What ‘ shape ’ is space - time ?

Some examples from the mathematics of shape are presented that question some of the almost hidden assumptions behind results on limiting behaviour of finitary approximations to space-time. These are presented so as to focus attention on the observational problem of refinement and suggest the necessity for an alternative theory of 'fractafolds' against which the limiting theory of C ∞-differential manifolds usually underlying (quantum) general relativity can be measured. Even from the start, the space-time 'manifold' was considered 'unphysical'. It involved numerous powerful mathematical concepts that were inherently beyond observation, although providing apparently essential tools for developing the physical theory. More recently this 'unphysicality' has stimulated attempts to use an 'observational' approach to model the differential, dynamic aspects of space-time (and eventually to quantise it) using discrete, algebraic or combinatorial models. One of the finitary approaches to discrete space-time was pioneered by Sorkin, [17]. This approach assumes space-time is modelled by a manifold, M , then assumes an open cover F of M is given. This F is thought of as corresponding to the set of observations being considered. If U ∈ F, then the events within U are thought of as being operationally indistiguishable by that observation. Of course, if x ∈ U and y / ∈ U , then the observation distinguishes x and y. If one considers the equivalence relation corresponding to 'operational indistinguishability' relative to F, the result is a T 0-space which is a 'finitary substitute for M ' with respect to the covering F. As has been remarked on elsewhere, e.g. in the papers of Raptis and Zapatrin, [15, 16], this construction of Sorkin is closely related to that of the nerve of the open cover F, a construction from algebraic topology usually attributed toČech, [5]. This yields a simpli-cial complex that will be denoted N (M, F) or N (F), if M is understood. The T 0 space