[FOM] category theory as the analysis of symmetry

[Paul Hollander]

[NOTE:  Includes corrections.]

Recent FOM discussion is evidence that category theory has a 
foundational role to play in mathematics.

For example, transformational grammar represents a sentence S in terms 
of conjoining a noun phrase NP and a verb phrase VP, i.e. (NP + VP) = S. 
  Here, (NP + VP) expresses a symmetric relation R such that R(NP, VP) 
mutually implies R(VP, NP).

The * in group theory is another example.

But another way to represent the fact that (NP + VP) = S is as NP --> VP 
--> S, which is distinct from VP --> NP --> S, as grammarians well know.

The difference is that the arrows --> express both inheritance and 
transformation, but not the symmetric representation (VP + NP) = S.

The arrow --> is DIACHRONIC, while symmetric operators like + and * are 
SYNCHRONIC.

Therefore, category theory is the general analytical tool or organon for 
interpreting symmetric relations like (NP + VP) into distinct maps (NP 
--> VP) and (VP --> NP), even when the analysis is syncategorematic, as 
with transformational grammar.

Using category theory, a diagram like the square of opposition becomes a 
logic of implication, contradiction and contrariety for A,E,I,O 
expressions, like Aristotle's syllogistic.

Cheers,

Paul Hollander