An A to Z of Theory | Alain Badiou: Ontology and Structuralism


Alain Badiou - Ceasefire Magazine - In Theory - 1Alain Badiou’s influential theory of the Event, revolutionary politics, and radical philosophy has taken the field of theory by storm, and even finds resonances in political texts such as The Coming Insurrection. In this new, ten-part series, Andrew Robinson summarises the political aspects of Badiou’s work, and explores their implications (both useful and problematic) for transformative action. From the third part onwards, the series will focus on Badiou’s theories of the state, the Event, and political action. Badiou arranges his work in such a way that political implications seem arbitrary without understanding his underlying philosophy.

Alain Badiou

Alain Badiou is one of the best-known French critical theorists today. He is best known for challenging the hegemony of poststructuralism within France and beyond (though his work also has poststructuralist overtones). He has revived questions of ontology (the nature of being) in a climate where such questions were deemed obsolete.

In many ways, Badiou is a systematising theorist. His ascent marks a less hesitant, more decisive style in philosophy – a change which has won him both supporters and critics.  Badiou terms his philosophy “austere”. It provides a minimum structure without much ornamentation. One could think of Badiou’s austerity as a kind of “cutting back” of philosophy. Among the cutbacks are the various qualitative aspects of other theories.

The political goal behind Badiou’s work is to reveal or create a subject, in a context where political action seems fruitless. He seeks to do this by theorising general, mathematical conditions for transformative action. His approach has inspired a wave of contemporary academic theorists, such as Peter Hallward, Nina Power, Bruno Bosteels, and Quentin Meillassoux. His influence on activism has been less notable – although certain of his concepts turn up in French anarchist works such as The Coming Insurrection.

Badiou’s Ontology

‘Maths is Ontology’

Badiou’s innovation in philosophy is his idea of mathematical ontology based on axiomatic set theory. Badiou advances the slogan: “Mathematics is ontology”. In other words, he believes that maths (specifically, set-theoretical maths) is the same as, or can explain, the nature of reality or being. This claim is a decision which is posited. Badiou does not claim to be able to prove it. It can be justified, if at all, only from its effects or consequences. And its consequences will unfold over time.

The status of this claim is debatable. Strictly speaking, Badiou does not declare that being is made up of mathematical objects. However, he believes that being, or at least its political dimensions, offers no significant barriers to its being treated mathematically. He argues that all discourses make claims about unity and multiplicity. And maths is the best, most rigorous way to talk about unity and multiplicity.

Badiou’s argument in ontology (the philosophy of what exists or has being) is that what exists are ‘presented multiplicities’. In other words, what exists is a multiplicity structured by maths. Prior to the structuring gesture performed by maths, what exists is an ‘inconsistent multiplicity’. (This means that being, at its most basic level, is always the same). Ontology, like maths, is a way of presenting an inconsistent multiplicity from the standpoint of a consistent multiplicity.  So Badiou’s ontology, from his own point of view, isn’t quite presenting reality as it is.  It’s presenting reality as it has to be seen in order to be treated conceptually.

Hence, Badiou does not actually claim that the world as such, beyond what can be thought, has the structure of set theory. Yet he operates in such a way as to give the impression that any such world beyond is an illusion. Everything which matters, about which we can think or speak, exists in set theory or its suspension in the Event. He does not actually say that being as such – “impure” being in his terms – is mathematical and non-qualitative. But he writes as if it is. In other words, Badiou writes as if nothing has any substance or existence prior to its being counted in a set. ‘Being’, as defined by mathematical ontology, is separate from lived existence. It is a quality related to belonging to a set.

Set theory can theorise the construction of collections (the issue of subsets, which is central to Badiou’s theory of the state). But it leaves aside the arrangement of relations among the parts which are collected. In effect, Badiou has simply decided that relations either don’t matter or don’t exist. At one point Badiou claims that there is as much or as little difference between any two people – say, a Chinese peasant and a Norwegian professional – as between any other two people. This prevents, for instance, nuanced analysis of class structures or global hierarchies of labour.

Hence, Badiou’s ontology does not study what exists within situations. It studies what are taken to be laws which govern whatever elements end up appearing in a situation. In order for something to “be” – to have being – in Badiou’s theory, it must be an organised multiple indexed by a count-for-one. This structural invariant is taken to be objective, and separate from how humans make sense of a situation.

Social analysis of the usual kind – mediated by intervening, middle-order categories such as class, gender, consciousness, personality, coloniality, and so on – is rejected in advance. It is seen as a way of depoliticising the world. This is because the mediations always belong to an existing order of knowledge, and because they focus on the substantive.

In his more recent work, Badiou suggests that Being is not all there is. Being defines whether something belongs to a set. But things also “exist” or “appear” to varying degrees. This distinction allows him to address some of the relational issues he had neglected earlier. In Logic of Worlds, a situation has a ‘logic’ as well as a count-for-one.  Each situation has a logic. But the logic is different across different situations. This partly allows Badiou to provide the kind of nuanced social analysis his earlier theory precludes.

However, he continues to maintain that belonging to a set is primary. Each situation is equivalent to a mathematical universe. Mathematical representations are always necessary for logical constructions. Without things being defined as ‘ones’ or ‘units’, we can’t make logical claims about them. The founding ‘count-for-one’ of a situation also founds its logic.

Badiou and Structuralist Marxism

Badiou’s philosophy has its origins in structuralist Marxism. In his early works, Badiou was a follower of Althusser. In Althusserian theory, a particular element within an existing totality is the motor of change. The location of such an element has always been a major concern of Badiou’s thought. His theory of the excluded part, the Evental site, and the Event is derived ultimately from Althusser’s (and Mao’s) theory of the ‘primary contradiction’.

He also adopts the Althusserian and Maoist view that Marxism (or more recently, transformative politics) is a political choice before it is a science of society. The partisan, divisive gesture comes first. Practice precedes knowledge. Struggle and division are permanent. In Badiou’s early work, theory even predicts its own obsolescence (or political irrelevance). This is very much in keeping with a Maoist view in which practical conflict is the source of knowledge.

Badiou’s quasi-realist approach to maths also follows from the Althusserian distinction between science and ideology. Badiou sees both maths and logic as different from socially-constructed language. They are mechanisms of production which produce regulated writings. Maths is a science rather than an ideology because it does not ‘suture’ (or construct the identity of) a subject. It is a kind of impersonal machine which produces signs, without requiring a subject or author. Crucially, Badiou takes a deductive rather than inductive approach to science. His claims are unfolded from axioms, rather than abstracted from concrete realities. Models are unfolded within a reality which is completely inside a particular set of claims.

However, in this approach, objectivity is assumed, not proven. The science-ideology distinction is posited by a particular science. For Badiou, this act of positing is formulated as “maths is ontology”. This is distinct from Althusser’s position that Marxism is the ultimate science. This shift is an attempt to keep up with what Badiou sees as a more advanced science. Science is constantly changing. One of the goals of Badiou’s work is to harness this change to a particular direction of social transformation.

Critique of Constructivism

Maths is to be preferred to any approach which uses language as its main tool. Like most poststructuralists, Badiou accepts the view that language is incomplete. The process of linguistic representation is politically problematic. Unlike most poststructuralists, however, he believes that maths is immune from this critique of language. For Badiou, maths is not a system of signs, which refers to things, the way a language is. Therefore, maths can be used to talk about reality in a way which does not succumb to the problems of representation.

This is why Badiou feels that his philosophy is different on a very basic level from most of its rivals. Badiou maintains that there are three possible ontological approaches, echoing three approaches to maths: constructivist, transcendent, and generic. (In earlier works, he referred to his own approach as ‘praxical’ instead of generic. ‘Praxis’ is the unity of theory and practice).

Each philosophy has its own structure of knowledge and its own figure of truth. Badiou’s philosophy has a theory of knowledge as practical, and truth as ruptural. One difference from transcendental theory is that Badiou does not posit reality, but unfolds it from axioms. The main difference, however, is in how Badiou deals with the relationship between elements and relations.

The theories differ in how they handle the excess of possible relations over elements in a set. Constructivists restrict maths/philosophy to formulaic, predictable kinds of relations. Transcendentalists create overarching sets which are meant to contain all the smaller sets. Generic theory creates a generic subset. Badiou advocates a fourth option, in which the excess can be measured in a process of transformation, through the Event and its ‘investigations’ and ‘truth procedures’.

Badiou criticises ‘constructivism’ – a category in which he includes diverse theorists, ranging from Derrida and Foucault to Wittgenstein and Leibniz, to Anglo-American analytical theorists. His criticism of these theorists is that they define truth in terms of conformity to a dominant discourse. Therefore, they don’t allow an excess over existing discourses. For Badiou, this is conservative. Constructivism is a pure “law and order” philosophy. It keeps the “state” in place. It prevents the work of destruction which is necessary for radical change.

Constructivists also demand evidence before they admit a truth. This excludes the Event in Badiou’s sense. In set-theoretical terms, the constructivist world seems impoverished because it limits thought to accessible elements. For Badiou, being exists before language: We have to assume something exists before we can say anything about it. Badiou claims that this is the case because, in maths, one has to assume a set before one can make claims about it.

The main difficulty with this critique is that it misrepresents constructivist politics. Some constructivists, such as Rorty, the Anglo-Foucauldians, legal positivists, and analytical followers of Wittgenstein, largely fit Badiou’s characterisation. But many constructivists derive from the constructedness of present discourse a position of critique. They argue that, since things could be constructed differently, there is no reason to accept oppressive, miserable or repressive social structures. And they argue for transgressive resistance practices – either to resist particular dominant essentialist orders, or to maintain a general state of openness and reflexivity.

Kant is another target of critique because, like Badiou, he offers a theory in which philosophy is autonomous from substance. The main difference between them is that Kant assumes subjective freedom to be always present. Badiou, in contrast, sees it as a rare and exceptional achievement of an Event. If we aren’t in an Event, we are trapped in habit, and we don’t have freedom. Hegel and Spinoza are criticised for identifying being with Truth, a position which, as we shall see, precludes Events. Hegel and Spinoza assume a continuity where Badiou insists that there is a decision and a discontinuity.

For more essays in this series, visit the In Theory pag

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