Announcement: Chris Langan has published a new paper, "An Introduction to Mathematical Metaphysics", in the journal Cosmos and History.

Topological-descriptive duality

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"Because states express topologically while the syntactic structures of their underlying operators express descriptively, attributive duality is sometimes called state-syntax duality. As information requires syntactic organization, it amounts to a valuation of cognitive/perceptual syntax; conversely, recognition consists of a subtractive restriction of informational potential through an additive acquisition of information. TD duality thus relates information to the informational potential bounded by syntax, and perception (cognitive state acquisition) to cognition.

In a Venn diagram, the contents of circles reflect the structure of their boundaries; the boundaries are the primary descriptors. The interior of a circle is simply an “interiorization” or self-distribution of its syntactic “boundary constraint”. Thus, nested circles corresponding to identical objects display a descriptive form of containment corresponding to syntactic layering, with underlying levels corresponding to syntactic coverings.

This leads to a related form of duality, constructive-filtrative duality." Langan, 2002, PCID pg. 26

Description means to take all relevant characteristics and qualities for describing an object. In the CTMU, "description" and "descriptive inclusion" are understood as synonymous. As for descriptive inclusion, we understand it as a descriptive function acting as syntax for the related topology. (Remember that "syntax" and "state" are relative attributes.) A topology serves as the dual or opposite of its corresponding description. The concept of a set is then related as the topology. In a topology, things are "unisected" to form the medium it is, a set. A "unisection" is then related as a syntactic product of differences.

Essentially, a topology is a collection of objects "extending in the differential parameter". The objects within a topology are therefore different from one another. Culminating in a greater object, unisection relates things "ecto-syntactically". Also, in terms of duality, the "outside" and the "inside" are known as the topology and the descriptor respectively. Generalizing any duality to the interior or exterior of it results in many corresponding forms of duality. For example: subject:object, in:out, stabilizer:user, enabler:designer, emotion:logic, etc..