Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 10 Jun 1999 10:03:42 +0300 (EET DST)
On Wed, 9 Jun 1999, Stephen P. King wrote:
> Dear Matti and Friends,
>
> In [time 395] Constructing spacetimes, Matti wrote:
>
> "There is also problem about information flow between different
> LS:s. How can one define information current between LS:s if
> these systems correspond to 'different spacetimes'?"
>
> There is much to be discussed here!
>
> If I am correct, "current" is defined as some quantity of change
> occurring through a boundary of some sort.
> (http://www.whatis.com/current.htm) It is usually assumed that some
> particle or fluid is being transferred from one location to another and
> a term "density" is associate with "Current per unit cross-sectional
> area". So we are thinking of the concepts: "flow", "boundary",
> "information", "different space-times", and "particle".
Yes. This definition works also in infinite dimensional case.
> We need definitions that are mutually consistent, I am proposing
to
> using graph theoretic concepts since we can easily generalize them to
> continua:
>
> http://hissa.nist.gov/~black/CRCDict/termsArea.html#search
>
> Flow: "A measure of the maximum weight along paths in a weighted,
> directed graph" We could consider the "weight" as the degree to which a
> given edge connects a pair of vertices, e.g. if a pair of vertices are
> identical relative to their possible labelings the weight is 1, the
> weight is 0 if their respective sets of labels are disjoint. (When
> considering spinors as labels of the vertices we use alternative
> notions.)
>
> http://hissa.nist.gov/~black/CRCDict/HTML/flow.html
>
> Boundary: I can not find a concise definition so I will propose a
> tentative one: the boundary of a graph B{G} is the minimum set of
> vertices |V_G| that have as incident edges that connect a pair of
> points, one of which is an element of ~{G} and the other which is an
> element of {G}; where {G} and ~{G} are a graph and its complement.
> I am not sure that this notion is appropriate. :( I am thinking of
>the
> way which traditional set theory defines a boundary of a set: "a point
> is in the boundary of a set iff every neighborhood of the point
> intersects both the set and its complement". So the boundary of a set of
> these points. It looks like the only element involved would be the empty
> set {0} in the usual way of thinking of sets in the binary logical
> sense, this relates to my discussion of the Hausdorff property...
>
Does this approach generalize simplicial cohomology? Simplicial complex
defines homology groups. It has simplices up to dimension D if simplicial
complex is D-dimensional. One can consider functions in the set
simplices of given dimension. One can define co-exactness and
co-closedness and cohomology groups. Info current would be a function in
the set of D-1 dimensional simplices. Info current would in general
correspond to an element of cohomology which is not coclosed. This makes
sense only in ordinary topology defined by norm but you are talking about
non-Hausdorff property.
> Information: Now here is the key problem: How to define "information"!
> What is Information? Is is "meaning" as in "the semantic content of a
> pattern of matter/energy"? Is it the bits that are recovered when a
> string of bits is encoded or compressed by some scheme and then decoded
> or decompressed by the scheme's inverse? Is it the value of a quantity
> present at some arbitrary point?
>
Very stimulating questions! While visiting at your homepage I
realized for the first time how many times 'information' appeared
there. For some mysterious reason I have managed to circumvent the
challenge of defining this concept until now. Perhaps my
strong opinions about computationalism explain this(;-).
a) I think that its is meaningful to talk about 'meaning'
only if one talk about *conscious* information. OK?
b) It is certainly impossible to characterize conscious experience
by a bit sequence. This was one of reasons I have been very sceptic
about 'information content of cs experience'.
One could however circumvent this problem! Conscious information could be
defined as *difference* of informations associated with initial and
final states of quantum jump. This would reduce problem to that of
associating information measure to quantum states! Since quantum states
correspond to a well defined geometric objects there are hopes of
associating information measures with them! This looks a clever
trick to me at least!(;-)
c) One can probably define several types of informations associated
with configuration space spinor field and assign
information measures to them. Perhaps one must give up the idea
about single information measure. Perhaps the essential question
is 'About what the information is about' and each question gives
different measure of information.
d) I realized that the information associated with configuration space
spinor field, about which I talked in previous postings,
is essentially *information about position in configuration space*
plus information about spin degrees of freedom relative to the
ground state which corresponds to Fock vacuum and contains no information.
Information is defined as the information gain involved in total
localization of the configurations space
spinor field to single point in Fock vacuum state. Single 3-surface
in configuration space into Fock vacuum state is selected and Shannon
formula defines the information gain. Same works for Schrodinger
amplutitude in nonrelativistic situation.
Critical Question: Does the information of the configuration space
spinor field provided about the position in configuration space
of 3-surfaces provide information about configuration space
and spacetime geometry? There are hopes since spin degrees of freedom
(which correspond to fermionic degrees of freedom in infinite dimensional
context) are involved and entanglement is associated with these degrees of
freedom. Recall that fermions describe 'reflective level of cs' in TGD
approach to cs (Fock state basis has interpretation as Boolean algebra).
e) You mentioned bit counting as a possible manner to define information.
Interesting possibility is that real to p-adic correspondence
could provide measure for the information content of the configuration
space spinor field based on counting of bits, or actually pinary digits.
i) Pinary cutoffs of configuration space spinor field provide a
sequence of more and more accurate discretization of configuration
space spinor field.
ii) The mapping of real configuration space spinor field to its
p-adic counterpart involves *minimal* pinary cutoff for which
continuation to smooth p-adic configuration space spinor
field is possible. Minimal pinary cutoff comes from
the requirement that the canonical image of the pinary cutoff allows
continuation to a *smooth* p-adic configuration space spinor field.
If pinary cutoff of the canonical image is too
detailed, completion is not possible.
iii) There would be thus some number N of pinary digits and
I(X^3) =N(X^3)
would serve as a measure for the information contained by
the value of the configuration space spinor field at given point of
configuration space.
iv) One could define the total information contained by configuration
space spinor field as sum of informations associated with
discretized configuration space.
N= SUM_i N(X^3_i).
This number is infinite as real integer but *finite as p-adic number*!
Real information is obtained as the canonical image of I
and would be finite. Higher pinary digits would
not be be given such importance as for low pinary digits in this
information measure. This is indeed very reasonable: lowest pinary
digits contain the essential and the rest is just details.
Note: the value of p-adic prime associated with entire universe is
very probably infinite so that N is probably infinite as
ordinary integer still. Note that infinities can cancel
in info content of cs experience defined as difference.
v) This information is obviously information about the construction of
the p-adic counterpart of configuration space spinor field from
its real counterpart by canonical identification mapping. Is this
information given by conscious experience? Perhaps! Conscious
experience always involves coarse roughening: higher pinary digits
do not have same importance as lower pinary digits. Conscious experience
forms abstractions. So, perhaps the contents of conscious experience
involve essentially the coarse roughening involved with reals to p-adics
map?
f) All geometric structures of real quantum TGD
are mapped to their p-adic counterparts using phase preserving canonical
identification map with minimal pinary cutoff.
i) This approach might work also at spacetime level
for spinor fields defined on spacetime surface. To each spacetime
time=constant section of spacetime surface one could associated
information I in similar manner and pinary cutoff
would provide the discretion of 3-surface making it possible
to define total information as sum over informations associated
with the points of X^3. CAnonical image would define real
information which would be finite.
ii) The mapping of real spacetime surface to its p-adic
counterpart involves this map and one can assign the real counterpart of
p-adic integer N of pinary digits to each point of real spacetime
surface as its information content. Again also total
information content could be defined as sum for the
minimal pinary cutoff of spacetime surface.
I think I must stop here.
Best,
Matti Pitkanen
> Different space-times: This statement implies a plurality, a multitude
> of configurations of distinguishable particles such that a basis of
> three orthogonal directions is definable in conjunction with a dynamic
> that alters the configurations in a uniform way.
>
> Particle: An entity that in a given reference frame or framing is
> indivisible. It should not be assumed that an entity that is indivisible
> in one framing need be so in another framing. I am thinking of a framing
> as a finite context or environment that acts as a "contrast" for the
> entity in question.
>
> The problem I see right away is that information is not a
> substance in
> the normal sense, since it has the properties of compressibility and,
> according to Bart Kosko, irrotability, which are in contrast with those
> properties of matter which is, usually incompressible and rotateble....
>
>
> But, I think that Peter's notions are the most relevant to this
> conversation of "information flows" between LS, so we need a way of
> bridging between the formalism of graph theory and the formalisms used
> in Peter's papers.
>
> We'll take that up after some discussion. :)
>
> Later,
>
> Stephen
>
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