All the real knowledge which we possess depends on
methods by which we distinguish the similar from the
dissimilar. The greater the number of natural distinctions
this method comprehends the clearer becomes our idea of
things. The more numerous the objects which employ our
attention the more difficult it becomes to form such a
method, and the more necessary.
-- Carolus Linnaeus, Genera Plantarum, 1737
We are exploring a new algebraic theory of knowledge representation based
on the isomorphism of "dimension" and "ordered class". A "synthetic dimension" is defined recursively, as an ordered class of variables, themselves described in synthetic dimensions. All abstract classes can be assembled from lowest-level empirical measurements through this method
All empirical measurement is defined by values in dimensions. All abstract classes are defined by composite assemblies of dimensions, and an ordered class can be shown to be algebraically isomorphic to a dimension. Thus, any high-level composite abstraction can be determinately defined bottom-up from lowest-level empirical/dimensional measurements. This approach creates a linearly recursive cascade across ascending levels of abstraction, through which any concept, category, classification, or data structure can be exactly defined, to any desired degree of accuracy/error tolerance (number of decimal places in measurement).
Synthetic Dimensionality is, at the same time, a model of the
foundations of mathematics (real number line), an intuitive theory of
natural language semantics, an algebraic theory of classification, and a general theory of all conceptual structure. Any idea, concept, mental model, or "information structure" can be constructed in perfect detail with synthetic dimensions.