Naturalizing Badiou

Mathematical Ontology and Structural Realism

Fabio Gironi

• Éditeur: Palgrave Macmillan
• Année: 2014
• ISBN:
  9781137463470
• Sujet: Philosophie
• Sous-sujet: Geschichte & Theorie der Philosophie

Crossing the boundaries between 'continental' and 'analytic' philosophical approaches, this book proposes a naturalistic revision of the mathematical ontology of Alain Badiou, establishing links with structuralist projects in the philosophy of science and mathematics.

1

Badiou’s Mathematical Ontology

In this chapter, I offer a reading of the central nodes of Alain Badiou’s mathematical ontology of multiplicity, highlighting those elements which I take to be most significant for the secularising naturalist, and paving the way for my attempt to link Badiou’s philosophy – his mathematical ontology (here) and his ‘generic’ conception of truth (Chapter 5) – with contemporary trends in analytic philosophy of science. It is my contention that Badiou’s metaphilosophical insights, properly reinterpreted and recast in the service of naturalism, can be of use for a fully secularised metaphysical worldview. In the course of this reading I will encounter an obstacle offered by Badiou’s philosophy, that of the unsettled relationship between the ontological and the ontic, empirical realms.

The strongest and philosophically most fertile claims of Badiou’s philosophy are the two main consequences of his equation of mathematics with ontology: that ‘the one is not’ (Badiou 2006a: 23) (a thesis about Being, or what there is) and that the infinite can be secularised (or de-romanticised, made thoroughly immanent) – and has been, thanks to the mathematical work of Georg Cantor – by rationally demonstrating its thinkability (a thesis about truth).¹ These two theses – two axiomatic decisions at the base of the set theoretical universe which Badiou adopts as an ontological model – while undermining both the infinitely other God of the prophets (or phenomenologists) and the self-present God of the metaphysicians, can deliver to the scientific realist two authentically post-theological metaphysical principles.

1.1 Death(s) of God: for axiomatic immanence

The goal of any anti-theological philosophy is immanence, that is, the equivalence of the ultimate ground of Being with the totality of its empirical instantiations. Indeed, the desired outcome of any contemporary philosophy (more so when committed to naturalist guidelines) must be the articulation of an immanentist metaphysics: I take the repudiation of a transcendental ground of sense as being not merely an historically contingent philosophical turn, or even timely, but rather an unavoidable feature (and an unrelinquishable achievement) of human intellectual development. As Deleuze and Guattari eloquently put it, ‘[i]mmanence can be said to be the burning issue of all philosophy because it takes on all the dangers that philosophy must confront, all the condemnations, persecutions, and repudiations that it undergoes’ (1994: 45). Atheism, now a fashionable label for any self-appointed defender of rational thought, must be philosophically articulated as conceptual immanentism.

Alain Badiou’s most explicit atheist pronouncements are to be found in the prologue of his Briefings on Existence (Badiou 2006c). Here, interrogating the equivocity of the referent of the Nietzschean statement ‘God is dead!’, Badiou draws a distinction between three different understandings of God. According to him, the history of metaphysics itself is the history of ‘the mortification of God’ (2006c: 25) since the metaphysical concept of God running from Aristotle to Descartes (and indeed Kant) is, precisely, a concept in light of which, the philosopher can make sense of things.² Such is the God of the philosopher, frozen, by Galileo and Descartes ‘in the trans-mathematical punctuality of actual infinity’ (2006c: 22), working as a Principle allowing for the ‘suture of mathematical truths to their being, or [as] a guarantee of judgments in the shape of the Other’ (2006c: 23). The infinite is linked to the One in the concept of a unique God of metaphysics.³

Such a God, Badiou argues, is a God who never lived, a far cry from the God of religion, the God of Paul, of Pascal and Kierkegaard, a ‘living God which is always somebody’s God’ (2006c: 23) met through personal, lived encounter and not through logical proof.⁴ As Pascal had it: ‘[t]he God of Christians does not consist in a God who is merely the author of geometrical truths and of the order of the elements’ (2004: 227). In aphoristic form, Badiou summarises this opposition claiming that ‘[w]ith respect to God, it is true that religion is vivifying and metaphysics mortifying’ (2006c: 25). Therefore, even though the death of God means that ‘religion is finished’ (2006c: 23), this death ‘leaves the question of the destiny of the God of metaphysics unresolved’ (2006c: 26). The evacuation of the latter from philosophy began with Nietzsche (whose announcement of the death of God was, in Badiou’s taxonomy, the paradoxical announcement of the death of a dead God, the God of Metaphysics), but was only accomplished by Heidegger, having identified Nietzsche’s reversal of Platonism – becoming over Being – as the swansong of Western metaphysics. His destruktion of onto-theology amounted then to the rejection of a univocal understanding of Being (and God) as a metaphysical concept of the highest of beings. Yet, for Badiou it is Heidegger himself who opens the space for the third kind of God (what Badiou calls ‘Heidegger’s aporia’ (2006c: 27)), the God of the Poets. This is the God with which the philosopher-poet stands in a ‘nostalgic relationship’ (2006c: 28) of epochal expectation. Through a ‘meta-poetic metaphorizing’ (1999: 43) the post-metaphysical philosopher hopes for a re-injection of meaning into the world and orients his own finite being towards an attentive but passive receptivity to the historical self-presencing of Being.⁵

In sum, while the living, personal God of Religion has been repeatedly killed off – in a dialectical evolution running throughout the history of Western philosophy – by the aloof God of Metaphysics, it remains for us to free our ontology from the latter’s constraints while being wary of the nostalgic invocation, by those pronouncing the end of metaphysics (and its God) under the banner of existential finitude, of a God of the Poets. This latter God – currently the most insidious one – is expected as the only hope of a re-enchantment of the world by those finite, death-ridden mortals incapable of autonomous production of meaning and of mastery over the world (nihilistic mirages, as Heidegger considered both Nietzsche’s philosophy of the Will and humankind’s techno-scientific ambitions). Truth has vanished with the God of Metaphysics, and the philosopher can only have faith in the (non-philosophical) promise of a meaning to come.⁶ It is at this historical juncture that it is necessary to reaffirm a contemporary form of atheism as the destitution of all three varieties of God, one that is ‘all in all ... about finishing up with promises’ (2006c: 29) and which decisively breaks with a philosophy ‘caught between the depletion of its historical possibility’ and ‘the conceptless advent of a salvational turnabout’ (1999: 114) that is, the poetic⁷ hope for a redeeming return of (a) God.

Such an atheism pivots upon the decoupling of meaning from truth,⁸ and of the one from the infinite, in order to ‘restitute the infinite to the banality of manifold-being’ (2006c: 30). What needs restating is the

great modern declaration: the infinite exists, and, what is more, it exists in a wholly banal sense, being neither revealed (religion), nor proved (mediaeval metaphysics), but being simply decided, under the injunction of being. (2008a: 82)

Elsewhere, Badiou states that it is ‘necessary to de-suture philosophy from its poetic condition. Which means that it is no longer required today that disobjectivation and disorientation be stated in the poetic metaphor. Disorientation can be conceptualized’ (1999: 74). His atheism, in other words, amounts to the proclamation of a philosophy of axiomatic immanence⁹ capable of mathematically articulating the infinity of Being without attributing to it any meaning-full, transcendent significance.¹⁰ The immanentist creed is thus encapsulated in the aphorism (both declarative and imperative) that ‘everything is here, always here’ (2006c: 31). The atheist philosopher rejects the metaphysical pathos of (a) reason without (poetically) forfeiting the power of Reason. Immanence signifies responsibility, rational accountability qua the abandonment of any heteronomous intervention in (or justification for) either our rational life or natural reality, and it informs the anti-dualistic acknowledgement that the former is part of the latter.

This immanentist, or simply philosophical¹¹ stance,

is in rupture with [Nietzschean] anti-philosophy because it both retains and develops, by means of a rational critique, the idea of truth. But it is also in rupture with religion, because it refuses to identify truth with sense; it even willingly declares that in any truth there is always something of the nonsensical. (2001c: 9)

To champion atheism means to defend an immanentist ontology and a secularised epistemology. And this, as we will see, is possible only for a philosophy developing between the twin poles of the axiomatically established existence of the void and the infinite.

1.2 Subtractive ontology

Axiomatic decisions, for Badiou, are the anti-historicist duty of contemporary philosophy.¹² His rejection of the poetic approach to philosophy and its nostalgia for the divine, accordingly, begins with a founding axiom: the equation of mathematics with ontology.

Herein lies the most delicate point of Badiou’s project: the equation mathematics = ontology is a decision, an unprovable, foundational axiom,¹³ which can be evaluated only by its consequences. Badiou adopts an axiomatic procedure in explicit opposition to the poetico-linguistic philosophical climate, perceived by him as dominating the contemporary (in late 1980s France) intellectual scene.¹⁴ Unlike poetry, for Badiou mathematics

disciplines thought through explicit rules, not through the singular genius of language, and offers to everyone a shared demonstration, whilst never giving up on ultimate clarity – as complicated as its construction may be. It informs the True without conceding anything to the trembling or existential doubt before that whose cruel necessity it unveils. (2006d: 40)

However, this methodological point is hardly enough to warrant the equation of mathematics with ontology, an equation expropriating philosophy of one of its historically most central branches and recasting the philosophical role as the non-self-sufficient compossibilisation of non-philosophical truths. Could the unsympathetic reader question the validity of this axiom and reject it, thus collapsing Badiou’s system upon itself? The answer, quite simply, is yes.¹⁵ Axioms, in mathematics as elsewhere, are a matter of pure choice, posing unquestionable logico-architectonic constraints on all that which inferentially follows, yet incapable of self-justification. Such is Badiou’s rationalist wager: the singling out of an event (the Greek ‘discovery’ of the matheme),¹⁶ and the identification of this event as the starting point of a truth procedure (pursuing the truth of the equation of mathematics and ontology) which will achieve full maturity only with Georg Cantor’s creation of set-theory and whose meta-ontological importance will be assessed in his own work.¹⁷ In this light, Badiou’s project from Being and Event onwards can be read as one long retroactive explanation of the validity of his original axiomatic choice. To this effect, Brassier and Toscano’s definition of Badiou’s thought as an ‘aleatory rationalism’ (2006b: 260) is appropriate: his method is that of an inaugural, ungrounded, contingent philosophical choice (a pure choice that is, before anything else, a rejection of traditional ways of mooring philosophy to either an empirico-phenomenal world or to a transcendental subject), whose consequences have nonetheless to be pursued according to the strictest standards of logico-mathematical deductive reasoning. Only an empirical grounding would seem capable of vindicating Badiou’s inaugural decision, but at the price of a reinterpretation of Badiou’s own views. Herein lies the hermeneutic challenge of this book: how, if at all, can Badiou’s radically rationalist (and indeed anti-naturalist) stance be reconciled with a scientific realism which hopes to be warranted by the empirical success of the natural sciences?

The inaugural axiom of Badiou’s philosophy, taking a stance on the traditional metaphysical quandary of the priority of the One or the Many, is that ‘the one is not’ (2006a: 23). This should be immediately coupled with its corresponding positive claim: ‘what is is multiple’, or better yet ‘Being is multiplicity’. The phenomenal appearance of unity is illusory. What there is, what is presented, are structured situations, collections of multiplicities organised by an impersonal operation¹⁸ of count-as-one which makes heterogeneous multiplicities consist in a composed unity. In its foundational denial of ontological one-ness such an ontology is intrinsically anti-theological (‘there is no God. Which also means: the One is not’ (2001a: 25)), and is on a continuum with the denunciation of onto-theology, insofar as Badiou’s confinement of oneness as the mere product of an operation of counting-as-one of a multiplicity is in accordance with the deconstruction of an eternal plenitude or self-presence of Being. The One, for Badiou, is always an effect, ‘whose fictive being is maintained sole...

Table des matières

• Cover
• Title
• Introduction
• 1  Badiou’s Mathematical Ontology
• 2  The Ontological and the Empirical: Naturalist Objections
• 3  Taking a Stance on Realism and Naturalism
• 4  Structural Realisms
• 5  Truth and Randomness
• Conclusion
• Notes
• Bibliography
• Index