Nick,
Lemme try to present three examples of these two orthogonal dimensions
(Organization/Disorganization dimension vs
Predictability/Unpredictability dimension).
It all boils down to what phenomena one chooses to be interested in.
(Even if both dimensions are arguably present in a particular
phenomenon, one can choose to observationally ignore one of them in the
analysis of that phenomenon.)
The first example will be exclusively interested in the
Organization/Disorganization dimension.
The second example will be exclusively interested in the
Predictability/Unpredictability dimension.
The third example will be jointly interested in both.
Example I: An interest in Organization/Disorganization (structure or
lack of), with no interest in Predictability/Unpredictability.
Lets say we are interested in observing Hydrogen and Oxygen atoms within
a small region of space.
These atoms are capable of combining into several possible bonding
configurations: O_2 , HO, H_2 O, etc.
Suppose we observe this region for some finite time and make a list of
any of these configurations that are observed.
This list captures an interest in Organization/Disorganization, but not
in Predictability/Unpredictability.
We could go further and even develop a metric for the "degree of
organization" observed. That would be a continuation of the same interest.
Example II: An interest in Unpredictability/Predictability, with no
interest in Organization/Disorganization.
Lets say we are interested, as in Example I above, in observing Hydrogen
and Oxygen atoms within a small region of space.
However, this time, we have no interest in whether these atoms occur in
a bonded or unbonded form.
What we are emphasizing instead this time is the probability of
selecting one of these two atoms at random from the region -
AND in whether or not the resulting probability distribution describes
an Unpredictable situation, a Predictable situation, or somewhere in
between.
(Assume that we will use the best experimental practices to arrive at an
estimate of the population parameters from sample statistics.)
Let's say that we conclude that the distribution results in a .75 prob
for H and .25 prob for O. (We "throw back" other atoms.)
Then, we can conclude that the Shannon entropy for this distribution is
-[(.75)*(log_2 (.75)) + (.25)*(log_2 (.25))] = .675
So, an interest in Unpredictability/Predictability can be measured by
Shannon entropy.
(The subject of "degree of dissipation" or of disorganization never
arises here.)
Example III: A compound interest in both dimensions (Organization X
Predictability) jointly.
Let's go back to Example I above, where we are interested in the various
ways that H and O can bond.
Suppose that we take that interest a little further an ask...
"What is the probability distribution of the observed molecules and ions
involving H and O?"
Now, we have combined our interest in both "organizations" of H and O,
as well as
the relative probabilities of their occurrences.
Thus, our probability (sample) space now, by definition, has the
following possible outcomes: O_2 , HO, H_2 O, etc.
And, each has its observed probability, and thus we have a joint
probability distribution that we can apply Shannon's entropy against.
Depending on the probabilities of each of these "H-O compounds", the
Shannon entropy may be high or it may be low.
HTH,
Grant
Nicholas Thompson wrote:
Grant --
Glad you are on board, here. I will read this carefully.
Does this have anything to do with the Realism Idealism thing.
Predictibility requires a person to be predicting; organization is
there even if there is no one there to predict one part from another.
N
*From:* [email protected] [mailto:[email protected]]
*On Behalf Of *Grant Holland
*Sent:* Saturday, August 07, 2010 2:06 PM
*To:* [email protected]; The Friday Morning Applied Complexity
Coffee Group
*Subject:* Re: [FRIAM] entropy and uncertainty, REDUX
Russ - Yes.
I use the terms "organizational" and "predictable", rather than
"structural" and "behavioral", because of my particular interests.
They amount to the same ideas. Basically they are two orthogonal
dimensions of certain state spaces as they change.
I lament the fact that the same term "entropy" is used to apply to
both meanings, however. Especially since few realize that these two
meanings are being conflated with the same word. Von Foerster actually
defined the word "entropy" in two different places within the same
book of essays to mean each of these two meanings! Often the word
"disorder" is used. And people don't know whether "disorder" refers to
"disorganization" or whether it refers to "unpredictability". This
word has fostered the further unfortunate confusion.
It seems few people make the distinction that you have. This
conflation causes no end of confusion. I really wish there were 2
distinct terms. In my work, I have come up with the acronym "DOUPBT"
for the "unpredictable" meaning of entropy. (Or, "behavioral", as you
call it.) This stands for Degree Of UnPredictaBiliTy.) I actually use
Shannon's formula for this meaning.
This all came about because 1) Clausius invented the term entropy to
mean "dissipation" (a kind of dis-organization, in my terms). 2) But
then Gibbs came along and started measuring the degree of
unpredictability involved in knowing the "arrangements" (positions and
momenta) of molecules in an ideal gas. The linguistic problem was that
Gibbs (and Boltzmann) use the same term - entropy - as had Clausius,
even though Clausius emphasized a structural (dissipation) idea,
whereas Gibbs emphasized an unpredictability idea (admittedly,
unpredictability of "structural" change).
To confuse things even more, Shannon came along and defined entropy in
purely probabilistic terms - as a direct measure of unpredictability.
So, historically, the term went from a purely structural meaning, to a
mixture of structure and unpredictability to a pure unpredictability
meaning. No wonder everyone is confused.
Another matter is that Clausius, Boltzmann and Gibbs were all doing
Physics. But Shannon was doing Mathematics.
My theory is Mathematics. I'm not doing Physics. So I strictly need
Shannon's meaning. My "social problem" is that every time I say
"entropy", too many people assume I'm talking about "dissipation" when
I am not. I'm always talking about "disorganization" when I use the
term in my work. So, I have gone to using the phrase "Shannon's
entropy", and never the word in its naked form. (Admittedly, I
eventually also combine in a way similar to Gibbs :-[ . But I do not
refer to the combined result as "entropy".)
:-P
Grant
Russ Abbott wrote:
Is it fair to say that Grant is talking about what one might call
structural vs. behavioral entropy?
Let's say I have a number of bits in a row. That has very low
structural entropy. It takes very few bits to describe that row of
bits. But let's say each is hooked up to a random signal. So
behaviorally the whole thing has high entropy. But the behavioral
uncertainty of the bits is based on the assumed randomness of the
signal generator. So it isn't really the bits themselves that have
high behavioral entropy. They are just a "window" through which we are
observing the high entropy randomness behind them.
This is a very contrived example. Is it at all useful for a discussion
of structural entropy vs. behavioral entropy? I'm asking that in all
seriousness; I don't have a good sense of how to think about this.
This suggests another thought. A system may have high entropy in one
dimension and low entropy in another. Then what? Most of us are very
close to the ground most of the time. But we don't stay in one place
in that relatively 2-dimensional world. This sounds a bit like Nick's
example. If you know that an animal is female, you can predict more
about how she will act than if you don't know that.
One other thought Nick talked about gradients and the tendency for
them to dissipate. Is that really so? If you put two mutually
insoluble liquids in a bottle , one heavier than another, the result
will be a layer cake of liquids with a very sharp gradient between
them. Will that ever dissipate?
What I think is more to the point is that potential energy gradients
will dissipate. Nature abhors a potential energy gradient -- but not
all gradients.
-- Russ
On Thu, Aug 5, 2010 at 11:09 AM, Grant Holland
<[email protected] <mailto:[email protected]>> wrote:
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
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
--
Grant Holland
VP, Product Development and Software Engineering
NuTech Solutions
404.427.4759
------------------------------------------------------------------------
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
--
Grant Holland
VP, Product Development and Software Engineering
NuTech Solutions
404.427.4759
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
