Being and Event, Part 2: Excess and Situations

Alain Badiou’s conception of politics at least partially hinges on a distinction between two kinds of relations in set theory: belonging (∈) and inclusion (⊂).  Every mathematical set has elements which are included in it, but do not belong to it; there is always some excess.  In the same way, political states include everyone, but not everyone properly belongs.  He uses this idea to offer a critique of Engels’ concept of the state and hints at the role of political activism, that is, the work of radical justice.

The Point of Excess

For set theory, the categories of the one and the many are irrelevant.  There are no versions of unity or totality; there is no set that unifies all the others, or totals them all up in some kind of absolute sum.  The only relations that are used are belonging and inclusion.  Yet this does not mean there are two kinds of set, just that there are only two ways of counting sets.

The distinction between belonging and inclusion arises from the power set axiom.  Let’s say that all the multiples included in α belong to ß.  So all the γ’s and ∂’s that are included in α belong to ß..  Set ß counts-as-one all the sets included in α; this is the power set of α, and there are two ways of writing it: p(α) and {α}.  If γ belongs to the power set of α, that is the same as saying it is included in α.  But the set p(α) is different from α itself.  The gap between α and p(α) will “take us all the way to truth and the subject.”

The key point is that a set’s power set is larger than itself; p(α) has at least one multiple that does not belong to α.  Take all the ßs that belong to α.  Consider that some of those ßs will belong to themselves and some will not: some can be written as ß∈ß, and some can be written as ~(ß∈ß).  Let’s call ~(ß∈ß) ordinary multiples, and ß∈ß evental multiples.

Take the set of α’s ordinary sets and call it γ. So γ is all the ß’s that belong to α, but not to themselves.  We can say γ is included in α, but if we say that γ belongs to α, we end up with a contradiction.  If γ does not belong to itself and so is ordinary then it belongs to the ordinary sets of α, which is γ itself.  This is a formal contradiction, so γ does not belong to α.  Set α cannot make a one out of everything it includes.  Inclusion is always in excess of belonging.

Naming the Void

As we saw last week, the world that is presented to us is made of multiple things, so we might say “presentation is multiple.”  Ontology is essentially about what is going on behind the scenes, or about the reality behind the appearances, so if presentation is multiple, the question is: what lies behind presentation?  A typical answer is that what underlies multiple presentation is just one thing, say atoms as a single species of thing, or the mind of God.  Badiou has discarded this answer and claimed that what lies behind the consistent world of multiple presentation is inconsistent multiplicity.  Badiou named this inconsistency the void, and explained it as the empty set which is included in every other set.

One of the ongoing problems for ontology is how exactly to speak about what lies behind appearances.  If there really is a difference between reality and appearance, then how are we supposed to talk about reality without cheating somehow, without reducing it to an appearance?  In Badiou’s terms, if the void is unpresentable, then why is he even talking about it?  Shouldn’t it be something absolutely hidden from view, inaccessible to rationality?  (Q3)

The idea of the power set is what saves Badiou’s ability to talk about the void.  The void is an empty set.  Nothing belongs to it, not even the void itself: ~(ø∈ø).  However, the void does have a power set: p(ø), or {ø}.  While nothing belongs to the void, something is included in it: the void itself, ø⊂ø.

It looks like we’ve filled the void, because of the metaphor “being inside.”  However, Badiou says “in reality, the statement ø⊂ø solely announces that everything which is presented, including the proper name of the unpresentable, forms a subset of itself, the ‘maximal’ set.  It is a reduplication of identity, {ø}.” The power set of the void is the multiple that only the void belongs to.  {ø} is not the void itself. It is the fact that the void is included in itself that gives us the ability to name the void; calling the unpresentable “the void” is not some illegitimate leap of faith.  Badiou manages to dodge this bullet by proving that while nothing belongs to the void, something is included in it: its name.


Given that the void that the void, or inconsistent multiplicity is included in every situation, there must be a function to obscure that void to maintain a veneer of consistency.  As Badiou puts it, “the ruin of the One” must be avoided.

The guarantee of consistency (“there is oneness”) cannot rely on the count-as-one alone to stop the void.  This is because something escapes presentation: the count itself.  Subtracted from the count, the count itself could be where the void is given.  “In order for the void to be prohibited from presence, it is necessary that structure be structured.”  All structure is doubled by a metastructure, securing the former against the void.  Because of a metaphorical affinity with politics, we will call this the state of the situation.

Multiples can have one of three relations to the state.  They can be normal, which means they are presented in a situation, and then represented at the level of the state.  There can be excrescent multiples, which are not presented in a situation, and only represented at the level of the state.  Finally, a multiple can be singular, which means it is only presented in a situation, and not represented by the state.

The State of the Socio-Historical Situation

The rest of Part II of Being & Event is a formal description of the political state.  This is where we first begin to see the relation of the ontological situation, as presentation of presentation, to presentation as such.  The mathematical terms are used to describe political situations and states, as they will later be used to describe love and art.  The relation, for now, seems to be one of metaphor.

Badiou begins by partly agreeing with Marx on the role of the state.  Basically, the state has no relation to any individual person; it only deals with individuals as re-counted and re-presented, say as members of a voting demographic.  In other words, the state deals with normal multiples.  Another point of agreement between Badiou and Marx/Engels is the reason for the state’s existence.  The state is not based upon an existing social bond or social contract, rather it is based on the possibility of that bond’s unbinding.  Just as a set’s metastructure obscures the empty set, the political state obscures a fundamental antagonism at the heart of all political life: the conflict between the rich and the poor.

This is what “the state is always the state of the ruling class” means: the state’s law is that of forming-one out of the parts of a situation.  The state is the law that guarantees there is Oneness, not the immediacy of the social bond (which is a non-state structure).  The state is not a tool possessed by the ruling class.  It always represents what has already been presented.  The definition of the ruling class is not statist, but economic and social.  The presentation of the middle class (the bourgeoisie) is not about the state, but possession of the means of production.  To say the state is of the bourgeoisie is to say the state represents something that has already been presented, historically/socially.

This has nothing to do with government being constitutionally representational; the state is defined by the representation of the terms of the situation.  It is tied to presentation: “the parts, whose one it constructs, are solely multiples of multiples already counted-as-one” by the situation.   So the state is historically linked to society.  Only capable of representation, the state can’t bring forth a null term or null multiple.

But because what is included cannot be reduced to what belongs, the state is separate from society.  It is subject to the point of excess, which “forces the state to not identity itself with the original structure which lays out the consistency of presentation, which is to say, the immediate social bond.”  So the bourgeoisie state is separated from both capital and from its general structuring effect.  It represents terms structured by capital, but as an operator it is distinct.

As said above, the state does not deal with individuals as presented in their own situations, but rather as included submultiples.  Not as a singleton, “Kevin Spencer,” the proper name of an infinite multiple presented in a situation, but as {Kevin Spencer}, an indifferent figure of unicity, constituted by the forming-into-one of the name, re-presented by a metastructure.  The voter “is not the subject John Doe, but is rather the part that the separated structure of the state re-presents, according to its own one; that is, it is the set whose sole element is John Doe and not the multiple whose immediate one is John Doe.”  Another relation the state has to individuals is that of coercion, or punishment.  Coercion is being held as being included in society, but not as belonging to it.  In all these cases,  “… when it is a matter of people’s lives – which is to say, the multiple whose one they have received – the state is not concerned.”

Here, the Marxist analysis has a fatal ambiguity.  For Marx, the state “is finally its bureaucratic and military machinery; that is, the structural visibility of its excess over social immediacy, its character of being monstrously excrescent.”  Excrescence: to the represented but not presented.  The ambiguity is in thinking the state itself is an excrescence, that it is only the bureaucratic and military machinery.  “By consequence, as a political programme, the Marxist proposes the revolutionary suppression of the state; thus the end of representation and the universality of simple presentation.”

For Engels, the state does not result form the existence of classes/parts, but rather from their antagonism.  Without a monopoly on structured violence by the state, there would be permanent civil war.  Once again, we see that “the state is not founded upon the social bond, which it would express, but rather upon un-binding, which it prohibits.  Or, to be more precise: the State is less a result of the consistency of presentation than of the danger of inconsistency.”  This is why governments regulate the “emblems of their void,” say an inconsistent or rioting crowd.  It declares its non-tolerance of the one of these parts.  The numbering of inclusions preserves consistent belongings.

But that isn’t quite what Engels said: he said that the bourgeoisie was normal (presented and represented), the proletariat was singular (presented but not represented) and that the state was an excrescence.  So the state’s excrescence does not refer to the unpresentable, but rather to differences in presentation.  By modifying these differences, the state could disappear; a pure reign of the singular.  Communism “would in reality be the unlimited regime of the individual.”

The void, for Engels, is reduced to the non-representation of the proletariat, so the unpresentable is reduced to a modality of non-representation.  The separate count of parts the metastructure, is reduced to the non-universality of bourgeoisie interests, and the count-as-one is entirely an excrescence because, as Badiou says, “he does not understand that the excess it treats is ineluctable, for it is a theorem of being.”

This makes politics only an assault on the state.  But re-securing of the one over the parts cannot be defeated so easily.  Politics, i.e. radical justice, is always bordered by the state, but not guided by it.  It is not the antagonism of the state, because the dialectic of the void and excess  is not antagonism  “No doubt politics itself must originate in the very same place as the state; in that dialectic.  But this is certainly not to seize the state nor to re-double the state’s effect.  On the contrary, politics stakes its existence on its capacity to establish a relation to both the void and excess which is essentially different from that of the state; it is this difference alone that subtracts politics from the one of the statist in-difference.”

“. . . the activist constructs the means to sound, if only for an instant, the site of the unpresentable, and the means to be thenceforth faithful to the proper name that, afterwards, he or she will have been able to give to – or hear, one cannot decide – this non-place of the place, the void.”


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