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Hi
Pete,
Thanks for the
reply. I'll hold off on most comments and wait for your more
complete post.
I'd just like to
remark that the distinction I make between algorithmic (computable) and
non-algorithmic (noncomputable), predicative and impredicative, is related
to the distinction between simple and complex, and is not
related to order vs. chaos. Complex systems have do have an order to them,
but sussing that out is what is difficult since it does involve noncomputable
models and nonreductionistic paradigms. Something like in the Wooden Plough
post: "Beyond doubt, pattern exists there, but it is the kind of pattern we
haven't yet learnt to see."
Regards,
Tim
Hi Tim:
Your
comments must've been quicker than you thought. You must have been thinking so
fast that you got some sort of relativistic time dilation or mass accretion
thing happening... or something. By the time your four comments got to me, they
had exgronificated themselves into six comments. (HEH)
;-)
Here are some "quick"
replies:
Tim Gwinn wrote:
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! In the
interim since my last post, I managed to glom onto
my own copy of FM, mostly due to a quasi-miraculous
process whose causative agent demonstrates operational characteristics roughly
akin to... uh, Maxwell's Demon in a
Skirt (MDIAS). Not that she, like, ever posts to this list or
anything...
Anyhow, now that I have access to RR's take on the QM
"paradox", I might as well equip myself with the same inputs that you
obviously already have, by which I mean that you may infer that I'd prefer to
defer further detailed comment until I've had the benefit of RR's perspective.
For now, I will just say that RR is completely correct in his
identification of the impredicativity within the
context of the existing quantum-theoretic paradigm.
Nevertheless, the paradox vanishes -- meaning that the impredicativity
resolves to coherent specificity -- when the epistemological quicksand on
which quantum theory is built is put on a more solid footing. IOW, the
quantum-theoretic paradigm itself is epistemologically crippled, and the
damage was done in the assumptions on which the fundamental, classical
mechanical paradigm was based. It's always been there, but it didn't break
until it was stretched beyond the limits of its applicability in the
microscopic domain.
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.
OK... thanks for fleshing
out the background for me. As it turns out, it appears that I'm on the right
track in my still-in-progress impending message, which includes a discussion
of the interrelationships among the concepts of time-fractionation, syntax,
and semantics. Based on what you've written immediately above, it appears that
I'm already in alignment with RR's perspective.
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.
That's only because there is no
consistent, integrated complexity-theoretic formalism... yet.
[Caveat: I say that from the perspective of a
physicist who is still in a state of massive ignorance of the bulk of RR's
work. For all I know, he might very well have moved complexity much closer
to such a formalism -- one that can be used in applied Rosennean-complexity-theoretic
science. I'm not a biologist, and up until now I must admit that I haven't
viewed those portions of RR's work that I've studied as complexity
cum biology. Biology is not my frame of reference.
That might sound a bit bizarre to you, especially since you have a
deep interest in biology. From your perspective, it might seem pretty weird
that anyone could not view RR's work as essentially
biological, but that's where I'm coming from. It's certainly not a negative
reflection in any way on the validity of RR's work in biological science;
far from it. In fact, from my perspective, it's a testament to how
profoundly on-track RR's scientific epistemology is, that his work is so appealing to a
physicist.
Judith tells me that this phenomenon has been a recurring
theme in the history of RR's work as it makes cross-discipline penetration.
Physicists tend to view him as a physicist, mathematicians view him as a
mathematician, biologists see him as a biologist, ...etc.]
I have my own views about the applicability of Chaitin's work --
including his work on computational complexity (OK... "complicatedness" -- I apologize for munging
up the semantics, but I thought I had made it clear that Chaitin's
"complexity" is not semantically equivalent to RR's complexity. That was my
intention in including the qualifier that "Chaitin doesn't appear to be referring to
Rosennean complexity."). Your point is well made,
and I appreciate your observation that Chaitin's work deals quite specifically
with syntactics.
Nevertheless,
there are important epistemological entailments in Chaitin's work, and those
entailments speak to the nature of system descriptions per se. I
fully agree that Rosennean-complex systems are non-algorithmic within the
constraints of "normal" algorithmic models & methodologies (i.e., the
constructivist, context-independent paradigm that dominates what RR calls
"contemporary physics", but I don't believe that RR accepted that we must
forever be bound by such constraints
in the construction of our models & methodologies.
In fact, isn't
that precisely the point in your October 29 "Wooden Ploughs"
message?
It's the same point that Eddington made in his famous example
in The Philosophy of Physical
Science: given a fish net with mesh of a certain size, you can
only catch fish that are larger than the mesh size. There might be fish that
are smaller, but you can't catch them because they just swim right through the
mesh. Therefore, you cannot make any conclusive statements about such
hypothetical fish, which statements are bereft of any empirical corroboration.
Now, generalize that principle: that's not just a limitation on the semantic content of what we can observe; it
also applies to syntactic structure,
by which I mean the kinds of things we can compute with our algorithms. Why?
Because it places fundamental syntactic limitations on the the kinds of
statements that we can make;
it limits the ways we can syntactically organize our
semantic referents.
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.
It's my fault for not
articulating it -- "...this middle ground of 'mutuality'...
" -- in a way that makes it as clear to you as it is to
me. From my perspective, the border between chaos and order is not anywhere
near as sharply defined as I infer you are saying it is in your statement,
"To me, either a system allows algorithmic translations
(relativity) or not (context-dependence)." There are
(relatively) context-independent elements in all systems. The question is,
"To what extent do those context-independent
elements constitute process constraints at the scale of the phenomena under
study?" For example, in macroscopic systems, to what extent do
microscopic phenomena contribute to the observed macroscopic behaviors? In
many cases -- maybe even most cases -- the answer is, "In no way that we can measure or quantify."
But scaling is continuous, despite the fact that we try to break it up
into relatively discrete domains, based on system parameters such as, say,
spatial dimensionality, or the magnitudes of applicable forces, or the momenta
of system elements, or the coherence of process kinetic energies... to name a
few. We simplify our models -- in physics, we idealize them -- to mitigate the intrinsic
complicatedness of the real-world processes we want to describe. It's that
process of idealization that enables the algorithmic models to work at all,
and then we say "Close enough for engineering,
dude." And as long as the bridge doesn't fall down, we're
"right"... but that doesn't mean the model is an infinitely precise
description of the dynamical parameters; it only means that the model works
for that purpose.
At some point, Rosennean complexity will prove its
value as far more than rhetorical grist for this discussion list. That's its
attraction for me, anyway. IOW, the theoretical models will be refined to the
point that they integrate with useful algorithmic representations, because
that's the way scientific knowledge penetrates its natural market. If those
algorithms don't exist yet, so what? They'll show up as the theory evolves.
Once again, your own "Wooden Ploughs"
message (which was superbly done, by the
way!) makes that point
elegantly.
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".
The point is
this: I believe that there are fundamental entailments in Chaitin's work as it
pertains to the downstream applicability of Rosennean complexity. That's a
matter of perspective based on my own intuitive take on its integrability as
syntactic structure with Rosennean complexity as semantic content. I can't
prove it at this point, and as a matter of speculation or hypothesis, I
wouldn't expect it to carry much weight. Fair enough.
But that's not
all I'm saying; I see a connection between Chaitin's clear
acknowledgment of the unknowable and the concepts of non-integrability, undecidability, and incomputability.
That connection transcends the vagaries of algorithmic considerations, which
always must involve an idealized methodology that
necessarily
constitutes an attempt to cram a context-dependent peg
into a context-independent hole. It's the nature of real-world applications
that they compromise precision to some degree, but they only have to work for
their intended purposes.
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 think I see
what you mean, but I'm not sure that we're talking about the same thing... or
at least we're looking at it from very different perspectives. Specifically, I
see algorithmic independence as being a more broadly applicable characteristic
from an information-theoretic perspective. Perhaps the differences in our
perspectives are due to differences in the way we each think about information
theory as it pertains to Rosennean complexity. Another possibility is that we
each understand RR's usage of "incommensurable time frames" differently. For
my part, now that I have access to FM, I intend to dig into it to ensure that my
understanding squares with RR's intended meaning.
I think you're
correct in your interpretation of the way Chaitin intended his stuff to be
interpreted, but I don't think that's the end of the story. I respect his work
immensely, and I don't make any pretense to being able to have done anything
like it myself. The guy's a genius in his own right. But every genius has his
limitations, and I don't expect that Chaitin is any exception. Very few
scientists who create significant work are able to foresee how their work may
be applicable to domains that are outside their idiosyncratic views. I once
tried to discuss the vast entailments of information theory with Shannon; what
a hoot! He listened politely for a minute or two, and then waved my comments
aside with the sweeping dismissal, "I just did
some work on telephone switches... it's really nothing more than
that." Yeah, right.
I have had similar experiences with
other scientific geniuses who created work of great significance. Is it
humility? Maybe... but that's hardly a universal trait among scientists. I
think it's more a matter of the narrow perspective that necessarily results
from focusing so intensely on a given problem that everything else fades into
relative insignificance. Yet, the clarity of vision in that one area, when it
is sufficiently acute, penetrates the nature of perceived reality so deeply
that it opens a window for others to look through from different perspectives.
In so doing, they see things that the original innovator was not looking
for.
RR probably had his own blind spots too, although I haven't found
any yet. He is the closest thing to a universal scientific genius that I've
ever seen.
Regards,
Pete
I look forward to your more complete
exposition!
Regards,
Tim
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