[FOM] geometry and mathematical logic
Jon Awbrey
jawbrey at att.net
Mon Aug 10 10:42:07 EDT 2009
Re: http://www.cs.nyu.edu/pipermail/fom/2009-August/013896.html
C.S. Peirce explored a variety of De Morgan type dualities in logic
that he treated on analogy with the dualities in projective geometry.
This gave rise to abstract formal systems where the initial constants
-- and later on their geometric or graph-theoretic representations --
had no fixed meaning but could be given dual interpretations in logic.
It was in this context that his systems of logical graphs developed,
issuing in dual interpretations of the same formal axioms that Peirce
referred to as "entitative graphs" and "existential graphs". It was
only the existential interpretation that he developed very far, since
the extension from propositional to relational calculus seemed easier
to visualize there, but whether there is some truly logical reason for
the symmetry to break at that point is not yet known to me.
When I have explored how Peirce's way of doing things might be extended
to "differential logic" I have run into many themes that are analogous
to differential geometry over GF(2). Naturally, there are many surprises.
Jon Awbrey
cc: Arisbe, FOM, Inquiry
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