On 10/21/2011 1:05 AM, Russell Standish wrote:
On Thu, Oct 20, 2011 at 08:00:55PM -0400, Stephen P. King wrote:
There has to be some form of identity thesis between brain and mind
that prevents the Occam catastrophe, and also prevent the full retreat
into solipsism. I think it very much an open problem what that is.
Hi Russell,
Would the conjecture that the Stone duality provide a coherent
version of this "identity thesis"? Minds, as per Comp, -> logical
algebras and Brains -> topological spaces. Not not, how so?
Onward!
Stephen
I have to confess to not having the slightest inkling what you're
saying here. I did briefly look at Stone duality on Wikipedia, but it
didn't help much. I assume that you're interested in some duality between
an algebra (perhaps one of Bruno's hypostases, if they're an algebra)
and a topological space that could stand in for physical reality, but
beyond that I'm totally lost :).
Hi Russell,
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:
http://boole.stanford.edu/pub/ratmech.pdf
If Bruno's UD is a logical algebra, then it would have a Stone
space as its dual. If the UD evolves, then so too does its Stone space.
This implies a nice identity thesis and avoids the Occam catastrophe
because of compactness. BTW, compactness requires a topological form of
finiteness, thus the measure problem is also solved. There are still
some open problems, such as the degenerasy into solipsistic systems,
that need to be addressed. I suspect that Tennenbaum's theorem might be
a place to start.
Onward!
Stephen
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