In this series of posts, I’m going to have a go at “motivating” the mathematical theory of the object that Badiou develops in Book III of Logics of Worlds, and relating it to some other currents in “object-oriented” thinking. In my view, Badiou does a strangely poor job of explaining what all of the mathematical machinery in Book III is actually for: what good it might do to think of objects in the way he proposes, what the philosophical import of these proposals might be. So I’m going to try to supply some answers of my own to these questions, although of course they may not be the same answers as those Badiou himself would give.

So far as the mathematical motivation is concerned, I’m not able to give an account of the development of the theory Badiou draws on, or of the particular kinds of problems it was framed to investigate. What I want to do instead is to “remotivate” it from the outside, to bring my own (philosophical) problems to it and see what it says about those. The outcome may well strike any passing mathematicians as bizarre, and I’m obliged at the outset to issue a very firm caveat lector concerning the likelihood of errors. For the most part, though, I’m going to steer away from a kind of technical mathematical demonstration for which I have little aptitude, and concentrate on getting some of the basic concepts clear.

To begin with, then, I’d like to talk about an “object” as a network of entities, something with an internal structure that can be diagrammed. Are the entities in the network themselves objects? Well, perhaps; but for present purposes, we’re only interested in their participation in the diagram of some other object’s interior, in which capacity they appear only as “elements” or as the “support” of the network that is strung out between them. We can draw this network as a graph of connected points, for example:

figure 1

A few notes on this figure:

• Any two “elements” (the labelled circles) are either connected or not connected. (Later we will consider greater and lesser degrees of connectedness). As a shorthand, for the moment we’ll write “a is connected to b” as a ∼ b.
• Connections are bi-directional: if a is connected to b, then b is connected to a. We can express this as a rule of symmetry: a ∼ b -> b ∼ a.
• Not all connections are shown. There are some “implicit” connections, which can be inferred to exist by following some simple rules that will be discussed in a moment.
• Of the two elements that are unconnected to any other elements, one (h) is connected to itself and the other (i) is not.
• Within this graph there are four distinct parts – groups of connected elements. (In fact, i, which is not even connected to itself, does not really form a part in the same sense as the others).

In addition to the connections explicitly shown in this graph, there are some “implicit” connections. First of all, if there is a path between any two elements, then those elements are implicitly connected: if a ∼ b, and b ∼ c, then a ∼ c (this is the rule of transitivity). Secondly, if an element is connected to any other element at all, then it must be connected to itself. This is the rule of reflexivity, and it follows directly from the rules of symmetry and transitivity:

• Rule of symmetry: if a ∼ b, then b ∼ a.
• Rule of transitivity: if a ∼ b, and b ∼ c, then a ∼ c.
• Rule of reflexivity: if a ∼ b, then by the rule of symmetry we have the path a ∼ b ∼ a, and by the rule of transitivity we then have a ∼ a.

These rules can also be used to define a particular type of order relation: an equivalence relation. Given a set of elements {a, b, c…}, an equivalence relation partitions the set into subsets like the distinct groups of connected elements in the figure above: all of the elements in the same subset are equivalent to all of the other elements in that subset. In an equivalence relation, the rule of reflexivity is unconditional: for all a in the relation, a ∼ a. An element that is not connected to itself cannot participate in the relation at all: it is inexistent from the point of view of the relation.

One way to understand this is to see the equivalence relation itself as a collection of statements of the type “a is equivalent to b“. Whenever such a statement is made, it immediately follows by the rules of symmetry and transitivity that a ∼ a and that b ∼ b: there never appears in the relation any a for which the rule of reflexivity does not hold true. In the graph above, on the other hand, we have an element i that is not connected to itself (or to anything else). It belongs to no part of the object (that is, to no group of connected elements). It’s just there.

An equivalence relation can be represented as a set of ordered pairs {(a, b), (c, d)…}, where the presence of each pair (a, b) means that a ∼ b. There is no way in this representation to show an element i that is not connected to anything. We therefore need a different representation for the graph in figure 1, one which holds information about the collection of elements {a, b, c…} separately from information about which elements are connected to each other.

One way to do this is to use the notation Badiou uses for an object, (A, Id), where A is the set of elements {a, b, c…}, and Id is a function A⊗A → Ω which takes each pair (a, b) of elements in A to a value in another set, Ω. In the case that Ω is the set {0, 1}, the function Id splits A⊗A into precisely two sets of pairs of elements of A. Those where Id(a, b) = 1 are connected; those where Id(a, b) = 0 are not connected. The equivalence relation containing all of the connected elements in the graph is the preimage (under Id) of {1} in A⊗A. But the function Id can also assign the value 0 to some (a, a) without violating the rule of reflexivity in this equivalence relation, provided that Id(a, b) = 0 for all b. It thus has the power, which the equivalence relation lacks, of naming the inexistent.

We’re now coming quite close to where Badiou’s account of this “object”, (A, Id), begins. As you can see, with Ω (the codomain of Id) = {0, 1}, we are already able to partition an object into connected parts, and to say which of its elements are connected to which of its other elements. Things really start to get interesting, however, when Ω is some larger, more structured set. In subsequent posts, I’ll talk about what kind of mathematical object Ω might be, and what consequences this has for our view of the object as “internally” a network of connected entities.

This entry was posted on Friday, July 16th, 2010 at 1:33 pm and is filed under General. You can follow any responses to this entry through the RSS 2.0 feed. You can skip to the end and leave a response. Pinging is currently not allowed.