9 Chapter 1:The Mathematics of the Virtual: Manifolds, Vector fields and Transformation Groups
Of all the concepts which populate the work of Gilles Deleuze there is one which stands out for its lonevity: the concept of multiplicity. This concept makes its appearance in his early books and remains one of central importance, with almost an unchanged meaning and function, until his final work. Its formal definition is highly technical, including elements from several different branches of mathematics: differential geometry, group theory and dynamical systems.
In this chapter 1 I will discuss the technical background needed to define this important concept but some preliminary informal remarks will prove helpful in setting the stage for the formal discussion.
In the first place, one may ask what role the concept of a multiplicity is supposed to play and the answer would be a replacement for the much older philosophical concept of an essence. The essence of a thing is that which explains its identity, that is, those fundamental traits without which an object would not be what it is.
If such an essence is shared by many objects, then possession of a common essence would also explain the fact that these objects resemble each other land, indeed, that they form a distinct natural kind of things.