It looks to me that there is currently deep confusion around the use of
both words "chaos" and "disorder" in the field of dynamical systems.
Sometimes the meaning "unpredictability" is evident in that usage. But
at other time the meaning "disorganization" is. These two ideas are
different and very often are in need of critical distinction.
An important usage of "disorganization" is the "disunion of constituent
parts". It pertains to structure, arrangement or configuration. It is a
phenomenon that can be observed statically, instantaneously, without the
passage of time. It pertains to the interrelationships among parts. It
also pertains to the relationship of the parts to the whole. It is
well-modeled by geometry, topology and graph theory. Any notion of the
"degree of organization or disorganization" would pertain to how rich
the interrelationships among parts are. Defining a measure for this
would be challenging, and none dominates.
On the other hand, "unpredictability" pertains to chance variation,
probability and even epistemology. It is defined independently from any
notion of parts/whole. It is well-modeled by probability theory and
stochastic processes. Any notion of "degree of predictability or
unpredictability" would be subject to some matter of probabilities. In
information theory, it is measured by statistical entropy and its
related entropic functionals (relative entropy, conditional entropy,
mutual information, entropy rate, etc.) It is worth noting that the
definition of statistical entropy has probabilities as its only
parameters. Thus, it is capable of measuring degrees of predictability
or unpredictability, but not degrees of organization or disorganization.
Of course, the two ideas can be combined. For example, one could be
interested how a collection of constituent parts are arranged,
configured or organized - and how that organization changes (subject to
chance) over time. (Such as particles in an ideal gas.) We could even
define some kind of stochastic dynamical system wherein (instantaneous)
system state could be its current organization (defined as, say, a
topology), and where the Markov chains (or more elaborate conditional
processes) can provide the stochastic dynamics.
This confusion further shows up in the use of "chaos" in dynamical
systems. One dictionary defines "chaos" as "where chance reins supreme".
But in nonlinear dynamics, "chaos" means "sensitivity to initial
conditions" and is strictly deterministic. (That is, chance is banished
from "Chaos Theory").
Food for thought.
Grant
On 12/1/14, 3:57 AM, Eric Smith wrote:
Hi Nick and Arlo,
Yes. What got me about this was the fact that the idea of gas as a
chaotic state of matter goes WAY back.
Here it seems the etymology goes the other way, though, right? The
notion of a "chaotic state of matter" is actually a new borrowing of a
term, with about as much connection to the original as Murray's
"color" in QCD has to the visual chromatic spectrum.
If original chaos meant mostly a void or gap (which seems to be what
Wikipedia -- the authority on all matters -- says, then it is not a
bad fit to most of our everyday experience of gases, and would work
even better for the vacuum.
It seems that the Term of Art "chaos", referring to tons of structure
that is merely recalcitrant to description, is the odd man out.
Although, perhaps the argument against my position is that when
artists need to represent chaos, they paint a lot of structure that is
meant to exist but to defy description. Since I don't know how
ancient Greeks actually handled these things, maybe, as you say, they
wouldn't tolerate a notion of "not-there", so they would have
conceived of gaps as having a substantive essence, but just beyond
tractability by perception, in which case modern chaos would indeed be
much older.
Thanks for those,
Eric
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