*This will be the reading for November 3*.

We have seen all along how any given set can actually produced larger sets from out of itself, and how a situation belongs to or is included in a larger state. A third way of putting this is that representation is “larger” than presentation; the question is, can any of these versions of “larger” actually be quantified? In other words, is there some kind of hard limit to the excess of representation over presentation? It turns out the answer is “no.”

There are two significant answers to this problem. The first is what he calls constructivism. It only counts those parts which allow themselves to be distinguished, so “thereby reduced to counting only those parts which are commonly nameable, the state, one hopes, will become adequate to the situation again.” (283) The other response is that of the generic. The description of constructivism is actually an anatomy of knowledge, while the generic corresponds to truth.

**The Discernible and Constructivism: Knowledge**

In order to solve the problem of the unquantifiability of being, Cantor wanted to show that “the quantity of a set of parts is the cardinal which comes directly after that of the set itself, its successor.” So p(ω) should be exactly the same as ω1, the first cardinal which measures the next infinite. This is the continuum hypotheses, and while it does not break with the axioms, it does mean that the axioms are subordinated to the limits of language. This hypotheses works within what Godel called a “constructible universe” and deals solely with “constructible sets.”

Take set α. Its power set p(α) is everything included in α, which is the origin of excess. If we say that p(α) must be constructible, then it will only admit “as part of α what can be separated out. . .by properties which are themselves stated in explicit formulas, whose field of application, parameters and quantifiers are solely referred to α itself.” (297)

D(α) is the set of definable parts of α, which is a subset of p(α). We can use these parameters and quantifiers to advance step by step, using only nameable and definable parts. Only what language is capable of controlling is admitted as existing. Being, in a very real sense, becomes quantifiable. There is no refutation of this in ontology, per se.

While the axiom of the point of excess means the state is certainly larger than the situation, constructivist thought is nostalgic for a solution, for a way to reconcile the state and situation, or presentation and representation. It attempts to base itself on the constraints of language. The only things that exist are the things that we can clearly know and describe. He describes constructivism this way:

“In its essence, constructivist thought is a logical grammar. Or, to be exact, it ensures that language prevails as the norm for what may be acceptably recognized as one-multiple amongst representations. The spontaneous philosophy of all constructivist thought is radical nominalism. . . .

What is at stake, in fact, is a mediation of interiority, complete within the situation. Let’s suppose that the presented multiples are only presented inasmuch as they have names, or that “being-presented” and “being-named” are the same thing.” (287)

Essentially, constructivism claims that only what can be rationally discerned actually exists. If something cannot be pointed at and clearly named or described, then it does not actually exist. It is a positivist epistemology, basically; clear facts are all that matters. Badiou thinks is most common way of thinking today: witness how politics is now almost entirely a matter of economic number crunching and poll taking. We can also see it in the cultural prestige of science – if something is not scientifically provable, then it is nonsense. It is even easy to see in love: how many people turn to Slate articles about evolutionary psychology to tell them about their desires?

The technical point is that language is thus interposed between presentation and representation. It is the link between presentation and unquantifiable, excessive representation. The limits of language make the state commensurable with the situation. This produces an odd paradox, however. On one hand, the state’s power is restricted; there are only nameable parts. But on the other hand, the state completely masters these parts. “What it loses on the side of excess it gains on the side of ‘right over being.'” (288)

There is no place for an event – and in this, it looks like ontology. But this is because constructivism has no need to decide upon the non-being of the event; self-belonging simply does not exist. The event is refractory to knowledge. But of course, change happens all the time; if there are no events, what is the source of change? And if only the clearly nameable exists – that is, if there is no gap between presentation and representation, why does the world have so many different situations in it?

“In the final analysis, the [constructivist] doctrine of the multiple can be reduced to the double thesis of the infinity of each language (the reason behind apparent change) and the heterogeneity of languages (the reason behind apparent diversity of situations). And since the state is the master of language, one must recognize that for the constructivist change and diversity do no depend upon presentational primordiality, but upon representative functions. The key to mutations and differences resides in the state.” (291)

Ultimately, constructivist thought is identifiable with the knowledge of the situation:

“Knowledge is the capacity to discern multiples within the situation which possess this or that property; properties that can be indicated by explicit phrases of the language, or sets of phrases. The rule of knowledge is always a criterion of exact nomination. . . .Discernment concerns the connection between language and presented or presentable realities.” (328)

Knowledge is the ability to judge properties, and to link these judgements together as a set of classifications. A key point here is that working entirely within the axioms of ontology, knowledge in this sense is all we have. As it stands, it appears as if only that which is clearly nameable and understandable can be said to exist, but there is an alternative: the indiscernible.

**The Generic and the Indiscernible: Truth**

The terms generic and indiscernible mean almost the same thing. But indiscernible carries with it a negative sense – the indiscernible is what is subtracted from knowledge or exact nomination. Generic, on the other hand, as a more positive sense as being the truth of a situation. The key point to be resolved here is the relation between a post-evental truth and a fixed-state of knowledge.

A fidelity is not about knowledge, or being an expert, because the minimal gesture of a fidelity is saying whether or not a multiple is connected to the event – rather than a classification of a thing as being this or that.

As far as a fidelity is concerned, there are only two relations a multiple can have to the event: connected or not connected. x(+) is positively connected, while x(-) is negatively connected. An enquiry, when looking at the situation, will have a string of reports: x1(+), x2(+), x3(-), etc. We get two sets: those multiples which are positively connected, and those negatively connected.

This looks like exactly the same sort of judgements that we said made up knowledge, and there is a resemblance of fidelity to knowledge. This resemblance is further supporting by the fact that every presented multiple being is nameable in the language of the situation. But if every part of the situation is already counted, then isn’t the procedure of fidelity redundant?

“In order to clarify this situation, we will term veridical the following statement, which can be controlled by a knowledge: ‘such a part of the situation is answerable to such an encyclopedic determinant.’ We will term true the statement controlled by the procedure of fidelity, thus attached to the event and the intervention: ‘Such a part of the situation groups together multiples connected (or unconnected) to the supernumerary name of the event.'” (332)

Knowledge (the group of discernible facts about the situation) can always trump the fidelity “with a peremptory ‘already-counted!'” (333) So we need a way to distinguish between the multiples reported by the fidelity – the true – and those already named by knowledge.

One criteria is “the true only has a chance of being distinguishable from the veridical when it is infinite. A truth (if it exists) must be an infinite part of the situation, because for every finite part one can always say that it has already been discerned and classified by knowledge.” (333) To say a procedure is infinite is to say that it carries out an infinite number of enquiries. Badiou’s statement of the problem runs like this:

“What we are looking for is an ontological differentiation between the true and the veridical, that is, between truth and knowledge. The external qualitative characterization of procedures (event – intervention – fidelity on one hand, exact nomination in the established language on the other) does not suffice for this task if the presented multiples which result are the same. The requirement will thus be that the one-multiple of a truth – the result of true judgements – must be indiscernible and unclassifiable for the encyclopedia. This condition founds the difference between the true and the veridical in being.” (333)

The infinity of a truth does not quite fulfill that requirement. But since math can deal with infinities purely as a matter of knowledge, a truth’s infinity is not sufficient to distinguish itself from knowledge. Marxism ran into this same issue. It died because it stayed too close to knowledge, and was undercut language and the state:

“[It] claimed that truth was historically deployed on the basis of revolutionary events by the working class. But it thought the working class as the class of workers. Naturally, ‘the workers’, in terms of pure multiples, formed an infinite class; it was not the sum total of empirical workers that was at stake. Yet this did not prevent knowledge (and paradoxically Marxist knowledge itself) from being for ever able to consider ‘the workers’ as falling under an encyclopedic determinant (sociological, economical, etc.)” (334)

The Generic Procedure

Any given statement, or determinant of the encyclopedia, also has its contradictory determinant. If we group all the multiples that have a particular property in one class, that implies a second class of multiples which do not have that property. So if x has a property, let’s say white, and y is not white, then we can make set {x, y} which is indifferent to the property “white.” “Knowledge considers that this finite part, taken as a whole, is not apt for discernment via the property [white]”. (335)

A finite part avoids an encyclopedic determinant if it contains multiples which both have the property and others which do not. In our case, the finite part {x, y} avoids the determinant white.

“The general idea is to consider that a truth groups together all the terms of the situation which are positively connected to the event.” (335) The positive terms are more important because the negative terms only repeat the terms of the situation.

But still, we need to distinguish the positive terms from those merely counted in the situation. We can’t just look and the infinite set of x’s, because being infinite, this set is always to-come, and moreover, the x’s a fidelity encounters are entirely random. So instead of looking at the x’s and showing how they are distinguished from knowledge, we need to look at the form of the enquiry.

“The crucial remark is then the following. Take an enquiry which is such that the terms it reports as positively connected to the event (the finite number of x(+)’s which figure in the enquiry) form a finite part which avoids a determinant of knowledge in the sense of avoidance defined above. Then take a faithful procedure in which this enquiry figures: the infinite total of terms positively connected to the event via that procedure cannot in any manner coincide with the determinant avoided by the x(+)’s of the enquiry in question.” (336)

The enquiry, at any one point in time, examined x1 and x2, and found them to be positive or negative. But given the infinite nature of a procedure, there will always be new x’s to be examined: Xn. The set of multiples to be examined in the future are both Xn(+1) and Xn(-). Some of our Xns will have the property white, and others will not. This is the difference between X and Xn: Xn cannot be the class of things defined in the language by the classification ‘all the multiples discerned as having this property [white].'” (337) (Everybody follow?)

To go a little further, let’s imagine that the above condition is satisfied for every property: that for whatever properties one cares to name – white, black, red, yellow, brown – there is at least one enquiry whose x(+)’s avoid that determinant.

“We shall therefore say: a truth is the infinite positive total – the gathering together of x(+)’s – of a procedure of fidelity which, for each and every determinant of the encyclopedia, contains at least one enquiry which avoids it. Such a procedure will be said to be generic (for the situation).” (338)

“An indiscernible inclusion has no other property than that of referring to belonging.” They have no other quality than that of belonging to the situation – their properties of being black or yellow are irrelevant to their place in the procedure.

“For what the faithful procedure thus rejoins is none other than the truth of the entire situation, insofar as the sense of the indiscernible is that of exhibiting as one-multiple the very being of what belongs insofar as it belongs. Every nameable part, discerned and classified by knowledge, refers not to being-in-situation as such, but to what language carves out therein as recognizable particularities.” (339)

Hi, I’m Kyu Don. Are we meeting this Saturday?

Yep!