If we're allowing ourselves a little informality, then I'd appeal to
the notion of observer moment. Within any observer moment, a finite
number of bits of the bitstrings has been read, and processed by the
observer. Since only a finite number of bits have been processed to
determine the meaning of reality at that moment, the
observer map O(x) is a prefix map. Hence at any point in time the
arguments in section 2 of the paper hold.

The meaning O(x) could also be called the "observer moment". If
observer moments are enumerable, one can inject OMs into the set of
natural numbers.

Observers find themselves embedded in a psychological time. I have not
been explicit about exactly what this time is, however I envisage it
to probably be what mathematicians call a "time scale", which is a
closed subset of the real numbers. Time could be continuous, or it
could be discrete (eg the set of natural numbers). It could be
something else, eg rational numbers or the Cantor set. All of these
are example time scales. The exact nature of time is something to be
settle later (if possible), but if you are more comfortable witrh
discrete time (as many are on this list), then you are welcome to use integers.

How this feeds back to our original observer map is that we'd expect
the map O(x) to be dependent on time, ie O(t,x). This is consistent
with time being "psychological". The description or "universe" x is
independent of time. It would correspond to what David Deutsch calls a
block universe.

Now perhaps section 3 makes some sense. What I call "robustness" of
the observer, ie that observers will not be fooled by a little noise
on the line - lions in camouflage are still observed to be lions for
instance constrains the form of time evolution of O(t,x). I haven't
formalised exactly what this constraint is, but it is something along
the lines of continuity of |O^{-1}(t,O(t,x))|, or continuity of the
observed complexity of the world. 

On Wed, Jun 08, 2005 at 09:09:04AM -0700, "Hal Finney" wrote:
> Russell Standish writes:
> > On Mon, Jun 06, 2005 at 01:51:36PM -0700, "Hal Finney" wrote:
> > > In particular, if "an observer attaches sequences of meanings to sequences
> > > of prefixes of one of these strings", then it seems that he must have a
> > > domain which does allow some inputs to be prefixes of others.  Isn't that
> > > what "sequences of prefixes" would mean?  That is, if the infinite string
> > > is 01011011100101110111..., then a sequence of prefixes might be 0, 01,
> > > 010, 0101, 01011, ....  Does O() apply to this sequence of prefixes?  If
> > > so then I don't think it is a prefix map.
> >
> > Yes I agree this is vague, and seemingly contradictory. I'm not sure
> > how to make this more precise, but one way to read the paper is to
> > treat observers as prefix maps for section 2 (Occam's razor), and then
> > for section 3 (White Rabbit problem) ignore the prefix property.
> >
> > It could be that the way of making this more precise is to assume
> > observers have some internal state that is constantly updated (a time
> > counter perhaps), so actually going through a sequence of prefix maps
> > in (psychological) time, but at this stage I don't have an answer.
> 
> Unfortunately I still don't understand this.  You agree that it is a
> seeming contradiction but that doesn't help me to see how to interpret it.
> 
> Here's an idea.  Would it be possible for you to explain how this
> page is meant to be understood, in an INformal way?  Often when people
> present concepts they do a formal writeup, but if they give a seminar
> or explanation they will depart from the formalism and explain what is
> really going on behind the scenes.  That's the kind of explanation I think
> I need.
> 
> Could you explain how these concepts relate to the actual experiences
> we have as human observers?  What are "descriptions" and "meanings"
> in terms of our sensory and mental experiences?  Which "descriptions"
> does an observer observe?  What are the "sequences of prefixes" and
> how do they relate to our day to day lives?  What is the point of the
> equivalence classes and what does that have to do with what we observe?
> 
> I think an informal explanation of these topics would help me, and
> perhaps Paddy, to better understand the structure that you formally
> describe.  At this point I am still failing to see how it all relates
> to my experience of the world as an observer.
> 
> Thanks -
> 
> Hal Finney

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