We now return to the suggestion made in the Introduction that at sufficiently small length scales the geometry of space-time might be better described by a noncommutative algebra. The physical hypothesis is that geometry based on a set of commuting coordinates is only valid at length scales greater than some fundamental length. At smaller scales it is impossible to localize a point and a new geometry must be used. We can use a solid-state analogy and think of the ordinary Minkowski coordinates as macroscopic order parameters obtained by ‘coarse-graining’ over regions whose size is determined by the fundamental length. They break down and must be replaced by elements of the noncommutative algebra A when one considers phenomena on smaller length scales. If a coherent description could be found for the structure of space-time which were pointless on small length scales, then the ultraviolet divergences of quantum field theory could be eliminated. In fact the elimination of these divergences is equivalent to coarse-graining the structure of space-time over small length scales; if an ultraviolet cut-off ∧ is used then the theory does not see length scales smaller than ∧-1. When a physicist calculates a Feynman diagram he is forced to place a cut-off ∧ on the momentum variables in the integrands. This means that he renounces any interest in regions of space-time of volume less than ∧-4. As ∧ becomes larger and larger the forbidden region becomes smaller and smaller but it can never be made to vanish.