NOTES FROM: Badiou, A (2011) ‘Ontology of Multiplicity: omega as event’ EGS video

There is no ‘one’ as such.  Everything finite is a multiplicity.  The finite does not exist inside a One, whether this is Nature  or God, and nor does it only exist outside but  as dependent upon the One.  The multiplicity is the fundamental term of ontology.  Of course we need a strict definition, and this is why we turn to set theory.  What is a set?

Intuitively a set arises when we put things together, but we need a more precise definition.  A set is something composed of elements, represented as X E Y, meaning X is an element of Y.  To exist at all is to belong to a set, to be an element in a multiplicity.  Everything that exists is an element of something different.  We also have to remember that the names of sets are not included among the elements of the set, that X E X is never possible.  This means an ‘obligation of the other’, that the other is ontologically necessary.

There are many consequences of this position.  Nothing can exist separately and alone, but is always an element of something.  We never exist alone, for example.  Difference is integral to existence, and so is a relationship with something other.  There is no solipsism.  There was always an exterior world.  So the ontology of the multiplicity is one of difference, and never pure identity.  In the political field, we can see that those who advocate pure identity not difference are nearly always reactionary, opposed to the other.  So radical rejection of the other is an ontological fault.  This is an important political issue today.  All attempts to develop a closed identity have led to terror and the destruction of the other.  For example the ‘Nazi real’ acknowledged the Jews as others, but saw their destruction as necessary, and then the same for all others.  This is a negative relation to the other, but there are positive ones as well.

What we need to do is develop our experience of the different rather than just explore our own identity.  This is an axiom of true politics, to assume there is an other, to assume difference, to agree that identity is not the first element of reality.  What is first is a complex of differences, and we only explore our identity by experiencing others.

So the notion of God as the pure One must be excluded from our analyses of being, since God is pure identity.  Actually, there are complications, and it might be the case that God creates the external world and difference to develop His true identity, that the external can be seen as the development of God, a development of the infinite.  Alternatively, God can be seen as the repository of all differences, as in Spinoza, or even as omega, as a recapitulation of finite differences, so that God is the name of differences.

If X is always an element of Y, it also means that X must be a multiple, arranged in a series—something else is an element of X, Y is an element of something else, and so on.  The series is stopped at either end by infinity and the void.  The void belongs to a singleton, but nothing belongs to the void, so the void is between nothing and something.  Further, there is only one void, unlike infinity, so there is no determination in  the void -- it cannot generate another void.

How can we explain the difference between two multiplicities, where X may belong to Y, but is not be equal to it?  X is different from Y if it contains an element that Y does not possess—let us call it Z.  This describes the property known as extensionality, where we can find a single point of difference in one element in X but not in Y.  This is a particular or local difference, but it can explain much larger global differences, the whole of the difference between X and Y. 

However there are other cases, the differences between colours, for example, where there is no single distinct element which can act as a point of difference.  We're talking here of qualitative differences.  These are differences outside ontology [assuming we mean outside set theory and its version of ontology].  For this reason, qualitative differences have been seen as quite different differences, not extensional and localized, but qualitative and global, different in their totality.  And this has appeared as a basic philosophical distinction and produced disputes about the qualitative as the only true difference—this is the claim of Bergson, Deleuze and Nietzsche who think that global differences, the quality of things, is the only proper difference, and they do not consider extensional or quantitative differences as being equally important.  However, it might be the case that we can explain the global from many local differences, which would render global differences as an illusion.  There may be infinite local differences in one multiplicity, and only one difference in another.  We might be able to explain the differences between yellow and blue in terms of different wavelengths, for example.  [This seems to be an important insight into Deleuze’s insistence on the intensive operating at the virtual, and his more or less total disregard of any quantitative differences in the empirical, which he sees as merely non-conceptual, the province of science and so on.  There also clearly links between the limited otherness for women in phallogocentrism as opposed to their virtual otherness, for example in Irigaray].

Issues arise with the notion of succession too [see the earlier lecture].  We have seen that omega emerges as outside finite succession, but then yields a succession of omegas.  Succession does not take place only inside the finite and cannot be used as a definition.  All finite numbers do succeed, but they also have successors in the infinite, in the new worlds opened by omega.  It follows that the infinite can not be defined as non-succession, since omega is succeeded.  Omega can be seen as a cut between the finite and the infinite, it is not produced by the finite but succeeds it, not as a consequence of the finite but as a recapitulation of the world as well as a cut [reminds me of Lyotard’s definition of the postmodern].  Omega is therefore like an event, something which is not precisely law governed and which opens a new sequence.  It can only be calculated afterwards.

In the new succession after omega, the laws of the world still apply [the laws of set theory]—the new world is not an illusion.  There is some repetition inside the new.  We never totally enter into a totally new, and the laws of succession and so on remain.  However, we are in another dimension.  Some people think this is chaos, but it is a new world on the other side of a rupture with the old world.  Omega is a good image for this [partial] rupture.

Omega as a point distinguishes the finite and the infinite.  Omega is both the successor and a limit.  What is the difference between these terms?  Is it not possible that the true content of the infinite can be omega itself? [No because] omega is an absolute particular point that does not succeed the finite—this is the limit.  Even new worlds can exist inside omega.  In this sense, the infinite is always a limit [of the finite].  However there is another sort of infinite, not as the limit but as a succession.  Are these identical?  Not if there is a major difference between rupture and succession [some very old themes in French philosophy here --epistemological breaks and all that!].  This second distinction is also inside the infinite, it is a second kind of both successor and limit.

In the infinite set, succession shares the law of succession in the finite, it is a kind of memory of the finite, a repetition of succession.  In the limit though, there really is a new beginning, and here it is like the void, which does not succeed.  The void is also a beginning, and omega can be a new beginning.  The differences between succession and limit are combined.

The homely analogy refers back to concrete life.  We can experience an omega.  We know it's a beginning, but we also recognize a certain familiarity in the succession after the beginning, certain familiar bits reoccur.  It's the same with infinity—there are differences inside it, familiar bits as well as revolutionary ones.  Only omega itself is the true change, and after it we return to common life.  There all sorts of implications for politics here, such as the routinization that follows the insurrection as the true political moment.  Even so, we are not returning to the finite and the old world—it is simply that something must continue into the new succession.

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