Glen is very close to interpreting what I mean to say. Thanks, Glen!
(But of course, I have to try one more time, since I've thought of
another - hopefully more compact - way to approach it...)
Logically speaking, "degree of unpredictability" and "degree of
disorganization" are orthogonal concepts and ought to be able to vary
independently - at least in certain domains. If one were to develop a
theory about them (and I am), then that theory should provide for them
to be able to vary independently.
Of course, for some "applications" of that theory, these
"predictability/unpredictability" and "organization/disorganization"
variables may be dependent on each other. For example, in
Thermodynamics, it may be that the degree unpredictability and the
degree of disorganization are correlated. (This is how many people seem
to interpret the second law.) But this is specific to a Physics application.
However, in other applications, it could be that the degree uncertainty
and the degree of disorganization vary independently. For example, I'm
developing a mathematic theory of living and lifelike systems. Sometimes
in that domain there is a high degree of predictability that an
organo-chemical entity is organized, and sometimes there is
unpredictability around that. The same statement goes for predictability
or unpredictability around disorganization. Thus, in the world of
living systems, unpredictability and disorganization can vary
independently.
To make matters more interesting, these two variables can be joined in a
joint space. For example, in the "living systems example" we could ask
about the probability of advancing from a certain disorganized state in
one moment to a certain organized state in the next moment. In fact, we
could look at the entire probability distribution of advancing from this
certain disorganized state at this moment to all possible states at the
next moment - some of which are more disorganized than others. But if we
ask this question, then we are asking about a probability distribution
of states that have varying degrees of organization associated with
them. But, we also have a probability distribution involved now, so we
can ask "what is it's Shannon entropy?" That is, what is its degree of
unpredictability? So we have created a joint space that asks about both
disorganization and unpredictability at the same time. This is what I do
in my theory ("Organic Complex Systems").
Statistical Thermodynamics (statistical mechanics) also mixes these two
orthogonal variables in a similar way. This is another way of looking at
what Gibbs (and Boltzmann) contributed. Especially Gibbs talks about the
probability distributions of various "arrangements" (organizations) of
molecules in an ideal gas (these arrangements, states, are defined by
position and momentum). So he is interested in probabilities of various
"organizations" of molecules. And, the Gibbs formula for entropy is a
measurement of this combination of interests. I suspect that it is this
combination that is confusing to so many. (Does "disorder" mean
"disorganization", or does it mean "unpredictability". In fact, I
believe reasonable to say that Gibbs formula measures "the
unpredictability of being able to talk about which "arrangements" will
obtain."
In fact, Gibbs formula for thermodynamic entropy looks exactly like
Shannon's - except for the presence of a constant in Gibbs formula. They
are isomorphic! However, they are speaking to different domains. Gibbs
is modeling a physics phenomena, and Shannon is modeling a mathematical
statistics phenomena. The second law applies to Gibbs conversation - but
not to Shannon's.
In my theory, I use Shannon's - but not Gibbs'.
(Oops, I guess that wasn't any shorter than Glen's explanation. :-[ )
Grant
glen e. p. ropella wrote:
Nicholas Thompson wrote circa 08/05/2010 08:30 AM:
All of this, it seems to me, can be accommodated by – indeed requires –
a common language between information entropy and physics entropy, the
very language which GRANT seems to argue is impossible.
OK. But that doesn't change the sense much. Grant seemed to be arguing
that it's because we use a common language to talk about the two
concepts, the concepts are erroneously conflated. I.e. Grant not only
admits the possibility of a common language, he _laments_ the common
language because it facilitates the conflation of the two different
concepts ... unless I've misinterpreted what he's said, of course.
I would like to apologize to everybody for these errors. I am beginning
to think I am too old to be trusted with a distribution list. It’s not
that I don’t go over the posts before I send them … and in fact, what I
sent represented weeks of thinking and a couple of evenings of drafting
… believe it or not! It seems that there are SOME sorts of errors I
cannot see until they are pointed out to me, and these seem to be, of
late, the fatal ones.
We're all guilty of this. It's why things like peer review and
criticism are benevolent gifts from those who donate their time and
effort to criticize others. It's also why e-mail and forums are more
powerful and useful than the discredit they usually receive. While it's
true that face-to-face conversation has higher bandwidth, e-mail,
forums, and papers force us to think deeply and seriously about what we
say ... and, therefore think. So, as embarrassing as "errors" like this
feel, they provide the fulcrum for clear and critical thinking. I say
let's keep making them!
Err with Gusto! ;-)
--
Grant Holland
VP, Product Development and Software Engineering
NuTech Solutions
404.427.4759
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