Lacunae | issue 11 | November 2015“I START OFF FROM THE LIMIT” On the first lesson of Seminar XX, Encore ADRIAN R. PRICE …this name of Inconstancy, which hath so much been poisoned with slanders, ought to be changed into variety, for which the world is so delightfull… John Donne The real, the same that we meet at every turn in the oeuvre that stands as a testimony to Freud’s enquiry, is undisputed in its status as the the crux of the psychoanalytic experience. Nevertheless, as it took on a more dominant place in psychoanalytic doctrine it proved necessary to account for this real in logical and structural terms. So it was that the teaching of Jacques Lacan followed the requirements of an effort that aims above and beyond that which is immediately manifest in the day-to-day course of an analysis, targeting a formalisation that would hold the status of a demonstration in the mathematical sense. However, the very condition of this effort is the fact that the psychoanalytic real, to be understood as the real of the symptom and, more widely, the real of jouissance, pitched as it is between the ontological and the discursive, has consistently proven to be incompatible with the real of science that readily allows of a description in mathematical laws and algorithms. This is precisely what led to Jacques-Alain Miller’s assertion that the effort of formalisation in psychoanalysis tends rather in the direction of a demonstration of the absence of knowledge in the real (Miller, 2012, pp. 115-16), a project that is neither of capital importance to the scientist, nor certain of achieving its aim, since demonstrating an absence is no easy matter. Voices in the fields of logic and philosophy were making themselves heard in the 1970s to refute the notion that the relation between the real and its 131 “I start off from the limit” nomination was as straightforward as hitherto reckoned. Kripke for one, in his Princeton lectures1 to which Lacan explicitly refers in 1975, drew a persuasive distinction between a description of the (imaginary) form of an entity and the act of naming that transforms this entity in its Being. Slightly earlier, at the end of Seminar XVIII, Lacan was already considering some of these paradoxes of naming in relation to the real. For example, in the penultimate lesson we read: In truth […] a proper name […] is only completely stable on a map where it denotes a desert. These are the only things on a map that don’t change name. […] A desert only gets renamed when it has been fertilised. […] This is not the case for sexual jouissance which science does not seem to win over to knowledge. (Lacan 2006, p. 148)2 Lacan goes on to observe that sexual jouissance assumed a preponderant place in psychoanalysis to the extent that this place was emptied out. Sexual jouissance functions as a dam, holding back the advent of sexual relation. An oft-repeated formula during this phase of his teaching, it also offers a first outline of what we meet in the opening chapter of Encore under the term faille, a fault, in the sense of a flaw or fault-line, of sexual jouissance.3 132 To speak in terms of a fault-line of jouissance is not to imply that there is no jouissance. On the contrary, jouissance lies on the side of that which there is. What there is not is sexual relation. Lacan’s effort at this late stage of his teaching, and especially here in Seminar XX, is to offer a finer account of the link between non-relation and jouissance. This is what would allow Jacques-Alain Miller in 1999 to affirm that the last of the paradigms of jouissance in Lacan’s teaching is the paradigm of non-relation, where jouissance is articulated to a point of impossibility; an affirmation that ought not to surprise us if we take seriously the formulation that jouissance belongs to the register of the real, and the real belongs in turn to the category of the impossible. Recall that Seminar XX is contemporary with Television, in which Lacan critiques the Freudian approach to jouissance as an apparatus of energetics (Lacan 1990, p. 9; pp. 18-19). Freud ends up positing “a substance, a fluidic myth titled libido”4 but the infinite reserve of possible cases of jouissance cannot be treated Lacunae | issue 11 | November 2015 by algorithms in the way that this reference to energetics is liable to suggest. This reserve is at once an accumulation and a fault-line of jouissance. As Lacan will go on to say in “La troisième”: jouissance fait défaut, fait dépôt, it “defaults” and by the same token “forms a deposit”. Nomination of this reserve of jouissance by means of the signifier thus proves to be fraught with intricacies, such that we are forbidden from thinking in terms of the absence or presence of a quantifiable substance when we turn to scrutinsing the oscillation between the fullness of the reservoir and the emptiness of the desert. Depending on the paradigm one adopts, its aspect and essence will alter.5 A name, a space, and its limits The hypothesis that Lacan introduces here in the first chapter of Seminar XX is that one can indeed treat and circumscribe the structure of sexual jouissance on the basis of a closed topology in which one proceeds incrementally, step by step as it were. The snag is how to get this process under way when one can foresee neither the nature nor the number of the steps that will be necessary for this treatment to carry through. Lacan opens his examination on the logical status of jouissance here in Seminar XX by specifying that the jouissance of the other’s body, the enjoyment one witnesses in the body of one’s other, is neither a necessary nor a sufficient response. The overriding evidence of this is that love demands more, and what comes in response is more love. Love is able to constitute a response for the subject. It constitutes a “sign” for the subject. To say that jouissance does not meet these conditions of a necessary and sufficient response is already a logical approach. The jouissance of the other’s body is indeed a response, only the response of love, if it is effectively present, is more concrete, more palpable for the subject to the extent that it enters the play of representations. This distinction between the sign of love and a potential anonymity of jouissance is a critical point of entry into this seminar. It signposts one of Lacan’s hypotheses that can be read between the lines throughout the seminar: jouissance is determined by language, by speech, but it is not fully articulated as such in distinct and precise signifiers. 133 we would meet sexual difference properly speaking. but only through the intermediary of a name that tells you: “not enough”. Lacan further proclaims that if this fault-line does carry a proper name. This demarcates the fault-line in the Other from which the demand for love sets out. we have a space: the fault-line. On page 12 of the Seuil edition we meet the term béance in reference to the gap between the One and Being. What interests Lacan is the body inasmuch as it symbolises the Other. of ascertaining its signifying logic behind what emerges through discrete signifiers associated with jouissance. therefore. 1975. 39). and the body provides this symbolic material. and which possesses at least one limit point: the point from which the demand for love sets out. One Being Just afterwards. One (sexual relation) Being (asexuated) . We have underlined the term faille. encore…” 134 Thus. That the body symbolises the Other and not vice-versa is a somewhat surprising formula that when first uttered in this opening lesson could give rise to some doubt as to just how deliberate it is. p. on the side of the One. Meanwhile.“I start off from the limit” It is a matter. Lacan will qualify this Being as “the jouissance of the body as an asexuated body”. 21. Here. defining the corporeal aspect of this amur in opposition to the biological body. Lacan will even identify this limit point with the wall of amur. were it not repeated in a more categorical fashion in the lessons of 19 December (Lacan. p. the locus of the Other is a logical fault-line that requires symbolisation in order to be tangible. “don’t stop”. which bears a name: Encore. 26) and 16 January (p. “keep going. Behind this Being lies jouissance. were there a relation by which this dimension could be established and confirmed. then this name is Encore. it may be possible to surmount the intrinsic condition of anonymity. In other words. if it is liable to nomination. we shall be looking at the interval [0. Lacan asserts that a woman. Each whole number constitutes a limit to the infinitude of points that reaches to the right towards the higher number. we find “faille… [comma] …béance”. n + 1. Later. just like men. and to the left towards the lower number. For our purposes. is subject to the condition of not being able to reach the limit. on page 14. a mid-point C that results from bisecting the [A. the jouissance of the body qua jouissance of the body of the Other falls under the sign of infinitude. In the French text. which is formed of the set of real numbers between 0 and 1. for example.Lacunae | issue 11 | November 2015 Lacan then speaks of a double characteristic of jouissance.1]. The limit is set as a limit point. but on no account can it be reached. n – 1. This stands in contrast. but rather to the interval between each whole number that is necessarily limited by the properties of the number itself. This 135 . On one side.B] segment. When you take any two points on the line that stretches from 0 to 1 – two points that you can label A and B – you can always set down another point between these two points. you will never reach the end. If you set out to list the totality of elements that are members of the set of real numbers between 0 and 1. we shall see the importance of having included the limit points within the set when we look at the theorem of compactness which accounts for the topological nature of such intervals. there will be an infinite number of them. Here. Here and elsewhere7 Lacan takes care to explain that this is not a reference to the sequence of whole numbers that stretch to infinity without meeting any limit. This infinitude is consistent with the infinitude that is a property of any whole number. which is formed of the real numbers that lie strictly between 0 and 1. but from which 0 and 1 are themselves excluded. 0 and 1 included. because it is an enjoyment of one’s own body at the level of the organ – the phallic organ – and not the jouissance of the woman’s body. The canonical example in mathematics is provided by the case of the real numbers between 0 and 1.6 On the other side. it constitutes an obstacle. to the interval (0. and so on and so forth. even though the real numbers that lie between 0 and 1 are “bounded” in the sense that they must be greater than zero but less than one. In both cases.1). in the sense that these limit points are included in the space thus defined. What jouissance is at issue now that it has been localised in this fault-line? Compactness. and then we have the proper name that treats it at the level of the demand addressed to the Other. one may select a finite number that cover it adequately. To realise this.“I start off from the limit” time the two terms. stand side by side. we shall say that just as topological space is a pair constituted of a set. This is the fault-line that is named Encore inasmuch as jouissance is neither necessary nor sufficient. now he is referring to the fault-line in jouissance. bounded. the space must meet two criteria. we can foreground the notion of a locus. It allows Lacan to treat the fault-line of jouissance in terms of compactness. We have a potential anonymity at the level of jouissance. This space is constructed such that. as a fault-line of jouissance. whilst on the other. Here he is speaking of a locus of jouissance that entails a certain dimensionality. that is. but ultimately what is at stake is the same fault-line. On the one hand. given an infinite collection of subsets that cover the space. The locus of the Other was a staple of Lacan’s classical teaching. This is where we pass from a fault-line in jouissance to the fault-line itself. and a collection of subsets that form the topology of X. we can lay the emphasis on the correspondence between Other and jouissance. the One of the signifier and Being (as the Being of jouissance). Where previously Lacan was speaking of the fault-line in the Other. It must be: 136 1. The space that Lacan presents is compatible . 2. but from a topology. we can say that the set named Encore is comprised of a collection of subsets of jouissance. in the sense that the elements that it contains may not exceed the limit points of this topological space. “fault” and “gap”. presented as a hypothesis If we transpose the above into the language of mathematics. X.8 This second condition is critical. which arises not from a physical organic structure. a certain architecture. and closed. 137 . but whether one approaches the space in terms of the union of open subsets or in terms of the intersection of closed subsets. p. Lebesgue and others generalised its application over the following years to encompass arbitrary coverings. 1966. whereas the set (0. which holds that a subset of a metric space is compact if it is complete and totally bounded. “the very definition of compactness”. To use the same example as above. There. Borel set out a first version of this theorem for countable coverings in 1895.1) is not. he is referring to this axiom. 1971) for his understanding of compactness. the Borel-Lebesgue theorem is introduced by means of a deduction via complementaries from the axiom that every family of closed sets of X whose intersection is empty contains a finite subfamily whose intersection is empty (Bourbaki. First posited by Bernard Bolzano in 1817. we shall limit ourselves to considerations on compact spaces in the frame of metric space.1] is compact. considering their intersection.9 For our purposes. from which one then seeks to establish the finite collection or “family” of subsets that covers the space. Since then. Thus. the result is in fact the same: these are two dual presentations that each account for the same infinite topological space. Lacan seems to be drawing above all else on the recent reprint of Topologie générale by the Bourbaki group (Bourbaki. It should be borne in mind that the Bourbaki presentation of compactness that is usually read today in the 1971 volume (or the 1966 English translation) actually dates from its first publication in 1940. when he says. 83). It is a perfectly valid introduction to compactness. this theorem would later be complemented by the Heine-Borel theorem (sometimes termed the BorelLebesgue theorem). rather than their union. we shall say that the set [0. it has become far more common to begin the other way around: with a topological space covered by an infinite number of open subsets. Lacan respects the ordering employed by Bourbaki in that he first establishes a characterisation of a compact space with the aid of closed sets. Today this strikes us as more than a little peculiar. but from today’s perspective looks somewhat idiosyncratic.Lacunae | issue 11 | November 2015 with the Bolzano-Weierstrass theorem that is valid for bounded sequences in metric spaces. We shall see both the nuance and the equivocation that Lacan adds when he approaches this space from the angle of woman. at this level. This move from logic to mathematics is indicative of the move from sexuation to jouissance. we have to understand that the intersection of the finite number of closed subsets contains an infinite number of points (the points of the set that thereby proves to be compact). and later he will introduce the hypothesis that woman’s position in relation to sexual jouissance possesses a structure that may be understood as the complement to the male structure that covers this same space of jouissance. and the faille is identified with its homophone. We insist on this fact since many commentators have spoken of this chapter as introducing the compact space of feminine jouissance. is phallic. constitutes a set that contains a multiplicity of constitutive elements. At no point does Lacan say anything of the like. the proper name Encore refers to the phallus. When he says that what results from this is that “the intersection exists in an infinite number”. the fault-line] having been accepted as existing in a finite number of sets”. what is at issue is the compact space of sexual jouissance. Another detail that ought not to be lost sight of is that Lacan is not defining the logical formulas of sexuation here. As such. he is drawing on a mathematical model in order to consider the field in which the formulas operate. the capital phi. qua sexual. According to this first characterisation. he is referring to the operation that consists in selecting closed sets that possess the finite intersection property. each case of jouissance may be considered to be sexual to the extent that it is localised with respect to the subject’s Being and converges towards the One of sexual relation.“I start off from the limit” The difficulties that this passage has presented to commentators over the decades are not due solely to Lacan’s drawing on the idiosyncratic Bourbaki ordering. In this first phase. being singular. 138 Lacan uses this manner of characterising a compact space to account for sexual jouissance qua phallic jouissance. Therefore. and here Lacan states perspicuously that jouissance. He says first of all that the space of sexual jouissance is compact. Each case. he is imprecise in his articulation. these cases needn’t be pursued in their infinite recurrence for certain key .10 When he says “the intersection of everything that is enclosed within [the faille. and Lacan’s too. trajectories and their respective relations” (Koyré 1922. p. Each step that Achilles takes corresponds 139 . or implicitly collapsed into the notion of a temporal “end”. is quite different. In the terms of Bergson’s critique. Zeno’s paradoxes have given rise to a number of commentaries and attempts at resolution over the centuries. they touch on movement only to the extent that it unfolds in time and space. p. we have disregarded the temporal dimension. according to these same terms. as did Koyré. Yet.Lacunae | issue 11 | November 2015 elements to be distinguished within them that cluster around a point of accumulation that incarnates the phallus. one has to take into account the steps she takes. in spatial terms. given that to catch up with the tortoise one needs to know where she is heading. For them. which would function as the upper bound. it is a matter of touching on the real of structure precisely by stripping it of everything that pertains to reality in the descriptive sense. that of Achilles and the tortoise presents as the most difficult to resolve since the end of the racetrack. Koyré’s understanding. which cannot be materialised spatially since. despite the fact that they are not “equal” (Koyré 1922.” Next. This characterisation is consistent with the more general identification of the finite intersection property with a centred system of sets. he takes “one step further” by effectively eliminating the dimension of time so as to consider the spatial dimension alone: “spatial distances. 25). Next. A synthesis of glosses from the turn of the twentieth century was presented and analysed by Alexandre Koyré in 1922. The locus of jouissance remains heterogeneous.11 The terms of the paradox foreground instead the point at which Achilles would catch up with the tortoise. their paths converge without ever intersecting. “the difficulties that arise [from these paradoxes] do not concern movement qua movement. but right across it one meets this element. Of all Zeno’s paradoxes. Lacan provides an illustration of this perspective in his reading of the race between Achilles and the tortoise. is more often than not omitted from the description. this point. we have just cast out the real of movement to retain nothing but a mere symbolic cartography. in order to focus solely on the spatial aspect of the paradox. 20). So. that carries a phallic denotation. we treat the “spatial distances” specific to the respective steps of Achilles and the tortoise as units that we hold to be “equivalent”. where he concludes that. Her stride. he will present what he calls the “complement” to this hypothesis. Lacan. Next. Before we turn to Lacan’s use of the union of open sets in the first lesson of Encore. when he turns to its complement. We will never find an algorithm or any other artifice capable of rendering them “equal”. He overtakes the tortoise in the sense that he takes larger steps at a faster pace. we need to remind ourselves of the preliminary work on the not-all which would have been . but rather a “hypothetical” application of compactness to the space of jouissance. that is. in the vocabulary he is using. which amounts to the same thing). Most commentators concur in supposing that the race would end when Achilles catches up with the tortoise. however. he can only overtake her. this provides him with a means of establishing a topological treatment of jouissance from the perspective of the logic of the pas-tout. Already. This is where Lacan introduces a detail that was unheard of in previous analyses of the paradox: Achilles cannot draw level with the tortoise (or only draws level with her at infinity. This aspect of the paradox is surely what inspired Carroll’s dialogue in which the tortoise leads Achilles into an infinite regression. in adding this detail that Achilles can overtake her. 140 The not-all and the problem of its existential import Whilst Lacan’s characterisation of the compact space by means of the intersection of closed sets was consistent with a universal “phallic” presentation of jouissance. accentuates the fact that there is no common measure between the stride of Achilles and the stride of the tortoise. Lacan presents this version of compact topology as a first “hypothesis”. meanwhile. “becomes shorter and shorter”. the union of open sets. but in so doing effectively overshoots the upper bound of the space to be covered. tracing her path towards infinity by marking out ever shrinking sets. She proceeds in much the same fashion as Cantor in his infinite intersection. we can hear that Lacan is undertaking not a direct treatment of the theorem of compactness (which had been well established for at least a century). Achilles sprints towards the point he believes to be the end.“I start off from the limit” to a step taken by the tortoise. the “not-all”. from “L’étourdit”. with all its inherent equivoques. to speak of logical entities that are not described or illustrated except in that language. and “not to all”. “belonging to some. in the very opening words of the Prior Analytics. to a lesser extent. in the opening sentence of the Prior Analytics. Thus. His isolation of a minimal particular and a maximal particular in the Prior Analytics. we find it defined as ἐν μέρει δὲ τὸ τινὶ ἢ μὴ τινὶ ἤ μὴ παντὶ ὑπάρχειν. Aristotle uses a “multiplicity of expressions for one same ‘logical constant’. p. whilst the negative particular embraces two definitions: “not to some”. or not to all”. In the case of the particular. 141 . [which] shows that what interests Aristotle is the unique signified that he is targeting through these expressions” (Brunschwig 1969. and their implications for two variants of the i-form and the o-form. thus predating the Brunschwig article by a full seven years. In the first reading. though in fact we meet an initial elaboration of the theme in the lesson of 17 January 1962 from Seminar IX. It has often been contended that Lacan’s development of the not-all was first instigated by Jacques Brunschwig’s article “La proposition particulière et les preuves de non-concluance chez Aristote”. the affirmative particular is defined just once. …ou pire. which had been delivered in July 1972 and was soon to be published in the long-awaited fourth issue of Scilicet. in Aristotle’s term logic.Lacunae | issue 11 | November 2015 familiar to Lacan’s audience from the previous year’s seminar. the positive and the negative particular are compatible. may indeed have provided Lacan with a fresh prompt to return to his earlier readings. though with a markedly different purpose12. of particular propositions. and. published in the tenth and final issue of Cahier pour l’Analyse at the end of 1969. Drawing heavily on Robert Blanché’s recently published work.13 The two possible readings14 of the particular in the Prior Analytics each lead to different consequences for the relations between the full set of terms as set out on the square of opposition. 6). and so a brief consideration of the article will not be unhelpful. Brunschwig’s study concerns the “thorny problem” of the existential import. or not to some. Brunschwig notes that the crucial difference between Aristotle’s formal language and the modern formalist approach to the logic he founded lies in Aristotle’s almost total reliance on natural language. what were hitherto relations of implication between a and i. 7). This reading yields the particular said to be maximal: maximally. p. in Seminar XIX. 9). only one of the statements i or o needs to be true. In this version. saying that i and e are compatible means that they may both be true. The important thing is that they not both be false. however. a particular which is on no account sufficient to sustain it. Likewise. In so doing.“I start off from the limit” This allows the traditional logical square to be maintained with its customary relations (that is. which is precisely why he [Aristotle] locates existence at the level of the particular. p. and between e and i. contrariety between a and e. and between e and o). This reading yields the particular said to be minimal: minimally. and between e and o. as Brunschwig explains. but they do not have to be. yielding subaltern relations of implication between a and i. The paradoxical result. which is its equivalent” (Brunschwig 1969. Lacan mentions the Brunschwig article just once. 142 Brunschwig argues that whilst Aristotle effectively rules out the maximal in favour of the minimal in his definition. “it has nevertheless to be acknowledged that in the concrete examples he gives of particular propositions Aristotle regularly uses terms that are linked by a maximal relationship of belonging” (Brunschwig 1969. is that “each of the universals cannot contradict one particular without contradicting the other particular. and between e and i. between i and o. The ensuing part of the article – which is actually its main thrust since the section we have paraphrased here corresponds to little more than Brunschwig’s introduction to the apparatus – argues how Aristotle progressively “liquidated the maximal connotations of the particular” as he worked his way through his different proofs. The second reading introduces implication between the affirmative and the negative particular. both statements i and o are true. transform into relations of contradiction. contradiction between a and o. paying tribute to how it perceives with great assurance how existence can in no way be established except outside the universal. can only be maintained at the cost of yielding a similar relation of implication between a and e. contradiction between a and o. even though it gives the illusion of doing so by virtue of the use of . a “response” and a “rejection”. And this is where the difficulties begin. 105) There you have it for the full extent of Lacan’s explicit reference to the Brunschwig article and the findings it presents. (Lacan 2011. namely that there are some who are not…. but “this doctrine is obviously connected with his assumption that universal statements entail their subalterns. it is impossible to extract such an affirmation from the not-all [du pas-toutes]” (Lacan 2011. 58-60). rather than a “contradiction”. he writes that this “saying no” is to be grasped as a “containment”.Lacunae | issue 11 | November 2015 the word some. the same is not true of universal affirmatives and particular negatives (Kneale & Kneale 1962 pp. 143 . or a “rectification” (Lacan 2001. 453). William and Martha Kneale consider the inconsistencies that can arise when one attempts to reform Aristotle’s logic by holding particular statements to be equivalent to statements about existence. inference to subalterns is not always valid. p. a “negated reprise”. which cannot be wholly identified with a particular one or some who negate the phallic function. p. In …ou pire. In “L’étourdit”. 25a) that the universal affirmative allows of partial conversion to the particular. In the key section on “The Four Forms of General Statement” that features in the chapter on the Organon from their 1962 book on The Development of Logic. Lacan plainly distances the negative particular from his version of the negation on the universal. whereas particular affirmative statements can be converted into universal negative statements with little change in meaning. For Lacan. p. it is essentially a matter of taking into account the act of “saying no” to the phallic function. the “saying no” at issue here is a contained act. Furthermore. the not-all: “Contrary to the function of the particular negative.” As we have seen from the Brunschwig article. Aristotle claims (in Book I: 2. 46). This implies that the position from which this act of “saying no” makes itself heard has also to be modulated with respect to the positioning of the exception: whereas the phallic function can be flatly negated in such a way as to ground an existential position (in which case we obtain the contradiction to the universal affirmative which is familiar to us from the male side of the table of sexuation). As we read in the “Kneale and Kneale”. is moving in the opposite direction from such clearly bounded domains and the deductive relations between premiss and conclusion that they enable. and nothing more. Lacan defines the not-all as follows: “It is reserved to this not-all [au pas-toutes] to indicate that somewhere. it might be surmised that Lacan draws inspiration from the rearrangement of the square of oppositions that Brunschwig’s maximal reading of the negative particular requires. p. and promotes a distribution of sexuation that is built around a relation of contradiction between the universal affirmative and the particular affirmative (just as Brunschwig’s second square of oppositions stipulates contradiction between a and i.15 At most. thus invalidating inference to subalterns). 207) and the second of Brunschwig’s squares shows that the coincidences end there. For our purposes here. Lacan. 202.“I start off from the limit” This would seem to account for the fact that Lacan makes such scant reference to Brunschwig’s 1969 article. however. p. Thus. In Seminar XIX. But a comparison between the foursquare table that Lacan provides in Seminar XIX (Lacan 2011. as in the various modi formalised by the mediaeval philosophers. we shall recall that the conventional o-type proposition allows of representation in a Venn diagram. woman has a relation . and indeed Brunschwig himself is said to have been somewhat bemused at commentators’ frequent matching of his text with Lacan’s developments from the early ’seventies. Some A does not belong to B would be represented as follows: A AB B 144 This kind of sharp demarcation between fields allows categorical syllogisms to be elaborated. Thus. 466) Written here in one word. This move is designed to free us of any notion of a closed “set of women”. p. This first definition anticipates the more elaborate version seven months later in “L’étourdit”: [F]or having entered as the other half through the saying of women. with the sequential effect.16 Note that this “somewhere” is implicated by an act of saying. promoting instead a domain that brooks no exception to the phallic function. p. This domain is therefore a “somewhere”. p. just as they will prove to be especially inadequate for describing the topology of jouissance. The circles of the Venn diagram prove to be inadequate in accounting for the domains described by the formulas of sexuation. even that which is of a provenance without reason. (Lacan 2001.Lacunae | issue 11 | November 2015 with the phallic function” (Lacan 2011. (¬∀ x) Φ x (Not all 145 . 466) This paragraph takes us from the first negated quantifier. But this is an all that lies outside the universe. unhyphenated. all may here be said thereof. which is read all at once from the quantifier as notall. they [“elles”] are notall. (Lacan 2001. but without the limit point that would ascribe to it the property of the universal. “like everywhere”. an “anywhere”. the subject is determined by the fact that. and an “everywhere”. and by virtue of the same. there is no means of ensuring anything whatsoever by way of a universe. since there exists no suspension in the phallic function. given the fact that nothing existent constitutes a limit to the function. as he develops it in the following paragraph: The subject in the half where there is determination by the negated quantifiers. that nor is any one of them all [toute]. there is a contrast between the universe of the all and the “everywhere” that only belongs to the register of the all insofar as it bears the negation that forbids it from functioning as a universal. grounding themselves in this half. the closing comme pastout resounds as comme partout. Lacan has just said that the thrust of this act of saying is not to be understood as “testifying to the existence of a subject through saying no to the phallic function.” On the contrary. but which cannot be totalised as a universe. 46). Thus. and not by any form of existence. p. Rather. 1972. 69). To say that at the level of singular existence there is no example here of an exception to the phallic function has the consequence of forbidding us from reading the negation of the universal as a negative particular applied “in extension” to a plural proposition. He has led us from a not-all that is specifically indexed to a plurality. being more advisable (Smith 1989. (¬ Ǝ x) ¬ Φ x (There exists no x such that not phi of x). contended that translating the affirmative παντὶ as “belongs to all” is an “unnecessary barbarism”. not all of a woman belongs to the phallic function. yet this is precisely what Lacan is taking great pains to circumvent. drawing on (Geach. This clarification is important because the formula (¬ Ǝ x) ¬ Φ x might at first blush seem to lend itself to a reading on the side of the universal. effectively forming a universe. 1962. But the latter feeds back into the former. This singular reading of the not-all does not contradict the suite mentioned in the passage (which we have rendered as “sequential effect”) and we shall see shortly how the extention of a suite or sequence of cases might correspond to the singular variable in question. 61). thus introducing the negated existential implication in the final twist to Lacan’s theory of sexuation. 146 This is the moment to remind ourselves that whilst using the singular all conforms to a long-standing tradition of Aristotelian term logic in English. p. allowing a logical deduction to be made as to the status of any one of the subjects who align themselves with the not-all. “belongs to every”. since in French the term pastoutes is both gendered and plural. it may also be remarked that Robin Smith. However. albeit in a negated form.17 . to the second. ix). without treating them as a totality. If the existence of a subject who says no to the phallic function cannot be grounded on woman’s side. It is not: “some are” and “some aren’t”. this opinion runs counter to that found in the Kneale and Kneale: “in some modern versions of Aristotle’s doctrine the difficulties of his account of opposition are unnecessarily aggravated by use of examples and formulae in the plural” (Kneale & Kneale.“I start off from the limit” x has the property phi of x). with the plural form. it is because such a subject would constitute a limit point to the domain and thus enclose it. p. to an existential proposition with a singular connotation. since whenever he uses the Greek vocabulary. he zooms in on the formula (¬ Ǝ x) ¬ Φ x to extrapolate the impact of the “saying” of woman at the singular level of her jouissance: How much easier. his frequent use of the plural pastoutes indicates that the condition of the not-all is to be inferred as the condition of each subject who could be described as a woman. leading as it does to the logical power of the notall being inhabited by dint of the recess of jouissance that womanliness tucks away. p. in On Sophistical Refutations. even a delight to pledge oneself. This term has proven much harder to track down. A more likely source is the passage from De Interpretatione to which Lacan alludes when he first mentions the not-all in his Seminar. It occurs just once in the Organon. is it not. then. X. and not of some women. What is the existential import of this confine? How does the not-all that defines 147 . Thus. we have seen that Lacan has read the negation on the existential quantifier from the angle of its consequence on the negation on the universal quantifier. and the “containment” we saw back on page 453. the singular of a “confine”. Here the non-univeralisable domain that knows no limit matches up. 170b. even on coming to conjoin with what maketh thomme…(Lacan 2001. confin comes from the Latin confina. presumably a nod to Luther’s commentary on Psalm 116. 466)19 What is at stake here is a domain endued with a “confine”. but a plural form for which he never provides a specific reference18: μὴ πάντες. Invariably plural in French. he does not cite the well-known singular form from the opening lines of the Prior Analytics. in this very deliberate use of the singular we have a tentative articulation between what we have so far encountered only as a “somewhere” or an “anywhere”. on 17 January 1962. with a “bounded and closed” topological space. this is where we meet one of Lacan’s less conspicuous departures from the classical term logic. “with end” or “with bound”. whatever Lacan’s source might be. Nevertheless. to charge to the account of the other quantifier.Lacunae | issue 11 | November 2015 Crucially. to page 466 of “L’étourdit”. Next. but this time the precise reference is obfuscated by Lacan’s choice of example: the Latinised (non) omnis homo mendax. To return. despite everything. 33. this does not exclude another version.21 148 The effort to theorise this inroad to jouissance in terms of compact space thus equates with Lacan’s search for a new approach to Being sexuated in the feminine. “L’étourdit” deals very little with the question of jouissance. From this angle there is admittedly some credence to be lent to the notion that the first chapter of Encore presents woman’s jouissance in terms of compactness. and therefore seeks to define neither the ambient topological space nor the limit points of the space. sharing key properties with the logic of the not-all. but in no case equal either to the point of departure or the point of arrival. until the opening lesson of Encore. “that which is defined as greater than one point and less than another. presented as a complement to the initial hypothesis Returning to the presentation of compactness in the first lesson of Encore.“I start off from the limit” each woman come to be “inhabited” by means of this “recess”? If there exists no woman who does not conjoin with sexual jouissance qua phallic jouissance. only the route that Lacan takes in unfolding the logic that would support it is far more subtle. but as noted above.” concerns these open subsets. Seminar XIX introduces the quantifiers of sexu- . how exactly does the topological space of sexual jouissance find itself assuming the status of conjoint to the logical condition of the not-all?20 These are the questions that “L’étourdit” was to leave hanging for another six months. that conforms to the conventional presentation of a compact topology. and obliges us to word it in the following terms: the presentation of jouissance in the opening chapter of Encore includes one version. 15). Aside from the afore-cited passage from page 466. that conforms to a less common presentation of a compact topology. but via what results from a logical exigency in speech” (Lacan 1975. one “which does not go via the body. the version that Lacan opens nominally as the “complement to this hypothesis of compactness” is consistent with the more conventional definition of compactness that we saw earlier: the covering formed by the union of open subsets. p. sharing key properties with the logic of the all. Compactness. We simply need to add that the parenthesis that reads. The domain that accounts for the feminine logic is not closed: it contains elements that are not and shall not be specified. Just like the subsets that cover the compact topological space. Within the bounds there is sexual jouissance. p. which exceed this space. not solitary jouissance. whilst union remains on the threshold. 20). within this space.” This “union […] on the threshold” is the One of sexual relation that bounds the topological space without constituting an accessible point. This offers a logical formalisation of what Lacan writes in “L’étourdit”. Constructing existence This is the first and last time in his Seminar that Lacan will mention the theorem of compactness in relation to jouissance. it can extend to infinity). making a partner of her solitude. but it is only here in the seminar Encore that we meet this first articulation between the logical quantifiers of sexuation and the topological space of jouissance. Compactness allows Lacan to take a step further by distinguishing between a woman. but jouissance whose partner is solitude. which like all sexual jouissance is bounded and closed within a space (even if. again on page 466: a woman is “the lone one in that her jouissance goes beyond. qua open set disjoined from its limit. each open set may contain points that are not included within the bounded space.” This formula is echoed shortly afterwards in the sentence: “the jouissance that one obtains from a woman divides her.Lacunae | issue 11 | November 2015 ation as possible ways of “writing a function of jouissance” (Lacan 2011. Lacan’s recourse to the two models of 149 . Nor will it find any elaboration whatsoever in his writings. the jouissance that is formed from coitus. This opens the question as to whether one woman is sufficient to account for the topological space of sexual jouissance. beyond the bounds. Further still. and her jouissance qua sexual. a question which did not arise within the perspective of the all where the operation of castration clearly demarked a uniformity between the range of cases of sexual jouissance and each x that conforms to the conditions of membership in the closed set. 150 Lacan goes on to specify that on this side – the side of the “complement to this hypothesis of compactness” – there is a “requirement” to count. 1975. . To conclude. 68). we can indeed start counting. an order has to be found. which is what we saw in Koyré’s Cantorian reading of the Achilles. If sexual relation is not realised for a woman. the jouissance of the phallic organ stands as an obstacle to the movement towards the Other. If we accept the argument of equivalence. Here. p. her partner’s or even her own. Applying a bijection establishes that the elements of a set are countable. is to affirm that the topology cannot be conclusively characterised as a compact topology. to posit a relationship of equivalence in the absence of a relation of equality. Lacan does not specify the existence of any such obstacle. I would like to consider an aside that Lacan voices in this first lesson and which from one perspective already seems to anticipate the clarification to come in February by introducing a certain restriction on the use of the classical theorem of compactness with respect to the not-all. “for them to be countable. thereby preventing sexual relation from being realised. by implication. on man’s side. he is effectively zooming in on the way that the finite family of open sets is selected from the infinite series in order to form a sub-covering. since speaking in terms of complementarity amounts to sliding back into a logic of the all (Lacan. further problematised in the lesson of 20 February when he stresses that womanly jouissance is “supplementary” and not “complementary”. it may be because she too is impeded by the obstacle of the organ. and we have to pass through a prior phase before supposing that this order can be found. but perhaps instead this means that the limit point is not included within the space of sexual jouissance. On woman’s side. but Lacan leaves open the possibility that we do not accept it. however. that is.“I start off from the limit” compactness built from complementaries is. perhaps. or that it does not fall under a covering. Lacan indicates that. an indispensible criterion when dealing with an infinite set. if she does not meet this limit point. He adds that. or cannot be covered.22 Recall that to affirm that the limit point is not enclosed within the topological space.” This “supposing” is the same that we meet in Seminar XIX to posit a Cantorian bijection. 2011. which this notall seems to tender us. To the extent that the points between 1 and Aleph-naught are denumerable (for example. 467) This is a condensed presentation of what had been developed in Seminar XIX (Lacan. The allusion here is to Cantor’s inaccessible cardinal. therefore. its enumerable being a sure thing. cannot be enumerated.Lacunae | issue 11 | November 2015 The perspective that takes shape here on the basis of this aside is confirmed to some extent when we refer back to “L’étourdit”. The “stronger” definition of inaccessibility given by Sierpiński and Tarski in 1930 runs as follows: an infinite cardinal number m is inaccessible if each product (and. Lacan uses this to read the inaccessibility of Aleph-naught as a repetition of the inaccessibility of the number 2. Lacan’s playful use of the expression “in sum” is an allusion to the summation of an infinite sequence. but the repetition that is. the transfinite Aleph-naught that serves to posit a limit to the set in a different manner from the existential exception. the curious circularity of the argument: we start off from the sexual difference that the not-all “seems” to offer. shows that what is involved is an inaccessible. Thus. its reduction becomes so as well. 1947. one sets off from the “illusion” to arrive at something “sure”: an enumeration that can then be reduced. but this is all that it takes to be able to start to enumerate the elements of the set by means of transfinite induction – elements that. however. only to conclude as 151 . pp. pp. “Reduction” here seems to be a nod to Russell’s axiom of reducibility. forms an illusion. to which Gödel was more sympathetic than many. 186-724). in sum. 175-923). by essence. Notice. also each sum) of fewer than m numbers < m is < m.25 In this context it would presumably entail reducing the series to the notall predicate. In 1947. so too are the points between 1 and 2. Gödel included an endnote to his text on Cantor’s continuum problem where he gives this definition. It is a paradoxical limit because it is an “inaccessible” limit. the transfinite. in the case of the positive integers). (Lacan 2001. on the basis of which. which produces a series. The following passage shows that what Lacan draws above all else from Cantor is the notion of an inaccessible: The support of the two [deux] that twains [faire d’eux]. p. and on this basis build an interval that in turn accomodates a manifold (eux). and goes on to open a parenthesis asserting that the same definition may be applied to finite numbers: “0 and 2 and no others are inaccessible in the strong sense” (Gödel. In appealing to a constructivist model. that is. 94). that his question pertains to “a jouissance that.“I start off from the limit” to the inaccessibility of this difference. to know how to find out where this existence is” (Lacan. Lacan specifically mentions. one also has to be able to construct it. “to posit a ‘there exists’. Intuitionism problematises to varying degrees the notion of a set for infinite sequences. or “spread”. He then proceeds to a more sophisticated formulation of the not-all that contrasts with the references he was using the previous year. where what is at stake is to extract a finite covering from any which infinite covering of a compact space. E. 1975. within this model. the question is as to whether a not-all grouping can even be posited as a set. 1975. . p. The operative word here is “construct”.26 152 In a certain sense. p. somewhat ambiguously. even went so far as to replace sets with his concepts of Spreiding. Its founder. Furthermore. J. is merely the other face of the same coin. p. Rather. as is done in intuitionistic mathematical logic. He tells his audience that although conventionally one cannot posit that the not-all entails the existence of something that is produced by a negation or by a contradiction. 94). belongs to the order of the infinite” (Lacan. At the close of the lesson of 10 April 1973 from Encore. This accounts for why the famous “dit-femmation” that Lacan singles out (Lacan. Another possible reading of this “reduction” is to see it as a foreshadowing of Lacan’s use of the compactness theorem. and “species”. Brouwer. every set has an order (if we take the Axiom of Choice). 79) is not restricted to the word of hurtful intent: playing spokesperson of the Athena doctrine. Lacan seems to be moving away from the classical Cantorian model of sets bounded by transfinites. 1975. because according to Cantor and Zermelo. one can posit it as an indeterminate existence. the Dutch mathematician L. with regard to everything that serves a purpose in the function of Fx. a more commonly met recourse in our day. the question is no longer as to whether the not-all set can be ordered. But moving from the logical formulas in Seminar XIX and “L’étourdit” to the mathematical model in Encore obliges us to consider how taking jouissance into account modifies the existential import of the not-all. Lacunae | issue 11 | November 2015 which accommodate Wahlfolgen. Georg Kreisel proposed an axiomatisation for lawless sequences ranging over fans (Kreisel. A fan is a finitely branching spread that is accounted for by “bars” that have been clipped from longer spreads. the sequences are performed contingently. or “choice sequences”.29 This so-called “bar induction” offers an effective analogue to “transfinite induction”. We might hazard an analogy between Brouwer’s “ideal mathematician” and the “ideal dream worker”.28 2. at whichever point the development of the sequence has reached at any given moment. each of the topological descriptions presented in the first chapter of Encore may be reproduced using intuitionistic characterisations. 96). and thus. 1900a. Brouwer’s fan theorem allows for a constructive substitute for compact sets. Thus. 1958). subject-dependent objects” (van Atten. p. These lawless sequences. but two aspects of Brouwer’s mathematical legacy are particularly pertinent to our considerations here: 1. a “creating subject”. Lacan does not develop his reference to intuitionism and constructivism any further. nor determination. step by step. 2007. or judge” (Freud. betray the same condition as described by the not-all quantifier: there is neither closure. Second. First. one finds there the subject who is responsible for them. in the flow of time. Lacan seems to be transposing – in all likelihood unknowingly30 – something 153 . aside from being “the strongest case of time-dependent. as opposed to algorithms or data strings. Freud’s unconscious subject that “does not think. This notion of a creating subject was devised as an ideal mind that would be performing and hosting the mathematical operations. calculate. much like the dreamer who occupies each of the places in the dream.27 The spreads are chains of signifiers that are generated subjectively. 507). Brouwer holds that choice sequences reflect the constant presence of a scheppende subject. without interference from individual “psychological” factors. In 1958. p. However. here transform into names of women. of incarnating the man whom she will use as a relay. but the singular Being of the enumerating subject. The myth of Don Juan allows for a kind of axiomatisation. in passing via this fantasmatic character who places her among other women in a series that can be enumerated. 154 We may note that there is nothing in this passage that contradicts or modifies what Lacan had developed on Don Juan in Seminar X: as an incarnation of this absolute object in a woman’s fantasy. It is not certain that he desires. the male sex. allows jouissance to be approached by setting down names. just as Lacan qualifies the fantasy as an axiom. whereby she “becomes this Other unto herself. as a man. Here. he is. but this is the exact opposite of what Lacan sets out. In effect. Don Juan is not the father of the horde who enjoys all the women. If. He is the one who. as some have chosen to read it. as we have suggested. What we singled out earlier as cases of jouissance. and second. the notion of the “countable” sounds more like an “enumerable”. This is one way. but at woman’s approach to something particular to her. nor even that he enjoys. as she is for him” (Lacan 1982b.“I start off from the limit” strongly akin to this constructive model into the field of psychoanalysis when he turns to “the feminine myth of Don Juan”. and resembles far more Brouwer’s concepts of spread and species than the open set of classical mathematics. a jouissance that would perhaps be anonymous were it not for the phallus. Lacan is tackling two intimately linked but nevertheless distinct issues: first. We are not looking at a recipe for the man in his approach to women’s jouissance. As we have seen. Note that the Being that is concerned by this naming is not some multiplicity of supposed female beings. It concerns what “the other sex. Many commentators have read the passage on Don Juan as describing man’s approach to woman. which is then enumerated. Lacan implicitly excludes one of the limit points . the status of the limit point that bounds the locus of sexual jouissance and its implications for an existential construction. quite to the contrary. and there are surely others. and allows of a nomination. Each woman takes the place of a case of jouissance. the inaccessible limit that is implied by the negation on the universal quantifier. once again. in a position of radical imposture. we need to be attentive to the nuances. p. A name is set down. is for women”. 92). one uses the One in a different way: it is no longer the “fusional One” that marks the place of (impossible) relation. p. 1982a. with the latter being founded on the former (van Atten.Lacunae | issue 11 | November 2015 from the space of sexual jouissance when he takes the inroad of the not-all. 65). the fantasmatic Don Juan). with its resulting “centrifugal tendency” (Lacan. It is precisely this rigorous requirement of naming and enumerating that distinguishes the feminine encore from the “persistent divergence” met in male sexuality. As Mark van Atten has argued in his analysis of choice sequences. to figure this differently. un(e) par un(e)) Being (Requirement of infinity) This is how one can. despite what might be an absence of limit point within the covering collection. In other words. We would nuance this observation by suggesting that the enumerating subject is not to be identified with the axiomatising operator that enables the enumeration (in the example given. p. 91-231). p. but is now identified with a “requirement of infinity” (Lacan. 155 . it is the One that repeats across the domain each time that it is enumerated under a different name. thus producing two readings of the not-all (Grigg. one singular creating subject is present across each step of the spread that is being generated. 2007. 84-85). within the space of the interval. in a certain sense. Or. and the condition of the subject as not-all is equivalent to the condition of the spread itself. which previously constituted the other boundary of the topology. Something similar also holds for Being. each case in the enumeration contributes to the description of one single variable that is thereby defined in (¬ ∀ x) Φ x. p. 15): (Requirement of the) One (Requirement to count. 2005. 469). Russell Grigg has argued that the reference to the enumeration of cases implicitly means taking the formula (¬ ∀ x) Φ x in extension. pp. 1975. the same that according to the terms of “L’étourdit” renders a man “clumsy” in imagining that “having two of them makes her (la) all” (Lacan. the One reaffirms its presence. In our reading. as a requirement. from the second perspective (the so-called “complement to the hypothesis of compactness”). 2001. start off from the limit. we have to keep in mind the distinction between the identity of the process and the identity of the sequence that is constituted in it. (1966). which allows the not-all to be written. J. Aristotle.net/en/articles/85-les-versants-masculinet-feminin-de-la-limite-par-henri-cesbron-lavau. available on UMI Microform. “La proposition particulière et les preuves de non-concluance chez Aristote”. is as close as we have come to a demonstration of an absense of knowledge in the feminine real. Don Mills. H. This. Book III: General Topology. ON: Addison-Wesley. Bourbaki.html. Zeno. this is how Lacan approaches the transfinite the following year in Seminar XXI. 10: 3-25. which he reads as dependent upon Cantor’s act of saying. Badiou. NJ: Rutgers. “Versants masculin et féminin de la limite”. Bourbaki. (2008). unpublished PhD thesis. Paris: Seuil. (2007). 287-305. “Sujet et infini”. And he will add. Cesbron-Lavau. The Racetrack and the Achilles: A Historical and Philosophical Investigation. A. namely Aleph-naught. on 13 November 2013. (1992). N. 156 References Allen.” The expression “allows to be written” is not innocent here: it denotes a specific mode of inscription that depends upon the contingency of the enumerating process as performed by a subject in time. The above text is a slightly revised version of the lecture presented at Barnard College. livre III: Topologie générale. in the lesson of 19 February 1974. In Cahier pour l’Analyse. . New Brunswick. (1971). Paris: Hermann. Elements of Mathematics.“I start off from the limit” Adopting the intuitionistic model with respect to the not-all brings with it the consequence that the inscription of (¬∀ x) Φ x will prove to be contingent. 2009. W. London/ Palo Ito/Reading. ought to be replaced by the sign for the countable. Brunschwig. Indeed. MA. “the little bar I’ve been placing above the inverted A. constitute a knowable entity.mathinees-lacaniennes. that. at the invitation of Maria Cristina Aguirre. New York. (1969). Chapitres 1-4. and not necessary. however. In Conditions. N. Columbia University.32 The inscription does not. Éléments de mathématique. B. It is an expanded version of the paper delivered in French to the Pont-Freudien association in Montreal on 11 September 2013 at the invitation of Anne Béraud. in any case. Available online at: www. W. Kierkegaard. (2005). Kneale. & Harman. D. C. The Semantics of Natural Language. M. “What is Cantor’s continuum problem?”. (1972). (1996).) 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A. audiovisual recording available online at: www.. Peirce and Contemporary Thought. “Commentaire sur l’intervention en forme de lettre adressée au docteur Lacan”. (2007). (1930). L. 4: 64-73. “Peirce’s Continuum”. Sutherland. G. In Aristotle. Aristotle’s Prior Analytics. London: Hamish Hamilton. Recanati. “A relationship between Lacanian theory of sexuation and Brouwerian intuitionism”. “Poetry and Subjective Infinity”. “Preface”. Springer. K. 76). Sierpiński. (1973b). (2014). 2 Here in the mid-seventies the vocabulary is specifically one of “naming”. com/sussexleverhulme/. M. Oxford University Press. Rinehart & Winston. “Sur une propriété caractéristique des nombres inaccessibles”. (2009). In Scilicet. 159 A transcription of Kripke’s lectures was published in 1971 in (Davidson & Harman. van Atten. (2013). Prior Analytics. (1970). A. & Tarski. Indianapolis: Hackett. Striker. Recanati. clearly influenced by Kripke’s work to which Lacan was exposed by François Recanati. L. New York: Fordham University Press. In Fundamenta mathematicae. Sinclair. It was reprinted in 1980 as the book Naming and Necessity (Kripke. This should not detract from the fact that the same theme had been present in Lacan’s teaching from a much earlier date. J. Putnam. 3 Except where noted. Sciacchitano. See for example Lacan’s 1960 letter to Winnicott: “the signifier marks the real as much and more than it represents it” (Lacan 1990. H. In Scilicet. Inc. all quotations from Seminar XX reference the French text published by 1 .Lacunae | issue 11 | November 2015 119-127. & Seebach. 1975). Some commentators strove to reply to this part of their critique by brushing up the passage. somewhat notoriously. concerning the definition of the whole number.“I start off from the limit” Seuil: (Lacan. the Other embraces the One plus the object a. Sokal and Bricmont attacked in particular its ostensible definitions of an open set and a limit.] this One. It’s the law we accentuated forcefully last year with regard to the recursive One. is called recursion. The following readings each take this inroad: (Krutzen. 10 Mention cannot be made of this passage without a brief review of the numerous and often contradictory commentaries it has spawned. 160 In 1962. but in so doing they did not manage to head off the renewed critique that Sorkal and Bricmont published in Metascience. 2012). failing as it does to respect the standard mathematical terminology. does not exhaust the function of the Other (Lacan. (Landmann. p. Sokal and Bricmont refute Krips’s commentary on open and closed sets.] – the whole number.. notably the mathematician and psychoanalyst Nathalie Charraud (Charraud. and (Cesbron-Lavau. 14) and on which the parenthetical remark that includes se définit comme plus grand qu’un point.. [. are reduced. 6 As Lacan had said ten years previously in Seminar X: “Since man will never bring the leading edge of his desire this far. but rather that of an open set. again in Seminar X. Notwithstanding the fact that the Book of Seminar XX was published by Éditions du Seuil during Lacan’s lifetime (1975). p. 14-15) does not seem to qualify the concept of limit. 2008). insofar as they are distinct. the English-language translation gives “…titrated for what he calls libido”.. p. plus petit qu’un autre… &c. Indeed. 1998) and the academic Henry Krips (Krips. Aside from what they perceived to be a general imprecision in the argumentation of this chapter. 9 For a much fuller account of the historical development of compactness. 2014. this passage incurred. Other commentators have tried to decipher this passage by referring to other sources.] This false infinity is linked to the kind of metonymy that. without there being anything in either the text or the original recording that would support this allusion to titration. one is able to say that man’s jouissance and woman’s jouissance will never conjoin organically” (Lacan. see (Pier. Lacan was already insisting on “finite desire” in opposition to a more circumspect “infinity of desire”: this pseudo-infinity is due to one thing alone [. the Other embraces the One plus the fault-line that is structured in accordance with a “true” infinity (like the infinity of a compact space).. 5 We have tackled this intricate interplay between nomination and the real of jouissance in (Price. 265). Lacan is indeed uncharacteristically rapid in his presentation of the elements necessary for a clear comprehension of the mathematical theorem at issue. it has . pp. (Lacan 1975. 2000). 1993). 1980). 26). In 1972. 2007). the scorn of Sokal and Bricmont in 1997. notably the recording of this lesson of the seminar on which one can clearly hear Lacan utter en un nombre fini d’ensembles and not sur un nombre infini d’ensembles (Lacan 1975. Such as it is reproduced on page 14 of the Seuil edition. as well as his elaboration on “finite” and “countable” coverings. 7 For instance. 4 Curiously.. to which at the end of the day the succession of signifying elements. But [. 2014. 8 The sole published English-language translation of Book XX lamentably offers the reader no means of finding his bearings in this distinction.. 1969. which I’m pegging out. we would refer the reader to (Allen. 2013). 17 Recent translators and commentators have tended to maintain the use of “all”. whilst the second ὴ conforms more to the usage of the English “or” that we find defined in the OED. 9-10). put her finger on it and swore to me earlier that she would find the place again for me (Lacan. for your use. rather than that of transmitting any precious in extremis observations from the historian. 2009). who is here today. 18 Cf. by using the μὴ πάντες. it is hard to offer a comprehensive reading of this application of topology to the space of sexual jouissance. sure enough that my daughter. as it were. 2008). 2008) and (Grigg. which is the opposition. 1969. have been defined in Lacan’s analogical use of them. 12 Blanché the logician and epistemologist was seeking to formalise further Aristotelian logic. I haven’t managed to find it again. 14 Actually. since the phallic function localises jouissance at a specific site or sites in the body as the result of castration. putting him to the question on this matter. “everyman”. and tome. pp. pp. in this case denoting the negative particular. The reader of this Hommage (Darmon. Note that Blanché expanded the square of oppositions into a logical hexagon that embraces six statements. 8). 15 Regrettably. so long as neither the points of the space. notably Gisela Striker in (Striker. the tomos of a cut or a section. but the third is proffered as a “recreational” amusement and is immediately discounted as bearing nothing that would allow it to be upheld. 2010) is thus left utterly in the dark as to Brunschwig’s actual reckoning of Lacan’s work. p. the same author who took the initiative of tracking down Brunschwig in the year prior to the latter’s death. It consists in dispensing with any relations of contradiction between universal and particular by transforming them into relations of implication (Brunschwig. however. 14). 2005. and which I picked out from the Organon. dismissed by Aristotle. with its author exposing himself in his chief mission of debunking contemporary commentators. 13 Furthermore. the opening lesson of Seminar XXIII: Hence my formula about woman. to the universal of the πάν. What we do take on board in their critique is that. 11 On this and related issues. see (Le Gaufey. 2005. For examples of explicit and extended matching between Lacan’s not-all and the maximal reading of the negative particular. 19 161 .Lacunae | issue 11 | November 2015 to be admitted that Lacan’s words as preserved on audio tape offer a sharply divergent version of the details that were to become the object of Sokal and Bricmont’s critique. p. since the first ὴ marks the disjunction between the affirmative particular and the negative particular. the status of the two occurrences of the conjunction “or” ought not to be accorded equal weight. as Brunschwig also observes. 4. but is dependent upon elements developed further on in the text (Brunschwig. Thomme is both tout homme.: “connecting two words denoting the same thing”. nor the open sets. that this inference cannot be established from the grammar alone. whereas Brunschwig the historian and philologist was concerned with identifying the particularities of Aristotle’s thinking in his time. by the very same – surely this is no coincidence – who floundered in his borné rendering. See also the note on “Every and all” from the chapter on “The Orthodox dictum Semantics” in (Malink. saw fit in his article to blur the boundaries between the remarks of the deceased and his own notions. 16 This has not stopped them from being used for just that. 53-65). but I did read it there. for which he was seeking higher authorisation. Brunschwig will add. he offers three.a. and nothing more. 21 In the Seuil version. alleging that Miller is incorrect in asserting that the universe of discourse in Aristotelian logic is finite.com/90546839. 25 Note also that in his 1944 article on Russell’s mathematical logic. 464). however far the series is pursued. 24 In Gödel’s revised version of the article in 1964. too. Gödel wrote of “this transfinite theorem of reducibility” (1944. p. whether maximal or minimal.. In section IV of his “Note threaded stitch by stitch” in the appendix to Le séminaire livre XXIII. read the Lacanian not-all as compatible with the negative particular. 20 A conjoint is also a “spouse”. 127). In the end. but he amplifies Miller’s diagnosis of an intuitionistic model of a potential infinite in Lacan. Lacan stresses that her fantasy is not the act of castration itself. 1949.“I start off from the limit” Woman’s conjoining with the man could be her sexual encounter with the man or her hysterical performance of faire l’homme. Recall that the fantasy of the woman in the film is to kill her partner. Lacan’s later reflection on Nagisa Oshima’s film In the Realm of the Senses: S of barred A is something altogether different from F. 208) The Australian philosopher Russell Grigg (Grigg 2005) takes issue with this reading of the not-all. however. contrasting it with the Cantorian model of the actual infinite. As such. p. p. Grigg does. If the law of the formation of the series (all As are B) has not been posited at the outset. See endnote 10 above. As for what the woman fantasises. 265). 34-5). Prynne’s remark to Keston Sutherland. Therefore. (Lacan. 22 Consider. he makes love with his unconscious. like Miller. Aristotle’s not-all plays only on lack and incompleteness [. and Sutherland’s reply. either way. p. the same note has been amended to include the observation that since (non strong) accessibility is not equivalent to strong accessibility for finite numbers. if this is really what the film presents us with. see J. see (Koyré. 23 We refer to the Seuil text with one proviso: Lacan’s remark that Aleph-naught se trouve réaliser le même cas. Lacan’s not-all is deployed on the contrary in an infinite universe. Jacques-Alain Miller argues that 162 Aristotelian quantification is inscribed into a universe of discourse that is finite. p. “playing the man” (Lacan.H. 2014). 136). has been expanded to read “se trouve réaliser le même cas que le 1”.. this definition is appended to “the limit”: la limite est ce qui se définit comme… &c. its equivalence is doubtful in the case of transfinite numbers (Gödel. 2001. 2005. pp. ℵ0 realises the same case as 2. 27 Brouwer’s creating subject has also been likened to Husserl’s version of the transcendental subject . Prynne’s remark is in response to (Sutherland. impedes the encounter (Lacan. On related issues of actualised subjective infinity and objective infinity. and it is constructed on the intuitionistic model of a choice sequence: the emphasis shifts to the impossibility of stating the universality of the predicate. She then cuts off his penis. it is something that. side with those commentators who. It is not that with which man makes love. but removed the formula from the 1964 and 1972 reprints of the text.]. For more on the infinite in antiquity. In our understanding. at: vimeo. it will be impossible ever to conclude that this is the case for all of them. the sequence is lawless. 26 This aspect of the not-all has also given rise to a recent disputation that is of some pertinence here. 2005. and even if from one moment to the next it has been confirmed that there is no A that is not B. 1964. but then its potentially is destroyed. Grigg reponds to Badiou by making a persuasive case for Lacan as a constructivist in his mathematics. p.Lacunae | issue 11 | November 2015 (van Atten. p. even if it looks like a subject” (Lacan. We understand the absence of calculation at the inter-subjective level: there is no calculation or judgement as to the reception of the work. and he sees this as a result of Lacan’s reliance on a model that is “pre-Cantorian” (Badiou. 15-16) and (Mayo-Wilson. impossibilities. thus the 2 has no rational explanation in Hegel’s sense. On the coincidences between Peirce’s notions and Brouwer’s. However. p. he is the dream itself ” (Lacan. 125). 93-4). (Recanati. A poem that is being written. 2011). p. Lacan’s comment on Finnegans Wake: “the dreamer is not any one character. 2006. 46). and so on. p. (Recanati. Our critique 163 . p. it is the general possibility of non-effectuated. 2005. pp. 1973a. p. p. Lacan matches Marx’s ideal worker with Freud’s dream-work in Television (Lacan. while still remaining classical in his logic (Grigg. 57)? 32 Akihiro Kanamori has argued that “a historical misrepresentation” has been perpetrated that constantly pits constructivity against Cantorian and post-Cantorian methodology. 16). and also: “I am not a poet. 296). verb. 1977. Only the potential embraces indistinctly all the points of a set. pp. 1900a. it has no necessary explanation. but the same author shows that “non-lawlike sequences in no way fit into Husserl’s picture” (p. it is interesting to compare the intuitionistic critique of the Cantorian model with the critique by C. Our hypothesis is that Lacan’s approach to jouissance as possessing a compact topology is an attempt to formalise this ground. 31 Van Atten compares the unfolding of the choice sequence to the unfolding of a melody (2007. 28 “All the places” is not to be understood as “all the characters”. S. non-inscribed. In the aforecited article. see (Petrakis. xl). 58) Recanati develops this in a letter addressed to Lacan on 18 June 1972. “grew out of work by analysts with a definite constructive bent” (Kanamori. it does not entail any necessity. 2013. 64) Recanati concludes with the observation that. but a poem. 179). 1990. 21). Cf. article. 30 Lacan’s reference to intuitionism features much further on in Seminar XX. 1992. only a brand of calculating that is restricted to “giving things a new form” (Freud. that is. which Lacan published in issue 4 of Scilicet (the same that includes “L’étourdit”): A potentiality is only ever realised individually. Alain Badiou criticised Lacan’s use of the not-all for what he perceived as a failure to deduce an affirmative existential premise from the negation on the universal. Could this be the same property to which Kierkegaard demonstrated such great sensitivity when he qualified the myth of Don Juan as “intrinsically altogether musical” (Kierkegaard. However. 1995. 19. “the ground forms the link between the potential and singularity”. 1987. Thus the potential has no common measure with the order of its individual realisations. In fact. p. pp. p. “it is somewhat ironic but also revealing” that pushing the mathematical frontier of the actual infinite past Aleph-naught. see (Putnam. preposition. as Malcolm Lowry would say of his Under the Volcano (quoted in Sinclair. in relation to the inscriptions that are produced there. 29 For more on Brouwer’s fan theorem. p. Recanati isolates the Peircean concept of potential. p. p. 1996. 15). which is the locus in which impossibilities are inscribed. but as every noun. 507). 14. 2007. Peirce that was presented by François Recanati in the penultimate lesson of Seminar XIX. adverb. 58-60). that is. 1973b. undated). 293). and Cantor. We have shown that Lacan uses compactness to inscribe the not-all as a covering on the bounded yet infinite locus of sexual jouissance. for which he posits two complementary characterisations. Tarski. Other jouissance is only implied as that which falls under the covering. though not within the bounded interval. 164 . p. seemingly unaware that his quarrel is with Sierpiński. and not the psychoanalyst. But not even this implication is mentioned by Lacan. he strays further off course by seeking to refute Lacan’s use of the definition of strong accessibility. and finds it in feminine jouissance (Badiou.“I start off from the limit” of Badiou’s article is slightly different: Badiou goes looking for the infinite field in which the notall operates. who carefully restricts his argument to the field of sexual jouissance. Then. 1992. It is Badiou’s identification of the not-all with feminine jouissance that leads him into his various divergences from Lacan’s use of the formulas of sexuation.
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