Double Dualization: Functions on vs. Figures in

Double Dualization: Functions on vs. Figures in

[Venkata Rayudu Posina]

Dear All,

The constructs of GENERALIZED POINT (Sets for Mathematics, p. 150) and
CONCRETE GENERAL (in the context of Functorial Semantics) are similar:
(i) both are encountered in the course of getting to know a given
object / graph / category; (ii) both begin with measurements
(functions on [the given object] as opposed to figures in; Conceptual
Mathematics, pp. 82-83); and (iii) both involve a two-step process
i.e. double dualization.  In light of these similarities, what exactly
is the relation between generalized points

A --> V

(where A is a set of maps B --> V) and concrete generals

A --> V

(where A is a category of functors B --> V)?  In other words, I'd
appreciate any pointers to literature that explicitly brings
functorial semantics to bear on physics (e.g. center of mass; Sets for
Mathematics, p. 101).  On a related note, one can get to know a given
B by way of figures in B, instead of the above functions on B.  Does
the figures-and-incidence (Conceptual Mathematics, pp. 249-253)
approach to knowing also involves two steps (like double dualization)?
  Can we think of modelling, for example, an irreflexive directed graph
G as a parallel pair of functions

source, target: Arrows --> Dots

by way of taking points of map objects

1 --> C

(where C is a map object of Dot- or Arrow-shaped figures T --> G in
the given graph G; Conceptual Mathematics, p. 150) as analogous to
double dualization (albeit in the opposite direction)?

Thank you,
posina
namingthegiven.wordpress.com


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