The Phenomenon of Science 0231039832

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THE PHENOMENON OF SCIENCE

.......

THE PHENOMENON OF SCIENCE V. F. TUR CHIN Translated by Brand Frentz

New York

■ Columbia University Press ■ 1977

Library of Congress Cataloging in Publication Data Turchin, Valentin Fedorovich. The phenomenon of science. Includes bibliographical references and index. I. Science-Philosophy. 2. Evolution. 3. Cosmol­ ogy. 4. Cybernetics. I. Title. 77-4330 501 Ql75.T7913 ISBN 0-231-03983-2

7f--D3 New York ■ Columbia University Press ■ Guildford, Surrey Copyright © 1977 by Columbia University Press All Rights Reserved Printed in the United States of America

Contents

Foreword� by Loren Graham \'ii ' Preface X\ One The Initial Stages of Evolution / Two Hierarchical Structures 22 56 Three On the Path toward the Human Being Four The Human Being 74 101 Five From Step to Step Six Logical Analysis of Language / 14 Seven Language and Thinking 140 Eight Primitive Thinking / 73 / 90 Nine Mathematics before the Greeks Ten From Thales to Euclid 212 Eleven From Euclid to Descartes 239 Twelve From Descartes to Bourbaki 263 Thirteen Science and Metascience 289 Fourteen The Phenomenon of Science 315 345 Index

Foreword VALENTIN TURCHIN presents in The Phenomenon of Science an evolu­ tionary scheme of the universe-one that begins on the level of indi­ vidual atoms and molecules, continues through the origin of life and the development of plants and animals, reaches the level of man and self-consciousness, and develops further in the intellectual creations of man, particularly in scientific knowledge. He does not see this de­ velopment as a purposeful or preordained one, since he accepts en­ tirely the Darwinian law of trial and error. Selection occurs within a set of random variations, and survival of forms is a happenstance of the relationship between particular forms and particular environ­ ments. Thus, there are no goals in evolution. Nonetheless, there are discernible patterns and, indeed, there is a "law of evolution" by which one can explain the emergence of forms capable of activities which are truly novel. This law is one of the formation of higher and higher levels of cybernetic control. The nodal points of evolution for Turchin are the moments when the most recent and highest control­ ling subsystem of a large system is integrated into a metasystem and brought under a yet higher form of control. Examples of such transi­ tions are the origin of life, the emergence of individual self­ consciousness, the appearance of language, and the development of the scientific method. Many authors in the last century have attempted to sketch schemes of cosmic evolution, and Turchin's version will evoke mem­ ories in the minds of his readers. The names of Spencer, Haeckel, Huxley, Engels, Morgan, Bergson, Teilhard de Chardin, Vemadsky,

FOREWORD

Vii

Bogdanov, Oparin, Wiener and many others serve as labels for con­ cepts similar to some of those discussed by Turchin. Furthermore, it is clear th�t Turchin knows many of these authors, borrows from some of them, and cites them for their achievements. It is probably not an accident that the title of Turchin's book, ''The Phenomenon of Science," closely parallels the title of Teilhard's, "The Phenomenon of Man.'' Yet it is equally clear that Turchin does not agree entirely with any of these authors, and his debts to them are fragmentary and selective. Many of them assigned a place either to vitalistic or to theological elements in their evolutionary schemes, both of which Turchin rejects. Others relied heavily on mechanistic, reductionist principles which left no room for the qualitatively new levels of bio­ logical and social orders that are so important to Turchin. And all of · them-with the possible exception of Wiener, who left no compre­ hensive analysis of evolution-wrote at a time when it was impossi­ ble to incorporate information theory into their accounts. The two aspects of Turchin's scheme of cosmic evolution which distinguish it from its well-known predecessors are its heavy reliance on cybernetics and its inclusion of the development of scientific thought in evolutionary development that begins with the inorganic world. The first aspect is one which is intimately tied to Turchin' s own field of specialization, since for many years he was a leader in the theory and design of Soviet computer systems and is the author of a system of computer language. Turchin believes that he gained in­ sights from this experience that lead to a much more rigorous discus­ sion of evolution than those of his predecessors. The second aspect of Turchin's account-the treatment of scientific concepts as "objects" governed by the same evolutionary regularities as chemical and bio­ logical entities-is likely to raise objections among some readers. Al­ though this approach is also not entirely original-one thinks of some of the writings of Stephen Toulmin, for example-I know of no other author who has attempted to integrate science so thoroughly into a scheme of the evolution of physical and biological nature. Taking a thoroughly cybernetic view, Turchin maintains that it is not the "sub­ stance" of the entities being described that matters, but their princi­ ples of organization.

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FOREWORD

For the person seeking to analyze the essential characteristics of Turchin's system of explanation, two of his terms will attract atten­ tion: ''representation" and "metasystem transition." Without a clear understanding of what he means by these terms, one cannot compre­ hend the overall developmental picture he presents. A central issue for critics will be whether a clear understanding of these terms can be gained from the material presented here. One of the most difficult tasks for Mr. Frentz. the translator, was connected with one of these central terms. This problem of finding an English word for the Russian tem1 predstavlenie was eventually re­ solved by using the term '·representation. n In my opinion, the dif­ ficulty for the translator was not simply a linguistic one, but involved a fundamental, unresolved philosophical issue. The term pred­ stavlenie is used by Turchin to mean ·"an image or a representation of a part of reality. ·' It plays a crucial role in describing the situations in which an organism compares a given circumstance with one that is optimal from the standpoint of its survival. Thus, Turchin, after in­ troducing this term, speaks of a hypothetical animal that ''loves a temperature of 16 degrees Centigrade'' and has a representation of this wonderful situation in the form of the frequency of impulses of neurons. The animal, therefore, attempts to bring the given circum­ stances closer and closer into correspondence with its neuronal repre­ sentation by moving about in water of different temperatures. This same term predstavlenie is also used to describe human be­ havior where the term "'mental image" would seem to be a more felicitous translation. If we look in a good Russian-English dic­ tionary, we shall find predstavlenie defined as ''presentation, idea, notion, representation." At first Dr. Turchin, who knows English well and was consulted by the translator, preferred the translation "notion. " Yet it seemed rather odd, even vaguely anthropomorphic, to attribute a '"notion'' to a primitive organism, an amoeba, or even a fish. On the other hand, the term Hrepresentation" seemed too rudi­ mentary for human behavior where "idea" or "mental image" was clearly preferable. This difficulty arose from the effort to carry a constant term through evolutionary stages in which Turchin sees the emergence of FOREWORD

ix

qualitatively new properties. The problem is, therefore, only second­ arily one C?f language. The basic issue is the familiar one of reduc­ tionism and nonreductionism in descriptions of biological and psy­ chological phenomena. Since the Russian language happens to possess a tenn that fits these different stages better than English, we might do better to retain the Russian predstavlenie. In this text for a wide circle of English readers, however, the translator chose the word "representation," probably the best that can be done. The difficulties of understanding the term "metasystem transi­ tion'' arise from its inclusion of a particular interpretation of logical attributes and relations. Turchin believes that it is impossible to de­ scribe the process by which a particular system develops into a metasystem in the terms of classical logic. Classical logic, he says, · describes only attributes, not relations. For an adequate description of relations, one must rely on the Hegelian dialectic, which permits one to see that the whole of a metasystem is greater than the sum of its subsystems. The Hegelian concept of quantitative change leading to qualitative change is thus not only explicitly contained within Tur­ chin's scheme, but plays an essential role in it. The behavior of human society is qualitatively different from the behavior of individ­ ual humans. And social integration, through the "law of branching growth of the penultimate level,'' may lead eventually to a concept of "The Super-Being." These concepts show some affinities to Marxist dialectical mate­ rialism, in which a similar differentiation of qualitatively distinct evolutionary levels has long been a characteristic feature. The British scientist J. D. Bernal once went so far as to claim that this concept of dialectical levels of natural laws was uniquely Marxist, when he wrote about ''the truth of different laws for different levels, an essen­ tially Marxist idea.'' However, many non-Marxists have also ad­ vanced such a view of irreducible levels o f laws; one should there­ fore be careful about terming a system of thought Marxist simply because it possesses this feature. Most Marxists would reject, at a minimum, Turchin's discussion of the concept of the Super-Being (although even in early Soviet Marxism "God-building" had a sub­ rosa tradition). In Turchin's case we are probably justified in linking

X

FOREWORD

the inclusion of Hegelian concepts in his interpretation of nature to the education in philosophy he received in the Soviet Union. Soviet Marxism was probably one of several sources of Turchin' s philo­ sophic views; others are cybernetics and the thought of such earlier writers on cosmic evolution as Chardin and Vemadsky. In view of the links one can see between the ideas of Turchin and Marxism, it is particularly interesting to notice that Turchin is now in political difficulty in the Soviet Union. Before I give some of the details of his political biography, however, I shall note that in this essentially nonpolitical manuscript Turchin gives a few hints of possi­ ble social implications of his interpretation. He remarks that the cy­ bernetic view he is presenting places great emphasis on ''control'' and that it draws an analogy between society and a multicellular orga­ nism. He then observes, ''This point of view conceals in itself a great danger that in vulgarized form it can easily lead to the conception of a fascist-type totalitarian state. '' This possibility of a totalitarian state, of whatever type, is clearly repugnant to Turchin, and his per­ sonal experience is a witness that he is willing to risk his own secu­ rity in order to struggle against such a state. As for his interpretation of social evolution, he contends that ''the possibility that a theory can be vulgarized is in no way an argument against its truth.'' In the last sections of his book he presents suggestions for avoiding such vulgar­ izations while still working for greater social integration. Turchin is wrestling in this last part of his interpretation with a problem that has recently plagued many thinkers in Western Europe and America as well: Can one combine a scientific explanation of man and society with a commitment to individual freedom and social justice? Turchin is convinced that such combination of goals is pos­ sible; indeed, he sees this alliance as imperative, since he believes there is no conceptual alternative to the scientific worldview and no ethical alternative to the maintenance of individual freedom. It is the steadfastness of his support of science that will seem surprising to some of his readers in the West, where science is often seen as only a partial worldview, one to be supplemented with religious or non­ scientific ethical or esthetic principles. Turchin, however, believes that humans can be explained within an entirely naturalistic framework.

FOREWORD

Xi

His belief that ethical and altruistic modes of behavior can emerge from an evolutionary scheme is, therefore, one that brings him in contact with recent writers in the West on sociobiology, physical an­ thropology, and evolutionary behavior. His emphases on information theory, on irreducible levels, and on the dangers of vulgarizations of scientific explanations of human behavior while nonetheless remain­ ing loyal to science may make contributions to these already interest­ ing discussions.

.......

Valentin Fedorovich Turchin, born in 1931, holds a doctor's degree in the physical and mathematical sciences. He worked in the Soviet science center in Obninsk, near Moscow, in the Physics and Energetics Institute and then later became a senior scientific re­ searcher in the Institute of Applied Mathematics of the Academy of Sciences of the USSR. In this institute he specialized in information theory and the computer sciences. While working in these fields he developed a new computer language that was widely applied in the USSR, the "Refal" system. After 1973 he was the director of a labo­ ratory in the Central Scientific-Research Institute for the Design of Automated Construction Systems. During his years of professional employment Dr. Turchin published over 65 works in his field. In sum, in the 1960s and early 1970s, Valentin Turchin was considered one of the leading computer specialists in the Soviet Union. Dr. Turchin's political difficulties began in 1968, when he was one of hundreds of scientists and other liberal intellectuals who signed letters protesting the crackdown on dissidents in the Soviet Union preceding and accompanying the Soviet-led invasion of Czechoslovakia. In the same year he wrote an article entitled ''The Inertia of Fear'' which circulated widely in samizdat, the system of underground transmission of manuscripts in the Soviet Union. Later the same article was expanded into a book-length manuscript in which Dr. Turchin criticized the vestiges of Stalinism in Soviet so­ ciety and called for democratic reform. In September 1973 Dr. Turchin was one of the few people in the Soviet Union who came to the defense of the prominent Soviet physi-

Xii

FOREWORD

cist Andrei D. Sakharov when the dissident scientist was attacked in the Soviet press. As a result of his defense of Sakharov, Turchin was denounced in his institute and demoted from chief of laboratory to se­ nior research associate. The computer scientist continued his defense of human rights, and in July 1974, he was dismissed from the insti­ tute. In the ensuing months Dr. Turchin found that he had been blacklisted at other places of employment. In the last few years Professor Turchin has been chairman of the Moscow chapter of Amnesty International, an organization that has worked for human rights throughout the world. When other Soviet scholars were persecuted, including Andrei Tverdokhlebov and Sergei Kovalev, Dr. Turchin helped publicize their plight. During this period, his wife, a mathematician, has financially supported her husband and their two sons. In 1974 and 1975 Dr. Turchin received invitations to teach at several American universities, but the Soviet government refused to grant him an exit visa. Several writers in the West speculated that he would soon be arrested and tried, but so far he has been able to con­ tinue his activity, working within necessary limits. His apartment has been searched by the police and he has been interrogated. Dr. Turchin wrote The Phenomenon of Science before these per­ sonal difficulties began, and he did not intend it to be a political state­ ment. Indeed, the manuscript was accepted for publication by a lead­ ing Soviet publishing house, and preliminary Soviet reviewers praised its quality. Publication of the book was stopped only after Dr. Turchin was criticized on other grounds. Therefore, that the initial publication of The Phenomenon of Science is outside the Soviet Union, should not be seen as a result of its content, but of the non­ scientific activities of its author after it was written. LOREN

Columbia University June 1977

FOREWORD

Xiii

R.

GRAHAM

Preface scientific knowledge of reality? To answer this question from a scientific point of view means to look at the human race from outside, from outer space so to speak. Then human beings will ap­ pear as certain combinations of matter which perform certain actions, in particular producing some kind of words and writing some kind of symbols. How do these actions arise in the process of life's evolu­ tion? Can their appearance be explained on the basis of some general principles related to the evolutionary process? What is scientific ac­ tivity in light of these general principles? These are the questions we shall attempt to answer in this book. Principles so general that they are applicable both to the evolu­ tion of science and to biological evolution require equally general concepts for their expression. Such concepts are offered by cyber­ netics, the science of relationships, control, and organization in all types of objects. Cybernetic concepts describe physicochemical, bio­ logical, and social phenomena with equal success. It' is in fact the de­ velopment of cybernetics, and particularly its successes in describing and modeling purposeful behavior and in pattern recognition, which has made the writing of this book possible. Therefore it would be more precise to define our subject as the cybernetic approach to science as an object of study. The intellectual pivot of the book is the concept of the metasys­ tem transition-the transition from a cybernetic system to a metasys­ tem, which includes a set of systems of the initial type organized and controlled in a definite manner. I first made this concept the basis of WHAT IS

PREFACE

XV

an analysis of the development of sign systems used by science. Then, however, it turned out that investigating the entire process of life's evolution on earth from this point of view permits the construc­ tion of a coherent picture governed by uniform laws. Actually it would be better to say a moving picture, one which begins with the first living cells and ends with present-day scientific theories and the system of industrial production. This moving picture shows, in par­ ticular, the place of the phenomenon of science among the other phe­ nomena of the world and reveals the significance of science in the overall picture of the evolution of the universe. That is how the plan of this book arose. How convincingly this picture has been drawn I propose to leave to the reader's judgment. In accordance with the plan of the book, many very diverse facts and conceptions are presented. Some of the facts are commonly known; I try to limit my discussion of them, fitting them into the sys­ tem and relating them to my basic idea. Other facts are less well­ known, and in such cases I dwell on them in more detail. The same is true for the conceptions; some are commonly recognized while others are less well known and, possibly, debatable. The varied nature of the material creates a situation where different parts of the book require different efforts from the reader. Some parts are descriptive and easy to read, in other places it is necessary to go deeply into quite specialized matters. Because the book is intended for a broad range of readers and does not assume knowledge beyond the secon­ dary school level, I provide the necessary theoretical information in all such cases. These pages will require a certain effort of the un­ trained reader. The book gives an important place to the problems of the theory of knowledge and logic. They are, of course, treated from a cyber­ netic point of view. Cybernetics is now waging an attack on traditional philosophical epistemology, offering a new natural-science interpreta­ tion of some of its concepts and rejecting others as untenable. Some philosophers oppose the rise of cybernetics and consider it an in­ fringement on their territory. They accuse cyberneticists of making the truth "crude" and "simplifying" it; they claim cyberneticists ig­ nore the "fundamental difference" between different forms of the

xvi

PREFACE

movement of matter (and thi s is despite the thesis of the world ' s u nity ! ). But the phi losopher to whom the possessive attitude toward v arious fields of knowledge is foreign should welcome the attacks of the cybemetic ists. The devel opment of physics and astronomy once destroyed n atural phi losophy , sparing phi l osophers of the need to talk approx i matel y about a subject which scientists could discuss exactly . I t appears that the development of cybernetics will do the same thi ng w i th phil osophical epi stemol ogy or , to be more cautious , with a sig­ n i ficant part of it . This should be nothi ng but gratifying . Philosophers w i l l always have enough concerns of their own ; science rids them of some , but g i ves them others . Bec au se the book is devoted to science i n toto as a definite method of i nteraction between human society and its env ironment , it c ontains practically no discussion of concrete natural-science dis­ c i pl i nes . The presentation remai ns entirely at the level of the concepts of cybernetics , logic , and mathematics , which are equally signi ficant for al l m odern science . The onl y exception is for some notions of modem phys ics which are fundamentally i mportant for the theory of sign systems . A concrete analysis of sc ience' s interaction with pro­ d uction a nd s oc i al l i fe was also outside the scope of the problem. This is a di stinct matter to whi ch a v ast literature has been devoted; i n thi s book I remain a t the level o f general cybernetic concepts . It is d angerous to attempt to combine a large amount of materi al from d ifferent fields of knowledge i nto a si ngle , whole picture ; de­ tai l s m ay become di storted , for a person cannot be a speci alist in ev­ erything . Because this book attempts precisely to create such a pic­ ture , it is very l ikely that speciali sts in the fields of science touched o n here w i l l fi nd omissions and inaccuracies ; such i s the price which must be paid for a wide scope . But such pictures are essential . It only remains for me to hope that this book contains nothing more than errors i n detai l whi ch c an be el i minated withou t detriment to the overall picture . V. F.

PREFACE

xvii

TU RCHIN

THE PHENOMENON OF SCIENCE

CHAPTE R O N E

The I niti a l Sta ges of Evol ution ■

THE BASIC LAW OF EVOLUTION

of the evolut ion of life , as far as we know , the total mass of living matter has always been and is now increasing and growing more complex in its organization . To increase the complex­ i t y of the organi zation of biological forms , nature operates by tri al and error. Exi sting forms are reproduced in many copies , but these are not ident ical to the original . Instead they differ from it by the presence of small random v ari ations . These copies then serve as the materi al for natural selection . They may act as individual living be­ ings , in whi ch case select ion leads to the consolidation of useful vari ­ ations, or element s of more complex forms , in whi ch case selection i s also directed to the structure of the new form (for example , w i th the appearance of multicellular organi sms) . In both cases selection i s the re sult of the struggle for exi stence , in which more vi able forms sup­ plant less v i able one s . Thi s mechani sm o f the development o f life , whi ch was disco v­ ered by Charles Darwin, may be called the basic law of evolution . It is not among our purposes to substanti ate or discuss thi s law from the point of v iew of tho se laws of nature which could be declared more fundamental . We shall take the basic law of evolution as gi ven .

IN THE PROCESS

I N ITIAL STAGES O F EVO LUTION

1

■

THE CHEMICAL ERA

before the appearance of the human being can be broken into two periods , which we shall call the " chemical " era and the "cybernetic " era . The bridge between them is the emergence of animals with distinct nervous systems , including sense organs , nerve fibers for transmitting information , and nerve centers (nodes) for converting this information . Of course , these two terms do not signify that the concepts and methods of cybernetics are inapplicable to life in the ' 'chemical ' ' era; it is simply that the animal of the ' 'cybernetic' ' era is the classical object of cybernetics , the ohe to which its appearance and establishment as a scientific discipline are tied. We shall review the history and logic of evolution in the precy­ bernetic period only in passing, making reference to the viewpoints of present-day biologists . 1 Three stages can be identified in this period . In the first stage the chemical foundations of life are laid. Mac­ romolecules of nucleic acids and proteins form with the property of replication, making copies or " prints " where one macromolecule serves as a matrix for synethesizing a similar macromolecule from el­ ementary radicals . The basic law of evolution , which comes into play at this stage , causes matrices which have greater reproductive inten­ sity to gain an advantage over matrice s with lesser reproductive inten­ sity , and as a result more complex and active macromolecules and systems of macromolecules form . Biosynthe sis demands free energy . Its primary source is solar radiation . The products of the partial decay of life form s that make direct use of solar energy (photo synthesis) also contain a certain reserve of free energy which may be used by the already available chemistry of the macromolecule . Therefore . this reserve is used by special forms for which the products of decay serve as a secondary source of free energy. Thus the division of life into the plant and animal worlds arise s . THE HISTO RY OF L I FE

1 I a m generally fol lowin g the re pon b y S . E . Schnoll entitled " The Essence o f Life . In­ vari ance i n the General Direction of B iolog ical Evolution , ' · in Materialy seminara · 'Dialektika i so vremennoe estesti·o:_nanie' ' ( M aterials of the · · Dialectics and Modern Natura l Science · ' Semi nar) . Dubna, 1 967 .

2

I N ITIAL STAGES OF EVO L UTION

The second stage of evolution is the appearance and develop­ ment of the motor apparatus in animals . Pl ants and animals differ fundamentally in the way they obtain energy . With a given level of illumination the intensity of absorption of solar energy depends entirely on the amount of plant surface , not on whether it move s or remains stationary . Plants were refi ned by the creation of outlying light catchers-green leave s secured to a system of suppo rts and couplings (stems , branches , and the like) . This de­ sign works very well, ensuring a slow shift in the gree n surfaces toward the light which matches the slow change in illumination . The situation is entirely different with animals , in particular with the most primitive types such as the amoeba . The source of energy­ food-fills the environment around it. The intake of energy is deter­ mined by the speed at which food molecule s are diffused through the shell that separates the digestive apparatus from the external environ­ ment . The speed of diffusion depends less on the size of the surface of the dige stive apparatu s than on the movement of this surface rela­ tive to the environment; therefore it is possible for the animal to take in food from different sectors of the environment. Consequently , even simple , chaotic movement in the environment or, on the other hand , movement of the environment relative to the organism (as is done , for example , by sponges which force water through themselves by means of their cilia) is very important for the primitive animal and , consequently , appears in the process of evolution . Special forms emerge ( intracellular formations in one -celled organisms and ones containing groups of cells in multicellular organisms) whose basic functio n is to produce movement . In the third stage of evolution the movements of animals become directed and the incipient forms of sense organs and nervous systems appear in them . This is also a natural consequence of the basic law . It is more advantageous for the animal to mo ve in the direction where more food is concentrated , and in order for it to do so it must have sensors that describe the state of the external environment in all direc­ tions (sense organs) and information channels for communication be­ tween these sen sors and the motor apparatus (nervous system) . At fi rst the nervous system is extremely primitive . Sense organs merely

I N ITIAL STAGES OF EVO LUTION

3

disti nguish a few situations to which the an imal must respond dif­ ferently . The volume of information transmitted by the nervous sys­ tem is sl ight and there is no special apparatus for processing the i n­ formation . Duri ng the process of evol ut ion the sense org ans become more complex and de liver an increasing amount of i nform ation about the external environment . At the same time the motor apparatus i s refined , whi ch makes ever-increasing demands o n the c arry i ng c apac­ ity of the nervous system . Special form ations appear-nerve centers which convert information rece ived from the sense organs i nto i nfor­ mation controlling the organs of mo vement . A new era begins : the "cybernetic " era.

■

CYBERNETICS

evolution in the cybernetic period and to di scover the laws govern i ng the organization of l i ving bei ngs in thi s period (for brevity we will call them " cybernetic animal s " ) we must i ntroduce certain fundamental concepts and laws from cybernetics . The term " cybernetics" i tself w as , of course , i ntroduced by Norbert Wiener, who defi ned it descri pti vely as the theory of rel a­ tionshi ps and control in the l i vi ng organi sm and the machi ne . As i s true i n any scientific discipl i ne , a more preci se definition o f cybernet­ ics requ ires the i ntroduction of i ts basic concepts . Properly speaki ng , to i ntroduce the basic concepts i s the same as defi ni ng a particular science , for all that remains to be added i s that a descri ption of the world by means of thi s system of concepts i s , i n fact , the particular, concrete science . Cybernetics i s based above all on the concept of the system , a certain materi al object which consi sts of other obj ects which are called subsystems of the given system . The subsystem of a certain system may , in its turn , be viewed as a system consisting of other subsystems . To be preci se , therefore , the meaning of the concept we have i ntroduced does not l ie i n the term " system " by itse lf, that i s , not i n ascri bing the property o f " being a system " t o a certain object (thi s i s quite meani ngless , for any obj ect may be considered a sys-

TO A N A L Y Z E

4

I N ITIAL STAGES OF EVO L UTION

tern) , but rather in the connection between the terms H system" and " subsystem , " which reflects a definite relationship among objects . The second crucial concept of cybernetics is the concept of the state of a system (or subsystem) . Just as the concept of the system relie s directly on our spatial intuition, the concept of state relies di­ rectly on our intuition of time and it canot be defined except by refer­ ring to experience . When we say that an object has changed in some re spect we are saying that it has passed into a different state . Like the conce pt of system, the concept of state is a concealed relationship: the relationship between two moment s in time . If the world were im­ mobile the concept of state would not occur, and in those discipline s where the world is viewed statically , for example in geometry , there is no concept of state . Cybernetics studies the organization of systems in space and time , that is , it studie s how subsystems are connected into a system and how change in the state of some subsystems influence s the state of other subsystems . The primary emphasis , of course , is on organi­ zation in time which , when it is purposeful, is called control . Causal relations between states of a system and the characteristics of its be­ havior in time which follow from this are often called the dynamics of the system, borrowing a term from physics . This term is not applicable to cybernetics , because when we speak of the dynamics of a system we are inclined to view it as something whole , whereas cybernetics is concerned mainly with investigating the mutual infl u­ ence s of subsystems making up the particular system . Therefore , we prefer to speak of organization in time , using the term dynamic de­ scription only when it must be j uxtapo sed to the static de scription which considers nothing but spatial relationships among subsystems . A cyberne tic de scription may have different levels of detail. The same system may be de scribed in general outline , in which it is broken down into a few large subsystems or ' ' block s , ' ' or in greater detail, in which the structure and internal connections of each block are described . But there is always some final level beyond which the cybernetic description does not apply . The subsystems of this level are viewed as elementary and incapable of being bro ken down into

I N ITIAL STAGES OF EVOLUTION

5

are all the parts of the bicycle and human body which are moving rel­ ative to one another : the wheels , pedals, handlebar, legs, arms, and so on. Their states are their positions in space . These positions are described by coordinates (numbers) which can assume continuous sets of values. If a system consists exclusively of subsystems with discrete states then the system as a whole must be a system with discrete states. We shall simply call such systems ' " discrete systems," and we shall call systems with continuous sets of states ' ' continuous sys­ tems . " In many respects discrete systems are simpler to analyze than continuous ones. Counting the number of possible states of a system, which plays an important part in cybernetics , requires only a knowl­ edge of elementary arithmetic in the case of discrete systems. Sup­ pose discrete system A consists of two subsystems a 1 and a 2 ; subsys­ tem a 1 may have n 1 possible states , while subsystem a 2 may have n 2 . Assuming that each state of system a1 can combine with each state of system a 2 , we find that N. the number of possible states of system A , is n 1 n 2 . If system A consists of m subsystems a i where i = 1 , 2, . . . , m, then From this point on we shall consider only discrete systems. In addition to the pragmatic consideration that they are simpler in princi­ ple than continuous systems , there are two other arguments for such a restriction. First , all continuous systems can in principle be viewed as dis­ crete systems with an extremely large number of states. In light of the knowledge quantum physics has given us, this approach can even be considered theoretically more correct . The reason why continuous systems do not simply disappear from cybernetics is the existence of a very highly refined apparatus for consideration of such systel]ls : mathematical analysis, above all, differential equations. Second, the most complex cybernetic systems, both those which have arisen naturally and those created by human hands, have in­ variably proved to be discrete. This is seen especially clearly in the­ example of animals. The relatively simple biochemical mechanisms

I N ITIAL STAGES O F EVOLUTION

7

that regu l ate body temperature , the content of various substances in the blood, and simil ar characteristics are continuous, but the nervous system is constructed according to the discrete pri nci pie . .

■

THE RELIABILITY OF DISCRETE SYSTEMS

W H Y DO DISCRETE SYST E M S prove to be preferable to continuous ones whe n it is necessary to perform complex functions? Because they have a much higher reliability . In a cybernetic device based on the principle of discrete states each elementary subsystem may be in onl y a small number of possible states, and therefore the system ordi­ narily ignores small deviations from the norm of various physical parameters of the system, reestablishing one of its permissibl e states in its " primeval purity � " In a continuous system , however, small dis­ turbances continuousl y accumulate and if the system . is too complex it ceases functioning correctly . Of course, in the discrete system too there is always the ·possibility of a breakdown, because small changes in physical parameters do lead to a finite probability that the system will switch to an ' ' incorrect' ' state. Nonetheless, discrete systems definitely have the advantage. Let us demonstrate this with the fol-

o�--------► Oi---------..Qt---------..o------+o T ra n s m i t t e r

R ec eiver

I n te rm e d i ate s t a t i o n s

F i g u re 1 . 1 . T ra n s m i ss i o n o f a s i g n a l in c o n t i n u o u s a n d d i s c rete syste m s . ( T h e s h ad e d p a rt s h ows the a rea of s i g n a l a m b i g u i ty . )

passes the I 00 stations the root -mean square magnitude of noise will be one volt (the mean squares of noise are summed) . Thus, average noise is equal to the maximum signal, and it is therefore plain that we shall not in fact receive any useful information . Only by accident can the value of the voltage received coincide with the value of the volt­ age transmitted . For example , if a precision of 0. I volt satisfies us the probability of such a coincidence is approximately l / l 0. Now let us look at the discrete variant . We shall define two " meaningful " states of the initial segment of the wire : when the volt­ age applied is equal to zero and when it is maximal (one volt) . At the intermediate stations we install automatic devices which transmit zero voltage on if the voltage received is less than 0 . 5 volt and transmit a normal one-volt signal if the voltage received is more than 0 . 5 volt. In this case , therefore , for one occasion ( one signal) information of , the " yes/no . type is transmitted (in cybernetics this volume of infor­ mation is called one ' " bit " ) . The pro bability of receiving incorrect in­ formation depends strongly on the law of probability distribution for the magnitude of noise . Noise ordinarily follo ws the so-called normal law . Assuming this law we can find that the probability of error in transmission from one station to the next ( which is equal to the prob­ ability that noise will exceed 0 . 5 volt) is 0. 25 · 1 0- 6 . Thus the proba­ bility of an error in transmis sion over the full length of the line is O . 25 · I 0- 4 • To transmit the same message as was transmitted in the

I N ITIAL STAGES OF EVOL UTION

9

previous case-that is, a value between 0 and I with a precision of 0. I of a certai n quantity lying between 0 and I -all we h ave to do is send four " yes/no " type signals. The probabi l ity that there w i l l be error in at least one of the signals is I 0- 4 . Thus, with the discrete method the total probabi l ity of error is 0. 0 I percent, as agai nst 90 percent for the conti nuous method .

■

INFORMATION

describing a concrete cybernetic system it was i m­ possi ble not to use the term information-a word fam i l i ar and under­ standable in its i nformal, conversational meani ng. The cybernetic concept of i nformation, however , has an exact quanti tati ve meaning . Let us imagine two subsystems A and B (see figure I . 2) , which are interconnected in such a way that a change in the state of A le ads to a change i n the state of B . Th is can also be expressed as fol lows: A i nfluences B . Let us consider the state of B at a certain moment i n time t 1 and at a later moment t2 • We shall sign ify the first state as S 1 and the second as S2 . State S2 depends on state S 1 . The rel ationshi p of S2 to S 1 is probabi l istic , however , not unique . This is because we are not consideri ng an ideal ized theoretical system governed by a deter­ mi nistic l aw of movement but rather a real system whose states S i are the resu lts of experi mental data . With such an approach we m ay also speak of the l aw of move ment , understandi ng it i n the probabi l istic sense-that is, as the conditional probabil ity of state S 2 at moment t2 on the condition that the system was i n state S 1 at moment t 1 • Now let us momentari l y ignore the l aw of movement . We sha l l use N to des­ ignate the total number of possibl e states of subsystem B and i magine that conditions are such that at any moment i n time system B can asWHEN WE BEGAN

B

A F i g u re 1 .2 .

10

I N ITIAL STAGES OF EVOLUTION

sume any of N states with equal probability , regardless of its state at the preceding moment . Let us attempt to give a quantitative expres­ sion to the degree ( or strength ) of the cause-effect influence of sys­ tem A on such an inertiale ss and " lawle ss" subsystem B . Suppose B acted upon by A switches to a certain completely determinate state . It is clear that the " strength of infl uence " which is required from A for this depend s on N, and will be larger as N is larger . For example , if N = 2 then B , even if it is completely unrelated to A , when acted upon by random factors can switch with a probability of . 5 to the very state A "recommends . " But if N = 1 0 9 , when we h ave noticed such a coincidence we shall hardly doubt the influence of A on B . Therefore , some monotonic increasing function of N should serve as the measure of the ' " strength of the influence " of A on B . What this essentially means is that it serves as a measure of the intensity of the cause-effect relation.s hip between two events , the state of A in the time interval fro m t 1 to t2 and the state of B at t2 • In cybernetics this measure is called the quantity of information transmitted from A to B between moments in time t 1 and t2 , and a logarithm serves as the monotonic increasing function. So, in our example , the quantity of information / passed from A to B is equal to log N . Selection o f the logari thmic function is determined by its prop­ erty according to which log N I N2

= log N I + log N2

S uppo se system A influences system B, which consists of two in­ dependent subsystems B 1 and B 2 with number of pos sible states N 1 and N2 respectively (see figure I . 3 ) . Then the number of states of system B is N 1 N2 and the quantity of information / that must be trans­ mitted to system B in order for it to assume one definite state is , owing to the abo ve-indicated property of the logarithm , the sum I = log N I N2

= log NI + log N2 = I I + /2

where / 1 and /2 are the quantities of information required by subsys­ tems B 1 and B 2 • Th ank s to t his property the information assumes defi­ nite characteristics of a substance ; it spreads over the independent subsystems like a fluid filling a number of vessels . We are speaking

I N IT I A L STAGES OF EVOLUTION

11

A

Figure 1 . 3 .

of the joining and separation of information flows , informat i on capac­ ity , and information processing and storage . The question of. information storage is related to the question of the law of movement . Above we mentally set aside the law of mo ve­ ment in order to define the concept of information transmission . If we now consider the law of movement from this new point of view , it can be reduced to the transmission of information from system B at moment t 1 to the same system B at moment t2 . If the state of the sys­ tem does not change with the passage of time , this is information storage . If state S 2 is uniquely determined by S 1 at a preceding mo ­ ment in time the system is called fully deterministic. I f S 1 is uniquely determined by S 2 the system is called reversible ; for a reversible sys­ tem it is possible in principle to compute all preceding states on the basis of a given state because information loss does not occur . I f the system is not reversible information is lost . The law of movement is essentially something which regulates the flow of information in time from the system and back to itself. Figure 1 . 4 shows the chart of information transmission from sys­ tem A to system C through system B . B is called the communication channel. The state of B can be influenced not only by the state of sys­ tem A , but also by a certain uncontrolled factor X, which is called noise . The final state of system C in this case depends not only on the state of A , but also on factor X (information distortion) . One more important diagram of information exchange is shown in figure 1 . 5 . This is the so-called feedback diagram. The state of system A at t 1

12

I N ITIAL STAGES O F EVOLUT ION

X A

B

C

Figure 1 . 4 .

Fig ure 1 .5 .

influences the state of B at 12 , then the latter influences the state of A at 13 . The circle of information movement is completed . With this we conclude for now our familiarizati on with the gen­ eral concepts of cybernetics and tum to the evolution of life on earth.

■

THE NEURON

of a nerve cell (neuron) is shown sche­ matically in figure 1 . 6 . A neuron consi sts of a fairly large (up to 0 . I mm) cell body from which several processes called dendrites spread , giving rise to finer and finer processes like the branching of a tree . In addition to the dendrites one other process branches out from the body of the nerve cell . Thi s is the axon, which re semble s a long , thin wire . Ax ons can be very long, up to a meter , and they end in tree­ like branching systems as do the dendrites . At the ends of the branches coming from the ax on one can see small plates or bulbl ets . The bulblets of one neuron approach clo se to different segments of the body or dendrites of another neuron , almost touchin g them . These THE E X TE R N A L A PPE A RA NCE

I N ITIAL STAGES OF EVO LUTION

13

C e l l body

Dendrites

:1 �

Dendrites

! Axon ·,

/ Axon � Sy n a p s e s

�J

- -�

'r--, //'._ _,_

·

-

F i g u re 1 . 6. D i a g ra m of t h e s t r u c t u re of a n e u ro n .

contacts are called synapses, and it is through them that neurons in­ teract with one another. The number of bulblets appro aching the dendrites of the single neuron may run into the dozens and e ven hundreds . In this way the neurons are closely interconnected and form a nerve net . Wh en one considers certain physicochemical properties (above all the propagation of electrical potential o ver the surface of the cell) one discovers the neurons can be in one of two state s-the state of dormancy or the state of stimulation . From time to time , influenced by other neurons or out side factors , the neuron switches from one state to the other . This process takes a certain time , of course , so that an investigator who is studying the dynamics of the electrical state of a neuron , for example , considers it a system with continuous states . But the information we now have indicates that what is es sential for the functioning of the nervous system as a whole is not the nature of switching processes but the very fact that the particular neurons are in one of the se two state s . Therefore , we may consider that the nerve

14

I N ITIAL STAGES OF EVO L UTION

net is a discrete system which consists of elementary subsystems (the neurons) with two states. When the neuron is stimulated , a wave of electrical potential runs alon g the axon and reaches the bulblets in its branched tips. From the bul blets the stimulation is passed acro ss the synapses to the corresponding sectors of the cell surface of other neurons. The behav­ ior of a neuron depends on the state of it s synapses. The simplest model of the functioning of the nerve net be gins with the assumption that the state of the neuron at each moment in time is a single-valued function of the state of its syna pses. It has been established experi­ mentally that the stimulation of some synapses promotes stimulation of the cell, whereas the stimulation of other synapses prevents stimu­ lation of the cell. Finally, certain syna pses are completely unable to conduct stimulation from the b ulblets and therefore do not influence the state of the neuron. It has also been established that the conduc­ tivity of a synapse increases after the first passage of a stimulus through it. Essentially a closing of the contact occurs. This explains how the system of communication among neurons , and consequently the nature of the nerve net 's functioning , can change without a change in the relative positions of the neurons. The idea of the neuron as an instantaneous processor of informa­ tion received from the synapses is , of course , very simplified. Like any cell the neuron is a complex machine whose functioning has not yet been well understood. This machine has a large internal memory, and therefore its reactions to external stimuli may show great variety. To understand the general rules of the working of the nervous sys­ tem, however . we can abstract from these complexities (and really, we ha ve no other way to go ! ) and be gin with the simple model outlined abo ve.

■

THE NERVE NET

of the nerve system of the Hcybemetic animal '' in its interaction with the environment is shown in figure I . 7. Those sensory nerve cells which are stimulated by the action of outside factors are called receptors (that is, receivers) because they A GENERALI ZED DIAGRAM

I N ITIAL STAGE S O F EVOL UT ION

15

E nvi ro n m e n t

Figure 1 . 7. N e rvo u s syste m of t h e " cybe rnet i c a n i m a l . "

are the first to receive inform ation about the state o f the environment . This information enters the nerve net and i s converted by it . As a result certain nerve cells called effectors are sti mulated . Branches o f the effector cells penetrate tho se ti ssues of the organi sm wh ich the nervous system affects directly . Sti mulation of the effector causes a contraction of the correspond ing muscle or the stimulation of the ac­ tivity of the appropriate gland . We shall call the state of all receptors at a certain moment in time the situation at that moment. (It would be more preci se-i f more cumbersome-to say the ' 'result of the effect of the situation on the sense organs . " ) We will call the state of all the effectors the "action . " Therefore , the role of the nerve net i s to con­ vert a situation into an action . It is convenient to take the term " environment" fro m figure I . 7 to mean not just the objects wh ich surround the animal , but also its bone and muscle system and generally everything that i s not part o f the nervous system. Thi s makes it unnecessary to give separate repre­ sentations in the di agram to the animal body and what is not the body , especially because thi s di stinction is not important in princi ple for the activity of the nervous system . The only th i ng that is impor­ tant i s that sti mulation of the effectors lead s to certain changes in the " environment . " With th i s general approach to the problem as the basi s of our consideration , we need only classify the se changes as "useful " or " harmful" for the animal without going into further de­ tail. The objective of the nervous system i s to promote the survival and reproduction of the ani mal . The nervous system work s well when sti mulation of the effectors leads to changes in the state of the envi­ ronment that help the animal survi ve or reproduce , and it work s badly when it lead s to the reverse . W i th its increasing refinement in the pro-

16

I N ITIAL STAGES OF EVOLUT ION

cess of evolution , the nervous system has perfom1ed this task increas­ ingly well . How does it succeed in this? What laws does this process of refinement follow? We will try to answer these questions by identifying in the evolution of the animal nervous system several stages that are clearly distinct from a cybernetic point of view and by showing that the tran­ sition from each preceding stage to each subsequent stage follows inevitably from the basic law of evolution . Because the evolution of living beings in the cybernetic era primarily concerns the evolution of their nervous systems , a periodiz ation of the development o f the ner­ vous system yields a periodization of the development of life as a whole .

■

THE SIMPLE REFLEX (IRRITABI LITY)

of the nerve net is when there is no net at all . In this case the receptors are directly connected to the effectors and stimulation from one or several receptors is transmitted to one or sev­ eral effectors . We shall call such a direct connection between stimu­ lation of a receptor and an effector the simple reflex. This stage , the third in our all-inclusive enumeration of the stages of evolution , is the bridge between the chemical and cyber­ netic eras . The Coelenterata are animals fixed at the level of the simple re flex . As an ex ample let us take the hydra , which is studied in schoo l as a typical representative of the Coelenterata . The body of a hydra has the shape of an elongated sac . Its interior, the coelen­ teron, is connected to the environment through a mouth, which is sur­ rounded by several tentacles . The walls of the sac consist of two layers of cells : the inner layer ( entoderm) and the outer layer ( ec­ toderm) . Both the ectoderm and the entoderm have many muscle cells which contain small fi bers that are able to contract, thus setting the body of the hydra in motion . In addition , there are nerve cells in the ectoderm; the cells located closest to the surf ace are receptors and the cells which are set deeper, among the muscles , are effectors . If a hydra is pricked with a needle it squeezes itself into a tiny ball . This is a simple reflex caused by transmission of the stimulatio n from the receptors to the effectors . THE SI M P L EST VA RI A N T

I N IT I A L STAGES OF EVO L UT IO N

17

Yo u n g , d eve l o p i n g hyd ra-----..,-Y ---- E ct o d e r m c e l l s

Figure 1 . 8. The s t r u ctu re o f t h e hyd ra .

But the hydra is also capable of much more comple x behavior. After it has captured prey , the hydra uses its tentacles to draw the prey to its mouth and then swallows the prey . This behavior can also be explained by the aggregate action of simple re flexes connecting ef­ fectors and receptors locally , within small segments of the body . For example , the model of a tentacle ( depicted in figure I . 9) explains its

Figure 1 . 9 . M od e l of a t e n tac l e .

18

I N ITIAL STAGES O F EVO LUTION

ability to wrap itself around captured object s . Let us picture a certain number of links connected by hinges ( for simplicity we shall consider a two-dimensional picture) . Point s A and B . A ' and B ' . B and C, and B ' and C ' , etc . are interconnected by strands which can contract (muscle s) . All these points are sensitive and become stimulated when they touch an object (receptors) . The stimulation of each point causes a contraction of the two strands connected to it (reflex ) .

■

THE COMPLEX REFLEX

relationship between the stimulated cell and the muscle cell arises naturally , by the trial and error method , in the pro ­ ce ss of evolution. If the correlation between stimulation of one cell and contraction of another proves useful for the animal , then this cor­ relation become s established and reinforced. Where interconnected cells are mech anically copied in the proce ss of growt h and reproduc­ tion , nature receive s a system of parallel-acting simple reflexes re­ sembling the tentacle of the hydra . But when nature has available a large number of receptors and effectors which are interconnected by pairs or locally , there i s a · " temptation' ' to make the system of con­ nections more complex by introducing intermediate neurons . This is advantageous because where there is a system of connections among all neurons , forms of behavior that are not po ssible where all connec­ tions are limited to pairs or localities now become so . This point can be demonstrated by a simple calculation of all the po ssible methods of converting a s ituation into an action with each method of intercon­ nection . For e xample , assume that we have n receptors and effectors connected by pairs . In each pair the connection may be po sitive ( stimulation causes stimulation and dormancy evokes dormancy) or negative (s timulation evoke s dormancy and dorm ancy causes stimula­ tion) . In alL therefore , 2 n variants are possible , which means 2n variants o f behavior . But if we assume that the system of connections can be of any kind, which is to say that the state of each effector (stimulation or dormancy) can depend in any fashion on the state of all the receptors , then a calculation of all possible variants of behav­ ior yields the number 2 2 " >n , which is immeasurably larger than 2n . THE SIM PLE RE FLEX

I N ITIAL STAGES OF EVO LUTION

19

Exactly the sam e calculation leads to the conclusion that joining any subsystems which join independent groups of receptors and effectors into a single system always leads to an enormous increase in the number of possibl e variants of behavior. Throughout the entire course of the history of life, therefore, the evolution of the nervous system has progressed under the banner of increasing centralization. But "centralization " can mean different things. If all neurons are connected in one sense lessly confused clump , then the system­ despite its extremely "centralized n nature-wi ll hardly have a chance to survive in the struggle for existence. Centralization poses the fol lowing problem : how to select from al l the conceivable ways of joining many receptors with many effectors (by means of interme­ diate neurons if necessary) that way which will correlate a correct ac­ tion (that is, one useful for survival and reproduction ) to each situa­ tion? After all, a large majority of the ways of interconnection do not have this characteristic . We know that nature takes every new step toward greater com­ plexity in living structures by the trial and error method. Let us see what direct app lication of the trial and error m ethod to our problem yields. As an example we shall consider a small system consisting of I 00 receptors and I 00 effectors. We shal l assume that we have avail ­ able as many neurons as needed to create an intermediate nerve net and that we are able to determine easily whether the particular method of connecting neurons produces a correct reaction to each sit­ uation. We shall go through al l conceivable ways of connection until we find the one we need. Where n = l 00 the number of functionally different nerve nets among n receptors and n effectors is This is an inconceivably large number . We cannot sort through such a number of variants and neither can Mother Nature. If every atom in the entire visible universe were engaged in examining variants and sorting them at a speed of I billion items a second , even after billions of billions of years (and our earth has not existed for more than I O billion years) not even one billionth of the total number of variants would have been examined .

20

I N ITIAL STAGES OF EVO LUTIO N

But somehow an effectively funct ioning nerve net doe s form ! And hi gher animals have not hundreds or thousands but mil lions of receptors and effectors . The answer to the riddle is concealed in the hierarchical structure of the nervous syste m . Here again we must make an excursion into the area of general cybernetic concepts . We shall call the fourth stage of evolution the stage of the complex reflex . but we shall not be able to define t h i s concept unt il we ha ve fami liar­ ized ourselves with cert ain fact s about hierarchically organi zed nerve nets .

I N IT I A L STAGES O F EVO LUTION

21

CH APTE R TWO

H ierarch i ca l Stru ctures ■

THE CONCEPT OF THE CONCEPT

L ET us LOO K at a nerve net which has many receptors at the i nput but j u st one effector at the output . Thu s , the nerve net di vides the set of all situations i nto two subsets: situ at ions that cause sti mulation of the effector and situations that leave i t dormant . The task bei ng per­ formed by the nerve net in thi s c ase is cal led recognition (discri mi­ nation) , recognizing that the situation belongs to one of the two set s . In the struggle for exi stence the an i mal is constantly sol v i ng recogn i ­ tion probl ems, for example , distingu i shi ng a dangerous situ ation from one that i s not, or disti ngui shing edibl e objects from i ned ible one s . These are only the clearest example s . A detai led analysi s o f ani mal behavi or l eads to the conclusion that the perform ance of any comple x action requires that the animal resolve a l arge n umber of · Small ' ' recogni tion problems conti nuou sly . In cybernetics a set of situations i s called a concept. 1 To make c lear how the cybernetic u nderstanding of the word H concept " i s related t o i t s ord inary meani ng l e t us assume that the receptors o f the nerve net under consideration are the l ight-sensitive nerve endings of the retina of the eye or, speaki ng i n general , some l i ght-sensitive poi nts on a screen which feed i nformation to the nerve net . The 4

Late r we shall gi ve a somewhat more ge neral defin ition of the concept and a set of situa­ tions shall be called an Ari stotel ian conce pt . At present we shall drop the adjective · · Aris­ totelian " for brevity . 1

22

HIERARC H I CA L STRUCTURES

receptor is stimulated when the corresponding sector of the screen is illuminated (more precise ly, when its il lumination is greater than a certain thresho ld magnitude) and remains dom1ant if the sector is not i lluminated. If we imagine a light spot in place of each stimulated receptor and a dark spot in place o f each unstimulated one, we shal l obtain a picture that differs from the image striking the screen only by its discrete nature (the fact that it is broken into separate points) and by the absence of semitones . We sha ll consider that there are a large number o f points (receptors) on the screen and that the images which can appear on the screen ( '" pictures" ) have maximum con­ tr asts-that is , they consist entirely of black and white . Then each situation corresponds to a definite picture. According to traditional Aristotelian logic, when we think or talk about a definite picture ( for example the one in the upper left comer of figure 2. 1 ) we are dealing with a particular concept. In addi­ tion to particular concepts there are general or abstr act concepts. For example , we can think about the spot in general-not as a particular, concrete spot (for example , one of those represented in the top row in figure 2. I ) but about the spot as such . In the same way we can have an abstract concept of a straight line, a contour, a rectangle, a square, and so on. 2 2 According to the term i nology accepted by many logicians, juxtaposing abstract concepts to concrete concepts i s not at all the same as juxtaposing general concepts to particular ones . In a logic textbook (Logika [ Logic] . State Publi shing House of Political Literature , Moscow, 1 956) we read the fo llowing: · · A concept by w hose properties an object is conce ived as such and as a given object is called concrete. A concept by whose properties what is conce ived is not the given object as such but a certain property of the object or relationship among objects is called abstract. " Thi s definition can hardly q uali fy as a masterpiece of clear thinking. Still, we may con­ clude from it that general concepts can also be considered abstract if they are formed not by list­ i ng particular objects included in them but rather by declaring a number of properties to be sig­ ni ficant and abstracting from the other. insigni ficant properties. This is the only kind of general concepts we are go ing to consider and so we shall call them abstract concepts also . For ex­ ample , an abstract triangle i s any triangle regardless of its size, sides, or angles , or its pos ition on the screening surface ; therefore thi s is an abstract concept. The term " abstract " is used th is way both in everyday life and in mathematics. At the same time , according to the logic text­ book , " triangle , " " sq uare , " and the like are concrete general concepts , but "triangularity " and " squareness , " which are inherent in them , are abstract concepts. W hat is actually happen­ ing here is that a purely grammatical difference is being elevated to the rank of a logical dif­ ference , for, e ven from the point of view of an advocate of the latter variant of termino logy , the possession of an abstract concept is eq uivalent to the possession of the corresponding general concept.

H IERARCH ICAL STRUCTURES

23

- -"' \

....-------- --- - -....---- - -

•

�

I

D

/

� _,...,,,,.

�

�

6J

ff

c:::::::J

I " V ' Z < - I . " The definition given above for the logical connective == can be written as follows : [ (A == B ) ::J (A &B ) V ( -A & - B ) ] & [ (A &B ) V ( -A & - B ) ::J (A == B ) ]

We will let the reader translate the follo wing statement into con­ ventional language : - 'The light is turned on'' & ' - the bulb is not burning" � " there is no electricity" V " the plugs have burned out " V "the bulb is burned out. " If we consider that statements can only be true or false, and con-

1 26

AN ALYS I S O F LANG UAGE

sider nothing else about them , then the connectives we have listed are enough to express all conceivable constructions made of statements. Even two connectives are adequate- for example, negation and con­ junction or ne gation and disjunction. This situation obtains , in partic­ ular, in relation to mathematical statements. Therefore other connec­ tives are not used in mathematical lo gic. But natural language reflects a greater diversity in the evaluation of statements than simply separating them into true statements and false. For e xample, a statement may be considered meaningless or implausible even though it is possible ( " " There are probably wolves in this forest ' '). Special branches of logic which introduce other connec­ tives are devoted to these matters. For modern science (unl ike clas­ sical mathematical lo gic) these branche s are not very important and we sha ll not deal with them.

■

PREDICATES

that associates a statement with certain objects is called a predicate. Predicates are divided into one-place, two-place , three-place, and so on according to the number of objects they require. Functional notation is used to represent them. The predicate can be written as a function with unfi lled places for variables , for example: A CON ST RUCTION

P( ) , L( , ), I( , ' )

or in the form P (x ) , L (x, y ) , l(x, y , z )

having stipu lated that x, y, and z are object variables, that is, symbols which must in the last analysis be replaced by objects-although which objects is not yet known. But the second form of notation, strictly speaking, no longer represents a predicate ; rather, it is a state­ ment containing object variables. In addition to capital letters we

A N ALYS I S O F LA NGUAGE

1 27

shall also use words and phrases within quotation mark s , for ex ­ ample : " red" (x ) or " between " (x , y, x ) and special mathematical signs such as < (x, y ) . The one-place predicate expresses an attribute of an obj ect , while a predicate with more than one variable expresses a relatio n amo ng objects . If the places for variables in the predicate are filled, then we are dealing with a statement which as serts the existence of the given attribute or relation . The statement " ' red " ( " ball" ) means that the " ball " possesses the attribute "red. " The construction < (a , b ) is equivalent to the relation (inequality) a < b . By joining predicate constructions with logical connectives we obtain more complex statements . For example we formerly wrote the I z l > I without breaking the statement down into elements , but now we write it

> (z, " I " ) V < (z, " - I " )

■

QUANTIFIERS

a large role is played by assertions of the universal­ ity of a given attribute and of the existence of at least one obj ect that possesses the given attribute . To record the se assertions the following so -called quantifiers are introduced: universal quantifiers V and the existential quantifi er 3 . Let us suppose that a certain statement S con­ tains a variable (indeterminate obj ect x; therefore we shall write it in the form S(x) . Then the statement (\fx )S (x) means that S (x) occurs fo r all x, while the statement ( 3 x )S (x ) represents the assertion that there exists at least one object x for which the statement S (x) is true . A variable included in a statement under the sign of a quantifier is called a bound variable , because the statement does not depend on this variable just as the sum IN MATHEMATICS

does not depend on the indexes i. The bound variable may be re­ placed by any other letter that does not coincide with the remaining

1 28

AN ALYS I S OF LAN G UAGE

variables and the mean ing of the statem ent will not change as a result. A variabl e whi ch is not bound i s called free . The statement depends enti rely on the free variables it contains. Here are som e examples of statements containing quant ifiers. I ) ( \fx ) ( \fy) [ "b rother" (t ,y ) & "man" (y ) :::J "brother" (y ,x ) ] . For every x and every y , if x is the brother of y and y i s a man then y is the brother of x. 2) If D (x,y ) i s used to represent the statement · ·x i s a divisor of y, ' ' then one of the relat ion sh ips cited above as an e xample of a state­ ment w ill be represented in the form n

(V n ) [ >(n 1 � • 1 ) => ( 3 p ) D (P 1 n )] ( 3 x ) W (x) -:J - (V x) - W (x)

The last relat ion i s true for any statement W(x) and shows that there is a connect ion between the universal and existent i al quan­ tifiers . From the exi stence of obj ect x for wh ich W(x) is true it fol­ lows that the assertion that · · W(x ) is untrue for all x' ' is not true. A quantifier is also, in essence , a lo gical connective. The at­ tribut ion of a quant ifier changes a statement into a new statement which contains one less free variable. The d ifference from the con­ nectives we considered above is that one must indicate , in addition to the statement , the free variable that must be coupled. The coupling of a variable m eans that concrete objects w ill be put in its place. If the numbe r of objects that can be subst ituted for the variable is finite then the quantifiers can be viewed simply as conveni ent abbreviat ions because they can be e xpressed by the lo gical connect ives of conjunc­ tion and disjunct ion. Suppose variable x can assume n values, which we shall desi gnate by the letters x 1 , x2 , . . . , Xn . Then the following equ ivalences will occur. (V x ) W (x) == W(x 1 ) & W(x2 ) & . . . & W(xn ) , ( 3 x ) W (x ) == W(x 1 ) VW(x 2 )V . . . V W(x n )

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THE CONNECTIVE "SUCH THAT"

of our table describes a construct ion that correlates an obj ect to a statement. In natural languages this construct ion is very

THE THIRD LINE

A NALYSI S O F LA NGUAGE

1 29

widely used . When we say " red ball, " we have in mind an object " ball " which possesses the attribute " red , " that is , it is such that the statement " red " ( " ball" ) i s true . We transfer the statemen t about the object to the adjective which modifi es the noun by which we de sig­ nate an object; in other cases this can be achieved by participles , par­ ticipial constructions , and constructions with the connectives "which " and " such that . " If we carry this analysis further we shall find that the noun, like the adjective , indicates first of all a definite at­ tribute or attributes of an object . Like the word " red , " the word " ball " depicts a certa in class of object s and may be correlated to a one -place predic ate , ' "is a ball " (x) , or simply ' ' ball " (x) . Theil " red ball " is such an object that the statement s " ball " (a ) and " red " (a ) are true ; in other words , the' statement " ball " (a ) & " red" (a ) is true . Notice that there are three independent elements operating in the logical notation : the letter a and the object s " ball " and " red , " while in writing in natural language there continue to be just two , " ' red ' ' and " ball. " But the letter a, which is introduced in logical notation to identify the given object and distinguish it from others ( and which is called the identifi er ) , does not completely d isappear in natural nota­ tion. It has been transferred to the concept " ball, " changing it from an attribute to an object.' Unlike the word ' " red , " the word ' " ball' ' identifies; you can say , ' ' This is the ball we lo st ye sterday' ' or ' ' I have in mind the same ball I was talking about in the previous sen­ tence . ' ' But what is an "obj ect " ?

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THE PHYSICAL OBJECT AND THE LOGICAL OBJECT

us that the world we live in is characterized by a certain stability and repetition (and also , of course , by constant movement and variation) . Suppose we see a tree . We walk away from it and the image of the tree on the retina of our eye change s in relationship to our movements . This change follows a definite law which is very familiar to us from observation of other object s . But when we return to our former place the image becomes almo st exE X PE RIE NCE TE A C H E S

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ANALYSIS O F LAN G UAGE

actly the same as it was before . Then we say , · "This is the tree , , ' having in mind not onl y the image of the tree-the mental photo­ graph-at the given moment in time but also the situations at nearby moments . If we are talking about classifying distinct si tuations by them selve s , w ithout considering their relations to other situations, then there is no difference at all between the noun and the adjective ; the concept "ball ," just like the concept "red , " is comple tely de ­ fined by indicating a certain set o f situations , and the discriminator (natural or art ificial ) of the se concepts need only be able to use the following sentence s correctly: ' ' Thi s is red ,'' · 'This is no t red ,'' "Thi s is a ball /' and . . This is not a ball. " It is different when we must classify time sequence s of situations rather than separate situations ; we shall repre sent them as if they were a movie film whose frame s each depict the situation at a given mo ­ ment. In the movie film ' ·ball ' · i s not simply a detail of the situation in one frame; i t i s a detai l that recurs in many. The discriminator of the concept "ball " cannot simply say , "Ye s, my friends , thi s is a ball ! ' ' It must identify the particular details in the frames , saying : " Here i s how the bal l look s in frame no. 1 3 7 ; here is the same ball in frame no. 1 3 8 ; here it is again in frame no. 1 39; and here is what it looked like in frame no . l 20,' · and so on. The detail of the situation which is cal led ' " the same ball' ' can change quite considerabl y be­ cause o f change in the po sition o f the eye relat ive to the ball or a change in the shape of the ball itself, but the ball itself is invariabl y and absolutely the same. This invar iability reflects the relative and temporal invariability we find in real ity. It is as if we were to draw a line in time connecting the details in the different frames o f film and dec lare that everything on thi s l ine is "the same" object. It is this line, in combination with a certain set o f attr ibutes (characteristics) , that forms the concept of the physical object. The logical concept of the object re flects a property of physical objects-they pre serve their identity. The object of logic is simply an identifier. Samene ss is its only attribute , as reflected in our imaginary connecting line . I f there are several different classes of objects , then various type s o f identifier s are ordinarily used to denote the objects in different classes. For example , line segments will be repre sented by

A NA LYSI S O F LA NG UAGE

1 31

small letters, points by capital letters, angle s by Greek letters, and so on. But more concrete attributes characteristic of objects are written in the form of distinct assertion s which include the introduced de sig­ nations. This make s it possible to get by without a con struction in­ volving the connective " 'such that . '' It is true that at the very begin­ ning of his famous treatise Elements de mathematique Bourbaki introduces the de signation rx[A (x)] for a certain object which po s­ sesses attribute A (x ) , that is, such that A {r.r[A (x ) ] } is a true state­ ment. After this, ho we ver , the designation disappears from his text . Thus a de finite name for the construction that associate s an object with a statement has not even been established and we are forced to leave a blank in our table. In the last analysis, a complete division of labor between identifiers and statement s is more convenient . For example let us take the sentence : ''The reddish-brown dog of Lieutenant Pshebyssky's widow killed the stray cat." When writ­ ten in the language of logic this sentence break s do wn into se veral statements which are implicitly contained in it and expressed by means of the grammatical category of attribution . They can be joined into one statement using the conjunction sign , but we can obtain a more con ventional and readable notation if we simply write out all the assertions being made-each on a new line separated by commas instead of conjunction signs . Assuming that the meaning of the attri­ butes and relation s being introduced is clear from the context , we re­ ceive the following equivalent of the above sentence : " dog" (a) , " reddish-brown " (a ) , " be longs " (a , b ) , " widow " (b , c ) , " Lieutenant Pshebyssky " (c ) , " ki lled " (a , d) , " cat " (d ) , " stray " (d ) .

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FUNCTIONS

the predicate, "Lieutenant Pshebyssky n (c ) , is the only one that is plainly not elementary. In the attribute H to be LieuIN THE EX A M PLE ,

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ANALYS I S OF LA N G U AGE

tenant Pshebyssky'' we distinguish two aspects: to have the rank of lieutenant and to have the surname Pshebyssky. That is why this predicate is expressed by two separate words. Of course, we could have put e ach of these word s in the form of a distinct predicate, but the fact that ''lieutenant '' is the rank of object c and ''Pshebyssky'' is the surname o f it would not have been reflected in thi s case, and therefore we considered such a separation meaningless. "Su rn ame " and 4' ,rank " are examples of a function of one free variable-o f a construction that juxtaposes the obj ect which is the meaning of the function to the object which is the free variab le. The function is written, as customary in mathematics, "surname " (x) , "rank " (x ) , and so on. If there are several free variables they are sep­ arated from one another by commas . after which we are dealing with the function of sever al variables. This construction associates an ob­ ject-value with a set of object-variables (their order is important !) . An example of a function of two free variables is ' 'the result of a game of chess " (x, y ) . Let us give examples of functions from mathemat­ ics . Functions of one free variable: sin (x ) , ! x i ; functions of two vari­ ables: arithmetic operations which may be written + (x, y ) , - (x,y ) , and so on; the distance (A , B ) between two points in space A and B : a function of three variables: the angle formed at point B by paths to points A and C; the designation