In order of appearance:
Badiou’s exceptional approach to the questions of contemporary philosophy is widely considered to have maintained (or reinstated) the position of militant subjectivity within and against the annihilative milieu induced by (post-)structuralism. With Badiou’s guidance, it seems, we are able eclipse the dialectic of subjectivity and the action of the structure in favour of a flexible structural model in which novelty can miraculously emerge.
Yet this approach is not without its problems. Being and Event‘s ‘functionalist’ approach has been criticised for being unable to reliably describe and investigate human reality because of its stalwart grounding in mathematical ontology. Consequently, it is not manifestly evident that functionally-defined terms, such as subject and event, refer in any substantive way to the material entities which ontology is typically understood as addressing. As much as the significance of this has been dismissed, it is of vital importance that such onto-mathematical terms are given substantial and demonstrably ontic iterations; without doing so Badiou’s ontology runs the risk of ambiguously defining the scope of subjectivity and potentially empowering the structure and undermining the militant subjectivity his ontology is supposed to restore.
Despite billing Logic of Worlds as a solution to these problems in the form of a supplementary and descriptive onto-logy of appearing, any substantive discussion of pure ontic reality – and thus the potential for individual militant subjectivity – remains allusive. Moreover, the worldly framework of a continuum of intensity of appearance inevitably relies on a single transcendental register and thus runs the risk of reintroducing a structural bias into his ontological model.
This paper will investigate a number of avenues in which Badiou’s ontology may discuss pure ontic reality – and thus the capacity of individuals within a structure – without recourse to an atavistic notion of subjectivity, democratic materialism or any model which would jeopardise Badiou’s fidelity to mathematical formalism.
In this paper I aim to present Badiou’s use of the Axiom of Choice as a formalised concept of freedom. This ties his philosophy to a powerful existential heritage, whilst freeing him from existentialism’s dependence on phenomenology and an ontology grounded on the subject.
This leads to his innovative yet problematic dependence on the axiom in both Being and Event and Logics of Worlds. I will give a brief history of the Axiom of Choice in set theory and focus on the different style of forcing used to prove the independence of this axiom. The Axiom of Choice does not fail in the generic extension, but between the generic extension and the original ground model, in a suitable sub-model.
If mathematical forcing provides an example of the formal rigour of a subjective truth procedure, within the genre of science, and, more broadly, a model for subjectivity in general, then an alternative style of forcing suggests a different form of subjectivity – a deviant or under-privileged subject. It also suggests a situation, or world, absent of freedom, independent of the Axiom of Choice.
Finally, I will show how Badiou tries to avoid the method of forcing that is used to prove the independence of the Axiom of Choice, and his arguments for its obvious inclusion in not only his set theoretical ontology, ZFC (Zermelo-Fraenkel with Choice) but also his category theoretic phenomenology. While its inclusion in ZF set theory is relatively uncontroversial its inclusion in any form of category or topos theory is far more problematic. Some of Badiou’s most direct and unguarded discussion of these topics can be found in the two works on category theory, written between Being and Event and Logics of Worlds, recently translated into English and published together as Mathematics of the Transcendental.
The background of my paper is the ongoing discussion in the reception of Badiou concerning the status of the event. It is argued by a lot of authors that the notion of the event in Badiou must be considered problematic because of its lack of substantial relation to being, that is, as some kind of miracle. (ao. Bensaïd, 2004; Hallward, 2012; Kouvelakis, 2003; Žižek, 2004) Although Badiou himself has made this explicit as a connotation of the event in his book on Saint-Paul, it can be doubted if this has to be considered more than a metaphor to think change in an ontological way.
As Bosteels keeps suggesting, the articulation of the event in L’être et l’événement was not Badiou’s first attempt to subordinate the question of change to the concept. In his 1982 Théorie du Sujet, he already discussed this in exclusively political terms with the notion of ‘force’. Therefore, the question I want to raise in this paper is what it means to shift from a dialectical setting to a mathematical setting, that is, from the exclusively political framework of Théorie du Sujet to the mathematical/ontological framework of L’être et l’événement.
It will be the hypothesis of my paper that the core Badiou has saved from Théorie du Sujet is the forced ‘passing of the subject’ from a point of ‘real’ impasse (as characterized by Lacan) towards new forms of (symbolic) consistency. It is exactly therefore that in his ontological work, he keeps putting emphasis on the consequential nature of the event, which may not be considered an achievement but a hypothesis, or as Zizek states: “the ‘positing of the supposition’ which opens the actual work of positing.” (Zizek 2004, p.176) Hence, I will argue that the latter emphasis has to be considered an ‘ontological translation’ of the former idea insofar as this idea was intimately related to the dialectical context of Théorie du Sujet.
This proposed paper examines the relation between the one and the many in Badiou’s univocal ontology. In his criticism of Deleuze, Badiou accuses Deleuze’s univocal ontology as being fundamentally an ontology of ‘the one’—as according to Deleuze, ‘a single voice raises the clamour of being… Being is said in a single and same sense.’ Yet Badiou is at the same time adamantly committed to univocal ontology himself. This raises many questions: Doesn’t the uni-vocity of being presuppose that there is only one single way/value of ‘being’ or perhaps what we may call ‘being-ness’? If so, wouldn’t all univocal ontology be of ‘the one’ rather than ‘the multiple’? How does Badiou’s set-theory ontology, known for its ‘axiomatic decision’ for the multiple over the one, be against ‘the one’ and uphold ‘univocity’ at the same time?
Badiou’s commitment to univocity is even more problematic when we consider Logics of Worlds in which Being and Event’s set-theory ontology is said to rely on a classical logic which presumes the law of double negation. According the ‘classical’ logical law of double negation, ontology can only be binary: Something either exists or doesn’t exist—there is no intermediary ‘middle’ value of halfway-existence (what Badiou calls the ‘excluded middle’). In other words, there is only one way/value of being—to re-quote Deleuze: ‘Being is said in a single and same sense.’ Here the question of the one univocal way/value of ‘being’ reappears.
This paper considers how Badiou addresses these issues by adopting the notion of ‘univocity of being’ from Duns Scotus’ doctrine of God into his very own conception of (the death of) God where he not only fulfils the Heideggerian ethos to overcome ontotheology, but also provides a reconciliation of the univocity of being with the ontological affirmation of ‘the many’.
Contrary to what seems to be an obvious choice of Being and Event’s most demanding part, it is not its final chapters that fit this role; it is its very beginning. The equation mathematics = ontology is not, in fact, the book’s real point of departure. It is rather chosen as a means to fit another speculative schema, based on two axioms: being inconsists; radical breaks happen.
Being, having no structure whatsoever, is not an object of ontology. If mathematics is ontology, it is not because of a type of its supposed object, but because of the way it acts, and of the result it constructs. Ontology is therefore chosen by philosophy not because of what its theory reflects upon, but because of a type of procedure it is.
But if being is not what ontology theorizes about, it is nevertheless what it presupposes as its absolute starting point, and to what it adjusts constantly the result of its activity: the Form (of pure multiplicity). This starting point is the moment of inconsistency as such, sutured to the mathematical discourse by means of a pure letter.
This way ontology serves philosophy’s goal: thinking the compossibility of truths. Being means inconsisting; and an event is nothing but the appearance, in a point of the situation/world, of its inconsistency proper. The name for what exactly happens in the moment of event is: subtraction. The name for what happens in the process of its continuation is: formalization.
Just like ontology subtracts consistency in order to momentarily suture being to the letter and continue with the process of its formalization, a truth-procedure subtracts the situation’s structure so as to suture the event to an axiom and continue with its proper type of formalization: political, amorous, artistic, scientific fidelity.
In recent years, there has been a return to considerations of formalism in contemporary European philosophy. Corresponding to this, there is the nascent possibility of a philosophical reinvestigation of the possibilities for thought opened up by contemporary mathematics. Such a project is, however, limited by the paucity of conceptual resources available following a century in which the ‘philosophy of mathematics’ was largely conceived as a specialized subfield of ‘analytic’ philosophy, wherein the majority of investigations were limited to set theory and logic – apparatus that in many ways are in principle surpassed by contemporary mathematics. The question of the relation between mathematics and philosophy must today be reposed on open ground. In this respect, it is necessary to pass through the work of Alain Badiou, whose reception has in many ways shaped the horizons of this formalist revival. The interest of Badiou’s work, in part, is that it attempts an asymptotic approach to contemporary mathematics from within set theory. In this paper, I will examine the general development of Badiou’s thinking on mathematics. Beginning with Being and Event and the equation ‘mathematics=ontology’, I will then trace Badiou’s general relation to contemporary mathematics, proceeding via an analysis of his rationale for reducing the contemporary geometric formalism of category theory to a ‘logic’. Specifically, I will seek to examine and problematize Badiou’s division of formal apparatus into ‘strong’ and ‘weak’ mechanisms. This examination will in part centre on an argument of Badiou’s contrasting the singularity of the void (as sign) in set theory with the ‘equivocal’ void of category theory: the empty diagram. I will then draw out some more general questions concerning the nature of ‘formalization’, contrasting Badiou’s approach with that of Gilles Châtelet, who conceived formalization as an operation of contamination.
Socrates asked how one should live. Badiou (2006: 35) replied: ‘to live is to participate, point by point, in the organization of a new body, in which a faithful subjective formalism comes to take root’. Is Badiou right?
To believe that a general philosophical system is right (or wrong), to afford it normative ‘fidelity’ (or not), the metaphysician must be persuaded of both its intrinsic consistency (or flaws) and completeness (or deficiencies) and its extrinsic strength (or shortcomings). A philosophical stance, which is both grand (ontologically totalizing) and right (epistemologically accessible), will usually be historically ‘great’ (axiologically influential). Following an exegetical summary of Badiou’s overall conceptual framework, this paper starts to discuss how it could be evaluated.
Badiou thinks his position is ‘great’ (1988: xi), that his own work constitutes an Event. He argues ‘fastidiously’ (2006: 3) that he is right, that his position truly transcends the reactionary pluralism of ‘democratic materialism’ (2006: 35). He claims that his system is ‘grand’: his radical, revolutionary ‘truths’ (Events) are absolute, they create objective value and they are mutually dependent upon forms of embodied subject; his axiomatic set-theory ontology (1988) and materialist dialectical processes (2006) are explanatory and exhaustive; and his (2000) mathematicism enables a subject to know the infinite, indeterminate and absolute.
Assuming realism, metaphysics must always be inadequate to the world. ‘Gaps’ between fallible theory and absolute reality can be manifested as intrinsic, systemic dogmata or blindspots, which need to be defended. Badiou’s admitted dogmata include his axiomatic mathematicism (1988) and his realist ‘postulate of materialism’ (2006). His formalist theories of object, subject and power (Bhaskar, Callinicos) frame conceptual blindspots.
Extrinsically, Badiou exemplifies Williams’ (1957) methodological combination of attempted destructive and constructive persuasion: conceptually, historically and mathematically. Following Williams, Moore (2012) has proposed three meta-metaphysical criteria for metaphysical adequacy – transcendence, novelty and creativity – against which Badiou performs well, but then so does Deleuze. Where does that leave us, methodologically, in our assessment of Badiou and his mode de vie?
In this paper I compare the diagrammatic logic/semiotic of Charles S. Peirce with the philosophy of Alain Badiou. First, I offer a novel interpretation of one of Peirce’s later complex typologies – a version of his classification of ten signs – which he sent to Lady Welby, known as the Welby Diagram. Peirce scholar, Nathan Houser, has described the interpretation of Peirce’s ten and sixty-six sign typologies as the most pressing problem facing Peircean semioticians, and my hope is that the interpretation I offer here will be a genuine contribution to Peirce scholarship. My interpretation of the Welby Diagram provides an example of diagrammatic reasoning which enables me to discuss several concepts central to Peirce’s logic/semiotics. These include: his three argument types abduction – deduction – induction; the pragmatic maxim; and his reduction thesis. I emphasis the difference between the tradition of algebraic logic to which Peirce belongs, and the tradition of mathematical logic which dominated logic for most of the twentieth century. I also highlight some recent scholarship which demonstrates how the range of Peircean logic/semiotic, when interpreted diagrammatically, may extend beyond the realm of human language.
This paper also engages in a critical discussion of John Mullarchy’s use of diagrammatic reasoning to analyse Badiou’s philosophy. I argue that there are strong reasons for thinking of Badiou as diagrammatic thinker along Peircean lines, and consequently disagree with certain of Mullarchy’s conclusions concerning Badiou.
This paper proposes to relativise Badiou’s discovery presented in the preface to his Being and Event by looking at some of his predecessors in French philosophy of mathematics. Badiou presents in this preface the thesis according to which the object of mathematical discourse is being qua being, thereby justifying the equation of mathematics and ontology, as well as the whole project developed in this most famous of his books. My hypothesis is that we can see in the French tradition of
philosophy of mathematics from Léon Brunschvicg to Jean-Toussaint Desanti via Jean Cavaillès and Jacques Lacan a shift from an epistemological perspective to an ontological perspective, Badiou being the who brings it to its conclusion. The paper thus sets out from Brunschvicg’s Neo-Kantianism showing how he is led to undermine the debarment of being in itself in order to respect science’s immanent creativity, which goes well beyond the philosopher’s prescriptions. It then looks at Cavaillès’ critique in Sur la logique et la théorie de la science of any endeavour to guarantee mathematics by indexing it to a transcendental subjectivity. Thereafter, it considers Lacan’s ‘antiphilosophical’ critique of philosophy, the latter, in his opinion, failing to come to terms with mathematics, too busy as it is with making sense of it, integrating it into a pre-established universe of meaning. Then, finally, we will see how Desanti in Les Idéalités mathématiques tries to abolish any extra-mathematical figure of reason by remaining attentive to the way in which a mathematical theory appears from the inside. I hope to show by this overview that those four philosophers have in common a critique of an epistemological posture of philosophy vis-à-vis mathematics. I hope to show also that this critique is a precondition of Badiou’s discovery and essential to understanding its meaning.