Fitness Landscape and other landscapes. LO27222

From: AM de Lange (
Date: 09/03/01

Dear Organlearners,

Greetings to all of you.

In a recent private E-mail discussion on the "cloning of stem cells", I
came under the impression how an acquaintance to the idea of a "fitness
landscape" can actually impair understanding the dynamics of free energy
and entropy production.

The notion of a fitness landscape was already introduced in 1930 by the
geneticist Sewall Wright. Stuart Kaufman with co-workers incorporated it
into his theory of Complex Adaptive Systems (CASs). They developed a set
of complex models (called NK-models) to study algorithms for solving the
problem how complex interconnected systems interact. The theoretical
NK-model requires an "element dimension" with a number N of system
elements as well as a "state dimension" with a number S of different
states which each element can be in. Interconnections between elements are
modelled by assigning each element to a "spillover set," consisting of K
elements. A "fitness function" is then specified which represents each
element's contribution to a system-wide variable. This fitness function
has to have a limit (maximum or minimum) with respect to the configuration
of the K elements in the spillover set.

Many kinds of theoretical functions with limits and many kinds of
theoretical algorithms for finding these limits are possible. An algorithm
which Kaufmann et al found particularly attractive is the simple
trial-and-error procedure known as the "adaptive walk." It is very
efficient at finding the highest point on the fitness landscape in systems
that have no interconnections or spillovers between elements (i.e., in
systems with K = 1) so that an element's fitness contribution is a
function only of its own state. But as the spillover effects increase (K >
1) so that the system becomes more complex, the algorithm performs
progressively less efficient. The adaptive walk becomes increasingly
trapped on what is called a "local fitness peak" (local optimum) from
which there is no escape towards greater fitness.

It seems that increasing the spillovers induces a ruggedness into the
fitness landscape based on the element dimension N and the state dimension
S. This ruggedness looks like a landscape with many mountains of various
heights of which one is the highest. That peak is called the "global
fitness peak" (global optimum). The more interconnected the individual
elements, the more likely the adaptive walk will end up in a local rather
than the global optimum from which there is no escape by means of the
adaptive walk. Thus Kaufmann uncovered a variant to he adaptive walk which
he calls the "patching algorithm". In essence the adaptive walk is allowed
to leave a local optimum if there is a higher local optimum close by, the
two local optima forming a "patch".

I have drawn for you fellow learners such a fitness landscape. See
Figure 1. Please Rick, archive it for us and supply the URL below in
your kind and efficient manner.

Figure 1 is a 3D graph. The two horizontal axes represent the elements N
and the states S. The vertical axis represents the fitness. This landscape
which I designed is not particularly rugged. The adaptive walk algorithm
here may lead to two local optima. The extended patching algorithm will
easily lead to the global optimum. However, imagine a hollow dishlike
basin with many hills in it, those towards the rim higher than those
towards the centre as well as a high mountain in the centre. For such a
landscape the patching algorithm will not find its global optimum at the
centre when beginning somewhere close to the rim.

I know of several such eerie places in the Richtersveld desert in the
north-west of South Africa at its "Big Bend". These places are eerie
because a person's sense for perspective and field depth gets completely
lost (and you will experience it when getting lost there ;-). Landscape
artists will find these places compelling. I had a late friend (lecturer
in art) who took his students to these places to become freed of their
metal models. But the most compelling such a place for me is the Namgorab
desert to the south-west of Namibia. Here the same dishlike basins can be
observed, but with less ruggedness because of the sand plains in between.
Consequently some perspective and depth of field is regained with such a
gentle effect as if one is now gazing upon heaven. This is where my dear
wife exclaimed upon her first visit to it "At, you are right, it is

In my own mind I have made like Goethe many interconnections between the
main subjects (branches) of knowledge with many spillovers from one
subject to all the others. It is something which baffles as well as
frustrates you fellow learners. Goethe did it to excel in his poetry and I
do it to excel in authentic learning, hopefully then knowing how to teach
it ;-) I know how you feel because I felt the same. How many times since
1970 had I not become trapped on a local optimum, not knowing how to reach
the next one. But, whereas Kaufmann discovered the "patching algorithm", I
learned a pattern which with difficulty I could much later articulate as
the MonCat (monadic categorical) pattern. I later found out that Goethe
already as far back as two centuries ago called it
"Steigerung"=staggering. In Afrikaans it would be "opstapeling" so that
"(up)stapling" in English will also do.

As the years passed, I became aware how this staggering enabled me to
wander from local optimum to local optimum in an endless manner. Among
other things, I learned how "entropy production" driven by "free energy"
changes drives this staggering while the 7Es (seven essentialities of
creativity) guides the fractal progression in the staggering.

When I think of the NK-models of Kaufmann, I perceive in terms of my own
system of thinking that the "element dimension N" corresponds to the
essentiality otherness ("quality-variety") while the "state dimension S"
corresponds to the essentiality spareness ("quantity-limit"). The five
other 7Es then interconnect these elements and states to cause spillovers
from almost any subject in the academical spectrum to almost any other
subject. These spillovers are one of the many kinds of a thing which I
often refer to by the phrase "one-to-many-mapping" in our LO-dialogues.
The source of these one-to-many-mappings is in the very LEP (Law of
Entropy Production) as I have explained in a number of dialogues.

In 1982-83 I discovered empirically that LEP also has a spiritual
dimension. In 1986, if I remember correctly, I was teaching university
students the intricate calculations concerning free energy in chemical
reactions. During that lecture I suddenly became aware how my mind was
rushing along two levels. The lower level was concerned with chemical
processes as a physical phenomenon. I executed this level almost
automatically. But the higher level of my mind was exploring free energy
in the process of knowing/learning as a spiritual phenomenon. I followed
this level with great curiosity. I think the students became aware that I
was rather absent minded that day, not perceiving my highermost thoughts.
That day marks the beginning of my continual exploring of the role of free
energy in the spirituality of humankind.

To manage both material and mental systems I had to reform my
understanding of the "Gibbs free energy" G and entropy S considerably
compared to that of 1967 (after five years of training in chemistry,
mathematics and physics). I learned since 1968, as a result of doing
research on soils which are surprisingly complex, that Classical
Thermodynamics (CT) cannot handle complex systems whereas Irreversible
Thermodynamics (IT), initiated by Prigogine, can do so. The first major
difference between the two is that in CT the focus is on "entropy S" as a
picture whereas in IT the focus is on "entropy production /_\S" as a
movie. The second major difference is that in CT entropy is interpreted as
chaos whereas in IT the causes and consequences of entropy production /_\S
get attention rather than trying to interpret it.

I am deeply under the impression of what I am about to write, will not
readily connect to your own experiences and tacit knowledge. But let me
articulate it because somehow it might express your own experiences.

Let me then give a quick summary of "free energy" F as I understand it
presently. Every system, material or mental, has patterns in it. When
these patterns are considered together, I think that they can be called
the system's organisation. The entropy S of the system is a measure of the
system's organisation, irrespective of the kinds of patterns (order or
chaos, structure or process, microscopic or macroscopic). The system's
organisation cannot change without its entropy S changing too.

The change (symbolised by /_\) in the entropy S gets the name "entropy
production" (symbolised by /_\S). The reason is that entropy is increased
(created) by what I prefer to call "entropic" (rather than thermodynamic)
"force-flux pairs". These entropic force-flux pairs emerge when some
forms of energy are converted into different forms of energy. Every form
of energy may have its own entropic force-flux pair. Every form of energy
may be expressed by the product XxY of an extensive quantity X and an
intensive quantity Y. When any system is scaled in size, all its extensive
quantities X get scaled while all its intensive quantities Y remain the
same. Differences in the values of an extensive quantity X is an entropic
flux while differences in the values of an intensive quantity Y is an
entropic force.

All the forms of energy of a system together is called its "total energy
E". Some of this total energy E is bound into maintaining the systems
present organisation, among others even those forms of energy! The rest of
the total energy E is called the system's free energy F. Since it is not
bound to keep up the system's organisation, it is readily available to
drive the transformation in energy forms of the system and hence the
associated entropy production /_\S.

When the system does not have sufficient free energy F to drive its
entropy production and thus organisational changes, there are two options
to obtain further free energy. One option is that it can be imported from
an external source like animals eating food or plants absorbing sunlight.
The other option is that it can be liberated from within by giving up some
of its organisation through a process which I call the "creative collapse"
(the complexity version of Heidegger's and Derrida's "deconstruction").
Taking animals or plants as example, they will then use their own tissues
as source of free energy rather than eating food.

The change of the system's free energy F during its course through
successive higher levels of complexity is itself very complex. That course
may be called evolution when it involves innumerous generations or
metamorphosis (development) when it involves one generation. Evolution and
metamorphosis can be observed in both material and mental systems. Assume
one level of complexity has been reached by the emergence of "kernels" or
the "infants" of that level. They now have to mature into "adults" by a
process called digestion. Their intensive (scaling independent) properties
Y do not change much whereas their extensive (scaling dependant)
properties X do change considerably. These extensive properties grow
during the digestion so that the total energy E, entropy S and thus free
energy F of the system increases. At first the increase in total energy E
is fast, but it slows down gradually as maturity is reached. Thereafter it
may even decrease gradually.

Processes happen spontaneously when the free energy F of a system
decreases. But during digestion the free energy of the system increases.
Is this not a contradiction? No, for digestion the system has to be open.
Thus we have to consider the system and all the surrounding systems upon
which it feeds as a whole. For this whole ecosystem (the system SY and the
surroundings SU) the free energy does decrease so that the digestion
happens spontaneously. However, the system's gain in free energy is made
possible by the extended loss in free energy of the sourcing systems. One
system gets rich while many systems get poor.

When maturity is reached, the system begins to use that free energy to
drive itself to the ridge of chaos where other kernels of the same order
or even kernels of a higher order can emerge. At first the entropy
production is low, but as the system approaches the ridge of chaos, enough
entropy has to be produced fast enough to actually reach that ridge. In
other words, the pattern of entropy production in the swing towards chaos
to acquire new qualities which are opposite to the pattern in the swing
towards order in the same qualities.

It is frequently claimed that systems have to be open during the swing
towards chaos. However, this is often detrimental to the system. What will
happen is that the system SY will lose to the environment SU the extra
entropy it produced rather than using it to change its internal
organisation. When this happens, the system begins to feed like a parasite
upon the free energy sources in its environment to supplement its own free
energy which ought to be liberated by means of a creative collapse. This
supplement becomes detrimental to the environment. The latter loses free
energy which it then cannot use self. Later on the environment gets
deluged by entropy which it has not produced self. This free energy
poverty and entropy pollution which degrade the environment will finally
cause the demise of even that too an open system.

The digestive swing together with the bifurcative swing form a cycle.
(Many biochemical and ecological cycles are essential to life.) This
peculiar cycle stays in form essentially the same. But in content it
increases the free energy F, entropy S and total energy E after each
cycle. In nature such a peculiar system would let to catastrophes (few
becoming rich at the expense of many becoming poor) were it not for
mutualistic symbiosis. Furthermore, somewhere something must act as a
source of free energy at such a vast time scale so that whatever uses it
without giving back, can indeed complexify through all levels in this time
scale. This source of pure free energy in the case of our world is nothing
else than an yellow star which we call "sun".

I have drawn for you fellow learners a "free energy landscape". See Figure
2. Please Rick, again archive it for us and supply the URL.

This landscape is a simplified version of the image in my mind. The
vertical axis represents free energy, the "Urphaenomen" or prototype of
all functions having limits. All fitness functions, how imaginative we may
create them, depend on free energy as the mother of them all. No change is
possible without free energy changing somewhere in the universe, whether
in the system SY or in the surroundings SU. The free energy F is not
merely a theoretical concept of the imagination. It is a quantity based on
innumerous measurements and calculations in the realm of physical
chemistry. It is a quantity of bewildering consequences, the nemesis of
many a student in physical chemistry.

The two horizontal axes in Figure 2 may be any two of the 7Es. In the case
of Kaufmann they are the elements N as an indication of the essentiality
otherness ("quality-variety") and the states S as an indication of the
essentiality spareness ("quantity-limit"). Should we try to depict all 7Es
rather than two of them, we would arrive at a 7D image which is impossible
to represent with our 3D geometry. In my own empirical discovery that LEP
has indeed a spiritual complement, the two horizontal axes of such a free
energy landscape were wholeness ("monadicity-associativity") and sureness
("identity-categoricity") rather than Kaufmann's N and S.

Please notice the two shaded regions, designated past and future
digestions. (Forget for a moment the thickest lines called A, B and AB as
well as the strange barrier in the unshaded region where the two lines
join.) The free energy in both shaded regions increases (the landscape
bulges upwards to a summit). The difference is that the "hill of the past"
is often lower than the "hill of the future". (I have actually drawn the
future hill much higher than the past hill so that you can easily observe
it.) Looking towards all the free energy hills of the future, there is a
gradual elevation along the future hills.

It is as if the system is gradually climbing the rugged landscape called
free energy F from sea level towards a high mountain "Everest" beyond the
horizon. "Steigerung"=staggering is necessary to do so. Specialization, on
the other hand, will cause the system to stay meandering within in a
"patch" (region) containing only some hills this side of the horizon. In
Namibia, should we begin to meander inland from the Skeleton Coast, any
specialisation (moving within a patch) would be deadly. That is why that
some 70km broad coastal plain got its name Skeleton Coast. But should we
stagger from patch of dunes to patch of dunes eastwards as I have done, we
will eventually cross patches of hills becoming gradually higher. Then,
suddenly the summit of the Brandberg ("Burnt Mountain") will appear on the
horizon. From foot (not sea level) it is the highest mountain in Africa.
Yet another 50 km is needed before arriving at its foot.

The entropy production /_\S decreases when going up the shaded region of
each hill so that at the summit of that hill it is very low. It means that
the entropy S becomes almost static at the summit. Thereafter the system
has to go down that hill along the unshaded region. (Again, forget for a
moment the thickest/highest lines called A, B and AB as well as the
strange barrier in the unshaded region where the two lines join.) But now
the entropy production /_\S increases because the free energy gets lower.
The order of the system cannot change until it has reached that strange
barrier. Hence this entropy production has to manifest itself as
increasing chaos rather than increasing order. We may think if it as the
"swing to chaos" which succeeds the "swing to order" in the shaded region.
Together they form a cycle.

Now please focus on that strange barrier of free energy in the valley of
the unshaded region. Perhaps I have drawn the barrier too high, even
higher than the summit of the hill to the past. But I did it to signify to
you its crucial importance. The system will not reach the next hill unless
it can "bump" over that free energy barrier. This barrier looks like a
"thin" hill itself, but its summit can never be reached by digestion with
decreasing entropy production /_\S so that the entropy S becomes almost
static. It has to be reached by a bifurcation with increasing entropy
production /_\S so that the entropy S changes rapidly on it. Here on the
barrier /_\S is high whereas at the summit of the ordinary hill /_\S was

Now also focus on the two thick lines A and B going up and over the hill.
They designate two subsystems (maturing kernels or organs) A and B of the
system. They mature by digestion as they go up the hill. As they go down
the hill, they begin to move towards each other so as to collide at the
bottom. But just before making contact with each other, they begin to
experience a barrier of free energy AS A RESULT OF the very contact which
they have to make. Should this barrier not be too high, they will connect
effectively at the summit. But when it is too high, the barrier will
bounce them back before they can make an effective collision with each
other. The height of this barrier is determined by nothing else than the
LRC (Law of Requisite Complexity)! The more complex (not confused ;-) the
system, the lower the free energy barrier.

I have to stress that the lines A, B and AB do not belong to the 3D. They
actually introduce a fourth dimension, namely fruitfulness
("connect-beget"). The decreasing "distance" between A and B indicates the
increasing fruitfulness between A and B.

The free energy barrier is most important to all chemical reactions and
especially so in biochemistry. It is here where the role of catalysts
(called enzymes in biochemistry) becomes crucial. Activating catalysts
lower the free energy barrier so that effective contact becomes easier.
Inhibiting catalysts make the barrier higher so as to prevent an effective
contact. Both kinds of enzymes are needed in living organisms. The
activating enzymes act like teachers should do while the inhibiting
enzymes act like the police should do. I suspect that when a totipotent
stem cell is transformed into a differentiated somatic cell, it is enzymes
which switch some genes on and others off, the total DNA remaining the

When you compare figure 1 (related to the work of Kaufmann) with figure 2,
you may easily get confused by the vast differences. However, it is
possible to understand how they correspond to each other. First of all,
remove in your imagination the free energy barrier shown and all others
not shown. What remains is a rugged landscape of hills going smoothly into
valleys between them. In South Africa close to Durban there is such a
place called the Valley-of-a-Thousand-Hills.

Secondly, take the smoothened version of figure 2 (all hills, but no
barriers) and invert by your imagination the landscape so that hills
become valleys and valleys become hills. (This inversion hinges on two
assumptions which we will discuss later on.) By this inversion you will
have transformed the vertical axis which represented the free energy F
into an axis which now represents the entropy S of the system. You will
then look at the entropy landscape. It is almost isomorphic (similar) to
the fitness landscape of NK-models.

When the system goes up a hill of the entropy landscape (down the hill of
a free energy landscape) with entropy production its chaos will increase
since it has not yet reached the summit where the new order begins. At the
summit or "ridge of chaos" a new order may emerge from the chaos upon the
present order. In short: "Order out of Chaos" as Prigogine articulates it.
Then, going down the hill of the entropy landscape (up the hill of a free
energy landscape) by digestion, the entropy production decreases. When it
reaches the "valley of equilibrium" the new order has become mature too. I
suspect that many of the constraints in the fitness landscape is a mature
order not capable of emerging into a new order -- a fixation of mental
models and paradigms.

Should we introduce the free energy barrier of the free energy landscape
once again into also the entropy landscape, what will we get? (Remember
that we have to invert all hills into valleys and valleys into hills.) The
free energy barrier in the free energy valley now becomes a cleft in the
summit of the entropy hill. This is the crucial difference between entropy
landscape and the fitness landscape of Kaufmann. The entropy landscape has
clefts in almost all of its hills. Sometimes these clefts are shallow and
thin like mere gulleys. But sometimes they are deep and wide like canyons.
When reaching the summit of a hill just to find a canyon in front rather
than a gully which can be jumped over, is a startling experience. It is
then when the "ridge of chaos" becomes the "edge of chaos".

I have drawn for you fellow learners a "entropy landscape". See Figure 3.
Please Rick, archive this last figure for us and supply the URL.

Canyons are usually features of deserts because of little erosion
happening there. In the Great Namaqualand desert to the south of Namibia
there is a magnificent canyon called the Fishriver Canyon of the likes of
the Grand Canyon in the USA. I have many stories to tell of my own
experiences in the Fishriver Canyon, some seemingly too fantastic to be

Occasionally the entropy gulleys are so immense that we may think of them
as a rift in a continent. One famous such a rift is the East African Rift.
When I visited it, I came under the impression that it is too big to
really comprehend. So I will not speak about it, but leave it to our dear
fellow learner Leo Minnigh the geologist to tell us more about continental
rifts and what important role they play in continental drifts. I want to
stress upon him to connect these rifts with those problems which plague
our organisations in a manner known as "paradigm shifts".

Let us look at the two assumptions made in this inversion. The first is
that the total energy E remains constant. However, systems may inflate or
deflate. When a system inflates (like in expansion or growth) the total
energy E gradually increases. The effect of this on the entropy landscape
is to make the lowering elevation less or even change it into a lifting
elevation. However, the relative distances between the summits of hills
and the valleys adjacent to them stay the same. But when the system
deflates (like in downsizing) because of decreasing total energy E the
entropy landscape drops gradually like moving from the inland to the
ocean. Many of the future hills are lower then the previous ones of the

The second assumption concerns the temperature (of material systems) and
the temperament (of mental systems). I will use the word "tempera" for
both. When the tempera (-ture or -ment) is hot (high), the effect is to
make differences in height of the entropy landscape less. Hills become
lower and valleys (and gullies) become shallower. It is as if an
increasing tempera has the typical "erosion effect" so common to
geographical landscapes. The entropy landscape becomes more smoothened.
But when the tempera is lower (cool), hills become higher and valleys
(even gullies) become deeper. Here the saying "cool down" is apt. As the
spirit cools down, the stark topographical features become clearer. The
hills become higher while the valleys (and even the gullies) become
deeper. The ruggedness of the entropy landscape becomes amplified,
allowing a better course to be planned.

Since as a kid I was aware of the abyss (super rift) between the material
and mental worlds. I tried to bridge this abyss, but could not. Yet I kept
on with my own "Steigerung" as Goethe did. Then during 1982-83 I
discovered empirically that LEP (Law of Entropy Production) applies to the
spiritual world as it applies to the physical world. My joy knew no
bounds. I have found the bridge between these two worlds with which to
cross the abyss between them. But all others thought I was crazy and would
not dare to publish my account.

To make categorically sure that I was not fooling myself, I set myself the
task to discover a second bridge over the abyss. The result was the 7Es
(seven essentialities of creativity). Since then I have learned that the
7Es are far more important as a bridge than LEP. LEP is dangerous when
used in ignorance. To control and guide the outcomes of LEP we need the
7Es. They connect us to the mountain "Everest" at the end of the free
energy landscape. What is this mountain "Everest"? For me it is nothing
else than love-agape.

To cross the free energy barriers in the free energy landscape, we need
catalysts when they are too high to lower them. Leadership is like acting
as a catalyst. Teaching is like acting as a catalyst. Sometimes God-love
was the only leader and Jesus Christ the only teacher left over for me. To
cross the gulleys, canyons and even rifts in the hills of the entropy
landscape, we need bridges. LEP, the 7Es, creativity, learning and
speaking are such bridges. When we act them in the
"Steigerung"="stapling"=staggering=fractal fashion, they also do something
else. They let us evolve in complexity. Thus, by way of the LRC, the rifts
diminish into canyons and the canyons into gullies so that we can cross
the clefts in the entropy hills on own feet without bridges made by
someone else.

Dear fellow learners, we need the LO-dialogue as the bridge to cross these
gullies, canyons and rifts which have cause so many turmoil and conflicts
among humankind. We did not make this bridge, but we can certainly learn
how to use it to cross rifts in the entropy landscape. Let us talk. Thank
you once again Rick for hosting a space where we can talk with each other
and learn together.

With care and best wishes,


At de Lange <> 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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