  A `resolution' of the interior singularity of the spherically symmetric
Schwarzschild solution of the Einstein equations for the gravitational field of
a point-particle is carried out entirely and solely by finitistic and algebraic
means. To this end, the background differential spacetime manifold and
Calculus-free purely algebraic (:sheaf-theoretic) conceptual and technical
machinery of Abstract Differential Geometry (ADG) is employed via Sorkin's
finitary (:locally finite) poset substitutes of continuous manifolds in their
Gel'fand-dual picture in terms of discrete differential incidence algebras and
the finitary spacetime sheaves thereof. It is shown that the Einstein equations
hold not only at the finitary poset level of `discrete events', but also at a
suitable `classical continuum limit' of the said finitary sheaves and the
associated differential triads that they define ADG-theoretically. We infer
that the law of gravity does not break down in any (differential geometric)
sense in the vicinity of the locus of the point-mass as the usual manifold
based analysis of spacetime singularities in General Relativity has hitherto
maintained. Various possible implications that such a total evasion of smooth
gravitational singularities, as well as some anticipations of the wider
significance that the general ADG-framework, may have for certain current `hot'
issues in both classical and quantum gravity research are briefly discussed at
the end.
