Tuesday, April 2, 2013

In excess

On the recent theme of how math is changed in a strange mereology beyond a Hegelian dialectic, I offer some excerpts from Bryant's The Democracy of Objects, chapter 6.2:. As to the nature and implications of the referenced excess, see this post and several posts and pages following for one of its manifestations as the objet a.

"Within the domain of formal reasoning, Z-F set theory shows the inconsistency of any attempt to form a totality or whole. Set theory provides a variety of resources for contesting the consistency of any totality or whole, however, here I'll focus on the power set axiom. As we've already seen, the power set axiom allows one to take the set of all subsets of an initial set....If the power set axiom spells the ruin of any whole or totality, then this is because it reveals the existence of a bubbling excess within any whole or collection.... What the power set reveals is the bubbling pluralism of 'the' world beneath any unity or totality. Any totality or whole, in its turn, is itself an object or One alongside all sorts of other ones.

"At the formal level, the real force of the power set axiom lies in the manner in which it reveals the possibility of a multiplicity of relations and objects within any collective. It will be recalled that any exo-relation between objects is potentially itself also an object. If we ask the strange question, 'when is an object?' we can answer this question with the hypothesis that an object is when exo-relations among other objects manage to attain operational closure such that their aggregate or multiple-composition becomes capable of encountering perturbations as information in terms of their own endo-consistency. On the one hand, the power set axiom reveals the possibility of a plurality of other objects within any collective. On the other hand, the power set axiom discloses the possibility of alternative exo-relations among objects, not present in the whole from which the subsets are drawn. Finally, the power set axiom reveals the possibility of withdrawing objects from their relations to collectives so that they might function as autonomous actors, either entering into other collectives, subsystems, or going it alone within the order of being.

"If, from the standpoint of formal reasoning, the Whole is not, the One is not, or the world does not exist, then this is precisely because these subsets, these other possible objects and relations populating the power set of the Whole or alleged One are neither counted nor countable within the Whole or One. In short, every Whole or One contains an excess within it that is not itself treated as a part of the Whole or One. Put differently, such subsets are included in the set from which they are drawn, without belonging to it. Yet it is precisely this absence of belonging or membership that spells the ruin of the Whole, One, or World."

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