Smooth Singularities Exposed: Chimeras of the Differential Spacetime   Manifold
  The smooth gravitational singularities of the differential spacetime manifold based General Relativity (GR) are viewed from the perspective of the background manifold independent and, in extenso, Calculus-free Abstract Differential Geometry (ADG). In particular, the inner Schwarzschild singularity is being `resolved' ADG-theoretically in two different ways. A plethora of important mathematical, physical and philosophical issues in current classical and quantum gravity research are addressed and tackled. 
Anastasios Mallios and Ioannis Raptis