  The purpose of this paper is to provide a new account of multiplicity for
finite morphisms between smooth projective varieties. Traditionally, this has
been defined using commutative algebra in terms of the length of integral ring
extensions. In model theory, a different approch to multiplicity was developed
by Zilber using the techniques of non-standard analysis. Here, we first
reformulate Zilber's method in the language of algebraic geometry and secondly
show that, in classical projective situations, the two notions essentially
coincide. As a consequence, we can recover intersection theory in all
characteristics from the non-standard method and sketch further developments in
connection with etale cohomology and deformation theory. The usefulness of this
approach can be seen from the increasing interplay between Zariski structures
and objects of non-commutative geometry.
