Dear Organlearners,
Jon Krispin <jkrispin@prestolitewire.com> writes:
>Herrnstein's matching law, in its general form, can be expressed
>in the following equation:
>
>Pr(A) = k( Ra/Ra+Re )
>
>Where
>
>Pr(A) = probability of behavioral response A
>k = highest # of possible behavioral responses available
> (total # of responses)
>Ra = Reinforcement experienced for response A
>Re = Reinforcement experienced for extraneous behavior (behavior
other than the response of interest)
and then later on:
>This brings us back to the general form of the matching law
>that I originally stated
(snip)
>and the Digestor model that At articulated in LO21272 in the
>form of free energy:
>
>/_\F(un) = -/_\n*mSU*[E(mSY, Msy) - E(mSU, Msu)]
(snip)
>I would propose that the matching law is the equivalent of the
>Digestor model as it would apply to behavior with the following
>"translation":
(snip)
>Or maybe the Digestor model is the matching law for behavior
>extrapolated to the world of crystallography. Is this obfuscation?
>IMHO, I don't think so. In both cases, the relationships were
>uncovered independently. Which relationship was "discovered"
>first? What should we do with this?
Greetings Jon,
This is my second reply to your lovely contribution. In my first
contribution I steered clear of Herrnstein's law and your "adjunction"
(patterns of correspondence) between it and the Digestor.
Before I go into technicalities, allow me some general remarks.
Firstly, the issue of "who made what discovery first"? The Digestor as
model for self-organisation close to equilibrium was discovered in the
middle eighties by me. It is based on the Ostwald digestion process of
producing larger and more perfect crystals. This process was
discovered and decribed early this century by Wilhelm Ostwald.
However, using large, perfect crystals metaphorically to indicate a
sure outcome of human behaviour, dates back much further. The earliest
references are in the history of ancient Mesopotamia. Much later,
Moses of Israel used precious stones in the seremonial clothing of the
high priest. About 1600 years later the Apostle John used crystals to
describe some of the Revelations of Jesus Christ. About Another 1600
years later and Gottfried Leibniz used a crystal in his Philospher's
Dream.
Secondly, note that in naming the process discovered by Ostwald the
"digestion" process, a metaphor (eating) from the animal kingdom has
been introduced into the inanimate world. But hold your breath -- the
Ostwald process is sometimes also called the "ripening" process. This
name introduce a metaphor from the plant kindom. This another name for
the model could gave been "Ripener". Is this obfuscation?
Thirdly, Ostwald is not only famous for the Ostwald digestion process.
For example, in 1896 he intoduced the unit "mole" to chemistry. It is
the primary unit which differentiates chemistry from phsyics. The
"mole" is a very large unit for counting. It is in the same class as
the units "dozen", "gross" and "ream". Its value is
1 mole = 6.02x10^23 (10 to the power 23)
1 gram of the carbon-12 isotope has exactly 1 mole of carbon atoms.
Another thing which I enjoy much, is to look at the movie made up by
all the contributions on numerous topics on this list. Does a LO
involve only the abstract concepts of learning and organisation? No.
An ongoing topic parallel to the Digestor as topic, is "Buildings,
Offices as LO enabler", initiated by Rick. Yes, even the physical
structure in which a LO has to operate, now receives our attention.
But what about the individual learners who make up a LO? Is the body
of a learner not a physical structure also? What happens inside this
body? Chemical reactions -- thousands of them. If the building
("body") which has to accomodate a LO deserves attention, why not the
body ("building") which has to accomodate all the chemical reactions
from which a LI (Learning Individual) will emerge?
I have already pointed out in a reply to Leo Minningh
("Transdiciplinary Thinking") that the reaction involved in the
Ostwald digestion process is an identity reaction, i.e no change in
chemical identity happens. For example, we begin with crystals like
barium sulphate crystals BaSO4(c) and we end up with the same kind of
crystals. This makes the Ostwald digestion reaction a difficult nut to
crack by the tools of normal chemical reactions. However, to
understand digestion reactions, it is now high time to look at the
tools of normal chemical reactions. This will also help us to
understand the Brusselator, a model for a chemical reactions linked
together into a loop with positive feedback.
Jon, above all, I am thankful to you for your bold step in pointing
out a relationship between the Digestor and Herrnstein's matching law
for behaviour. The Scientific Method itself is a loop with primarily
three stages: observation, speculation, falsification. Stage 1 feeds
on any information. Thus it is primarily an evolutionary (digestive)
phase. Stage 2 (speculation) introduces a positive feedback while
stage 3 (falsification) introduces negative feedback. Thus stages 2
(emergent) and 3 (immergent) are primarilly the revolutionary phase.
You have already completed stage 1 (observing the Matching Law long
ago and the Digestor recently) and probably have completed stage 2
(proposing a relationship in general and specifically an "equivalency"
between the two). Thus I have to continue with stage 3.
However, I want to begin with stage 2. There are many kinds of
relationships between two seemingly unrelated phenomena. The
"equivalency" relationship propose by you is one of them. This
relationship says that they behave the same with respect to truth, i.e
they lead to the same true statements as well as false statements. In
other words, they are indistinguishable with respect to logic. In my
reply to John Gunkler ("Junk Science") I mentioned that logically
there are still 15 other logical relationships possible. For example,
the Digestor may IMP (implication) the Matching Law. Or perhaps we
should work with both together and not stress causality. In that case
the logical relationship would be AND (conjunction). In any case,
should I proceed to stage 3, I will have to weed out 15 of the logical
relationships is the relationship is indeed a logical one.
What we shoud bear in mind is that our speculation in stage 2 should
be as creative as possible before we proceed to stage 3. The less
creative we are in stage 2, the more we have to run complete cycles of
all three stages. Creativity is not restricted to logics. In other
words, we have to speculate on other relationships than merely logical
ones in step 2. We need real brainstorming as Osborne might have said.
For example, chemical relationships and biological go far beyond
classical logic. Thus why not speculate on chemical and biological
relationships between the Digestor and the Matching Law? Here is a
chemical relationship as example, namely the "leveling effect". It is
one which students in chemistry have great difficulty with because it
goes beyond classical logics. Ot what about "commensalism" (symbiosis)
as a biological example?
Should we speculate on a chemical relationship, we will have to get
our chemical acts in order when we proceed to stage 3 (falsification).
My own gut feeling is that chemistry in general has much to do with
the Matching Law and not merely the Digestor (digestion process). Thus
I want to offer to you and fellow learners a short summary of the soul
of chemistry, namely the chemical reaction. With this I intend to meet
you half the way on your bold assertion that the relationship between
the Digestor and the Matching Law is an "equivalency". In other words,
I want to help you in stage 3, weeding out the false speculations of
step 2 in case they are of a chemical nature.
The following balanced chemical equation refers to a specific chemical
reaction, namely
BaCl2 + Na2SO4 = BaSO4 + 2NaCl
Here, for example, NaCl is the chemical formula of a compound with the
chemical name "sodium chloride". It also has a common name -- table
salt! Our problem is that there are already several million of
chemical compounds known. With them we can set up billions of chemical
reactions! How are we going to refer to any one of these billion
possibilities in a general manner?
The general equation for any chemical reaction is
aA + bB + .... = cC + dD + ....
Here A, B, .... represent the formulae of chemical compounds called
the reagents. They are reagents because they occur on the left hand
side of the equation. Products occur on the right hand side,
represented by C, D, .... In the example of the previous paragraph,
A=BaCl2, B=Na2SO4, C=BaSO4 and D=NaCl. The numbers a, b, ...., c, d,
.... balance the equation for its elements. They are called the
"stochiometrcal coeffcients". In the example above Ba, Cl, Na, S and O
represent the elements barium, chlorine, sodium. sulphur and oxygen.
Furthermore, a=1, b=1, ...., c=1, d=2, ...... (Note one of the
idiosyncracies of chemists -- they never write the coefficient when it
has the value 1 (one).)
Now what exactly does the chemical equation represent? It is exactly
here where most pupils and students get lost forever in the subject
chemistry, especially if they have little sensitivity to the
essentiality liveness ("becoming-being") of creativity. It repesents
both "being and "becoming". Let us first consider the BEING of the
chemical reaction.
The equation tells us that at any MOMENT IN TIME, it consists of
species which we can summarise by the formulae A, B, ...., C, D,
...... Perhaps the species is not the formula as a whole, but parts of
it. For example, BaCl2 will dissolve to form Ba++ and Cl- ions. But at
this stage it does not matter. Of species A there will be n(A) formula
units, measured in moles. Likewise for B, ...., C, D, ..... we have
amounts (collections) n(B), ...., n(C), n(D), ....., expressed in
moles. Ordinarily, when we begin with a reaction, n(A), n(B), ....
will have maximum values while n(C)=0, n(D)=0, ....., (lowest of
minimum values). In other words, we begin a chemical reaction with
only reagents. But nothing prevents us from adding one or more or all
products C, D, ..... also to the reaction. In fact, WE ARE FREE TO
BEGIN ANY CHEMICAL REACTION WITH ANY AMOUNT OF EACH OF ITS
PARTICIPATING COMPOUNDS.
Yes, I want SHOUT it out. There are billions of billions of
possiblities how we can start the specific chemical reaction
BaCl2 + Na2SO4 = BaSO4 + 2NaCl
by using different amounts for BaCl2, Na2SO4, BaSO4 and 2NaCl. Since
there are billions of chemical reactions possible, the soul of
chemistry concerns billions of billions of billions ..... possible
beings. These possibilities staggers the mind. (This mind boggling is
so staggering that even probability theory becomes useless. I will
come back to it.) Herein lies a very important lesson for Learning
Organisations. It can begin with any number (more than one) of any
kind (more than one) of individual learners. A LO need not to be a
business organisation. It can be any kind of organisation such as in
schools, hobbies, lobbies, churches, politics and nations.
For the general chemical reaction we will write underneath it
aA + bB + .... = cC + dD + ....
n(A) n(B) n(C) n(D)
where n(A), n(B), ...., n(C), n(D), .... represent the amounts
(collections in moles) of the species A, B, ...., C, D, .... involved
at any MOMENT OF TIME. Usually, we write down the values of n(A),
n(B), ...., n(C), n(D), .... down for the initial (beginning) moment.
Now how do chemists deal with such a staggering number of
possibilities in the BEING of a chemical reaction? By also having a
BECOMING in the same chemical reaction. This becoming will be written
in the second line underneath the chemical reaction as
aA + bB + .... = cC + dD + ....
n(A) n(B) n(C) n(D)
z(A) z(B) z(C) z(D)
(I would have prefered to use the symbolic expression /_\n or "delta
n" rather than the one single symbol z because this is actually what z
stand s for, namely the "change in n".) It tells us that during any
INTERVAL OF TIME an amount z(A) of A, an amount z(B) of B, ..... will
disappear (immerge) while an amount z(C) of C, z(D) of D, .... will
appear (emerge). Furthermore, whereas we are free to choose the
amounts n(A), n(B), ...., n(C), n(D), ..... when beginning the
reaction, the amounts z(A), z(B), ...., z(C), z(D), .... have to
match. Thus chemistry has it own "matching law". Now for the great
surprise, using * for multiplication, z(A)=z*a, z(B)=z*b, ....,
z(C)=z*c, z(D)=z*d, ......
By rewriting these matching laws below the general chemical equation,
we get
aA + bB + .... = cC + dD + ....
n(A) n(B) n(C) n(D)
-z*a -z*b +z*c +z*d
The sign - means disappear (immerge) and the sign + means appear
(emerge). The quantity z (or rather /_\n) is most important. It is
called the "advancement" of the reaction. (This z and the /_\n of the
Digestor equation is one and the same thing.) The z may be though of
as the flow (flux) of the reaction. In a simple reaction it increases
uniformly. In a complex reaction consisting of many simple reactions,
it increases steplike. At every step it is determined by the slowest
of all the simple reactions at that step.
So where does the chemical reaction "advance" to? Is there also
billions of billions of possible end (output) states just as there are
possible beginning (input) states? Yes, yes, yes. Each unique input
state has its corresponding unique output state after every interval
of time. In other words, the chemical reaction is mathematically not a
relation, but a function. Is all these staggering possibilities not a
source for great dispair or imidation? No. When the output state has
reach the final end along the creative course of time, we have
K = n(C)^c*n(D)^d...../n(A)^a*n(B)^b.....
where * mean "multiplication", ^ means "power" and / means "division".
Thus n(A)^a means the amount (collection in moles) n(A) of substance A
raised to the power a (its stoichiometrial coefficient). It is very,
very important to note that n(A) refers to the amount of A at the very
very end state.
What is so peculiar about K? Surely, anybody can think out any kind of
expression which involves n(A) and a, n(B) and b, etc. But of all
those many possible expressions, only the one above has two unique
properties which I will soon come to. On the other hand, all those
expression like the one for K above have one other property in common.
The value of each of them, like the value for K above, becomes
constant. Why? Because the values of n(A), n(B), ...., n(C), n(D),
..... do not change any more when the final end state has been
reached. This final state is better known as the "equilibrium state".
Should we have begin with the chemical reaction with values for n(A),
n(B), ...., n(C), n(D), ..... equal to that which they will have in
the equilibrium state, the values for each of z(A), z(B), ...., z(C),
z(D), .... would have been zero. In other words, there would not have
been any advancement z for the reaction. Thus we can think of the
equilibrium state as the "attractor state". Any state not being the
eilibtium
The one unique property of K is the following. Except for the
influence of tempertaure, the value of K is fixed for any specific
reaction. In other words, although the values for n(A), n(B), ....,
n(C), n(D), ..... can have billions of billions of variations at the
equilbrium state because the reaction began with billions of billions
of variations in the beginning state (remember functionality), once we
have combine these values into the expression for the constant K, it
is one and the same fixed value. In other words, the billions of
billions of variations in the right hand side of
K = n(C)^c*n(D)^d...../n(A)^a*n(B)^b.....
is forgotten by K on the left hand side. Each chemical reaction has
its own unique equilbrium constant K. Now is that not extraordinary
order in what seems to be utter chaos!
The other unique property of K is even more astounding. Every chemical
compound X has its own free energy F(X). It is unique for each
compound. F(X) increases/decreases as the amount of X increases, the
amount being expressed by n(X). Thus we can imagine what happen during
the chemical reaction. If the advancement z is in a forward direction
(from left to right), the free energy of each reagent and thus the
free energy of all reagents decrease while the free energy of each
product and thus the free energy of all products increase. The free
energy of the reagents do not change with the same amount as the fre
enegy of the products as the reaction adavnces. Thus there is a
gradual change of the free energy for the reactive system as a whole
(reagents and products). Now for exactly one mole of change in the
advancement /_\n (or z) of the reaction, the change in free energy
/_\F for the system can easily be calculated from tables of the
formation free energy of the compounds. It has become clear that each
reaction has its own associated /_\F. So what has /_\F and K to do
with each other?
Once again, J W Gibbs, the genius among geniusses, provided the
answer. The relationship is
/_\F = -R*T*lnK
or
K = e^(-/_\F/RT)
where "e" is the natural number, "ln" the natural logarithm, "T" the
temperature and "R" the universal gas constant. Incredible, is it
not -- such a simple order in the chaos of billions of billions of
billions of possibilities? In other words, the molar (1 mole of
advancement) change in free energy /_\F for every reaction determines
its attractor state expressed by K in a perfect manner. This
relationship has been tested a zillion of times. All zillion chemical
laboratories and industries are designed and run upon it being the
actual relationship between BEING and BECOMING.
Jon, can you imagine with what shock I suddenly realised that chemists
in their chemical reactions do not work with Probability Theory (PT).
The great physicist C Maxwell created PT and introduced it to physics
in his epic work "Theory of heat". Sadly, among other things, he
wanted to reduce the Law of Entropy Production to the Law of Energy
Conservation by combining the latter with PT. The far majority of
physicists actually believe that he his dream has been fulfilled. If
that dream gets fulfilled, then entropy is purely a measure of chaos.
But maybe it is just a dream which will never become actualised.
In the mean time, Gibbs accomplished something which became far more
valuable for chemists than PT for physicists. He extended
thermodynamics to include chemistry. He gave them the equation
/_\F = -R*T*lnK
so that they need not work with probabilities in the billions of
billions of possibilities which they have to manage. This very
equation is the reason why PT did not flourish in chemistry. No wonder
that when the genius Maxwell read that great (phsyically and
intellectually) paper of Gibbs, he exclaimed to the effect that "here
is a man who has forgotten what we still have to learn".
Jon, this contribution itself is already too long. I have not yet
given attention to the "identity peculiarities" of the Digestor. I
will have to do so at a later time. But I do hope that I have given
you enough to rethink the relationship between the Digestor and the
Matching Law of behaviour psychology. The equation
/_\F = -R*T*lnK
establishes an equivalency between the change in free energy /_\F for
any chemical reaction and the equilibrium (attractor) constant K for
the same reaction. Thus I do not suspect a direct relationship between
/_\F and the probability Pr for the Matching Law
Pr(A) = Ra/(Ra+Rb)
However, since K is related to the "numbers" (amounts, collections in
mole) n(A), n(B), ...., n(C), n(D), ..... by the equation
K = n(C)^c*n(D)^d...../n(A)^a*n(B)^b.....
I strongly suspect that the probability Pr will be connected
indirectly through K to /_\F since Pr is laso related to numbers. In
other words, your gut feeling is in the right direction. We only have
to refine it!
Fortunately, you have taken the first bold step so that something has
emerged which we can refine. By denying (even by using cliches such as
obfuscation) such bold steps in stage 2 (speculation) of the
Scientific Method, how will we ever get to stage 3 (falsification)?
Furthermore, has obfuscation ever worked as a valid falsifier in the
history of science? I would be glad to learn of it, even if it is
merely one single case.
Jon, you end your contribution with:
>I, for one, find these parallels to be compelling, and I have
>invested a considerable amount of time articulating the
>connections that I have made for others to examine. To me,
>the obvious next step is to see how much further the model
>can be extended. Is this really a step into oblivion?
No.
Best wishes
--At de Lange <amdelange@gold.up.ac.za> Snailmail: A M de Lange Gold Fields Computer Centre Faculty of Science - University of Pretoria Pretoria 0001 - Rep of South Africa
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