Badiou's Being and Event and the Mathematics of Set Theory

Burhanuddin Baki

• Publisher: Bloomsbury Academic
• Year: 2014
• Print ISBN: 9781474288002
• eBook ISBN: 9781472578716
• Edition: 1
• Topic: Philosophy
• Subtopic: Logic in Philosophy

Alain Badiou's Being and Event continues to impact philosophical investigations into the question of Being. By exploring the central role set theory plays in this influential work, Burhanuddin Baki presents the first extended study of Badiou's use of mathematics in Being and Event. Adopting a clear, straightforward approach, Baki gathers together and explains the technical details of the relevant high-level mathematics in Being and Event. He examines Badiou's philosophical framework in close detail, showing exactly how it is 'conditioned' by the technical mathematics. Clarifying the relevant details of Badiou's mathematics, Baki looks at the four core topics Badiou employs from set theory: the formal axiomatic system of ZFC; cardinal and ordinal numbers; Kurt Gödel's concept of constructability; and Cohen's technique of forcing. Baki then rebuilds Badiou's philosophical meditations in relation to their conditioning by the mathematics, paying particular attention to Cohen's forcing, which informs Badiou's analysis of the event. Providing valuable insights into Badiou's philosophy of mathematics, Badiou's Being and Event and the Mathematics of Set Theory offers an excellent commentary and a new reading of Badiou's most complex and important work.

1

Mathematics = Ontology

How does one make sense of this militant equation by which the entire conceptual framework and philosophical topos of Being and Event is delimited? Before we pursue the philosophical implications of Badiou’s wager, it would be appropriate to instruct ourselves as to what it says and what it means. Our goal is not the closed security of an etymologically exhaustive or analytically precise translation, which could only be realized and verified as an infinite truth procedure. But it would be pedagogically beneficial, by way of a preliminary instruction, to initiate ourselves into the meaning of each side of this equation, sketch their respective relations to philosophy, and try to understand what is being proposed in the commitment to treat the two terms as identical. Granted, the act of equating revises, reconfigures and redistributes the meanings of mathematics and ontology. But a first point of departure must be provided and we shall be open towards correcting ourselves as we go along.

Mathematics, ontology and philosophy

Any answer to the question of defining the first side of the equation is guaranteed to be complicated and controversial. The question has constituted the most dominant and contentious topic for almost the entirety of what has been called the ‘philosophy of mathematics’ for the past one hundred years and more, particularly involving the project of defining mathematics by proposing a foundational philosophical definition of its ‘objects’, of ‘entities’ deemed ‘mathematical’. In the process of formulating such a foundation, a huge menagerie of philosophical orientations, oftentimes competing, cooperating and cross-breeding with each other, have been proposed, some dating back to the very beginning of mathematics as an investigative discourse:

antifoundationalism

atomism

computationalism

conceptualism

constructivism

fictionalism

finitism

formalism

foundationalism

holism

idealism

inflationism

instrumentalism

intuitionism

logicism

Meinongism

neutralism

nominalism

phenomenalism

Platonism

predicativism

psychologism

Pythagoreanism

realism

reductionism

social constructivism

social realism

structuralism

verificationism

and so on.

Badiou’s equation purports to be not only a solution but also a dissolution of the matter, for a problem can be resolved by either providing a correct answer or showing the speciousness of its aims. He proposes not just another definition – which equates mathematics with ontology – but also the beginning of a demonstration that the questions ‘How can mathematics be founded?’ and ‘What is a mathematical object?’ are actually pseudo-problems at best.

However, most of us would easily recognize what is meant by mathematics: it is simply the investigative discourse comprising arithmetic, geometry, algebra and calculus. Depending on our degree of erudition, we might add various other advanced subfields to the list: topology; combinatorics; probability theory; the theory of computation; differential equations; numerical analysis; dynamical systems; category theory; statistics; mechanics; set theory; and so on. Mathematics is what mathematicians study, practise and do. This is a sufficiently instructive way to begin and for the moment we will not expand further.

In the historicity of its respective discursive threads, mathematical thought has often penetrated, informed and inspired philosophical thought. To use the Badiouian vocabulary, we say that mathematical thought has often constituted a ‘condition’ that ‘forces’ philosophical thought. A cursory inspection of the history shows that this has been the case ever since the latter’s inception going back at least to the time of Thales. We examine the history of what has been called continental philosophy for the last one hundred years and observe such a mathematical conditioning at work, to various degrees of success, controversy and thoroughness, within the oeuvres of Albert Lautman, Jules Vuillemin, Jean Cavaillès, Gilles Deleuze, Jacques Derrida, Friedrich Kittler, Gaston Bachelard, the ‘Speculative Realists’, Michel Serres, Jean-Toussaint Desanti, François Laruelle and, in particular, Jacques Lacan, whose employment of logical and topological mathemes motivated Badiou’s philosophy and whose identification of mathematics with the science of the real – the real that is the impasse of formalization – can be compared to Badiou’s militant equation. But I would say that, without doubt, the most fruitful, methodical and influential ‘forcing’ took place in the early decades of what we call the analytic school of philosophy through the paradigmatic roles played by mathematical rationality following from certain developments by Gottlob Frege, Georg Cantor, members of the Vienna Circle, and Cambridge logicians such as Bertrand Russell and Ludwig Wittgenstein. The mathematical revolutions from the late nineteenth to the early twentieth century provided not just new spaces for the circulation of philosophical thought but also a new epoch of science where the very ‘axiomatic’ basis of mathematical rationality revealed itself.

As for the task of formulating a sufficiently instructive definition of the right-hand side of the equation, a more straightforward answer based on the simple etymology of the term is available: ontology is the study of Being or, more precisely, of Being-qua-Being. The matter in question involves examining not the particularity of beings, not specific entities or the specificities of their being-present, but investigating, at the most general level and with the most direct attention, into what is and only insofar as it is.¹

While subtracting itself from any particular presentation, Being forms, by definition, the substantivation of the presentative basis and immanent presentativity for every being. So ontology is linked with the study of first causes, the primum movens and the fundamentum absolutum. However, it is necessary for Badiou that this study be scrupulously laicized and its ‘captivating aura’ be thoroughly exorcized. Theology is one thing, ontology another. When it comes to ontology per se, the halting point of investigations must be Being itself. Jean-Toussaint Desanti is correct to call Badiou’s ontology ‘intrinsic’ (2004) because the focus here is pure and immanent Being, without any recourse towards unifying or undermining it in favour of some originary sovereignty, primordial exteriority, grounding monism, or distinct theos, be it God, Geist, language, the Other, the Idea, the Self-Same Subject, the Real, the Big Bang, the Unconscious, Capital, Vital Energy, or the ever-shifting vicissitudes of some substance. Badiou takes this rejection seriously and at the most uncompromising point, so much so that he recruits the propositions ‘The one is not’ and ‘Being is pure multiplicity’ as a second wager to supplement his militant equation. This rejection can be understood as a more radical continuation of Heidegger’s deconstruction of metaphysics, particularly in its rejection of any onto-theological approach towards understanding Being.

In lieu of everything that is implied by Heidegger’s name, we can provide yet another description of ontology by repeating what was often understood, since at least the time of Aristotle, to be its fundamental relationship to philosophy. Ontology is the ‘first philosophy’, and Badiou agrees with Heidegger’s return to the question of the meaning of Being as the inaugural question of philosophy. ‘Along with Heidegger,’ as Badiou writes at the beginning of Being and Event, ‘it will be maintained that philosophy as such can only be reassigned on the basis of the ontological question’ (BE, 2). Ontology is one field within philosophy – in fact, the most central and most essential field, the kernel of philosophical discourse that frames, conditions and secures the site for all the others such as epistemology, philosophical logic, philosophy of mind, political philosophy, philosophy of morality, aesthetics, philosophy of science, philosophy of law, and so on. The study of Being-qua-Being has often been understood as the field most native to philosophy, and any other field could be recomposed to take the form of ‘philosophy of X’, a philosophy conditioned by the investigation into something that is, strictly speaking, outside of ontological considerations per se, be it knowledge, reason, the subject, politics, ethics, art, science or law. At the very least, ontology is the core branch within metaphysics, another field that is often understood to be most intrinsic to philosophy. But the difference is a matter of semantics: sometimes the word ‘metaphysics’ is meant to be synonymous with ontology; sometimes ontology is called ‘General Metaphysics’; and sometimes metaphysics, following the well-known Heideggerian line of thought, is said to be the vulgarized version of a study that has forgotten the inaugural question of Being by seeking some causal, theological or quasi-theological explanation to undermine Being-qua-Being. Nevertheless, much of what is given in Being and Event has direct implications, as we shall see, for many of the central issues in traditional metaphysics – questions involving identity, predication, modality, universals, reality and so on. But our two main points remain:

1. Ontology is a branch of philosophy
2. The question of Being-qua-Being forms the very basis of philosophy as such.

Badiou preserves only the second of these two points. Since what mathematicians do is different from what philosophers do, and since ontology equals mathematics, then ontology cannot be philosophy. But ontology remains the kernel for philosophy. It is not the first philosophy, but the first condition for philosophy, albeit one condition among other non-primary conditions. When recognized as being equivalent to ontology, mathematics is the literal inscription of Being into discourse. Philosophy still has Being as its central question, except that it cannot study it directly at the first-order level – that particular task is reserved for ontology, that is, for mathematics. Badiou appropriately names ‘metaontology’ [métaontologie] as the part within philosophy that immediately concerns and is directly forced by ontology. So we correct ourselves by saying that the first philosophy is not ontology but metaontology. The role of what, following the Heideggerian return, used to be conceived as ontology is now played by a field involving a second-order thinking that is conditioned by ontology. Philosophy is at most the study of Being at the second-order level. Philosophy no longer deals with first-order questions as the letter of Being enters into discourse directly as mathematics.

The other conditions of philosophy and the compossibilization of truths

One upshot of Badiou’s wager is that the possibility and possible meaningfulness of metaphysics and metaphysical knowledge (or at least the part of metaphysical philosophy that now belongs to the separate discourse of ontology) can be understood in terms of the possibility and possible meaningfulness of mathematics itself. Badiou’s equation delegates and merges such questions with those involving the epistemology of mathematics itself. For example, if one accepts Badiou’s equation and agrees that mathematical knowledge is partly a priori, then one can be led to conclude that some metaphysical knowledge is partly gained outside the realm of experience and sense-perception. The well-known problem posed by Rudolf Carnap (1950) regarding the possible futility of metaphysics has been partly converted to a problem of how to make sense of the strange meaningfulness of mathematical knowledge that has no concrete referent.

We must nevertheless remember that mathematics has not been the sole condition for philosophy, for it is trivial that philosophers are informed by other forms of activities, investigations and experiences. First, there is the internal condition: the history and the textual archive of philosophy itself, an archive to which every philosopher must relate and respond. Second, every branch within philosophy concerns itself with extra-philosophical realms. Epistemology, philosophy of mind, aesthetics and political philosophy have as their principal objects something that, strictly speaking, lies at least partly outside of philosophy proper. For example, epistemology studies knowledge and is informed by recent discoveries in psychology, neuroscience, cognitive science and even the socio-political science of knowledge. As implied by their own names, the philosophical fields of aesthetics, political philosophy and the philosophy of science study art, politics and science, respectively. So every philosophy is always a philosophy of something outside philosophy. ‘Almost all our “philosophers”,’ writes Badiou, ‘are in search of a diverted writing, indirect supports, oblique referents, so that the evasive transition of a site’s occupation may befall to philosophy’s presumably uninhabitable place’ (MP, 28).

Badiou’s wager posits that the study concerning the most intrinsic focus of philosophy, namely Being-qua-Being, belongs wholly to the mathematicians, who are not philosophers when they are doing mathematics. ‘[A]ffirming that mathematics accomplishes ontology unsettles philosophers because this thesis absolutely discharges them of what remained the centre of gravity of their discourse, the ultimate refuge of their identity. Indeed, mathematics today has no need of philosophy, and thus one can say that the discourse on [B]eing continues “all by itself”’ (BE, 10). Absented from within itself, hollowed from its own essence, philosophy is fundamentally a cross-, inter- and trans-disciplinary investigative discourse that simultaneously invents new disciplinary classifications outside of the university and the encyclopaedia. Philosophy resides in the neutral in-between spaces of disciplines and lives in the heterogeneous times of truths. Every philosophy takes the form of a program for the ‘compossibility’ of truths. Badiou writes:

I have assigned philosophy the task of constructing thought’s embrace of its own time, of refracting newborn truths through the prism of concepts. Philosophy must intensify and gather together, under the aegis of systematic thinking, not just what its time imagines itself to be, but what its time is – albeit unknowingly – capable of. (TW, 15)

One of the most controversial and well-known propositions in Badiou’s philosophy – although he does not sufficiently elaborate it in Being and Event – is that every external philosophical condition must belong to four very specific but general domains: science, politics, art and love. Badiou places mathematics under the domain of science (which is a controversial move in itself) and takes the former as the rational and paradigmatic basis for the latter. If physics is the scientific study of physical matter and chemistry is the scientific study of chemical reactions, then mathematics is the scientific study of Being. Mathematics is what is left of science when it is without any object.

We read Being and Event and find nothing in the philosophical commentary on the mathematics that justifies Badiou’s decision to group the external philosophical conditions into these four domains. But we observe that, despite constituting a finite number, the domains are not as restrictive as they appear, particularly when we note that the word ‘science’ could be understood to cover a huge territory that includes not just the natural sciences (physics, chemistry, astronomy, and so on), the social sciences (economic science, political science, linguistics, and so on), but any systematic field of investigation that conserves the figure of mathematical rationality as a paradigm for thought.² Nevertheless, scientific thinking rejects certain modes of knowing and certain forms of knowledge such as mythology, alchemy and religion – unless, of course, it was possible to render such modes under a different condition, such as art or love.

One fundamental common denominator that defines each of Badiou’s domains is that it must allow for the possibility for events, for ruptures that completely reconfigure the individual situations corresponding to each domain on an ontological level. For example, amorous encounters occur in the domain of love, and revolutions erupt in the domains of science, art and politics. The eruption of the event allows for the possible emergence of new truths, with each truth realized as an infinite truth procedure that is contemporary with the weaving of a new subjectivity that is militantly committed to it. Philosophy, for Badiou, proposes a conceptual framework in which the contemporary compossibilization of its conditions can be grasped in the rupture of an evental truth. Philosophy conceptually seizes and houses the site of heterogeneous truths by circulating between the procedures that arise from science, art, politics and love.

So philosophy can be forced only by scientific knowledge, political activity, artistic practices and amorous experiences. Despite its essential relation to these four domains of truth, philosophy has its own discursive sovereignty that is fundamentally independent of its conditions. A philosopher, when he is doing philosophy, is not a scientist, a political activist, an artist or a lover. Philosophy must not be sutured to its conditions. The domains themselves do not wholly pre-determine the individual manner in which a condition is philosophically seized. There is nothing ‘necessary’ about the way Badiou understands and makes use of mathematical set theory to construct a new metaphysics of Being. The task o...

Table of contents

• Cover-Page
• Half-Title
• Series
• Title
• Contents
• List of Figures and Tables
• Acknowledgements
• Note on Abbreviations, Citations and Translations
• Introduction
• 1 Mathematics = Ontology
• 2 Ontology of Axiomatic Set Theory
• 3 Metaontology of Situations and Presentation
• 4 Metaontology of the State and Representation
• 5 Ontology and Metaontology of the Cardinal and Ordinal Numbers
• 6 Ontology and Metaontology of the Constructible
• 7 Ontology of Forcing and Generic Sets
• 8 Metaontology of the Subject, Truth, the Event and Intervention
• Epilogue
• Notes
• References
• Index
• Copyright