29 Kasım 2009 Pazar

To Look at Spaces

To look at the space is different by its definition from looking at an object. To look at a space is by definition to look at a space inside an area or volume. Because there exists at least the observer.

Our eyes focus when we look at an object. Our attention concentrate on that object.

When we look at a panaroma-general view we use a special technique of looking. We do not look at individual objects but view the general view. In fact this may be the closest way to look at space. Indeed when we view a panaroma we do not look at the elements that form the general view but the abstract volume that they create…

What is abstract volume then? Abstarct volume is the entity that is created in multiple dimensions by a set of abstarct entities.

If we reutrn back to the beginning once more… Looking at space should trig the opposite of looking functions that are based on focusing (and the other cognitive functions such as attention and concentration). For example, to look at individual trees in a sparsely populated woods vs. looking at the spaces between trees and comparing them etc. may help to difuse concentration and get relaxed. Looking at nothing or not dpecific things may help to relax.

25 Eylül 2009 Cuma

The Right of Time

Time has its own rights. Humans, animals, nature, children, plants, almost everything has some basic rights. Time has its basic rights also…

At the moment that an event takes place between two opponents, justice gets set as one is right the other is wrong or at least to a degree or on the basis of some limiting conditions. Among all these conditions time has a special privilege, it has a right also in establishing the justice.

After the initial conditions that establish the ‘right’ on the basis of justice, time begins to tick. If the conditions related to the dispute are dependent on time, at the end of a certain period of time the ‘winning’ side at the beginning may find itself as ‘losing’…

An other example from engineering: To solve a technical problem, it is not enough to know the subject… You must do research and then analyse the problem, diagnose and locate the problem, etc… You must put your time into solving the problem. In the beginning some problems seem impossible to solve but if you put the necessary time and may be some more it is certain you will solve the problem. It is only a matter of time. Because it is the right of time.

27 Ağustos 2009 Perşembe

The perception and assignment of meaning to randomness

Somewhere on the Aegean coast of Turkey, two miniscule particles of sand lies together on the beach. Infact millions of them lie together there… Is it a coincidence or contingency that they lie together? It is not even a coincidence. There is not any meaning to it. If you name it as coincidence it is because you percieve things a intentional or coincidence etc… it is how you percieve things nothing else… Also the words ‘sand particle’, ‘beach’, ‘coast’ would not exist and we would not name those ‘things’ as as such if we did not have a vacation there, if we did not come to meet these things… There were not any coasts, sand dooms etc on the Mars or any planet till we, the human beings come and ‘discover’ them and gave them names…

Perceptions lie at the bottom of every mental activity – including cognition in general. Nature builds intelligence based on perception. Once we meet the sand particles, and many of them, we simple see or feel the warmth of them. We do not assign any meaning to them at this phase. The necessity to tell them or to remember them easily causes us to desribe them. Finally it is easier to give them a name. How we name things depend on how we relate to them rather than to their internal attributes.

The naming process causes us to use deeper levels of meaning as we get to learn more about things and the interaction between things develop. While there is a minimum common ground of the meanings of words, there still are different views to things which bring different meanings to their names.

Different levels of meanings create abstraction and the abstract names and concepts. This creates the possibility to use basic names with different meanings in different contexts.

The way we name sand particles on the beach displays our tendencies to name randomness. Some times we name random things as if they are absolute entities (accompanied with a strong sense of belief). Sometimes we do not care whether it is random or not and name the whole thing or process – as beach. Sometimes we get so high in abstraction that we name it as if from somewhere high in the atmosphere – as coast…

The perception and assignment of meaning to ramdomness is a critical mental process which displays the strength and weaknesses of the individual human mind. It is no coincidence that the initial signs of many behavioral or clinic mental problems are related to the deficiencies in this area.

4 Temmuz 2009 Cumartesi

The Necessity of Noise

What is noise? It is a threshold. A threshold at which point a signal loses its identifiability. Here, signal means meaningful activity which has a purpose or value. A group of people sitting at a bar, makes noise because of talking with each other relative to the silent state of the bar or relative to the music being played at that moment.
Noise may carry abstract meanings and may carry various threshold functions such as noise in public decision enquettes or unrelated information on a subject you are trying to make a decision.

If we think of neural networks in our brains, these circuits make correct decisions although they get some garbage(noise) in their inputs, namely not very clear situtions.

Mysteriously, noise may decide what is wrong what is right because it sets the environment to make a decision.
It eliminates what is relevant what is not according to the subject, it sets the boundaries of the elemnts that will make the decision.

Noise sets the threshold for making a decision. In situations where the input namely our observations of the phenomena around us are not clear, we try to clear things by changing or adjusting to the environment before we make decisions. In fact we solve the problem by adjusting the noise threshold. Noise may be necessary in order to make healthy decisions when the inputs are not completely clear.

7 Haziran 2009 Pazar


Miller’s rule states that a human can handle seven plus minus one things in his/her mind at the same time. If the number increases the human mind groups these items so that the number of groups remain less than seven plus minus one(attention span).

Grouping things according to a certain point of view is called abstraction.You can group things first than you can group the groups you have formed and so on. Every step of grouping in this process is called an abstraction level.

Whenever the human mind tries to handle things that overpassess its capacity to handle things simultaneously
it does abstraction. Air traffic control(ATC) is no exception of this fact. ATC systems are complex discrete event dynamic systems. These systems are huge in the sense that they can handle the whole air space of large countries such as Germany or Turkey.

An air traffic control system ensures that airplanes move safely from one geographical point to another. In order to do this each airplane’s safety is given to the responsibility of an air traffic control officer (ATCO) at each point of time in its journey.

The complexity arises from the fact that you have to establish areas of control responsibility for each controller. An ATCO can handle upto 12-13 airplanes at the same time depending on the size of its control area and density of the air traffic. So, in order to control an air space as large as Turkey there must be quiet a number of control sectors. Actually, you generally have first, a number of air spaces depending on the geography and the height, such as eastern Turkey, western Turkey air spaces and lower air space and higher air space which may be divided at flight level 280. The more the number of sectors the more difficult it is to assign tasks to these sectors in coordination with others.

It is inevitable to use abstractions when one just imagines the complexity and the size of an ATC system and the risks involved with it. The first abstraction is the types of control, such as area control, approach control and tower control. The type of the traffic specifies the character of the system that handles it. For the sake of simplicity, and due to my experience I will mention only the area control from now on.

The human-kind expressed itself with symbols long before he/she found the letters and writing[1]. Symbols provide a visual representation of an idea or word as can be observed in Far Eastern alphabets and languages. The symbols played a vital role in the development process of alphabets. One can only imagine a little bit the difficulty people experienced in inventing a language first, then writing and then creating an alphabet. But it should be simple even for a dummy to appreciate the importance of using symbols in our life to fight against the difficulties that we meet.
“Symbols provide a visual representation of an idea or word. Children who find difficulty in reading can be helped to visualise the meanings of words by seeing a symbol.” Symbols In Education - Why we use symbols [2].
“Unlike things, feelings and ideas are difficult to exchange. People wishing to exchange physical objects may simply hand them to each other. Feelings and ideas, however, are without physical substance. They cannot be handed directly to another person. Rather, they must be exchanged through the use of symbols–things that represent or stand for other things.”[3]
The origins of our existence, the meaning of life und so weiter are no simple things to handle and this situtaion is not an exception to our ways of handling difficulties. For example : “The concept of using images in worship finds its origins in the Old Testament. The Temple contained numerous visual images, including the cherubim on the Ark of the Covenant. The Temple Solomon built for the Lord contained many carvings of trees, gourds, flowers, and angels (1 Kings 6). It is clear that God did not forbid images used in the Sanctuary to glorify God[4].”
Abstraction and the use of symbols are vital elements in solving math problems or others in science. The use of arbitrary symbols and the process of symbolisation have made possible the discourse of modern mathematics and logic. “Mathematics uses symbols in creative ways. Two such methods, one dealing with the process of `alphabetisation' and the other based on the notion of `formal similarity', are described. Through these processes, originally meaningless symbols get embodied and coded with meaning through mathematical writing and praxis[5].”
We use symbols when things are difficult to rxpress such as feelings, like a red rose for a lady. Arts is a wide spread application area of this use of symbols. “In the play Macbeth, Shakespeare uses many symbols to add to his story. His use of blood, water, light, dark, rampant animals, and even the witches are examples of how he used symbols to add depth to his play. These symbols were often times recurring and they were all related to the central plot of the play[6].”

1- Symbol extracts what is important – relevant and preserves it
2- Symbol makes it reacheable against all odds its a KEY
3- Symbol provides insulation – better cooperation at the related levels.
4- Symbol provides preservation.
5- Symbol increases processing speed.
6- Symbol decreases mental workload and the amount of relevant speech.
7- Symbol provides space for future enhancements by making abstractions.

The air space used by the airplanes have to be abstracted so that a human being can easily manage the airplanes flying in his own area of responsibility. Simply this means a map but that does not suffice alone…

You must abstract the height, the third dimension in a humanly controllable manner: This means flight levels. The height difference between FL’s are something around 300Ms. Breaking the third dimension to constant and seperate flight levels gives us many facilities: the ability to fly at a certain direction at each level, given a certain filight level you do not have to worry on vertical seperation, limitation references for ascends and descends and many more…

After you have the airspace map, you should choose reference points to fly from and to fly to, also reference points where certain actions have to be taken such as entry and exit points where adjacent centers or sectors have to be acknowledged…

If you think about the size of the German air space, and the number of planes flying concurrently, you can quickly appreciate that a single controller is vital but totally insufficient to do the job. A single ATCO can control 12-15 aircrafts(A/C) at the same time. So you must divide the airspace into enough small sized sectors to reduce the workload to humanly manageable levels. Once more we made an abstraction and created sectors.
Each sector has a symbol, a few characters long to symbolize the section of the airspace they stand for.

An other abstraction is done to facilitate the route declaration in the voice and electronic communications.
Instead of telling each point the A/C is going to fly over, the pilot says uppergreenone (UG1), this indicates a sequence of many points in direction depending on the departure and arrival airport. Each route has a symbol like UG1 which identifies it.

All the symbols used in ATC, points, routes, and their attributes such as entry(NP),exit(XP) etc. are identified in a static data storage or Static Data Bank(SDB). SDB is the symbol of the Static Data Bank. The marvelous character abstraction is you can do abstractions to create the entities that are used to make abstractions of the real life. The Term SDB is used by both ATC interface people who change the static properties of the system and also by the engineers who maintain the software system that supports the operations. The world of the ATC engineers is a complete abstraction full of symbols, actually they use online dictionaries while it is impossible to remember everything.

SDB handles the static properties of the air traffic. The dynamic properties of the air traffic is handled in the Dynamic Data Bank – DDB. Flight level changes, mid air entries, category changes, speed changes usw. all are recorded in the DDB. DDB is an abstraction of the dynamic attributes of the air traffic. It is defined via many symbols that reflect these attributes.

Actually the ATCOs are generally not aware of what is in DDB or SDB. They use other abstractions and many symbols. A flight plan is an abstraction of what a pilot tends to do with his airplane indicating departure and arrival aerodromes, speed, height over all the route points, route, time schedule etc. Strips are small strip of paper which outlines this information in this small area with many symbols. The area must be as small as possible because of the spae limitation on the ATCOs control table. The controller has also the keyboard and display system, the radar screen and ATN messages to use. All of these are highly abstract modellings of the real life air traffic. All of them use many symbols.

From the point of an air traffic controller using symbols is not only being able to identify them but also having the right feel of them. Aviation is not only a science based technological area but it is also an art and air traffic control is no exception to that.

The use of symbols makes things more identifiable and increases comprehension by the use of phrasepology in the pilot – controller voice communication. An air plane appears as a small symbol and a vector indicates the direction and speed it flies on the radar screen. There is height, speed etc data all written as symbols + numeric values… The static control points, airports etc. all are indicated as symbols. There is even a book of ICAO which you can find these values.

But, the controller is not supposed to see all these symbols and overabundance of information, in all this mass, he simply has to keep the mental picture of the traffic in his mind. Similar to a theatre artist, he should not try to remember the text or what movements he should do, he has to keep his audience going according to the point where they are in the act.

This is also reflected in the evolution of a controller. He begins with the easiest sector and like performer he rehearses his role many times and gets to know the stage he is acting. If you remember my example about the use of symbols in Sheakspear plays, symbols help the controller to get hold of the reality, the actual reality at that moment of time during the play… Symbols play a crucial role in helping the air traffic controller to do so.

The use of symbols reduce the mental load of the air traffic controller. Unfortunately, the homeostatic
tendency and ever increasing traffic load causes the air traffic controller get more load in this situation.
At the end of the day, the use of symbols causes the air traffic controller to get more traffic load.

Similarly spatial processing increases the load of the controller instead of decreasing it. Large radar screens provide ample space to hold many more symbols, tables, etc.

One important point to remember; symbols contribute to our subconscious. Anything we do not understand goes to our subconscious. When concentrating on a traffic situation both on voice and radar a controller’s subconscious gets busy with the other things, and these are symbols which he does not have attentional resources to deal with. So, they go to the subconscious. Just like watching and art work that you do not understand.
You try to make an abstraction to overcome the difficulty you face by pushing that thing into your subconscious.
If you know how to interact with your subconscious then your subconscious will pop up the problem back to you, and in percieving the problem once more you will have formed abstraction of it, namely ah that flight that ı skipped.

In fact, symbols are the key to understanding the character of air traffic control. Phraseology reminds me abstract languages, the rule based structure the logic, all the symbols on the radar screen and the slow stepwise movements of airplanes has a certain lyric effect almost poetic, and the strength of symbols has a religious tone, with all the controllers and engineers walking slowly in and around the control center like the Branchid priests serving at the antic oracle’s Didyma.

[1] The Park Ridge Center,
“Before the written word, human beings used symbols as the primary means of self-expression. Hope and fear, joy and sorrow, sickness and health, love and hate, good and evil, yin and yang, feminine and masculine — all found early expressions as symbols.”
[2] Symbols In Education - Why we use symbols
“Symbols provide a visual representation of an idea or word. Children who find difficulty in reading can be helped to visualise the meanings of words by seeing a symbol.”
[3] Britannica Student's Encyclopedia
“Unlike things, feelings and ideas are difficult to exchange. People wishing to exchange physical objects may simply hand them to each other. Feelings and ideas, however, are without physical substance. They cannot be handed directly to another person. Rather, they must be exchanged through the use of symbols–things that represent or stand for other things.”
[4] The use of Images, Signs, and Symbols in Anglican Worship By
Rev. Rebecca
“The concept of using images in worship finds its origins in the Old Testament. The Temple contained numerous visual images, including the cherubim on the Ark of the Covenant. The Temple Solomon built for the Lord contained many carvings of trees, gourds, flowers, and angels (1 Kings 6). It is clear that God did not forbid images used in the Sanctuary to glorify God .”
[5] The Use of Symbols in Mathematics and Logic by Sundar Sarukkai
“Abstract. It is commonly believed that the use of arbitrary symbols and the process of symbolisation have made possible the discourse of modern mathematics as well as modern, symbolic logic. This paper discusses the role of symbols in logic and mathematics, and in particular analyses whether symbols remain arbitrary in the process of symbolisation. It begins with a brief summary of the relation between sign and logic as exempli_ed in Indian logic in order to illustrate a logical system where the notion of `natural' sign-signi_ed relation is privileged. Mathematics uses symbols in creative ways. Two such methods, one dealing with the process of `alphabetisation' and the other based on the notion of `formal similarity', are described. Through these processes, originally meaningless symbols get embodied and coded with meaning through mathematical writing and praxis. It is also argued that mathematics and logic di_er in the way they use symbols. As a consequence, logicism becomes untenable even at the discursive level, in the ways in which symbols are created, used and gather meaning.”

“In the play Macbeth, Shakespeare uses many symbols to add to his story. His use of blood, water, light, dark, rampant animals, and even the witches are examples of how he used symbols to add depth to his play. These symbols were often times recurring and they were all related to the central plot of the play.”

15 Ocak 2009 Perşembe


Many times that I have stood in front of a toy seller’s shop, had this small boy in me wished to transcend the glass shield and reach out to one of the toys on the display. Although light can pass through the glass we can not. This feeling that you feel against the seperating property of a shield of glass, the feeling of powerlessness because of the inability to reach something you yearn for hurts me deeply. On the other hand, “one should not reach out to the things he is not able to catch” says our parents.

As the years pass by, notices one that our lives are divided with glass shields, a compartment next to another… Sometimes there exists nothing, not even a glass shield between. Somebody new born to this system of life does not notice these ‘glass’ compartments in the beginning, only till he tries to reach something he does not deserve…

To speak is a privelege of this type. When you look around everybody talks. It’s just as simple as happening by itself without any effort… But, when it is up to you to speak to a foreigner, to a teacher, to a supervisor or to one of your students or a farmer, just try and see what happens… One notices that it is not as simple as it looks to speak to others, just like does a small kid who tries to reach something higher than his own height…

Imagine a child aged around two years old! He can understand what’s going on around him. He keeps track of every close thing . He is even aware that he is a seperate being which owns its own life. But he cannot speak… An unseen shield of glass stops his voice to be heard when tries to speak… He is aware that he has a voice. He does not know what he cannot do. He tries to reach out to and touch things that only his words could touch. He fails. He does not succeed because there is an unseen transparent thing between. Indeed he tries to find what that is. At the end, he does find it. A few words come out of his mouth. Something like “mom, dad”… The foundations of the structure which have been laid underneath will help him stand up for the rest of his life. This is the name of the thing that stands between him and others: SELF.

Each person who succeeds to speak lives similar difficulties a couple of more times in their life. The most striking example of these situations is learning the first foreign language. Specially if one goes to a country of which language he does not know well and learns it slowly day by day like a small kid, this situation becomes quiet similar to the process in his very childhood. Psychological problems that may appear in people living abroad may have substantial relation with this phenomenon. If scrutinized, schizophrenia and other similar problems have some roots in this difficult period of life around 2 years old.

Many have English as a second language in Turkey. Unfortunately, as one of my visiting European colleagues has mentioned ‘Everybody speaks Turkish in Turkey, but all does so badly.’ Hence we are a society who can speak the second language not so good. OK, what happens if one tries to learn the 3rd and even 4th languages as many do in Europe? Unfortunately, the number of people who knows this, who has tried and succeeded are very few in Turkey compared to key European countries. When pushed to learn the 3rd language your second language begins to waver, you begin to forget some words etc… When pushing the 4th language the grammar difficulties in every language including your mother tongue may appear… You forget words, or mix languages using French words in English etc. Even worse is,you think you are speaking your native tongue when speaking an other language, your students although benefiting from this, politely make fun of it. The worst, because of not knowing which language to listen while passing people are speaking on the street you may think some Germans are speaking quiet good Turkish. The rest of your life, you hear French, German, English words in the noise that you hear…

Foreign language education is a strategically important subject in our country. Around our country lies Greek, Bulgarian, Serbian and other ex-Yuogoslavian, Romenic, Hungarian, Russian, Ukranian, Moldovian, Checnic, and other ex-Russian, Armenian, Persian, Arabic speaking countries. From the European perspective we have to count English, French, German, Italian and Spanish to name a few… Turkey can succeed against this great challenge by only good organization, planning and specialization, not by acts of good luck.

The complexity of the Turkish geography surrounded with oceans has created the obligation of a closed culture in a single umbrella language and culture. The physical largeness and the neessity to keep everything in order has caused the development of practices that may not be in parallel with Europe for many centuries…

Turkey tries to improve her relations with the European Union primarily for economical obligations. But similar to the child learning to speak for the first time Turkey has communication difficulties with Europe. The struggle to express her own concerns and convince her counterparts to give her deserved rights, is driving Turkey to redefine and find herself anew in the 21st century world. When mixed with the effort to reown her own culture coming from the past, the effort to transcend the invisible glass between Europe and Turkey, pushes the Turkish society to the limits of her cultural and sipiritual strengths. An adventure initiated by economical obligations is having much wider and unforeseen effects on our society driving her to the limits of healthy development.

3 Ocak 2009 Cumartesi


Fighting Techniques II

To the memory of our father Hasan SARAL.

“the chemist Kekule came upon one of the most important discoveries of organic chemistry, the structure of the benzene ring, in a dream. Having pondered the problem for some time, he turned his chair to the fire and fell asleep:’Again the atoms were gamboling before my eyes…. My mental eye…could now distinguish larger structures…all twining and twisting in snkae-like motion. But look! What was that? One of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke.” The spontaneous inner image of the snake biting its own tail suggested to Kekule that organic compounds, such as benzene, are not open structures but closed rings[1].”

Everyday, we fight with many difficulties of many kinds. Some of these are as simple and short as loosening of the shoelaces. Some of them are as difficult and long as making an invention and some of them as abrupt and serious as a traffic accident… If you have a look at the many problems that we fight with you will notice that we can categorize them and their solutions in different groups. Although these groups may carry similarities in their quantity, quality and content attributes, their use by individual persons and the importance assigned to them may differ. Also, even though the problems and their solutions may be similar the personalities of the individuals that they interact may cause them to appear different.

Studying the nature of problems makes it possible for us to solve similar problems easily and also understand ourselves better We can categorize problems in various ways. For example, problems that repeat endlessly are called chronic in medicine. I am afraid, some of our political problems may be called the same. On the other hand some problems are seasonal. For example, opening the streets to transportation by cleaning the snow. These problems repeat with more or less a certain period. Acute problems happen suddenly and are serious to handle. For example, a sewage pipe gets broken in your house… Some problems are ubiqutious. You meet them in many areas of life. Some are wide spread with in a limited area. Your computer program does not work. When nothing works at all you have made a main mistake with wide results. Focused problems affect a certain functionality and the system recovers as soon as you fix it. Like a tooth ache…

Some problems are light but persist for a long duration. In fact, we can group the problems by their durations that they sustain, short – long etc. or according to their largeness. The way the problems happen may be classified also. Few or many in quantity. Fighting with more than one problem at the same time increases the total difficulty. The way we categorize problems is not constant. It changes according to the subject of the problem and the context it happens. To put a tiny piece of thread through a sewing needle may be percieved as difficult while a much more concentration demanding computer programming task may be percieved easier, just like a technical problem may be percieved much more difficult after midnight than at noon.

The fact that our perception of the problems is variable makes our categorization of the problems more difficult and hence reduces the benefit that arises from the categorization. If you are throwing anything you get hold of to your target in a chaos that you can not apprehend you have come to that point where you have to take a deep breath and try to categorize the problems correctly. If you can categorize the problems as convenieint as possible to reach the target your chance of hitting your target via a similar solution. Solving problems is basically a problem of classification.

I had mentioned the chronic problems in the beginning. The word ‘chronic’ means ‘: marked by long duration or frequent recurrence’ by Merriam-Webster. The most apparent characteristic of a chronic problem
is repetition or continuity. Reduction of the repetition intervals indicates that the problem gets severe or light. For example, a severe crisis give way to the lighter ones but more frequent or irregular problem periods, or the increase in the severity of problems and the increase of the frequency may indicate a worsening.

The repetition of the problem in chronic problems may happen in various ways:

1-The problem arises in a flow of random events. The crux of the issue here is; the events other than the chronic problem are random and carry no relation with the problem.

2- The chronic problem happens after a chain of happening events. After each occurence things repeat the same iteration of events. In this case, repeating event is not only the chronic problem but the events that prepare it.
The events’ iteration does not have to be constant array of events. The presence of some events may be obligatory, but two seperate iterations may be composed of completely different events. But most of the events that form the iteration belong to the set of events that form the reasons of chronic problem.

If we look at a chronic problem from a closer point of view, we may observe that a single crisis begins at a moment in time, continues and finishes, or the problem has begun at a moment and continues for a long duration without any interruption. Aset of the general conditions that prepare the outbreak of the crisis, a set of the causes that push the events to happen and a set of the triggers which may come just before the crisis may be observed. In the case of the repeating crisis, there is a set of conditions that prepare, a set of causes that make it happen and a set of triggers that initiate the end of the crisis.

One component of the equation that gives rise to the chronic event is a function which may change by time but may also be accepted as constant for relatively short periods of time. This function may be related to the events in the past or to the material dependent on the nature of the interacting elements/participants of the problem.

When studided closely, the reasons, the causes and the triggers that lead to the outbreak of the crisis may be related to their own previous values and also may be related to the previous crises values. For example, the seriousness of a previous crisis, its duration, its attack/sustain/release durations. Sometimes the slowness of the evolution of the problem, namely slowness in its attack period, mathematically its 1st and 2nd derivatives being small, or precautions that balance and slow sudden changes may stop the forming of crisis episodes. The dependence of the reasons that form the chronic crisis to the characteristics of previous crises and to its own evolution leads to the unmanageable repetition of a chronic problem.

Above, I had mentioned a chronic problem that occurs among unrelated random events. There may be such problems that may be dependent on only themselves and their own past iterations. These self triggering chronic problems are recursive[3] in nature. In fact they may be evaluated as a special case of functions mentioned in the 2. item.

If I may return back to the beginning of this article, if we have a closer look at the story of the chemist who found the benzene ring, we may now come to appreciate the value of the symbol ‘the snake which bites its own tail’. The metaphor of ‘the snake which bites its own tail’ or ‘the scorpion biting itself’ is a method utilized in solving problems that are very difficult. The problem is so difficult that it can not be solved or cured by external effects such as, force, medicine etc. It becomes inevitable that the energy at the source of the problem may be used to kill itself. This kind of solutions may not be present or evident. I believe, chronic problems have the ability to be solved by using the metaphor of ‘the snake which bites its own tail’ on the premises of their own definition.


Note: My simple article ‘A Mathematical Model of Chronic Problems’ is available at my blog
http://tekne-techne.blogspot.com in Turkish.

[1] Rober H. McKim, Experiences in Visual Thinking , Brooks/Cole Pub. Co. Monterey, California, s. 11.
[2] Merriam-Webster Dictionary
Main Entry: chronic
Etymology: French chronique, from Greek chronikos of time, from chronos
Date: 1601
1 a: marked by long duration or frequent recurrence : not acute b: suffering from a chronic disease the 2 a: always present or encountered ; especially : constantly vexing, weakening, or troubling
Medical Merriam-Webster:.
1 a : marked by long duration, by frequent recurrence over a long time, and often by slowly progressing seriousness : not acute Chronic her b : suffering from a disease or ailment of long duration or frequent recurrence chronic
2 a : having a slow progressive course of indefinite duration -- used especially of degenerative invasive diseases, some infections, psychoses, and inflammations chronic-- comparre ACUTE 2b(1) b : infected with a disease-causing agent (as a virus) and remaining infectious over a long period of time but not necessarily expressing symptoms
[3] Merriam-Webster Dictionary
Main Entry: re•cur•sion
Pronunciation: \ri-ˈkər-zhən\
Function: noun
Etymology: Late Latin recursion-, recursio, from recurrere
Date: 1616
1: RETURN 2 : the determination of a succession of elements (as numbers or functions) by operation on one or more preceding elements according to a rule or formula involving a finite number of steps 3 : a computer programming technique involving the use of a procedure, subroutine, function, or algorithm that calls itself one or more times until a specified condition is met at which time the rest of each repetition is processed from the last one called to the first — compare ITERATION

Main Entry: it•er•a•tion
Pronunciation: \ˌi-tə-ˈrā-shən\
Function: noun
Date: 15th century
1: the action or a process of iterating or repeating: as a: a procedure in which repetition of a sequence of operations yields results successively closer to a desired result b: the repetition of a sequence of computer instructions a specified number of times or until a condition is met —compare RECURSION 2: one execution of a sequence of operations or instructions in an iteration