Inclusion, belonging and the count-as-one

In the four Meditations that make up Part II Badiou sets out some basic set-theoretical concepts and procedures before explaining their wider (extra-mathematical) pertinence and then, typically, engaging with a past thinker – in this case Spinoza – whose con-trasting claims can be seen to throw his own into sharper relief.

What none the less gives this part its strong sense of a developing and tightly structured argument is Badiou’s constant circling back to the relationship between being and event, or the domain of ontology (with its basis in mathematics) and the domain of events (taken as denoting whatever exceeds the bounds of any pre-existent ontology and establishes new terms for the conduct of future investigation). I shall therefore focus on the salient themes – in particular the set-theoretical concepts along with their emer-gent political implications – and also offer some background commentary on Badiou’s project in the wider context of present-day philosophical (including Anglo-American analytic) thought.

Meditation Seven is entitled ‘The Point of Excess’ and takes us directly to the heart of Badiou’s mathematically based concep-tion of ontology as applied to issues in the formal, physical, social and human sciences. In other words it carries forward the discussion that began with his intensely dialectical staging of the difference between Platonist and Aristotelian ontologies

and that was all along rehearsed – though now more explicitly – in relation to developments in post-Cantorian set theory.¹ Most fundamental here are the relations of belonging (symbolized ∈) and of inclusion (symbolized ⊂), the former indicating that

‘a multiple is counted as an element in the presentation of another multiple’, whereas the latter signifies ‘that a multiple is a sub-multiple of another multiple’ (p. 81). That is to say, as con-cerns the relation of belonging it is here a matter of some multi-ple α that forms an element of some other multiple β such that α is ‘presented’ by the count-as-one or the existing ‘situation’ as prescribed or dictated by β. Thus ∈ is the ‘unique foundational sign of set theory’ since it establishes the possibility of all those relations (among them inconsistent, anomalous, contradictory or suchlike problematic relations) that constitute both an obsta-cle to thought and the means by which thinking typically achieves its most decisive stages of advance. In the case of inclusion, con-versely, multiple α is taken to include all the subsets (i.e. constit-uent multiples) of β and β is thus defined as itself a subset of α and yet – as Cantor showed with respect to the different ‘sizes’ of infinity – a subset that must be thought of as equinumerous with α or as capable of having its members paired off one-for-one with the members of α.² For clarity’s sake Badiou uses the term

‘element’ to signify belonging and ‘subset’ to signify inclusion, although these should not be taken to mark any further, that is, ontological distinction. His principal concern – here and through-out Being and Event – is to show how such seemingly abstract considerations in the realm of pure mathematics can have a direct (not merely suggestive, oblique or analogical) bearing on matters outside that realm.

So it is with the dualism of belonging and inclusion which, Badiou says, ‘directs, step by step, the entire thought of quantity and . . . the great orientations of thought, prescribed by being itself’ (p. 82). And again, more specifically, ‘[i]n one case (the case ∈), the multiple falls under the count-as-one which is the other multiple. In the other case (the case ⊂), every element pre-sented by the first multiple is also prepre-sented by the second’ (p. 82).

If the former (in Badiou’s clearly specified terminology) ‘pres-ents’ certain elements as ‘belonging’ to a given ‘situation’ while others are thereby excluded from it, the latter should be thought of as ‘representing’ all the subsets included in a given ‘state of

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the situation’, that is to say, as involving no such selective mem-bership conditions. It is precisely through the constant possibil-ity of a rift, a mismatch or breakdown of structural reciprocpossibil-ity between these two basic conditions of belonging and inclusion that there also emerges the potential for significant change – for revolutions of thought or theoretically informed practice – in the various spheres to which Badiou applies his dialectic of being and event. Above all, as we shall see shortly, it has to do with that breakdown in the order of commonsense-intuitive mathe-matical grasp signalled by the discovery of the power-set axiom, namely that ‘if a set α exists (is presented) then there also exists the set of all its subsets’, a set whose numerical value must clearly exceed that of set α by an order of magnitude that increases exponentially with the size of set α itself and which generates multiple orders of infinity as soon as mathematics goes trans-finite in the wake of Cantor’s revolution.

This means that the restrictive conditions on belonging which defined the membership of α must now be lifted or redefined so as to acknowledge the existence of β – the power-set of α – whose numerical value far exceeds anything admissible under those same restrictive conditions. What the power-set axiom requires us to think is the effect of that strictly ubiquitous ‘point of excess’ that will always signal the existence of a gap between belonging and inclusion, the situation and the state of the situation, or consis-tent multiplicity (as presented by the dominant count-as-one) and inconsistent multiplicity (as re-presented by all those supposedly non-belonging but none the less included subsets). This can also be phrased in terms of the rift between structure and meta-structure, or again (in Badiou’s chosen terminology) between the elements of a set and its numerically ‘larger’ multiple of subsets. That those quote-marks are required around the term

‘larger’ is one consequence of Cantor’s showing that seemingly different ‘sizes’ of infinite set – like ‘all the integers’ and ‘all the even integers’ – could be counted off one-for-one against each other without limit and could not, therefore, be thought of as larger or smaller in any such straightforward, self-evident or intuitive terms. However, it follows from the power-set axiom that in a different, mathematically definable sense the subsets of any given multiple will be larger (numerically greater) than the multiple itself and that when this axiom is extended to the

realm of the infinite (or transfinite) it goes beyond the utmost scope of calculation.

Badiou terms this the ‘theorem of the point of excess’ and regards it as occupying a central place not only in the structure, history and genesis of set theory but also in his own elaborations of a set-theoretically based ontology with far-reaching philo-sophic, political and ethical consequences. ‘This is a crucial the-orem’, he writes, ‘which leads to a real impasse: it is literally impossible to assign a “measure” to this superiority in size. In other words, the “passage” to the set of subsets is an operation in absolute excess of the situation itself’ (p. 84). Hence the need to distinguish ‘situation’ from ‘state of the situation’, the latter taken to include all those subsets whose number – by this theo-rem – absolutely exceeds that of the elements which belong to the situation according to the prevalent count-as-one. It is here – at this point where the resources of ontology are pressed to the limit and beyond – that philosophy finds itself equipped or com-pelled to conceive of the event as an ‘ultra-one’ or as a strictly

‘supernumerary’ item vis-à-vis the existing order of things, that is, an occurrence whose advent marks a decisive break with that order. Such would prototypically be instances of – in the proper as distinct from the debased or everyday usage of these terms – invention in science, creation in art, revolution in politics and passion in love.³ Each of these has its negative counterpart, according to Badiou: culture in place of art, management in place of politics, technique in place of science and sex in place of love.

Moreover, it is chiefly on the strength of his set-theoretical elabo-rations – his formal rendering of the process whereby truth-events come about in excess of any prior reckoning, predictive capacity, or power of ontological grasp – that Badiou is able to draw these distinctions and to specify what counts as a genuine event in each of those subject-domains. On his account, ‘no mul-tiple is capable of forming-a-one out of everything it includes . . . [since] inclusion is in irremediable excess of belonging’ (p. 85).

And again, ‘the included subset made up of all the ordinary ele-ments of a set constitutes a definitive point of excess over the set in question. It never belongs to the latter’ (ibid.).

So despite his extreme care to distinguish the evental and the ontological (since the former is here defined as that which can-not possibly be deduced, predicted or allowed for in accordance

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with some pre-existent state of knowledge or conceptual scheme) still there is a clear sense in which Badiou’s whole project rests on ontological foundations and indeed requires them just in order to make that same distinction. More precisely, what he sees as philosophy’s proper task is not that of making ontolo-gical discoveries or exploring new ontoloontolo-gical regions on its own account – since this is a role best left to the mathematicians – but rather that of pursuing a ‘meta-ontological’ enquiry that expounds, clarifies and draws out the consequences (some of them decidedly extra-mathematical) of any results thus obtained.

One can therefore see why he lays such stress on the claim, contra Wittgenstein and Heidegger, that ‘mathematics thinks’ insofar as it involves a creative, inventive and truth-disclosing activity of thought that cannot be reduced either (following Wittgenstein) to a mere assemblage of vacuous since purely self-confirming logical tautologies nor again (following Heidegger) to a mere expression of the techno-scientific-metaphysical will-to-power over nature and humankind alike.⁴ Indeed one fascinating aspect of Badiou’s work is the way he pursues a selective exegetical path among the many thinkers – from Parmenides, Plato and Aristotle, via Descartes and Pascal, to Frege, Russell, Heidegger, Wittgen-stein and others – against whose projects he measures his own with varying degrees of perceived affinity or (very often in the same thinker) perceived differences of view. In each case his basic argument is that we can and should read these thinkers differently in light of the epochal advance brought about by Cantor’s inaugural discoveries in set theory and their subsequent development by mathematicians and some (very few) academic or professional philosophers.

Most crucial here is the power-set axiom since it establishes the principle – the point of departure not only for Badiou’s mathe-matically based critical ontology but also for his thinking on mat-ters of political, scientific and ethical import – that no instance of the count-as-one, whatever its claim to universal inclusivity, could ever contain (or purport to represent) those endlessly proliferat-ing subsets of multiples revealed by a grasp of that axiom. Badiou follows his ‘technical’ rendition of the case as concerns set theory and its formal structure with a sentence that effectively re-states that case in a register whose normative moda lity seems to waver between laying down the necessary (non-negotiable) terms and

conditions for any adequate address to these matters and pre-senting something like an ethical injunction to observe, respect, or act upon those terms and conditions. ‘Once this is admitted’, he writes, ‘one is required to think the gap between simple presen-tation and this species of re-presenpresen-tation which is the count-as-one of subsets’ (p. 85). That requirement clearly derives its imperative force from the various formal demonstrations, from Cantor down, of how set theory achieves its most signal advances by a procedure of thinking through-and-beyond the various obstacles – mostly of a commonsense-intuitive kind, like that initially provoked by the power-set axiom – which have risen against it. To this extent it is normative in the sense that it pre-scribes what properly counts as a correct or valid application of the formal procedure concerned. That is to say, it belongs to the domain of mathematical truth where the issue of fidelity has nothing to do with ethics or the moral virtues and everything to do with domain-specific requirements such as consistency, rigour, demonstrative force, logical explicitness and so forth.

However it also belongs to that other dimension where the term

‘fidelity’ does have a strong ethical toning and where issues of truth cannot be held entirely apart from matters of truthfulness or intellectual virtue.

Not that I would wish to place Badiou in the company of those who have argued for a virtue-based epistemology with its roots in the tradition of ethical thought descending from Aristotle and taken up lately by philosophers in quest of some alternative to the current range of often sharply conflicting (e.g. deontologi-cal versus consequentialist) views.⁵ In brief, this is an approach that envisages no possible solution to the ‘problem of knowledge’

in its various forms except by instancing the sorts of jointly moral and cognitive-investigative qualities, dispositions or intel-lectual traits that best, most reliably conduce to the advance-ment of human understanding. These would typically range from epistemic virtues like perceptual acuity or sensory-intuitive

‘feel’ for the physical items or properties under investigation to epistemologically salient aspects of intellectual character such as dedication, perseverance, open-mindedness, respect for the evidence, willingness to test even one’s most cherished or firmly held beliefs against that evidence, and a well-developed capacity for self-criticism. That Badiou has the utmost regard for those

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virtues is clear enough from his writing about certain exemplary figures – such as the mathematicians Jean Cavaillès and Albert Lautman, both of whom sacrificed their lives as members of the French Resistance – whose actions and work can be seen to have manifested them in the highest degree.⁶ All the same he is very careful not to confuse truth with truthfulness, or issues of the kind: ‘is statement x true or justified according to the strictest standards of evidence or of formal (e.g., set-theoretical) proce-dure?’ with issues of the kind: ‘has statement x been arrived at by someone (or some community of like-minded enquirers) possessed of all the relevant, truth-conducive or knowledge-promoting virtues?’. That these two conditions may often be satisfied by the same statement – since the latter is defined as one that should predictably give rise to the former – is everywhere implicit in Badiou’s account of those particular discoveries or stages of advance, in set theory and other domains, that have typically occurred through a combination of rigorous thinking with the courage to defend them against the weight of established doc-trine or commonsense-intuitive belief. Still they cannot be run together or this distinction effectively collapsed – as advised by some proponents of a virtue-based epistemology – without consequently opening the way to all manner of sceptical, social-constructivist or cultural-relativist ideas. For it is no great distance from this invocation of the intellectual virtues (however carefully or strongly specified) to the idea that any such appeal is reliant on the existence of a certain socially accepted or culture-specific conception of just what constitutes a virtuous epistemic practice.

The next pair of Meditations, Eight and Nine, bear the somewhat forbidding titles ‘The State, or Metastructure, and the Typology of Being (normality, singularity, excrescence)’ and

‘The State of the Historical-Social Situation’. All the same read-ers should not be tempted to skip since there is, as those titles suggest, substantive ethical and socio-political as well as ‘purely’

philosophic content in this formal demonstration of his central claim as concerns set theory and its implications for the various domains of applied ontological enquiry. Indeed they should if possible be read at a sitting since they run to just 18 pages in all and between them offer the clearest account in Being and Event of how the three main dimensions of Badiou’s work (crudely put: the mathematics, the ontology and the politics) should rather

be seen as so many aspects of a single, strictly indivisible project.

After all, Badiou is among the most committed – that is to say, the least ‘repentant’ or shiftily backsliding – Marxist intellec-tuals of our time, and it has always been a central Marxist thesis that any genuine project of political emancipation must achieve the overcoming of this false dualism through an active, that is, practically engaged yet also theoretically informed pursuit of specific political goals.⁷ His response to this vexed question of the relationship between theory and practice is remarkable chiefly for the fact that it pushes the dualism to what looks like a point of extreme – even irreconcilable – antinomy yet does so precisely in order to expose that falsehood and thereby make his case for the practical-political relevance (indeed the indispensabi-lity) of certain, on the face of it highly ‘abstract’ set-theoretical axioms and proof-procedures. Not that we should take this relevance-claim as adequately borne out just because there can be shown to exist an analogical relation between on the one hand set-theoretical terms such as ‘class’, ‘state’, ‘inclusion’, ‘belonging’,

‘member’, ‘part’, ‘subset’, or ‘count’ and on the other hand iden-tical or kindred terms that typically figure in the lexicon of polit-ical theory and the thinking of those whose primary aim is to transform or undermine existing structures of privilege and power. Badiou enters this cautionary note – albeit in a muted way – when he remarks during Meditation Eight that it is ‘due to a metaphorical affinity with politics’ that he will henceforth deploy the phrase state of the situation to signify ‘that by means of which the structure of a situation – of any structured presen-tation whatsoever – is counted as one, which is to say the one of the one-effect itself’ (p. 95). This affinity will be explained, he promises, in Meditation Nine when the focus switches more explicitly to politics and when these so far ‘metaphoric’ connec-tions or suggestive cross-domain analogies acquire a more detailed working-out in conceptual-analytic terms.

The link is accomplished chiefly through that same technical coinage, ‘state of the situation’, which Badiou variously defines as

‘count-of-the-count’, ‘metastructure’, or ‘that by means of which the structure of a situation is, in turn, counted as one’. His point is that this involves a ‘second count’, a further operation – of the kind that has its place in all formal, logical or set-theoretical sys-tems – whereby the first count is subject to a duplicate reckoning

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so as to confirm its consistency and comprehensive grasp, or so as to ensure that nothing has gone uncounted by the prior opera-tion. To this extent, he claims, ‘concrete analysis converges with the philosophical theme’ since in both cases the thesis

so as to confirm its consistency and comprehensive grasp, or so as to ensure that nothing has gone uncounted by the prior opera-tion. To this extent, he claims, ‘concrete analysis converges with the philosophical theme’ since in both cases the thesis