|
Hi
Pete,
Four quick
comments:
1) "There is no
quantum measurement paradox." !!?? So, the impredicativity Rosen described
will not stand? I am intrigued - please comment further on this, as this is a
distinct topic from the time topic!
2) For others
who may not be sure what I was referring to, here is the quote regarding
"incommensurable timeframes" is as follows:
"In a nutshell, we
find that impredicativities (i.e., complexity) and pure syntax are incompatible.
More specifically, complexity and an ontology based on a single syntactic time
frame (the ordering of purely syntactic operations into individual steps) are
incompatible. In the present context, in which we have identified
impredicativities with nonfractionabilities, we cannot build nonfractionable
systems by purely syntactic means either. We must accordingly either invoke
semantic elements transcendental to syntactics (e.g., taking of limits) or (what
may be equivalent) utilizing two or more incommensurable time frames."
[EL 294-295]
Certainly, in
this quote Rosen refers to context-dependence, as you pointed out; in this
case, that context-dependence comes in the form of nonfractionability that Rosen
is discussing. Rosen offers two alternatives: semantic aspects that cannot be
reduced to syntax or incommensurable time frames. It is not clear from the
discussion whether these are two sides of one coin, or
two non-interchangeable strategies.
3) "Relative
complexity" and similar notions that rely on algorithm (or combinations or
sums or size of algorithms) all relate to systems that are inherently "simple
systems" in Rosenspeak. They are manners of comparing complication, not
complexity. In general, I have not found Chaitin's work to be relevant
for Rosennean complexity, since Rosennean complex systems are, by their nature,
non-algorithmic, and so too would any measure of their complexity not be based
in such algorithmic measures.
4) Regarding
relativity, mutuality and context-dependence.
I do not know what
you mean by this middle ground of "mutuality". To me, either a system allows
algorithmic translations (relativity) or not (context-dependence). The
bifurcation between Rosennean simple systems (which must
be context-independent) and complex ones is quite precise in the
distinction between algorithmic and non-algorithmic. I can see no middle ground
here. Chaitin's work allows such a concept of mutuality
because it is all situated within the realm of algorithmic systems,
but it is thereby irrelevant for Rosennean complex
systems.
5) I would
consider non-integrability, undecidability, incomputability, etc. to be
perhaps examples of incommensurability. But I would not say that
they are the same as incommensurability.
I take
incommensurability as the dictionary indicates: "having no common measure
or divisor or standard of comparison".
6) More
importantly, as I see it, algorithmic independence (as the sum of two other
algorithmic complexities) is NOT at all "semantically
equivalent" to incommensurability. Alg. indep. refers to comparisons of
information content of the smallest program H(x,y) in comparison to the
size of smallest programs for H(x) + H(y). Algorithmic independence
refers to optimality conditions for computational strategies,
not incommensurability. (http://www.cs.umaine.edu/~chaitin/dijon.pdf)
I look forward to
your more complete exposition!
Regards,
Tim
Hi Tim:
This is a
brief interim message, by way of immediate response to your October 27 post
(excerpted below), with my preliminary comments interposed in sequence. I
intend to reply more fully in my next post, integrating my complete response
with the content of my next message. I have identified the "problem" with the
way we view time in contemporary physics, and I know the
solution-in-principle; now I must figure out how to articulate it in a way
that would convince me if I were walking in on this discussion cold, and were
just seeing it for the first time.
Tim Gwinn wrote:
Continuing thoughts on time and impredicativity,
with note of Judith's 10/26 remarks.
First of all, I would be surprised if no one
challenged my assertion that the study of time involves the impredicativiy
I described. Well, I
intend to reply to your assertion, but I hope you won't be dissapointed if it
doesn't show up as a challenge, as much as it does as a clarification. I don't
see it as a matter of agreement or disagreement. You're talking about one
thing, and you're doing it from a certain perspective that seems to dead-end
in an impredicativity. RR has already resolved it, as I will demonstrate. I
talked around it in my last post -- not intentionally, but rather because I
hadn't thought my way through it yet. It's much clearer to me now.
This impredicativity is something that occurred to me
some time ago but which I had kept setting aside because it was
personally unsatisfying in the kinds of limits it implied for any study of
temporal phenomena. It may well be that someone
will notice that there are observables, or combinations of
observables, relevant to time that do not fall within the confines of the
impredicative situation and that therefore we have other ways to
study time and (more importantly, perhaps) to ask deeper questions
about time. I think you're
on the right track here.
That being said, if the impredicativity I have
described stands, it would hardly be a death-knell to the study
of temporal phenomena. As you may recall, the point of departure for
Rosen's book "Fundamentals of Measurement" was another impredicativity:
the measurement problem in QM. From examination of that problem flowed all
the insights and formal representational systems which comprise that
book. I'm at a
disadvantage here, because I still haven't succeeded in glomming onto a copy
of Fundamentals of Measurement.
Nevertheless, I don't think it will matter; the impredicativity does not stand. There is no quantum
measurement paradox.
Likewise, the impredicativity with regard to time
seems to me to offer many avenues for the investigation of time. For
one, the question "what is a clock?" spawns an entire field of
inquiry. We can ask things like "can the system itself be the
clock?" which can lead to Rosen's notion of "age"[AS 4.8]; "Can a system
have multiple clocks moving at different rates?" which can lead to
anticipatory systems; "Are the units of time (the ticks of the clock)
always the same?" shows up in his discussion of Hamiltonian systems[AS
4.5]; and on. Perhaps complex systems may always involve "two or more
incommensurable time frames"? [EL
295]. Perhaps... but I
don't know whether that's a necessary
constraint for all complex systems -- at least with respect to time. Some systems have relative complexity; that is, their
relative complexity is the size of the smallest program (algorithm) it takes
to calculate the complexity of System
A if we already have an algorithm for calculating the
complexity of System B. (Yes, that's
a big " if
", but Chaitin proved it in 1975 with his
decomposition theorem.) [Note: Chaitin doesn't appear to be referring to Rosennean
complexity.
indeed, in such a syntactic approach as the one in which he
excels, semantic content is low. Nevertheless, there's an
important equivalency to other concepts that integrate with RR's
work, which concepts I'll illustrate presently.]
An entailment of the same theorem is another concept called the mutual complexity of two given systems,
which is defined as the extent to which the complexity of the systems is less
than the sum of their individual complexities. But there's a third type of
complexity called algorithmic
independence, in which the total complexity of the pair is
equal to the sum of their individual complexities. That would be semantically
equivalent to your concept of incommensurability.
Your
mention of incommensurability is interesting, but it might be nothing more
esoteric than the inclusion of context-dependent system constraints, as
distinct from merely relative ones.
That was an important distinction that you yourself made in your October 22 post, and
I immediately recognized it as a critical distinction. It suggested a
particular line of inquiry that put me on the track of the real nature of the
problem.
The distinction, of course, is that in the case of relativity, the differences between any two
given systems may superficially appear to be great, but in fact they are related in that one is algorithmically
convertible to the other. In other words, they have equivalent encodings, in
RR-speak. Midway between relativity
and context-dependency is the case of
mutuality, in which there is a degree
of overlapping complexity. Finally, we have the case of context-dependency, in which there is no
way to map one system to the other; that is, the systems do not have
equivalent encodings, nor can any method be devised whereby the encodings can
be equilibrated. That case is the semantic equivalent of your concept of
incommensurability.
There is a long history of gradual refinement of
the concept of incommensurability. Poincaré called it non-integrability; Godel
called it undecidability; Turing called it incomputability; Prigogine called
it non-isomorphism; and Chaitin called it algorithmic independence. Same
principle... different languages. From my perspective, it's remarkable to find
that principle identified in such a broad cross-disciplinary sampling. But
even more remarkably, every one of those threads leads me directly back to
Rosennean complexity. For me, none of those examples, individually, has
sufficiently penetrating semantic content in its original context to imply the
sense of integrability that they all have -- in the aggregate -- in context
with RR's take on the same principle: RR called it
non-equivalent encodings. From my perspective, there's your
concept of incommensurability, and I don't believe that it has anything to do
with time, in its most fundamental interpretation & manifestation. More on
this in a later post.
Further, the relation
of clocks to morphogenesis in biological systems is similarly then
less obvious and deserving of study on its own. Is it sensible to use one
common clocktime to describe the various morphogenetic processes (and
their relationships) which by outward appearance (i.e., by a common
clocktime) seem to be
"simultaneous"?
Maybe... that is, it
might be sensible for, say, the purpose of fleshing out some of the process
details reductionistically; I mean, that kind of work ultimately must be done
as a precursor to the development of practical (technological) applications ,
but it's not apparent to me that such an approach will provide a theoretical
description that will have the broadest applicability. I think there's a more
fundamental approach for the purpose of creating a useful theoretical model.
(Later for this one, too.)
(snip)
P.S. - When I think about "time", one formalism that
keeps nagging at me is "projective geometry". I haven't found a way to
make use of it......yet. But it keeps coming to mind - particularly,
the notion of "homogenous coordinates" which the topic uses. I just
thought I'd mention it to see if it triggers something in one of you
out there. :)
It triggers my curiosity,
but I don't get very far with it... probably due to my own ignorance of the
subject. If you're comfortable about indulging your own nagging thoughts
(something I would strongly encourage you to do!), I'd be interested in seeing
where you go with it. Since you're not claiming to have an integrated theory
and it's still in the cooking stage, that sufficiently qualifies it as
hypothetical musing, hypothesizing, speculation... or other suitable
qualifier. Go for it. It sounds intriguing.
Regards,
Pete
After
thinking yet some more about "time", I am inclined to think that the
situation we are in with regard to time is somewhat akin to the limitation
in physics we know as "the measurement problem" in quantum
mechanics.
Rosen
describes the key aspect of the measurement problem as
follows:
"The problem here is
that the very acquisition of data, the very cognition of phenomena
(phenotype) in a material system, requires one to consider a
larger system ("system + observer") and not to consider smaller
ones, as reductionism (or context-independence, or objectivity) requires.
This in turn creates a chicken-egg situation, an impredicativity;
specifically, one must know the larger system to characterize the smaller,
but one cannot know the larger until the smaller is characterized."
[EL 106]
With time, I
feel we are in a similar situation. In this case the "smaller system" is
some system which exhibits the apparent quality of "time"
we want to describe using terms like
dynamism/change/process/action/etc. But, we cannot know this quality
of this smaller system except by considering a larger system,
consisting of the ("system + observer + clock-reference-system"). Here the
clock-reference-system is some system which acts to generate the
time labels or references. This can be an actual clock, or the sun
moving across the sky, or even our internal awareness of our changing
thoughts (as Mach noted). Without some such reference
system we cannot even have a sense of the "passage of
time".
If this is
so, then it seems to me that the scientific investigation of time will be
circumscribed by this impredicativity in the same way that scientific
investigations of quantum systems are circumscribed by the impredicative
nature of the measurement problem. As a result, it seems to me that the dynamical
qualities of systems can only be studied with
reference to some clock system. This
impredicativity also thereby sets limits on the sensibility of
questions about time that are outside of that circumscribed
realm.
Note that
the clock system need not always be separate from the system itself. In
the discussion of "Time and Age" [AS 4.8], Rosen demonstrates that "age"
is perhaps most accurately portrayed not by reference to some
external common clock time, but by a dimensionless quantity in
which the changes in state of the system are themselves the basis for
the "units of time" used to detemine the system's "age". With his
example of a system having radioactive decay, he notes "If we measure
time in units proportional to the rate of decay (e.g., in
half-lives), then all systems obeying [an equation of radioactive
decay] decay at the same rate." [AS 273] However, this kind of
measure is often unsatisfying for most of our investigations of
dynamical qualities: we generally want to contrast their dynamical
qualities with those of their environment and/or other particular systems
(e.g., ourselves!), and thus we generally invoke some system external
to the one under study to act as the clock reference
system.
Regards,
Tim
|