On 2/9/2012 3:40 PM, acw wrote:
[SPK]
We must consider the entire range of possible observers and
technological abilities. We cannot limit ourselves to humans with their
current technological abilities. Therefore we cannot put a pre-set limit
on the upper bound. I agree that the machine must be finite, but my
reasoning follows from mathematical considerations. My conjecture is
that the content of experience - the sequence of OMs - of a generic
observer is constrained to be representable by a sequence of Boolean
Algebras of propositions or "Free Boolean Algebras
<http://en.wikipedia.org/wiki/Free_Boolean_algebra>". This restriction
ties the contraints that exist on Boolean Algebras to being countable
(and the compactness of the topological spaces that are their dual) to
the finiteness of what can be observed by an observer. So we do not have
to postulate finiteness separately iff we take the Stone duality as it
has finiteness built in.
To explain this reasoning further, I would like to point out that for a
large number of entities to be able to communicate with each other, it
is necessary that whatever the means of communication might be, it must
be such that what is true for one will be true for all otherwise we get
a situation where "The Tree is tall" is true for some observers pointing
at a giant redwood while it is false for some other observers pointing
at the same giant redwoods. Communication requires mutual consistency of
propositions and this can only happen if the logic of their means of
communication is bivalent with respect to truth values. Now we can
quibble about this and discuss how in Special Relativistic situations we
can indeed have situations there "X caused Y" is true for some frames of
reference and "Y caused X" for some other frame of reference, but this
dilemma can be resolved by considering the effect of a finite speed of
light whose "speed" is an invariant for all observers, e.g. general
covariance.


Mostly agreed, although my category theory knowledge is limited, so I don't know what intuitions led you to that particular Boolean Algebgra conjecture about the OMs. One thing that might be worth considering is the machine which keeps expanding: consider an AI running on an actual Turing Machine (unbounded memory), the actual implementation shouldn't matter (be it running directly in some UD or actually living in a physical universe where it constantly harvests resources to increase its memory), how does your FBA conjecture deal with such self-modifying, self-improving, self-extending observers (humans are not yet there, obviously we're very good at working with limited resources and finite bounded memory at the cost of forgetting).

Hi ACW,

I have to break the "Ontological Problems of COMP" up into pieces to respond to your important questions. Please remember that this is just an embryo of a theory. It has not yet made it to the "half-baked" stage. ;-)

My thought is that the FBAs are not restricted in the number of prepositions that they include thus can grow to include new data. It is the means by which they are modified that goes to the answer of your question. This is conversed by the process of "residuation" explained in http://boole.stanford.edu/pub/ratmech.pdf It is important to note the way that dynamics are treated by Pratt. What I am trying to do is to explicitly deal with the problem of time within the conjecture. I will try to explain more of this in subsequent mails.

Onward!

Stephen

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