Cosmos and History: The Journal of Natural and Social Philosophy, Vol 2, No 1-2 (2006)

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chiesa

Count-as-one, Forming-into-one, Unary Trait, S1

Lorenzo Chiesa

Abstract: While a significant amount of research has recently been carried out that investigates the similarities and differences between Alain Badiou and Jacques Lacan’s theories of the subject, less attention has been paid to the direct relationship between the latter and Badiou’s set-theoretical ontology. This article applies some of the most important conceptual propositions advanced in the first two parts of Being and Event to the key psychoanalytic issue of the identification of the conscious and unconscious subject as expounded by Lacan in his ninth Seminar, L’identification. More specifically, this article aims to show how Badiou’s notions of the “count-as-one” and the “forming-into-one” can profitably be put to work in order better to understand Lacan’s notions of the “unary trait” and the S1, the “master-signifier”. What is at stake in both cases is the relationship between structure and metastructure, presentation and representation. Furthermore, this article provides an outline for a set-theoretical formalization of the relation between consciousness and the unconscious as developed by Lacan in L’identification. Lacan’s breaking of the solidarity between unity and totality allows him to work with parts: from the inexistence of totality as a one follows the possibility of thinking the part as “partial system”. Lacan identifies this system with the unconscious. Applying a number of set-theoretical axioms, this article argues that the existence of the unconscious as partial system ultimately relies on the in-existence of the void, or, more specifically, the existence of the void as part that in-exists as element.

Keywords: Badiou; Lacan; Ontology; Psychoanalysis; Set-theory; Subject; Identification; Number

‘Let us give Lacan his due: he was the first to make a systematic use of numericality’

—Alain Badiou

Introduction

According to Alain Badiou, psychoanalysis thinks the amorous ‘procedure’, one of the four procedures for the sake of which the ‘abstract categories’ of his Being and Event have been formulated.[1] Badiou invites psychoanalysts and psychoanalytic theorists to ‘practice’ these categories in their own field.[2] Against the background of such an invitation, in this paper, it is my intention to outline a possible application of some of the most important conceptual propositions advanced in the first two parts of Being and Event to the key psychoanalytic issue of the identification of the (conscious and unconscious) subject as expounded by Jacques Lacan in his ninth Seminar, L’identification. More specifically, I aim to show how Badiou’s notions of the ‘count-as-one’ and the ‘forming-into-one’ can profitably be put to work in order better to understand Lacan’s notions of the ‘unary trait’ and the S1, the ‘master-signifier’. What is at stake in both cases is the relationship between structure and metastructure, presentation and representation, starting from the common premise that ‘the one, which is not, solely exists as operation’ (BE 24) (Badiou), as an ‘instrument’ (Lacan) which is not ‘the one of Parmenides, nor the one of Plotinus, nor the one of any totality’.[3] It should be noticed that, although L’identification arguably remains one of Lacan’s most abstract Seminars, it is nevertheless the case that the topics it discusses have vast repercussions for basic technical questions concerning the cure, such as the handling of the transference and the emergence of anxiety. Applying Being and Event to the practice of psychoanalytic thought will thus also implicitly indicate its relevance to the practice of psychoanalytic treatment.

In Being and Event, Badiou endeavours to think philosophically, that is meta-ontologically, what ‘circulates’ between the modern theories of the subject, in primis the Lacanian one, and ontology understood as axiomatic set-theory (BE 3). As Justin Clemens and Oliver Feltham rightly remark, Badiou does not merge ontology into the theory of the subject, ‘rather, the tension between the two drives his investigations’.[4] This tension is what appears to be annulled when one ‘practices’ Badiou’s categories across Lacanian psychoanalysis. While a significant amount of research has recently been carried out that carefully investigates the similarities and differences between Badiou and Lacan’s theories of the subject,[5] less attention has been paid to the direct relationship between the latter and Badiou’s set-theoretical ontology. Badiou’s ontology of the One and the Multiple—the ‘a priori conditions of any possible ontology’—relies on the ‘law that the one is not’ (BE 23, 28). By Badiou’s own admission on page 1 of Meditation One of Being and Event, this law is closely associated to Lacan’s pathbreaking principle according to which ‘there is [symbolic] Oneness’ (BE 23)—or better, there is only symbolic Oneness. What Badiou fails to emphasize in this context is that this principle is, for Lacan, confined to a theory of the subject: in spite of proposing important hypotheses about being, Lacan never really developed any ontology independently of his notion of subjectivity.[6]

My considerations should therefore always be measured against the threat of a short circuit in Badiou’s magnum opus, which I do not intend to investigate any further in this occasion. On the one hand, Badiou’s theory of the subject-event in Being and Event may rightly be labelled as ‘beyond Lacan’—as the title of Part VIII of the book suggests—due to his rigorous philosophical appropriation of Paul Cohen’s mathematical notion of forcing. On the other hand, in spite of its reliance on the radical thesis according to which ‘ontology […] is nothing other than mathematics itself’ (BE xiii), Badiou’s solid ontological edifice is itself amply anticipated by Lacan’s own theory of the subject. The latter is indeed based on the principle that there is only symbolic Oneness—or, adopting a formula closer to Badiou’s own terminology, ‘there is no One except in mathematics’[7]—albeit in an often hesitant and imprecise manner. Thus, the least we can say is that, in practising the ontological categories made available by Being and Event across Lacanian psychoanalytic notions, we will not be surprised to discover a high degree of compatibility between them. The ideal result of such a practice would be nothing less than an accurate set-theoretical formalization of the relation between consciousness and the unconscious, succeeding there where Lacan’s courageous attempts to demonstrate that mathematical topology is structure—and meta-structure—failed.[8]

Count-as-one, One, Phantom of Inconsistency

‘Everything turns on mastering the gap between the presupposition (that must be rejected) of a being of the one and the thesis of its “there is”’ (BE 23). For Badiou, the one is not, yet it exists as an operation, the count-as-one. The count-as-one is not a presentation either: what presents itself, a situation, is multiple. However, every situation is structured by means of the operation of the count-as-one. Thus, the relation between the multiple and the one is retroactive: the multiple will have preceded the one only after having necessarily been structured by means of the count-as-one. As Badiou puts it, ‘the count-as-one (the structure) installs the universal pertinence of the one/multiple couple for any situation’ (BE 24). This amounts to saying that, with regard to presentation, the one is also an ‘operational result’ (BE 24, my emphasis). A concomitant splitting occurs on the side of the multiple: inconsistent multiplicities, multiples that are retroactively understood as non-one ‘as soon as being-one is a result’, are to be distinguished from consistent multiplicities, ‘multiple[s] as “several-ones” counted by the action of structure’ (BE 25). We can thus conclude that the law that ‘the one is not’ is at the same time the law that ‘the one is a law of the multiple’ (BE 25).

It is vital to stress that, according to Badiou, what is normally presented in any situation is the fact that the one is: all that is presented in a situation is counted-as-one, which is to say that the principle ‘the one is not’ cannot be presented in it. At this level, inconsistent multiplicity is ‘solely the presupposition that prior to the count the one is not’ (BE 52). Having said this, we should also keep in mind that the one is an operational result, and that for this reason, there must be ‘“something” of the multiple [that] does not absolutely coincide with the result’ (BE 53). In other words, in situations, which are as such always structured, a remainder exceeds the one of consistent multiplicities, and this can be nothing other than the very operation, the law of the count-as-one, from which the one results. With regard to a given situation, inconsistent multiplicities, the pure multiple, are therefore included as an exclusion: adopting a quasi-psychoanalytic terminology whose Lacanian affinities, as we shall later see, are remarkable, Badiou suggests that this inclusive exclusion is what ‘causes the structured presentation to waver towards the phantom of inconsistency’ (BE 53, my emphasis). This phantom, a retroactive by-product of the count-as-one, cannot itself be presented, yet it is included in the situation ‘in the name of what “would be” the presentation itself, the presentation “in-itself”, if what the law does not authorize to think was thinkable: that the one is not’ (BE 53). Put differently, from the structured situational standpoint for which the law is ‘the one is a law of the multiple’, the phantom of inconsistency amounts to the excluded law that ‘the one is not’.

More specifically, the pure multiple, unpresentable insofar as it is excluded by the law ‘the one is a law of the multiple’, is nothing from the standpoint of the situation. As Badiou observes, being-nothing is different from non-being: ‘There is a being of nothing, as form of the unpresentable. The “nothing” is what names the unperceivable gap […] between […] the one as result and the one as operation’ (BE 54, my emphasis). But being-nothing is not, just as the one is not: the there is of being-nothing does not instigate any search for the nothing, and thus avoids falling back into an ontology of presence: ‘The nothing is neither a place nor a term of the situation. For if the nothing were a term that could only mean one thing: that it had been counted as one’ (BE 54). Rather, the nothing is the non-one of any count-as-one; or, the phantom of inconsistency is the name ‘nothing’, which is not a-nothing [un-rien]. At this stage, it is important to emphasize that, for Badiou, the nothing is both the pure unpresentable multiple, the name of unpresentation in presentation, and the operation of the count, that which exceeds the one-result.[9] The nothing, or better the void as its local—yet unlocalizable—occurrence, has a dual status.

Let us consider this question further. Being qua being is neither one nor multiple; although being is certainly presented as multiple, being indeed occurs in every presentation, being does not present itself (see BE 27): being qua being ‘is what presents (itself)’ (BE 24), and, it is as such, ‘in being foreclosed from presentation’ (BE 27), that it is sayable. Thus the void is the name of being insofar as the void indicates precisely that nothing is presented; by means of the void, presentation gives us the non-access to an unpresentable (see BE 56). However, in addition to naming being as an unpresentable, a non-one ‘that wanders in the presentation in the form of a subtraction’, ‘the subtractive face of the count’ (BE 55, my emphasis), the void also concomitantly names being in the very operation of the count-as-one which, in exceeding the presentable one-result, sutures a situation to its being (a suture is quite literally an operation). Again, the void is the name of being in two inextricable ways. Both exclude the possibility that the void may be localized and thus encountered in the normal regime of structured situations: Badiou believes that, from the situational standpoint, the void as name of being is equivalent to an ‘absolute “unconscious” of the void’ [‘inconscience du vide] (BE 56). The phantom of inconsistency cannot be conscious.

Unicity, Forming-into-one, Anxiety of the Void

Badiou states that ‘there are four meanings concealed beneath the single signifier “one”’ (BE 89). The first two distinguish the count-as-one from the one: as we have already seen, the one, which is not, can only be the retroactive and fictive effect of a structural count, the count-as-one. Since being is always presented as multiple, multiple of multiples, what is really counted as one through the ‘nominal seal’ [sceau] of the count-as-one is the multiple-of-multiples (BE 90): multiples are counted by the count-as-one as ‘one-multiples’, consistent multiplicities, ‘multiples as “several ones”’. In other words, the couple one/multiple installed by the count-as-one qua structure should ultimately be understood as the couple one-result/one-multiple.

The third meaning of the signifier ‘one’ is, for Badiou, unicity. Unicity is not a being, ‘but a predicate of the multiple’ (BE 90). Multiples are unique: this simply means, ‘a multiple is different from any other’ (BE 68). What differentiates a multiple from all other multiples is its proper name, that is, being counted as one-multiple by the nominal seal of the count-as-one. A notion of unicity that has done with any filiation from the being of the one and only accepts the one as result, is what allows us to think the relationship between the same and the other in a new way: given that the one is not, ‘it is in regard to themselves that the others are Others’ (BE 33). A multiple is Other than any other multiple only due to its unicity. Or, ‘the Other […] cannot designate the gap between the one and the others-than-one [autres-que-l’un], because the one is not’ (BE 33 trans. modified). Put simply, the Other is coextensive to the unicity of the others, not the one.

Finally, there is a fourth meaning of the signifier ‘one’, which Badiou designates as ‘forming-into-one’ [mise-en-un]. This is basically a ‘second count’, ‘a count of the count’ (BE 83-4 my emphasis), which should be understood in two inextricable ways according to the two sides of the couple one-result/one-multiple installed by the first count, the count-as-one. Indeed, the forming-into-one indicates the concomitant possibility of both ‘count[ing] as one an already counted one-multiple’ and ‘apply[ing] the count to the one-result of the count’ (BE 90). Such an operation is possible insofar as, after the first count, the one is not really distinguishable from the multiple: given that the one is the result of the structuring count that makes the multiple consist, it remains immanent to presentation, which, as such, can present only multiples. Differently put, the one-multiple results from the count-as-one, and for this very reason the one-result can only itself be a multiple.

If we now consider that the count-as-one is, as we have already remarked, a law that produces a name—the proper name of each multiple as unique—it also follows that the forming-into-one will be nothing other than ‘submitting to the law the names that it produces’ (BE 90). It is important to emphasize that the resulting ‘multiple of names’ (BE 91), the product of the forming-into-one, is itself a multiple: even after the second count has taken place, the one is solely a retroactive fiction, albeit a more elaborate one, since it now transcends presentation into representation. At this level, the one as representation can be distinguished from presentation as multiple, yet it remains a re-presentation of a multiple and thus a fiction. On the other hand, notice that retroactive representation will necessarily have a retroactive effect on retroactive presentation: it is also in this sense that I understand Badiou’s suggestion according to which ‘forming-into-one is not really distinct from the count-as-one’ (BE 91).

The relation between the two counts, the counting-as-one of presentation and the forming-into-one of representation, is to be conceived of in terms of a relation between structure and metastructure, situation and the state of the situation. Although Badiou insists on differentiating the two counts—they are ‘absolutely distinct’ (BE 83)—he also affirms that the ‘reduplication’ of the count is necessary—‘every structure call[s] upon a metastructure’ (BE 84)—and consequently structure and metastructure, situation and its state, are not really distinct. The reason for this necessity is ‘countering the danger of the void’, warding it off from structured presentation (BE 84): all situations are thus defined by an inevitable ‘anxiety of the void’ [angoisse du vide] (BE 93). As we have seen, the unpresentable and unlocalizable character of the void as the name of a situation’s (inconsistent) being is what guarantees the consistency of this very situation, the emergence of consistent multiplicity. That is to say, it is only insofar as a structured presentation does not encounter ‘its’ own void that the situational one is not ruined. However, we have also seen that, within presentation, something exceeds the count: the very operation of the structural count-as-one qua nothing. This means that the errant void could fix itself in the guise of structure: ‘It is […] possible that, subtracted from the count, and by consequence a-structured, the structure itself be the point where the void is given’ (BE 93). In order to counter the danger of the void, it is therefore necessary to structure the structure or ‘that the “there is Oneness” be valid for the count-as-one’ (BE 93).

Significantly, according to Badiou, this redoubling, the second count, should be understood as an imaginarization of the first: if the count-as-one as a (symbolic) operation retroactively produces a fictional (imaginary) one-result, the count’s ‘undergoing, in turn, the operation of a count’ is equivalent to ‘the fictionalizing of the count via the imaginary being conferred upon it’ (BE 95). Put simply, while the first count symbolically produces the one, the second count, the count of the count, imaginarily is one.

Unary Trait as Trait Unique, or ‘La Multiplicité Actuelle’

The central notion of Lacan’s ninth seminar is arguably what, in an explicit attempt to echo the function of the one in set theory, is designated as the unary trait [trait unaire].[10] Generalizing and elaborating on Freud’s notion of the einziger Zug,[11] Lacan believes that identification is ultimately based on identification with the signifier, and the unary trait is ‘what all signifiers have in common’, their ‘support’.[12] More precisely, the one as unary trait is the ‘instrument[13] by means of which identification is made possible: the unary trait is not a one but an operation, a count, that constitutes ‘the foundation of the one’ of identification with the signifier.[14] Simply put, the unary trait should be understood as what produces a ‘stroke’, /, not a unity, let alone a totality:[15] Lacan openly denies that he is taking into consideration any of the many significations of the one proposed by philosophical tradition, rather ‘it is a question of the 1 […] of the primary teacher, the one of “pupil X, write out a hundred lines of 1s for me!”, namely strokes [which have] always been sufficient for minimal notation’.[16]

In describing the unary trait as a count, and even as a first count, the count-as-one, that as such is to be distinguished from a second count, I am far from forcing Lacan’s own terminology. ‘The unary trait begins the function of counting’: this initial ‘activity of counting [that] begins early for the subject’ should not be confused with the activity of ‘establishing collections’.[17] In other words, the count-as-one of the unary trait is what produces the one, many ones, as strokes ///; however, at this stage, there is no second count, or addition, that can count the strokes as 1s, or put differently, that can count the operation of the first count as an operation. As Lacan observes, ‘the unary trait […] supports […] one plus one and one again, the plus being meant there only to mark well [a] difference, where the problem begins is precisely that one can add them together, in other words that two, that three have a meaning’.[18] In order to stress how / + / + / is not the same as 1 + 1 + 1, Lacan goes as far as suggesting that a child may well be able to count up to two and three without being able to operate with numbers: two and three are in this case nothing but a repetition of the / produced by the unary trait, and should be distinguished from the number 2 and 3 understood as 1 + 1 and 1 + 1 + 1. This ‘early’ counting is ineffective when dealing with numbers higher than 3: we should therefore not be surprised ‘when we are told that certain so-called primitive tribes along the mouth of the Amazon were only recently able to discover the virtue of the number four, and raised altars to it’.[19] What is at stake in the gap that separates these two counts is nothing less than the birth of the subject’s identification as modern Cartesian subject split between consciousness and the unconscious.[20]

In what precise sense does the + of the / + / + / ‘mark a difference’ between the strokes produced by the count-as-one of the unary trait? This question certainly has to do with the fact that, throughout Seminar IX, Lacan indiscriminately alternates the phrases ‘trait unaire’ and ‘trait unique’: put simply, a unary trait is a single trait. If the unary trait, as instrumental operation, is the ‘most simple structural trait’ in the sense that it presents ‘no variations’,[21] its sole property will be its unicity. That is to say, the + separating / from / denotes the singleness of the trait as such, the absence of any ‘qualitative difference’ in it,[22] and thus works as an indicator of ‘signifying difference’, ‘difference in the pure state’.[23] Or, every count, every / is absolutely different from any other / without its ever being a question of counting the trait as a qualitatively differentiated 1: as we have just seen, the unary trait precedes number stricto sensu.[24]

Two crucial specifications should be made. Firstly, the unary trait marks ‘difference as such’,[25] which does not mean it is difference as such. This is a straightforward way to distinguish the unary trait from the full-fledged signifier which it ‘supports’. While the unary trait is a stroke, and, significantly enough, it is as ‘letter’ that it can be differentiated from all other strokes,[26] the signifier is ‘the one as difference’, that is, following Saussure, ‘simply being what the others are not’.[27] Thus, unlike the unary trait, the signifier ‘implies [the] function of the unit […] qua pure difference’.[28] According to Lacan, the one as difference, the emergence of ‘the one [which] as such is the Other’,[29] that is the fictional big Other, necessitates the second count.

Secondly and most importantly, ‘at its first appearance, the one manifestly designates actual multiplicity [multiplicité actuelle]’.[30] The count-as-one of the unary trait as trait unique produces nothing other than consistent, that is ‘actual’, multiplicity: Lacan also refers to it as a ‘distinctive unity’ [unité distinctive].[31] In order to illustrate this last point, he evokes a scene of everyday pre-historic life:

I am a hunter […] I kill [an animal], it is an adventure, I kill another of them, it is a second adventure which I can distinguish by certain traits characteristic of the first, but which resembles it essentially by being marked with the same general line. At the fourth, there may be some confusion: what distinguishes it from the second, for example? At the twentieth, how will I know where I am?[32]

Like a child who ‘counts’ without numbers, our primitive man can initially distinguish the second adventure from the first by certain imaginary—intuitively qualitative—traits that are then symbolically presented as a stroke / on an animal rib-bone. Yet, as soon as this occurs, his two adventures are marked by ‘the same general line’, the same kind of stroke which leads to ‘signifying sameness’, //. Although qualitative difference is never eliminated completely, the fact that each ‘adventure’ is, for a ‘limited time’, ‘intuitively’ experienced as new proves to be all the more secondary inasmuch as quality is precisely what is overshadowed by the signifying in-difference of the traits //.[33] From a slightly different perspective, all this amounts to saying that the distinctive unity of the unary trait is still immanent to the situation it counts, and thus runs the risk of becoming indistinguishable from the non-situation of a primitive man who is still lacking any ‘method of location’.[34] Thus, marking signifying difference as such results in nothing other than signifying sameness, in-difference, if the count is not itself counted, if the ‘actual multiplicity’ /, the one-multiple that presents the hunter’s adventure, is not itself represented as 1. We can then understand why Lacan pays so much attention to the later appearance in pre-history of ‘a series of strokes’ carved on an animal rib-bone—‘First two, then a little interval and afterwards five, and then it recommences…’.[35] It is only at this level, that of the count of the count, where 1 + 1 retroactively replaces //, that the properly human symbolic dimension begins: this is the subject’s own identification with the signifier.

Finally, it is quite remarkable that, in this context, Lacan himself draws a comparison between the use of the / made by the primitive hunter and the notion of the one with which set theory operates. Against what we are taught at school—‘You cannot add up oranges and apples, pears with carrots and so on’[36]—the primitive hunter counts as one a multiple ‘adventure’ made by irreconcilable ‘objects’ and ‘things’. In the same way, in set theory, ‘you can very well add up what you want’.[37] More technically, Lacan acknowledges that ‘in what one calls the elements of sets, it is not a matter of objects, or of things’, it is rather a question of the multiples of a multiple. In other words, at the level of presentation, the set is a one-multiple, what Lacan refers to as ‘actual multiplicity’: the count-as-one of the unary trait presents a multiple-of-multiples, or to put with Badiou, a ‘multiple as “several ones”’ (BE 25).

In order to appreciate better the proximity between Badiou’s consistent multiplicity and Lacan’s distinctive unity we should not lose sight of the following convergence. Badiou’s consistent multiplicity, the one-multiple, is ‘initially’ determined solely by its unicity, in all cases; unicity is the property of consistent multiplicity qua counted-as-one, independently of any other possible property of a situation (or set). What matters at the level of presentation without representation is the proper name alpha, a letter that ‘seals’ the multiple, rather than the extension of what is being presented—the terms or elements of the situation. Indeed, extension is not properly defined before the second count takes place and the state of the situation is established retroactively: only at that stage, the one-multiple will have been counted as a situation (or, ontologically, all the parts of a set will have formed the elements of a set as the powerset). A situation is not identical to what is being presented in it. Following Ray Brassier, another way to put this would be to say that presentation as such, presentation without representation—the prehistoric hunter’s ‘adventure’—is an anti-phenomenon.[38]

Letter and Proper Name, or ‘A is not A’

The most basic formula of identification is ‘A is A’. Lacan believes that its apparent simplicity conceals a number of problems. It is therefore only insofar as we question this formula that we can really grasp the difficulties involved in identification. This questioning is strictly related to ‘the [signifying] function of the one’ and, conversely, ‘the extended use of the signifier in mathematics’.[39] More specifically, for Lacan, ‘A is A’ presupposes first of all the existence of A, the emergence of the letter, which, as we have seen, should be understood as a unary trait, a first count. Lacan’s bold propositions according to which ‘“A is A” is a belief’ and ‘There is no tautology’ are thus always to be considered against the background of the dimension of the letter. ‘It is not insofar as the first A and the second A mean different things that I say that there is no tautology, it is in the very status of A that there is inscribed that A cannot be A’;[40] ‘A is not A’ means that A is not identical to itself, or, to use a well-known Lacanian locution, A is barred, not that A is actually B: more precisely, the letter A as unary trait counts as one but is not a one.

Lacan’s theory of the proper name and his theory of writing aim to show how the true nature of the proper name is the letter as unary trait, which in turn is inextricable from the written mark. The proper name cannot be understood as a ‘word for particulars’, a definition proposed by Bertrand Russell: this would soon lead us to paradoxes such as ‘Socrates’ not being a proper name since, for us, it is no longer a particular but an abbreviated description—‘Socrates’ is indeed Plato’s master, the man who drank the hemlock, etc.—or, conversely, the demonstrative ‘this’ is a particular and could therefore be designated as ‘John’. Relying in part on the linguist Alan Gardiner, Lacan believes that a proper name functions on the basis of the distinction between meaning and signifying material (signified and signifier); however, departing from him, he specifies that it should not be identified with a distinctive sound to which the subject pays particular attention as sound. From a structuralist standpoint, it is indeed a matter of fact that all language is based on the differentiality of distinctive sounds, or phonomes; what is more, ‘it is absolutely not true […] that each time we pronounce a proper name we are psychologically aware of the accent put on the sonant material as such’.[41] In other words, the problem with Gardiner’s notion of proper name is that he relies on a psychologically substantialist idea of the subject: the subject is for him simply someone who pays attention to signifiers when they are proper names. On the other hand, for Lacan, the subject can be defined only ‘with reference’ to signifiers, not as someone underlying their functioning;[42] the central role played by proper names in the subject’s identification should thus be explained solely in terms of the signifier, especially according to its most basic appearance, the unary trait of the letter. ‘There cannot be a definition of the proper name except in the measure that we are aware of the relationship between the naming utterance and something which in its radical nature is of the order of the letter’.[43]

Most importantly, the proper name’s relation to the letter as unary trait is itself dependent on the logic of the written mark. Lacan plainly points out that ‘the characteristic of the proper name is always […] linked to […] writing’.[44] As Aaron Schuster remarks in his elegant commentary on the fourth lesson of Seminar IX:

The crucial point for Lacan is that writing emerges first as ‘the isolation of the signifying trait’ (unary trait) which then becomes—again, retroactively—the basic support for the phoneticization of language, i.e. the treasury of signifiers proper. Far from being simply the translation of a more ‘original’ speech, it is speech itself that ultimately finds its basis in the exteriority of the written mark. ‘What results from this’, Lacan adds, is that the proper name qua brand […] ought to be linked not with sound à la Gardiner, but with writing. The proof of this is found in the decipherment of unknown languages: one always begins by looking for proper names since they remain the same across all languages […]. In the proper name, one thus rediscovers within the synchronic order of language a signifier in its ‘pure state’, a state represented in diachronic (pre-)history by the primitive hunter’s notched bone.[45]

Let us dwell on these issues. It is doubtless the case that man’s vocal utterings preceded writing, chronologically speaking; however, language stricto sensu as determined by the function of the signifier is, for Lacan, ultimately retroactively dependent on writing. Conversely, the letter as writing potential ‘was waiting to be phoneticized’: this is what we have at a first stage in prehistory, the simple traits, or strokes, of primitive bone etchings. Lacan is careful in specifying that, as we have already seen, the letter as unary trait always involves an initial imaginary dimension—‘something figurative’—that is soon ‘effaced’:[46] this is valid both for the simple stroke / carved on an animal’s rib—which originally marked the image of an ‘adventure’—and, even more so, the more sophisticated traits used in ideographic notation—the schematic representation of the head * of the animal I killed during my adventure still functions as a unary trait.

At a second stage, the writing of the trait—that designates something imaginary—is accompanied by the utterance of a phoneme; but, for the time being, the utterance ‘m’ is not as yet detached from the image represented by the trait *. Put differently, phoneticization here depends on the designation of an object via the marking of the trait. In a third and final stage, which determines the retroactive passage from prehistory to history, we witness the reversal of this relation: now the marking of the trait depends solely on phoneticization. This is writing proper: the letter is retroactively transformed into a signifier and, being an element in a differential structure of other signifiers, acquires a life that is completely independent of the object it used to designate.

From a slightly different perspective, we can say that writing proper—and language stricto sensu with it—only really begins when the marking of the trait * is phoneticized as trait, that is, named as such. At that point, * becomes the support of the phoneme ‘m’ which was previously the mere ‘sound’ of the object designated by *—the mooing head of the animal I killed. We retroactively move from the—ultimately animalic—sound ‘m’ to the—human—phonematic signifier ‘m’ only when ‘m’ can even be regarded as a proper name ‘M’. As Lacan has it, ‘It is a fact that letters have names’, ‘a’ is named ‘alpha’.[47] We should pay particular attention to this apparently trivial remark which, in its expanded form, reads as follows: it is only insofar as ‘a’ has a name, insofar as ‘A’ is a proper name, that the letter ‘a’ can be said to be ‘a’, that ‘a’ is identical to itself (albeit as part of a differential ‘sonant structure’). The ‘idiotic character’[48] of the proper name—its meaninglessness, the fact that, as already noted by John Stuart Mill, ‘it is not the meaning of the object that it brings with it’[49]—is nothing less than the precondition of identification: Lacan is Lacan only if ‘Lacan’ is a proper name.

More specifically, ‘the proper name […] specifies as such […] the rooting of the subject’ precisely insofar as it is ‘more specially linked than any other, not to phonematicization as such, the structure of language, but to what in language is already ready […] to receive this informing by the trait’.[50] The proper name is closer to the letter than to the symbolic proper: it approaches the unary trait by redoubling its operation, the idiotic in-difference of its count, and in this way guarantees the consistency of the structure of language, the differentially phonematic chain of signifiers. In other words, the proper name ‘make[s] us question ourselves about what is at stake at this radical, archaic point that we must necessarily suppose to be at the origin of the unconscious’, that is, primary repression.[51] And this in two complementary ways: the proper name as the redoubling of the letter, the unary trait, raises the issue of ‘the attachment of language to the real’,[52] as well as that of negation as directly involved in ‘the genesis of language’ in the guise of an ‘existential relationship’.[53] If the letter as unary trait is that which retroactively makes the real object exist as negated (be it the killed animal or the mother’s breast), the proper name is that which, operating retroactively on the letter, allows the subject’s own identification by naming this very negation.

ø = ‘There exists a negation’

Although the origins of writing lie outside the concerns of Badiou’s general ontological edifice, it is nevertheless profitable to begin to accommodate Lacan’s reflections on the proper name to the notions made available by Being and Event. Using Badiou’s terminology, we could suggest that, for Lacan, the proper name can be situated on two different levels, that of the situation and that of the state of the situation, while preserving the same ‘sealing’ function. The proper name as letter, the stroke / on the primitive hunter’s bone that counts as one the multiple of the hunter’s ‘adventure’ works exactly like the proper name stricto sensu, the name ‘Lacan’: indeed, the latter forms-into-ones the ‘multiple[s] of names’ (BE 91) (bluntly put, the multiples ‘p.s.y.c.h.o.a.n.a.l.y.s.t’., ‘d.o.c.t.o.r’., ‘b.u.f.f.o.o.n’., ‘f.r.e.n.c.h’., ‘b.o.u.r.g.e.o.i.s’., ‘s.q.u.e.a.k.y’., ‘s.m.o.k.i.n.g’., ‘e.t.o.u.r.d.i.t’. etc.) made of proper names as letters. We could also suggest that the proper name stricto sensu is equivalent to structure in the metastructure, presentation in representation. It amounts to the insistence of the unary trait of the first count in its meaningless unicity, the insistence of the letter, within the state of a situation—where number and meaning as such are now possible. Consequently, the proper name accounts for the fact that ‘the state of the situation can either be said to be separate (or transcendent) or to be attached (or immanent) with regard to the situation and its native structure’ (BE 98).

At this point, there is a question we cannot postpone any longer: how does Lacan account for that which is being counted, and thus named, by the unary trait of the letter? What is involved in ‘early counting’—with regard to both the phylogenetic ‘adventure’ of the primitive hunter and the ontogenetic emergence of number in the child—is first and foremost ‘the functioning of the sensorium’.[54] This means that it is only with the / of the unary trait that something ‘really exists’ for the subject, that ‘the judgement of existence begins’.[55] Yet, one should note that the unary trait is always necessarily associated with the retroactive effect of negation: the in-different notch on the bone presents the primitive man’s adventure as effaced—significantly enough, under the sign of a killing—just as the ‘early counting’ of the child marks a proto-symbolic relation with an object insofar as he has been frustrated of it. In opposition to what he terms Bergson’s ‘naïve realism’,[56] Lacan believes that negation is not the negation of a primordial affirmation which would affirm the existence of a real that is immediately given. It is doubtless the case that negation ‘supposes the affirmation on which it is based’ but this does not in the least entail that such an affirmation is ‘the affirmation of something of the real which has been simply removed’;[57] affirmation does not precede negation; negation and affirmation occur concomitantly by means of negation. Put differently:

There is no more, and not at all less, in the idea of an object conceived of as not existing, than in the idea of the same object conceived of as existing, because the idea of the object not existing is necessarily the idea of the object existing with, in addition, the representation of an exclusion of this object by the present reality taken as a whole.[58]

To cut a long story short, according to Lacan, what is being counted by the unary trait is the possibility of the real (its affirmation) through the preservation of the ‘rights of the nothing’.[59] In opposition to the false axiom for which anything real is possible, one should always start with the axiom for which ‘the real takes its place only from the not possible’: the possibility of affirming that something exists always relies on a law that safe-guards the nothing. But if the real, or better reality, only originates in the not possible, this not possible, presented as such by negation, is the real. Here, Lacan uses the term ‘real’ in two ways: a) as the possible that follows the not possible; b) as the not possible that originates the possible. But it is in fact possible to think these two acceptations together: the real qua reality ‘takes its place’ only from the possibility of the not possible, the possibility of the real as such. Indeed, ‘this real exists’, Lacan says, as ‘exception’ or ‘exclusion’.[60] In other words, ‘there is not only the not possible at the origin of any enunciating’, but also the possibility of the not possible: the origin of any enunciating is the ‘enunciation of the nothing’,[61] the affirmation of negation accomplished by the unary trait.

Badiou’s philosophy thinks meta-ontologically a set theoretical ontology which relies on the very same axiom, the axiom of the empty set, which formalizes existence at its most basic level. As he writes in Being and Event, ‘the axiom of the empty set states, in substance, that there exists a negation’ (BE 86 my emphasis); it is necessary that the ‘absolutely initial existence be that of a negation’, the existence of an inexistent (BE 67). What negation as the absolutely initial existence negates is belonging: no elements belong to the void-set, not even the void. If, for Badiou’s set theoretical ontology, the void presents the unpresentable as that which ‘alone in-exists’ (BE 69), for Lacan’s theory of the subject, the possibility of the not possible affirms the real that ex-sists as exclusion. If for Badiou, the in-existent void subtractively sutures a situation to its being (inconsistent multiplicity), for Lacan the ex-sistent, or ex-timate, real—the real-of-the-symbolic—retroactively reminds a subject of the undead (an inconsistent real which was and will be not-one, barred in itself, before and after the presence of the symbolic).[62] If, finally, for Badiou, the void as set is absolutely ‘in-different’ in the sense that ‘nothing differentiates it’, its unicity is not based on a difference that can be attested (BE 68), for Lacan, the real-of-the-symbolic is the other side of a ‘distinctive unity’ whose unicity precisely resolves itself, as we have seen, into in-difference. (It is important to bear in mind that the count of the unary trait is two-sided. Moreover, this two-sidedness is somehow ‘unbalanced’ towards one side, that of the nothing/void: just as, for Badiou, the void as name is both unpresentation in presentation and the operation of the count, which as such exceeds the one-result, so for Lacan the ‘enunciation of the nothing’ is both the real-of-the-symbolic as the possibility of the not possible and the very operation of the unary trait, which as such exceeds the distinctive unity).

We must then take seriously Lacan’s provocative remark according to which ‘“A is A” signifies nothing’.[63] Initially, there exists a negation, non-A, no element belongs to the empty set ø: in order for the set alpha to exist, in order for A to be A as a set to which at least one element belongs, the nothing must first be enunciated, ‘signifierized’. At this stage, it is crucial to emphasize that if the ‘primal fact’[64] is the enunciation of the nothing—the void as name, the void-set—then we witness here nothing less than the collapse of the traditional categories of unity and totality. Lacan explains this point quite clearly: ‘Unity and totality appear in the tradition as solidary […] totality being totality with respect to units [and] unity being […] the unity of a whole’; such a solidarity is what is being ‘shattered’ by the ‘other meaning of unity’ he proposes, that of distinctive unity, the -1 brought about by the enunciation of the nothing.[65] From now on, any possible semblance of totality (or unity for that matter) can only be based on the -1, since the primal fact is that the one is not. Note that this is exactly what is ultimately at stake from a philosophical, or better, metaontological, perspective in the revolutionary contribution of axiomatic set-theory. As Badiou remarks:

It would not be an exaggeration to say that the entirety of speculative ontology is taken up with examinations of the connections and disconnections between Unity and Totality. It has been so from the very beginnings of metaphysics, since it is possible to show that Plato essentially has the One prevail over the All whilst Aristotle made the opposite choice.

Set theory sheds light on the fecund frontier between the whole/parts relation and the one/multiple relation; because, at base, it suppresses both of them. The multiple […] for a post-Cantorian is neither supported by the existence of the one nor unfolded as an organic totality. The multiple consists from being without-one (BE 81).

Interestingly enough, in this context, it is Badiou himself who bends his fundamental ontological thesis according to which ‘the one is not’ towards a formulation, ‘being without-one’ [sans-un], that is reminiscent of Lacan’s -1, the ‘one-less’. Unity and totality, the particular and the universal—or better their semblances—can only be conceived of if one begins from the multiple, which initially un-presents itself in the void-set as being without-one. The fact that both elements and sets are multiples-of-multiples and thus become indistinguishable collapses the traditional distinction between unity as an element of a totality and totality as a set of unities. Using natural language paradoxically, we could suggest that a ‘particular’ unity is always already a one-multiple whilst being an element of a ‘universal’ totality which is never as yet a one-multiple. Thus, there is only one possible relation between sets and elements, belonging, ‘which indicates that a multiple is counted as element in the presentation of another multiple’ (BE 81). Besides this, all we can do is count the multiple according to its parts. This is the relation of inclusion, ‘which indicates that a multiple is a sub-multiple [or part] of another multiple’ (BE 81). Such a relation is dealt with by the axiom of the powerset, the set of subsets, among others: this affirms that between belonging and inclusion ‘there is at least the correlation that all the multiples included in a supposedly existing alpha [the initial set] belong to a beta [its powerset]; that is, they form a set, a multiple counted-as-one’ (BE 82 my emphasis). Let us dwell on this last point, which is crucial for Badiou. According to axiomatic set-theory, the following can be stated:

  1. inclusion is derived from belonging as the sole primitive relation between sets and elements, yet belonging and inclusion are distinct;
  2. the fact that inclusion and belonging are distinct entails that there is an excess of inclusion over belonging, the powerset over the set; this excess is an excess in belonging: there is always at least one element of the powerset which does not belong to the initial set;
  3. nothing belongs to the void, not even the void itself;
  4. the void is a subset of any set: by the very fact that nothing belongs to the void, the void is included in everything;
  5. the void possesses a subset, the void itself; hence, the powerset of the void must also exist;
  6. the powerset of the void is the set to which the void alone belongs, since everything included in the void belongs to the powerset of the void; the void, or better its name, is therefore an element of the powerset it forms while it is not an element of itself.

Badiou can thus conclude that the powerset of the void, the set to which the name of the void alone belongs, is the first set that is able to count-as-one the result of the first count, the relation of belonging. Thus, the powerset of the void is what gives us the forming-into-one: indeed, it is only ‘once […] the forming-into-one of ø […] is guaranteed via the power-set axiom applied to the name of the void [that] the operation of forming-into-one is uniformly applicable to any multiple supposed existent’ (BE 91 my emphasis). If, on the one hand, what is presented by the forming-into-one is always the multiple—the effect of its operation is again a one-multiple, the same as on the level of the count-as-one—on the other hand, it is nevertheless the case that the powerset of the void accomplishes something quite remarkable, namely counting the name of the void, the -1, as an element, a 1. In this way the powerset of the void operates against what Badiou calls the ‘errancy of the void’, the fact that, after the first count, the void is included in all sets without belonging to them. As a consequence of this, it is inevitable that we consider the second count as an operation which, by turning the -1 into an element, representing the name of the void, somehow preserves the semblance of the distinction between unity as an element of a totality and totality as a set of unities, even though what is being counted are multiples-of-multiples.

This in no way means that, after the second count has taken place, the void does not continue to err on the level of the first count; after all, the state of a situation can be said to be ‘separate’ (or ‘transcendent’) with regard to the situation. While the retroactive effect of the forming-into-one on the count-as-one definitely makes the void ‘take place’ in a ‘part [that] receives the seal of the one’ (BE 97), its errancy is far from being interrupted within this circumscribed ‘partial’ place. Here, it would certainly be reductive, if not misleading, to regard the situation as a mere part of the state of the situation, since, in a sense, the state is ‘attached’ (or ‘immanent’) to its structure (the powerset is still a set); rather, we should acknowledge the following: the situation as situation characterized by the errancy of the void takes place in a part of the situation as state of the situation. From a slightly different perspective, we can propose that both the initial counting of the multiple in the set and the second counting, that of the parts of the set as elements of the powerset, both structured presentation and metastructured representation, ultimately rely on the void-set—the ‘initial multiple’ as ‘absolutely initial point of being’ (BE 48)—which should always remain errant. As a matter of fact, what should be avoided at all costs as ‘the catastrophe of presentation’, is a ‘fixation of the void’, the presentation’s encounter with its own void (BE 93-94).

It should be stressed that Badiou himself seems implicitly to distinguish the errancy of the void in a situation as such, ‘the pure errancy of the void’ (BE 96 my emphasis) from the errancy of the void at the level of the situation after the state of the situation has been established: this second, impure errancy is nothing other than what he refers to as the ‘unconscious of the void’. Given Badiou’s deliberate choice to employ psychoanalytic terms to describe the basics of his meta-ontological edifice, I do not think I am forcing his argument in finally suggesting that the ‘unconscious of the void’ amounts to the unconscious status of the situation under state control, or put simply, the state’s unconscious. The unconscious of the void, or, significantly enough, the ‘phantom of inconsistency’, is the name retroactively imposed on the name ‘void’, the letter ø, by state repression. Having said this, it must be observed that Badiou fails to emphasize the following: as long as the state of a situation (consciousness) remains both separate from the situation (the unconscious) and attached to it, repetition is the movement that prevents the taking place of the void in the phantom of inconsistency from degenerating into a fixation of the void.[66]

The Subject as an Error of Counting

Lacan’s breaking of the solidarity between unity and totality allows him to work with parts. ‘Repudiat[ing] the reference to totality does not prevent one speaking about the partial’; rather, from the inexistence of totality as a one follows the possibility of thinking the part as ‘partial system’. This system is, for Lacan, the unconscious.[67] At this stage, it would not be exaggerated to suggest that the Lacanian unconscious can properly be understood according to a third set-theoretical axiom, that of separation (‘For any multiple supposed given, there exists the sub-multiple of terms which possess the property expressed by the formula λ(a)’) (BE 46). Paraphrasing Badiou’s explanation of this axiom, we could propose that, for Lacan, language separates out, within a supposed given existence—the undead real as not-one—the existence of a sub-multiple, the unconscious as partial system. This partial system is constituted from terms which ‘validate’ language, that is follow its metonymic and metaphoric laws—the famous thesis according to which ‘the unconscious is structured like a language’. Against common accusations of idealistic structuralism, for Lacan, ‘language cannot induce existence, solely a split within existence’; his notion of the unconscious ‘breaks with the figure of idealinguistery’ and is therefore materialist (BE 47). We are now able to see why Badiou himself briefly refers to Lacan’s notions of the symbolic and the real as an exemplification of the axiom of separation: the supposed given existence of the undead real as not-one anticipates what language, the symbolic, retroactively separates out from it as implied existence, the unconscious partial system. Such an implication concomitantly entails conscious reality, a semblance of existence which, rather successfully, attempts to totalize the partial unconscious, turning the system—the structure—into the mirage of a one/whole.

Applying both the axiom of separation and that of the empty set, it is important to emphasize that the existence of the unconscious as partial system ultimately relies on the in-existence of the void, or, more specifically, the existence of the void as part that in-exists as element. Indeed, the most basic sub-multiple that language—the unary trait as first count—separates out from the undead real as not-one is the void which un-presents itself as the part object, the object a. Unsurprisingly, Lacan identifies the void as part with the breast; the ‘primal fact’, which, for what we have seen, should also be conceived of as the primal existence, is the ‘enunciation of the nothing’ as the -1 of the absent mamma. In this way, a fundamental un-presentation functions as the ‘radical support’ for any relationship of inclusion. More precisely, Lacan explains how we can formulate a definition of the traditional category of class—‘if you really want to guarantee it its universal status’[68]—only by means of the un-presentation of the -1: the mammalian class can only be postulated on the basis of the absence of the mamma.

There is first of all the absence of the mamma and [then] one says: it cannot be that the mamma is missing, here is what constitutes the mammalian class. […] The zoologist, if you allow me to go this far, does not carve out the mammalian class in the assumed totality of the maternal mamma; it is only because he detaches the mamma that he can identify the absence of the mamma.[69]

The in-existence of the un-presented mamma which nevertheless exists as void-part determines both the particular existence of the mamma and the representation of the possible absence of the mamma with regard to the mammalian class taken as a whole. However, Lacan immediately specifies that, if the construction of the whole relies on the un-presentation of the -1, then it is the product of an ‘error of counting’, and consequently universality can be regarded only as a semblance; in order to obtain the universal, the ‘enunciation of the nothing’, the void-set as -1, must necessarily be primally repressed.

More precisely, Lacan believes that it is the subject who necessarily makes an error of counting: there is a ‘constituting necessity [that] the subject should make an error in the count’.[70] This count is a second count for the subject since, on an initial level, he is nothing other than what dis-counts itself by means of the unary trait, the very un-presentation of the -1. Put differently, initially, ‘the subject as such is minus one’[71] insofar as he identifies with the absent object, first and foremost the mamma—‘In the [first form of the] identification relationship […] what the subject assimilates […] is him in his frustration’.[72] It is only at a second stage, which works retroactively on the first and carries out identification proper, the splitting between the unconscious and consciousness, that ‘we are going to rediscover the subject [as] first of all established as minus one […] as [himself] verworfen’, primally repressed.[73] Identification proper is then the subject’s retroactive counting of himself, a - 1, as a 1. More specifically, the second count concomitantly brings about in a retroactive way the conscious subject’s primal repression of himself as the un-conscious un-presented -1 and his unconscious ‘seeking’ (or, desiring) himself as that very same un-conscious un-presented -1, that is, the ‘enunciation of the nothing’, the void-set. (Strictly speaking, what precedes the second count is not unconscious: the unconscious, just like consciousness, is a retroactive effect of the second count on structure qua count of the unary-trait).

Lacan clearly states that, with regard to the subject, the fact that should most interest philosophers and psychoanalysts is that his inaugural mistake is what allows him to express, or name, himself as a subject. Thus, it will not be a matter of simply ‘rectifying the means of knowing’ in order to avoid the mistake: what is ultimately involved in it is the subject’s conscious access to reality (the re-presentation of structure) and, at the same time, his endless unconscious search for ‘the real qua not possible’[74]—since, as we have seen, the real is precisely what in-exists as ‘enunciation of the nothing’. The second count has therefore a retroactive effect on the original counting of the un-conscious unary-trait ///; more specifically, the latter should now be understood in terms of unconscious repetition, in the precise sense of a compulsion to repeat something which is as such unrepeatable.[75] Consequently, repetition is characterized by unicity, ‘the unicity as such of [each] circuit of repetition’, just like the counting of the un-conscious unary trait ///.[76] As Lacan puts it, ‘repetition in the unconscious is absolutely distinguished from any natural cycle, in the sense that what is accentuated is not its return’, the sameness of the cycle; what is accentuated is rather the original unary trait /—the initial enunciation of the nothing as the real qua not possible—which ‘has marked the subject’ as -1.[77] Each circuit of repetition is unique since repetition, the making of / always anew, amounts to the impossibility of repeating the signifying uniquity of the first /, the un-presentation of the part-object.

The subject’s conscious access to reality, his knowledge [connaissance], presupposes a mistake, an error of counting, about which he knows nothing, and which moreover forces him into an endless unconscious search for what ‘preceded’ it. It is important to remark that this mistake, bluntly put, the turning of -1 into 1, originates at the very moment the initial enunciation of the nothing, the void-set, is turned into the absence of the part-object, the void as part. In Being and Event, Badiou clarifies precisely this point when he discusses the operation from which the property ‘the void is a subset of any set’ is obtained, the fact that the void is omnipresent in all structured presentation, its errancy. As Badiou has it, this fundamental ontological theorem is deduced as a particular case of the logical principle ‘ex falso sequitur quodlibet’: ‘if a statement A is false (if I have non-A) and if I affirm the latter (if I posit A), then it follows that anything (any statement B whatsoever) is true’ (BE 86-87). The void as part which is universally included in all sets supposed given follows from a falso; it relies on the negation of the true negative statement advanced by the axiom of the empty set, that is, ‘there is a negation’, or ‘nothing belongs to the void’, not even the void itself. The void as included part tacitly presumes the existence of an element that belongs to the void.

This kind of negation of negation is precisely the error, or falso, on which Lacan’s symbolic structured like a fiction, the big Other qua ‘one as difference’, is based. As we have seen, for Lacan, initially we have non-A, which is why taking ‘A is A’ as the basis of identification is so problematic; the A of l’Autre is barred and tautology is possible only at the price of making a mistake. More specifically, in Lacan’s theory of the subject, the void as part of all sets supposed given—whose existence as formed-into-one is itself affirmed only starting from the in-existence of this part as universally included—should be located on the level of what he calls the ‘symbolic object’. During the dialectic of frustration between the mother and the child at the beginning of the Oedipus complex—the time of the un-conscious counting of the unary trait ///, of structure without metastructure—the symbolic object is the object which the child demands beyond the object of need, the object ‘as grasped in what it lacks’. Remarkably enough, Lacan also specifies that the symbolic object ‘is not nothing since it has the property of being there symbolically’;[78] the part-object as ‘there is the nothing as part’, the void-part, results from the falsity of the negation of the initial true ‘enunciation of the nothing’, the void-set. The mistaken falsification of truth as the real qua not possible is what allows any affirmation whatever to be symbolically true, first of all that which proclaims the existence of the void as part. This error will then be aggravated retroactively by the second count, an operation (a metaphor) which Lacan refers to as the Name-of-the-Father; through the Name-of-the-Father, the part-object as the errant, and thus potentially dangerous, void-part itself receives the seal of the one, taking its place as a part in the phantasy -a. The Name-of-the-Father operates on the extimate part-object in which the subject identifies himself as vanishing in the same way as the forming-into-one operates on the in-existent part of a situation that initially does not belong to it.

‘Nous Nous Comptons Comptant’

While Badiou only hints at the distinction between the metastructured state of the situation and the structured situation as a distinction between consciousness and the ‘unconscious of the void’, Lacan attempts to delineate the two concomitant sides of the second count, the Name-of-the-Father, in a more elaborate manner. He does this precisely by thinking consciousness as both immanent and transcendent with regard to the unconscious: just as the situation will have been the ‘unconscious of the void’ of the state of the situation, so the un-conscious—the count of the unary trait as structure—will have been consciousness’s unconscious—the phantasy -a as the repressed structure of repetition. The phantasy -a to be read as ‘the subject split by the signifier in relation to the object a’ is the unconscious result of the operational metaphor of the Name-of-the-Father. Insofar as it seals as one the phantasy as unconscious structure, the Name-of-the-Father can also be designated as the S1, the master-signifier. Concomitantly, the subject’s proper name, which is equivalent to the possibility of saying ‘I’, having an ego, will be nothing other than the conscious (metastructural) side of the Name-of-the-Father.

Note that the S1 as the metastructure that structures the unconscious signifying chain amounts to a resumption of the unary trait at another level. Put differently, the un-conscious unary trait as structure will have been the S1, the structural, that is unconscious, side of the metastructure. It would also be correct to suggest that the S1 is the unary trait as repressed. Lacan himself stresses the similarity between the unary trait and the S1 when, in Seminar XI, the first seminar to introduce the notion of the master-signifier, he openly refers the S1 to the notch made by primitive hunters on sticks in order to signify the killing of an animal.[79]

Let us conclude with the following remarks:

  1. Initially, the subject in-exists as -1, it is what dis-counts itself by means of the original unary trait as ‘enunciation of the nothing’; more precisely, at this level, the subject should be regarded as the gap, or cut, between the structured presentation of signifiers, or more precisely letters, which signify the subject’s in-existence, and ‘their’ inextricable void, the symbolic object. After the operation of the Name-of-the-Father, the second count, has taken place, the subject as -1 counts himself as 1; this 1 should rather be understood as a new gap between the structured presentation of letters, now turned into the signifiers of the unconscious, and metastructured representation, that is, conscious discourse.[80]
  2. The first gap, between structure and ‘its’ void, which causes the latter’s errancy, designates the metonymic dimension of demand, the unstoppable sliding of the symbolic object (the object of love) beneath the objects of need. On the other hand, the second gap, between structure and metastructure, designates the metaphoric dimension of desire. In order to pass from the gap of demand to the gap of desire the subject must carry out a positivizing organization of the void: the void must ‘take place’ within the phantasy -a by means of the metaphor of the Name-of-the-Father.
  3. The subject that counts himself as 1 is equivalent to the subject as the gap of desire who is represented in the unconscious phantasmatic object a as ‘enunciation of the nothing’; that is, the subject is 1 in the unconscious insofar as he appears there as not-one, -1.[81] More precisely, the subject continues to make 1 in the unconscious phantasy precisely because, as ‘enunciation of the nothing’, he is not-one. Differently put, the subject can name himself ‘I’ in consciousness—and thus value himself [se compter]—only because he repeats the act of counting himself [se compter] as not-one in the phantasy—there where in fact the object a functions as a ‘lost name’.[82]
  4. The subject’s naming of himself as ‘I’ is what allows him to count numbers, 1,2,3,4,…; this counting is nothing other than the conscious side of the unconscious repetitive circuit traced by the phantasy and sealed as one by the S1. While the latter manages to accomplish an organization of the void, this by no means amounts to saying that the void is eliminated: the organization of the void is thus repressed and this operation can be considered as an error of counting. At the synchronic level of the unconscious, the identifying representation of the subject as 1 in the object a necessarily preserves the -1 and thus gives rise to the repetitive series ////… in which each ‘count’ is started anew, each ‘go’ is absolutely unique. On the other hand, at the level of consciousness, the subject’s naming of himself as ‘I’ mistakenly adds 1 + 1 + 1 + 1… and obtains 2, 3, 4…, which is to say, the diachronic ‘temporal’ continuity of his lived experience.
  5. The fact that the subject as 1 is, at the same time, a subject as gap means nothing other than that the subject is himself a one-multiple.[83] The two counts retroactively differentiate three ‘levels’ of the multiple: the inconsistent undead real as not-one; the consistent multiplicity given by the metonymic slide of the objects of demand (marked as letters); the subject as split between conscious signified and unconscious signifier. The split subject’s multiplicity is an empirical fact attested by the existence of the formations of the unconscious, such as symptoms, jokes, and slips of the tongue. However, the subject’s multiplicity is repressed by the second count, just as the inconsistency of the multiple is un-presented by the first count. The second count both symbolizes the void and carries out, through repression, an imaginarization of the first count, that of the letters as ‘pure’ signifiers; the void is symbolized as phantasmatic desire, but desire is itself repressed and can be ‘approach[ed] [consciously] only by means of some sort of demand’.[84] Finally, the phantasy where the void takes place should also be considered as a ‘picture’ in which anxiety is ‘framed’ and thus ‘tamed, placated, admitted’.[85]

Lorenzo Chiesa
School of European Culture and Languages,
University of Kent, UK

REFERENCES

Badiou, Alain, Un, Deux, Trois, Quatre, et aussi Zéro, unpublished.

Badiou, Alain, L’être et l’événement, Paris, Seuil, 1988.

Badiou, Alain, Infinite Thought: Truth and the Return to Philosophy, ed. and trans. Justin Clemens and Oliver Feltham, London, Continuum, 2003.

Badiou, Alain, Saint Paul: The Foundation of Universalism, trans. Ray Brassier, Stanford, Stanford University Press, 2003.

Badiou, Alain, Being and Event, trans. Oliver Feltham, London, Continuum, 2005.

Bosteels, Bruno, ‘Alain Badiou’s Theory of the Subject: The Recommencement of Dialectical Materialism? (Part I)’, Pli: Warwick Journal of Philosophy, no. 12, 2001, pp. 200-29.

Bosteels, Bruno, ‘Alain Badiou’s Theory of the Subject: The Recommencement of Dialectical Materialism? (Part II)’, Pli: Warwick Journal of Philosophy, no. 13, 2002, pp. 173-208.

Brassier, Ray, ‘Presentation as Anti-Phenomenon in Alain Badiou’s Being and Event’, Continental Philosophy Review, 2006.

Chiesa, Lorenzo, Lacan and Subjectivity: A Philosophical Introduction, Cambridge, MIT Press, 2007.

Chiesa, Lorenzo and Alberto Toscano, ‘Ethics and Capital, Ex Nihilo’, Umbr(a): A Journal of the Unconscious, no. 1, 2005, pp. 9-25.

Freud, Sigmund, ‘Group Psychology and the Analysis of the Ego’, The Standard Edition of the Complete Psychological Work of Sigmund Freud, vol. XVIII, London, The Hogarth Press and the Institute of Psychoanalysis, 2001.

Lacan, Jacques, Seminar IX (1961-1962): ‘L’identification’, unpublished.

Lacan, Jacques, Seminar VI (1958-1959), ‘Le désir et son interpretation’, unpublished.

Lacan, Jacques, Seminar XIX (1971-1972), ‘…Ou pire’, unpublished.

Lacan, Jacques, The Seminar of Jacques Lacan. Book I, Freud’s Papers on Technique, 1953-1954, Jacques-Alain Miller (ed.), trans. John Forrester, 1st American ed., New York, W.W. Norton, 1988.

Lacan, Jacques, Le séminaire livre IV. La relation d’objet, 1956-1957, Paris, Seuil, 1994.

Lacan, Jacques, Le séminaire livre V. Les formations de l’inconscient, 1957-1958, Paris, Seuil, 1998.

Lacan, Jacques, The Four Fundamental Concepts of Psychoanalysis, London, Vintage, 1998.

Lacan, Jacques, ‘L’étourdit’, Autres écrits, Paris, Seuil, 2001.

Lacan, Jacques, Le séminaire livre X. L’angoisse, 1962-1963, Paris, Seuil, 2004.

Pluth, Ed and Dominiek Hoens, ‘What if the Other Is Stupid? Badiou and Lacan on ‘Logical Time’’, in Peter Hallward (ed.), Think Again: Alain Badiou and the Future of Philosophy, London, Continuum Books, 2004, pp. 182-90.

Safouan, M., Lacaniana: Les séminaires de Jacques Lacan * 1953-1963, Paris, Seuil, 2001.

Schuster, A., Commentary on Lacan Seminar IX L’Identification, 20 December 1961, unpublished.

Žižek, Slavoj, The Ticklish Subject: The Absent Centre of Political Ontology, New York, Verso, 2000.

Žižek, Slavoj, ‘Foreword to the Second Edition: Enjoyment within the Limits of Reason Alone’, For They Know Not What They Do: Enjoyment as a Political Factor, 2nd ed., London, Verso, 2002, pp. xi-cvii.

Žižek, Slavoj, ‘Badiou: Notes From an Ongoing Debate’, International Journal of Žižek Studies, vol. 1, no. 2, 2006, http://ics.leeds.ac.uk/zizek/article.cfm?id=21&issue=3

[1]. I would like to thank Ana Alvarez Velasco, a true ‘non-working’ mathematician, for her invaluable advice. I am also grateful to Alberto Toscano for his Badiouian comments on an early draft of this article.

[2]. Alain Badiou, L’être et l’événement, Paris, Seuil, 1988, p. 10. The English translation, which is otherwise excellent, curiously omits this reference to psychoanalysis, replacing it with a reference to art—which does not appear in the original, Alain Badiou, Being and Event, trans. Oliver Feltham, London, Continuum, 2005, p. 4 (henceforth BE).

[3]. Lesson of 29/11/61 from Jacques Lacan, Seminar IX (1961-1962): ‘L’identification’, unpublished (henceforth Seminar IX).

[4]. Alain Badiou, Infinite Thought: Truth and the Return to Philosophy, ed. and trans. Justin Clemens and Oliver Feltham, London, Continuum, 2003, p. 6.

[5]. See, for instance, Slavoj Žižek, The Ticklish Subject: The Absent Centre of Political Ontology, New York, Verso, 2000, pp. 127-70.; Slavoj Žižek, ‘Foreword to the Second Edition: Enjoyment within the Limits of Reason Alone’, For They Know Not What They Do: Enjoyment as a Political Factor, 2nd, London, Verso, 2002, pp. lxxxi-lxxxviii. See also Bruno Bosteels, ‘Alain Badiou’s Theory of the Subject: The Recommencement of Dialectical Materialism? (Part I)’, Pli: Warwick Journal of Philosophy, no. 12, 2001, pp. 200-29, pp, 220-9. and Bruno Bosteels, ‘Alain Badiou’s Theory of the Subject: The Recommencement of Dialectical Materialism? (Part II)’, Pli: Warwick Journal of Philosophy, no. 13, 2002, pp. 173-208, pp. 197-208.; Ed Pluth and Dominiek Hoens, ‘What if the Other Is Stupid? Badiou and Lacan on ‘Logical Time’’, in Peter Hallward (ed.), Think Again: Alain Badiou and the Future of Philosophy, London, Continuum Books, 2004, pp. 182-90.

[6]. For a discussion of Lacan’s ethics as an ontological ethics, see Lorenzo Chiesa and Alberto Toscano, ‘Ethics and Capital, Ex Nihilo’, Umbr(a): A Journal of the Unconscious,, no. 1, 2005, pp. 9-25. And Chapter 5 of Lorenzo Chiesa, Lacan and Subjectivity: A Philosophical Introduction, Cambridge, MIT Press, 2007.

[7]. Lesson of 17/5/1972 from Jacques Lacan, Seminar XIX (1971-1972), ‘…Ou pire’, unpublished.

[8]. ‘Topology is not “made to lead us” to structure. Topology is this structure’ (Jacques Lacan, ‘L’étourdit’, Autres écrits, Paris, Seuil, 2001, p. 483.).

[9]. ‘The law of the count as condition for existence, which renders presentation possible by precluding the presentation of inconsistent multiplicity (i.e. being itself), is ultimately indiscernible from the ontological inconsistency whose presentation it forecloses’, Ray Brassier, ‘Presentation as Anti-Phenomenon in Alain Badiou’s Being and Event’, Continental Philosophy Review, 2006, [available on-line at: http://www.springerlink.com/content/k3r6782060171279/?p=1661625d348044a1b73f87e8d0c3fd49&pi=4]

[10]. Seminar IX, lesson of 6/12/61.

[11]. According to Freud, in some cases, ‘identification is […] partial […] and only borrows a single trait from the person who is its object’, Sigmund Freud, ‘Group Psychology and the Analysis of the Ego’, The Standard Edition of the Complete Psychological Work of Sigmund Freud, vol. XVIII, London, The Hogarth Press and the Institute of Psychoanalysis, 2001, p. 107.

[12]. Seminar IX, lesson of 22/11/61.

[13]. Seminar IX, Lesson of 29/11/61.

[14]. Seminar IX, Lesson of 22/11/61.

[15]. See Seminar IX, lesson of 29/11/61.

[16]. Seminar IX, Lesson of 29/11/61.

[17]. Seminar IX, Lesson of 7/3/62.

[18]. Seminar IX, Lesson of 7/3/62.

[19]. Seminar IX, Lesson of 7/3/62.

[20]. For a description of the Cartesian God as the cogito’s unary trait, see lesson of 22/11/61.

[21]. Seminar IX, Lesson of 22/11/61.

[22]. Seminar IX, Lesson of 6/12/61.

[23]. Seminar IX, Lesson of 6/12/61.

[24]. ‘Lacan affirms that the signifier as such serves to connote difference at its purest, and, we can add, it is all the purer because it precedes even number’, M. Safouan, Lacaniana: Les séminaires de Jacques Lacan * 1953-1963, Paris, Seuil, 2001, p. 193.

[25]. Seminar IX, lesson of 6/12/61.

[26]. Seminar IX, Lesson of 6/12/61.

[27]. Seminar IX, Lesson of 29/11/61.

[28]. Seminar IX, Lesson of 29/11/61.

[29]. Seminar IX, Lesson of 29/11/61.

[30]. Seminar IX, Lesson of 6/12/61.

[31]. Safouan even names this concept ‘distinctive one’ [un distinctif], Safouan, Lacaniana: Les séminaires de Jacques Lacan * 1953-1963, p. 202. To the best of my knowledge, Lacan never uses this expression in Seminar IX. In the lessons of 13/12/61 and 20/12/61 he also refers to the ‘distinctive trait’, which I take to be synonymous with ‘unary trait’.

[32]. Seminar IX, lesson of 6/12/61.

[33]. Lesson of 6/12/61. Lacan concedes that, while the function of the unary trait ‘is linked to the extreme reduction […] of qualitative difference’, even just at the level of the imaginary appearance of the stroke itself, ‘it is quite clear that there will not be a single [trait] like another’. Commenting on this point, Safouan rightly observes the following: ‘It is clear that the function of these notches is not more related to their [qualitative] differences than it is to the elimination of these differences. It is not because the traits are different that they work differently, but because signifying difference is different from qualitative difference’, Safouan, Lacaniana: Les séminaires de Jacques Lacan * 1953-1963, pp. 192-3.

[34]. Seminar IX, lesson of 6/12/61.

[35]. Seminar IX, Lesson of 6/12/61.

[36]. This amounts to a very advanced ‘definition of addition which supposes a number of axioms which would be enough to cover the blackboard’ (Seminar IX, Lesson of 6/12/61).

[37]. Seminar IX, Lesson of 6/12/61.

[38]. See Brassier, ‘Presentation as Anti-Phenomenon in Alain Badiou’s Being and Event’.

[39]. Seminar IX, lesson of 29/11/61.

[40]. Seminar IX, Lesson of 6/12/61.

[41]. Seminar IX, Lesson of 20/12/61.

[42]. Seminar IX, Lesson of 20/12/61.

[43]. Seminar IX, Lesson of 20/12/61.

[44]. Seminar IX, Lesson of 20/12/61.

[45]. A. Schuster, Commentary on Lacan Seminar IX L’Identification, 20 December 1961, unpublished (my emphasis).

[46]. Seminar IX, lesson of 20/12/61.

[47]. Seminar IX, Lesson of 10/1/62.

[48]. Seminar IX, Lesson of 20/12/61.

[49]. Seminar IX, Lesson of 20/12/61.

[50]. Seminar IX, Lesson of 10/1/62.

[51]. Seminar IX, Lesson of 10/1/62.

[52]. Seminar IX, Lesson of 10/1/62. ‘But what is it that inscribes [the fact] that a real exists? It is the symbolic as such. Thus, we will say that 1 is the digit of the symbolic. The 1 is the unary trait, that is to say, the minimal possible Other for the pure letter of the real’, Alain Badiou, Un, Deux, Trois, Quatre, et aussi Zéro, unpublished.

[53]. Seminar IX, Lesson of 10/1/62.

[54]. Seminar IX, lesson of 7/3/62.

[55]. Seminar IX, Lesson of 7/3/62.

[56]. Seminar IX, Lesson of 17/1/62.

[57]. Seminar IX, Lesson of 17/1/62.

[58]. Seminar IX, Lesson of 17/1/62.

[59]. Seminar IX, Lesson of 7/3/62.

[60]. Seminar IX, Lesson of 7/3/62.

[61]. Seminar IX, Lesson of 7/3/62.

[62]. Although the notion of the undead has been employed profitably by Žižek in a number of ways, Lacan appears to delimit it within a particular domain: the undead refers to the ‘closed world’ of the animal as that which is always ‘already dead’ from the perspective of the individual and ‘immortal’ from that of the species or nature, see for instance Jacques Lacan, The Seminar of Jacques Lacan. Book I, Freud’s Papers on Technique, 1953-1954, trans. John Forrester, Jacques-Alain (ed.), 1st American ed., New York, W.W. Norton, 1988, pp. 121, 137.

[63]. Seminar IX, lesson of 6/12/61.

[64]. Seminar IX, Lesson of 7/3/62.

[65]. Seminar IX, Lesson of 7/3/62.

[66]. Badiou’s failure to account for the function of repetition in the ‘phantom of inconsistency’ gives rise to terminological ambiguity when he describes the difference between the ‘taking place’ of the void—which wards it off—and its ‘fixation’—that is, ‘the ruin of the One’ (BE 93): how does the ‘fixation’ of the void, its ‘becom[ing] localizable’ (BE 56), differ from its ‘taking place’ if one does not specify that the latter still entails (repetitive, circular) movement? In Saint Paul, Badiou seems to suggest that repetition should rather be associated with fixation, a fixation of the subject’s desire which is, however, a fixation of the law (and not aimed against it): ‘The law is required in order to unleash the automatic life of desire, the automatism of repetition. For only the law fixes the object of desire […]’, Alain Badiou, Saint Paul: The Foundation of Universalism, trans. Ray Brassier, Stanford, Stanford University Press, 2003, p. 79. Beyond terminological confusion, should we not identify such a repetitive ‘fixation’ of the law with what Being and Event defines as the ‘taking place’ of the void?

For a recent critique of Badiou’s unsatisfactory notion of repetition with regards to the political subject, see Slavoj Žižek, ‘Badiou: Notes From an Ongoing Debate’, International Journal of Žižek Studies, vol. 1, no. 2, 2006, http://ics.leeds.ac.uk/zizek/article.cfm?id=21&issue=3 [available on-line at http://ics.leeds.ac.uk/zizek/article.cfm?id=21&issue=3].

[67]. Seminar IX, lesson of 24/1/62.

[68]. Seminar IX, lesson of 7/3/62.

[69]. Seminar IX, Lesson of 7/3/62.

[70]. Seminar IX, Lesson of 7/3/62.

[71]. Seminar IX, Lesson of 7/3/62.

[72]. Seminar IX, Lesson of 24/1/62.

[73]. Seminar IX, Lesson of 7/3/62.

[74]. Seminar IX, Lesson of 7/3/62.

[75]. Seminar IX, Lesson of 7/3/62.

[76]. Seminar IX, Lesson of 7/3/62.

[77]. Seminar IX, Lesson of 7/3/62.

[78]. Jacques Lacan, Le séminaire livre IV. La relation d’objet, 1956-1957, Paris, Seuil, 1994, p. 155 (my emphasis).

[79]. Jacques Lacan, The Four Fundamental Concepts of Psychoanalysis, London, Vintage, 1998, p. 141.

[80]. ‘The gap between a (which counts-as-one the belongings, or elements) and ℘(a) (which counts-as-one the inclusions, or subsets) is […] the point in which the impasse of being resides. […] I said that a and ℘(a) were distinct. In what measure? With what effects? This point, apparently technical, will lead us all the way to the Subject and to truth’ (BE 83-84, my emphasis).

[81]. See especially lesson of 3/6/1959 from Jacques Lacan, Seminar VI (1958-1959), ‘Le désir et son interpretation’, unpublished.

[82]. Seminar VI, lesson of 3/6/1959.

[83]. Put differently, ‘negation is irreducible’ (Seminar IX, lesson of 24/1/62).

[84]. Jacques Lacan, Le séminaire livre V. Les formations de l’inconscient, 1957-1958, Paris, Seuil, 1998, p. 330.

[85]. Jacques Lacan, Le séminaire livre X. L’angoisse, 1962-1963, Paris, Seuil, 2004, p. 91. For a detailed analysis of the three logical times of anxiety with regards to phantasy, see Chapter 5 of Chiesa, Lacan and Subjectivity: A Philosophical Introduction.