### Randomness – is it useful?

Articles which were published in the NKS forum

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

Cellular automata (CA) provide simple models for evaluating interesting philosophical questions. Here is a simple teaser:

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}.

Why do I dislike randomness?  As a physician I am flooded with medical statistics based on the Normal Distribution and its derivatives. The Normal distribution is the hallmark of randomness in science.  However, in Medicine (like in Love) nothing is really random. Life and randomness is a medical oxymoron, which nurtures medically induced diseases known as Iatrogenesis.

More teasers in my site: http://www.what-is-cancer.com/papers/ca/ca01.htm

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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)

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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.

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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.

A meaningful model of Life cannot be based on Randomness.

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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 Darwin, variation was regarded as  God’s creation. Some even believed that God whose nature  is incomprehensible, presents himself to us as variation.

In 1859, Darwin published “The Origin of Species” in which he explained variation in a novel way. Variation evolves. He distinguished between three kinds of variation:
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 Darwin’s theory gained the status of a religion: “Nothing in Biology makes sense except in the Light of Evolution” said Theodosius Dobzhansky.  Modern Darwinists ignore  that there may be other more sophisticated ways  to model evolving variation.

The crudeness of  Darwin’s model is evident in Genetic Algorithms (GA) which apply it for classifying and generating various solutions,. They manipulate their objects in the same way as Darwinists would suggest. Despite some impressive achievements, GA  hardly ever generalize. Above all they are not creative.

They better be called Creationistic GA, since their proper function requires a programmer god (demiurge), whose task is to select proper fitness spaces, define fitness measures, and crossover modes. GA illustrate the major weakness of Darwin’s theory, it is based on Randomness which by its very nature cannot be creative.

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

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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.

Bias in medical statistics

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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.

Randomness exists solely in the eye of the beholder

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.

You may regard my site as  CA-comix illustrating interesting ideas.

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?

Shouldn’t we shy away from Randomness  when breading CA?

Randomness is a property of a set (collective)

One may argue  that Randomness is most certainly transformable  I can take one random variable and transform it into another random variable with different properties.
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.

Randomness is a property of a set (collective)  and not of a single number or a variable. You may call your  variable  random and I may regard it as non-random, and we shall both be right since talking about a single number.  You don’t have a randomness test for a single variable. Even when programming you don’t have the option to define a variable as random. It might be an integer, a real, or complex, but never random. Even the Random[] function is not random. It is pseudo-random!

Randomness (noise) is not transformable!

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.

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.

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.
Statistics started as an experimental science, and gradually was formulated analytically.  NKS and its CA, introduce a new era in statistical experimentation.

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?
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When you add structure to noise it becomes more complex

Q: If you take noise and mirror it then you have something that is not pure noise.

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

Your example consists of three steps:

1. You take (infinite) noise and create a demarcation.  Result: The new structure is more complex than just noise.
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.

Once you add structure to noise you can make it more complex. Border is a structure.
By placing noise within a rectangle you created a demarcation which makes the new structure more complex than just noise.

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Randomness and Creativity

In a previous section we have realized that randomness cannot generate complexity. Creativity is somehow linked with complexity. It may be regarded as an unexpected rearrangement of complexity, or a creation of a novel structure. Since randomness cannot generate complexity it cannot be creative.

Randomness does not exist as such in nature. It is applied by the exact sciences to understand nature and contributed to some important theories, like statistical thermodynamics. It fails however to account for creativity which is the hallmark of life. Take the  theory of evolution, whose cornerstone is random variation. However,  when observing the complexity of our brain, one starts wondering whether  it really evolved by random events?

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.

Randomness can perturb a complex system to make it creative. By itself it is not creative.

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.

Creationism bans randomness altogether which makes it so appealing: “In the beginning of creation, when God made heaven and earth, the earth was without form and void, with darkness over the face of the abyss. . .” (Genesis). The abyss was not at all random. Neither was  the formless earth. They served as an initial condition for life.

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.
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Randomness and Determinism

What we observe and perceive is change. We wonder weather this change has a cause, or not? The first kind of change  we call deterministic and the other random. We tend to regard them as complementary. If it is random it does not have a cause. Yet with cellular automata  we can generate change which appears to be random and nevertheless it is deterministic. This change is called pseudo-random. It is somewhat frustrating that we cannot distinguish between genuine and pseudo randomness. Pseudo random numbers meet all statistical requirements of randomness, and  Mathematica software applies a cellular automaton to generate random change (numbers).

Life also presents itself to us as a change which seems to be  random, yet differs from the above mentioned kinds of randomness. Since 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.

Heraclitus (500 B.C.) created a powerful metaphor for change. He said: ”You never step twice into the same river.” We, his humble followers, add: “You never meet twice the same individual”.