Here's an argument which seems to underlie a whole lot of Badiou's metaphysics of "the void," or his name for the "being of beings."
Badiou's claim is that inconsistency (what he means to be "difference" or the non-being which is being) is never presented (encountered by human beings) as such, even in set theory (52). This is “subtractive” given that it merely draws a distinction in being already presented pre-linguistically (48). It is on the basis of the consistent count as a "result" - counting "voids" in Badiou's theory - which seems to lead Badiou to see inconsistency as pointed to by any count (53). This, he claims, unhinges "counting" as consistent, given that the pre-count cannot be counted qua pre-count. To translate this into metaphysical language: there is an act of understanding which results in knowing beings. Prior to this act, no beings are known by the Subject. Therefore, nothing exists prior to the act of understanding. But this is a clearly fallacious shift in sense; it does not prove that only nothingness precedes a count or an act of understanding/encountering beings.
Here's an analogy of shaping a statue from clay:
Unshaped clay precedes my shaping of the statue.
I cannot shape unshaped clay qua unshaped, as it is definitionally impossible.
Therefore there is no clay preceding my shaping of the statue.
But this is clearly just a silly shift in senses. I do shape something into something else, with varying properties over the change; the clay moves from a state of unshaped to that of a shaped statue. But Badiou, invalidly, analogically draws a stronger thesis - no clay existed prior to the shaping. Which just goes to show you how careful one needs to be about our littlest logical mistakes.
Angelo Codevilla on the Helsinki Summit - Required reading. Yes, kiddies, this will be on the final.
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