It is a quite general feature of theories that try to characterize the limits of our cognitive abilities to think, describe, grasp, that they end up implying that they themselves cannot be thought, described, or grasped. Yet it would appear that they can be thought, described, and grasped. Otherwise, what on earth is the theory doing?
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The contradictions at the limits of thought have a general and bipartite structure. The first part is an argument to the effect that a certain view, usually about the nature of the limit in question, transcends that limit (cannot be conceived, described, etc.). This is Transcendence. The other is an argument to the effect that the view is within the limit—Closure. Often, this argument is a practical one, based on the fact that Closure is demonstrated in the very act of theorizing about the limits. At any rate, together, the pair describe a structure that can conveniently be called an inclosure: a totality, Ω and an object, o, such that o both is and is not in Ω.
On closer analysis, inclosures can be found to have a more detailed structure. At its simplest, the structure is as follows. The inclosure comes with an operator, δ, which, when applied to any suitable subset of Ω, gives another object that is in Ω (that is, one that is not in the subset in question, but is in Ω). Thus, for example, if we are talking about sets of ordinals, δ might apply to give us the least ordinal not in the set. If we are talking about a set of entities that have been thought about, δ might give us an entity of which we have not yet thought. The contradiction at the limit arises when δ is applied to the totality Ω itself. For then the application of δ gives an object that is both within and without Ω: the least ordinal greater than all ordinals, or the unthought object.
OR, in OOO-ese: objects appear, and thus they are within Ω, but they also withdraw, and so they are not. The appearance of a (withdrawn) object is precisely Priest's δ.
Inclosure is now my favorite withdrawal substitute. Along with secret.
2 comments:
Jarrod, Priest holds Badiou's math in rather low esteem. He and I are working on a critical assessment of Badiou's math. Badiou relies on a castrated version of Cantor (Zermelo-Frankel) which enables him to adhere to the (never proved) law of noncontradiction.
And I have a visceral negative reaction to this idea that life has four easy to remember flavors.
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