  A fourth source of concern relates to the intuition, implicit in Aristotle's definition, that boundaries are lower-dimensional entities, i.e., have at least one dimension fewer than the entities they bound.  The surface of a (continuous) sphere, for example, is two-dimensional (it has no “substance” or “divisible bulk”), the Mason-Dixon Line is one-dimensional (it has “length” but no “breadth”), and a boundary point such as the vertex of a cone is zero-dimensional (it extends in no direction). This intuition is germane to much of what we ordinarily say about boundaries. But it is problematic insofar as it contrasts with several independent intuitions that are of a piece with both common sense and philosophical theorizing. For instance, there is a standing tradition in epistemology (from Moore 1925 to Gibson 1979) according to which boundaries play a crucial role in perception: we see (opaque) physical objects indirectly by seeing their surfaces. Yet it is not clear how one could see entities that lack physical bulk.  Likewise, we often speak of surfaces as of things that may be pitted, or damp, or that can be scratched, polished, sanded, and so on, and it is unclear whether such predicates can be applied at all to immaterial entities. In such cases, it would rather seem that surfaces (and boundaries more generally; see Jackendoff 1991) are to be construed as “thin layers” that are schematized as having fewer dimensions than the wholes to which they apply.