Continuing with the summary of the video, from 22m to 31m.
Spinoza attempts to think this immanent infinity in the figure of the One Badiou will do it in terms of the thought of the multiple. Badiou maintains that this does not constitute a decisive difference.
Let us begin with the idea that « the being of any thing is to be a multiplicity, i.e. the One is not, there are only multiplicities ». However, we must be more precise than that, as we do not just have « multiplicities in general » but diverse « forms of the multiple » (there exist a whole range: finite, infinite, different sorts of infinite, with different relations between elements, etc.).
« We shall define the Absolute as the place of thought where are deployed the totality of the possible forms of the multiple ».
In militant politics today it is often question of « forms of life ». These forms of life are in fact possible forms of the immanent multiplicity that constitutes every life. So one could make use of this ontology to for an effective thought of the demand for new forms of life. This thought would be absolute in that it would participate in the existence of the totality of forms of life as power of life.
Returning to Spinoza’s terms, we can say that for our intellect mathematics is our mode of access to the Absolute. In the precise sense that mathematics is (for our intellect) the rational mode of thought of the totality (for us) of the possible forms of the multiple.
This place of the thought of the totality of the possible forms of the multiple is not itself a possible form of the multiple, it does not itself have the form of the multiple, and so it is subtracted (note: one could say « withdrawn ») from the access we have to the forms of the multiple.
In other words the Absolute is not immediately « inside » itself. This is not a question of reflexivity but of interiority. Mathematics gives you access to the place for thinking the forms of the multiple, but the place itself is withdrawn from that access, as the place is not inside the place, it is not an element of itself.
This leads us to distinguish sets, which are the possible forms of any existing multiplicity, and classes, which are families of sets, which are generally assembled by an explicit property, but which are not normed as possible forms.
This distinction opens up the possibility of affirming of certain classes that they assemble possible forms of the multiple in conditions that would allow us to say that they are in a certain sense attributes of the Absolute, that they express the place of thought of the forms of the multiple in a particular manner, i.e. inside a specific property.
In the same way for Spinoza, extension and the understanding are « classes », one could almost say, of the Absolute, they are differentiated figures of the Absolute in that the elements of the understanding are not the same as the elements of extension, although the structure is the same, although all that expresses a possible mode of access to the Absolute.
As usual mathematicians have made an interesting choice of symbol to designate the general class, i.e. the Absolute: V, the void, the vacuum.