Computable scalar fields: A basis for PDE software

Given computable scalar fields, the operations on tensor fields will also be computable. As a consequence we get computable solvers for PDEs. The traditional numerical methods for achieving computability by various approximation techniques (e.g., finite difference, finite element or finite volume methods), all have artifacts in the form of numerical inaccuracies and various forms of noise in the solutions. We hope these observations will inspire the development of a theory for computable scalar fields, which either lets us understand why these artefacts are inherent, or provides us with better tools for constructing these basic building blocks.