Milton Harvey Sanchez Pag. 44

Revista Universidad Eafit. Julio - Agosto - Septiembre 1997

ethology

A Theory of Learning and Knowing vis vis Undeterminables, Undecidables,Unknowables(*)Heinz Von Foerstern nirst of all I want to thank the Authorities of the Comprensorio di Primiero, the sponsors and organizers of this Symposium on Cognition as Education, who gave us a unique opportunity not only to learn about the many perspectivas of learning and knowing, but also to meet old friends and colleagues from different parts of the globe, and to make new ones here at this magnificent place.

Heinz von Foerster. Profesor de la Universidad de Ilinois. Investigador en Ciberntica, Sistmica y Filosofa. (*) This article is an adaptation of an address given on April 26,1990, in San Martino di Castrozza at the Seminario Internationale Conoscenza come educazione. 13

was very impressed by the courage of the organizers to invite Americans, namely my colleagues and me, to this symposium on education, though it is well known that the United States of America have one of the worst educational systems of the Western world. Let me read from the May 1989 Edition of the Review Draft by the Curriculum Commission to be submitted to the California State Board of Education (1): ..."In 1983, A Nation at Risk declared that American education had become victim to a rising tide of mediocrity. The National Science Boards Commission an Precollege Education in Mathematics, Science and Technology confirmed that the situation in science education was particularly critical and recent studies have placed Americas students last among their internacional counterparts in understanding science. In 1988, the National Assessment of Educational Progress of the Educational Testing Service issued The Science Report Card, and noted that while the responses in the intervening years since 1983 have resulted in some progress, average science proficiency across the grades remains distressingly low." Perhaps we have been invited to find out how not to set up an educacional system. Why indeed does function so poorly? American education

I

Lethology: A Theory of Learning and Knowing vis vis Undeterminables, Undecidables, Unknowables

cultural roots from Europe, Africa, Asia, and Latin America. But many other causes are invoked as well, rightly or wrongly, for explaining the failure of the American school system: too much emphasis on sports; overburdened and underpaid teachers; decisions on how and what to teach left to local school boards; politization of textbooks; and so on and so forth. May be all of the above contribute to the appalling results, but I think it is the choice of an epistemology, a theory of knowledge, that is counterproductive, even inhibitory, to the cognitive processes of appreciation, fascination, enthusiasm, curiosity, etc., that are prerequisite for learning and understanding. Let me take as an example again the report to the California State Board of Education I mentioned before. On its almost 200 pages there is not a single sentence that addresses the questions of how do students learn, what takes place in the mind of the learner and what this gigantic machinery we call "schooling" is all about. Indeed, what is learning? If this question is asked in an academic context say, in departments of education or psychology there will be many answers. However, if this question is asked in an operational context, there are no answers at all: we have not the slightest idea of what is going on within us when we say we have learned something. 14

To start with, there is great difficulty in setting up an educacional strategy in a pluralistic society with a population of diverse


I mean by that that here we are all walking talking and socializing ever since we were about two years of age, although we never took courses in our mother tongue nor in the art of locomotion. There were no curricula regarding these faculties, and we have no idea how we acquired them.Indeed, what is learning? If this question is asked in an academic context say, in departments of education or psychology there will be many answers. However, if this question is asked in an operational context, there are no answers at all: we have not the slightest idea of what is going on within us when we say we have learned something.

explain our mathematical abilities, perhaps with special organelles for addition, subtraction and multiplication. These and similar notions fall into the category of "explanatory principles", a category which was invented by Gregory Bateson to answer "What is..." -questions from his daughter (2). When she asked: "Daddy, what is an instinct?", he answered: "An instinct, my dear, is an explanatory principle." And when she wondered what does it explain, he said: "Anything - olmost anything at all. Anything you want it to explain." When she protests that it could not explain gravity Bateson retorts; "But that is because nobody wants instinct to explain gravity. If they did, it would explain it. We could simply say that the moon has an instinct whose strength varies inversely with the square of the distance...", whereupon she stops him: "But this is nonsense!" and he: "But it was you who mentioned instinct, not I", etc. I leave it to you to follow up on these charming "Metalogues" as Bateson called them; my point here is to draw your attention to such stopgaps of inquiry of which there are, besides "language organs", "instincts" many others, e.g., "drives", "mind", "memory" (3), etc., etc. They always come handy when we do not know what is going an. With the following example, however, I would like to invite you to come with me to the brink of the abyss of our fundamental ignorance, and to stand with me in awe before the vastness of this void. I take this example from a book by the psychiatrist Oliver Sacks The Man Who 15

The denotative school of language acquisition will argue that we understand very well how we learn to speak, namely, by imitating those who point to things and make the appropriate noises. But I learned from Margaret Mead, the anthropologist, who easily picked up colloquial language of the many different tribes she worked with, that this is not so. Once she used this method by pointing to different things hoping to learn how to call them. To her dismay she always got the same answer "chumumula". First she thought they have a very primitive language until she found out that "chumumula" means pointing with ones finger. The Noam Chomsky school of thought will argue, that we know very well how we learn to speak, namely, by activating a "language organ" that is grown into our body. Following this train of thought I would propose a mathematical organ" to


Mistook His Wife for a Hat (4). Among the many cases of the astounding functioning of "dysfunctional" minds, the most fascinating for me is his report about a pair of twins he once met in a state hospital, John and Michael, who were variously diagnosed as autistic, psychotic, or severly retarded. Here is his description: "They are... unattractive at first encounter, a sort of Tweedledum and Tweedledee, indistinguishable, mirror images, identical in face, in body movements, in personality, in mind, identical too in their stigmata of brain and tissue damage. They are undersized, with disturbing disproportions in head and hands, high-arched palates, higharched feet, monotonous squeaky voices, a variety of peculiar tics and mannerisms, and a very high, degenerative myopia, requiring glasses so thick that their eyes seem distorted, giving them the appearance of absurd little professors, peering and pointing, with a misplaced, obsessed, and absurd concentration." When he first met them, they were already known as having a remarkable "documentary" memory, that enabled them, for instance, to say at once on what day of the week a date far in the past or future would fall. However, he did not think about them, until he had another encounter of which he writes: "I forgot [them] until a second, spontaneous scene, a magical scene, which I blundered into, completely by chance." "The second time they were seated in a corner together, with a mysterious, secret smile on their faces, a smile I had never seen before, enjoying the strange pleasure 16

and peace they now seemed to have. I crept up quietly so as not to disturb them. They seemed to be locked in a singular, purely numerical, converse. John would say a number, a sixfigure number. Michael would catch the number, nod, smile and seem to savour it. He, in turn, would say another six-figure number, and now it was John who received and appreciated it richly. They looked, at first, like two connoisseurs wine-tasting, sharing rare tastes, rare appreciations. I sat still, unseen by them, mesmerised, bewildered." "What were they doing? What on earth was going on?" Oliver Sacks asked himself. But since he is not only a psychiatrist but also a numbers buff, he could provide at least a clue: while the twins were playing their game with numbers, he wrote them down and looked them up later at home in a book that lists all prime numbers up to nine-figure primes. Prime numbers are those peculiar islands floating in the infinite sea of numbers that do not evenly divide by any number but by themselves or one. To his amazement, his hunch was correct: all the six-figure numbers the twins exchanged were primes! This persuaded him to join them the next day, and equipped with his prime number book he presented them with an eight-figure prime: "... They both turned towards me, then suddenly became still, with a look of intense concentration and perhaps wonder on their faces. There was a long pause - the longest I had ever known them to make, it must have lasted a half-minute or more - and then suddenly, simultaneously, they both broke into smiles." Now all three were playing the game, with the


prime numbers getting larger and larger, until the twins were coming up with numbers much larger than those in the book. When they moved on, swapping twenty-figure numbers, Oliver Sacks could only sit in amazement, watching an unfathomable prime-numbers-ping-pong game, and contemplating unfathomability. From this example I learned at least one thing, namely, that we generally do not appreciate our own miraculous faculties when they work. We are, however, surprised when they dont work in the usual way and manifest themselves in other forms. Since there is not the smallest handle in sight with which to grasp the enigmatic behavior of the twins, I claim we are precisely in the same situation when we wish to grasp our own. Vis vis this enigma and vis vis our ignorance and, paradoxically, vis vis our sense of knowing, I thought about an epistemology, a theory of knowledge, that is cognizant of the vastness of our ignorance, a tip-of-the-iceberg epistemology that is aware of its floating state of affairs. May be one could call this a development for a calculus of un-knowables, or a theory of un-knowledge, but I was unhappy with the negative connotation implied by the prefix "un-", and I looked for a word that would refer to the absence of a faculty in a positive sense, as blindness is "un-seeing" or deafness is "un-hearing". Neither in Greek nor in Latin I could find what I was looking for, and I was on the verge of giving up my search, when I remembered 17

that "truth", believed by many today to have a positive sense, had a negative connotation in ancient Greece: "Aletheia", or "that which is not obscured", with the prefix "a" for "not", and "letheia" from "lanthano" to "hide" to "obscure". It is the river Lethe, you may recall, one crosses to enter Elysium and all memories vanish, while crossing the river Acheron all memories are reinforced before entering Hades, so that they haunt you ad infinitum. Thus Lethe offeres itself naturally for naming a calculus of unknowables "Lethology", and I am going to use this calculus as a rigorous platform for discussing the problems I mentioned before. I see within unknowables two components: undeterminables and undecidables; thus I shall address these in the following two points: l. How to principle 2. How to principle deal with systems that are in undeterminable; and answer questions that are in undecidable.

Thus Lethe offeres itself naturally for naming a calculus of unknowables "Lethology", and I am going to use this calculus as a rigorous platform for discussing the problems I mentioned.

l.

UNDETERMINABLES

"Causality determines the flow of events in the universe" is one of the central beliefs in our Western culture. It is the belief that if we were to understand the Laws


of Nature we would understand the world: our quest, therefore, is to determine these Laws. While skeptics argued for over two and a half millenia against this belief, I would like to report about some other arguments in the same direction that were developed over only the last fifty years by logicians and mathematicians who studied the fundamental principles, functions, and operations of systems in general, because the results of these studies have a direct bearing on the central themes of our symposium, namely, cognition, education, and learning. Although I shall discuss theoretical aspects of these notions, there is no need to go through logico-mathematical acrobatics that would be hard to follow by the uninitiated. Thanks to an elegant intellectual twist invented by the British mathematician Alan Turing (5), we can leave all the cumbersome derivations, deductions, inferences, etc., to a (conceptual) "machine", and can comfortably sit back and watch the machine grinding out answers for our illumination and contemplation. A "machine" in this context is a set of rules by which some state of affairs are transformed into some other state of affairs. For our purpose it is sufficient to distinguish only two kinds of such machines: one, usually referred to as "trivial machine", has only one fixed rule that operates without change on the various state of affairs; the other one, the "non-trivial machine", having rules which, however, change the rules that operate an the state of affairs: a machine within a 18

machine, a "second order machine", so to say. To make these notions more tangible let me first construct, or synthesize, a typical trivial machine whose "state of affairs" consists of only the first four letters of the alphabet A, B, C, D, and whose "rule of transformation" is to associat (anagrammatically) each state (letter) with one in the opposite sequence D, C, B, A. If one presents this "anagrammor with, say, a "B", it will respond wih a "C", and mutatis mutandis- will do so in perpetuity. As you will see at once, the trivial machine is one of the central pillars of Western thought (6). Take, for instance, the mini-universe of the four letters of before, together with the anagrammatic rule of transformation as metaphor for a universe with four states of affairs and the transformation rule as its Law of Nature. Then with A, B, C, D, as causes and D, C, B, A as effects, causality is now determining the flow of events in this Universe; or take a transformation rule as the property of an organism, then certain stimuli will elicit the appropriate responses; or take the character of a person as a transformation rule, then his or her motives will entail their corresponding actions; or look at computer science where the transformation rule is, of course, the program computing from its inputs the appropriate outputs. The underlying triadic structure of all these examples is that of logical syllogisms with their two premisses and their inescapable


conclusion. This structure is also embedded in our language through the words "because", "in order to", and "for", usually with an unspoken reference to an immutable rule. However, remember when I proposed in my earlier example a partircular transformation rule, a Law of Nature, it was my choice. In other words, I was playing God for this Universe, for I could have chosen other Laws, for instante replacing each letter with its follower (and D with A); or pulling the four resulting letters out of a hat, letting chance determine the Laws of Nature (in contrast to other beliefs (7)). etc., etc. It is important to see the considerable freedom we have in synthesizing these machines. But it is also important to see that if we do not know their workings we can through examination identify the operations of such machines. One has simply to go through all available states (A, B, C, D) and pair them with their responses (say, D, A, B, C): then the pairs AD, BA, CB, DC, identify this anagrammatic machine, for they are the "machine". In other words, trivial machines are not only determined through their synthesis, they are also determinable through analysis. Morover, since their operating rules remain unchanged, i.e., they are historically independent, thay are also predictable! lt was apparently this insight that prompted Laplace almost 200 years ago to make his paradigmatic statement (8) that if for a superhuman intelligence the present conditions of all particles in the universe would be known "... nothing would be uncertain and the 19

future and the past would be present to his eyes." Today Laplace would rejoyce: "The Universe: a trivial machine!" I am sorry to say, mon cher Pierre Simon Marquis de Laplace, you rejoyced too early. This is the appropriate moment to turn to non-trivial machines. As I described them before, they are changing their operating rules according to a "second order" rule, a "program". Such a program may be seen as operating on "internal states of affairs", the operating rules of before, namely, those that operate on the "external states of affairs". To stay with the earlier example of the "anagrammor", we are now changing the anagrams according to the program that specifies a particular non-trivial machine. If we contemplate changing one anagram into another, we have to know the anagrams at our disposal. Referring again to our 4-letter universe (A,B,CD), then, depending on whether we allow different letters (say, A, B) to be translated into like letters (say, A-C; B-C) or not, we have with 4-Letter anagrams for the two cases (see Appendix (Apx)): Nlike = 256 anagrams Ndiff = 24 anagrams For the purpose of constructing now a nontrivial machine, I have listed (Apx) and numbered all 24 anagrams possible with all four letters different; and for the purpose of demonstrating the workings of such a machine I picked for the "internal states of affairs" the four anagrams #10, 17, 19, 24, where #24 you may recognize as the one


used in the example for the trivial machine: 10 A C D A 17 C D A B 19 D A B C 24 D C B A

D instead of C as before! Since the machine has moved now into #19, B again produces A, etc., etc. Below is for a repeated sequence of A, B, C, D (first line) the sequence of responses (second line): A, B, C, D, A, B, C, D, ..... B, C, A, A, D, A, B, B, ..... I hope these examples are sufficient to demonstrate the fundamental difference between these machines and their trivial sisters. However, when the Transition Table of above is given, the determination of any sequence is now trivial. Why then calling these machines non-trivial? This will become obvious when we do not know the program or transformation rules, and have to identify them through experimentation. Before planning such an experimental procedure it would be wise to estimate the effort that has to go into, solving the identification problem. This effort depends, of course, upon our knowledge of the system. Let us assume we know that the program works (as in our example) with exactly 4 anagrammatic rules and the alphabet consists of precisely 4 letters; then the number of different machines, one of which is the one we want to identify, is precisely (Apx): N4 = 4,294,967,296. This looks like a large number, but with large number-crunching computers that may test one million of our possible machines in one second, it takes perhaps not more than 1 hour and 15 minutes to have our machine identified. But let us assume now that we do 20

In fact, whenever our to be constructed machine stays in any one of these internal states, it acts as a trivial machine according to that anagrammatic rule. Now we are in a position to select the program that will run our machine; this program determines the letters X and the rules (anagrams) R that follow after the machine operated on letter X and under rule R. For this demonstration I chose the following program of operations: TRANSITION TABLE R R = 10 R = 17 R = 19 R = 24 X R X R X R X R B X B C D B C D A 10 17 19 24 C D A B 17 19 24 10 D A B C 19 24 10 17 D C B A 24 10 17 19

Example: Assume the machine is in a state in which it computes for A, B, C, D the anagram #10 (B, C, D, A), and is presented with the letter B (X=B); of course, it will produce C (X=C), but at the same time it will change the anagrammatic rule from #10 (R=10) to #17 (R=17). Hence, if given B again, we have to look under R=17 and find the response to be


not know that only 4 anagrammatic rules (i.e., Laws of Nature) are operative, but we do know that this universe has the property that two different "causes" will never result in like "effects", then the number of possible universes, of which ours is only one, is (Apx): N24 = 6.3 x 10 . Since the large computer of before can test 12 only about thirty trillion (i.e. 30x10 ) machines per year, and the universe we are living in is at most only 20 billion (i.e., 9 20xlO ) years old, it is much too young to have tested only a fraction of the possibilities large enough to be mentioned. But our ignoranse may run deeper. Since distinguishing difference in causes and difference in effects is a question of cognitive skills, we cannot be certain that the canon "different cause/different effect" is valid for the universe under investigation. Hence we have to be prepared to look for the one sample out of (Apx) N256 = 5 x 10616 57

machine configurations whose identity cannot, in principle, be established by a finite sequence of experiments: the machine identification problem is in principle unsolvable! While non-trivial machines can be synthetically determined, they are analytically undeterminable, history dependent, and unpredictable. It takes a long time for this insight to sink in, for it contradicts all the intuitive notions we have of Natures magnificent order, of the reliability of our friends and of a coherent sense of ourselves. Shall we doubt the nontriviality of all this? When I ask my friends whether or not they consider themselves to be trivial or non-trivial "machines", they unequivocally opt for nontriviality, although when asked of their opinion about others, the answers are mixed. This should not surprise, because in comparison to the fickle, unpredictable, and unanalyzable non-trivial machine, the trivial machine with its reliability and predictability appears to be a gift from Paradise. We pay considerable sums of money for garantees that the machines we buy are not only trivial when we buy them but maintain their triviality for a long period of time. When one morning our car refuses to start, its history dependent, non-trivial, true nature comes to the fore, and we have to call a professional trivialisateur who, with his tools re-establishes the cars apparent trivality. It is clear that we as children of our culture are infatuated with trivial systems and whenever things do not go the way we 21

.

This is a number with 616 zeros following 5. Clearly, the identification problem of nontrivial machines is non-trivial or, as it is put in the language of computer scientists, it is "transcomputational". Translated back into common language: it cant be done!. Optimists, nevertheless, may argue that sooner or later we will have the theoretical or technical means available to tackle this problem. Unfortunately, however, this hope is unfounded. It can be shown (9) that there are


think they should go we will trivialize them: then they become predictable. I have discussed this point at length, because at some uneasy moments I sense that in the absence of an understanding of how to deal with one of the most non-trivial, inventive surprising, unpredictable creatures I know of, our children, some educational systems confuse learning with trivialization. In learning, the number of internal states grows and the semantic relational structure (the "program") becomes richer. Trivialization, on the other hand, means amputation of internal states, blocking the evolution of independent thought and rewarding prescribed, hence predictable, behavior: "6" is the answer to the question "What is 2x3?"; unacceptable is: "an even number", "3x2", "my age", and others (10). This begs the question of the meaning of "tests". Are tests designed to establish the workings of another mind, the mind of the student? Then tests try to establish the impossible, for as we know now, the mind of a non-trivial student is analytically indeterminable. Are tests designed to establish the degree of success an educational system had in trivializing its students? Then the results do not reflect upon the malleability of students but upon the educational system and the tests it designs. That is: tests test tests, (and not those supposedly tested) examinations examine the examiner, not the examined. This becomes evident when we see students study exams in order to pass exams, 22

which is not the same as to study the subject matter to know the subject matter. But then the question arises of how do we know what they know of the subject matter? Indeed, how do we know? Or, in this context, how do we know vis vis unknowables? Or, using the concept of non-trivial systems, what is it that we may know about them, if we cannot know about their workings?

This begs the question of the meaning of "tests". Are tests designed to establish the workings of another mind, the mind of the student? Then tests try to establish the impossible, for as we know now, the mind of a non-trivial student is analytically indeterminable. Are tests designed to establish the degree of success an educational system had in trivializing its students? Then the results do not reflect upon the malleability of students but upon the educational system and the tests it designs. That is: tests test tests, (and not those supposedly tested).

25 years ago this question was either not asked, rejected when asked, or one thought about developing methods of trivialization. However, within these years the pursuit of two ideas changed fundamentally the approach to this question, One of these ideas evolved in mathematics where the notion of non-triviality was translated into nonlinearity, which brought forth a colorful bouquet of surprises and amazements under the names of non-linear dynamics and chaos theory; the other one grew within systems and computer science, where the cybernetic notion of circularity and closure brought forth a wealth of new insights and


perspectives. Let me first and address closure, the for notions these of are

will sooner or later, converge to a stable behavior. Because of the short history of this development, these dynamic equilibrio are by different people given different names: "fix points", attractors", "strange attractors", and "eigen-behaviors" (11). I shall demonstrate these dynamic stabilities in a moment, after I have accounted for the other fascinating results of these studies. One of them is that the initial condition of some systems determine their behavior in a crucial way: they may under one condition assume one, two, or more different forms of eigenbehaviors or under another condition, will go on and on without ever going into any form of stability: they become chaotic! One of the surprising observations here is that frequently the two initial states which separate the systems behavior into

circularity

notions that were avoided by orthodox science. What is meant by circularity is that the outcome of the operation of a system initiates the next operation of that system: the system and its operations are a "closed system". This is to allow that an experimenter considers her- or himself as part of the experiment; or that a family therapist perceives of him or herself as a partner of the family; or that a teacher sees her- or himself etc. To see the unorthodoxy of closure, i.e., including the actor into the actors universe, one should remember Laplace, who did not think of himself being a part of his universe, otherwise he would not have declared it to be a trivial machine, or think of "objectivity": it demands that the properties of the observer shall not enter into the description of his observations. I ask how can this be done? Without him there would be no description nor any observation. I shall now report on the fast developing field of non-linear dynamics, where circularity and closure are essential ingredients. There are three results that are here of fundamental significance. The one is that if one lets a closed system operate recursively on its outcome, it as participant in the learning/ teaching process in the classroom, etc.,

convergence to stability or into divergence to chaos can be infinitesimally apart: even systems whose rules of operation are known may be unpredictable! An example shall clarify these points. Let us operate recursively the non-trivial machine of before offering it the letter "A", and asuming it to be in the initial state of computing anagram #10 (A/10). From the Transition Table (page 8) we see the response to be B/10. B/10 now recursively re-enters the machine to produce C/17 and, recursively A/24, and so on and so forth. Below is the sequence of the first 17 events (top row letter/bottom row number of state): A, B, 19 C, 19 A, 17 D, 24 A, 24 D, 19 C, 19, ...

A,D, 10

A, 10

D, 17

C, 24

A, 24

D, 19

A, 19

D, 17

... 24 23

24


As can be seen, after a transient period of only two steps (A, B), the machine converges to a dynamic stability that manifests itself in producing periodically the sequence CADAD, the eigen-behavior of this machine under the chosen initial condition. Let us try another experiment by starting with a condition that did not appear in the previous run, say C/24: C, 24 B, 17 D, 19 C, 17 A, 24 D, 24 A, 19 D, 19

participants in such a network will converge to a stable dynamics, to the eigen-behavior of this network. The third step, or should I call it a leap, now follows: Let the interacting participants be the participants in a social network, then their eigen-behavior manifests itself in the language spoken, the objects named, the customs maintained, the rituals observed. Embedded in this network are the "teachers" and the "students" who, through their dialogue, establish an understanding, not of each, but of each other where a subject matter may be the vehicle for this understanding, for learning how to learn.

There, after 3 transient steps (C, B, D) the system assumes a stable dynamics, the eigen-behavior of before: CADAD, CADAD appears, so to say, to be the manifestation of this machines inner workings which, for those who do not know it, will remain an unknowable for ever. I leave it to the curious among you to find out whether the machine so assembled is capable of other dynamic stabilities, or whether CADAD is the only thing it can say of itself. I shall go on to generalize these observations in three steps. The first is to give without proof the essence of a theorem concerning these machines. It says that an arbitrarily large closed network of recursively interacting non-trivial machines can be treated as a single non-trivial machine operating on itself, as, for instance, the machine in our example. This insight entails the second point, namely, that the dynamics between all interacting 24

How this comes to pass is unknowable; but that it comes to pass is because of our doing it together in a recursive dialogue. Here is what Martin Buber has to say (12). "Contemplate the human with the human, and you will see the dynamic duality, the human essence, together: here is the giving and the receiving, here the agressive and the defensive power, here the quality of searching and responding, always both in one, mutually complementing in alternating action, demonstrating together what it is: to be human. Now you can turn to the single one and you recognize him as human for his potencial of relating; then look at the whole and recognize the human for his richness of relating. We may come closer to answering the question: what is a human?, when we come to understand him as the being in whose dialogic in his mutual present two-getherness, the encounter of the one with the other is realized at all times.


2.

UNDECIDABLES

There are among propositions, problems, questions, etc., those that are decidable and those that are in priciple undecidable. Decidable, for instance, is the question whether 3,536,712 is, without remainder, divisible by 5. The answer is clearly a "No"; however, if we had asked "divisible by 2?", the answer clearly is "Yes". One could, of course invent more difficult questions, very difficult questions, extraordinary difficult questions that may take years to decide, but in the pursuit of answering them we are assured of their decidability because of our choice of the rules how to climb from one node in this crystalline structure of logico-mathematical relations to the next one. This is why for example mathematicians are, after 250 years, still trying to "proof", that is, to give explicit instructions to the climbers of how to procede, a conjecture that Christian Goldbach wrote in 1742 in a letter to Leonard Euler. Goldbach had the hunch that every even number can be representad by the sum of two primes, as, for instance, 12 = 5+7, or 16 = 13+3, etc., etc. Indeed every even number tried so far can be decomposed into two primes but this is, of course, not a proof! In other words, the question: "Is Goldbachs conjecture provable?" is not (yet) decided. By inserting the three letters "yet" in the previous sentence decidability is stipulated. But with Kurt Godels observation in 1931 that within our 25

mathematical system there are undecidable propositions (13), the suspicion arises that Goldbachs conjecture may be one of them (14). But there is no need to participate in logicomathematical somersaults to appreciate the appearance of in principle undecidable questions for everyday language and lore is laced with them. Take the question of the origin of our Universe: how did it come about? Clearly, this question is in principle undecidable, for there could not be any witnesses, and if there were, who would believe them. Nevertheless, there are many answers to this question. Some say it was the union of Chaos with Darkness that brought forth all there is; others say it was a singular act of creation some 4000 years ago; others insist that there was no beginning and there is no end because the universe is in a perpetuos dynamic equilibrium an "eigen-universe"; still others argue that the whole thing began 10 or 20 billion years ago with a Big Bang, whose noise can still be heard as a whisper over large radio antennas; I have not yet accounted for the answers Hindus, Arapesht Massaist Nubas, Khmersp Bushmen, etc., etc., would give when asked this question. In other words, tell us how the Universe began, and we tell you who you are. The distinction between decidable and in principle undecidable questions may be now sufficiently clear, that I can present to you the following thesis (15): "Only those questions that are in principle undecidable, we can decide".


Why? Simply because decidable questions are already decided by the choice of the framework within which they are asked. The frame work itself may however, have been an answer we chose to an in principle undecidable question. This observation sharpens the distinction between these two kinds of question: answers to decidable questions are forced through necessity, while for those to undecidable questions we have the freedom to choose. But with this freedom of choice we must assume the responsibility for our choice. This sharpens further the distinction between these questions: procedures for arriving at answers to decidable questions may be faulty, hence, here arises the notion of truth; ethics however is the domain within which we assume responsibility for our decisions: the antonym for necessity is not chance (16), it is freedom, it is choice.The distinction between decidable and in principle undecidable questions may be now sufficiently clear, that I can present to you the following thesis: "Only those questions that are in principle undecidable, we can decide". Why? Simply because decidable questions are already decided by the choice of the framework within which they are asked. The frame work itself may however, have been an answer we chose to an in principle undecidable question. This observation sharpens the distinction between these two kinds of question: answers to decidable questions are forced through necessity, while for those to undecidable questions we have the freedom to choose.

How do these considerations affect our perspectives on cognition, on learning, and on "Cognition as Learning"? I think in a crucial way. Here some examples: Mathematicians dwell in two distinct worlds that are irreconcilably separated by deciding differently the in principle undecidable question "Are the numbers, the formulas, the theorems, the proofs, etc., of mathematics discoveries or are they our inventions?". Here is a report (18) about the way a citizen of the world of discoveries sees how we know mathematics: "A deity he fondly calls the Supreme Fascist has a transfinite book of theorems in which the best proofs are written. And if he is well intentioned, he gives us the book for a moment. Like a medium at a seance it is said, a good mathematician is one who is especially adept at communicating with this Platonic realm where abstractions and symmetries sit waiting to be discovered by the properly prepared mind. And here is the confession of a citizen of the world of inventions (19): "I for my part believe that what a mathematician does is nothing but the derivation of statements with the aid of certain, to be enumerated and in various ways choosable, methods from certain, to be enumerated and in various ways choosable, statements and all what mathematics and logic can say about the mathematicians activity,... is contained in this simple statement of the state of affairs. Let us consider children growing up in these two different worlds: In the world of discoveries they must learn to repeat what 26


others were by the Supreme Fascist permitted to glance from "The Book"; in the world of inventions they are invited to play a game in which they write the rules, invent their mathematics, from which mathematicians may learn one thing or another (20). Here another example: Ever since the French psychologist Alfred Binet invented a century ago a test for intelligence, the belief in the ability to test for intelligence became very popular indeed, surprisingly more so in Brittain and in the United States than in the country of its origin. It is therefore understandable that when electronic computers became more and more sophisticated, the question of whether these "chaps" are intelligent, and how to decide whether they are, was first raised by an Englishman, our friend Alan Turing, the inventor of the non-trivial machine. The test he proposed to establish whether or not computers can "think" has now become the credo for those who believe in Artificial Intelligence, or AI. The test consists of having an "X". which can be a human being or a computer, placed behind a curtain, and having examiners bombarding "X", with questions to find out what is behind that curtain: man or machine? lf they erroneously conclude "X" is a human being, or if they cannot decide and give up, it is said that the computer has tested positive an the Turing Test, that is, this computer is intelligent, this computer can think. AI is justified (16)! It is always surprising and amusing to me that it is not plain to everyone that it is not 27

the machine that has passed the test, but that it is the examiners who have failed it by making wrong judgements or by accepting defeat. This surprises me the more, since the problem of "The Other Mind", that is, "Are there other minds besides me?" is, with a few Continental exceptions, a problem confined to the Isle (17), and those philosophers who pursued this problem would not have accepted the Turing Test to decide it. At this point you may have guessed that I would like you to see "The Other Mind" and related questions as in principle undecidable, hence for us to decide and to take the responsibility for our decisions. lf you have followed me so far, I ask you to stay with me through the next points, though they may hurt first before they take shape. I take the metaphor of seeing the question of "The Other Mind", that is, "Does X have a mind?", to let the answers to the questions "is X incompetent?", "is X a criminal?", "is X insane?", etc., to be seen as being the responsibility of those who decide these questions: the examiners, the jurors and judges, the psychiatrists, etc. This points to the ontological trap where attention is placed on the is in the question "is X insane?", instead of directing the attention to Y who decides (for her or himself) what "is". Ontology, and objectivity as well, are used as emergency exits for those who wish to obscure their fredom of choice, and by this to escape the responsibilty of their


decisions. Here is Jos Ortega y Gassets observation (21): "Man does not have a nature, but a history... Man is no thing, but a drama... His life is something that has to be chosen, made up as he goes along, and a man consists in that choice and invention. Each man is a novelist of himself, and though he may choose between an original writer and a plagiarist, he cannot escape choosing... He is condemned to be free." Indeed, we are condemned to be free! Let us rejoice in this freedom by joining the chorus in Beethovens Ninth Symphony with the new version of Schillers words where "Freude" (joy) is now sung all over the world as "Freiheit" (freedom) (22): Freiheit schner Gtterfunken Tochter aus Elysium Wir betreten feuertrunken Himmlische dein Heiligtum.01 02 03 04 05 *A *B *C *D A B D C A C B D A C D B A D B C 06 07 08 09 10 A D C B B A C D B A D C B C A D B C D A

APPENDIX(i) The number of ways in which n different objects can be put into n different boxes, called permutations is: Ndiff = n! = 1, 2, 3 . . . . (n-1), n; In our case, n=4: Ndiff = 4! = 1, 2, 3, 4 = 24. (ii) The number of ways in which n symbols (say, numbers) can be written in strings of p places (digits) is: Nlike = n If p = n: Nlike = n ; In our case, n=4, hence: Nlike = 4 = 2 = 256. (iii) The 24 Four-letter Anagrams of A, B, C, D.4 8 n p

11 12 13 14 15 16 17 18 19 20 B D A C B D C A C A B D C A D B C B A D C B D A C D A B C D B A D A B C D A C B

21 22 23 24 D B A C D B C A D C A B D C B A

(iv) The number of distinct non-trivial machines NS (X, Y) that can be synthesized with S internal, X input, and Y output states is (23): NS (X, Y) = YSX, In our case, X=Y=4. For S=4: 4x4 32 N4 = 4 = 2 = 4,294,967,296. 28


For S=24: N24 = 424x4

9.

= 2

192

, or about 6.3 x 10 .

57

Gillt A.: Introduction to the Theory of Finite State Machines, McGraw-Hill, New York (1962).

For S=256: N256 = 4256x4

=2

2048

, or about 5 x 10

616

.

REFERENCES1. Curriculum Commission of the State Board of Education, State of California: Science Framework Field Review Draft. Science Subject Matter Committee, University of California, Berkeley, (May,1989). 2. Bateson,G.. "Metalogue: What is an Instinct?" in Steps to an Ecology of Mind.Balantine Books, New York, 38-60 (1972). 3. Ashby, R.: "The Brain of Yesterday and Today" in Mechanisms of Intellicience,Roger Conant (ed), Intersystems Publications, Seaside, 397403 (1981). 4. Sacks, O.: "The Twins" in The Man Who Mistook His Wifw for a Hat, Harper and Row, New York,195-213 (1987). Turing, A.: "On Computable Numbers, with an Application to the Entscheidungsproblem", Proc.London Mat.Soc.,2-/42:230-265 (1936). von Foerster, H.: "Kausalitaet, Unordnung, Selbstorganisation" in Grundprinzipien der Selbstorganisation, K. Kratky und F. Wallner (eds)t Wissenschaftliche Buchgesellschaft, Darmstadt (1990). Eigen, M. und R.Winkler: Das Spiel: Naturgesetze steuern den Zufall. Piper, Mnchen (1975). Laplace, P .S.: Essai Philosophique sur le Probabilits, Paris (1814). 29

10 I was seeing a family when their 6 year old boy came home from school l/2 hour late and in tears: "I had to stay over in school"; "Why, what happened?"; "The teacher said I gave a fresh answer"; "What did you say?"; "She asked what is 3x2, and I said 2x3, and everybody laughed; then she put me in the corner". Now I interfered: "l think you gave a correct answer but can you prove it?" At once he drew on a piece of paper three columns with 2 dots each and said "Thats 3x2:

Then he rotated the paper 90o and said "Thats 2x3.":

11. Von Foerster, H.: "Objects: Tokens for (Eigen-) Behavior" in Observing Systems, Intersystems Publications, Seaside, 273-286 (1984). 12. Buber, M.: Das Problem des Menschen, Lambert Schneider, Heidelberg (1961). 13. Gadel, K.. "Ueber formal unentscheidbare Saetze der Principia Mathematica und verwandter Systeme. I Monatsh. Mat. Phys. 38: 173-193 (1931). 14. It came to my ears that significant steps for a proof of Goldbachs conjecture have been made. However, I could not confirm this rumor at the time of writing. 15. von Foerster, H.: "Wahrnehmen wahrnehmen" in Philosophien der neuen Technologien, Merve Verlag, Berlin, 27-38 (1989). 16. Searl, J.R., Churchland, P .M., Churchland, P .S.: "Artificial Intelligence: A Debate", Scientific American, 262: 25-39 (January, 1990).

5.

6.

7.

8.


17. Shorter, J.M.: "Other Minds, in The Encyclopedia of Philosophy, Macmillan, New York, 6: 7-13 (1967). 18. Johnson, G.: New Mind, No Clothes" The Sciences, New York Acad. of Sc., New York, 45-48 (August 1990). 19. Menger, K.: "Vorwort" in Einfuehrung in das Mathematische Denken. Friedrich Waismann, Gerold & Co., Wien, v-viii (1936). 20. This is a reference to Reference #10: The teacher apparently was not aware of the importance of the commutative law of multiplication, a consequence of the commutative law of addition. This elegant proof of commutativity in multiplication,

using the invariance of area under rotation, could serve well as a tutorial device. 21. Ortega y Gasset, J.: Historia como sistema. Madrid (1941). 22. Translation M.v.F. Freedom, flash from heaven, Daughter from Elysium Drunk with fire, heavenly We enter your sanctuary. 23. von Foerster, H.: "Molecular Ethology, An Immodest Proposal for Semantic Clarification" in Molecular Mechanisms in Memory and Learning, Georges Ungar (ed), Plenum Press, New York, 213-248 (1970).

30

na Aplicacin de la Teora de Sistemas al Desarrollo de ProductosMilton Harvey Snchezn n

l enfoque de la teora de sistemas ha sido aplicado en muy diversas reas cientficas contribuyendo esencialmente a la aparicin de fundamentos y de movimientos tericos en las reas que han visto ampliados sus estudios con esta teora universal e interdisciplinaria. La aplicacin de los sistemas en el desarrollo de productos pretende suministrar un instrumento metodolgico y terico en este campo y as, permitir la construccin de modelos que sirvan de base en la adopcin de decisiones en procesos de diseo. La importancia de la teora de sistemas radica en que all, en donde el discurso terico-sistmico ha sido aplicado, los fundamentos y terminologa de estaMilton Harvey Snchez. Diseador Industrial. Universidad Nacional. Santaf de Bogot. Doctorado PHD Bergische Universitt Alemania.

Una Aplicacin de la Teora de Sistemas al Desarrollo de Productos

teora son abordados e interpretados de diferentes maneras, de acuerdo con el objetivo de la investigacin o del campo cientfico y tecnolgico que buscan en los sistemas un marco terico. En nuestro caso el diseo de productos es, por un lado, un componente ms del complejo desarrollo de la produccin industrial; por tanto, es esperado del diseo actualmente, la realizacin en el producto a desarrollar de una variedad de exigencias relacionadas con el usuario, con formas de produccin y mercadeo, as como con factores ambientales. Por otra parte, un producto desarrolla formas de relaciones comunicativas y procesos de observacin, en este sentido, los objetos se encuentran determinados por formas especficas de comunicacin que los delimita unos frente a otros. De este modo, es evidente, que para poder realizar el diseo de un producto, ste solo puede llevarse a cabo de una manera metdica y sistemtica. Cabe mencionar que en este escrito no se pueden considerar todos los aspectos relacionados con el diseo de un producto; como intereses especficos del usuario o de mercado; no obstante, supone servir como modelo de aplicacin en otros factores que all intervienen. El trabajo que aqu se presenta responde a la necesidad de darle la posibilidad al diseador, especialmente al que se encuentra en formacin, de proceder tanto en el anlisis, como en la valoracin y formalizacin de problemas y tareas de diseo. En este sentido, la aplicacin de la teora de sistemas en el desarrollo de productos hace posible en una delimitacin sistema/entorno, al definir el producto como sistema, crear incisiones en el anlisis de un objeto o producto de origen industrial que no2

eran posibles con otros mtodos tericos. El objetivo ms importante de este trabajo es poner a disposicin un marco para el desarrollo de mtodos en torno al diseo de productos y adquiere gran significado como un intento de formar unos fundamentos cientficos en este campo, especialmente en el mbito colombiano. La teora de sistemas ms avanzada tiene muchas races, dentro de las cuales se pueden aqu destacar la ciberntica de Norbert Wiener, la teora de la informacin de Claude Shannon y la Teora General de Sistemas de Ludwig von Bertalanffy. El gran valor del discurso de los sistemas es su universalidad, su dinmica y la gran capacidad de desarrollo y de aplicacin, en donde los ms diversos temas son emprendidos con trminos semejantes. El mdico y bilogo chileno Humberto Maturana, por ejemplo, ha generado en los ltimos aos dentro del desarrollo de la teora de sistemas la expresin autopoiesis como explicacin a la autoorganizacin en los sistemas, particularmente en los seres vivos. Estos planteamientos de Maturana han tenido una importante resonancia en el desarrollo de la teora de sistemas.

La aplicacin de la teora de sistemas en el desarrollo de productos hace posible en una delimitacin sistema/entorno, al definir el producto como sistema, crear incisiones en el anlisis de un objeto o producto de origen industrial que no eran posibles con otros mtodos tericos

El punto de vista terico-sistmico en el caso del proceso de diseo de productos, es poder observar funcionalmente un objeto o producto; por ejemplo una silla, en donde una posible


solucin al problema de sentarse cmodamente descubra y determine principalmente, no la composicin mecnica de las partes de la silla que haran posible el sentarse, sino ante todo, otras soluciones equivalentes funcionales al mismo problema; como butaca, banco, cojn, etc. Por otra parte, la perspectiva de los sistemas permite crear ciertas bases y cortes en las relaciones formales de un objeto y, as, determinar posibles estructuraciones en las relaciones y en la organizacin (orden) existentes entre los elementos del objeto.

aquello de que hablamos y especifica sus propiedades como ente, unidad u objeto (Maturana/Varela 1987).

Qu es Sistema?La palabra de origen griego sistema tiene muchas connotaciones. Segn Bertalanffy 1950/Danzer 1976, sistema es un conjunto de elementos recprocamente relacionados para alcanzar un fin. Bajo esta definicin, cualquier objeto puede ser observado como sistema. Un sistema es de cualquier manera un todo ordenado. No es suficiente, para poder establecer una diferencia o distincin, simplemente separar algunos elementos. Para que los elementos de un todo puedan ser diferentes y diferenciados de otros, deben constituir en determinada forma un orden. Ordenar tiene igualmente un orden, ste ltimo se compone de procesos de seleccin, de relacionar y de un regulamiento. Para poder sealar una diferencia debemos seleccionar algunos elementos de la totalidad y en determinada forma relacionarlos entre s. Si se tienen estas dos condiciones, seleccionar elementos y relacionarlos, se puede hablar de que tenemos un sistema. Los sistemas se constituyen y se conservan a travs de la creacin y mantenimiento de una diferencia frente al entorno y utilizan sus lmites para regular dicha diferencia. Un sistema est caracterizado por una cantidad determinada de elementos relacionados entre s, cuyas relaciones hacen posible determinados procesos. Para que yo juzgue a este objeto como una silla es necesario que yo reconozca que ciertas relaciones se dan entre partes que llamo patas, respaldo, asiento, de una manera3

Fundamento de la teora de sistemas Punto de partida: La DiferenciaCualquier anlisis terico y de aplicacin de la teora de sistemas al desarrollo de productos debe tener como punto de partida la diferencia entre sistema y entorno. La DIFERENCIA es la mxima del discurso de la teora de sistemas. Algo existe siempre y cuando pueda ser diferenciado de otro algo; es decir, slo cuando este algo pueda ser diferente. Draw a distinction sirve de punto de partida a Spencer Brown 1977 para su propia construccin de la teora del conocimiento. Segn Spencer Brown (Laws of form) para poder reconocer algo, es necesario que el observador pueda establecer diferencias, a lo que le debe seguir, otorgarle a lo diferenciado una denominacin (nombre) para poder darle una capacidad de empalme. En este sentido, observacin es una operacin de diferenciar y de denominar. El acto de sealar cualquier ente, objeto, cosa o unidad, est asociado a que uno realice un acto de distincin que separa lo sealado como distinto de un fondo. Cada vez que hacemos referencia a algo, implcita o explcitamente, estamos especificando un criterio de distincin que seala


tal que el sentarse se haga posible (Maturana 1987). La delimitacin del sistema y sus elementos depende de la perspectiva de observacin del objetivo especfico y del objeto a analizar. Para el diseador de una cafetera elctrica, el motor de sta es un elemento de su sistema. Para un ingeniero elctrico el motor de la cafetera es su sistema. En cada sistema, los elementos o subsistemas tienen igualmente propiedades y se pueden establecer categoras segn su significado o percepcin: tamao, color, peso, material. Las propiedades del sistema dependen igualmente de las propiedades de sus elementos. Hay productos cuyas propiedades tienen fundamentalmente una funcin esttica: joyas, artculos de moda; tambin hay productos orientados por una funcin prctica: bienes de capital o beneficio.

Los sistemas se constituyen y se conservan a travs de la creacin y mantenimiento de una diferencia frente al entorno y utilizan sus lmites para regular dicha diferencia. Un sistema est caracterizado por una cantidad determinada de elementos relacionados entre s, cuyas relaciones hacen posible determinados procesos.

Delimitacin sistema/entornoAl crearse un sistema a travs de sealar una diferencia o distincin, se debe diferenciar este sistema de todo aquello que no pertenece al sistema. De ese modo, todo eso que no pertenece al sistema constituye el entorno. No existe sistema sin entorno o entorno sin sistema. Un sistema se diferencia en primer lugar y principalmente del entorno. Adems,4

no hay ningn sistema que est fuera de un contexto especfico y que no est condicionado por dicho entorno. En este sentido, todos los sistemas son abiertos, ya que tienen, en mayor o menor grado, algn tipo de intercambio con el entorno. El producto como sistema es de la misma forma un sistema abierto, ya que los objetos estn estructuralmente orientados a un contexto y, sin l, no existiran. La diferenciacin entre cerrado y abierto en los sistemas es una cuestin relativa. Sistemas cerrados no presentan un intercambio de materia, energa o informacin. Sistemas cerrados existen nicamente, si estos se pueden abstraer o separar de las relaciones con el entorno. En todo producto como sistema se pueden diferenciar unas variables determinadas cuya composicin nos describir un sistema especfico. Estos parmetros o variables son los siguientes: Entradas (input) son los recursos necesarios para el funcionamiento del sistema; Salidas (output) es el producto final para cuya obtencin se han seleccionado los elementos y debe concordar con el objetivo del sistema; Proceso de transformacin (throughput) o proceso de diseo, es la conversin de entradas en salidas y est formado por todos los elementos, propiedades y relaciones entre s; Retroinformacin (feedback) es la confrontacin de la salida con un criterio previamente establecido; Entorno (environment) es el conjunto de elementos externos que influyen al sistema o pueden ser influidos por l.

Reduccin de Complejidad (Negentropa) o formacin de los sistemasUn estado, en donde no se ha establecido ningn tipo de diferencia o delimitacin, donde


encontramos una disposicin en la cual todo es posible y donde no se presenta ninguna forma de seleccin, de relaciones, de regulamiento o de lmite representa la complejidad absoluta. A esta clase de estado, en donde todo tipo de probabilidades y casualidades pueden, se le denomina Caos, o para utilizar un trmino de la termodinmica o de la teora de la comunicacin: entropa (tendencia hacia el desorden); es decir, el caos comprende la complejidad absoluta. Cuando un sistema ha sido conformado al establecer una diferenciacin, tenemos necesariamente este sistema como menos complejo, ya que el sistema, por estar de cierta forma ordenado, tendra menos elementos y menos posibles relaciones que un estado tan complejo como el caos. Se puede entonces, reducir la complejidad de ese estado inicial (caos) negando la entropa con la formacin de un sistema (negentropa o tendencia a una mejor organizacin).

sistema, que servir de base para aplicar, a manera de ejemplo, el enfoque de la teora de sistemas y un anlisis funcional en un problema de diseo (cocina). Al definir y observar cualquier objeto como sistema, se debe tener en cuenta que un sistema est caracterizado a travs de:

1. 2. 3. 4. 5.

Cantidad determinada Elementos Propiedades Relaciones Orden/estructura

De ese modo, al definir un sistema como reduccin de complejidad, estamos llevando cierta cantidad de elementos relacionados recprocamente a un orden u organizacin determinada. Segn Fuchs (1973) las relaciones entre los elementos de un sistema existen a travs del intercambio de energa, materia y/o informaciones entre los elementos o subsistemas. Esto puede ser aplicado directamente en el anlisis de un producto; para ello tomaremos y analizaremos la cafetera de la fotografa (figura 1) como sistema que consta de los siguientes elementos:

De qu se compone un sistema? Objeto como sistema -CafeteraDe todo lo anterior y a partir del concepto de sistema se puede proceder a un anlisis y a una valoracin de problemas de diseo. Inicialmente el recurso para ello es, merced al anlisis de un objeto (cafetera), definir este objeto como

FIGURA 1 Elementos de una cafetera como sistemae1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e125

Recipiente de agua Tapa superior Vlvula Tapadera del filtro Sujetador del filtro Tapadera de la jarra Jarra Parrilla caliente Dispositivo del cable Interruptor, lmpara Cable Calentador


principio modular, en donde una determinada Las relaciones funcionales entre los elementos cantidad de mdulos origina una gran variedad de la cafetera arriba enumerados se presentan de combinaciones, dndole al usuario la a travs del flujo de energa, materia y/o informaciones, as, la relacin entre el posibilidad de una presentacin individual y recipiente de agua (e1) y la vlvula (e3) resulta permitindole al producto facilidades en el a travs del intercambio de materia: agua empaque, transporte y reparacin. caliente. La relacin entre sujetador del filtro (e5) y la tapadera de la jarra (e6) est Las relaciones funcionales resultan a travs determinado por el flujo de la estructura funde materia: caf. cional de los elementos Un estado, en donde no se ha establecido ningn Entre la jarra (e7) y la parrilla caliente (e8) se origina una relacin por intercambio de energa: calor. Aqu se trata de relaciones activas, ya que los intercambios all presentados tienen un valor; es decir, no son iguales a cero. En las relaciones inactivas el flujo es igual a cero; como por ejemplo en la relacin entre el recipiente del agua (e1) y el dispositivo del cable (e9). En el diseo de productos son interesantes las siguientes relaciones (figura 2): FIGURA 2 Tipo de relaciones)250$/(6 (VSDFLR 7LHPSR 5(/$&,21(6 )81&,21$/(6 0DWHULD (QHUJtD ,QIRUPDFLyQ

tipo de diferencia o delimitacin, donde encontramos una disposicin en la cual todo es posible y donde no se presenta ninguna forma de seleccin, de relaciones, de regulamiento o de lmite representa la complejidad absoluta. A esa clase de estado, en donde todo tipo de probabilidades y casualidades pueden se le denomina Caos, o para utilizar un trmino de la termodinmica o de la teora de la comunicacin: Entropa (tendencia hacia el desorden)

y determinan un principio de orden del sistema. Elementos relacionados entre s que ejecutan una funcin especfica pueden presentar, como lo veremos ms adelante, una estructura en cadena, paralela o circular. La grfica (figura 3) muestra esquemticamente en la cafetera, las relaciones formales y funcionales de los elementos entre s y, entre stos y el entorno.

Las relaciones formales entre los elementos de la cafetera resultan a travs de las propiedades de estos. As por ejemplo, la relacin entre los elementos e2 tapa superior y e1 recipiente de agua est determinada por las propiedades formales de cada elemento (dimensiones), esto significa que al encontrarse dos elementos relacionados formalmente; estos elementos deben tener una propiedad comn que pueda construir esta relacin. Las relaciones funcionales entre los elementos de la cafetera resultan a travs de flujos o intercambios de materia (agua, polvo de caf, caf preparado, polvo de caf ya filtrado y filtro)6

Las relaciones formales comprenden, por una parte, la correspondencia de elementos entre s; por ejemplo, la relacin entre una lmpara y la mesa respectiva, por otra parte definen un


FIGURA 3 Flujo y Relaciones en cafetera como sistema

Entrada: polvo de caf, Filtro de papel

e3 e2

e4

e 5 Salida: Caf, filtro e6

Entrada: Agua

e1

e7

Salida: Caf preparado

e12

e8e11

Entrada: energa

e9 e10

Las relaciones entre elementos de un sistema existen a travs del intercambio de energa, materia y/o informaciones entre los elementos. Relaciones Formales _____________ Relaciones Funcionales -------------

Entrada: seal

y energa. Estas relaciones funcionales determinan el Orden o estructura de los elementos entre s y frente al entorno. Debido al flujo de energa, el elemento e1 recipiente de agua debe estar siempre ubicado junto al elemento e12 calentador. Esta correspondencia permite plantear varias alternativas de orden que hacen posible la ejecucin de la misma operacin para el cual han sido relacionados los elementos (figura 4). Sin la correspondencia entre los elementos e12 y e1 de la cafetera no habra transmisin de energa de e12 hacia e1 y el sistema como tal y como unidad no funcionara. Por lo tanto, la relacin funcional es una condicin esencial para establecer un principio de orden de los elementos recprocamente y frente al entorno y, as, permitir que el sistema con su organizacin pueda ejecutar una operacin especfica. FIGURA 4 Alternativas de orden entre los elementos e1 y e12

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En este sentido, se puede afirmar que el principio de orden de un sistema que selecciona, relaciona y permite ejecutar una operacin, es un cdigo, mediante el cual el sistema reconoce las operaciones y procesos que le son propias y las deslinde del entorno o de otros sistemas.

nos permite comprender el efecto global de un sistema (cocinar como funcin general en una cocina); por un lado, a travs del anlisis de cada uno de los elementos o subsistemas (almacenar, pelar, adobar, servir, etc.) y, por otro, al observar la interaccin entre los elementos que hace posible al sistema presentarse como unidad. La funcin general de un producto resulta de la relacin global entre el producto y el usuario, as como entre estos y su entorno. Con el enfoque de los sistemas se pueden definir la cantidad de funciones o subsistemas propias del problema a solucionar, precisar las propiedades de stas, determinar las posibles relaciones entre cada uno de los elementos y, de ese modo, poder reconocer el orden o estructura funcional. Segn Maser (1982) el anlisis como sistema de la funcin general y del portador de funcin; el objeto que ejecuta la funcin, se deben llevar a cabo correspondientemente, como un anlisis de lo que es (producto real) y un anlisis de lo que debe ser (producto a proyectar). Un anlisis funcional puede llevarse a cabo con xito tomando como recurso, el proporcionar a cada una de las funciones (verbo) un portador de funcin (sustantivo), que permite que se realicen las funciones del sistema, en la cual las propiedades de las funciones (adverbio) deben ser transformadas en correspondientes propiedades del portador de funcin (adjetivo). Por ejemplo: funcin (verbo) llega a ser

El sistema de las funcionesSegn la teora constructivista de Pahl/Beitz 1986 se entiende bajo funcin la relacin entre entrada (input) y salida (output) de un sistema (Black Box), que tiene como objetivo poder solucionar una tarea o cumplir un propsito determinado. Siegfried Maser (1982) define como funcin todo lo que hace un algo (funciones activas) o con lo que ese algo puede ser hecho (funciones pasivas). Idiomticamente son expresadas las funciones a travs de verbos; es decir, relaciones entre un sujeto y un objeto pueden ser representadas como el efecto entre funciones activas y pasivas: cortar, preparar alimentos, cocinar, lavar vajilla, beber, etc.El principio de orden de un sistema que selecciona, relaciona y permite ejecutar una operacin, es un cdigo, mediante el cual el sistema reconoce las operaciones y procesos que le son propias y las deslinde del entorno o de otros sistemas.

Todo sistema y en consecuencia un producto como tal, puede ser descompuesto en subsistemas que poseen algunas caractersticas bsicas del sistema del cual proceden. Por lo tanto, un anlisis funcional8

propiedad (adjetivo) del portador de funcin (sustantivo) calentarcalefaccin/caliente.


Esta representacin tiene como ventaja el poder analizar de forma unificada y funcionalmente el producto como objeto, as como en sus relaciones internas. Tjalve (1978) define al portador de funcin como la parte de un sistema o un elemento que hace posible ejecutar la funcin deseada del sistema; sin embargo, se puede agregar que un portador de funcin tambin puede ser un sistema, como es el caso del automvil o el de una mquina de afeitar. Esto significa que un producto puede ser, por un parte, monofuncional; ya que el objeto est reducido a realizar a una sola funcin determinada, en nuestro ejemplo de la cafetera solo se puede preparar caf en ella; as como un abrelatas solo sirve para abrir latas. Por otro lado, un producto puede ser polifuncional, si se pueden

ejecutar con este objeto varias funciones, como es el caso de un ayudante de cocina, con el cual se pueden realizar varias funciones: agitar, mezclar, licuar, cortar o el de un equipo de sonido, que se puede utilizar como radio, grabadora o tocadiscos. Al observar, que un portador de funcin puede ser o es un sistema, debe ser ste igualmente analizado como tal (figura5).

Con el enfoque de los sistemas se pueden definir la cantidad de funciones o subsistemas propias del problema a solucionar, precisar las propiedades de stas, determinar las posibles relaciones entre cada uno de los elementos y, de ese modo, poder reconocer el orden o estructura funcional.

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Anlisis funcional como Sistema Problema de diseo: CocinaEl planteamiento anterior permitir en el proceso de diseo de una cocina, reducir la complejidad del problema a travs del anlisis de esta tarea como sistema y, as, alcanzar un inventario completo9


de los elementos del sistema cocina, de sus relaciones y propiedades que harn posible desarrollar alternativas y soluciones funcionales equivalentes al problema formulado. Este ejercicio no pretende proponer una forma concreta de diseo de cocina, es, nicamente, una manera ejemplar y general de crear unas bases de procedimiento para formalizar a partir de un punto de vista tcnico y de un principio de orden. Partiendo de lo visto anteriormente, podemos descomponer la funcin global del problema (cocinar) en elementos o funciones parciales, para ello, representaremos la funcin general en forma de una caja negra (Black Box), que solo hace referencia a las entradas y salidas del sistema. Una representacin Black Box no indica de que manera la salida (output), como funcin de la entrada (input), puede ser alcanzada (figura 6). FIGURA 6 Funcin cocinar en forma de caja negra%ODFN %R[

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Solo se puede comprobar en esta representacin que algo entra en la caja (funciones; causa) y que algo sale de all (funciones; efecto). Hay que observar que una cocina es un objeto polifuncional, es decir, en la cocina se llevan a cabo varias funciones. Se puede comprobar, por un lado, que tipo de entradas se encuentran a disposicin (tipo de energa o vveres) y, por otro, en qu clase de salidas debe ser transformadas las entradas. En esta descripcin, la funcin general del problema representa la relacin entre entradas y salidas. Al descomponer la funcin general cocinar (figura 7) en elementos o funciones parciales (almacenar, adobar, repartir, comer.) permitir darle claridad gradualmente a esta caja negra luego de un anlisis de las funciones comprometidas y, as, comprobar explcitamente los efectos y relaciones funcionales internas. FIGURA 7 Funcin general Cocinar y sus funciones parcialesApvyBrrhy

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Segn Maser (1982) las relaciones funcionales internas de un sistema pueden presentar una estructura en cadena, paralela o circular (figura 8); y se constituyen, por ejemplo, de forma temporal (coccin se ejecuta antes de servir) o surge de una relacin espacial (rea de coccin cerca del rea para servir). FIGURA 8 Estructuras en las relaciones funcionales de un sistema

El enfoque de la teora de los sistemas descompone sistemas globales complejos en elementos, de la misma manera, un anlisis funcional hace posible descomponer una funcin general en funciones parciales y determinar la estructura de sus relaciones internas; en nuestro ejemplo del problema de la cocina, la funcin general que se lleva a cabo es cocinar (figura 9).

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FIGURA 9 Transformacin de caja negra a travs del anlisis funcional como sistema7yhpx7

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Un recurso en el anlisis funcional es otorgarle a cada una de las funciones un portador de funcin (sustantivo) encargado de ejecutar la funcin (verbo) deseada en el sistema. Para ello es igualmente necesario observar como sistema cada uno de los portadores de funcin del sistema global cocina. As, portadores de funcin como depsito o comedor deben ser definidos y analizados tambin como sistemas (figura 10). El sistema depsito como portador de la funcin almacenar nos permitir, a manera de ejemplo, ilustrar este ejercicio.

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FIGURA 10 Anlisis del portador de funcin como sistemaApvyBrrhy TvrhBrrhy

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Un examen detallado del portador de funcin Depsito como sistema determina racional y sistemticamente la organizacin entre las relaciones del sistema (depsito) y entre ste y el entorno. El depsito es un vnculo entre el sistema global (cocina) y el entorno. A travs del depsito entran en el sistema artculos y mercancas como utensilios y vveres e igualmente salen materiales como basuras y empaques por ejemplo. Un depsito en una cocina es necesario por razones econmicas e higinicas, pero tambin el factor esttico cumple un papel importante en el depsito

como elemento de la cocina. En el depsito son almacenados materiales y artculos con diferentes propiedades; lo que obliga clasificar al depsito en diferentes elementos de depsito y, as garantizar, por un lado una conservacin ptima de los artculos o vveres, y por otro, definir un orden y correspondencia funcional entre cada uno de los elementos de depsito. Bsicamente podramos establecer segn este planteamiento, los siguientes elementos de depsito (figura 11) depsito para temperatura normal, depsito fro y un depsito para refrigeracin.

FIGURA 11 Elementos de depsito segn sus propiedades

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En el depsito normal estaran almacenados materiales secos, vveres y alimentos, utensilios o aparatos, as como envases y material de empaque (figura 12 ) FIGURA 12 Depsito normal y sus elementos'HSyVLWR QRUPDO

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En el depsito para material seco son almacenados materiales como harina, cereales, pan, azcar arroz, sal, caf, etc. Estos vveres deben ser protegidos de la humedad, prdida de aroma. Segn Burckardt (1981) los depsitos de temperatura deben mantener una temperatura mxima de 20 C. Por factores econmicos, ecolgicos y estticos surge la necesidad de tener en este tipo de depsito un rea reservada a envases y material de empaque, como por ejemplo, botellas, recipientes, cajas. Este mismo ejercicio puede efectuarse en los otros elementos de depsito de la funcin almacenar; en donde se debe tener en cuenta, que la tarea principal de un depsito fro es la de mantener frescos y a corto plazo determinado tipo de vveres perecederos. El depsito fro debe disponer de reas para almacenar carnes, lcteos, frutas y verduras, bebidas, as como desperdicios. 14

Segn Pieper (1981) un depsito para carnes debe proteger contra cambios de temperatura, luz, humedad y corrientes de aire y, as evitar cambios en la calidad, apariencia y peso. Por otra parte, lcteos deben ser almacenados con temperaturas promedio de 0-4C, igualmente un depsito para lcteos debe garantizar que estos no entren en contacto con otros vveres; dada la tendencia en este tipo de alimentos de absorber aromas y sabores de otros alimentos. Al almacenar frutas y verduras en el depsito fro se debe tener en cuenta una temperatura media de almacenamiento entre 3 y 5C y que estas reaccionan fcilmente a la temperatura, luz y oxgeno. El tercer elemento del sistema de almacenamiento (depsito para refrigeracin) debe albergar y conservar por un espacio de tiempo prolongado a temperaturas bajo cero (entre 0 y -18 C) vveres y alimentos como carnes, pescados y verduras. El tamao y dotacin del sistema de depsito debe ajustarse lgicamente de acuerdo con las condiciones especficas del usuario.


Las relaciones entre elementos: Preparacin de VerdurasDado que las relaciones entre elementos de un sistema existen a travs del intercambio de energa, materia y/o informaciones podemos determinar esta corriente entre los elementos tomando como ejercicio la preparacin de verduras dentro de las actividades que se llevan a cabo en una cocina. Observamos que en esta fase, las verduras se colocan, lavan, pelan, limpian, cortan y se disponen de tal manera, que puedan continuar siendo preparadas para

una presentacin y consumo final. Para ello, la preparacin de verduras debe estar asociada funcionalmente en el sistema integral de la cocina, por un lado, con el depsito y la recepcin de verduras (con el entorno para obtener verduras frescas) y, por otro, con una preparacin especfica y adobo, as como con el fregadero. Esto significa que en el momento de definir la estructuracin del orden del sistema cocina, los elementos portadores de las funciones comprometidas deben estar juntos y vinculados directamente (figura 13).

FIGURA 13 Relaciones a travs del flujo de material de vveres, verduras, utensilios, basuras y envases.

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Las relaciones funcionales internas en la preparacin de verduras (disponer, lavar, pelar, limpiar, cortar, colocar) deben ser estructuradas en cadena, ya que deben ejecutarse temporalmente una despus de la otra (figura 14). 15


FIGURA 14 Estructura en cadena de relaciones parciales en la preparacin de verduras'HSyVLWR

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La organizacin de las funciones internas o del orden de los portadores de funcin en la preparacin de verduras depende fundamentalmente del plano general del rea

de preparativos, para ello, se podra de acuerdo al anlisis anterior, proponer las siguientes soluciones en la estructuracin de los portadores de funcin para la preparacin de verduras (figura 15).

FIGURA 15 Posibles soluciones para el orden en la preparacin de verduras

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La cocina calienteDenominaremos como cocina caliente a la preparacin de vveres y alimentos para ser consumidas a travs de un tratamiento trmico; siendo esta actividad el punto central de todo sistema de cocina. La cocina caliente est funcionalmente relacionada con la

preparacin de verduras, con la preparacin de carnes, con el racionar y hacer porciones, con la entrega de alimentos preparados, con el fregadero, con la cocina fra, as como con las diferentes clases de depsitos. Funcionalmente presentaran las relaciones internas de los elementos de la cocina caliente un intercambio de materia de la siguiente manera (figura16).

FIGURA 16 Relaciones en la cocina caliente a travs del flujo de materiaWrqh rhhqh

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Podramos establecer que los diferentes procesos trmicos que se llevan a cabo en la cocina caliente, se reducen bsicamente a cocer, asar y hornear. Las funciones en la preparacin de un solo alimento determinado presentan en general, al igual que en la preparacin de verduras, una estructura en cadena; a la preparacin de una salsa especfica se le agrega luego la verdura o la carne.

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Sin embargo, en la preparacin de varios tipos de comida de la cocina caliente aparecen o pueden aparecer las relaciones internas de las diferentes actividades que all se ejecutan de forma simultnea; presentando una estructura paralela de las relaciones funcionales internas. De esta manera, se alcanza un ahorro de tiempo y se disminuye la prdida de vitaminas en los alimentos. De all, una estructuracin en el orden de los portadores de funcin (aparatos, horno, parrilla, utensilios, etc.) puede presentarse y disponerse de manera libre e independiente unos de otros, lo que permitira una configuracin de estructuras paralelas, seriadas o una combinacin paralela-seriada (figura 17). FIGURA 17 Posibles soluciones en la estructuracin de los portadores de funcin de la cocina caliente

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La entrega de comidas tiene un significado especial al ser un vnculo entre la preparacin de comidas fra y caliente, entre el comedor y finalmente entre el sistema de cocina y el entorno. Aqu debe observarse la estrecha correspondencia entre cada uno de los elementos involucrados al definir una configuracin en el orden de los portadores de

funcin y en la definicin de la forma del sistema (figura 18). El comedor por otra parte es un vnculo entre el sistema de cocina y el entorno. El comedor debe estar funcionalmente en correspondencia; por un lado con el entorno y, por otro, asociado con la entrega de alimentos, con el fregadero y con el depsito de basuras (figura 19).

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El ejercicio de analizar y clasificar la cocina como sistema a travs de sus relaciones funcionales y segn la importancia de sus elementos, hace posible una estructuracin racional en el ordenamiento de los elementos del sistema. Para ello se tomaron y analizaron en este ejercicio slo algunos ejemplos de las diferentes fases y funciones que se efectan en la cocina. El enfoque de los sistemas al formular un problema de diseo significa; reducir la normal complejidad que representa iniciar un problema, creando un sistema a travs de un principio de diferenciacin. En este escrito se ha intentado representar esquemticamente

con un ejemplo, la importancia de las relaciones funcionales recprocas de los elementos del sistema cocina; entre ms estrecha sea la relacin entre dos o ms elementos, se debe tener en cuenta esta asociacin y estar juntos o directamente vinculados en el momento de la configuracin del orden de los elementos. En la cocina, existe entre la entrega de comidas y el comedor una relacin importante a travs del intercambio de materia como comida preparada, vajilla y comensales, por esto deben encontrarse estrechamente agrupadas, permitiendo igualmente descubrir soluciones funcionales equivalentes (figura20).

FIGURA 20 Soluciones funcionales equivalentes al problema cocina como sistema

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El ejercicio de analizar y clasificar la cocina como sistema a travs de sus relaciones funcionales y segn la importancia de sus elementos, hace posible una estructuracin racional en el ordenamiento de los elementos del sistema. Para ello se tomaron y analizaron en este ejercicio slo algunos ejemplos de las diferentes fases y funciones que se efectan en la cocina.

La clasificacin de las relaciones funcionales segn su importancia representa una valoracin global del buen o mal funcionamiento de los procesos generales de trabajo en el sistema de cocina y, as, observar como un sistema a travs de una diferenciacin interna puede operar exitosamente; en el caso de la cocina fueron considerados, para ilustrar la aplicacin de la teora de sistemas, nicamente factores tcnicos y de organizacin. Una valoracin de un caso concreto o de un problema especfico de diseo depende lgicamente de las otras variables involucradas, es decir, del objeto y sujeto concreto a valorar. Al formular una situacin concreta se debe considerar: qu debe ser valorado, quin valora y para quin debe ser valorado y analizado. El

ejercicio representado a lo largo de este escrito son propuestas generales que sirven como herramienta para una aplicacin concreta del punto de vista de la teora de sistemas. Una estimacin de las posibles soluciones solo puede ser indagada en asocio con condiciones especficas, particularmente con los requerimientos del usuario, del constructor del medio ambiente, etc. Dado que la teora de sistemas se ocupa con problemas generales o con tipos de problemas, se puede sostener que las soluciones generales desarrolladas desde un punto de vista terico-sistmico, sirven en casos concretos como herramienta y mtodo, as como de marco terico. Por este motivo es el objetivo aqu propuesto, indicarle al diseador de productos una forma de proceder sistemticamente en el momento de encontrarse frente a problemas complejos de diseo. De all, un anlisis sistemtico y funcional debe ser interpretado como un instrumento para el diseo industrial; en donde los problemas planteados en el desarrollo de productos puedan ser formulados de manera diferenciada; permitiendo por una parte, llevar a cabo una valoracin explcita de las variables comprometidas y, por otra, poder desarrollar alternativas y soluciones funcionales equivalentes.

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Anlisis y desarrollo funcional de un producto(Ejemplo tomado de Begena

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