean clinic yields a real unconscious in final Lacan. In this article I will mobilize the implication of the Borromean knot as a space and connect it with a real spatial logic. When I call this real space, a ‘logical space’, it does not mean that there cannot be Imaginary and Symbolic spaces that are logical in their own rights. I will spell them out as we proceed. However, given the special connection between logic and the real, I am more interested in real logical space than the avatar of logic in the spatiality of the other two orders. In his “Commentary on the graphs” at the end of Écrits, Jacques-Alain miller articulates a certain logic of space for the unconscious subject. He reflects: “there is no longer any occultation of the symbolic in the topology that Lacan establishes because this space is the very space in which the subject’s logical relations are schematized” (Lacan, 2006, 858; emphasis original). In this remark, there is a nestled notion of ‘logical space.’ This topological space schematizes the subject’s logical relations. It does not hide the Symbolic anymore. As miller suggests, in its Symbolic over-determination, this logical space “prohibit[s] imaginary capture” (858). miller’s logical space is primarily a reference to Lacan’s graphs. It cuts out the Imaginary through the Symbolic. miller is quick to differentiate this “symbolic space” from a Kantian “space of intuition” (862), but in his thinking, this space is dominated by the Symbolic and he remains silent on the real here. Is the logical space just Symbolic then? What about its purchase on the real? Stated differently, how does logical space relate to the real? As we know, for the Lacan of Seminar XXI, logic is Arka Chattopadhyay: Logical Space in Lacan S13: 121 the unmistakable “science of the real” (session of 12.2.1974)? In what follows, I will trace a real logic of space in Lacan. Just as there is a Symbolic dimension of space that becomes dominant in the graphs, there is also another, i.e., Imaginary aspect of space, which Lacan thinks at some length across many essays in Écrits. This is the space of the mirror, of course. Lacan calls this a “kaleidoscopic” space “in which the imagery of the ego develops and which intersects the objective space of reality” (2006, 99). This notion of space emerges from mirrorical projection. But, as Lacan contends, this imaginary space is not the be-all and end-all of the subject. He underlines how the mirror may not reflect anything to the subject on certain occasions: [w]hen man, seeking to empty himself of all thoughts, advances in the shad- owless gleam of imaginary space, abstaining from even awaiting what will emerge from it, a dull mirror shows him a surface in which nothing is re- flected (2006, 153; emphasis mine). How does this absence of reflection alter the Imaginary space? does it not introduce a bit of real into it? When the subject is at the limit of its thinking, an emptying out of thought installs the real as that which cannot be thought. It is interesting to note that Lacan moves from this mirror without reflection to an enigmatic glimpse of “unextended” and “indivisible” space that speaks back to logical time (153).1 Though Lacan still thinks through this space in gestalts or images, one can see a cut of the real in it precisely because this space interrupts imaginary reflection or mirroring. It is the real dimension of the Lacanian unconscious that has been foregrounded in recent studies as well as in the practice of 21st century Lacanian psychoanalysis.2 As I see it, the later-Lacan’s project of situating the unconscious in and through Borromean logic and topological geometry bespeaks a real logic of space. This spatial aspect of the real in the logic of the unconscious has remained underexplored and hence this attempt to construct a real logical space in Lacan. For instance, in the essay, “position of the Unconscious,” Lacan grounds the unconscious as a real logical space in topological terms. This is a space, built on gaps and holes, rather than any solid substance. talking about this “open sesame” of the unconscious that opens and closes in the same breath, Lacan observes: “[t]he structure of what closes [se ferme] is, indeed, inscribed in a geometry in which space is reduced to a combinatory: it is what is called an ‘edge’ in topology” (2006, 711). This topological space is reduced to a combinatorial logic of the “edge” that flattens out the 1. Lacan’s expression—a “kind of time that is caught between expectation and release, a time of phases and repetition” (153)—gestures towards the logical time, caught between hesitation and haste. 2. For example, see Colette Soler’s book, Lacan—The Unconscious Reinvented in which she discusses the real unconscious at length. Arka Chattopadhyay: Logical Space in Lacan S13: 122 distinctions between inside and outside. We will return to this liminal notion of the edge through the geometry of point as hole. II. Space in “Seminar on ‘The Purloined Letter’” Let me turn to the dynamic of space in Lacan’s reading of poe. to state that “The purloined Letter” is a story about spatial displacement is to speak the obvious. Lacan himself highlights that the story demonstrates how “the signifier’s displacement determines subjects’ acts, destiny, refusals, blindnesses, success, and fate” (21). Though this sounds self-evident, space has been tackled primarily as a signifying position for the subject in this story and not so much as an independent entity in itself. In other words, the critical move has been from the signifier to its changing loci and not space in its real dimensions. In his reading of poe’s story, Lacan famously distinguishes between the Symbolic and the real functions of spatiality: For it can literally [à la lettre] be said that something is not in its place only of what can change places—that is, of the symbolic. For the real, whatever upheaval we subject it to, is always and in every case in its place; it carries its place stuck to the sole of its shoe, there being nothing that can exile it from it. (17; emphasis original) While Lacan’s reading has elicited a lot of discussion on spatial displacement through the logical metonymy of the signifier, I would focus on this other functionality of space as real wherein stasis rules over motility. This motionless real space that remains immune to the signifier’s gymnastics will be our object of attention. In Seminar X, Lacan makes a telling distinction between lack and hole to talk about the real in spatial terms. He says that “the real is teeming with hollows” and in the same breath, he clarifies that “the real doesn’t lack anything” (185). What is this real space that is full of holes and yet does not lack anything? Lack belongs to the Symbolic order while the real is unlacking, so to speak. on the other hand, this unlacking space of the real, is full of holes, drilled by the letter as it writes itself in the real. The fact that the real does not have any lack ensures that it remains immovable. If there were lacks in the real, they would have triggered a movement, akin to the metonymy of desire that runs through the lacking chain of signifiers. This static space is full of holes but they do not allow it to generate motion in any way. to address the theme of space in some of the existing literature on Lacan’s reading of “The purloined Letter,” we find it only as a passing reference. For norman Holland, the story is a “study in the way we use spatial metaphors for states of mind” (320). He considers the movement of turning the inside into the outside vis-à-vis the letter as the principal operation of the story. But he does not mark this out as a topological operation. We shall see how Lacan’s topological externalization of the inside and vice versa ultimately erases the distinction between interiority and exteriority and generates an “extimacy.” It is key to this spatial dynamic. While this might seem Arka Chattopadhyay: Logical Space in Lacan S13: 123 to contradict the immovability of the real, it is not a contradiction because the topological movement that flattens out the partition of inside and outside makes the real space unlacking. It is this topological movement of mathematical inscription that enables us to arrive at this real space that is undifferentiated, unlacking and unmoving. We will come back to Lacan’s mobilization of this topological space as a non-euclidian space of the real in his later teachings. to return to the existing figuration of space in “Seminar on ‘The purloined Letter,’” Barbara Johnson’s famous essay on Lacan’s reading of the poe-story touches upon the question of space in terms of what she calls “frame” and “framing.” She treats the frame as a borderline space and appreciates Lacan’s project of testing “limits of spatial logic” and “breaking out of” the euclidean model (481). As Johnson acknowledges, “breaking out of” spatiality is itself a spatial metaphor and hence, what we are looking at is not so much a flat renunciation of spatial logic but, more of an attempt to find a new logic of space. We will see how Lacan’s late thoughts on a real logic of space, if not a real logical space, grapples with its ontological dimension. does this ‘logical space’ of holes have any being (in the ontological sense)? We will return to this question through Lacan’s recourse to the Borromean chain as a support for topological space. discussing the police’s search in the minister’s apartment in “The purloined Letter”, Lacan thinks through the question of space in the register of exhaustion. For the police, it is important to cover every nook and crevice. Their Imaginary programme is premised on this totalistic “exhaustion of space.” It is interesting that Lacan calls this spatial exhaustion more than “literal.” He calls it a “theoretical” concern (16). His question about space at this point triggers the movement from a depth model of psychic space to a psyche of surfaces: The division of the entire surface into numbered “compartments,” which was the principle governing the operation, is presented to us as so accurate that “the fiftieth part of a line,” it is said, could not escape the probing of the investigators. Are we not then within our rights to ask how it happened that the letter was not found anywhere, or rather to observe that nothing we are told about a higher-caliber conception of concealment ultimately explains how the letter managed to escape detection, since the field exhaustively combed did in fact contain it, as dupin’s discovery eventually proved? (16; emphasis original) The principle of spatial exhaustion, operating at the level of an Imaginary-Symbolic complex fails when an encounter happens with the real space as inexhaustible surface, rather than exhaustible depths of compartmentalized space. Lacan draws our attention to poe’s meticulously detailed descriptions of this spatial structure of investigation: from the division of that space into volumes from which the slightest bulk cannot escape detection, to needles probing soft cushions, and, given that they cannot simply sound the hard wood [for cavities], to an examination Arka Chattopadhyay: Logical Space in Lacan S13: 124 with a microscope to detect gimlet-dust from any holes drilled in it, and even the slightest gaping in the joints [of the furniture]. (17) Lacan gives a certain independence to space as an entity here: “As their network tightens to the point that, not satisfied with shaking the pages of books, the police take to counting them, don’t we see space itself shed its leaves like the letter?” (17). When space sheds its own leaves, we realize “the imbecility of the realist” that restricts space to a realistic idea of hiding where something is hidden in unfathomable depths. As opposed to this realist’s space, we encounter the real space wherein the hidden is “not in its place.” The letter has been hidden by making sure that it is not in its place. This elsewhere is no impenetrable depth but an evident surface which displaces depth. Just as there is no distinction between inside and outside in the real, there is no difference between depth and surface in the real. In the real logic of space, depth is transformed into surface. The letter is in the same place. The police scan exhaustively and yet they cannot find it because there are two different subjective approaches to space at work within the same locus. miller spots this in his notes to Seminar XXIII: “a euclidian metrical space, the only space that the police in ‘The purloined Letter’ move about in. The police fail to spot the paradoxical relationships, even the singularities, that are authorized by topology” (2016, 200). As he rightly observes, topological space is a non-euclidian entity and the Borromean nodality that Lacan adopts in his later teachings is characteristic of his emphasis on this other kind of space. The letter is right there and elsewhere at the same time. It is in a real point on that space that has zero dimensionality. Hence it does not exist for the policeman’s Symbolic- Imaginary framework which involves a linear compartmentalization. For them, space starts from the line which has one dimension but in the real, as we shall see, space concerns the zero-dimensionality of the point. III. Real Borromean Space: Point to Hole in Ontology Let me sketch out here, the real ‘logical space’ in the later-Lacan’s emphasis on the true hole in the real. We will investigate this real logic of space by considering whether or not it has a solid ontological dimension. In Seminar XXI, Lacan talks about the “three dimensions of the space inhabited by the speaking being” as “dit-mansions” (constructions/mansions of what is said/dit) of the Imaginary, the Symbolic and the real (session of 13.11.1973). This means there is an Imaginary, a Symbolic and a real space for the speaking body of the subject. Lacan also points out in the same session that this is neither the intuitive space of greek geometry, nor is it the Cartesian coordinates. This is a new geometry of Borromean knots—the so-called “rings of string.” He calls this a space of points and this is how he defines the Borromean point: “if you pull somewhere on any one of these rings of string, you see that there is a point, a point which is somewhere there where the three are squeezed” (session of 13.11.1973). Arka Chattopadhyay: Logical Space in Lacan S13: 125 In Seminar XX, Lacan has another defi nition of the point: What cuts a line is a point. Since a point has zero dimensions, a line is de- fi ned as having one dimension. Since what a line cuts is a surface, a surface is defi ned as having two dimensions. Since what a surface cuts is space, space has three dimensions. (1998, 122) Lacan calls this a cut-centric approach to space—a “sawing technique” (131) and qualifi es the point which does not have a dimension as a mark of inexistence or, shall we say, real ex-sistence: “It is even refl ected in the notion of the point, for the fact that it qualifi es as one that which has, as is clearly stated, zero dimensions - that is, that which doesn’t exist - says it all” (131). Th e real logical space thus takes aft er the knott y topology of points where the principle of spacing lies in what Lacan calls “wedging.” Th e real point is not just a point. In the Borromean space, the real point is a real hole. When the three rings are not squeezed together to form a knot, we can see three points: Th ese three points are wedged together when the rings are squeezed and we get a “threefold point” in the Borromean knot. Th e important detail is that this point does not hark back to the one-dimensionality of the line. As Lacan clarifi es, this triple-point is not just an ordinary point: “Th is point is not constituted here by the convergence of three lines, if nothing else because there are two diff erent points – a right and a left ” (132). Th is is a point made of other points. Th is is a space of holes that are points and vice versa. In Seminar XXII, it is this triple-point that Lacan goes on to call the “inviolable hole” (session of 11.3.1975). He makes a further distinction between true and false holes (see Seminar XXIII, 67). A true hole is that which an infi nite line has passed through while a false hole is not constituted by an infi nite line. In other words, the Borromean holes are true holes insofar as the knot cannot be formed without infi nite lines and circles. For me, this true hole that helps constitute the Borromean knot through an equivalence of the infi nite line and the circle is the logical space of the real. It is a space of points that lead to further points, wedged together. most importantly, these points are also holes, ratifi ed by an infi nite straight line, running through them. As Lacan says in Seminar XXII, without a hole, one cannot make a Borromean knot (session of 17.12.1974). In the Seminar XX, 132 Arka Chattopadhyay: Logical Space in Lacan S13: 126 same seminar, he also draws a necessary connection between the hole of the real and the order of ex-sistence. As rona Cohen, reading Lacan’s spatial being, suggests, “[w]ith topology, Lacan addresses this dimension of space, as real” (219; emphasis original). In agreement with her derivation of real space, let me state that while her focus is on the functions of object and being in this real space, I am more concerned with the logic of the real that governs this spatiality vis-à-vis the hole and the point. The hole of the triple-point in the Borromean link is a space that has no solid being. It is not a space shared by the three rings and yet it is a twist around the central hole that makes the knot of three. It is not a space in a foundational and ontological sense because it has no presence without the rings and their knotting. The squeezing tightens the three points into one hole but none of the three rings can ontologically claim that space to be its own. Without that hole, the Borromean or Brunnian link will cease to exist but at the same time, this hole-point of real space does not have any ontological status as a space in itself. It has no being but as the logical cause of the entire Borromean structure, this real space has an inexistence that is etched out nevertheless. Being devoid of being, it is a motionless space. Having no existence, it cannot move. The things that move in and around this hole-point are the infinite lines that have formed rings while the hole-point as an index of real space remains immovable. Having no being, this space cannot lack anything. For it to lack something, it has to have some substantive being. This space is unmoving and unlacking, not to mention that it does not have an ontological dimension. It is the zero-dimensionality of the hole-point. This is the fundamental difference between Cohen’s and my critical positions on the matter. While she is interested in developing Lacanian topology as a consolidation of spatial ontology, for me, it is the topological or logical space of the Borromean chain that creates an impasse in ontology.3 In a spatial sense, topology creates warps in ontological solidity. Cartesian and euclidian space might have a certain degree of ontological solidity but topological space introduces heterogeneity into that consistency. to say this in another way, ontological space is more Imaginary than anything else while topological space is dominated by the real. The locus of the hole-point, generated in this non-euclidean space, jettisons the affirmation of space that has any solid ontological being. The real logical space that makes a hole of a point and a point of a hole does not have any being. It does not exist. The only thing, it has, is ex-sistence. In my argument, it is the point as hole that stands for a real locus of ex-sistence and unlike the Symbolic-Imaginary space of volumes, this is a space of zero-ontology. The point that enables the twist around the central hole of the Borromean knot does not have a being because that space only comes into being as a hole when the twisted third ring goes through the points on the two other wedged rings, to form a knot. This hole is not a point, shared by any of the three rings insofar as 3. one may consider Alenka Zupančič’s arguments on sexuality as an impasse in ontology in Freud and Lacan as a parallel and complementary approach to the complex question of Lacanian ontology. For more, see her book, Why Psychoanalysis? Three Interventions. Arka Chattopadhyay: Logical Space in Lacan S13: 127 they do not go into one another. In the Borromean knot, the three constituent rings only lie side by side or, in other words, one on top of the other. The point-hole of the real logical space thus has no shared being. This is a space that bores holes in being. This is the logical, rather than onto-logical space where the minister keeps the letter hidden in poe’s story. It will never be found in any examination that is premised on a solid ontology of depths. It can only be approached through the logic of the zero, operative in the logical ex-sistence of real space as a locus of hollows. In Seminar XX, Lacan defines “pure space” as a space “based on the notion of the part, as long as one adds to that the following, that all of the parts are external to each other – partes extra partes” (23; emphasis original). This is not the Cartesian space of extensionality where one part extends into another. As Lacan clarifies, all the parts are extraneous to one another. This is a fragmentary space where parts have replaced any notion of wholeness. Lacan calls this real space, a “mathematician” who knows how to go beyond an intuitive understanding of spatiality (135). This is the logical space of the real which knows how to count. Counting numbers are connected to counting holes in the rings of string that make up a Borromean space of chains and knots. 4 Seminar XXI evokes “vector space” and “fibred space,” taking the plot from Seminar XX where Lacan discusses the space of jouissance as a “compact” topological space. In Seminar XXI, he emphasizes the heterogeneous character of topological space by dwelling on the notion of ‘neighbourhood,’ intrinsic to topology. As he suggests, this is not a homogeneous neighbourhood. It is marked with an ‘axiom’ of otherness: “everything that forms part of a topological space, if it is to be put in a neighbourhood, implies that there is something else in the same neighbourhood” (session of 15.1.1974). This something else is the trace of heterogeneity in real topological space. In Seminar XX, Lacan continues to talk about this set-theoretical heterogeneity in terms of cracks, faults and holes in topological space: nothing is more compact than a fault, assuming that the intersection of everything that is enclosed therein is accepted as existing over an infinite number of sets, the result being that the intersection implies this infinite number. That is the very definition of compactness. (1998, 9) We are back to the space of “faults” or “holes” and these holes indicate an infinite locus of the real. Lacan notes that this is not a homogenous space but an “intersection extending to infinity” (9). This logico-mathematical heterogeneity is a pointer that there is no singular or uniform logic of the real. The logical space in its real incarnation is all about fragmentation, ambivalence and antinomy. This space is intersectional as well. It is poised on cusps, edges and holes. The extensional presence of the infinite line makes this into a true hole of the real. Lacan goes on to call this a space of open sets that exclude their own limits and constitute a 4. The zero-dimensionality of the Borromean point connects with the number zero and what Lacan makes of it by mobilizing the work of Frege and the logic of the plus-one where the originary zero of the number series comes back to haunt numerical succession. Arka Chattopadhyay: Logical Space in Lacan S13: 128 finite series that can only be counted one by one (10). Approaching each of these open spaces one by one in the domain of sexual jouissance is not what this article is concerned with. But the movement of the one-by-one echoes Lacan’s topological evocation of the Borromean knot which constitutes three ones that are alone in themselves, i.e., without any one-on-one rapport. The Borromean structure highlights the agency of this one, insofar as releasing any one ring dissolves the entire chain. This space is a space of the one in the sense that each of the rings is one-all-alone. Thus, we have to go one-by-one in the Borromean space. This one is also the one of the speaking body. IV. Logical Space and Body In what follows, I will connect the real logical space with corporeality. extending Freud’s point that the subject is not aware of psychic space, rona Cohen observes that the subject of the unconscious lacks knowledge about its embodiment and spatiality (216). She makes an argument about being that is homologous to my own, regarding the body. For Cohen, being does not occupy a place. Being is itself a place. She addresses the body as a structure that goes beyond the Imaginary specularity of the mirror and charts real space. The connection between body and space is given a thought in Seminar XX: “[i]n their complexity, knots are well designed to make us relativize the supposed three dimensions of space, founded solely on the translation we give for our body in a solid volume” (1998, 133). Solid geometry provides a corporeal modality that goes well with the Borromean spatial logic of the real. For Lacan, the three dimensions of space are founded on embodiment in terms of how the body occupies a solid volume in space. But the Borromean knot relativizes this dimensionality and changes the notion of embodied space as well. In Seminar XXIV, Lacan makes a gesture towards this other kind of space that the body inaugurates: “Space seems to be extension when we are dealing with descartes. But the body founds for us the idea of another kind of space” (session of 16.11.1976). He grounds this corporeal space in a torus which produces a tube when turned inside out. torus, in this seminar, becomes Lacan’s preferred topological shape to discuss the real of the body. A tube appears when the torus is cut open and topologically turned inside out through continuous deformation and transformation. The tube disappears when the torus is reconstituted with its inside on the outside and vice versa. This tube is a figure for the body in its real structure. This is the real spatial structure of the body that is susceptible to cutting and “extimacy.” Let me highlight the intersectional nature of this real body. If the tube presents the real of the body in some sense, it is precisely because it is a result of the topological operation of cutting and turning inside out. The real of the body appears and disappears in these corporeal transitions, i.e. the operation of cutting open the torus. With the torus, we are looking at multiple holes: the hole at the heart of the torus and the hole inside it through which we can run a series of other tori (see fig. I-1, Seminar XXIV, session of 16.11.1976, opposite). Arka Chattopadhyay: Logical Space in Lacan S13: 129 Seminar XXIV, session of 16.11.1976 Arka Chattopadhyay: Logical Space in Lacan S13: 130 According to Lacan, the human body has an asymmetrical relation with space. The asymmetry of the signifier and the signified translates into the asymmetry of the container and the contained, which has a particular corporeal function (session of 21.12.1976). The speaking body as container contains space that is asymmetrical vis- à-vis the body and vice versa. Instead of thinking embodied space, we are thinking of body itself as real space here. It has its hollows, surfaces and topological movements of internalization and externalization. In his “Seminar on ‘The purloined Letter,’” Lacan connects space with body when he compares the letter with “an immense female body, sprawl[ing] across the space of the minister’s office when dupin enters it” (2006, 26). Without going into the sexual identification of the body as female which relates to the letter’s feminizing features, let us continue to think the human body which has a real structure, i.e. a logical space in itself. V. Reading Valdemar: The Space of Death Let me now turn to the other poe story, “The Facts in the Case of m. Valdemar” to show how the real logical space of the human body reacts to ‘logical time.’ to clarify, this article is less about logical time which is an established Lacanian notion and more about establishing a new notion from Lacan, i.e., logical space. I am not claiming any simplistic space-time coordinate here but only registering the kind of reaction logical space might have on logical time. I will do this by going back to poe whose “The purloined Letter” was a story that helped Lacan in thinking through logical time. The story in analysis is “The Facts in the Case of m. Valdemar” and it is a story Lacan was more than aware of. It is interesting to note how the standard interpretations of this 1845 gothic story have almost inevitably centred around the protagonist’s impossible articulation of his own death and complications of identity.5 roland Barthes reads into the story the relation between death and language, among other things, while Jacques derrida considers death/exclusion of the ‘I’ as a pre-requisite for speech act. In a significantly different reading, Hannah murray approaches the story from a medical humanities perspective and reads into it the politics of control over the other’s body in the context of medical processes. There are queer as well as deconstructivist readings foregrounding ‘telegraphy.’6 The final state of the human body as mess and the torsions it produces in space as well as time have been largely ignored in these readings. This is the aspect I will try and unlock through the Lacanian installation of speaking body as ‘logical space’ here. We will see how the subject-body of Valdemar, caught in a limbo between life and death and uttering the linguistically impossible sentence, “I am dead” speaks 5. See Barthes’ essay “textual Analysis of poe’s ‘Valdemar’” or derrida’s reference to the story in Speech and Phenomena (1973, 97). They dwell on the structural impossibility of the sentence “I am dead” and ground their notions of subjectivity on its basis. See also der- rida’s comments on the story in The Structuralist Controversy (155-156). 6. For a queer reading of Valdemar, see Suzanne Ashworth’s piece and for a derridean reading of writing in the story, see Adam Frank’s article. Arka Chattopadhyay: Logical Space in Lacan S13: 131 to Lacan’s notion of logical time and the disintegrating corporeal space as a real refers back to temporality. ‘The Facts in the Case of m. Valdemar’ is a story that draws Lacan’s attention as early as 1954-55, during Seminar II. In the story, the narrator who practices hypnosis, is able to freeze his friend Valdemar in the borderline zone of life and death in the body. He mesmerizes him on the verge of death and thereby stalls the decay of the body. In the shocking final moment, when the ethically troubled narrator withdraws the hypnotic spell, Valdemar’s body becomes a putrid pool of liquid within a second. The arrested post-mortal decay of the body is expedited in this time warp that is hardly chronological but logical. After hypnotizing Valdemar, the narrator keeps asking him whether he is asleep and the three answers he gives at three different points in time are as follows: 1. “yes; —asleep now. do not wake me! —let me die so!” 2. “yes; still asleep—dying.” 3. “yes; —no; —I have been sleeping —and now —now —I am dead ” (79-81; emphasis mine). These three answers chart a movement from consciousness to the most radical point in a paradoxical state where death has eliminated consciousness and yet, due to the arrested time of hypnosis, a distinctly dead voice remains in the body. We could compare these three statements with the three moments of Lacan’s logical time: instant of the glance, time for comprehending and moment of concluding. Lacan’s subjective logic of time finds a linguistic mode of expression here in Valdemar’s three sentences. They bring us back to the body as a speaking-body and logical space as a space bodily “inhabited by the speaking-being”, to echo Lacan’s aforementioned quote from Seminar XXI. poe evokes the image of distance and touch to give a peculiar attribute of deadness to this voice that can say, “I am dead.” to quote his analogies: In the first place, the voice seemed to reach our ears –at least mine –from a vast distance, or from some deep cavern within the earth. In the second place, it impressed me (I fear, indeed, that it will be impossible to make my- self comprehended) as gelatinous or glutinous matters impress the sense of touch. (81) The voice is said to come from some hellish under-earth and it feels like jelly on touching. These images create a sense of space (‘cavern’) and materiality (‘glutinous matters’) for the voice. They also have a paradoxical implication of corporeality. Though the voice does not have a physical body, the aforementioned images of touching and gelatinous feel create an impossible sense of corporeality for it. Here we have the climax of a triadic scansion of logical time. to place the three Symbolic acts in logical time, the first articulation that said Valdemar is sleeping and wants to die in sleep is, in Lacan’s language, “the instant of the glance” that frames the predicament. The second utterance solidifies “the time for comprehending” when Arka Chattopadhyay: Logical Space in Lacan S13: 132 the subject surmises that he is still asleep and will soon die. The final speech presents the “moment of concluding” in encapsulating the previous two temporal scansions. It hesitates to articulate the state of death in language, the linguistic impossibility that, for Barthes, is the psychoanalytic core of this story.7 Instead of dwelling on speaking the unspeakable sentence “I am dead” which introduces a bit of real into the Symbolic structure of utterance, I would argue that the speech-sequence whereby Valdemar announces his changing state is located in logical time. Hypnosis has put an end to chronos for Valdemar. In other words, for hypnotized Valdemar, there exists no chronological time anymore. But time still remains in a strange subjective way. Valdemar keeps saying things that are located in time. These articulations are logically frozen in inter-subjective temporality—a time that remains in the form of a gap between the narrator and Valdemar. Valdemar initially says that he feels no pain; so, the narrator feels less troubled about his experiment from a moral perspective. But, as we proceed and he asks the question again and again, Valdemar’s voice twists in suffering and implores the narrator either to restore his consciousness or end it altogether. It is at this point that the narrator offers his reverse passes and the temporal warp is neutralized. Logical time returns to chronological time and we encounter the carrion — the scrap that the human body is and that which it becomes: his whole frame at once – within the space of a single minute, or even less, shrunk – crumbled – absolutely rotted away beneath my hands. Upon the bed, before that whole company, there lay a nearly liquid mass of loathsome – of detestable putridity (83). In Seminar II, while referring to the Valdemar story, Lacan dwells, on the real and unnamable aspect of this corporeal waste. This is a liquefaction of the subject- body into a real mass of unspeakability. poe himself links space with time in the expression “the space of a single minute” and his use of the word “shrunk” presents a specifically topological movement in the matrix of space and time. Lacan reflects on this final state of the body in the following way: m. Valdemar is no more than a disgusting liquefaction, something for which no language has a name, the naked apparition, pure, simple, brutal, of this figure which it is impossible to gaze at face on, which hovers in the back- ground of all the imaginings of human destiny, which is beyond all qualifi- cation, and for which the word carrion is completely inadequate, the com- plete collapse of this species of swelling that is life – the bubble bursts and dissolves down into inanimate putrid liquid (1988, 231-232). Lacan makes the real character of this dead matter quite obvious in this passage. to note the fundamental but important point, this real “putrid liquid” is a 7. Barthes reads this sentence as a paradox of affirmation and negation, i.e. “I am dead and I am not dead.” He also reads it as a return of death as the primally repressed into the order of language (see 1981, 154). Arka Chattopadhyay: Logical Space in Lacan S13: 133 transformation of corporeal space. It is nothing but a topological transformation. What was inside the body and invisible has become visible by coming out. This externalization of corporeal inside lends a topological dimension to this decay. The body has become this real space for which the metaphysical-ontological language does not have a signifier. It is a space that cannot be named by the Symbolic- Imaginary semantics. This is the space of the real that is strictly senseless and silent. The silence of the putrid liquid is a memory of corporeal solidity. It is the terminal limit of the body that speaks. This is a once-body, suffused with the silence of unspeakable real. It is a corporeal counterpart to the telling speech of “I am dead.” It is interesting to note that poe wrote another piece on mesmeric practices in 1849, titled “mesmeric revelation.” It involved a discursive and philosophical dialogue between the mesmerist and his subject, someone whose name reminds us of the tale in analysis: Vankirk. The text goes into meditations on matter and space among many other things. At one point in the dialogue, Vankirk says to the mesmerist: There are many things on the earth, which would be nihility to the inhabit- ants of Venus – many things visible and tangible in Venus, which could not be brought to appreciate as existing at all. But to the inorganic beings – to the angels – the whole of the unparticled matter is substance that is to say, the whole of what we term “space” is to them the truest substantiality; – the stars, meantime, through what we consider their materiality, escaping the angelic sense, just in proportion as the unparticled matter, through what we consider its immateriality, eludes the organic. (poe 2005, 73) The above passage on space from a text that rewrites Valdemar in a different form of the essay bolsters my argument about body as space. As we can see here, Vankirk, the spectral double of Valdemar, relativizes the ideas of matter and space, claiming that what we human beings call “space” might have the material substantiality of a body for non-human beings. Following the same logic, we can say that the human body is, from another perspective, nothing but space. to make the connection with logical time clearer at the level of troping, let us look into the passage from Lacan’s essay on logical time: […] we witness the reappearance of the objective time of the initial intuition of the movement which, as though sucked up between the instant of its be- ginning and the haste of its end, had seemed to burst like a bubble. owing to the force of doubt, which exfoliates the subjective certainty of the moment of concluding, objective time condenses here like a nucleus in the interval of the first suspended motion, and manifests to the subject its limit in the time for comprehending that, for the two others, the instant of the glance has passed and that the moment of concluding has returned (171; emphases original). Let me emphasize the figurative similarity between Lacan’s two passages—the one above from Écrits and the aforementioned one from Seminar II. They both deploy Arka Chattopadhyay: Logical Space in Lacan S13: 134 the same metaphor of bubble-bursting. When Lacan talks about the return of chronological time after a logical scansion, the time that was logically sliced in between, bursts like a bubble. This is the same metaphor Lacan uses to describe the final transformation of Valdemar’s body into the real space of dead matter. The bubble-burst in Lacan’s essay signals the return of chronological time. It is the same return of temporal chronology that allows us to see the transformation of Valdemar’s body-space into unnameable matter. Valdemar’s body had already undergone the putrid liquefaction in the chronological order of decay in the human corpse. But thanks to the narrator’s hypnotic spell, this condition was frozen in time. With this logical time of hypnosis withdrawn, we could see the carrion in the resurfacing of chronological time in the final moment of the story. But what I want to emphasize here is the mutation of logical time into space as Valdemar’s body in its real fragmented formlessness. The real body-space thus becomes the locus to subtly situate logical time in this story. As poe’s description implies, the body that “within the space of a single minute, or even less, shrunk–crumbled—absolutely rotted away beneath my hands” is inextricably knotted with space and time. We come back to a moment in Seminar XXI in which Lacan melts logical time into space: “time is perhaps nothing other, precisely, than a succession of instants being pulled out […] time is, it is perhaps that, finally, the trinities of space; what emerges there from a squeezing without remedy” (session of 11.12.1973). This dissolution of time into space has to be counterpointed with any notion of time-space coordinate. moreover, we can spot the Borromean figure of squeezing in this passage. Lacan here subordinates logical time to the three dimensions of space that lead us to the squeezed point in the knot. This is the point of Valdemar’s body as dead matter. The unnamable real body is produced due to a hypnotic arresting of time that stops chronology and makes time logical in a subjective sense. Like the tube we saw forming while the torus is cut and turned inside out, this is a real body that does not stay forever. It is fragmented and transitional. It appears but only to disappear. In fact, in Valdemar’s case, this is the final remainder of the body, soon to melt into thin air. once the hypnotism is taken out, what remains as corporeal space of a dead real matter is the ruins of the body as logical space. This putrid liquid hardly occupies the space that the solid body of Valdemar would. The carrion is thus body turned inside out. This body-space is homologous to a chronological time that was squeezed into a zero-point earlier. The chronos that had burst out, bursts back in, but, not by way of temporality. It comes back in and through space. This is time’s becoming into space. The logic of time speaks in to a logical space of death here. The radical decomposition of the body makes the latent, patent, i.e., the body does not simply occupy space; body is space. It is a space from which dead matter reveals itself in an expression that is impossible to be captured through the Symbolic order. Ironically, this is a body that ceases to exist as body and becomes pure space. does this space have a being? I would argue that this is a space of holes. As the body bursts, what is left is a real locus, teeming with hollows. It does not have an affirmative existential value more than a hole. The body as a porous entity melts Arka Chattopadhyay: Logical Space in Lacan S13: 135 into a space that is suspended between embodied and disembodied loci. death as a signature of real non-knowledge is key to this figuration wherein the unnamable remains of a body situate logical space as a homology of logical time. This is also a space of absolute stasis as deathly immobility dwells here. Lacan in Seminar XXIII, is more than aware of this real aspect of death: “this imponderable is death, whose real grounding is that it cannot be pondered” (106). The real of death is not only the unthinkable or the inexpressible, it is also a real aspect of bodily transformation as it turns into dead matter. It is this movement that makes it an incarnation of real logical space. As Valdemar’s decomposition into uncountable but fragmented corporeal matter indicates, logical space is nothing uniform or singular. Its logic is one of hollows, tensions and fragmentary multiplicity. to conclude, what I have done in this article is to foreground space as a logical incarnation of the real in Lacan’s reading of poe’s “The purloined Letter.” I have established a connection between this static spatiality and Lacan’s multiple evocations of a real logical space in the later seminars. We have seen how the topological space of the Borromean chain that dissolves the distinction of inside and outside formalizes a real logical space of the point as hole with zero-dimensionality. As opposed to the euclidean space of point-line-surface, this is a space of points that lead to further points. These points are holes and vice versa. Building on this, I have zoomed in on Lacan’s late thoughts on the structure of the human body as real space, rather than simply an item that occupies Cartesian space. I have then taken this corporeal space of hole-points through yet another poe story on Valdemar’s hypnotized state between life and death, and his subsequent liquefaction once the hypnotic spell is withdrawn. my reading has situated the real logical space of the body as a parallel to Lacan’s notion of logical time. This is a real corporeal space that becomes an inscription of the movement from chronological to logical time and vice versa. This real space bores holes into ontology by remaining grounded in a hole-point that defies positive existential status. death as a perennial figure of the unknowable adds to the real of the situation as we see the deathly transformation of corporeal space. Acknowledgements I would like to thank the peer reviewers and the editors, Sigi Jöttkandt and dominiek Hoens, for their careful reading and robust suggestions that significantly helped this essay in the process of revision. 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