*This is the reading for June 16. We will be meeting at Bless U at 3:30.*

Ontology is traditionally the field of philosophy that deals with existence as such. Variously, it has attempted to describe what kinds of things exist, under what conditions these things exist, and what “to exist” means in the first place. For some time now, philosophy has been mostly concerned with the third point, which could also be described as asking what being is. The question of being could even be described as philosophy’s central question, especially given that science has basically taken over the role of telling us what kinds of things exist. So while it is a question for physics whether or not multi-dimensional strings exist, a philosopher might insist that describing the underlying being of those strings belongs entirely to their field.

Alain Badiou presents a strange and disconcerting thesis, at least to a philosopher’s ears: ontology is not, and never has been, an element of philosophy. The question of what being is is entirely a matter for mathematics. For Badiou, ontology begins with the question of the one vs the many/multiple. The problem is, how can we think the multiple without making it just a sum of ones? In other words, how can a multiple be presented as subtracted from the one? An axiomatic, formal system can solve this problem in a way that normal language can not.

**The One vs the Many**

If we look around us, we can see many different things: multiple chairs, people and walls are presented to us. But we also see individuals: this chair is just one chair. The multiplicity of the situation around us is a sum of individual things: 1 chair + 1 chair + 1 person, and so on. On a deeper level, it is quite easy to believe that all these multiple things are, at bottom, one: the presented multiplicity of things is actually made out of a fundamental term from physics such as atoms, or presented things are a manifestation of the singular mind of God.

Let’s put it this way: what **presents** is multiple, while **what** presents is one. Multiplicity is presented to us, but what actually lies behind the presenting is one. From this, we get another fairly common thesis: if underlying reality is one, then the multiplicity around us is not actually real. In other words, if the one is, then the multiple is not. However, there are two problems with this. First, we do see a presented multiplicity of things; presentation is at least partly multiple. If presentation as multiple is unreal, then how could it be correct to say that being or underlying reality is presented in any way? Second, it is entirely unclear how someone could get outside presented multiplicity to see this illusive one.

We could try to solve the problem by claiming the opposite thesis: the multiple is, therefore the one is not. But this also gives us an obvious problem: all the multiples that are presented to us, such as the many individual things that make up this room, are individuals. The multiplicity of this room is the result of a counting, a sum of individual things: of ones. It seems there is no way to talk about a pure multiplicity: we always come back to the one.

Badiou’s solution to the problem will involve a serious of axiomatic decisions, and the first is a strange idea: there is oneness, but the one is not. What can we make of this odd, seemingly contradictory formulation?

**The Count: Consistent and Inconsistent Multiplicity**

We said above that the multiplicity around us is always the result of a count: we add many individual things together, and the result is one situation. This is the key point: the one is a result. This is why we can say there is oneness; we see the sum of individuals all around us. This meeting is the result of many ones counted together: people, drinks, chairs, words. A situation as a collection of ones, that is, individual things, is the result of an operation: the count-as-one.

The count-as-one results in a multiplicity of individuals, all summed up: let’s say that a multiplicity of individuals is a consistent multiple. Now, the question is: what is counted? Do we stop at saying that a consistent multiple is entirely the sum of other consistent multiples? If we take seriously the second half of the above phrase, “the one is not,” then a consistent multiple cannot just be the result of counting ones. There must be something else: the inconsistent multiple.

Above, we said that the idea “the one is not” is a decision. It is on the basis of this decision that we infer the existence of the inconsistent multiple. If consistent multiplicity is not the result of a counting of ones, because the one is not, then it must be the result of a counting of inconsistent multiples. The inconsistent multiple is what exists before the operation of the count-as-one, and the consistent multiple is what exists after the count. Every situation is split into consistent and inconsistent multiplicities, or in other words, every situation is both a sum of ones and pure multiplicity. A situation is one, but the being of this one is multiplicity.

Every situation we find ourselves in is a consistent multiplicity. Everything is counted, in some way. Now we have an obvious problem: if all that is ever presented to us is the result of a count, how on Earth are we supposed to talk about that which is before the count? If we simply talk about our supposed inconsistent multiplicity, what we will end up talking about is a particular example of an inconsistent multiplicity: let’s say the inconsistent multiplicity of this room. In other words, we will always speak about one inconsistent multiple; we will include what is supposed to be before the count in the results of the count. This appears to be a hopeless contradiction.

**Ontology is a Situation**

Everything that is presented is counted, and certainly when we do ontology, we are working within presentation. So ontology itself is a situation; that is, it operates according to a count. If the goal of ontology is to talk about being, then it must grapple with the above problem: how to talk about inconsistent multiplicity without reducing that inconsistency to consistency? How are we supposed to describe a multiple as inconsistent, that is, not as a sum of ones? What could the count-as-one that governs ontology be?

One way to dissolve the problem would be to claim that ontology is not actually a situation: that it is possible to escape the law of the count and just encounter pure, inconsistent multiplicity. Badiou thinks this is actually what drives mysticism (and Heideggarian phenomenology, in a sense). Mysticism is the attempt to escape the world in some way and gain a pure intuition or radically personal experience of being as such. He dismisses this path because of his commitment to transmissible knowledge and rationalism. It is not saintliness he desires, but rather rational knowledge.

Given that the being of any situation is inconsistent multiplicity, and that inconsistent multiplicity is not presented, he will say that being is subtracted from the count. This includes the ontological situation: ontology is not, contrary to much tradition, the presentation of being as such. Rather, ontology is the presentation of presentation. It is the presentation of what comes before the count. Once the mystical route has been abandoned, what is left?

**Paradox and the Axiom of Separation**

Two of the original pioneers of set theory, Georg Cantor and Gottlieb Frege, attempted to develop formulas that would both define and prove the existence of multiplicity. Frege, in particular, thought it was possible to show that a well constructed language could prove that a given multiple, or set, could correspond to certain properties. The hope was to be able to take a free variable such as ∂(a) and prove that it corresponded with ß, or in other words to be able to link existence directly to language.

Bertrand Russell threw a monkey wrench into this with what has come to be known as his Barber’s Paradox. He was able to take a well constructed formula and show a fundamental problem. Let’s say ∂ is a set which is not a multiple of itself: ~(∂∈∂). From that simple statement, we run into a paradox, which we will just call P, for paradoxical: P = {∂ / ~(∂∈∂)} That reads: “all ∂’s such that ∂ is not an element of itself.” If P contains itself, P∈P, then it must have the property which defines its element: ~(P∈P). If P does not contain itself, ~(P∈P), then it has the property which defines its elements; so it is an element of itself: P∈P. Therefore, we get P∈P –> ~(P∈P), if P is an element of itself, then P is not an element of itself.

Any attempt to provide a definition of what a multiple is will run into these sorts of paradoxes. In a nutshell, what he showed was that no well defined language could gather all variables together without incoherency. The alternative is to work without a definition of a multiple/set: instead of saying that the free variable ∂(a) proves the existence of ß by definition, we can only say for any multiple supposed given, there exists the sub-multiple of terms which have the property ∂(a). In other words, we can only infer, not prove, the existence of a free variable on the basis of an assumed multiple. This is what is known as the axiom of separation.

The upshot of the axiom of separation is that math cannot directly prove the existence of a given multiple; it can only assume multiplicity as already given. The axiom of separation is not existential; it does not show something exists. Frege gets ß directly from a; the axiom of separation does not. As Badiou says, “Language cannot induce existence, solely a split within existence” and “Language does not institute the ‘there is,’ only that there are distinctions within the ‘there is.'”

Perhaps now we have a better idea of why ontology is actually mathematics: only an axiomatic system, through rules imposed on pain of contradiction, can prevent itself from explicitly or implicitly defining what a multiple is. It is this lack of a definition of a multiple that prevents it from becoming a sum of ones. Mathematics, unlike normal writing, can reveal a world that is multiplicity all the way down. An axiomatic math system transforms inconsistency into consistency without a one. Inconsistent multiples and consistent multiples are inverted; instead of non-ontological presentation of consistent multiples, there is ontological presentation of inconsistent multiples.

**The Void**

In every situation, there is oneness. A non-ontological situation, that is, every non-mathematical situation, necessarily identifies being with what is presentable, and thus with the possibility of the one. There is only the one, but there is still a slight remainder. The question is, what is the status of inconsistent multiplicity, if the two statements “every situation has oneness” and “inconsistent multiplicity is not one” are both true?

From the point of view of the situation, unpresentable inconsistent multiplicity is nothing. This is not quite the same thing as saying it is non-being. Just like the status of the one hangs between the true statement, “there is oneness” and the false statement “the one is,” the status of inconsistent multiplicity hangs between the true statement “inconsistency is nothing” and the false statement “inconsistency is not.”

In a non-ontological situation, there is no escape from the one. There is no intuition of being. But there is a gap between the one as a result and the one as an operation. We need to name this gap, without counting that name as one. We need an unpresentable figure, which indicates both that every situation is sutured (connected) to being as inconsistent multiplicity and that there is something (nothing) which is subtracted from the count. The void is this suture.

From the point of view of a situation, the void is the name of being, or of inconsistency. There are two key points to be made about the void. First, in order for the void to become “localizable” at the level of presentation, in other words for some assumption of pure being to occur, a “disfunction” of the count is required: this disfunction of the count-as-one is the event, which we spoke about last meeting as the sign that change is possible within a situation. Change, on a fundamental level, takes place because the count-as-one somehow fails in some slight way, allowing the void to short-circuit the rules of a given situation.

Second, given that ontology can have no definition of “multiple” with which to begin weaving its compositions, then it must begin with the void. In fact, ontology can only be a theory of the void; if it presented terms other than the void, if it distinguished between the void and other terms, then it would authorize the count-as-one of the void. if there is no first one, then ontology must be about multiples of nothing. So a problem is solved: if being is presented as multiple, being qua being is neither one nor multiple.

Re: Question 3 and the notion of “decision”

“it begins with a decision, rather than some fundamental proof. Do you think this is a good way for philosophy to procede? Do you think we need to make decisions in order to begin, or must ontology always be about unquestionable first principles?”

We should make distinctions about what the term “decision” means. Badiou is not referring to mere whim, in the sense that somebody simply, unaccountably “feels” some way and acts accordingly. That is, just because philosophy begins in a decision does not mean that decision is without some justification. Rather, it seems that Badiou is pointing to something that might be called, paradoxically, a “compelled decision.” Deciding that language cannot fully represent being (which seems to be the gist of the axiom of separation) is a result of encountering a by-pass that cannot be resolved within the resources available. That is, the axiom seems to almost force itself on the thinker because no other alternative is imaginable.

Thus, we should be careful not to make any decision-logic binary, since the decision grows out of logic; however, it is not exhausted by logic.