Shape Theory. III. Comparative Theory of Backgound Independence (1812.08771v3)

Abstract: Background Independence is the modern form of the relational side of the Absolute versus Relational Debate. Difficulties with its implementation form the Problem of Time. Its 9 facets - Isham and Kucha\v{r}'s conceptual classification - correspond to 9 aspects of Background Independence as per the Author's classical-or-quantum theory-independent upgrade. 8 are local. The most significant arena for these is brackets algebra, the first 5 involving canonical constraints as follows. 1) Handling instantaneous gauge invariance. 2) Resolving its apparent timelessness. 3) Closure of 1-2)'s constraints. 4) Expression in terms of observables: commutants with constraints. 5) Reconstructing spacetime from constraint algebra rigidity. 6) is the spacetime counterpart of 2-4), and 7) is spacetime's foliation independence. 8) handles nonuniqueness. 9) renders 1-8) globally sound. We show how Shape(-and-Scale) Theory's mastery of 1) for N-point-particle models extends by placing a mechanics over shape(-and-scale) space to model 1-4). For flat-space Euclidean and similarity models, this gives a local resolution of the Problem of Time. This is moreover consistent within a global treatment if its reduced spaces are Hausdorff paracompact, which admit a Shrinking Lemma. GR's superspace is also Hausdorff paracompact. While 1-9) are poseable for all relativistic theories, and 1-4), 8), 9) for all theories - to all levels of mathematical structure: affine, projective, conformal, topological manifold, topological space... - resolution is on a case-by-case basis. Among N-point-particle theories, then, Article II's Hausdorff paracompact reduced space guarantee selects a very small subset. In particular, affine and projective shapes are precluded and conceiving in terms of shapes in space is preferred over doing so in spacetime. A substantial Selection Principle for Comparative Background Independence is thus born.