Subset of Theoretical Practice

One can conceive of the phenomenological object as a space for the game "20 questions"^[1]. In this game, one player (let us call her the oracle) privately commits to some ontological entity (typically restricted to an agreed-upon type), while the other asks a series of yes or no questions, up to 20, to find the entity. Each question determines a set: "is it man-made?" yields the set of "man-made" things. If the answer to a question is yes, the entity in question lies in the set, and if the answer is no, the entity lies in the complement of the set. The set of questions leading to an answer constructs what mathematicians call a "filter" (c.f. "net"), the collection of subsets containing some element, which can be ordered by inclusion (c.f. "restriction"). The space of such a game consists of all ontological entities of a certain type, along with a certain topology to be discovered by both players. To win against the oracle, one must choose those questions which together most approximate a filter (of height ≤ 20) for your target. In logical terms, the predicate π which uniquely singles out the entity is a conjunction of predicates π_1, π_2, ... π_n (where n ≤ 20). The value of each question then is determined by how much it excludes, i.e., the size of its complement-set - which is in turn determined by the questions and answers that came before it. In some cases, questions may not yield a true or false answer, but nevertheless yield information. Consider the entity "human" and the question "is it man-made?". Or "virus" and "is it alive?". Or "silence" and "is it part of language?". The underlying premise is that every being of a particular world has an address of at most 20 predicates. Likewise, to win as the oracle, one should commit to an entity that is not addressable, that most resists any anticipated line of questions, something which might be "virtual" with respect to the type of world agreed upon by the two players. To be fair, the world should be restricted to no more than 2^20 entities. If the domain is that of rational numbers for example, the oracle should have no problem winning. But even in a finite world, one cannot prevent diagonalization by a skilled oracle: "the being predicated on the negation of every predicate in the space". Conversely, every entity in the world can be said to generate its filters. Just as the conjunction of predicates produces a more precise answer, their disjunction will eventually lead to the entirety of the world. But there are multiple paths to generalization. One can form a "universal" way of getting to the answer from the particular filters, i.e. by collecting the set of all predicates from all filters based on the entity. We call this an "ultrafilter". With finite sets, every ultrafilter is also "principal", meaning it contains a least element - in our case, the object of the game. With infinite sets, things are not so simple ^[2]. In topology, spaces in which every ultrafilter converges to a point are "compact" and "Hausdorff" ^[3]. This is also, in essence, the "localic" (or Grothendieck) property of a topos. And in model theory, the process of generating a filter from a point (or set of points) is what lies at the heart of creatively extending a model^[4]. In a sense which can be made infinitely more precise, we can play the game in reverse: we can produce a new world by adjoining something alien to the existing one. The game of 20 questions is therefore a toy version of the game of countably-infinite questions, that is, mathematics (ontology) itself. The productive interplay between analytic specification and synthetic generation cuts across Badiou's project. It informs every one of the formal choices that he makes. In light of this, we can reconsider the following from Theory of the Subject ^[5]

"We welcome those vicinities of the vague in which the partitive multiplicity is dissolved, considering them to be the proof, administered by those who would desire the exact opposite, that there is a wager on the real. If, in this wager, the number inscribed on the dice is the result of a consecution, it cannot link up into a chain that which, in the thrower's gesture, produces the incalculability of its reach."