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Logics of Inexistence: ‘Aporetology or Aporetography’ in Derrida and Badiou

Simon Clarke


In recent years I have sought to emphasise areas of correspondence between the respective projects of Jacques Derrida and Alain Badiou, despite their manifest discontinuities and markedly contrasting affiliations. Certainly, Badiou sets himself against so-called “idealinguistry” from the outset, and later famously declares that “there are only bodies and languages, except that there are truths” (Badiou, Logics 4). And if we’re inclined to consign Derrida to the “dust-bin of some late-twentieth-century linguistic idealism and subjectivist constructivism that just plays with language” (Crockett 3), then the scope for comparison may immediately seem somewhat limited. Indeed, this impression is likely to be compounded if we perceive a deep “affinity between […] Heideggerian orthodoxy and its Derridean progeny, which consists in their mutual refusal to renew the project of ontology after the ‘end of metaphysics’” (Wolfendale 222). Badiou’s commitment, by contrast, to the Cantorian transfinite and to his own set-theoretic ontology, wherein “ontology’s compositions without concept weave themselves” (Badiou, Being 60), is explicitly formulated in opposition to Heideggerian finitude.1

Although there have nevertheless been high profile studies directly evaluating aspects of their work in conjunction with one another (e.g. Calcagno, Badiou and Derrida, Norris, Purcell), this has evidently not seemed a fruitful line of enquiry for many. And this despite Badiou himself (Logics and Pocket) tantalisingly raising the possibility of a very real convergence in declaring that his own concept of “inexistance” amounted to the same thing as différance. But whilst this may simply be downplayed as a fond adieu rather than a genuinely substantive claim (see Calcagno, “The Transcendental”), there is more to be said here. Indeed, it is by no means necessary to ignore entrenched (often fundamental) conceptual and methodological differences between Derrida and Badiou to suggest that there are nonetheless very real shared concerns. The approach that each therefore takes, where such commonalities obtain, may thus surely have the potential both to clarify the stakes of the issue at hand whilst also enriching our understanding of both thinkers’ work in the process.

It is in this spirit then that I have argued before now that inexistence may more convincingly be likened to iterability. From the perspective of a given world, which may be defined for the time being as “any discursive imposition whatsoever” (Badiou, Logics 545), an inexistent is nothing which is to say that its intensity of appearing is minimal. But yet, “being nothing is by no means the same as non-being. To be nothing is to inexist in a way specific to a determinate world or place” (Badiou, Pocket 141). This nothing, however, is the significant absence that defines the world in question; it is what this world must exclude in order for its structuration to be as it is. This nothing, though, cannot be understood or even acknowledged from within this world.

Iterability, meanwhile, as the “essential and ideal limit of all pure idealization, the ideal concept of the limit of all idealization […] this singular concept that renders possible the silhouette of ideality, and hence of the concept” (Derrida, Limited 119), is simply the necessary possibility of othered repetition. Now, an existent may appear in an indefinite number of worlds with varying intensities of appearing, whilst “units of iterability” (Ibid. 10) are subject to othered repetition to a similarly indefinite extent. An existent, however, is a being whose self-identity is assured (indexed to its multiple-being), whilst a “unit” of iterability is constitutively denied any such status. The difference in ontological terms here is fundamental.

But yet, since iterability necessarily entails a non-present remainder or supplement, the otherness in repetition that renders the elements of a given context “themselves” iterable in the first instance, a significant absence is presupposed (something I have proposed to call an “iterable”). Thus, although its ontological status differs to that of an inexistent (which latter is indexed to its multiple-being), from the perspective of (or “within”) a given context this significant absence is nothing in that it doesn’t appear as such. It is undecidable or unrecognisable on the context’s terms but yet is for repetition; from the context’s perspective, it is a pure “unit” of repetition much like, mutatis mutandis, an inexistent for a Badiouan world.

I will not pursue the implications of this argument further for the moment, but it’s the “discursive imposition” to which I alluded above that is at stake in what follows. To what extent can a Badiouan world be considered analogous to a Derridean (con-)text? Are there sufficient correspondences to warrant the sort of comparisons I have outlined above? The two concepts are by no means identical, certainly, but borders or limits (in the broadest of senses) are nevertheless crucial to both Derrida and Badiou. To what extent, then, are there shared concerns here, and where must division remain? Where must their respective projects be demarcated and distinguished and on whose terms? Where does this border lie, and how might this delimitation edge away from or inform our reading of both, particularly at their frontiers where their opposition is fiercest?

With this in mind, then, I will begin by focusing attention on the closures, borders and de-finitions of Derrida’s Aporias, before then evaluating the “transcendental indexings” of Logics of Worlds and drawing these sketches together in presenting them around the figure of absence.

Derrida’s Aporias

A concern for borders (or edges or limits) pervades Derrida’s work but, in order to focus this study and respect certain limits that Derrida himself imposes, I will concentrate to begin with on the “aporetic typology” of his 1992 Cerisy-la-Salle address (published as Aporias: Dying – awaiting (one another) at the “limits of truth”). So what, firstly, are the aporias to which Derrida is referring? Meaning of course literally “non-passage” (ἄ-πορος), Derrida is nevertheless careful to distinguish aporias from antinomies:

the word antinomy imposed itself up to a certain point since, in terms of the law (nomos), contradictions or antagonisms were at stake. However, the antinomy here better deserves the name of aporia insofar as it is neither an “apparent or illusory” antinomy, nor a dialectizable contradiction in the Hegelian or Marxist sense, nor even a “transcendental illusion in a dialectic of the Kantian type,” but instead an interminable experience. Such an experience must remain such if one wants to think, to make come or let come any event of decision or of responsibility. (Derrida, Aporias 16)

Indeed, Derrida speaks here of experience as indicating “passage, traversal, endurance, and rite of passage”, but also “a traversal without line and without indivisible border” (ibid. 14–15), and thus the experience of the aporia is “both an impossible and a necessary passage” (ibid. 17).

In what ways, then, is this experience of the non-passage to be understood as both possible and impossible? Derrida in fact speaks of a “plural logic of the aporia” whereby figures don’t so much oppose as ‘haunt’ one another (or, rather, this haunting installs the one “in” the other [installe la hantise de l’une dans l’autre]). In the first instance, an aporia would be where a border is uncrossable, though this border may be understood as national/cultural, conceptual or between different intellectual, discursive or scientific disciplines. Such borders are closed, opposing domains wholly. A non-passage as aporia may also, however, arise from an inadequate border in that the limit is scarcely (“not yet or no longer”) apparent, the distinction between “others” being indeterminate. Finally, an aporia may be “the impossible” because there is

not even any space for an aporia determined as experience of the step or of the edge, crossing or not of some line, relation to some spatial figure of the limit. No more movement or trajectory, no more trans- (transport, transposition, transgression, translation, and even transcendence). There would not even be any space for the aporia because of a lack of topographical conditions or, more radically, because of a lack of the topological condition itself. ( Ibid. 21)

The transgressivity in accordance with which the above should be understood, however, is explicitly figured according to a “double concept of the border” whereby decisions concern “the choice between the relation to an other who is its other (that is to say, an other that can be opposed in a couple) and the relation to a wholly, non-opposable, other, that is, an other that is no longer its other. What is at stake therefore is not the crossing of a given border” (ibid. 18). In this particular address, in fact, Derrida is predominantly concerned with what may be seen, after Heidegger, as the ultimate (or in another sense primary) aporia that is death. More specifically, Derrida is edging towards Division Two of Heidegger’s Being and Time and the “absolute uniqueness, from which every uniqueness is defined, particularly every Jemeinigkeit of expecting death” (ibid. 72). We won’t continue down that particular rabbit hole here, however, so let’s instead begin to unpack this “aporetology or aporetography” (in which Derrida has not “ceased to struggle”) in more detail.

Helpfully, Derrida (ibid. 15) at this point traces something like a lineage of the “interminable list of all the so-called undecidable quasi-concepts that are so many aporetic places or dislocations”. From the “paradoxical limitrophy of ‘Tympan’ and the margins [marges], the levels [marches], or the marks [marques] of undecidability” to the “step [pas] and paralysis in Parages”, and from the “birth date that ‘only happens by effacing itself’ in Schibboleth” to “iterability, that is, the conditions of possibility as conditions of impossibility, which recurs almost everywhere”, the implication is that Derrida’s work has always in some sense been an “aporetology or aporetography”. Each allusion could profitably be pursued at length, and we’ll return to iterability a little later, but perhaps the most instructive hint is then given with reference to “the gift of the impossible (Donner le temps)”. The gift, after all, “if there is any, would no doubt be related to economy […]. But is not the gift, if there is any, also that which interrupts economy?” (Derrida, Given Time 7). This, then, immediately puts us in mind not only of the ‘twin impossibilities’ of restricted and general economies but also of the aneconomic. Where restricted economy involves calculable expenditure, general economy demands expenditure without economy, without reserve. As Simon Critchley (Ethics 169) puts it, economy “is the thought of the limit, it is the attempt to master that limit and control the thought of the unlimited, the apeiron that would overstep the peras.” According to this logic, however, the gift, if there is any, would be impossible, indeed the impossible; and yet, “the chance for the gift itself” is “[t]he chance for the event” (Derrida, Given Time 158).

In a certain sense we may say that general economy haunts the calculable as what must breach and exceed it, but yet it itself can also only fall back into exchange and restitution. And in the same way, free passage as non-passage across porous borders is forever destined to be curbed by the imposition of uncrossable, fixed and immovable boundaries, whilst the most impenetrable of frontiers must always yet be traversed and endured. But where there is “not even any space for an aporia determined as experience of the step or of the edge” this is because “an event implies surprise, exposure, the unanticipatable” (Derrida, Saying the Event 441). Indeed, the “event’s eventfulness depends on this experience of the impossible. What comes to pass, as an event, can only come to pass if it’s impossible. If it’s possible, if it’s foreseeable, then it doesn’t come to pass [emphasis mine]” (ibid. 450). Non-passage in this third sense is thus (im-)possibility as (un-)foreseeability – there is seemingly no border to cross because one (fore-)sees no border or anything beyond a given horizon of expectation.

Badiou and the Transcendental

Badiou’s 2006 sequel to Being and Event, Logics of Worlds, unfolds the “logic of appearing” of the pure multiplicities of his set-theoretic ontology in worlds, their being-there. The primary concepts here are worlds, the objects that appear in worlds, and the relations that obtain between these objects, but everything nevertheless centres on Badiou’s conception of the transcendental.

A world is a revised or more nuanced version of what for Being and Event was a situation, which is to say the structuration of any consistent presented multiplicity, any regime of the count-as-one. Presented in its multiple-neutrality, however, which only affirms the ontological non-being of relation, the relational consistency of “appearing-in-a-world” cannot be accounted for. A Badiouan world, therefore, is a situation wherein the relative intensities of appearing of its objects are evaluated (a sort of objective phenomenology). And the transcendental of a world, modelled on the sub-object classifier of category theory, is an “immanent given of any situation”: “it is what imposes upon every situated multiplicity the constraint of a logic” (Badiou, Logics 101), namely the logical operators and organisational set of the order structure of a world’s variable intensities of appearing.

This subset accordingly assigns an intensity of appearing to each of the objects of a given world, ranging from maximum to intermediate to minimum. An object, meanwhile, is an objectivated multiple, which is to say a multiple in ontological terms but inasmuch as it appears there, in this world. But being thus ontologically a multiple (the object’s “support-set”), it is itself accordingly a set of other multiples, and as such its elements are also assigned intensities of appearing on the same basis. The conjunction (or co-appearance) of the identity of each “apparent” to every other is then likewise indexed to this same transcendental, as are their localisations.

Crucially, any given multiple-being may appear with varying degrees of intensity in an indefinite number of worlds, figuring strongly in one and more peripherally in another; inexisting in or for a given world, however, is not to be confused with non-being. Now, whilst this may all appear somewhat static, a more dynamic dimension is introduced through a subsequent typology of “becoming and true change” (also a refinement of ideas initially expounded in Being and Event).

Both worlds and objects2 admit of maximum and intermediate intensities of appearing and conjunction amongst their elements, but it is the minimum that is crucial since ‘not-being-there’ is fundamental to their very possibility. Without it, indeed, we would either have to accept the being of the Whole or One-all Universe (a set of all sets being categorically ruled out by Badiou’s set-theoretic ontology), or that being-there is “identical to the thinking of being-qua-being” (ibid. 122). But there may nevertheless be only one such minimum, the inexistent, proper to any given world or object of a world, which serves to determine it as its characteristic or founding exclusion, and it is only an inexistent thus understood that can effect real change within a world.

The weakest form of change, barely meriting the name, is modification. Here, appearances may superficially be recast, e.g. a different member of the ruling elite might be elected President or Prime Minister of a Western democracy, but this “change” would amount to little more than “intensive variation” within the same transcendental order structure, the same intensities of appearing and conjunction. Anything further, for Badiou, requires a “site”, which is to say the emergence of an object which as a multiple-being belongs to itself (something axiomatically forbidden in Badiou’s ontology).3 From the perspective of being-there, meanwhile, a site is an object that assigns to itself its own degree of intensity of appearing.

A fact is a site on the basis of the above, but its intensity of appearance nevertheless falls short of the maximum. Change is effected though this leaves little impression (technically because the object’s multiple-being doesn’t fully belong to itself). A weak singularity, meanwhile, is a site whose intensity of appearing is maximal; but the change it engenders remains weak because the object’s proper inexistent is not itself raised to the maximum intensity therein. The site’s long-term implications for the world in question are consequently limited.

Only a strong singularity, an event, actually subverts the order of appearing of a world. Although a site as a singularity accords to itself the maximum degree of intensity, this is only ever fleeting, as the “laws of being close up again on what tries to except itself from them. Self-belonging annuls itself as soon as it is forced, as soon as it happens” (Ibid. 391). The site’s inexistent, however, having been “absolutized”, is no longer minimal in its “post-evental figure” and the “maximally true” consequence of the event is the existence of this inexistent. The transcendental order of the world, the relative intensities and conjunctions of its apparents, is thus forever altered.4 But there can be “no composition of that which is by a single event. On the contrary, there is [only] de-composition of worlds by multiple event-sites” (ibid. 385).

Logics of Inexistence

Various aspects of Badiou’s thinking here would seem incompatible with the thrust of Derrida’s work generally, not least the inviolability of multiple-being. Furthermore, the ordered regularity of worlds’ transcendental structures, and the precise conjunctions of existence and identity in appearing they entail, would seem far removed from the infinitely elusive differings and deferrals of deconstruction (for Badiou, “no time is implied in the transcendental indexing of being-there. Time here is only a parasite introduced by the metaphorical or didactic usage of vulgar phenomenology” [ibid. 202]). But there is here, nevertheless, a fundamental (indeed foundational) point of correspondence between them, something like a shared border in the crossing of which (in either direction) they part ways.

Throughout Logics of Worlds Badiou refers to inexistence primarily with reference to “objects”, formalised as ØA (where A is an object, Ø is its proper inexistent). Later, however, in an endnote concerning the relation of his work to Derrida’s, he clarifies this by defining A as “any discursive imposition whatsoever [une imposition discursive quelconque]” (ibid. 545). Meanwhile, it should be remembered that Badiou (Being 8) is at pains to remind us that being is not itself mathematical, “which is to say composed of mathematical objectivities”, and that his ontology is thus “not a thesis about the world but about discourse [le discours]”. And since this ontology amounts to a structured presentation (ibid. 27) of the pure multiple (or the implicit prescriptions by which it is structured), “discursive imposition” here simply means structuration of anything whatsoever (anything that “is”) as it is imposed or “localised” (“the pure multiplicity that constitutes its generic or indeterminate being […] in so far as it is there” [Badiou, Pocket 127]). Understood this way, we have already adverted to Badiouan worlds and objects as being founded on what inexists for them – a structural necessity if there can be no world of all worlds, or Universe, or set of all sets (we recall the role of the void in Being and Event as the “phantom of inconsistency” or “remainder” which “is the base of there being the ‘whole’ of the compositions of ones in which presentation takes place” [Badiou, Logics 56-7]). So, at stake here in the definition (and bordering) of a world (as set) or object on the one hand is structuration, and on the other a founding exclusion. To reinforce this, although Badiou refers to appearing “in” worlds, he is quick to point out that he writes

‘in’ a world (in quotation marks) to indicate that we are dealing with a metaphor for the localization of multiples. As a situation of being, a world is not an empty place – akin to Newton’s space – which multiple beings would come to inhabit. For a world is nothing but a logic of being-there, and it is identified with the singularity of this logic. (Ibid. 102)

This doesn’t, of course, preclude a given world having physical borders, but it is nevertheless primarily a logical matter that may also be topographical (e.g. “Paris in the spring of 1871”). On a similar basis, Hubert Robert’s The Bathing Pool, one of Badiou’s favoured examples of a world, is defined by its logical operators, by its transcendental degrees of intensity and conjunction, not by its borders or frame. And yet, the equivocation remains since not only does this particular world have evident limits, but as with any world or object it is “founded” on an exclusion, an inexistent that both is and isn’t “there”, “in” the world. Without venturing a questionable Badiouan metaphorics5 (still less a parergonality), whatever appears “in” a world may necessarily also appear “in” a multiplicity of other worlds, and thus this equivocal status and porousness of “borders” is everywhere writ large. Indeed, the “hors-lieu in any lieu” has been with us in Badiou’s work since at least Theory of the Subject, reference to which Badiou (Pocket 136) seems self-consciously to make in the Derrida chapter of his Pocket Pantheon.6

We have already seen that Derrida in referring to “aporetology or aporetography” through Aporias strategically equivocates on a similar basis. He goes on (Aporias 40–1) to “problematise” three “forms of limit” (to which we alluded in passing above), starting with the “problematic closure” which

assigns a domain, a territory, or a field to an enquiry, a research, or a knowledge. All of this is ordered in relation to a thematic object, more precisely to an entity, to a modality of the entity whose identification is presupposed by the unity of this space, which in principle can be closed.

Referring to what he takes to be the strictest or most common sense of “border”, Derrida then designates as “anthropological” the “spacing edge […] that is not natural, but artificial and conventional, nomic” and which serves to separate “two national, state-controlled, linguistic, and cultural spaces”. Finally, we have “conceptual demarcation” or “logical de-finition” which, “if it were possible [emphasis mine], would tend to oppose rigorously two concepts or the concepts of two essences, and to purify such a demarcating opposition of all contamination, of all participatory sharing, of all parasitism, and of all infection”.

But we know already that according to Derrida’s “aporetology or aporetography”, any such borders inevitably involve non-passage. Whether the border is closed and the other is other, or the border is open to the extent that the other is the same, or again where there is no space for an aporia at all and the other is thus wholly other, these distinctions and their traversals are themselves traversed again and again. Let’s, though, take a step back for a moment.

Accepting that a “world” is in fact an order structure (whereby even “localization” may potentially seem misleading if it isn’t understood topologically), can the same be said for Derridean contexts? It might seem perverse to attempt to assign precise degrees of intensity and conjunction in given (con-)texts to the same extent as Badiou here (though how much more restricted could an economy be?), but hierarchies are of course very much at stake. This is particularly true with respect to familiar early texts as they contend with the oppositions of the Western metaphysical tradition. Amongst the most celebrated examples here would be Of Grammatology’s reading of Rousseau (“… That Dangerous Supplement …”) and the outright priority given to speech over writing (and nature over culture). These are rigorously opposed concepts, putatively purified of “all contamination, of all participatory sharing, of all parasitism, and of all infection”. The border between them would appear to be closed, yet the work of deconstruction proves it to be all too crossable in drawing the favoured term from the subordinate one which thus becomes prior as “more originary”. Ultimately, though, the opposition itself is destabilised by the emergence of an “undecidable” term that cannot be understood according to the logic of the text, from “within”, even though it is nevertheless of it. There is no space for this “supplement”; since it is on neither side of the border, there can be no step in either direction.

Later texts certainly, starting perhaps with Force of Law, go on to foreground aporias increasingly, most notably Aporias itself, but we saw above the extent to which Derrida avers continuity on this point through his work. And this logic of the supplement as “a sort of blind spot in Rousseau’s text” (Grammatology 163) is where a genuine correspondence between Badiouan worlds and Derridean contexts becomes apparent. Indeed, as the “not-seen that opens and limits visibility” the blind spot’s role founds the logic of the text, as do “hymen”, “trace”, “pharmakon” etc. “in” their respective texts,7 just as the inexistent is the necessary foundation of a Badiouan world (though, as indicated above, I have argued elsewhere that iterability best exemplifies this correspondence). These foundational exclusions cannot be “perceived” from “within” (on the text or world’s terms), but they are nonetheless there. They are what stands to breach the economy of the world or context’s assigned values.

But it is at this intersection, this crossing, that Derrida and Badiou part company.8 An event for Badiou involves the emergence of a subtractive, self-relating and self-supporting truth, something eternal and universal that may be re-subjectivated on an infinite basis. For Derrida, however, an event is an impossible possibility even should it have taken place, since its impossibility continues to haunt its possibility (as a sort of spectral revenance). Thus, even though an event must be singular and irreducible (e.g. a gift cannot be registered as such within the circularity of exchange) it will already have been “neutralized” by its iterability (hence the “centrality” of this latter, everywhere and always), by the necessary possibility of othered repetition. Both the order-structure of a world and the logical economy of a context must be founded on exclusion if they are to be non-totalizable, and thus their borders (be they disciplinary, nomic, conceptual) are determined by the values they assign and this way maintain. At stake for both, then, are the logics of (potential) transformation through this exclusion and the overturning of given orders and evaluations. For Badiou, this is a rupture where something of being itself subverts the order of appearing, but for Derrida this is an “interminable experience”. Indeed, “the concept of text or context” that guides him, Derrida insists (Limited 137), certainly “does not exclude the world, reality, history”. But to say that they “always appear in an experience, hence in a movement of interpretation which characterizes them according to a network of differences and hence of referral to the other, is surely to recall that alterity (difference) is irreducible”.

Simon Clarke, Royal Northern College of Music (UK)

Works Cited

  • Badiou, Alain. Being and Event. Trans. Oliver Feltham. London: Bloomsbury, 2005.

  • ---, Logics of Worlds. Trans. Alberto Toscano. London: Bloomsbury, 2009.

  • ---, Pocket Pantheon. Trans. David Macey. London: Verso, 2009.

  • ---, Theory of the Subject. Trans. Bruno Bosteels. London: Bloomsbury Publishing, 2009.

  • Calcagno, Antonio. Badiou and Derrida: Politics, Events and their Time, London: Continuum, 2007.

  • ---, “The Transcendental and Inexistence in Alain Badiou’s Philosophy: A Derridean Similarity?” Philosophy Today 58.2 (2015): 257-268.

  • Critchley, Simon. Ethics-Politics-Subjectivity: Essays on Derrida, Levinas and Contemporary French Thought. London: Verso, 1999.

  • Crockett, Clayton. Derrida after the End of WritingPolitical Theology and New Materialism. New York: Fordham University Press, 2017.

  • Derrida, Jacques. “A Certain Impossible Possibility of Saying the Event.” Trans. Gila Walker. Critical Enquiry 33.2 (2007), 441-61.

  • ---, Aporias: Dying – awaiting (one another) at the “limits of truth”. Trans. Thomas Dutoit. Stanford: Stanford University Press, 1993.

  • ---, Given Time. Trans. Peggy Kamuf. Chicago: Chicago University Press, 1992.

  • ---, Limited Inc. Trans. Samuel Weber & Jeffrey Mehlman. Evanston: Northwestern University Press, 1988.

  • ---, Of Grammatology. Trans. Gayatri Chakravorty Spivak. Baltimore: The John Hopkins University Press, 1997.

  • Livingston, Paul. “Derrida and Formal Logic: Formalizing the Undecidable.” Derrida Today 3.2 (2010): 221-39.

  • Norris, Christopher. Derrida, Badiou and the Formal Imperative. London: Bloomsbury Publishing, 2012.

  • Purcell, Lynn Sebastian. ‘Two Paths to Infinite Thought: Alain Badiou and Jacques Derrida on the Question of the Whole.” Cosmos and History: The Journal of Natural and Social Philosophy 8.1 (2012), 151-176.

  • Wolfendale, Peter. Object-Oriented Philosophy: The Noumenon’s New Clothes. Falmouth: Urbanomic Media, 2014.

1 .

“Heidegger still remains enslaved, even in the doctrine of the withdrawal and the un-veiling, to what I consider, for my part, to be the essence of metaphysics; that is, the figure of being as endowment and gift, as presence and opening, and the figure of ontology as the offering of a trajectory of proximity. I will call this type of ontology poetic; ontology haunted by the dissipation of Presence and the loss of the origin.” (Badiou 2005, p. 10)

2 .

Note that there is a degree of fluidity here on the one hand between elements and multiples, and on the other between objects and worlds. Ontologically speaking, elements are themselves of course multiples, but inasmuch as a multiple exists in a world as an object, ontologically distinct elements may be grouped together in appearing as ‘one’ component within this object. This ‘atom’ (actually a function) is the characteristic operator of maximal intensity for that component, e.g. for the object ‘temple’ of Hubert Robert’s painting The Bathing Pool, the most distinct column would be the atom or count-as-one for all the less distinct columns. As regards worlds, which are themselves also multiples, although the temple may be an object for the world ‘The Bathing Pool’, this latter would be an object for the world ‘The Metropolitan Museum of Art’, which itself would be an object of the world ‘Cultural Conservation in New York City’ (and so forth). Note that a world must nevertheless be an ontologically closed set, which is to say of an inaccessible cardinality.

3 .

Viz., amongst the nine axioms of ZFC (Zermelo–Fraenkel set theory), the axiom of foundation.

4 .

A characteristic example here is the world ‘Paris in Spring 1871’. ‘March 18’ is a site for this world since it marks the initiation and point of reference for the political authority of the Paris Commune; as such it bestows upon its proper inexistent, ‘worker-being’ (or ‘the possibility of a proletarian and popular governmental power’), the maximal degree of intensity of appearing. Although this political sequence is destined to collapse in a little over two months, its long-term consequences are that the political capacity of workers exists and cannot be effaced.

5 .

Cf. Book III, Section 3.7 of Badiou 2009a, ‘Atomic Logic, 1: Localizations’. Here Badiou proposes to move away from metaphors towards a ‘more fundamental determination, the one contained in the expression ‘being-there’, whose character is topological’ (p. 252). This ultimately consists in ‘retrogressing from the transcendental constitution of appearing towards the ontological constitution of what appears […] to something like the structuration of the multiple as such’ (p. 254).

6 .

See also Bruno Bosteels’ Badiou and Politics for a discussion of this continuity in Badiou’s work, from Theory of the Subject and Logics of Worlds, despite the rupture that Being and Event seems to represent.

7 .

See Livingston 2010 for the ‘position’ such terms occupy with respect to their texts and undecidability.

8 .

We may also note that one border in particular that would seem to remain fixed for Badiou would be that of the ‘problematic closure’ between fields of enquiry. Indeed, although there is a ‘retroaction of the logic of appearing on the ontology of the multiple’ (2009a, p. 261) and this achieves a ‘real synthesis’, and although the inexistent as the ‘reserve of being’ of a world ‘sutures’ this latter to the pure multiple, the domains are nevertheless distinct.

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