  The essence of the method of physics is inseparably connected with the
problem of interplay between local and global properties of the universe. In
the present paper we discuss this interplay as it is present in three major
departments of contemporary physics: general relativity, quantum mechanics and
some attempts at quantizing gravity (especially geometrodynamics and its recent
successors in the form of various pregeometry conceptions). It turns out that
all big interpretative issues involved in this problem point towards the
necessity of changing from the standard space-time geometry to some radically
new, most probably non-local, generalization. We argue that the recent
noncommutative geometry offers attractive possibilities, and gives us a
conceptual insight into its algebraic foundations. Noncommutative spaces are,
in general, non-local, and their applications to physics, known at present,
seem very promising. One would expect that beneath the Planck threshold there
reigns a ``noncommutative pregeometry'', and only when crossing this threshold
the usual space-time geometry emerges.
