Discrete Spatial Models

1. Introduction By " discreteness " of a spatial model we generally understand that in any bounded neighbourhood, or (bounded) region, there are only finitely many elements of the carrier of the model. Thus the model should be (at least) locally finite. Another way of putting it is that the bounded regions should not be infinitely subdivisible. As is well known, Aristotle argued (Physics: III,6; VI,2) that space and time intervals are potentially infinitely subdivisible. In 20th century mathematics, a similar position was adopted by Brouwer, in opposition to the actual infinities of Cantorian set theory. In proposing discrete spatial models, we embrace the possibility that space is (locally) neither actually not potentially infinite, but actually finite. We mention three streams of motivation for adopting such a point of view. The first of these is found in the " tolerance space " tradition, going back to Poincaré, 1905. The idea here is that any perceptual (as opposed to mathematical or physical) " continuum " must be finite and,