Hi Richard!
Wonderful! Another pair of eyes looking at Pratt's work. This is
progress! There are a couple open problems, such as how to model large
networks of bisimulations but from my toy model study I think I have a
solution to that one. The only technical problems are the formulation of
a tensor product rule for arbitrary Monads (whose bodies/minds are the
logical algebra and topological space "couples" that Pratt models using
Chu_k spaces) and the "forgetful" version of residuation. I have some
ideas on those too...
By the way, the entire question of particles/strings/etc. is
reduced to a phenomenology/epistemology question that can be addressed
using computational simulation modeling and considerations of
observational bases. We only need to recover/derive the data not the
"stuff". The mereology of monads would follow the entanglement scheme of
QM (for Chu_k ; k = complex number field) and allow us to use the
pseudo-telepathy idea from quantum game theory to model bisimulation
networks in a different basis. What I like about this the most is that
it offers a completely new paradigm for investigations into physics and
philosophy. See http://boole.stanford.edu/pub/ph94.pdf for even more
discussions.
On 8/22/2012 4:04 PM, Richard Ruquist wrote:
Stephan,
Many thanks for this wonderful paper by Vaugh Pratt
http://boole.stanford.edu/pub/ratmech.pdf
Pratt theory appears to replace Godellian theory.
But Godellian theory manifests consciousness, so some think.
And Pratt theory seems to apply to the interaction of physical particles
with each other and with the monads
Its axioms seem reasonable- but who am I to say.
1.A physical event a in the body A impresses its occurrence on a
mental state x of the mind X, written a=|x.
2.Dually, in state x the mind infers the prior occurrence of event a,
written x |= a.
3.States may be understood as corresponding more or less to the
possible worlds of a Kripke structure,
and events to propositions that may or may not hold in different worlds
of that structure.
4.With regard to orientation, impression is causal and its direction
is that of time.
5.Inference is logical, and logic swims upstream against time.
/ "Prolog’s backward-chaining strategy dualizes this by viewing logic
as primary and time as swimming upstream against logic, /
/ but this amounts to the same thing. The basic idea is that time and
logic flow in opposite directions."/
6.The general nature of these inferences depends on the set K of
values that events can impress on states.
7.Our first distinction between body and mind will be the trivial one
of using different variables to range over these sets: A, B over
bodies, X, Y over minds.
8.The second distinction will be in how the two kinds of sets
transform into each other.
9.Later we make a third distinction within the objects themselves
by realizing the two kinds as Chu spaces with dual form factors: sets
tall and thin, antisets short and wide.
10.We regard each point of the interval as a weighted sum of the
endpoints, assuming nonnegative weights p, q normalized via p + q = 1,
making each point the quantity p − q.
11.We shall arrange for Cartesian dualism to enjoy the same two basic
connections and the two associated properties, with mind and body in
place of −1 and 1 respectively.
12.Minds transform with antifunctions or antisets, and "sets are
physical".
13.Mental antifunctions/sets copy and delete, whereas physical
functions 'identify and adjoin'.
14. "For K the set (not field) of complex numbers, right and left
residuation are naturally taken to be the respective products ...
corresponding to respectively inner product and its dual outer product
in a Hilbert space"
That "The numbers ±1 are connected in two ways, algebraic and geometric"
suggests how the spatial separation of the monads is equivalent to an
algebra.
This also sounds much like a straight line with points along the line
having the properties P,Q such that P+Q=1
Now this is interesting: "Points have necessary existence, all being
present simultaneously in the physical object A.
15.States are possible, making a Chu space a kind of a Kripke
structure [Gup93]:
only *one state at a time* may be chosen from the menu X of alternatives.
Seems that divine intervention may be an assumption. I wonder who does
the choosing. May I suggest Godellian consciousness?
16. the spaces A and B play the interaction game A ⊗ B, their tensor
product.
17. The structure of ChuK is that of linear logic [Gir87], which can
be understood as the logic of four key structural properties:
it is concrete, complete, closed, and self-dual (which therefore makes
it also cocomplete and coconcrete).
The following implies some sort of entanglement in order to
interrogate all entities.
"When we unravel the primitive causal links contributing to secondary
causal
interaction we find that two events, or two states, communicate with
each other
by interrogating all entities of the opposite type."
It has been my supposition that the physical brain connects to the
human mind by way of entangled BECs.
The mind could connect to itself that way since it seems to be purely
a BEC.
So the physical brain must contain a BEC, I imagine, for this theory
to work.
But I am more interested in the connection of the mind to physical
particles/strings.
Particles can become entangled, but they are not BECs.
Elsewhere I have proposed that every physical particle is connected to
a (or many) monads.
It appears that Pratt theory may work for a particle connected to many
*or all* monads.
Thanks again,
Richard
--
Onward!
Stephen
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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