On 10/22/2011 10:44 PM, Russell Standish wrote:
On Fri, Oct 21, 2011 at 02:01:40AM -0400, Stephen P. King wrote:
The Stone duality was first found as an isomorphism between
Boolean algebras and totaly disconnected compact Hausdorff spaces.
Generalizations are being studied. Consider what these topological
spaces "look" like... What does a Cantor set look like, for example?
The idea is to shift from thinking of algebras and spaces as purely
static and consider them as evolving systems, ala Hintikka's game
theoretic semantics for proof theory. The idea that I am studying
was first proposed by Vaughan Pratt using Chu spaces. See:
Maybe I should take a look. The trouble is it'll require some study,
and I'm rather time poor, at present :).
Its a pity Bruno hasn't had more time to look into it, as it seems a
closer match for his ontology...
Aye, I wish you would have some time to look into it. I found
Pratt's Chu space construction to be equivalent (in a limited sense) to
the "topological system" discussed in Steve Vickers' book, Topology via
Logic. Vickers' discussion of "Continuous maps" is another form, albeit
a bit more simple, of the "dynamics" that I am considering. Vickers, as
far as I have studied, only seems to consider his own version of Chu_2,
which is the 2-valued logic version of the Chu construction.
Bruno seems to want a purely ideal monist ontology at the primitive
level. I am assuming something more like Bertrand Russell's neutral
monism at the primitive level and a "vanishing in the limit" dualism
that supervenes on it. Like a dual aspect dualism + property bundle
theory on a neutral monism. Bruno's ideas would map into and onto the
"abstract algebras" side of the duality, thus if Bruno's result is false
then so to is the idea that I am exploring as it seems to be a
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