So the problem for archi-metaphysics in its contrary delimitation of dogmatic or classical metaphysics is what to do with the indeterminate that dogmatics simply assumes to be there for it. The power of dogmatics lay in its use of this indeterminacy; the correction of archi-metaphysics is to determine this unknowable unknown, to replace it with an archi-metaphysics, that is, with the suspension of sense to an undetermined that is purely and simply left to the historial indeterminacy of its coming. Archi-metaphysics is the replacement of a necessary undetermined with a contingent one, or: the established power of an unknown master is opposed by the poetics or prophetics of the to-come (181).
This is exemplified in Kant – to which Wittgenstein and Heidegger and Comte pay their own particular homage (181) – by the shifting of God to regulative idea, from knowledge to ‘matter of faith.’ We thus recover religion within the limits of reason alone, while leaving God as indeterminate to his own devices. Once again, the limits of reason are coordinated with those of experience, enabling a closed world without any place for what is in exception to it.
What Kant regulates is the place of the indeterminate as such, such that it be at any time God, man, rule or law: whatever must name that space as off limits to thought as saving reason. This, for Badiou, is the great terror of critical, positivist or hermeneutical archi-metaphysics – the ruin of the concept. The sophistry of it – which in a certain sense is also a conceit – can be put in this way: to know what cannot be known; to have the knowledge of what cannot be known to knowledge.
Thus, as ever, in thrall to this indeterminacy at the limit of a conceit, all thought is reduced to a form of expression or a making sense or a language game which requires that it never overstep the mark marked out for knowledge beforehand, so that it never strays into ‘metaphysics’. To reverse a favourite citation of Badiou from Mao: for archi-metaphysics, ‘we will not come to know all that we do not know.’ Or: we must not because we cannot.
It is no coincidence that what is common to the three creatures of archi-metaphysics is a reductive approach to mathematics. Badiou, taking Kant as exemplar, compares them unfavorably to Leibniz, Spinoza and Descartes in terms of the proof of God (or whatever master signifier you like). For Badiou, the rationalism of these latter three, derived in good measure from their specific, extensive and knowledgeable interrogations of what mathematics thinks, trumps the former insofar as this rationalist metaphysics ‘blunt[s] indeterminacy and exposes transcendence to a rational control more rigorous than could ever be exerted by ‘positivism’s Humanity, Kant’s moral subject or the poet of hermeneutics’ (181-2).
Certainly Hegel, Badiou argues, recognises this rationalist advance in dogmatic metaphysics over archi-metaphysics. Hegel in fact recognises and takes as fundamental for thought itself that axiomatic alignment of thought and being, conceived by Parmenides. In other words, the subject/object dichotomy – the subject of thought and the object of its thought qua unknowable being – essentially concedes in advance and militates against what it supposes as the mark of man and subject as such – its very subjectivity. Or, at least, one side of subjectivity, that which is not so much subject as subjective; which is to say, that which would be subject not to the limits of language and world but to what is in exception to it and thus become a maker of its world as such. In Hegel’s terms: that ‘thinking in its immanent determinations and the true nature of things form one and the same content’.
If Hegel points the way, he is not the answer for Badiou. What this means is that Hegel recognises an essential aspect of dogmatic metaphysics that archi-metaphysics cannot see, and that is the exceptional nature of the indeterminate. As exceptional, then, thus in some form of thinkable relation to thought itself: not excluded from, but immanent to it. As Badiou writes: ‘A being, philosophically accessible as a name, can be said to be essentially undetermined if amongst the predicates that permit its definition is the claim that this being exceeds, in its very essence, any predicative determination available to an understanding such as ours (182).’ And: ‘The name of ‘metaphysics’ will then be given to that discursive disposition which claims that an undetermined being, as we have just defined it, that is, a being whose determination exceeds our cognitive power, is required in order to complete the edifice of rational knowledge’(182).
There are three things to note: Hegel does not endorse classical metaphysics, but recognises in it the power of the concept – to make a ‘predicate of the impredicable’ (183). But only insofar as this determines the question or marks the site from which the question of being thought must take its orientation. We can call this site negation, though that is only indicative of an operation at this site. This site is what is nothing for the efforts of predication and can thus have no bearing on reason but for Hegel – and Badiou in another way again – this being which exceeds determination in its essence (not its substance) is what must not be excluded from thought as such or else an integral aspect of the thought of being, that which is in exception to it, cannot be thought.
In other words, rational knowledge, classical metaphysics, would be that which takes on its own aporia as itself, that admits a thinking exists capable of working through what exceeds it without either reducing its essence to knowledge or knowing its essence to be unthinkable as such. Rather it constructs a discursive framework capable of supporting and articulating what is nothing to knowledge as real. Thus, as Badiou says, ‘that it be able to place, within a discursive framework available to all – an argumentative and not a revealed framework, in other words a rational framework – a point of indeterminacy that may, from that moment on, harbour any signifier of mastery whatsoever’(182). In Badiou’s determination of a set-theory ontology, this role is taken by the void. The nothing as name of being, that is!
But before we return to this metaphysics without metaphysics we must see what dogmatic metaphysics admits which archi-metaphysics – the metaphysics of contemporary philosophy – refuses to know as knowledge. Badiou recognises in this classical schema the sense given to what it pursues by Aristotle – metaphysics as the science of being qua being. Let’s note first that this makes metaphysics the same as ontology for Badiou. Ontology, he says elsewhere and everywhere in his work, and citing Aristotle, is the ‘science of being qua being.’ We must also note that ontology for Badiou thinks also the exception, in terms of the place or site of the coming to be of that which is not being qua being.
What is not being qua being for Badiou is what the event names within a situation of being, and as such marks the place of the possible coming forth of a new truth of that situation as in-exception to it – an immanent exception. All truth such that it must come to be is subjective, the subject being the finite support of an generic, infinite truth (infinite in its being). For Badiou, the subject is the meta-physical category par excellence, being what is between what is not being qua being and its being a body in a determinate world: as such, having ‘no place to be’. It requires a meta-physics because ‘of the subject, there can only be a theory. ‘‘Subject’ is the nominal index of a concept that must be constructed in a singular field of thought, in this case philosophy’. Thus: To think is to be.
But this is to get ahead of ourselves in the sense that for Badiou what forms the framework of the rationality of a classical metaphysics – that one may prove an existence without thereby determining what exists – is correlated to the notion of the indeterminate as One, while for Badiou, adhering to this same determination as to the power of a metaphysics qua the concept, rationality – mathematics (as for post-Aristotelian rational metaphysics) – dictates that being is not One. For the classical world, the One – in Plato, the Good which is not an Idea – serves as the determination of the indeterminate such that a thought can think it. That is to say, its existence is thinkable while its essence remains indeterminate. Or: ‘that one may prove an existence without thereby determining what exists is the core of metaphysics as power. Metaphysics is classical, or dogmatic, when it grants the undetermined point of its apparatus the rationality of its existence’ (183).
What classical metaphysics after Plato borrows from mathematics is the demonstration of existence purely on the basis of the concept. Metaphysics is at base the recognition of a pure existence. Meaning that this existence, which cannot be empirically attested, and the being of whose content is beyond the measure of our cognition, can nonetheless be rationally demonstrated (183).
For Badiou, then, this is what is crucial to classical metaphysics or what a classical metaphysics under the condition of the rational force of mathematics shows us to be crucial for thought as such: that existence is rationally shared between the undetermined and the determined, the infinite and the finite (184). In other words, that the transition from the finite to the infinite is ‘by way of existence,’ the decision that existence is not reducible to known knowledge, or that ‘there exists’ is the recommencement and not the end of thought. A ‘thought’ that is, in Badiou’s words from Being and Event, ‘nothing other than the desire to finish with the exorbitant excess of the state’. That is, to finish with a predication in excess of itself – what Plato called a false conceit of knowledge – precisely the knowing of what must not be known.
‘In the end nothing is more corrosive for philosophy than to separate itself from this [rational] regime, which creates, beyond that which can be empirically attested, the real of a simple possibility, and destines thought to the only thing that matters, its absolute identity with the being that it thinks (184).’
For Badiou this ‘subsumption of the existential’ by the mathematical – which Hegel has pointed to – is both what is common to Plato, Leibniz, Spinoza and Descartes and what Kant and by extension the positivists and hermeneuts miss in dogmatic metaphysics.
However, Hegel thinks that this rationality is lacking in terms of the absolute, that this rational apparatus lacks, if you like, the form of its rationality — which has to be given by speculative dialectics. As Badiou notes and laments, Hegel was himself not shy in deprecating mathematics. But it is with respect to the infinite that Hegel doesn’t fall into line with the anti-metaphysics of archi-metaphysics which, for Badiou, prides itself precisely on reducing knowledge to the dimensions of the finite alone; of what, in other language, can be demonstrated to be constructible relative to any posited existence.
This is also to link Hegel to dogmatic metaphysics, which, as we have seen, gives us a ‘rational treatment of the existence of the infinite’, thus holding at bay the finitist demands of constructivist-empiricism, which render death as the horizon of the knowing subject. Dogmatic metaphysics is a discourse of the effective proof of the infinite – proof as what assures the ‘mathematicitiy of existence’ (184). Its proof is in its discourse, that infinite being is sayable beyond knowledge as what we will (have) come to know.
Thus the anti-metaphysics of archi-metaphysics must separate out the infinite from what can be thought, from the subjective capacity for thought. Denying the discursive capacity of mathematics is one step in this deposition, returning us, Badiou says, to an empirical finitude that ‘Plato would not have failed to consider as anterior to any philosophy whatsoever’ (185). Thus in these terms archi-metaphysics is a sophistry: at least insofar as it is hostile to what mathematics effects as real with regard to the infinite (and so what isn’t real with regard to the finite).
Badiou notes here that Kant recognises another feature of metaphysics that treats it less in terms of it being an operation of thought than of it being something integral and indeed natural for thought itself. This ‘biological metaphor’, Badiou notes, is important. Hence Kant can recognise metaphysics as an existence, of nature such that it underpins cognition – the always there – and he can at the same time displace it from the subjective framework of this same cognition. Hence it is ‘always there’, in the nature of thought, as that which must be overcome or maintained in its proper place as excessive to reason, relative to the faculties available to the transcendental subject.
‘Kant is very close in the end to collapsing his critique of dogmatic metaphysics into an equally dogmatic metaphysics of the nature of thought and of the ultimate ends of the contradiction between the transcendental organisation of the understanding and reason's urge-to-transcendence’ (185).
In concentrating his attention on the faculties of cognition and the determination toward transcendence wherein the nature of this thought is annulled as, again, without knowledge, in the literal not relative sense, Kant exacerbates or even, as Badiou suggests here, ‘dogmatises’ the separation of thought and being all over again. Thus in recognising existence as qua metaphysics – the natural thought of ‘what is’, so to speak – and separating it off from what is the subject’s cognitive capacity qua subject, Kant ‘augments rather than decreases the part played by the undetermined, and consequently the recurrent possibility of a veritable metaphysical obscurantism’ (185). ‘Augments,’ because Kant determines its existence, and ‘obscurantist,’ because there must be an existing part of thought unable to be thought by thought as such. Previously the indeterminate had no existence and thought was limited by it: now an indeterminate is posited to exist such that thought itself must render it inexistent.
The dialectical challenge to this, which points the way out, is to propose a real determination of the undetermined that endows metaphysics with its power – the power to ‘infinitise the finite’ (186). So dialectics seeks this answer as the means to have done with the transcendental indeterminacy that organises and orients classical or dogmatic metaphysics; that is the form of its existence so to speak, while not of course lapsing back into what it considers worse. Thus, to be ‘neither Kantian, nor empirico-positivist, nor phenomenologico-hermeneutic’ (186). Badiou names in this neither-nor vein – besides and in debt to Hegel – Lenin contra the double injunction empirio-criticism; Freud and Lacan with regard to the ‘cunning power’ of negation and its realisation in terms of the subject of the unconscious. The power of the theory of negativity in each, thus that which marks out what inexists as real for any possible knowledge of being as such, maintains discursively this to-and-fro between the finitude of a being and the infinity to which it owes its determination.
However, Hegel’s praise of classical metaphysics in the sense of its rational relation to existence, opens onto what is for him the problem of how the conceptual apparatus it uses to grasp or name the existence of the indeterminate are themselves constructed. Thus its forms of [pure] thought, pace Kant here, are themselves uncritically deployed; that is to say, what metaphysics brings to bear as thinking itself is pure determination. Metaphysics is indeterminate in actu we could say, and not just its object. Indeed, the [life of the] object is precisely as such what must be thought for Hegel, such that being and thought are the same. Being must be thought, in other words, such that we can come to know what thought is – the rational determination of its concepts and categories.
This entails for Hegel, Badiou argues, that ‘each and every category, whether it be being, nothingness, becoming, quality, quantity, causality, and so on, ultimately consists of a definite time of determination, if only one has the patience to follow the true movement of transformation whereby each category takes place as the exteriorisation and dialectical truth of the preceding ones’ (187).
This is, then, logic: the logic of determination replacing dialectics; a move Hegel says he owes to Kant. The point being that dialectics is destined for higher things while the destitution of metaphysics is carried out by logic. As Badiou describes it: ‘a regulated process of determination, whereby the undetermined absolute (for example being, being as such) lets integral singularity take place as the ultimate immanent specification of itself. Logic is here the logic of determination, which leaves no indeterminacy behind, and which, in this sense, abolishes metaphysics’ (187). But in this form it clearly has its roots in Aristotle. One of the ironies of Kant’s claim against the science of metaphysics not changing since Aristotle is that the logic Kant has recourse to is itself unchanged since Aristotle and so he is, as Badiou has suggested, in the manner of repetition: despite himself, nothing new.
Determination here means to count what shows itself as tending toward its proper end: there being only one. As there is nothing indeterminate for knowledge, knowledge being the extent of determination, metaphysics has no proper end and so by extension there is no knowledge of it: or what it speaks of cannot be known and so is not. Metaphysics is an empty discourse, outside logic, nothing. But in a sense this is an auto-abolition, at least if we ascribe to Kant the nomination archi-metaphysician because the indeterminacy he invokes as nothing is the one that sustains his philosophy as object – being as such or the thing-in-itself. This is the case Badiou argues, following Hegel, because Kant’s critique of classical metaphysics, ostensibly that it begins with ‘special objects’ – soul, god, the world etc., and ‘forgets’ the categories that allow for the capture of these objects as objects, pushes so far against the object, that the categories obtain ‘an essentially subjective signification’ (188). The object becomes then almost absolutely indeterminate – thus an ‘infinite obstacle’ as Hegel put it.
‘It is this operation’, Badiou asserts – thinking of what he elsewhere calls Kant’s ‘obscurantist attachment to pious moralism’ – ‘that creates the radically unknowable. It allows the placing of all signifiers of conformism and of moralising oppression in the beyond of the suprasensible’ (188). Hence what Kant calls knowledge is reconciled to a faith that what cannot be thought – qua radical indeterminacy – must be, for this very reason, the site of the Good to whose wisdoms we logically submit.
It’s a perverted Platonism insofar as for Plato, under the sign of the Idea, thought names the commensurability of the known and unknown. What enables an other thinking of the indeterminate possible is mathematics, which, moreover, allows that a situation be re-thought beyond what logically constrains it. Referring to Plato ‘in passing’, Badiou notes that this is the courage of thought which amounts to, then, as dialectical in the first instance, to break with both classical and archi-metaphysics: to the objectivity of the undetermined and to subjective finitude which stands in critical-archi-metaphysics ‘alone against the undetermined absolute’ (188).
Essentially, dialectical argument poses that a category of thought is only such on condition that it exhausts without remainder that which is thought in thought through this category. Or, to quote Hegel, if the category remains a form of absolute thought, there cannot also be the surplus of ‘a thing-in-itself, something alien and external to thought’ (188).
Badiou reduces the principles of Hegel’s argument – indeed that argument is at stake – to two points, which I’ll summarise:
First, that it is by the movement of thought itself that any undetermined will come to be determined or that the ‘gap’ between finite and infinite is the locus of thought itself, the kernel of its procedure as such. In Badiou’s own ontological formalisation this locus is centred on the first infinite set – that of all ordinals and thus the concept of a limit, which can be marked as such and traversed, is critical to it.
Referring to Hegel, Badiou remarks that this is what the real is rational means and moreover this means that to the extent that thought is thinkable, it is thinkable absolutely. So thought as absolute and not the absolutely indeterminate. This thought, Badiou remarks, takes time, being the labour of the concept – what Plato referred to, speaking of hard things being worth doing, as ‘the long detour.’
Second, and now contra archi-metaphysics (and still classical qua objects) dialectics claims that the categories of thought are not simply, singularly, subjective: rather dialectics is a form of thought adequate to its objects as such. In other words, its categorical determinations are those adequate to that which it thinks, which is to say it can only think categorically with regard to what it thinks. Categories are not, then, ‘a priori’ and then addressed to an object thus making of the object, which cannot be thought, a knowledge. In this way dialectics is that form of thought which is conceptual and as Badiou avers, absolute: no indeterminacy remains over on either side. There is then a category for every determinate content and that ‘the becoming of concepts exhausts the real’ (189). Thus: ‘Not only, and contrary to what Hamlet declares, is there nothing in the world which exceeds our philosophical capacity, but there is nothing in our philosophical capacity which could not come to be in the reality of the world’ (189).
For Badiou, this is what philosophy is constrained to think, the thought of the absolute; which is not, as we can see here, the thought of the One or the whole as such, but of the Two. As Badiou notes, the change in the form of the transcendental under positivist and hermeneutic direction, from subjectivity to language, changes nothing in terms of this schema. Rather, ‘we are dealing here with a reinforcement, by means of a synthesis between criticism and positivism, and soon, via cognitivism, with a hermeneutics of intentionality, of all that which for the past two centuries has taken place in the way of archi-metaphysics’ (189).
Now as we have said, dialectics points the way – it opens up these determinations of the (being of the) One to the Two which founds them in order to rethink entirely what is thought to being or as Badiou says, referring again to Plato beyond Hegel – which is of course where he wants to get to recommence philosophy for today – ‘between the absoluteness of the concept and the creative freedom of negation.’
The problem is that while dialectics opens this question to thought, dialectics itself is behind the game in regard to what is thinkable of this relation between the finite and the infinite. ‘Hegel himself underestimates the link between finitude, infinity and existence within a mathematical paradigm’, Badiou argues, and if we were thus tasked to re-examine the ‘axioms of classical metaphysics’, to re-intervene on the question posed there of the rationality of the indeterminate, ‘we would learn that, as Descartes once glimpsed, it is possible, in light of contemporary mathematics, and namely of the Cantorian treatment of the infinite, to begin purely and simply with the infinite’ (190).
Thus the form of the relation that has hitherto underpinned ‘speculative ontology,’ Badiou says in Being and Event – and so also classical and archi-metaphysics – which comes in two ‘dialectical couples’, the one and the many or as whole and part, is no longer thinkable. ‘Set-theory ontology, contemporary mathematics, has substituted for them a wholly other double relation, one based in the actuality of the infinite, and which woven from the void, thinks no objects whatsoever: belonging, ‘which indicates that a multiple is counted as element in the presentation of another multiple' and inclusion, 'which indicates that a multiple is a sub-multiple of another multiple'.
‘Set theory sheds light on the fecund frontier between the whole/parts relation and the one/multiple relation; because, at base, it suppresses both of them. The multiple-whose concept it thinks without defining its signification-for a post-Cantorian is neither supported by the existence of the one nor unfolded as an organic totality. The multiple consists from being without-one, or multiple of multiples, and the categories of Aristotle (or Kant), Unity and Totality, cannot help us grasp it.’
Badiou’s notion of a ‘metaphysics without metaphysics’ is then subject to this contemporary mathematical condition. That the infinite can be thought undermines the necessary object of an archi-metaphysics and posits by this thought the absoluteness of the concept. Thus it has no need to posit the indeterminate at all given that mathematics renders such a notion superfluous to the thought of being – indeed ‘metaphysical’.
But of course this mathematical materialism of the infinite, to wax rhetorical, would also break with dialectics. The axiom schema of set theory while historical in terms of its invention has no recourse to what Badiou refers to here as ‘the theme of a historial auto-determination of the undetermined’ (190). That is to say, set theoretical ontology has no recourse to a notion of immanent becoming to account for being, being thought. As the discourse of presentation as such, set-theory thinks infinity directly and is the means of its coming to be. Hence we have our Platonic gesture or affirmation: ‘in a style bereft of any hyperbolic transcendence of the Good (and therefore outside of metaphysics) that for everything which is exposed to the thinkable there is an idea, and that to link this idea to thought it suffices to decide upon the appropriate axioms’ (190).
As we have said, this ‘demand to the world that it adjust its dread to rich and numbered postulates’, a propos and contra Heidegger, the ‘last universally recognisable philosopher’, is the task Badiou as Platonic gesture takes up – as any philosophy must, confronted with the inventions and interventions of the forms of thought that are its conditions. Badiou’s anti-metaphysical metaphysics is thus what he calls the return of philosophy to itself – which means also that philosophy is integrally divorced from ontology per se. Mathematics, we might say, bequeaths philosophy the freedom of thought it had erroneously supposed as its alone – which is to think again the complex of being, truth and subject.