Duality Indicating Unity

Peircers,

“Dualities Indicating Unities” is how I'll dub another of one
of those Peircean themes that a Peirce-attuned ear might hear
“in the air” of late.  As it happens, it strikes a chord with
the more pervasive theme of “Triadic Relation Irreducibility”,
as I hope to see an occasion to develop further in the future.

Re: Richard J. Lipton
At: http://rjlipton.wordpress.com/2013/01/26/making-primes-more-random/

A formal duality points to a higher unity — a calculus of forms
whose expressions can be read in two different ways by switching
the meanings assigned to a pair of primitive terms.

I just ran across an old post of mine on the FOM List where
I touched on this theme, so I think I’ll copy that here until
I get a chance and the concentration to comment further.

Re: Rupert McCallum
At: 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 consequently their geometric or graph-theoretic representations —
had no uniquely fixed meanings 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.

Regards,

Jon

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