Granular mathematics foundation and current state

The keynote contains two parts, one is the formal theory, the other is the current state. Though we have used the full title, this paper contains only the formal theory. Neighborhood system generalizes topological neighborhood system by simply dropping the axioms of topology. In this paper, we use it to define Zadeh's Granular Mathematics (GrM). The main results are: 1) GrM, though defined locally, can be axiomatized by global concepts. This axiomatization is significant, it has completed what Sierpinski had done partially in his book (1952). 2) GrM is a stable concept in the sense that a fuzzification of GrM is, up to the labels α, just another GrM. 3) GrM unifies rather trivially pre-/topological spaces and generalized rough sets, even the variable precision rough sets (though not so obviously).