  Noncommutative geometries generalize standard smooth geometries,
parametrizing the noncommutativity of dimensions with a fundamental quantity
with the dimensions of area. The question arises then of whether the concept of
a region smaller than the scale - and ultimately the concept of a point - makes
sense in such a theory. We argue that it does not, in two interrelated ways. In
the context of Connes' spectral triple approach, we show that arbitrarily small
regions are not definable in the formal sense. While in the scalar field
Moyal-Weyl approach, we show that they cannot be given an operational
definition. We conclude that points do not exist in such geometries. We
therefore investigate (a) the metaphysics of such a geometry, and (b) how the
appearance of smooth manifold might be recovered as an approximation to a
fundamental noncommutative geometry.
