Mathematical Primitives in Point-Free Topology

The mathematical equivalent of the Madhyamaka (and especially Gelukpa) view of svalaksana (“the real”) requires an alternative to point-sets as the mathematical basis for space: these are known as “point-free” topologies. Mathematically, these theories avoid several paradoxes associated with point sets (which involve the distinction between open/closed intervals). We nominate the following tenets for a psychologically-valid, mathematical theory of conceptual and non-conceptual mind:

• Space is represented as an open-dimensional, or multi-dimensional, manifold.
• Objects are demarcated by boundaries. A boundary is an imputed object which has zero width (it is a thing of dimensionality N-1 in an N-dimensional space).
• The division of space into objects is not set-like: it has the relationship of parts to wholes, and is transitive. The complete definition of an object entails both its larger context and its boundary (a separating hyperplane or hyperspace).
• The boundary of an object may be characterized by a function.  That function is not necessarily characterized intrinsically (i.e. by the contents of that object), since it can also be specified extrinsically.
• Objects, whether they are regarded as real or merely perceptual, serve as the elements of sets (or the ultimate contents of concepts).
• Sets may be labeled, and collected into higher-rank sets.