Abstract

We organize things and places in our world according to spatial relations such as ...is next to..., ...is close to..., ...is above..., ... is to the right of..., ...is twice as far from ...as from..., and ...is three feet from.... These concepts range from the qualitative, imprecise, and context-dependent to the quantitative, precise, and context-independent . They are logically inter-related in the following sense. If we know that a pair of objects satisfies one of these relations, then we can infer that the pair satisfies or fails to satisfy certain other relations. For example, if we know that a wall is fifteen feet from a house and that the house is closer to a certain tree than it is to the wall, then we can infer that the house is less than fifteen feet from the tree. If we know, in addition, that both the tree and the wall are directly behind the house, then we can infer that the tree is in between the house and the wall. ;It is tempting to assume that the inference structure of common-sense spatial reasoning can be analyzed within a classical mathematical theory, such as algebraic geometry. However, classical mathematical theories assume that we have already identified a domain of zero-dimensional points and perhaps other mathematical objects, such as lines and planes. I argue that the mathematical concept of a point demands some spatial sophistication and that quite a lot common-sense reasoning can take place without introducing the concept of a dimensionless point. The purpose of this dissertation is to construct an axiomatic theory of spatial concepts that are applied directly, not to points or sets of points, but to the three-dimensional spatial extents occupied by material objects. I use this theory to define analogues of common-sense spatial concepts in a way that preserves the inference structure of common-sense spatial reasoning