Articles which were published in the NKS forum

http://forum.wolframscience.com/showthread.php?s=&threadid=167

**:**
The use of random numbers during CA generation

We ought to distinguish between Randomness and Pseudo-randomness

CA don’t walk randomly

Randomness stands for ignorance

CA lack Entropy

Darwinism: A crude model of Life driven by Randomness

In
the world of CA, Randomness is meaningless

In
the world of CA the Central Limit Theorem
fails

Medical statistics are based on
Randomness and therefore unreliable

CA
and Randomness are mutually exclusive processes

Randomness exists solely in the eye of the
beholder

The Thing in Itself is neither random nor nonrandom

No transformation generates Randomness

Randomness is a property of a set (collective)

Two kinds of randomness

Algorithm to examine the Central Limit Theorem

Randomness (noise) cannot generate complexity

When
you add structure to noise it
becomes more complex

Randomness and Creativity

Randomness and Determinism

A new Oxymoron: The use of random numbers during CA generation

The definition of a cellular automaton (CA) excludes randomness. Nevertheless
S. Wolfram applies random numbers as initial states for his CA. What an
inconsistency! Rather he ought to use CA for generating initial states,
and ban the term “Randomness” from his models. You might apply the term
“Pseudo-randomness” to a given state provided
that you specify the CA (or CA set) which generated it, and the number of iterations
to reach it. Wolfram uses a CA
for generating pseudorandom sates, yet does not specify their two crucial
attributes {CA set,
number of iterations}.

**We ought to distinguish between Randomness and Pseudo-randomness**

We
ought to distinguish between Random and Pseudo Random initial conditions.
The first obviously cannot be reproduced. Pseudo Random initial conditions
meet my requirements provided that one specifies how they were generated.
I don’t recall that in his examples Wolfram specifies how the pseudo random
initial conditions were generated.

While
disliking Randomness, Pseudo Randomness is closer to my heart since I
know how it is generated. While Randomness and CA is an oxymoron, pseudo
randomness and CA, is not. My point is that in the world of CA, Randomness
is meaningless, and should not be used. The same applies to biology and
medicine.

I
agree with you that one can define terms in various ways. However in medicine
such terms have profound implication on therapy and may harm the patient.
This issue is dealt with in my site under the heading: A
New Kind of Medicine. (you might abbreviate it
as NKM)

**CA don’t walk randomly **

The Random Walk is another manifestation of the Randomness
concept. It is a stochastic process like Brownian motion, and serves among
other to describe the stock market and exchange rates. The Efficient Market Theory says that the prices
of many financial assets, such as shares, follow a random walk.

Random walk may explain why the stock market won’t make
you rich. However it fails to explain why some brokers got rich. No wonder,
since the stock market is more than a sum of random walks. It is a living
system, and as such cannot be modeled by random walks. Economists don’t
like this idea because they do not know how to model living systems. They
reduce and simplify the humans which make the stock market tick, to faceless
points, until they meet the prerequisites of random walks.

Yet all these amazing guys who made a fortune,
are not at all faceless as economists suggest. They are simply creative,
and since economists do not know how to model creativity, they ignore
it. It seems as if CA might be a good tool for modeling
creativity.

I wonder whether a CA model of the stock market would
help me to reduce my overdraft.

**Randomness stands for ignorance**

Nature presents itself to us as change. We distinguish
between two kinds of change: Change
which is explained by a theory, and unexplained change, which we call
Randomness. Randomness is not an inherent (ontological) property of nature.
It is our way to describe it.

In linear models randomness
is represented by a separate term, called error, which has
to be minimized. Here R-square
expresses the adequacy of a linear model. In other words, R-square stands
for the fraction of the observations explained by the theory
, and 1-Rsquare stands for randomness or error.

Neural Nets start from a random initial state and converge to a (non random)
solution. Thus Neural Nets are processes (algorithms) which eliminate randomness. On the other hand, chaotic
CA systems lack this property. If
they start from a random
initial state they will propagate randomness, and even amplify
it. Above all, such models can not be reproduced, since being sensitive
to initial conditions. For this reason their initial states ought to be
non random and uniquely specified.

Life never starts from randomness. Two cells, the sperm and the ovum unite to form a (non random) zygote, which evolves into what we are. This is why Life and Randomness is a medical oxymoron.

**CA lack Entropy**

The concept of entropy was introduced in 1865 by Rudolf Clausius According
to the second law of thermodynamics the total entropy of a thermally isolated
system can never decrease. In 1877, Boltzmann defined entropy as a function of the possible microstates
in a system. It is a measure of the system’s disorder. In this context
the second law of thermodynamics states that the disorder in an isolated system tends
to increase.

Which caught the imagination of doom
prophets. Since the universe is an expanding closed system, its
total disorder increases. Ultimately it will reach a state in which its
thermal energy will be homogenously distributed, and die a “heat death”.

Might a rising entropy account
also for human death? Obviously
not, since the organism is an open dissipative system. In his book “What
is Life” Erwin Schrödinger
suggested that the entropy of our organism remains low since it feeds
on negentropy.

Following Boltzmann’s model,
Claude E. Shannon
defined entropy as a measure of uncertainty. In the state of randomness,
entropy and uncertainty are maximal. Which brings us
back to a statement made here in a previous section, that Randomness is
ignorance.

Entropy is meaningless when applied to CA for two reasons. 1. In statistical
thermodynamics, entropy varies between 0 and 1. Since CA and Randomness
are mutually exclusive,
CA entropy will never vanish (be zero). Unlike in Information
theory or Statistical Thermodynamics, CA entropy is not defined over the
entire [0 , 1] interval. 2.
CA may be regarded as an open system. An isolated string of numbers will never change by
itself. It has to be
driven to its next state
by a processor. While the string may be regarded as isolated, together
with the processor it is an open system in which entropy is meaningless.

**Darwinism: A crude model of Life driven by Randomness**

Nature presents itself to us as change. We may distinguish between rapid
change, like a torrent, and a slow change, which is called variation.
Prior to

In 1859,

1. Spontaneous, known today as genetic mutation, or crossover.

2. The outcome of
competition between species.

3. Resulting from the selection of entities which will survive by the
environment, known as “Survival of the Fittest.”

Evolution is a random process. Its objects are powerless
to alter their fate. They are shuffled in the hyperspace representing
nature like dice. Yet life is more than that! It is creative, exclaimed
Henri Bergson in his book “Creative Evolution”.

He was ignored as an esoteric Vitalist. Today

The crudeness of

They better be called

I dislike Darwinism for two main reasons: 1. Social Darwinism promotes
discrimination, and 2. Medicine applies Darwinism to describe cancer progression.
As cancer evolves, it becomes fitter than its host (the patient),
and gradually destroys him . Yet Cancer is more than that! It
is a creative process operating in a creative host.

Decline of Darwinism

**In
the world of CA the Central Limit Theorem
fails**

The Central Limit Theorem (CLT) states, that any sum of many independent identically distributed random variables is approximately normally distributed. http://www.worldhistory.com/wiki/C/Central-limit-theorem.htm

For instance if dice are rolled repeatedly, the frequency distribution will resemble more and more the Normal
Distribution. You may check it experimentally at the following site:

http://www.stat.sc.edu/~west/javahtml/CLT.html

The CLT is the hallmark of Randomness,
which underlies many statistical. Does it apply also to the world
of CA? Create a pseudorandom set of initial conditions, and let the CA
evolve. When small they may obey
the CLT, yet when larger they do not. More precisely, when the distance
between the CA is such that they remain isolated, CLT works. When overlapping,
it fails, and in chaotic CA it is useless.

CLT works only in linear systems whose elements do not
interact and are isolated. In other
words CLT thrives on Randomness, which CA lack. Farewell to linear statistics.

Life also lacks
the two prerequisites of CLT. Neither are its elements isolated nor independent.
Unfortunately, epidemiologists ignore this common wisdom and base
their statistics on the CLT. They take the human being, and simplify his
attributes until they meet the requirements of the CLT. Yet this simplified creature is a far cry from
that which was created in Genesis. Epidemiology
thus nurtures
medically induced diseases known as Iatrogenesis. By now you might understand
why I dislike randomness.

Bias in medical
statistics

Back

**Medical statistics are based on Randomness unreliable**

The issue at stake is whether a model that one applies
is consistent. To my mind constructing or analyzing CA models with tools based on randomness leads
to conflicting conclusions. This precisely is happening daily in Medicine,
and I use CA to illustrate what I mean. For instance, most statements
by medical epidemiologists suffer from this inconsistency. You can’t trust
most medical statistics.

Back

**CA
and Randomness are mutually exclusive processes**

You
obviously may use any tool for any kind of model, yet when choosing CA
as your model you are somewhat limited. Regard modeling as a game. You
start with certain rules, and play. The rules of the “CA game”, and Randomness,
don’t go together. In CA the present state determines what the next state
will be, and in random processes the present state has no effect on the
next one. Either you stick to the CA rules, and keep away from using Randomness,
or you include Randomness in the “CA game”, whereupon it ceases being
a “CA game”. In other words, no
CA rule generates Randomness, and vice versa. Inclusion of Randomness
in CA models
leads to a contradiction.

In my opening statement I wrote:
“Cellular automata
(CA) provide simple models for evaluating interesting philosophical questions.
Here is a simple teaser . . .” I use CA to illustrate a fundamental property
of Life. Every state depends on the previous one. Since
statistical tools, like the Central Limit Theorem, or Random Walk require that the states of a process be independent
from each other, they are inadequate
for studying Life (and CA). New tools have to be invented.

Please remember that the discussion is essentially philosophical. The
question is whether Randomness exists as such in (biological) Nature.
I claim that it does not. This concept is applied by us for describing (understanding) Nature. Randomness exists
solely in the eye of the beholder.

You have to distinguish between reality and the tools we apply to study
it. To my understanding, traditional
statistical tools, which you mentioned, fail in (non trivial) CA models.
Therefore, if
we wish to model life with CA, we have to abandon this concept. In the CA universe, Randomness
is meaningless. Medicine enters
a new era in which tools based on Randomness will have to be replaced with better ones.

By the way I really don’t dislike randomness, I simply ignore it. After
all, there might be a take home lesson even for you. If you happen to
be treated by a doctor who applies randomness, you might ask another
one for a second opinion. This is the main message of “The
New Kind of Medicine”

**The Thing in Itself is neither random nor nonrandom**

Since we cannot comprehend the Thing in Itself we do not know whether it is random or not. All my arguments center about our perception of the Thing in Itself. As you have noticed I am an amateur philosopher (but a good physician), and use the narrative to convey my ideas. Like the philosopher Epictetus, I am not interested in the truth as such, but what people think of it.

**No transformation generates Randomness**

Randomness (noise) has an **unpleasant mathematical
property**. By definition, it cannot be generated by a program,
otherwise it would be called pseudo-randomness. In other words there does
not exist a transformation from noise to noise.
Which distinguishes it from other mathematical objects which may be generated
by a transformation.

Randomness reminds of another object with unpleasant mathematical properties,
the zero. One is not allowed to divide by it. 0/0 is indeterminate, and
1/0 is undefined. Might Randomness be the “zero” of complexity?
Since no transformation generates noise, what is Noise/Noise?

**Randomness is a property of a set (collective)**

One may say: You are dealing with
observables. If you have a variable and you don't know what it is now
but you will know what it is later, then it’s
a random variable.

By the way, who is talking about reality? I regard mathematics as a language,
its models as narratives with which you can spread illusions and just
so stories, like the notion that your variable which you just conjured
is random.

additional reading:

**Two kinds of randomness**

The random variable is a somewhat obscure object. Take for instance the definition in Wikipedia: “A **random variable** can be thought of as
the numeric result of operating a non-deterministic mechanism or performing
a non-deterministic experiment to generate a random result.
For example, rolling a die and recording the outcome yields a random variable
with range { 1, 2, 3, 4, 5, 6 }. Picking a random
person and measuring their height yields another random variable.” http://en.wikipedia.org/wiki/Random_variable

You perform the random experiment and are told
that there are two kinds of distribution, discrete and continuous. You
prefer the continuous and become confronted with the **central limit
theorem (CLT)** which states that whenever
a random sample is taken from **any** distribution with a mean and variance, then the sample mean will be approximately
normally distributed. The larger the sample size, the better the approximation
to the normal.

You satisfy yourself that it works for dice, coins and roulette. Then you start measuring the height of randomly
selected people and become somewhat uneasy. **The distribution is
skewed.** You continue sampling and it remains so for ever. Height
is obviously randomly distributed, and you chose the persons randomly.
What went wrong?

In fact, in medicine **all observed randomly distributed variables
are skewed**, and nothing is distributed normally! Nevertheless
medicine ignores the skew, attributes it to chance,
and regards all its phenomena as normally distributed. Which introduces bias in all
its statistical (epidemiological) statements.

Further reading:

Beware of the gene

Iatrogenic medicine

**But**** the skew is real**?
Why not say that random variables may have **two kinds of distribution**,
non-skewed, and skewed ? This is a mathematical blasphemy, since it undermines
the generality of the CLT. By
analogy with geometry you might say that there are at least two kinds
of Randomness: So called “**Euclidean-Normal**”, and “**Non-Euclidean-Skewed**”. Both with equal rights!

With all these heretic thoughts rushing through my mind I turned to Wolfram’s book, and
lo and behold this Gaussian Randomness is everywhere. However, **in
the world of CA randomness is meaningless**, and even if you regard
CA as random variables, they disobey the
CLT.** CA are “Non-Euclidean-Skewed”.**

I decided therefore to present this issue from a somewhat unusual perspective,
e.g., Oxymoron, Randomness is a zero of complexity, etc. It seems to me
that** in order to really grasp complexity you have to get used
to the notion of many kinds of
randomness**, whatever it means.

**Algorithm to examine the Central limit Theorem**

I doubt that you will find
an adequate data set of heights, or other biological
variables in order to find out the inadequacy of the Central Limit Theorem.

You might examine this issue on Wolfram’s class-4 CA.

1. Generate a set of class-4 CA.

2. Iterate one step.

3. As long as CA are not chaotic go to 2.

4. Generate “iid” sample (Independent and Identically Distributed), and sample ‘n’ CA. Compute the mean and store it.

5. Iterate one step.

7. Compute the mean of the set of means and store it.

8. go to 4

Each new mean will oscillate chaotically. Farewell to
the Central Limit Theorem.

**Randomness (noise) cannot generate
complexity**

In a previous
section I mentioned that noise (randomness) cannot be
generated by a transformation. However noise has an additional property
which might be useful for evaluating
complexity. Noise * Noise = Noise. Which is also the property of
the digit one. 1 * 1 = 1;

Whenever we multiply noise by itself it does not become more complex. Thus noise cannot generate complexity. We may therefore regard noise as a unit of complexity. Noise has an additional property. Its components are independent from each other, or uncorrelated. Let r be the correlation coefficient of an auto-correlation function of a set of random numbers. By definition r[noise] = 0.

One may generate complexity by making numbers dependent on each other. Dependency is proportional to correlation. We may thus express complexity of a set by correlating it with noise. In this context ‘r’ becomes a complexity measure.

What happens if we add (accumulate) noise? Still noise + noise = noise. No complexity is gained (r = 0). Might this measure improve the classification of CA complexity?

Back

**When
you add structure to noise** **it
becomes more complex**

**Q:**

For example, geometrically speaking (of a 2d rectangle filled with random noise) just mirror the rectangle about the far end (axis)... now you have one rectangle placed right next to the other with an axis of symmetry down the two touching edges. Now with the symmetry it is no longer noise.... (mind you i am thinking loosly, abstractly, and artistically).

The symmetry adds some kind of complexity (because now there is relation due to the symmetry axis) wouldn't you say?

**A: **When adding structure to noise it
may be made more complex

2. You take a rectangle full of noise and mirror it. Result: A
new rectangle full of noise. The same complexity as before.

3. You place the two rectangles side by side. Result: The new system is
more complex, since you got two rectangles.

By placing noise within a rectangle you created a demarcation which makes
the new structure more complex than just noise.

**Randomness and Creativity**

The more you think about it the greater your doubts despite the
fact that life had about ten billion years to evolve. Such doubts were
raised by the philosopher Henri Bergson in
his “Creative Evolution”. http://www.what-is-cancer.com/papers/Bergson.html
Unfortunately this important treatise was dismissed
by enlightened reductionists which still dominate
biomedicine today.

Genetic Algorithms (GA) illustrate the inadequacy of Darwinism. Since
applying randomness they are destined to approach their solutions asymptotically.
GA hardly ever generalize, and above all they
are not creative.

I prefer another version of creationism: In the beginning there
was a cell which arrived from outer space (panspermia).
From then and on, life evolved
by recreating itself and its own environment known as
Gaia.

Back

**Randomness
and Determinism**

**cells always
emerge from cells** (*omnis**
**cellula**
ex **cellula*) each cell inherits its complexity form
its predecessor which includes
also the change which seems to us to be random.
Yet it is neither random nor pseudo-random. It is a change typical of
life. It is somewhat frustrating that we cannot distinguish between
the three kinds of change, randomness, pseudo randomness, and the change
generated by life. There is however a profound difference between
the change created by life and the other two. Life is creative, while
the other two are not.