OBJECT-a AND OBJECT-u
- Towards a Materialism of Desire and Love -
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One of the most provocative theses in Alain Badiou's philosophy is that love comes to supplement the non-rapport of the sexes. Love escapes the primordial narcissism of desire by initiating a productive exploration of the world in the form of manifold decisions and enunciations made from the immanent perspective of the Two. Against the purely 'segregative' vision of the sexes, in which the individual is merely tethered to the circuitous loop around the impossible object-cause of desire (which Lacan famously called objet petit a), Badiou proposes thus to rescue the imaginary of Love within the universal scope of the material production of a new Truth. In order to advance the possibility of love's immanent exception, however, Badiou must challenge the purely disjunctive positioning of objet a, introducing a "minimal object" which is common to the two sexed positions, and which he calls object-µ. Love interrupts the narcissism of desire in the immanence of the Two; objet µ as the bare inscription, within the non-connection of the sexes, of a common element which indirectly links the two positions, and whose only ground is finally the void of being. If love is a truth-procedure, capable of being an index of the material transformation of the world, then this is possible only insofar as the sexed subject achieves universality through this worldly endeavor, emancipating itself from the alienation of desire around its impossible object-cause. Parallel to this attempt, we find Slavoj Žižek's (1993) startling crossbreeding of Lacan and Hegel, advancing a transcendental theory of the subject, in which objet petit a and the disjunction of desire finally coincides with the (Hegelian) vision of the universal as being always in excess to itself. Under Žižek's reading, the antinomies expressed in the disjunction of the sexes illuminate the universality of the "barred subject" as a kind of 'libidinal reservoir' of energy which, Žižek provocatively suggests, is needed even for the material void of being to 'come-into-being'. Explicitly re-philosophizing the Lacanian theory of the subject, Žižek thus proposes to elevate object petit a to a kind of universal exception which grounds the ontogenesis of the Real, material ontological order.
This paper attempts to elucidate these two philosopher's attempts to escape from the vacuity of the narcissism of desire in a reallocation of materiality: either in the form of a philosophical supplementation of desire by Truth (Badiou), or a reintegration of desire into philosophy (Žižek). In the first section we tentatively follow Žižek's intricate reading of the Lacanian formulae of sexuation, which builds on the work of Joan Copjec, around the question of the Kantian antinomies of pure reason. We show how Žižek attempts to reconcile the barred desiring subject circulating around the wound of primordial loss, with the Hegelian vision of universality as self-relating negativity; so that objet petit a may come to stand in for the material "crack in the universal" which potentiates even the void of ontology. In the second section we briefly survey the supplemental vision of Love advanced by Badiou, where the common objet µ indexes the material production of a subject in post-evental fidelity. We illuminate our discussion by assessing certain objections to each of the two proposals in relation to the question of change and becoming, focusing on how each vision attempts to explain how novelty comes to emerge in the apparent stability of the world and the ontological order.
I - Philosophical Reintegration: Objet a As Index of Materiality.
The second chapter of Žižek’s (1993) Tarrying With the Negative, indebted to the work of Joan Copjec, develops the thesis that Lacan’s positioning of the sexes stands in structural isomorphy to the famous Kantian antinomies of pure reason. What is of interest for us here is to understand how this isomorphic relation is meant to elucidate the relation of the subject to the universal; which in Žižek's Hegelian view, elucidates how contradiction is not merely "regional", but ontologically fundamental. If the Kantian localization of the antinomies of reason were originally paired to the famous 'cosmological' objects by way of the schematic application of the categories of the understanding, it remains to be seen how the isomorphy to the Lacanian sexed positions de-regionalizes the antinomies and distributes contradiction uniformly across all possible ontico-ontological strata. This is no small matter, as Hegel will tell us, since it places immediately before us also that inadequate treatment of contradiction which led Kant to simply develop schemas on the basis of the categories of the understanding, in order to construct possible logical instantiation of the contradictory objects of experience. This inability to dialectically deduce the possible content of the object from the Concept itself inhibits the critical project from realizing that contradiction is the base ontological principle of the entire process of becoming. Antinomies thus reduce contradictions to mere possible schemas of objects, within the specific domain of what Kant called 'cosmological objects'. These, however, remain relatively independent, and merely form distinct classes designating possible instantiations of contradictory objects. Against this, contradiction must be for Hegel developed so that we can see that its “…true and positive significance is that everything actual contains opposed determinations within it, and in consequence the cognition and, more exactly, the comprehension of an object, amounts precisely to our becoming conscious of it as a concrete unity of opposed determinations.” (E Logic, Pg. 93).
Without getting sidetracked, we might simply recall that the first moment of the logical development of the Concept in the Hegelian dialectic is precisely that of the overturning of Being into Nothingness, where this (indistinguishable) opposition is sublated as the unity of Becoming as such. That Being 'passes-over' into its Other installs from the beginning the 'crack in the universal', the inadequacy of the Universal to itself, which will be of outmost importance for Žižek's argument. We may summarize Hegel’s contention as saying that contradiction cannot merely be a set of possible, purely logical schemas on the basis of a presupposed categorical framework. The object is not merely contradictory insofar as it finds itself in possible "antinomic" individuation with respect to certain categories, offered to mere logical thought, but never to the realm of being-in-itself. Rather, the primacy of contradiction entails that the latter is not just logical but also constitutively metaphysical or ontological; that the gap between the universal and the particular indexes that which is metaphysically real. Contradiction comes to install itself from the start and all-pervasively, so that it will not be a possibility of logical reason alone, but a real necessity, shown by the dialectic, as well. The Kantian impasse is finally that of subordinating contradiction to the categories and thus to the understanding. But let us simply recall in passing the following crucial enunciation by Hegel himself:
”It may also be remarked that, as a result of his failure to study the antinomy in more depth, Kant brings forward only four antinomies. He arrives at them by presupposing the table of categories just as he did in the case of the so-called paralogisms. While doing this he followed the procedure, which became so popular afterwards-, of simply subsuming the determinations of an ob-ject under a ready-made schema, instead of deducing them from the Concept…[but] antinomy is found not only in the four particular ob-jects taken from cosmology, but rather in all objects of all kinds, in all representations, concepts, and ideas.” (Hegel EL, Pg. 92)
We must be clear to differentiate the scope of the manifold concepts “within which” antinomies are found in pure reason, from the general claim according to which contradiction functions as a general principle of the Concept, i.e. the logico-metaphysical motor of the dialectic which leads from the immediacy of Being to the full historical movement of the Spirit. But in any case, we bear witness to an interesting limitation that Hegel sees in the Kantian use of contradiction. The crucial question, for Žižek, is how to reconcile the Hegelian inscription of contradiction as ontological principle which affects “all objects”, with Lacan’s indexing of the sexed positions of the subject as antinomic, like Kant. Žižek’s solution / general strategy is roughly as follows: the four Kantian antinomies are grouped into two basic kinds, and the latter instantiate the two basic sexed positions for all subjects. If the subject is the very possibility of being positioned with respect to the basic disjunction of the sexed antinomies, then it will follow that all mediated objective content for experience will be distributed along the two axes that express the non-relation of the two sexes. The two groups of 'cosmological' objects of the Kantian antinomies are then to be re-inscribed across the two positions of the sexed subject, insofar as the latter's desire exhibits a distinct relation to the universal, which it fails to ever become equal to. Finally, this failure of the subject to ever find itself in adequation to the universal, exhibits the fundamental ontological gap and the inconsistency of its being, revealing the subject's monotonous circuit attempting to heal the wound of self-alienation. Behind every metonymic stand in for the promise of a restored universal, the Lacanian loop of desire thus unveils for Žižek the formally empty lure of the non-self-identical remainder of the Real: object a, cast onto infinite displacement by the desiring subject. Enveloping the void of desire with transient semblance of positive content, the phantasy of the subject's 'localization' of the phallic lost-object only masks its repetitive vocation; the mark of the truly universal, the non-self identical, or finally the negativity of being as such. This complicity between being and the failure of the completed universal to totalize itself is finally what allows Žižek to identify the subject's desire towards the impossible object-cause as the universality of its own proper void, with an index of the materiality of annexed to subjectivity itself: "...the existence of material reality itself bears witness to the fact that the Concept is not fully actualized. Things 'materially exist' not when they meet certain notional requirements, but when they fail to meet them - material reality is as a such a sign of imperfection." (SOI: Pg. xx) It is here that Hegel and Lacan meet again; the crack in the universal is finally rendered in the failure of desire to ever match its "phallic" semblance of completion.
In what follows, we shall attempt to clarify the connection between the two sexual positions and the basic monotony of desire in general, in relation to the thesis that the incomplete universal reveals objet a as the remainder of the Real. One might worry at this juncture, however, that under strict Hegelian terms the sexuation of the subjective positions would already constitute an schematic objectification of the subject, as opposed to the fully fledged Concept which knows nothing of sexuation. Since Hegel thinks of the categorical form which generates these antinomies as limited in its restriction to the four cosmological possible objects, there is a sense in which we can say that sexuation and desire seems to be “cosmologized”.
Or is it the other way around? If indeed the cosmological antinomies can be explained in their more basic dyadic division into dynamic and mathematical antinomies, proper to all subjects, then we may understand how they do not only constitute a “ready-made” schematic framework for schematized objects. Rather, the antinomies would actually stand as inherent formal iterations of the subjective self-relating negativity or loop around the void of its own desire, and the Universal as the irremediable errancy or failure which guarantees the impossibility of its full actualization. The Universal is then in a way always undermined upon exemplification, leaving a remainder void of any semblance of positive content. We obtain thus the basic crossbreeding: the Kantian cosmological antinomies, the Hegelian primacy of contradiction as the objectively pervasive logico-metaphysical motor, and the Lacanian sexuation of the subjective positions across a masculine and feminine axis.
But let us return briefly to the Kantian antinomies. As said above, Žižek separates the four possible resulting antinomies which devolve from the conditions for ‘cosmological objects’ into two classes: dynamic antinomies and mathematical antinomies. These correspond to two possible contradictory relations which emerge in consideration of how the phenomenal relates to the noumenal. The first case, that of dynamic antinomies, involves finding an exception within the phenomenal order to the rule established by the universal (categorical) framework of causes and effects, which in Kant defines the field of nature as such. So there is a possible contradiction which obtains in the logical interconnection of the elements in the phenomenal order as thinkable exceptions to the rule, which transgress the limits of the Natural order. They thus involve admitting of an “antinomic” presentation of a phenomenon in contradiction to the boundaries set forth by the categories. And this, in the Kantian account, is what precisely opens the space for the radical possibility of freedom qua noumenal act, in exception to the phenomenal order of causal nexuses and connections. The dynamic antinomy thus designates the negation or contradiction to the phenomenal order in the form of an exception which is situated “beyond” the phenomenal; a positivized occasion of something whose being appears as absolutely contingent, and truly “free”: "dynamical antinomies... are antinomies of universality: logical connection of the phenomena in the universal causal nexus necessarily involved an exception, the noumenal act of freedom which "sticks out", suspending the causal nexus and starting a new causal series out of itself" (Ibid: Pg. 55) The emblematic entities which fall under this category are allotted thus to an "ontological realm" in exception to the universal submission to the causal order of phenomenal necessity: God and the Soul being the prime examples of this sort.
Mathematical antinomies are, on the other hand, a negation of the possibility of ever conceiving of the totality of phenomena as such. The phenomenal realm is limited by the partial determination of disjunctive categorical schematic instantiations, and can thus only be encountered finitely under certain determinations. This amounts to saying that no object is ever given as the Infinity or Totality of all possibilities, since the phenomenal order is always partially given under certain determinations which excludes others. There is an impossibility of grasping a comprehensive phenomenal occurrence which would give itself as a single object or presentation under all possible determinations. Certain schema themselves contradict each other structurally, and thus constitute a limit to phenomena, rather than a beyond indexed to a noumenal outside the phenomenal. The emblematic example here is the thought of “the universe as a Whole”, the Infinity of the phenomenal as such. It is this ‘Whole’, this Totality, which is precisely inscribed under the mathematical antinomies; it establishes a threshold wherein exceptions would localize themselves as excluded. This is of particular interest, since Žižek will argue that there is a certain ontological priority of the mathematical antinomy, in determining the limit then to be 'filled-in' by the semblance of the dynamic beyond. But for now let us simply grant this basic construal of the Kantian position and assess the Lacanian account on sexuation, as read by Žižek. The following diagram presents the two sexed positions for all subjects (Lacan: SXX, Pg. 78):
Although no reference to Kant is made on this point by Lacan himself, he does nevertheless associate the ‘antinomic’ positions of the sexes to the Aristotelian conception of enstasis, an ‘obstacle’ of some kind, as something which doesn't quite fit into the universal submission to the law, but for this very reason founds or establishes it (Lacan, SXX, Pg. 69-70). Keeping in mind thus the dynamic 'obstacle' of the noumenal as a Beyond, and that of the mathematical obstacle as a phenomenal Limit, let us proceed to read both sides in the Lacanian formulas. The two formulas on the left column designate the so-called masculine position, while the two on the right column designate the female position. These are not to be associated with biological determinations, but may be occupied by members of the same sex; they are purely formal determinations and should be read thus. In order to read these formulas, we should first attempt to explain them elementarily:
The Masculine Position
("x): "man-as a whole..." (The universal)
(фx): "...is castrated" (function of incompleteness-castration, "the phallic function")
("x) фx "man as a whole is castrated" (the "incomplete universal")
We can alternatively read the formula:
("x) фx: "All men are submitted to the phallic function."
The Universal totalizing of "Man as a whole" is here inscribed through the determination of the phallic function; which designates that whatever falls under it is affected by the primordial alienation or loss resulting from inclusion into the symbolic order of language. Thus "sexuality is the effect on the living being of the impasses which emerge when it gets entangled in the symbolic order, i.e. the effect on the living body of the deadlock or inconsistency that pertains to the symbolic order qua order of universality." (Žižek, Ibid. Pg. 56) In other words, the phallic function designates the alienation occasioned by the subject's inclusion into the societal 'big Other', which is also the source of the (superego) injunctions to conform to some decentered standard for Ideal adequacy. We thus obtain the peculiar inscription of a Universal which is in its very determination 'severed', castrated, torn apart from that which would endow its being with full consistency. 'Man as a whole' is inscribed, but affected by the function of castration, which is why we call it here 'the incomplete universal'; it is in a sense a universal never adequate to itself, every exemplification falls short of it, failing to endow the subject with a fixed identity. However, as Lacan stresses, this incomplete Totality can be inscribed only on condition that there is also a position of exception. To inscribe the role of the exceptional supplement, we must observe the second formula from the masculine side: (Зx): "The father..." (The complete universal)
⌐ (фx): “...is not castrated” (function of completeness, negation of the phallic function)
(Зx) ⌐(фx): “The father is not castrated / The father possesses the phallus” (the complete universal)
We can alternatively read this as:
"There exists someone who is not submitted to the phallic function."
As we saw above, the phallic function designates a universal inadequate to itself, i.e. which being incomplete, is never fully equal to its instantiation / is non-self identical. We can propose that the universal is already then in exception to itself. So that, if the incomplete universal can only be inscribed on condition of there being an exception to it, then this must be in a sense an exception of the exception. But of course this 'double negation', this exception 'to the second power', can only come to negate the very exceptional position of the Universal to itself. It must be then the semblance of a truly complete position, outside the pervasive incompleteness of the universal exception. The exception to the incomplete universal is then indeed the father-figure, who possesses that which makes All men incomplete and lacking, i.e. he is not castrated, he has the phallus, he is equal to itself. The Oedipal logic of the insertion of the subject into the dialectic of desire around a lost object cause is transparently evinced here: the subject can only assume a full, ontologically consistent identity by being submitted to the injunction of trying to recuperate itself from a primordial loss occasioned upon the insertion into the symbolic. This “lost object” is positivized in the symbolic order as the missing phallus which indexes the 'little piece of the Real', or the imagined source of the ‘crack’ inherent to the universal. Piercing the Real by the Law of the father, the phallus thus designates the metonymic localization of the lost-object which organizes the empty circuit of desire around a fundamental impossibility, in hopes to achieve ontological consistency. As Lorenzo Chiesa explains: "The signifier primordially holes the Real; such a hole transforms the 'neutrality' of the Real into a lack which is then inextricable from the Symbolic as such; it cannot be "filled in", despite the fact that the phallus manages to "organize" / mark it." (SAO: Pg. 122) Here we find the divide between phallus and objet petit a: the former corresponds to the metonymic semblance of positive content which 'quilts' the emptiness of the formal emptiness of desire which repeats itself.
Just like in the Kantian dynamical antinomy, the phallic father-function thus lends consistency to a “Beyond” in exception to the Universal rule or principle of castration, in proper excess to that which the universal alienates by inclusion to its Law. Objet-a, as the remainder or index of the Real thus designates the irreducible ontological incompleteness of the subject inscribed in any totality, and how the latter only operates by radically not-coinciding with its exemplifying content. The phallic exception to the complete universal is thus in an important sense merely an attempt to organize the void of the Real, or objet a as the impossible object of desire. The non-coincidence of the subject to itself is thus negated in the position of the exception, so that if every one man is “non-coincident” with all men (men as a Whole), the father-function organizes the position of a fully self-identical being which, not missing the phallus, can assume free and unrestrained enjoyment. The masculine phantasy thus involves the nomination of such a decentered Other who really "is where it thinks", that suffers no alienation from its place of enunciation. This is what, in conjunction to the primary alienation of the phallic-function, fuels the desire of the subject around the recuperation of the lost phallus. The father function then will index the fundamental impasse of desire; the circuit of desire into which the subject is submitted, and which imagines through metonymic semblances the phallic object of lack as restoring the ontological homeostasis of form and content, of the universal with its particular instantiations. It designates the transcendental space of a (noumenal) subject no longer aghast before the sublimity of its beyond, but fully occupying his place, fully capable of enjoyment. That every such phallic semblance ultimately fails to restore the crack in the universal, that this object be strictly speaking impossible, filled by metonymic impostures and perpetually displaced, unveils desire's underlying formal vacuity behind any apparent symbolic-imaginary punctuation of meaning into the subjective enunciation.
In this regard, Badiou’s (2009) formula from Theory of the Subject apropos Hegel's logic of places articulates perfectly this tensional status of the universal as “tormented” by its incompleteness, by its exceptional being, in the vanishing of force from its positional straightjacket. It reads, in a sentence, "everything that belongs to the whole is an obstacle to this whole insofar as it is included in it" (TOS: Pg. 4). This is already the core of negativity and contradiction in the Hegelian dialectical process: the universal is never equal to itself, it is always being subverted by and into its opposite. The excess of what excludes the universal in the particular, the "failure" or inadequacy in actual reality itself, reveals the vector of desire's infinite objective demand; that is to say, the circulation around objet a as that which always exceeds its symbolic placement, always an obstacle to "the Whole" by virtue of being included in it". The partial object cause, objet a, is as such in its phallic investments within masculine desire an "organ" wrested from the body, an inscription in the symbolic of a primordial lack which forever inhabits it, the ground of an fundamental self-alienation: "The objet a is something from which the subject, in order to constitute itself, has separated itself off as organ. This serves as a symbol of the lack, that is to say, of the phallus, not as such, but in so far as it is lacking. It must, therefore, be an object that is, firstly, separable and, secondly, that has some relation to the lack". - (Lacan, FFC, Pg. 112). The non-castrated father is thus at once the index of noumenal freedom and the injunction of the paternal Law to become equal to itself; just like for Kant the space of noumenal freedom bombards the subject with the impossible demands it can never fulfill. In other words, the primordial father evinces how "I am a free and autonomous subject, delivered from the constraints of my pathological nature, precisely and only insofar as my feeling of self-esteem is crushed down by the humiliating pressure of the moral law." (TWN: Pg. 47). Of course, the twist here is that far from enacting an ascetic separation of the subject from the core of enjoyment, it is only in this morbid superego enjoyment of failure against the Law that the Universal moral address coincides with a kind of Sadistic self-flagellation. Desire wants itself, onto infinity, it is designed to succeed in always failing to 'satisfy' itself. This superego enjoyment underlies every symptomatic designation of the “phallus” within the masculine logic, just like for Kant the universal demand of the moral Law weighs on the shoulder of the noumenal subject with the pressure to follow its command. Thus the phallus is not just a bare "imaginary semblance", but it embodies desire already experienced as the Other's desire, i.e. from the symbolic address of the Law which seals the enunciated content with the semblance of meaning. As Lorenzo Chiesa (2007) puts it: “This feature [of identifying the subject with the symbolic Big Other] is the one which, according to the Lacanian definition of the signifier, “represents the subject for another signifier’; it assumes concrete, recognizable shape in a name or in a mandate that the subject takes upon himself and/or that is bestowed upon on him.” (SO, pg. 116) Here we can see once again the "return to Hegel": there is always more in the particular than in the universal itself; which means that the universal is never just a bare and isolated "in-itself". Never fully coincident with itself, it has to come-out-of-itself in mediation by its Other in order to constitute itself (as in-and-for-itself): “A figure of consciousness is not measured by an external standard of truth but in an absolutely immanent way, through the gap between itself and its own exemplification/staging.” (PV: Pg. 167) We may formalize this supplementary content inherent in the universal by the masculine position thus in the following “schematic” formula along set-theoretical strictures.
Function of incompleteness:
( (фx) / ("x) (Зz) (y ε x ® z ε y & ⌐(z ε x) )
This can be read:
"For all x, if x is under the phallic function, then if y belongs to x, then there exists a z such that z belongs to y but z does not belong to x."
We will return to set-theory in our assessment of Badiou below; but for now let us simply grant Lacan this undoubtedly Hegelian moment: everything that belongs to a universal brings 'more' to it than itself; objet a is then the formal index of the impossibility of the subject caught in the 'defiles through the signifier', caught amidst the comedic phallic spectacle of the prohibiting paternal Law. Or in its Lacanian version: the universal is always missing that which would make it fully coincide with itself. The phallus is the metonymic inscription of the imagined possibility of such a coincidence, masking the vacuous intentionality of desire against the infinite displacement of the non-self-identical object, which can never be identified with any positive content: object petit a.
Isn't then the impossibility of objet petit a nothing but the non-existence of the cosmological universal 'Whole' of material reality which, under the lure of the phallus, imagines its restoration? Desire's monotonous stupidity succeeds only in circling around the wound caused by the spear of primary alienation. The subject is crucially not an individual since, forever stapled to the lure of objet a, it chases after the confirmation of its own void. Without the phallic fiction of fullness in such a noumenal “Beyond”, the logic of desire disintegrates. But it is also the non-coincidence of the phallus with objet a, of the gap between metonymic stand-ins prescribed by the symbolic big Other, and the "little piece of the Real" that always remains after the Universal's voracious attempt to reintegrate itself, that finally indexes the unbridgeable gap between Ideality and Materiality.
Again, the phallic second order exception expresses the inconsistency of the universal so that "...by being the remainder of the Real, object a will also be its remainder, that which reminds us of the loss of an always-already-lost Unity... the remainder actually reminds us of something that never existed." (SAO: Pg. 122) Woman then is precisely one of the "Names-of-the-Father", one such inexistent and decentered place of inscription which, being imagined as fully consistent, can only be the object of phantasy for the masculine desire. She becomes the object of desire, a first mover in the Aristotelian sense: "If I base myself now on the inscriptions on the blackboard, it is assuredly revealed that it is in the opaque place of jouissance of the Other, of this Other insofar as Woman, if she existed, could be it, that the Supreme Being is situated - this Supreme Being that is manifestly mythical in Aristotle's work, this unmoving sphere from which all movements" (SXX: Pg. 67)
We have now seen how Lacan articulates the dynamic, masculine position, and how Žižek attempts to locate an index for the Real as remainder in the stain of loss, the indivisible remainder for the Gaze of phantasy, the infinite object called objet petit a. What about the female position, the side of the “mathematical antinomy” which, as we hinted to above, acquires a certain ontological precedence, and which first sets the Limit for the Beyond itself? Let us follow the same procedure as before, analyzing the two formulas in their contradictory placement.
The basic move for the inscription of the feminine position consists in denying the universal as such; that is, in denying the Wholeness that man imagines is possible. The Woman, Lacan writes, “…inscribes itself there, it will not allow for any universality -it will be a not-whole” (SXX; Pg. 78)
The female position.
⌐( "x): “Non-all...” (the non-all - the inconsistency of being)
фx: "... are submitted to the phallic function."
⌐( "x) фx “Not-all are submitted to the universal” (the impossibility of Real totalization)
This can also be read alternatively:
"There is no Totality which is submitted to the phallic function"
The trick to this formula consists in acknowledging that the fact that not-All are x does not imply some are not-x. If the universal can only be accepted on condition of its exception (the Father), by the same token, the non-all entails in turn that there is no exception (to the phallic function). Thus the second formula reads:
⌐(Зz): “No one” (the non-being of the exception, the Limit)
⌐фx “is not castrated”
⌐(Зz) ⌐фx “No one is not castrated" This can be read as:
"The “primal-father” does not exist"
The conjunction of both propositions give the following formulation: "there is no Totality that is castrated, and so no exception to the function of incompleteness - Being is inconsistent." That "the Woman does not exist" is therefore nothing but the marking of ontological inconsistency, or the non-being of Totality. Woman stands for the 'Nothingness' which exceeds the Universal 'big Other':
"Woman's exclusion does not mean that some positive entity is prevented from being integrated into the symbolic order: it would be wrong to conclude, from "not-all woman is submitted to the phallic signifier, that there is something in her which is not submitted to it; there is no exception, and "woman" is this very non-existent "nothing" which nonetheless makes the existing elements "not-all." (TWN: Pg. 58)
This finally means that the female position is placed outside the “phallic jouissance” under which man hopes to recuperate the lost object; there is no symbolic inscription of this radical phallic 'all' (thus Lacan’s formula denies the symbolic inscription of the phallic big Other: S (O)). Here the logic of the Kantian antinomies allows one to see a certain priority of the feminine position: it is not the inscription of the Beyond to the phenomenal, but the Limit to the phenomenal, the very delimitation of the inconsistency of the field, which is not totalizeable. In other words, insofar as Woman is not-all she designates a Limit which then man ‘fills out’ in the image of the phallic paternal superego Law which demands the restoration of the Universal.
There is never, strictly speaking, a relationship between the two sexes, since there is never a “shared” object of desire: phallus stands on the one hand as the dynamic promise for the universal to rejoin itself, while in the case of Woman, who doesn’t exist as such, jouissance can only come from being inconsistently filled in by the phantasy of the knowledge of man, of this Other whose discourse castrates him within the intentional vector of phallic jouissance. Lacan’s formula that "...it is impossible to make love without castration" condenses thus this feature of the sexual non-rapport. Woman thus at once accepts no exception to the phallic function, but because of this, she is forced to deny the possibility of the dreamt Beyond, of the fatherly space wherein the consistent big Other enjoys globally; she is the Nothingness indiscernible from Being which is positivized into an object of phantasy by man, the prohibited but existing space of the Beyond occupied by the Father. This inscription of inconsistency which denies the exception of the exception, thus attests to the non-existent of the fully consistent Other, the "Other of the Other" which would endow the subject with ontological consistency and bring the frustrating vector of desire to a halt in unmediated enjoyment:
" ...this not-all does not mean that woman is not entirely submitted to the Phallus; it rather signals that she sees through the fascinating presence of the Phallus, that she is able to discern in it the filler of the inconsistency of the Other. Yet another way to put it would be to say that the passage from S(A) to the Φ is the passage from impossibility to prohibition: S(A) stands for the impossibility of the signifier of the Other, for the fact that there is no "Other of Other", that the field of the Other is inherently inconsistent, and the Φ reifies this impossibility into the exception, into a sacred, prohibited/unattainable agent who avoids castration and is thus able really to enjoy (the primordial
Father, the Lady in courtly love)... in its original [Limiting] dimension, Beyond is not some positive content but an empty place, a kind of screen onto which one can project any positive content whatsoever-and this empty p ace is the subject. Once we become aware of it, we pass from Substance to Subject, i.e., from consciousness to self-consciousness. In this precise sense, woman is the subject par excellence. " ( Žižek, HWL)
The feminine delimitation of the inconsistency of the material thus subtracts objet a from every alleged "consistency" enjoyed by the phallus-wielding Father; she privileges the agency of thought over the stasis of substantial being, and because of it, becomes the "subject par excellence". Withdrawn from the Universal mandate for completeness, she in-consists as a pure Nothingness without semblance of depth; qua subject she is embodies the void of the lack of being ($ against the semblance of the Ego-Ideal). This is the priority of the inconsistency of the big Other which, irremediably fractured, has nowhere but outside itself to go to find itself. It remains confined to the iterations of the symbolic defiles where no Totality imparts meaning or endows the subject with the fullness of substantial being. It destroys the transcendence of the father to circumscribe the material field of immanence wherein being in-consists. Žižek can thus equate the Hegelian truth in the Concept, spiraling out-of-itself, with the immanent inconsistency of the feminine non-all: "With regard to truth, this means that, for Hegel, the truth of a proposition is inherently notional, described by the immanent notional content... in Lacanian terms, there is a non-All of truth... Hegelian truth is precisely without any external limitation/exception that would serve as its measure or standard, which is why its criterion is absolutely immanent: one compares a statement with itself with its own process of enunciation" (Žižek, SOI: Pg. xx) Radically decentered from itself, the non-all of being constitutes that non-Totality which always resists appropriation into an alleged symbolic fixity of places; objet petit a is nothing but the mathematical inscription of this material indivisible remainder without place, the non-being of the pure immanent inconsistency of the big Other, which is why just like woman and the subject, the big Other is finally barred: S(O).
In a sense then the phallic function which castrates Man as a Whole, inscribed through the functional symbol of the phallus, designates the inconsistency of the big Other itself. The difference is that, unlike the ascription of this place in the semblance of the Universal exceptional outplace, the feminine non-all deposes the possibility of assigning a place to the lack itself; of localizing the phallus in its symbolic place to heal the wound of primary narcissism which severed the Whole of Man. The impossible object of desire is thus impossible not because it lies in the phallic ephemeral Beyond, but in the non-place of that which lingers on every time as the hard kernel of the Real, and which marks no possible exception to our dwelling within the confines of the symbolic order.
This means that finally the indetermination of objet a as the Real remainder always functions as the necessary condition for the circuit of desire to mobilize the entire wealth of the ontological imaginary; even the void of being is thus tethered to the subtraction of a from every phallic semblance, that is, from every symbolic inscription or proper name. It thus comes to function as an irreducible 'reservoir' of the inconsistency of being, the materiality so inconsistent it falls only to the spiral of the libidinal subject facing its traumatic kernel, as the condition of possibility for all ontological becoming as such. The psychoanalytic thinking of the desiring subject becomes thus of a piece with the thinking of a proper philosophical materialism, tracking the gap of the universal to itself ad infinitum. The subject can do nothing but circulate around this Real stain, at the same time attempting to flee its reduction to it by subverting it. The veil of phantasy orients the subject to withhold its transcendence from ever fully coinciding with the Real by the phallic displacements of objet a; which amounts to saying that although the subject is generated by the Real, this is a Real that emerges paradoxically as the remainder in the production of the subject. The subject resists the finitude set by the Limit of the inconsistency of being, quilting the Real remainder in the beyond, as it attempts to alleviate the wound of loss in the reification of a completed universal. This is why the circulation of phallic jouissance serves as a perpetual reminder of the Real's persistence, set against the subjective 'passion' for the Real. In this sense the subject can never become equal to this Real materiality which always eludes it, which is precisely what endows the subject its 'transcendental' being; that is, its object-oriented intentionality. As Adrian Johnston puts it:
"The subject's anxiety in the face of anything that threatens to strip it of its seemingly transcendent, immaterial status through a reduction to its brute corporeal condition isn't a mysterious, inexplicable phenomenon. Only a form of subjectivity that constitutes itself as inherently incompatible with its own finitude experiences the prospect of being plunged back into its fleshly materiality, the inevitably occluded ground of its mortal being, as a horrible danger to be avoided no matter what...
The transcendental materialist theory of the subject is materialist insofar as it maintains that this thus generated ideal subjectivity thereafter achieves independence from the ground of its material sources and thereby starts to function as a set of possibility conditions for forms of reality irreducible to explanatory discourses allied to traditional versions of materialism." (Johnston, ZO: Pg. 275)
II - Philosophical Supplementation: Objet-µ as Index of Materiality
Rather than attempting to appropriate the narcissistic disconnection between the sexes in order to unearth an philosophical index for materiality from the void of the subject's desire, Alain Badiou (2009) proposes that there is a minimal common element which is common to both sexed positions, and which founds the possible non-narcissistic evental supplement of Love. This latter supplement is the minimal incorporation of the bare individual into a 'Truth-body' which, in the case of Love, initiates a common exploration of the local situation from the perspective of an 'immanent Two'. In this regard, against the vacuity of the subject's circuitous desire around objet petit a, Badiou proposes instead that materiality is to be found in the productive fidelity which composes a truth-body, and in which the subjectivated individual rises onto the infinite productivity of the Idea. The generic Truth that Love proposes "...enters into the defiles of desire, but love does not have the object of desire as cause." (WL, Pg. 46) Against the pre-Cantorian inconsistency of the 'actual infinity' advocated by the Lacanian non-all, Badiou's set-theoretical ontology avows the fictive anticipation of the completion of the Infinite Truth that the Event initiates; as a real process in the world. Thus, against Žižek's ascription of the "inconsistent" Universal potentiated by the generative reservoir of objet petit a, Badiou will claim that it is only in the post-Evental production tethered to objet-µ that the index of materiality is subjectively accessible, beyond the alienating impasse of desire. There is an "immanent exception" which is not merely the imaginary semblance of the primal father organizing the traumatic kernel of the subject's castration, but a common element upon which the Two anchors the process of fidelity to the Truth-event. In order to be as clear as possible, let us first raise the question about what exactly the purported 'immanence of the Two' amounts to in Badiou's account. That is, does the Two immanently adequate to the inconsistency of the pure multiple of being, and does not merely amount to an "ontotheological" reification of the One? Against all stakes of counting the Two from an external purview, Love is for Badiou like the minimal dillation of irreducible multiplicity to the effects of the count of a local situation. This is why Love is a matter of post-Evental truth as a kind of exception to the laws of the world; of sex as well as the family. The stakes are thus to advance thought within a properly generic conception of the Truth, and in accordance to the strictures set forth by the theory of the pure multiple (set-theoretical ontology) in order to ground Love's immanent exception to desire's narcissistic non-rapport. Badiou's systematic explanation begins thus by differentiating the purported immanence of the Two from a) the "fusional" notion of Love (the Two which make a One and sacrifice the pure multiple of being), b) the ablative conception of love in which the One disintegrates in the grace of the Other, and c) the reduction of love to the narcissism of desire (WL: Pg. 39). Love, finally, is not a relation, but is "...a production of truth. The truth of what? That the Two and not only the One, are at work in the situation." (Ibid.) What then, would the proper 'axiomatics' of this peculiar truth-production be like? Let us present the Badiouean 'axiomatics of love' systematically, linking them in their crucial elements to the Lacanian thesis on desire and sexuation, and to the Hegelian thesis of the universal's self-undermining. This tracing is of peculiar interest since, as we shall see, the initial thesis for Badiou concerns also a fundamental agreement with Lacan with respect to basic division of the sexed singular subject into two basic positions. It also disavows from the start the possibility of a third position from which the Two could be externally counted and supervised. Such a figure constitutes, Badiou argues, the figure of the "angel" who transcendentally fixes the space of the Two, thus annexing it to the Three: "What is it then which makes it possible for me, here, to pronounce on the disjunction without recourse to or without fabricating an angel? Since the situation alone is insufficient, it requires supplementation. Not by a third structural position, but by a singular event. This event initiated the amorous procedure: we will call it an encounter." (WL: Pg. 41) If a truth is to be generic and transpositional, subtracting itself from any predicative enunciation of the state of the situation, then this must also imply thus that there is no such external third position which would potentially include the terms of both so as to distinguish each position in its singularity as two separate science, but the two is precisely in subtraction of any such determinations. This means that "...we hold simultaneously that the disjunction is radical, that there is no third position, and also that the occurrence of truth is generic, subtracted from every positional disjunction." (Ibid; Pg. 42). The positivization of the two roles which form disparate knowledges compromises the immanent genericity of the Two which, as every truth, must be subtracted from the count of the state, i.e. it must resist positional distribution by the predicative resources of the state of the situation. Thus "the staging of the sexual roles, the enrollment of terms x into two apparent classes which we will call mx and wx, is not the expression of the disjunction, but its cover up, the obscure mediation administered by all sorts of distributive rites and access protocols." (Ibid; Pg. 43) Love is never equal to the couple, since the latter always counts the immanent uncounted Two from "the outside"; the State then knows nothing of Love (which is also the elementary disjunction between Love as a truth-event, and the family as the 'State of Love') (Ibid). What this means is that the Two remains subtracted from all calculation; in the ontology of love there is one position and another position, but no operation counts the Two immanently for what it is (Ibid). How are we to understand this enigmatic articulation of the place of love in relation to the purported incompleteness of the Universal order of being; the claim according to which there is 'always more' to be counted of being? A cursory glance a the formal apparatus of Badiou's materialist dialectic reveals that the ontology of the pure multiple localizes the spurious infinity of the situation in the expansive power of the state of the situation, i.e. the metastructure which guarantees an excess of parts over elements in every situation. Let us simply recall here that given the restriction set by the axiom of foundation, it is impossible for a set to belong to itself; a set is never an element presented to itself: ("x) ¬(α e α). This is what grounds the necessary difference between a set and its singleton: while the former presents whatever elements belong to it, the latter is the Statist meta-count which 'counts as one' the former's immediate presentation and thus re-presents it as such. To be as simple as possible let us take what would be intuitively the figure of the Two, a set with two elements:
α: {β, g}
3 1 2
Since the set that presents or "counts" the Two is itself not-counted within the same (set α as such), it is obvious this count operates from the transcendent perspective of the Three, which remains implicit in such a count and gives the set its consistency. That is to say, since α forms the 'total part' which is included in the count without belonging to it, the errancy of the State fixes itself always in excess from the terms presented/counted within. We can see straightforwardly from the axiom of separation and from the axiom of foundation that the set which "counts-as-one" this transcendent one is a 'Third' which is not itself one of the terms of the initial count: the singleton of α, that is, the set {α}, having only a single element, falls short from the "immanent" count of the Two as such. But let us follow Badiou's explicit allocation of the figure of the Two within the order of the natural numbers, which belongs to ontological order of Nature, and which he identifies in Being and Event across the denumerable infinity of the "series of ordinals" (BE: Meditation 10-14). To briefly recall, an ordinal will be a transitive set composed of other transitive sets, i.e. a set will be transitive iff every element that belongs to it is also a part: (β e α) ® (β e p(α)). The order of the natural numbers will then be obtained by applying the operation of succession for every ordinal in the sequence, which for any given set will result in the minimal expansion of this set by uniting it with its singleton. Thus, we will get for S(α) to proceed by α È {α}, which 'adds α' to the elements of α. An ordinal β will thus be a successor ordinal of an ordinal α if β = α È {α}. Finally, using the axiom of replacement we can generate on the basis of the repeated expansion of ordinals, the infinity of the Natural numbers, which is also the first denumerable infinity: wo. Having laid these preliminaries, let us locate the Two within the set-theoretical succession of ordinal series, beginning from the primitive existential inscription of the name-of-the-void:
0 = Æ, 1 = S(Æ) = {Æ}, 2 = S({Æ}) = { {Æ}, Æ }... w0
1 , 2 1 , 3 2 1 ... w0
The moment one 'begins counting', the errancy of the State's 'metastructure' comes to install itself, so that ontology reveals how the moment that the void is counted-as-one, it immediately splits into Two. And just like in the base dialectic of the Hegelian passage of Being into Nothingness already implied the third figure in the unity of Becoming, the Two, as counted, is already affected by the Statist agency of the Three, lurking to usurp its immanent inconsistency. The passage from the void to its singleton (Æ ® {Æ}), or from 0 to 1, is thus in a sense camouflaging the true stakes of the count: either one doesn't count and the inconsistency of the void fixes itself as being's inertia; or one begins counting and immediately the One divides into Two, the Two into Three... ad infinitum. There were the Hegelian infinity saw its truth in the self-sublation of its finite moments, Badiou observes only the spiraling of the metastructure's anti-void functions. The state's meta-count thus prevents the multiple inconsistency of being, ontologically indistinguishable from the void, to appear in its immanence. This is why the ontological situation, as the one situation founded by the void itself, proliferates the successive stratifications of the marking of a pure proper name (the 'name of the void) by the state without ever attaining closure, becoming thus a discursive figure of the pure multiple itself. The state expands the count indefinitely since, in the trajectory of the pure multiple, there is never anything like a 'positive determination' of the Being of multiplicity; the empty recursive operation of the axioms always exceeds the immediate presentation of the terms immanently positioned. There is thus a fundamental ontological indiscernibility between discursive non-being of the One, or the perpetual necessity of the metastructure's infinite expansion on the one side, and the very inconsistency of Being which is tethered to nothingness alone on the other. In counting the void from the sole operation of belonging and through the axiomatic enclosure of ZF well-founded set-theory, ontology thus enacts the Parmenidean indiscernibility between Thinking and Being that prevents the latter to be located transcendentally with respect to the former; as a kind of non-latency awaiting to be appropriated by subjective operations. This 'passing over' or indiscernibility between the nothingness of being and the non-being of the count, thus reinscribes structurally and formally necessary the Hegelian indiscernibility of immediate Being into its opposite, and so the gap between the immediacy of Being and the implicit mediation which discursively sets the Concept in motion.
The axiom of foundation in fact guarantees that this irremediable gap between a set and itself is ontologically founded on the immanent void of being, as that which is never presented. The axiom prescribes: for every given set which is not void there belongs at least one set such that the two have no common element. Therefore, a self-belonging element, presenting exactly the same elements, would 'close the gap' between α and {α}, leaving no room for the void to locate itself at the intersection and thus to be the distinct seal of the multiple. This can easily seen if we realize that if α e α then α Ç {α} ¹ Æ (BE: Pg. 186). From the perspective of love, shattering the pervasiveness of the representational state (ontology having itself no state), the axiom of foundation thus expresses something like the demystification of the angel, of the Third position which, subtracted from the disjunctive rule of the pure multiple, sought to restitute in its asexual omniscience the rule of the One. The relation of total disjunction brought forth by the axiom of foundation thus inaugurates something like a 'primal alterity' in the order of being, a self-alienation at core since: "... a non-void set is founded inasmuch as a multiple always belongs to it which is Other than it." (BE: Pg. 186)
The site of the double enunciation of the name of the void that marks the love-declaration (the two disjoined terms pronouncing "I love you") takes place precisely where the void intersects between the Two, marking the abyss separating them. As we shall see, this seems to suggest that there is indeed a common element between the Two: the counting-of-the-void which has only the void as its element. Or to anticipate Badiou: there is a set that belongs to both Man and Woman, but if anything belongs to this set, it is void. To return to our earlier set-theoretical formulation of Žižek's reading of the phallic function, we can easily see that the axiom of foundation, in its forbidding of self-belonging, then already implies the constitutive excess of that which is included in a set to itself: since for every non-void set there will be an element whose intersection with the initial set is void, then it plainly follows that every non-void set must contain at least one element whose elements are not elements of the initial set. Formally, we express:
("x) [(x ¹ Æ) ® (Зy) [(y ε x) & (yÇ x = Æ)]] .®. ("x) (Зz) (y ε x) ® [(z ε y) & ⌐(z ε x)]
Where the Hegelian dialectical process, avowed by Žižek, sees the potency of change in the unquantifiable, material excess of the Infinite actual which subverts the Universal, Badiou sees the immeasurable errancy of the State, where only the evental supplement can subtract itself. The axiom of foundation thus, guaranteeing the gap of pure presentation with the State, is the ground of the prohibition of the Event which is precisely an 'immanent exception' to the count of being; even to the ontological count which expresses the generic being of truth. Knowledge knows nothing of the event. As Badiou puts it: "The axiom of foundation de-limits being by the prohibition of the event. It thus brings forth that-which-is-not-being-qua-being, and it exhibits its signifying emblem: the multiple as such as it presents itself, in the brilliance in which being is abolished, of the mark-of-one." (BE: Pg. 190) Thus the immanent Two of love can only come to install itself apart from any count; slipping away from the metastructure of the state and the panoptical gaze which thwarts the immanence of the void of the disjunction of the sexed positions already described by Lacan. The nominal supplement which initiates Love then fixes the immanence of the disjunction on the basis of a singular element, whose only property is that if anything belongs to it, this be void. That is, an minimal element whose only requirement is that it be "on the edge of the void".
The count-of-the-count, which is the power of the state, thus always counts that which the first count itself 'misses', always aggregating 'one more', since the gap between a multiple and its count-as-one, a set and its singleton, presentation and representation, remains forever unbridgeable. Given the inscription of the name of the void as the ground of non-presentation, the state achieves its first proper presentation in the singleton it soon leaves behind; the first set which presents something, however pale, indistinguishable from nothing. One counts then, in the peculiar situation that is ontology, not the void, but its bare marking, this 'brilliance in which being is abolished'. It is in this sense that the immanent Two of the disjunction does not coincide even with the ontologically regulated inscription or set-theoretical knowledge of the Two, it instead is delivered over into the vast expanse of the world, of being, in the unique singularity of its composition and its militant production. The immanence of the Two, puts two enunciations well on the edge of the void, united by nothing but the workings of the fidelity which unfolds from the chance encounter: "This stage of the Two is not a being of the Two, which would suppose three. This stage of the Two is work, a process. It only exists as a track through the situation, under the supposition that there are Two. The Two is the hypothetical operator, the operator of an aleatory enquiry, of such a work or such a track." (WL: Pg. 45) The immanence of the Two is therefore outside the Natural cohort of being, and is properly Historical; it requires the supplementation of an event. In ontological terms, the "numerical schema" which composes the amorous procedure is then that between the two Ones (isolated sexed positions), the Two (the Evental supplement declaring the void of the disjunction, immanent to the encounter), and Infinity (the post-evental generic procedure organizing the Two after the Event). This productive transformation carried forth by the fidelity of the Two can thereby structure, in its productive dimension, the material becoming of a new situation, subtracted from the voyeuristic transcendence of the state (WIL: Pg. 45). Since the organization around this obscure nomination which counts the void organizes the Two immanently, fracturing the isolated One of individual desire, the love-encounter sets itself against the agency of narcissistic fantasy, even if Badiou acknowledges that desire's circulation around its proper being, objet a, is inseparable:
"Thus love, which marks on bodies, as matter, the supposition of the Two that it activates, can neither elude the object cause of desire nor can it arrange itself there any longer. This is because love treats bodies from the bias of a disjunctive nomination, whereas desire is related to bodies as the principle of the being of the divided subject...
In the night of bodies love attempts to expand to the extent of the disjunction, the always partial character of the object of desire. It attempts to cross the barrier of stubborn narcissism by establishing... that this body subject is in the descent of an event, and that before the unveiling of the brilliance of the object of desire it was (this body, the supernumerary emblem of a truth to come) encountered." (WL: Pg. 47)
Given the basic supplementation that the encounter implies, the declaration of love organizes the post-evental fidelity sutured to the void of the disjunction, that is, to the lack of a positive term presented that would be non-void, and which would connect the two sexed positions. That the encounter draws from the void alone is crucial, since the 'common object µ will have to be composed of only void, preserving the disjunction. That is, without risking the sexual non-rapport announced by Lacan, the Two which follows the respective declarations "I1 love you2" only operates by fixing for the fidelity the void which separates the disjunct positions. This is why it is only "...a name, drawn from the void of the disjunction, and a differential marking of bodies, that thus compose together the amorous operator." (Ibid; Pg. 46) The sexed disjunction is not separated in its being by any statist/angelic operator, i.e. it supplements the ontological order avoiding the respective (extensional) determination proper to each position by predicative attributions. In this regard the anti-biological formal exigency of the Lacanian sexual non-rapport is preserved under set-theoretical strictures.
The disjunction of the Two is finally "only a law, not a substantial delimitation." (Ibid. Pg. 46) The intersection, reduced to the formal emptiness of a nomination of the void, the minimal object µ presents the sharing of a nothingness indiscernible from inconsistent being, from the pure multiple of uncounted void. Put somewhat less bombastically, there is never a presentation of the Two (for which the Three is needed), nor is there a set of positive terms awaiting to be allotted or aggregated to it. There is only the mere marking of its immanent non-being. Thus against the objectual fixity of objet petit a which persists in the intentional vector of the individual phantasy and interrupts the ontotheological vocation, the Love process, proceeding from the void of the disjunction, interrupts the narcissism of desire. It is "...only in Love that bodies have the job of marking the Two... it is thus very true that there is no sexual relation, because love founds the Two, not the relation of Ones in the Two... there will have been a single truth of Love in the situation, but the procedure of this unicity stirs in the disjunction by which it makes truth." (Ibid. pg. 47). Without the nominal supplement, the material productivity under which the Two organizes its becoming disintegrates into the anonymous vacuity of the disjunction, of which desire knows nothing. Only in Love is the sexual disjunction capable of sustaining fidelity to the nomination of its proper void. Love is then, escaping the masturbatory alienation of the sexes, like an expansion of the void which disjoins the Two into the expanse of generic Universality.
Having established the basic ontological features of the amorous truth-procedure in its material subtraction, Badiou's formal axiomatics of Love may be now presented. The central feature which grounds the entire set of operations will be of course the possible inscription of the sexual disjunction between the two positions, such that the counting-for-two can preserve the immanence of the void which separates them. This is the problem, since as we have seen the disjunction is never something presented to either term positively; so that the minimal difference between the two positions marks how "the sexual disjunction is simultaneously its material and its obstacle." (ST: Pg; 45). The disjunction imposes, first, the impossibility of a symmetrical relationship between the two positions. Formally, we express: ¬ [(a º b) ® (b º a)]. This means that, given the disjunction of two terms, 'if the term a enters a place given in rapport with the term b, supposedly different, it will not in any way cede its place to this term. (Ibid. Pg 46). We hereby enter the question of the worldly inscription of the Love process into a determinate situation. We first obtain the basic three axioms of the 'transcendental of the world', which we list schematically as follows:
Basic Axioms of the Transcendental
2) a £ b and b £ c ® a £ c (Transitivity)
3) a £ b and b £ a ® (a = a) (Antisymettry)
From this, we then inscribe the non-relation between the two positions Man and Woman, expanding on Badiou's formalism for clarity, as follows:
Formula of the Non-Relation
M ^ W .=. ¬(M £ W) and ¬(W £ M)
Does this disjunctive claim mean after all that love is, like the pessimistic "French Moralists" claim, merely a fiction or a semblance without reality? An imaginary veil which, like the lure of the big Other, masks the void of the inexistence of the sexual rapport and which, as we saw, Lacan brutally inscribed by crossing out the inscription of the Other, signaling its inconsistency: S(O)? Could it be that in the end sex comes to install itself as the unique "truth of love"? Badiou labels this hypothesis the "segregative" thesis, and advances along these lines the possibility of an integral disjunction, marking the emptiness of the gulf that in-consists between the two positions (Ibid: Pg. 47). That is, as we saw in ontological terms, the negation of the possibility of a positive common term to both sexed positions. Or, in other words, the inscription of the void as that which in-consists, between the Two, and which articulates the disjunction. It is clear this follows from the enunciation of the non-rapport:
Formula of the Vacuity of the Intersection
[(t £ M) and (t £ W)] ® t = 0
Yet Badiou argues, setting itself against the purely segregative thesis, for the existence of at least one non-void term which is common to both positions, a term which will be strictly minimal, ubiquitous and unique. This term is precisely possible, as we have seen, in terms not of a presentation of the void of being, but of its bare marking, of a term 'edging it', as it were. As we have anticipated, Badiou designates this object with the letter µ. This object-µ, however, is the one and only term that accepts inclusion into both sexed positions. This means, against the "Humanist thesis", that there does not exist a wealth of predicative contents which are common to the Two, and which would establish a rapport between them. Yet because this term is singular and indecomposable, such that no other positive term can be linked to it, then it a completely indeterminate element: "...it is certain that this element is absolutely indeterminate, indescribable, uncomposable. It is in fact, atomic, in the sense in which nothing singularizable enters into its composition." (Pg. 48). That this term be non-void and at the same time decomposable ultimately means that if anything belongs to it, of course, this cannot be anything but a 'name of the void'; presenting nothing but the marking of primary inconsistency into the effects of the count.
We can therefore surmise that this element, which only presents the void as such, is the presentation of a singleton of the void, of a multiple which finally stands 'on the edge of the void' for the situation of the void of being. This can be expressed ontologically by the formula: ($µ) ("α) (α e µ ® α = Æ). It transparently follows, via the axiom of foundation, that the intersection of µ with its singular element α is void: µ Ç α = Æ. Given that α is itself void, we can establish without problems for µ that ("β) (β e α) ® ¬(β e µ). This means that α 'founds' µ or that it is 'on the edge of the void' with respect to µ. This can be stated as follows: only the void founds the term common to the Two. The formula which expresses this in logical strictures and which Badiou offers is the following one:
Axiom of the common-element: object µ
(µ £ M and µ £ W) ® [(t £ µ) ® t = 0
Badiou proceeds to distance himself from the 'Fusional Thesis' which holds that the two positions, being disjointed, nevertheless reconstitute in their union the whole of humanity. This preserves the earlier thesis that there is no 'third space', but threatens the immanence of the Two by reincorporating it into the aegis of the One. This thesis holds, formally, that: ("t) t £ M or t £ W, and that M È W = 1. In continuity with the Lacanian-Žižekean avowal of the fundamental inconsistency of being and its non-totalizability (the 'non-all' of the feminine position), Badiou holds rather that there always exists a term which escapes the two positions, and which thus avoids the reconstitution of Humanity "as a Whole" from their union. Formally, this is expressed:
($t) [¬(t £ W) and ¬ (t £ M)], and from this that M È W ¹ 1. In conjunction to the non-totalizability of Nature or of Infinity in the ontological order, the axiom of the common-term therefore is also a logical prescription against the phallic desire for a completed Universal, a totalized figure of being provided by the consistency endowing count. We can easily see that in this respect Badiou is also continuous with the presumed "ontological priority" of the feminine, insofar as the latter came to stand in the sexed positions, for the very Limit to the phenomenal order.
Yet as we have seen, Badiou also challenges the segregative thesis, so that the 'common term' µ, being common to the Two positions, in a sense restores the feminine opening to the public space, and her participation in the material becoming of Universality inside the Truth process initiated by Love. Let us see how this is so since, in this sense, we can perhaps see a crucial divergence between the Žižekean-Lacanian indexing of materialism to the individual subject of narcissistic desire, from Badiou's indexing of materiality to the production of Truths. While the feminine was excluded, in her ontological priority, from the 'public space' under which the phallic defiles of masculine desire addressed the Universal, Badiou assigns the common element µ to the two positions all the same, marking its indifference to how the individual desire, in its sexed split, comports itself towards the traumatic kernel of the Real. In fact, Badiou diagnoses as part of the same phenomenon both the expulsion of Woman from the public space, and the ontic elevation of the 'feminine mystery'. There is, in the hypostasized duality of the masculine desire, both an scission of µ from the feminine, and a stipulation of the integral unity of Woman as such. This can be formally expressed as follows:
Formula for the Scission-Expansion of Woman
W ® (W - µ) ® (W = 1?)
excision dilation
Badiou thus marks for us the position that raises Woman to the position of a 'Name-of-the-Father', and finally to being nothing but the object of the masculine phantasy:
"On the one hand, the common atom µ is excised from its feminine inclusion, which states: a woman has nothing indeterminate that destines her to the public space. Or again: woman is essentially a private creature. On the other hand, since nothing atomic joins a woman to a man, there is no masculine knowledge of the space occupies by a woman. Thus the supposition of a potentially infinite expansion of the feminine, which would equal the whole, having nul commonality...
A woman is then at the same time a destitute being (in regard to what has public value and an emphatically overvalued being (in regard to the infinity of the situation)." (Ibid; Pg. 50).
Badiou then clarifies that this common term can be dealt with in two manners: the Two revolves around the incomprehension around it. First, as the non-analyzable term that it is, µ wavers as the incomprehensible void of the link, as that which doesn't shed the light of any presentation for the Two to anchor their mutual exploration of the world and which for that reason becomes the source of a mutual misunderstanding of the world (Ibid; Pg. 51). Alternatively, the term is subtracted from the two positions, paired side by side, unveiling thus that which supports the Two in being subtracted from each position. For this reason, the two possibilities result in that either "µ is the One from which the Two slips away, or is undetermined" or "µ is the separated common One from which the Two is positioned in the universe." (Ibid). The abyss of incomprehension or the source of an infinite expansion towards the infinite, objet u designates that ambivalent point at which the generic process that is Love wavers between the solitary reminder of the void of being which separates the Two sexes, and the immanent participation in the Universal by the transformation of the world thus inhabited. It is in this sense that there is something of a tension between the sexes and Love, precisely in the juncture which reveals that µ, unlike objet a, ordains the Two to the possible emancipation from the narcissism of desire. The truth that Love constitutes is therefore in a sense always placed before the vacuity of the sexed disjunction that sustains it, and µ is nothing but that marking, that place "edging the void" on which the alienated lure of one's desire offers no "common ground". Love then transforms the externality world rather than establishing a connection between the Two on the basis of the common object; the latter is only the support for this possible transformation, and becomes the index of the material becoming which makes up the process of fidelity to the encounter-event. That is, of the immanent exception which subtracts itself from the order of places and from the solipsistic vacuity of the sexual. Therefore, even if cannot do strictly away with the first disjunctive dimension which places Love all too near the trauma of desire, the second function in Love attests to the Infinite potential of expansion that every Truth, in its genericity, ordains to the subjectived body. Love therefore oscillates between two extreme poles, each of which seeks to efface one of the two constitutive functions ordained to Love by object µ:
" On one side that which, reduced to the scheme of misunderstanding of the object, can be named sexual adventure... And on the other extremity, that which, only assuming the Two, without sharing of the object, can be named sublime love, or Platonic love, which has, if I may say so, no marching orders but proposes imaginarily that the segregation itself, or the sexual mystery, be singularized as the encounter." (Ibid; Pg. 52). Neither the trivial evanescence of the adventure, nor the sublime love of the Platonist, the essence of Love is to find itself suspended between the two poles, oscillating between both ends, in utter undecideability and indeterminacy. Love is the work to preserve the procedural imperative to continue, in spite of the setbacks always present, either from the state, from the trivialization of the sexual, or from the sublimation of the imaginary.
This means that Love, dealing with the external, material transformation of the world, must in a sense operate on the focalization of that which remains for each position after the µ is assumed subtracted from each of the two, i.e. the wealth of positive terms which make up the totality of the world for the subject once the empty presentation of the void which makes up the two is taken as the uncounted reservoir of infinite being. Put more simply, to see the world as that which surrounds the difference of the Two. Formally, this is expressed as follows, where t comes to stand for a term precisely after the excision of µ has been carried out from each respective position:
Axiom of the Assignment of the Two to the World
(W - µ) £ t and (M - µ) £ t and ¬ (µ £ t)
This is not to say, let it be noticed, that the term t is common to both Man and Woman as such, since this would obviously run counter to the thesis according to which µ and only µ belongs to the Two positions. It means that each position, subtracted from that which engulfs them in atomic non-analyzability, become ordained to the multiplicity of terms which precisely do not constitute their common ground, the minimal marking of difference. This time, not the vacuity of the void captured in the edge of a singular term, but the wealth of positive terms under which Love can achieve its infinite expansion, turning itself outwards. This is what Badiou calls, using a theatrical metaphor, the "Scene of the Two", and writes: "The limping rhythm of love can be described as the diastole of its expansion around the conjoined excision of u, and the systole of what, irresistibly, leads to the central atomicity of what was subtracted... love prescribes the aura which its atomicity lacks" (Ibid; Pg 53). Love thus comes to 'saturate' the sexual non-rapport by way of a construction which excises from each position that which would limit them to a rumination around the void of a name, or confine them to the stain of objet petit a and its alienating lure. Love is suspended between the two indeterminations: that of the displaced position of the non-rapport of the sexes, and that of the exteriority of the material which constitutes the theatric scene on which the amorous encounter plays out. Finally, Badiou writes "Between µ and what, from a term t subsuming W - µ and M - µ, returns on µ, the difference of the two determinations is opened." (Ibid) Either on the lack of the non-rapport positively marked in its void rather than ordained by the desire, or on the constitutive excess which supplements the void, as the excision of the common object µ that opens the Scene of the world for the exploratory ruminations of the Two.
Against the Žižekean assignment of the libidinal stain of objet petit a as the remainder of the Real, concentrated in the ontogenesis of the subject indexing the stain's irreducible inconsistency and which precedes even the void, the form of Love inaugurates thus a productive vision of materiality. That is, a vision under which only the post-Evental expansion of the truth the subject embodies in fidelity by interrupting the 'bad infinity' of inconsistency of being can attest to the transformability of the Real. This is finally the separation between the Lacanian vision Žižek espouses, and Badiou's materialist dialectic: the first subtracts the Real from the symbolic as the constitutive lack of the object's inconsistent being. This is precisely the 'dephilosophizing' of the Lacanian vision, according to Badiou, which falters in the dream for the Absolute. In doing so, Lacan "...takes a step too far in the direction of finitude... our defense is to situate the singularity of the human animal one notch further... It is only in the transhuman body that a subject takes hold of the divisible body of the human animal. The breach is then on the side of creation, not of the symptom..." (LOW: Pg. 481) That is to say, there is a certain stasis in the conception of the Real as the stain of the remainder, which tethers it to the reservoir of being, or as point of impossibility. Badiou's wager against the primacy of objet a, and more importantly the necessity to supplement individual desire with subjectivized love, is centered around the possibility of not simply displacing the Real (qua symptom), but of radically transforming it. And insofar as the Lacanian Real remains the hard kernel against all symbolic inscription, Badiou argues, that it becomes effectively impossible to discern its effects from those of the skeptic hyperbolic doubt, which results in mere inertia, and the likely cynical submission to the imperative of desire. The disagreement with the Lacanian is laid thus: "My debate with Slavoj Žižek concerns the real. Following Lacan he has proposed a concept of it, which is so ephemeral, so brutally punctual, that is impossible to uphold its consequences. The effects of this kind of frenzies upsurge, in which the real rules over the comedy of our symptoms, are ultimately indiscernible from those of skepticism." (LOW: Pg. 563)
The Žižekean re-philosophizing of objet petit a is therefore the required rehabilitation of the Hegelian incomplete universal, animated by the lure of the lost object of desire which fascinated the barred subject in its solitary interstices. For the latter, in turn, the 'breach of the Event' can be nothing but the proto-Kantian suspension of the phenomenal which, as we seen, could perhaps at best constitute a vision of the dynamic sublimity of noumenal freedom, accommodated to the plurality of Worlds and situations. This is in fact the Žižekean wager against Badiou: "[One] can even define Badiou’s systematic philosophy (developed in his last masterpiece Logics Of Worlds) as Kantianism reinvented for the epoch of radical contingency: instead of one transcendentally-constituted reality, we get the multiplicity of worlds, each delineated by its transcendental matrix, a multiplicity which cannot be mediated/unified into a single larger transcendental frame; instead of the moral Law, we get fidelity to the Truth-Event which is always specific with regard to a particular situation of a World." (ST: Pg. 203) The choice laid to us is the following one: either the noumenal intrusion of an exceptional decision where the individual rises into the subject, or the internal periodicity of the Universal thwarting itself by reverting into its opposite, spiraling down onto the infinite derision in the futile projection towards the wound that demands, always, more of its being.
The Real as the impossibility of a stain, displacing itself in its untamable hyperactivity, or the nearness of a Real open to the transformation which the Event ordains, in exception to the Laws of being and the logical ordering of local appearance, but rising above desire's dejecting failure around the object of lack. Woman as the place of inscription for the 'non-all' of being, wavering in her inconsistent hysteria and excised from the symbolic, or Man and Woman ordained to the chance event that summons then in the encounter to participate in the immanence of Truth. The self-splitting of the Universal from itself, charging towards its failure, traumatized by the stain of the Real where the actual rises up in subversion; objet petit a as index of matter. Or the errancy of State, casting the veil of its compulsive count, obsessed with its own representational transcendence, in its ablative straightjacket before the immanence of presentation. The insufficiency of the infinite situation itself, which can only be interrupted by being summoned to the chance event under which the Two organizes a fidelity to transform the Real. The mark of the impossible, or the Event which, in its miraculous imperative, realizes only that which is impossible. Objet µ as mark of the eternal void of being, which lurks under every transient form of presentation, but also which assigns to the subject the fate of being the harbinger of the new truth of which it is capable, if only rarely, to become something other than a mere individual of the situation's circuitous infinity.
Bibliography:
- Badiou, Alain. Being and Event, translated by Oliver Feltham, Continuum Press, 2006.
- Badiou, Alain. Logics of Worlds translated by Alberto Toscano, Continuum Press, 2006.
- Badiou, Alain. What is Love?, translated by Sam Gillespie, Semiotexte, 1999.
- Badiou Alain, The Scene of the Two, translated by Barbara P. Fulks, Lacanian Ink, issue 21, 2009.
- Badiou, Alain. Conditions, translated by Steven Corcoran, Continuum, 2009.
- Lacan, Jacques, Seminar XX: Encore, translated by Bruce Fink, Norton Press, 1999.
- Lacan, Jacques, Seminar: The Four Fundamental Concepts of Psychoanalysis, translated by Bruce Fink, Norton Press, 1998.
- Lacan, Jacques, Ecrits, translated by Bruce Fink, Norton Press, 2007.
- Chiesa, Lorenzo, Subjectivity and Otherness: a Philosophical Reading of Lacan, MIT Press, 2007.
- Žižek, Slavoj, The Parallax View, Verso Books, 2006.
- Žižek, Slavoj, Tarrying With the Negative, Verso Books, 1993.
- Žižek, Slavoj, The Sublime Object of Ideology, Verso Books, 1997.
- Žižek, Slavoj, Woman is One of the Names-of-the-Father, Lacanian Ink 10, 1995.
- Johnston, Adrian, Žižek's Ontology: A Transcendental Theory of Subjectivity, Northwestern University Press, 2008.
- Hegel, G.W.F. The Encyclopaedia Logic, translated by Theodore Garaets, H.S Harris, W.A Suchting, Hackett Pub, 1991.
Index of Abbreviations
LOW = Logics of Worlds
PV: The Parallax View
BE: Being and Event
WIL: What is Love?
SOT: Scene of the Two
SXX: Seminar XX - Encore
SAO: Subjectivity and Otherness
TWN: Tarrying With the Negative
SOI: Sublime Object of Ideology.
Appendix I - Annotations on Meditation 26 in Alain Badiou’s Being and Event
The other likely candidate for the count of Two might have been the notion of the ordered pair, which occurs in Badiou's edifice around the exposition of the concept of function. Although I will not attempt to address this question here, which I believe should be answered to in the negative and transparently, some technical concepts are worth elucidating in closer detail. This holds even if, as Badiou admits that the formal exposition of the ontological operations can exceed philosophical (and therefore meta-ontological) interest. In particular, it is easy to overlook Badiou’s explicit clarification of the explication of the concept of a ‘function’, since it is delegated to a short (but doubtlessly crucial) Annex at the end of the book. There, a function is described straightforwardly as a particular kind of multiple, in unproblematic continuity with the strictures of set-theory and the pure multiple. In what follows we’ll try to elucidate the surrounding notions, since the prose in the book lends itself to easy confusion.
A function f of a given set α to a set β, which can be written f(α) = β, establishes a one-to-one correspondence between the two sets, where it is understood that:
- For every element of α there corresponds via f an element of β.
- For every two different elements of α there corresponds two different elements of β
- For every element of β there corresponds via f, an element of β.
At this point, the set-theoretical grounding becomes quite necessary to follow Badiou’s argumentation, since the concept of ‘function’ outlined above is defined, after all, as simply a particular kind of multiple. What kind of a multiple is at stake here? Here we must move to Appendix 2 of the book, which provides the sought for clarification.
Badiou begins by describing multiples which constitute relations between other multiples. These are structured as series of ordered pairs, and are written as follows:
Let’s assume the existence of a relation R between two given multiples α and β: R(α, β). Badiou describes relation as getting behind two ideas: that of the pairing of the two elements, and that of their sequence or order. This second condition guarantees that even if R (α, β) obtains in a given situation, it is possible that R(β, α) does not. The first condition entails that all relations can be expressed as consisting of two element multiples, written in the form <α, β>, so that to say that there exists a relation R between two existing elements α and β finally amounts to no more than saying: <α, β> ε R. Given that for any two existing elements α and β there exists necessarily the set which has α and β as its sole elements {α, β}, although we will see right away that this set is not identical to <α, β> . The only problematic aspect pending is finally that of order, and thus of the stipulated asymmetry between R (α, β) and R(β, α): Interestingly enough, the ‘ordered pair’ solicited by Badiou is not simply the pairing of α and β, but actually the pairing of the singleton of α, and the pairing of α and β. So we get:
<α, β> ↔ { {α}, {α, β} }
This set must exist, given that the existence of α and β guarantees the existence of their respective singletons, as well as their pair. Therefore the union of either of the first terms with their conjunction must also exist. In other words, for any given two multiples α and β there exist two different possible ordered pairings, which are not identical:
<α, β> ≠ < β, α> .↔. { {α}, {α, β} } ≠ { {β}, {β, α} }
Notice, however, that both ordered pairings are completely indifferent with respect to order in the terms of the set {β, α} / {α, β}; which are transparently identical sets. The asymmetry between the two orderings laid above occurs in the difference occasioned by the choice between {α} or {β}. This must mean that an ordered pair always consists, for any two elements supposed existent, of the two-element set consisting of the singleton of one of the two elements and the two-element set consisting of the two already given elements. Additionally, it is implied that:
<α, β> = <г, g> .¨. (α = г) & (β = g)
Finally, to say a relation R obtains between two given sets α and β entails:
<α, β> ε R or <β, α> ε R
Having established that a relation is a multiple composed of ordered pairs, Badiou proceeds to explain how a function may be described a particular kind of relation. The trick here consists in grasping adequately the abovementioned idea of ‘correspondence’. Let us assume a function f that makes a multiple β correspond to α: f(α) = β. Having established functions are relations, and relations are sets of ordered pairs, it plainly follows that functions are sets of ordered pairs. If we then allow Rf to stand for the function of α to β:
<α, β> ε Rf
But the peculiarity of the function resides here on the uniqueness of β, so that no other element can correspond to α by the function. This means that for any two multiples β and y that correspond to α via a function R, it must be the case that β and y are identical. Formally we write:
[f(α) = β .&. f(α) = g] ¨ β = g
Or, alternatively:
(<α, β> ε Rf .&. <α, g > ε Rf) ¨ β = g
If we want to unpack this formula, we write:
({{α}, {α, β}} ε Rf .&. {{α}, {α, g}} ε Rf) ¨ β = g
With this Badiou completes his reduction of the concept of relation to pure set-theoretical constructed multiplicities. The next step is to ground the comparison between sets in the series of ordinals (natural multiples). With respect to a multiple’s ‘size’ or ‘magnitude’, there always exists an ordinal which is equal to it (which is not to say only natural multiples exist; we know this isn’t true given the existence of Historical multiples). Badiou claims that thus ‘nature includes all thinkable orders of size” [BE: Pg. 270]. Here things turn a confusing, since Badiou doesn’t really provide an example until later. We can, however, give a very simple case to illustrate how exactly this happens. First, recall that the series of ordinals are woven from the void alone, as the structured sequence or passage from the void into its singleton, and thus consecutively in serial manner. If we repeat the basic example laid above where Rf stands for the function of α to β. We got: R(α, β)] ↔ [f(α) = β] ↔ [< α, β> ε Rf]
Or, more explicitly:
R(α, β)] ↔ [f(α) = β] ↔ { {α}, {α, β} } ε Rf
However, we can easily see that the multiple thus produced has the same power as the ordinal which composes the Two, and which is guaranteed given the sequence of ordinals:
Π: {{Ø}, {Ø, {Ø}}
Notice, however, that although this ordinal certainly has the same power as the given set, there’s an infinity of ordinals with the same power as the laid set: we can easily imagine the ordinal: {{{Ø}}, {{Ø}, {{Ø}}} and successive variants, all with the same power. The requirement is merely that there will be at least one ordinal with the same power.
Appendix II - The origin of the singularity, the state's double anti-void functions, and the distinction between ontological and non-ontological situations
Assume the following sets obtain:
S = {α,β,g}
α = {β,g}
β= {g,d}
g = {d, p}
As we can see, α is included in S, α Ì S; while α belongs to P(S), α e P(S).
P(S) = {α, {α,β,g}, {α,β}, {β,g}, {α,g}, Æ}
β and g are singularities with respect to S (sets presented but not represented; they belong but are not included). These are what Badiou calls potential evental sites, since g is 'on the edge of the void' or foundational with respect to S. In other words, g founds α. They are not parts of S, since they present merely themselves, and not their elements, i.e. they present themselves alone and what constitutes them is not separately presented in the situation. These multiples are 'carried over' by P(S) and its indexing of parts, to which β and g belong, without counting as a part β and g irrespectively. These remain thereby existent for the situation, but unread by the state. This is the locus for the singularity needed for the evental site of the transcendental, eventful break constituted by the subject's effectuation of the generic. These occur transparently in non-ontological situations, where the recursive operations of the situation are not those founded by the void, and therefore subject to the state's singular excess (as opposed to the non-existence of the ontological sense which counts belonging alone, i.e. the metastructure is not a representation of ontology, but an inherent function which counts elements alone, ensuing the axiomatic consistency of the theory as a theory of inconsistent multiplicity as such. This is obviously related to the anti-void functions of the state, and in ontology in particular the metastructural power of the state to always expand to infinite to seize the count of that which the original presentation misses. The thesis Badiou advances in Meditation 8 is the following one: the 'danger' of the void is countered both locally and totally:
1) Totally: the total part {a, β, g} is counted as an element in P(S), thereby countering the fixation of the void in the non-counted 'totality' of the situation, and thus counting the situation itself as one from the lingering inconsistency; onto infinite. For any set there is the set of its parts, and this set will always be greater than the original set.
2) Locally: the metastructural count of subsets by the State counters the danger of the void fixing itself in uncounted parts (guaranteed by the theorem of the point of excess) uncounted by the initial structure. This is marked in the minimal expansion of a given set, also called its successor, by adding uniting it with its singleton. Given the set of parts, and given that the intersection of a set with its union is a possible property, it is certain such a set exists (in ontology, for the series of ordinals, given the existential axiom of the void-set). In non ontological-situations, the state cannot ward off, however, the singularities of the situation, insofar as these contain unrepresented elements in its representation of the situation. This is the potential reservoir for truth-events.
As should be obvious by now, since the relation of contrariness is excluded, the double negation in ⌐(Зz) ⌐фx does not entail ("z) ⌐фx. The justification Lacan gives, whose assessment is beyond the scope of this paper, concerns the alleged separation of the universal quantifier with the range of the finite: the All/Whole is not equivalent to "every one". If the latter equivalence is established, then the inference clearly follows, just like it would follow that ⌐("z)фx ® (Зz)⌐ф. Dealing with the infinite instead, the non-all and the universal no longer pertain to what the existential quantifier ranges over, in the form of an indeterminate existence which here would coincide with the feminine inconsistency of course. Lacan thus claims:
"Now, as soon as you are dealing with an infinite set, you cannot posit that the pas-tout implies the existence of something that is produced on the basis of a negation or contradiction. You can, at a pinch, posit it as an indeterminate existence. But, as we know from the extension of mathematical logic which is qualified as intuitionist, to posit a ‘there exists,’ one must also be able to construct it, that is, know how to find where that existence is." (Lacan, SXX: Pg. 102-103).
Badiou's contention against Lacan's 'intuitionism' and the intuitionist reluctance against the 'actual [determinate] infinity' of the post-Cantorian world needn't occupy us here. For the details see Badiou's Sujet et Infini, in: Conditions, Seuil, Paris 1992, pp. 287–305.
The axiom of separation determines basically that for any α, the set of elements given for any property l(α) also exists. The axiom of subsets/powerset states that for any set α there exists a set whose elements are parts/subsets of α; the set designated thus by p(α). Since in every case α Ì α, the 'total part', it plainly follows that α e p(α). Furthermore, the theorem of the point of excess establishes that for every set there will be more parts than elements; so that in every count one may always apply the power-set axiom to count the parts and obtain a larger set. This 'implicit' excess of the State over the explicit count of presentation is what impedes the Two to be immanently counted without, like in the Hegelian dialectic, itself 'passing over into another', mediated by its Other. Finally, the axiom of foundation establishes the impossibility of a self-belonging element by establishing that the intersection of a set with at least one of its elements must be void, making thus α Ç {α} ¹ Æ) follow from ("α) ($β) ( (β e α) & (g e β) & ¬(g e α) ). See Badiou, Being and Event, Meditations 3-6, 12.
