Continuing with the summary of the video, from 31m to 54m.
Badiou has just identified the Absolute with V, i.e. the class of all sets (possible forms of the multiple), which is not itself a set (as then it would have to be a member of itself).
Badiou remarks on the peculiar aptness of certain mathematical symbols, such as V, as if chosen with an unconscious philosophical intelligence. The choice of the symbol « V » for the class of all sets opens on to the acknowledgement that the only veritable Absolute is the void, « the place of all places ».
The Place posited as the place of all the forms of the multiple corresponds to substance in Spinoza’s system, the sets (the possible forms of the multiple) would correspond to the modes (particular things), and one is entitled to think that certain classes correspond to the attributes.
So we would have
1) The Absolute, Substance – the general Place that allows us to explore set theory as the theory of the forms of the multiple
2) Modes -the particular forms of the multiple that we manage to know and to explore, and which are also the forms of the multiple of empirical objects
We may ask ourselves how mathematics can apply to empirical objects, one has only to see that mathematics is the general theory of all the possible forms of multiplicity. So it is only to be expected that it will come up in the study of particular multiplicities.
In studying a particular multiplicity one begins by fixing the possible form of multiplicity and so convoking the Absolute in the thought of a singular, modal, figure of existence.
The problem arises of what corresponds to the attributes. An attribute cannot be a set, for then it would be a mode, a singular thing. It cannot be the Place, the Absolute, or the class of all sets, for this corresponds to Substance. An attribute must be a class that is not a set. If we try to think the attribute as something that expresses Substance without being identical to it, then we can clarify what a class is.
3) Attributes – an attribute is a class belonging to V that without being a set and without being V has properties very similar to V’s properties, and so « expresses » V, i.e. a class in V that expresses the most important immanent possibilities of V.
Take the expression x V.
This is strictly an incorrectly formed expression, as can be affirmed only between sets, and V is not a set, but a class. It can be read philosophically as
« x is a possible form of the multiple, identifiable as a set, and from this point of view figures in V, the total collection of all the possible forms of the multiple »
If we want a Class C similar to V it will be composed of sets, and will itself be a part, or subclass of V: C V.
Given the definition the question is can there exist something like C, a class contained in V, composed of sets of V, that would be similar to V. What could be the conditions of such a resemblance? The resemblance cannot be quantitative, because C is a piece of V
A set x will have certain specific properties that distinguish it from other sets (finite, infinite, well-ordered or not, etc.)
Let φ (x) mean x has the property φ.
C can be said to « resemble » V if whenever a set x in C has a property φ in V it has a corresponding property in C. In this case we may say that C expresses V. The property delegated by the Absolute can be found in the attribute, and the attribute will in that sense « express » the Absolute, in Spinoza’s sense.
Between C and V there must exist a certain relation. It is not enough to say that C is in V. Similarly in Spinoza to distinguish two attributes it is not enough to say that both are « in » Substance. The infinite multiplicity of the attributes requires that the representation of the Absolute by the attribute be also a transfer of the characteristics of the object under discussion from Substance to the attribute, internal to Substance itself.
For C to resemble V, between V and a subclass C there must exist a correlation that is stronger than the subclass relation, which is too weak a relation to differentiate between attributes. Let us call this correlation j (called in mathematics an « embedding » of V in C, an embedding of the Absolute in the attribute). This embedding is « elementary » because it correlates elements of V with elements of C.
We can now specify our correlation: if we have φ (x) in V then we have φ [j (x)] in C. Thus the correlation of C and V will be not only quantitative (C V) but also truth-preserving as what is true in V, φ (x), is projected as true in C, φ [j (x)].
This definition of attribute is interesting only if the correlation j is non-trivial, i.e. if it does not correspond to a simple relation of identity, where we would have j (x) = x for all x element of V. We need that at least for one point a we have j (a) ≠ a.
We will then be able to differentiate attributes from each other by means of the system of their differences from the absolute, given that the truths will be preserved.
The attribute amounts to the introduction of a principle of novelty and differentiation, as the attribute introduces novelty immanent to Substance at the same time as it expresses Substance. This is the dialectic of difference as supreme form of identity. It is because the attribute is different that it is particularly expressive of the absolute identity of the thing.
To sum up, we have the formal table of the attribute:
V – the Absolute, general form accessible to mathematical thought of the class of configurations of the multiple
C V – C a class internal to V
x V ↔ φ (x) – x an element of V defined by the property φ
j – the correlating relation, or embedding
j (x) C – the j-correlate of x in C, such that φ [j (x)]
j (x) ≠ x, for some x – j is non-trivial (it is not equivalent to the relation of equality)
C is an attribute if all these conditions are satisfied, i.e.
C is interior to Substance, it is immanent to Substance yet quantitatively different from it (it does not cover all of it), and it is in expressive truth-preserving correlation without it preserving truth by repetition but truth preserved in difference by a systemic operation of correlation of the two levels.
In conclusion, we have reached the following definition of an attribute:
an attribute of the Absolute is a class of this Absolute such that between the Absolute and this class there exists a non-trivial elementary embedding.