By Alain Badiou
Explores the hyperlink among arithmetic and ontology.
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Additional resources for Briefings on Existence: A Short Treatise on Transitory Ontology
But the status of multiplicities, the thought of difference and thus on what is thought beneath the a n o n y m o u s n a m e of Being. • N o w we have to reexamine logic from the perspective of ontological decision itself. H e n c e , we need to think the possible from the basis of the Real. T h i s also m e a n s we have to invest definitions from the standpoint of a x i o m s , and not vice versa. Finally, we have to posit that, as a thought apparatus, philosophy is essentially a x i o m a t i c — and not definitional or descriptive.
Being, pure Being and purely empty Being, is letter G, which is but the literal index of the O n e that-is. Rest is id(G), inert identity to oneself as the stopping-point of all action. Movement is the arrows, noninert isomorphisms plaiting G in the active manifestation of its identity. Last in Plato's list is the dialectic of the S a m e and the Other, which is made explicit in the difference of arrows. For this difference, in terms of difference, vouches for the Other. But this Other is also only the differe n t i a t i n g work of the S a m e b e c a u s e every "other," being an isomorphism, is nonetheless a figure of the S a m e .
As I said, its internal logic can be identified. The fact that this is well known stems largely from the immanent and local data of a topos. To give a couple of elementary examples, the true and false are local data of a topos, that is, "simple actions" that are part of the universe. By the same token, negation, conjunction, or even implication is also part of the universe. All of these terms designate relations that have been identified, and are present "in person" in the topos. There are no syntactical preambles or semantic interpretations.