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McGRAW-HILL PUBLICATIONS IN PSYCHOLOGY
CLIFFORD T. MORGAN, CONSULTING EDITOR

PRINCIPLES OF
TOPOLOGICAL PSYCHOLOGY

McGraw-Hill Publications in Psychology

CLIFFORD T. MORGAN

CONSULTING EDITOR

Barker, Kounin, and Wright CHILD BEHAVIOR AND DEVELOPMENT

Brown PSYCHOLOGY AND THE SOCIAL ORDER

Brown THE PSYCHODYNAMICS OF ABNORMAL BEHAVIOR

Cattell PERSONALITY

Cole GENERAL PSYCHOLOGY

Crafts, Schneirla, Robinson, and Gilbert RECENT EXPERIMENTS
IN PSYCHOLOGY

Davis PSYCHOLOGY OF LEARNING

Dorcus and Jones HANDBOOK OF EMPLOYEE SELECTION

Dunlap RELIGION: ITS FUNCTIONS IN HUMAN LIFE

Ghiselli and Brown PERSONNEL AND INDUSTRIAL PSYCHOLOGY

Gray PSYCHOLOGY IN HUMAN AFFAIRS

Guilford FUNDAMENTAL STATISTICS IN PSYCHOLOGY AND EDUCA-
TION

Guilford PSYCHOMETRIC METHODS

Hurlock ADOLESCENT DEVELOPMENT

Hurlock CHILD DEVELOPMENT

Johnson ESSENTIALS OF PSYCHOLOGY

Krech and Crutchfield THEORY AND PROBLEMS OF SOCIAL PSY-
CHOLOGY

Lewin A DYNAMIC THEORY OF PERSONALITY

Lewin PRINCIPLES OF TOPOLOGICAL PSYCHOLOGY

Maier FRUSTRATION

Maier and Schneirla PRINCIPLES OF ANIMAL PSYCHOLOGY

Miller EXPERIMENTS IN SOCIAL PROCESS

Moore PSYCHOLOGY FOR BUSINESS AND INDUSTRY

Morgan and Stellar PHYSIOLOGICAL PSYCHOLOGY

Page ABNORMAL PSYCHOLOGY

Pillsbury AN ELEMENTARY PSYCHOLOGY or THE ABNORMAL

Reymert FEELINGS AND EMOTIONS

Richards MODERN CLINICAL PSYCHOLOGY

Seashore PSYCHOLOGY OF Music

Seward SEX AND THE SOCIAL ORDER

Stagner PSYCHOLOGY OF PERSONALITY

Wallin PERSONALITY MALADJUSTMENTS AND MENTAL HYGIENE

John F. Dashiell was Consulting Editor of this series from its
inception in 1931 until January 1, 1950.

PRINCIPLES

OF TOPOLOGICAL

PSYCHOLOGY

KURT LEWIN

Professo r of Child Psychology, Iowa Child-
Welfare Research Station^ University
of Iowa

TRANSLATED BY

FRITZ HEIDER

Assistant Professor, Department of Education, Smith College

AND
GRACE M. HEIDER

FIRST EDITION
SDCTEC IMPRESSION

McGRAW-HILL BOOK COMPANY, INC,

NEW YORK AND LONDON
1936

PRINTED IN THE UNITED STA1ES OP AMERICA

parts thereof j may not be reproduced

in any form without permission of

the publishers.

THE MAPLE PRESS COMPANY, YORK, PA,

THE HEBREW UNIVERSITY
Jerusalem

PREFACE

DR. WOLFGANG KOHLER
Swarthmore College
Swarthmore, Pa.

DEAR KOHLER:

This book is the result of a very slow growth.

I remember the moment when more than ten years ago it
occurred to me that the figures on the blackboard which were to
illustrate some problems for a group in psychology might after all
be not merely illustrations but representations of real concepts.
Much interested in the theory of science, I had already in 1912
as a student defended the thesis (against a then fully accepted philo-
sophical dictum) that psychology, dealing with manifolds of coexist-
ing facts, would be finally forced to use not only the concept of time
but that of space too. Knowing something of the general theory of
point sets, I felt vaguely that the young mathematical discipline
" topology" might be of some help in making psychology a real
science. I began studying topology and making use of its concepts,
which soon appeared to me particularly fitted to the specific problems
of psychology.

However, this undertaking expanded rapidly, forcing me to
consider wider and wider fields of psychology and to face more and
more involved problems. That is the reason why this book has
seen quite a number of unfinished and unpublished editions, and why
it does not yet contain the "vector psychology." The main diffi-
culty has not been the mastering of the mathematical problems as
^uch, at least insofar as the topological problems are concerned.
After several attempts to employ the more complicated concepts
of topology, I found it both sufficient and more fruitful to refer to
the most simple topological concepts only. Vector psychology will,
of course, require a more elaborate mathematical setup and will
in all probability even make it- necessary to enter a somewhat
undeveloped field of mathematics. But the main difficulty was
the dealing with problems which lie, so to say, between psychology
and mathematics.

vii

viii PREFACE

We know, since the theory of relativity at least, that empirical
sciences are to some degree free in defining dynamical concepts or
even in assuming laws, and that only a system as a whole which
includes concepts, coordinating definitions, and laws can be said
to be either true or false, to be adequate or inadequate to empirical
facts. This "freedom," however, is a somewhat doubtful gift.
The manifold of possibilities implies uncertainty, and such uncer-
tainty can become rather painful in a science as young as psychology,
where nearly all concepts are open and unsettled. As psychology
approaches the state of a logically sound science, definitions cease
to be an arbitrary matter. They become far-reaching decisions
which presuppose the mastering of the conceptual problems but
which have to be guided entirely by the objective facts.

Theoretical p&ychology in its present state must try to develop
a system of concepts which shows all the characteristics of a Gestalt,
in which any part depends upon every other part. As we do not yet
have the knowledge of facts which really suffices to determine
this system of concepts and as, on the other hand, this knowledge
of "facts" cannot be acquired without developing this system of
concepts, there seems to be only one way open: to proceed slowly
by tentative steps, to make decisions rather reluctantly, to keep in
view always the whole field of psychology, and to stay in closest
contact with the actual work of psychological research.

Such an undertaking, if any, needs the cooperation of a group.
I have always found myself rather unable to think productively
as a single person* I hope that this handicap may, in this case,
turn out to be of some advantage, for it has made this book the
result of the work of a group. Those who are acquainted with you
know that you are not interested in "psychological schools," and
one of the main incentives of this book is to help develop a psycho-
logical language generally understandable and independent of
schools, (By the way, I have tried my best to destroy the myth
that Gestaltists do not attack each other.) Yet collectives have
had and will, I think, always have their place in scientific work.
The group which was called the Psychological Institute of Berlin
has been, I think, such a collective of friends, working together
for many years, interested in all fields of psychology, and concerned
as much with experiments as with theories. Whether it was
valuable, history will show; but at least it was happy and
lively.

PREFACE ix

May this book prove to be somewhat worthy of the spirit of this
collective and of the leading influence you have had on each of its
steps. For the friends scattered throughout the world this feeling
of cooperation seems to continue and the circle steadily to widen.
I would enjoy nothing more than to have contributed to this broad
cooperation.

I dedicate this book to a young scientific center at the meeting
of the East and the West where I hope new productive collectives
will arise.

KURT LEWIN.
IOWA CITY, IOWA,
May, 1936.

ACKNOWLEDGMENTS

Dr. Fritz Heider and Grace Heider have not only undertaken the
laborious work of translating this book, but have improved its form
and contributed much to its content. I am deeply indebted to the
productive help and criticism of Dr. Tamara Dembo. She, Dr.
Roger Barker, and Dr. Herbert Wright have spent much time in
improving the text. Dr. W. W. Flexner was good enough to
read the part dealing with the topological concepts and to give
valuable suggestions. I gained much by discussing several points
with Dr. Herbert Feigl, Dr. W, A. Hurwitz, Dr. E. H. Kennard, and
Dr. E. C. Tolman.

Harcourt, Brace & Company has kindly permitted the use of a
selection from Anne Morrow Lindbergh, North to the Orient. Figure
6 is taken from Charlotte Biihler, Zwei Grundtypen von Leben-
sprozessen; Fig. 7 from Kurt Koffka, Principles of Gestalt Psychology.

KUE.T LEWIN.

IOWA CITY, IOWA,
May, 1936.

CONTENTS

PAGE

PREFACE vii

ACKNOWLEDGMENTS .

PART I. THE TASK OF PSYCHOLOGY AND THE FOUNDATIONS
OF TOPOLOGICAL AND VECTOR PSYCHOLOGY

I. THE PRESENT STATE OF PSYCHOLOGY 3

II. FORMULATION or LAW AND REPRESENTATION OF SITUATION. . . . 8
Law and individual case The constructive representation of the
situation Person and environment; the life space Ways in
which the lif e space is represented.

III. GENERAL CONSIDERATIONS ABOUT REPRESENTING LIFE SPACE. . . 14

The life space as the totality of possible events Inventories and
systems of behavior Constructive procedure: Summary Good
and poor abstraction; the method of approximation.

IV. CONTENT AND EXTENT OF THE PSYCHOLOGICAL LIFE SPACE .... 18

Appearance and reality in psychology Experience and risycho-
logical existence What is real is what has effects Phenomenal
facts and physics Life situation and momentary situation
Quasi-physical, quasi-social, and quasi-conceptual facts within
the life space The quasi-physical facts The quasi-social facts
The quasi-conceptual facts Influences by way of perception
and "gross somatic" influences.

V. CAUSAL INTERCONNECTIONS IN PSYCHOLOGY 30

The historical and the systematic concept of causality Exist-
ence, temporal and causal relationships The principle of "con-
creteness" The relational character of causal facts The
principle of "contemporaneity" Past and future; the unreal
and the indeterminate in the life space Existence and temporal
determination of a psychological fact and its content Content
as a property The indeterminate.

VI. THE PSYCHOLOGICAL LIFE SPACE AS SPACE IN THE SENSE OF

MATHEMATICS 41

Examples of space-like relationships in the life space Space of
free movement; locomotion, forces Regions within the person;

iv CONTENTS

PAGS

Summary The mathematical concept of space Topological
space Metrical space The application of the concept of space
and physicalism.

VII. PSYCHOLOGICAL SPACE AND PSYCHOLOGICAL DYNAMICS 59

Problems of pure mathematics and problems of coordination
The instability of psychological situations Necessary condi-
tions for the application of metrical and topological concepts
to the life space Space and dynamics. The history of the con-
cept of space in physics and psychology The fundamental
concepts of dynamics.

Vm. THE PSYCHOLOGICAL WORLDS AND THE PHYSICAL WORLD 66

Physical and psychological space The singleness of the physical
world and the plurality of the psychological worlds The physical
world as a dynamically closed unity and the psychological worlds
as dynamically unclosed unities.

IX. MATHEMATICAL REPRESENTATION AND PSYCHOLOGICAL THEORY. . 76
Concept, picture, and symbol Concept and model Physio-
logical theories of psychological processes Representation and
explanation.

PART II. TOPOLOGICAL PSYCHOLOGY
A. CONCEPTS or TOPOLOGY WHICH ARE FUNDAMENTAL FOR PSYCHOLOGY

X. CONCEPTS OF TOPOLOGY FUNDAMENTAL FOR PSYCHOLOGY 87

The concept of region. The connected region Closed and open
regions Limited and unlimited regions Simply and multiply
connected regions Jordan curve, boundary, path Foreign
regions.

B. TOPOLOGY OF THE PSYCHOLOGICAL ENVIRONMENT

XI. PSYCHOLOGICAL REGIONS, LOCOMOTIONS, AND COMMUNICATIONS. . 93
Coordinating definitions The psychological region Psycho-
logical locomotion Being inside or outside of a region The
inner structure of a psychological region Determination of the
connectedness of regions Nonconnected regions Multiply
connected regions Limited and closed regions Representation
as path or as more than one-dimensional region Representation
as point or as more than one-dimensional region Locomotions
of a surrounding field Thing and medium.

XJH. BOUNDARIES OF PSYCHOLOGICAL REGIONS 118

Definition and determination of psychological boundaries -
Sharpness of a boundary; boundary zones Dynamic properties
of psychological boundaries Barriers Boundaries which affect

CONTENTS XV

PAGE

communication Boundary zones which can be passed only with
difficulty Zones of undetermined quality.

XIII. THE RELATIVE POSITION OF Two REGIONS 136

Foreign regions Overlapping regions; the relative weight of
situations Difficulties in representing the relative position of two
regions The two principal possibilities for the representation of
the inaccessibility of a point Topological and dynamical aspects
of the representation of limitations Discrete paths and their
totality Homogeneous and differentiated barriers; approach
and withdrawal Barriers and adits.

XIV. STRUCTURAL CHANGES 155

Differentiation, integration, and restructuring Changes of
structure and locomotion Changes of magnitude and of distance
Dynamic conditions of structural changes; fluidity, elasticity,
plasticity.

XV. THE LIFE SPACE AS FINITELY STRUCTURED SPACE 163

C. TOPOLOGY OF THE PERSON
XVI. THE PERSON AS A DIFFERENTIATED REGION IN THE LIFE SPACE. . 166

XVII. FUNDAMENTAL CONCEPTS AND COORDINATING DEFINITIONS FOR THE

REPRESENTATION OF THE PERSON 168

Coordinating definitions for environment and for person
Dynamic dependency Boundaries and boundary zones
Remarks about strong and weak gestalten; Gestalten with differ-
ent degrees of dynamic unity Dynamic properties of personal
regions Tension Groups of tension systems Structure of the
person Inner-personal regions and the motor-perceptual region;
Central and peripheral inner-personal strata Individual differ-
ences in the structure of the person The degree of differentiation
of the person The kind of structure and the function of the part
regions The connection between dynamic and topological
factors.

D. DIMENSIONS OF THE LIFE SPACE

XVIII. THE DIMENSIONS OF THE LIFE SPACE 193

Mathematics of dimensions The dimensions of the psychological
environment Reality Degrees of irreality Problems of dimen-
sions of the person The differentiation of the life space in the
dimension reality-irreality.

BIBLIOGRAPHY 207

GLOSSARY 213

AUTHOR INDEX 219

SUBJECT INDEX .221

PART ONE

THE TASK OF PSYCHOLOGY AND THE

FOUNDATIONS OF TOPOLOGICAL

AND VECTOR PSYCHOLOGY

CHAPTER I
INTRODUCTION
THE PRESENT STATE OF PSYCHOLOGY

In its present state of development psychology must be
thought of as a young science. There is only one field in which
it is relatively well established and in which it has advanced
steadily: this is the psychology of sensation and perception.
The scientific character of this field is fully recognized. Its
.findings are based almost entirely on experimental evidence, and
even when its theories are in conflict one feels that as far as
method is concerned it stands on relatively firm ground. The
situation is different with the psychology of will, of needs, and
of personality despite the fact that these fields have always
attracted popular interest. As recently as fifteen years ago it
was assumed that they, by their very nature, were not ame-
nable to scientific methods. The little experimental work that
had been done seemed too artificial and abstract to give an
insight into the real processes. It was generally accepted that
experimental investigations of these elusive and highly compli-
cated processes were intrinsically impossible, at least in so far
as human beings are concerned. Thus in Europe these prob-
lems were treated in a half-literary, half -philosophical way, and
in America the tendency was to study individual differences by
means of tests.

The only approach to deeper problems was the brilliant work
of Freud. However, the attempt of the psychoanalysts to base
general laws entirely on case studies and therapeutical work
seemed methodologically unsound to most scientists.

This skeptical atmosphere and the undoubtedly great
technical and conceptual difficulties have blocked the develop-
ment of an experimental psychology of will and needs. On the

4 TASK OF PSYCHOLOGY

other hand a number of branches of psychology have reached a
stage which makes their unification increasingly urgent. Child
psychology, for instance, has collected a great number of facts
about speech, play, and other forms of behavior at different
age levels. Animal psychology has to a large extent passed
beyond the more elementary questions and has begun to study
more inclusive and in many respects more " human" problems.
Psychopathology has brought together a great number of facts
that ought to be directly related to facts of normal psychology.
Finally problems of social psychology which lie across all these
fields are becoming more and more urgent.

From all these sources we possess a great amount of valuable
material. At the same time, especially in recent years, we have
become much more critical of what we have done. In America
there seems to be an increasing distrust of purely statistical
methods, and the indiscriminate use of tests is criticized in a
way which would have seemed impossible a few years ago.

Investigators are coming to feel that a mere piling up of facts
can only lead to a chaotic and unproductive situation. The
simple collecting of facts is indispensable at certain stages of a
science; it is a wholesome reaction against a philosophical and
speculative building of theories. But it cannot give a satis-
factory answer to questions about causes and conditions of
events. Only with the help of theories can one determine causal
interrelationships. A science without theory is blind because it
lacks that element which alone is able to organize facts and to
give direction to research. Even from a practical point of view
the mere gathering of facts has very limited value. ' It cannot
give an answer to the question that is most important for
practical purposes namely, what must one do to obtain a
desired effect in given concrete cases? To answer this question
it is necessary to have a theory, but a theory which is empirical
and not speculative. This means that theory and facts must
be closely related to each other.

Psychology needs concepts which can be applied not merely
to the facts of a single field like child psychology, animal
psychology, or psychopathology^ but which are equally appli-

INTRODUCTION 5

cable to all of them. One should be able to use the same con-
cepts for problems of emotional life as for problems of
behavior; or for problems concerning the infant, the adolescent,
and the aged; the healthy and the sick; animals and human
beings; the personality and the environment. Does this mean
that we are to return to the making of speculative "systems"?
Yes and no. Yes, in so far as we should not content ourselves
with a blind collecting that splits the field of psychology into a
number of unrelated branches. No, in so far as we must not
try to derive all psychological facts neatly from one single
concept such as association, reflex, instinct, or totality.

The system of concepts capable of bringing together the
different fields of psychology in an empirical manner would
have to be rich and flexible enough to do justice to the enormous
differences between the various events and organisms with
which it must deal. It would therefore have to be oriented in
two directions, namely, toward theoretical connectedness and
toward concreteness. In other words it would have to be
equally suitable for the representation of general laws and of
the characteristics of the individual case.

The unification of the different fields of psychology seems
quite hopeless until we have an adequate psychology of will
and needs and of personality. Fortunately, however, we need
not feel pessimistic about the possibility of developing these
central regions of psychology. Within recent years a great
number of studies have shown that in spite of the general
skepticism an experimental attack on fundamental problems in
those fields, including problems of Freudian psychology, is
quite possible.

We have come to see that in investigations of this kind we
must deal with persons as wholes to a much greater extent than in
the psychology of sensation. In the psychology of sensation the
individual's ideals, ambitions, and his social relationships play no
role at all or only a subordinate one. But an experimental inves-
tigation of needs, of action, or of emotions cannot be carried out
without taking into account the characteristics of the person,
his momentary state, and his psychological environment.

6 TASK OF PSYCHOLOGY

This shows again that the concepts of which psychology is
now in need have to meet the requirements which we suggested
above: the system of concepts must be broad enough to be
applicable to the most primitive bodily behavior as well as to
the emotions, thought processes, values, and social relation-
ships. It must be capable of representing these processes not
as single isolated facts but in their mutual dependence as
expressions of a concrete situation involving a definite person in
a definite condition. These concepts must unify without
undue simplification; they must include both person and
environment, both law and individual case.

These requirements can be fulfilled only if one turns from the
prevailing methods of "abstractive classification" and tries to
build constructive concepts.

The concepts which are discussed in the following chapters
have been developed and tried out in the course of the last ten
years. They are based on both experimental investigations and
case histories. In presenting them we are not promulgating a
new "system" limited to a specific content, but rather we are
describing a "tool," a set of concepts by means of which one can
represent psychological reality.

As I see it, the outstanding characteristics of this undertaking
are:

1. It tries to build up a framework for the constructive
representation and derivation of psychological processes which
is logically consistent and at the same time adapted to the
special properties of the "psychological life space/'

2. It includes both the characteristics of the environment and
of the person.

3. It makes no more assumptions than are required,

4. It proceeds by a method of successive approximation.

The concepts developed in the following pages are "opera-
tional" in so far as a univocal relation between concepts and
observable data is consistently maintained. Although the con-
cepts always extend from the level of phenomena into the level
of causal relationships, they are "descriptive" in the sense of
Newton's dictum: Hypotheses non jingo. That is, they express

INTRODUCTION 7

the nature of certain relationships and at the same time avoid
that type of " explanation" which is characteristic of specula-
tive theories and which is at present, as it seems to me, a real
handicap to our science.

The concepts that we here offer will certainly have to be
revised in the course of time. But I am optimistic enough to
believe that they, independent of all schools of thought, will
prove themselves to be fundamental to psychology in that the
later additions and changes will leave the validity of these con-
cepts, as first approximations, intact. Such a stability is after
all the only one possible in science.

The purpose of the following chapters is not that of a text-
book of psychology. There is no attempt to give an account
of the psychological data which have been found with the meth-
ods discussed herein, or to present the specific theories developed
with these concepts. I have merely tried to comply with
requests to give in extenso the definition of the concepts and the
mathematical background used in the research which is pub-
lished under the title Untersuchungen zur Handlungs- und
Affektpssychologie, I to XX, edited by Kurt Lewin in the
Psychologische Forsckung, 1926 to 1937. Concerning the fruit-
fulness of these concepts, one should turn to these particular
studies or to Dembo and Hanfmann (19) and Lewin (51, 55, 58).

The term "topological psychology"' is used to refer to that
part of theoretical psychology which is based upon concepts
of mathematical topology. It is to be complemented by
"vector psychology." In actual research, of course, both types
of concepts have to be used together.

The general, rather extensive first part of this book is an
introduction to both of these fields of theoretical psychology.

CHAPTER II

FORMULATION OF LAW AND REPRESENTATION OF

SITUATION

From the viewpoint of theory of science, the recent develop-
ment of psychology corresponds in magnitude, extent, and
character to the transition in physics from medieval Aristotelian
to modern Galilean concepts. 1 It is one of those advances
which are typical of certain stages in the development of sciences
and which narrow the gap between a still half-speculative theory
and concrete reality in a decisive way.

LAW AND INDIVIDTJAL CASE

One of the most striking features of this development is that
the opposition between universal concept and individual event
is overcome. Law and single occurrence enter into intimate
relationship. Thereby, the representation of single concrete
cases gains a new fundamental meaning for science. Hereto-
fore the single event could be thought of as only a chance
occurrence and its representation could be valued merely as a
curiosity. Only an average of many cases seemed to possess
general significance. But if one considers the single event also
as governed by law, one has to obtain scientific evidence from
concrete "pure cases" and not from averages of a great number
of historically given events. Thereby the representation of
single cases gains new scientific meaning. It has a direct
bearing on the determination of general laws.

Tables i and 2 compare main features of three developmental
epochs, which we designate briefly as "speculative," "descrip-

1 There is a discussion of the methodological and conceptual aspects of this
change in K. Lewin (57, 59). Also J. F. Brown (8, 9, 10).

The numbers in parentheses refer to the items in the Bibliography in the
back of the book.

FORMULATION OF LAW

tive," and "constructive/' (It may be emphasized that this
comparison is a rough schematic simplification.)

TABLE i. CHARACTERISTICS OF CONCEPTS AND METHODS IN DIFFERENT EPOCHS

OF PSYCHOLOGY

Epoch

Speculative
("Aristotelian ")

II
Descriptive

III

Constructive
(" Galilean")

Goal

To discover the essence
of things and the cause
behind all occurrence

To collect as many
facts as possible
and to describe
them exactly

To discover laws. To
predict individual cases

General character-
istics of concept
formation

Psychological concepts
are not separated from
non-psychological

Elimination of non-psy-
chological concepts

Dividing psychology intc
with different laws

> independent fields

Psychological phenom-
ena treated as one field
governed throughout by
the same system of laws

Friendly to theories
(speculative type)

Hostile to theories

Friendly to theories
(empirical type)

Historical and sys-
tematic problems

Problems of occurrence
and of quality are not
separated

Problems of occurrence
and of quality are sepa-
rated

Historical origin and
cause not clearly differ-
entiated

Historical origin and
cause are differentiated

Type of system

All-inclusive system de-
rived from a single
concept or from a few
dichotomic concepts

Descriptive classi-
fication by ab-
straction

Constructive system
based on a group of
interrelated concepts.
Concepts which permit
gradual transition be-
tween oppositions

THE CONSTRUCTIVE REPRESENTATION or THE SITUATION

In addition one has to consider the following fact. As late
as the end of the nineteenth century the question was still
debated whether psychology should only describe or whether it
should also try to determine the conditions and effects of
psychological/processes. At the present time we find that
questions about the "why," or in other words dynamic ques-

TASK OF PSYCHOLOGY

tions, claim the center of interest in both theoretical and applied
psychology.

In order to answer these questions it is necessary to find out
the laws which control psychological events. This means that

TABLE 2. LAWFULNESS AND DYNAMIC CONCEPTS

Epoch

III

A Law a rule. Indi-

A law & a rule. All

vidual case not lawful.

events are lawful in-

Lawfulness exists only

cluding those which oc-

Nature of lawfulness

where there is a regu-

cur only once.

larity of occurrences

An empirical proof that

an event is lawful is not

necessary

Demonstration of the

Investigation of indi-

frequency of similar

vidual "pure cases."

events, disregarding in-

Comparison of different

dividual differences.

cases (systematic vari-

The rule is the more

ation) ; no abstraction

The technique of

certain the greater the
number of cases and

from individual pecu-
liarities. The validity

proving a partic-
ular law

the greater their simi-
larity. "The excep-

of the proof depends
upon the purity of the

tion proves the rule"

case and not upon the

frequency of its occur-

rence. Experiment *

deliberate creation of

pure cases

Classification by ab-

Classification ac-

Concept formation

straction from differ-

cording to the

through construction

ences (statistical aver-

phenotype

(as opposed to classifi-

Logical properties of

age). The concept of

cation). Genetic defi-

concept formation

thing predominates

nitions. The concept

of event predominates;

functional, conditional-

genetic concepts

Causes are directed fac-

tors (tendencies). The

tors. Only relations

essence (general class)

between several facts

of the thing itself is the

can be causes of events.

Dynamics

cause of its behavior.

Every event depends

The behavior is deter-

upon the totality of the

mined by the past or

contemporary situation

the future (teleology)

one must determine under which conditions the different kinds
of psychological events occur and what effects they have. But
knowledge of the laws alone does not answer the question of

FORMULATION OF LAW 1 1

why in a particular case a given individual behaves in a given
way and not otherwise. Even if all the laws of psychology
were known, one could make a prediction about the behavior
of a man only if in addition to the laws the special nature of the
particular situation were known. The laws define functional
relationships between different characteristics of an event or
situation. The application of the laws presupposes the compre-
hension of individual cases. One can apply a law only if one
knows the nature of the concrete case with which one is dealing.
Considered from this point of view the laws are nothing more
than principles according to which the actual event may be
derived from the dynamic factors of the concrete situation.

This relationship can be made clear by the following formu-
lation : If one represents behavior or any kind of mental event
by B and the whole situation including the person by S, then B
may be treated as a function of 5 : B = f(S) . In this equation
the function /, or better its general form, represents what one
ordinarily calls a law (84, p. 366). If one substitutes for the
variables in this formula the constants which are characteristic
for the individual case one gets the application to the concrete
situation.

The determination of the laws is therefore only one side of
the task of explaining mental life. The other side, which is of
equal importance and inseparably connected with the determi-
nation of the laws, involves the task of representing concrete
situations in such a way that the actual event can be derived
from them according to the principles which are given in the
general laws. The usual description of a situation does not
make this possible. It can be done only by means of a con-
structive representation of a situation. We shall discuss the
necessary characteristics of such representations in detail.

PERSON AND ENVIRONMENT; THE LIFE SPACE

As far as the content is concerned, the transition from Aristo-
telian to Galilean concepts demands that we no longer seek the
"cause" of events in the nature of a single isolated object, but
in the relationship between an object and its surroundings. It

12 TASK OF PSYCHOLOGY

is not thought then that the environment of the individual serves
merely to facilitate or inhibit tendencies which are established
once for all in the nature of the person. One can hope to
understand the forces that govern behavior only if one includes
in the representation the whole psychological situation. 1

In psychology one can begin to describe the whole situation
by roughly distinguishing the person (P) and his environment
(J3). Every psychological event depends upon the state of the
person and at the same time on the environment, although their
relative importance is different in different cases. Thus we can
state our formula B = f(S) for every psychological event as
B = f(PE) . The experimental work of recent years shows
more and more this twofold relationship in all fields of psy-
chology. Every scientific psychology must take into account
whole situations, i.e., the state of both person and environment.
This implies that it is necessary to find methods of representing
person and environment in common terms as parts of one
situation. We have no expression in psychology that includes
both. For the word situation is commonly used to mean
environment. In the following we shall use the term psycho-
logical life space to indicate the totality of facts which determine
the behavior of an individual at a certain moment.

WAYS IN WHICH THE LITE SPACE Is REPRESENTED

At present we have no adequate scientific method for repre-
senting the psychological life space. In accord with the general
methods of psychology, the study of environmental influences
began with classification and statistics. For instance, the
average achievement of the "only child, 7 ' or of the "second
child in a family of three, " has been investigated by these
methods. In medical case studies, one usually finds more con-
crete detail of the psychological environment. Thus they
have given us excellent descriptions of the home environment. 2
The method of representation is partly akin to that of the

1 For the concept of field and the history of its use in psychology, compare
Kofka (47> PP. 54/0 and KoeMer (44, PP- 3of)-

2 For example, cf. A. Homburger (33, p. 242).

FORMULATION OF LAW 13

novelist, i.e., one tries to make as lifelike a picture of the situ-
ation as possible by choosing expressive words and bringing out
significant traits with examples. In general, the descriptions
that have been most valuable to science have not been those
made by scientific methods. Where theoretical concepts have
been introduced with the concrete description, they often stand
out as something alien. Instead of scientific description they
are nothing more than speculative interpretation.

The most complete and concrete descriptions of situations
are those which writers such as Dostoevski have given us.
These descriptions have attained what the statistical character-
izations have most notably lacked, namely, a picture that
shows in a definite way how the different facts in an individual's
environment are related to each other and to the individual
himself. The whole situation is presented with its specific
structure. This means that the single factors of the situation
are not given as characteristics which can be arbitrarily com-
bined in a "summative" way (88, 89, 90). If psychology is to
make predictions about behavior, it must try to accomplish
this same task by conceptual means. In selecting methods and
concepts we must use a pragmatic criterion: we have to find
concepts on the basis of which predictions can be made. In
other words our concepts have to represent the interrelation-
ships of conditions. This point of view will determine the
procedure of this book.

CHAPTER III

GENERAL CONSIDERATIONS ABOUT REPRESENTING

LIFE SPACE

THE LIFE SPACE AS THE TOTALITY or POSSIBLE EVENTS

If we are to accomplish the task of deriving the behavior
of the person (in more general terms: the psychological events)
from the life space, we have to characterize it as the '" totality
of possible events."

We shall later discuss in detail what sort of events are funda-
mental for the representation of the environment and of the
person. Here we shall only mention that from both the theo-
retical and practical point of view the most important character-
istics of a situation are what is possible and what is not possible
for the person in this situation. CEach change of the psycho-
logical situation of a person means just this certain events are
now "possible" (or " impossible ") which were previously
"impossible" (or "possible").

For instance, when an employee of a company is dismissed,
the important change for him is that he can no longer give orders
to the office boys, that he can no longer make purchases for the
firm, and that all other possibilities of action which he enjoyed
as a member of the firm are taken away from him. These may
include the privilege of using a certain entrance to the office as
well as any sort of behavior toward other persons to which the
prestige of the firm gave him a right. On the other hand, he
can now do many things which were not possible before. He
can snub his former employer, he can read books because he has
plenty of time, he can sleep late in the morning, etc.

Also, the difference between the situation shortly after the
dismissal and that after a long period of unemployment can be
characterized by changes in the possibilities of action. As time
goes on, the lack of money makes a good meal or a journey

GENERAL CONSIDERATIONS ABOUT LIFE SPACE 15

impossible. He may still be able to dress respectably, but he
can no longer afford to dress elegantly. Also he has lost the
courage to go out every day looking for work.

In a similar way, the difference between the rich and the poor,
between the youth, the adult, and the man who is almost too
old to hold a job, is in each case fundamentally determined by a
range of possibilities. The same is true for the difference
between the healthy man and the sick one, between people of
different educational level, and people in different political
situations. A dynamic psychology has to represent the person-
ality and the state of a person as the total of possible and not-
possible ways of behaving.

INVENTORIES AND SYSTEMS OF BEHAVIOR

Another fact leads to the same point of view, namely, that
the situation is to be regarded as the total of possibilities. One
of the chief difficulties which psychology has to face is the
following: If one investigates, for instance, the causes and effects
of anger, one is confronted with a great variety of responses
(20, pp. 27-30), although one may start with a definite experi-
mental set-up, which roughly at least guarantees a consistent
psychological structure of the situation. There appear anger
effects of very different degrees, and at the same time a great
number of other kinds of behavior: substitute actions, short
cuts, changes of level of aspiration, aggressive behavior, etc.
It is possible to classify and describe these processes. The
most one can accomplish in this way is to make a catalogue
of types of behavior and to amplify and refine it. The same
is true if one investigates success and failure, the boundaries
of the ego, punishment, etc.

Such a collection of facts is indispensable and has its scientific
value. However, the real task of scientific, especially of
experimental psychology, lies beyond such a collection. It is
necessary to understand why this and only this behavior occurs.
In place of a catalogue which gives no reason why under given
circumstances just these and no other forms of behavior occur,
there must be built up a framework of concepts which does not

16 TASK OF PSYCHOLOGY

have the arbitrary character of a mere list. This means
creating a system of deduction. The different kinds of behavior
that occur in a certain situation are to be understood as belong-
ing to a coherent system of "possible" events that are in their
totality an expression of the particular characteristics of this
situation.

The more we succeed in determining the details of the situ-
ation in this sense, the more the actual possibilities are limited.
A complete determination of the life space would show which of
the possibilities, given by its general structure, will be realized
at the moment.

CONSTRUCTIVE PROCEDURE: SUMMARY

To si.iimTnfl.rize what we have said about the representation of
person and environment :

1. The fundamental constructs which we use in representing
the situation must consist of concepts from which one can
derive, unambiguously, certain events as "possible/' others as
"not possible." Instead of dassificatory concepts one has to
use constructive ones which have a direct relationship to laws.

2. It should be possible to derive from such a representation
all forms of behavior which actually occur. This stringency of
the derivation of the totality of possible cases is valid not only for
the behavior of the person within the situation but also for the
possible changes of the person or of the situation itself.

3. Such a derivation of the totality of possibilities can only
be accomplished if one proceeds from the life space as a whole.

4. The center of interest shifts from objects to processes, from
states to changes of state. If the life space is a totality of
possible events, then "things" that enter the situation, espe-
cially the person himself and psychological "objects," have to
be characterized by their relationship to possible events.

GOOD AND POOR ABSTRACTION; THE METHOD OF
APPROXIMATION

In addition to the reasons given before, there are the following
advantages in starting with the life space as a whole in making

GENERAL CONSIDERATIONS ABOUT LIFE SPACE 17

an analysis. In psychology, as in every other science, the
investigation and representation of each single case is an infinite
task in itself, and one whose solution would presuppose a full
knowledge of the laws of psychology (7, p. 33). In comparison
with this ideal, every actual representation of a concrete case
is incomplete and simplified. There are two ways of meeting
this difficulty, and from the point of view of research they are
very different. One method, which one can call abstracting
classification, begins by taking into account important single
facts and then makes classifications according to one or another
of these facts. The individual peculiarities of each situation are
thus disregarded. Since there are almost always several such
significant facts, such a classification is usually open to attack.
It is ambiguous in itself and very often vacillates between
opposite characterizations.

In contrast to this the second method begins with the life
space as a whole and defines its fundamental structure. The
procedure in this case is not to add disconnected items but to
make the original structure more specific and differentiated.
This method therefore proceeds by steps from the general to
the particular and thereby avoids the danger of a "wrong
simplification" by abstraction. In such abstractive classifi-
cation, the second step often destroys the characterization of
the first. The "right simplification" implies a schematization
too, but it is a procedure of "gradual approximation." The
representation given in the first approximation will not be
destroyed but only made more articulated by the second
approximation since the whole situation is taken into account
from the beginning. To what point the approximation shall be
carried depends upon the particular problem under consider-
ation. It is important that even the representations in the
first approximation are of value in their own right. In this
way the formation of concepts becomes essentially similar to
that in mathematics.

CHAPTER IV

CONTENT AND EXTENT OF THE PSYCHOLOGICAL LIFE

SPACE

What is meant by psychological life space and what must one
take into consideration in order to represent it?

APPEARANCE AND REALITY IN PSYCHOLOGY

Certainly one will have to represent the physical environ-
ment of the individual to a certain extent, for instance the room
where he is and the position of the furniture and other objects
that are important for him at the moment; in certain cases also
the house in which the room is, the city, and even the country.
One will have to represent his^ocial environment, his relation-
ships to other persons, their positions and personalities, and his
own place in society, for instance his vocation. At the same
tune, his longings and ambitions will play an important role,
his fears, thoughts, ideals, and daydreams, in short everything
that from the standpoint of the psychologist exists for this
person.

Experience and Psychological Existence. It is, however,
not always easy to determine what things exist psychologically
for a given person. The most obvious method might seem to
be the use of consciousness as a criterion. This would mean
that the physical and social environment would be treated as
psychological environment in so far as the person is conscious
of them. Such a formulation is doubtful, however, even if one
uses the concept of consciousness in a very broad manner.
There is no question, for instance, that when a person is in a
familiar room, the part of the wall which is behind him belongs
to his momentary environment. Furthermore, the fact that
such and such other rooms are near this one, that the house
stands in a lonely settlement by the sea or on a busy thorough-

CONTENT OF THE PSYCHOLOGICAL LIFE SPACE 19

fare of a great city can be an essential part of the psychological
situation. This can be true even when one is not looking at
the landscape but is deep in his work and for the moment is not,
or at least in no clear way, aware of this wider environment.

It is likewise doubtful whether one can use consciousness as
the sole criterion of what belongs to the psychological life space
at a given moment in regard to social facts and relationships.
The mother, the father, the brothers and sisters are not to be
included as real facts in the psychological situation of the child
only when they are immediately present. For example, the
little child playing in the garden behaves differently when he
knows his mother is at home than when he knows that she is
out. One cannot assume that this fact is continually in the
child's consciousness. Also, a prohibition or a goal can play
an essential role in the psychological situation without being
clearly present in consciousness.

The same is especially true for the general social atmosphere,
its friendliness, unfriendliness, or tension. Doubtless, just these
general properties of the social atmosphere are of the greatest
significance for man's behavior and for his development. And
yet, one often realizes what the atmosphere has been only when
it changes.

What Is Real Is What Has Effects. Here, as in many
other cases (57) it is dear that one must distinguish between
"appearance" and the "underlying reality" in a dynamic
sense. In other words, the phenomenal properties are to be
distinguished from the conditional-genetic characteristics of
objects and events, that is, from the properties which deter-
mine their causal relationships. From the standpoint of
dynamics one must consider the whole situation as the total of
what has effects for the individual under consideration. As far
as the conceptual derivation is concerned, one may use effective-
ness as the criterion for existence: "What is real is what has
effects." 1

1 Thus far this view is in line with that of the New Positivism (cf. Feigl, 240,
p. 422), although we are here less concerned with the problem of "the reality
of mind." Our criterion rather serves as a tool for making practical decisions

20 TASK OF PSYCHOLOGY

Phenomenal Facts and Physics. The distinction between
phenomenal and conditional-genetic properties must not be con-
fused with the distinction between psychological and physical
objects. In psychological and philosophical discussions it is
common to identify the psychological with the "directly given."
Koffka (47, pp. 46/.), by calling attention to unconscious proc-
esses and reflexes, has clearly demonstrated that the experi-
enced world (behavioral environment) does not suffice to
explain behavior. Nevertheless he seems to hold to the above-
mentioned interpretation in so far that he does not object to the
identification of conditional-genetic and physical facts.
According to this point of view the physical is only indirectly
comprehensible; 1 it has to be inferred from psychological
experience. The psychological is not limited to the phe-
nomenal, but on the other hand everything phenomenal is
something psychological. This conception has far-reaching
consequences. It implies that the conditional-genetic, dynamic
facts belong to the physical world, even in psychology, and
therefore that explanation of psychological events in the last
analysis has to be based upon physical facts.

These and similar conceptions are widely accepted. How-
ever, they seem to me erroneous, both from an epistemological
and from a psychological point of view. The objects of all
empirical sciences, including the objects of physics, can be
experienced no less directly than those of psychology. This
direct experience concerns first of all the appearance of objects,
i.e., their phenomenal properties. In order to understand
causal relationships one has to proceed to the conditional-
genetic properties. But this progression to deeper levels takes
place within one and the same field of science. The conditional-
genetic properties of a piece of iron that physics finds remain
properties of this same piece of iron, which one perceives
directly and uses, however far the concepts of physics may

in psychology. We do not presuppose, as New Positivism generally does, the
reducibility of Psychobiology to Physics.

1 He says: But every datum is a behavioral datum; physical reality is not a
datum but a constructum (47, p. 35).

CONTENT OF TEE PSYCHOLOGICAL LIFE SPACE 21

progress and however indirect physical methods may be.
Otherwise the scientific analysis would be meaningless, both
from a practical and from a theoretical point of view.

In the same sense one can distinguish in psychology between
the properties which are more phenomenal and more directly
accessible and the conditional-genetic properties of its objects. 1
In psychology as well as in other sciences an explanation of
events is only possible if one succeeds in advancing to the
dynamic properties. And again the phenomenal and the
dynamic properties are properties of one and the same psycho-
logical event.

Such a point of view recognizes in both physics and psy-
chology phenomenal as well as conditional-genetic properties.
This is not the place to give detailed proof of our argument.
Certainly it has great methodological advantages. The point
of view according to which all psychological explanations
must finally rest on physics is based essentially on the philo-
sophical Utopia of a single universal science. But this means
that in advancing from description to explanation psychology
is forced to make a sudden jump into an alien field. If one
disregards this philosophical Utopia one can represent a life
space in a continuous progression and take into account all
necessary dynamic facts, whether they are determined directly
or indirectly. This method seems to me the only one that
makes it possible to include in one representation everything
which is and only that which is necessary for the conceptual
derivation of actual behavior.

A thoroughly worked out dynamic representation of person
and environment will have the character of a construction
and it must have this "conceptual" character if it is to serve
as a means of deriving actual behavior. But such a construct
(10; 7, p. 3; 84) must not be confused with a general class con-

1 Tolrnan (84) points to the fact that one cannot derive behavior directly from
behavior. Instead, one has to introduce some "intervening" variables between
the behavior to be derived and the observable facts indicative of its causes. So
far as I can see, these intervening concepts are the same as our "dynamical"
or " conditional-genetic " concepts. The term "intervening concept " may prove
to be quite convenient.

22 TASK OF PSYCHOLOGY

cept because it represents the characteristics of a concrete
individual case. 1

LIFE SITUATION AND MOMENTARY SITUATION

By psychological situation can be understood either the
general life situation or more specifically, the momentary
situation.

A woman stands at the loom in a big noisy factory, next to the last in
the eighth row. A thread is broken. She is about to stop the machine
to see what has happened. It is shortly before the lunch hour. She has
accomplished very little during the morning. She is annoyed.

These are a few data of the momentary situation of this
woman. About her life situation, one can say:

She has been married for three years. For a year and a half, her hus-
band has been unemployed. The two-year-old child has been seriously

1 The necessity of psychological theories which permit logically strict deriva-
tions of the concrete facts is stressed by C. L. Hull (35). Hull formulates four
postulates of a " sound scientific theory ' J (3 50, pp. 495/0 . These postulates con-
cern its logical characteristics and the way a theoretical system should be proved.
He stresses "that the deduction or proof of each theorem is a complex multiple-
link logical construct involving the joint action of numerous principles or postu-
lates, as contrasted with simple syllogistic reasoning where but two premises are
employed" (p. 499).

We fully agree with Hull's four principles, which are in line with our own
postulate i (pp. 6 and 16). We too want to stress particularly that no single
concept suffices as a basis for derivations. But it may be well to say a word about
the limits of a purely formalistic approach. Hull probably would agree that a
theoretical system might be logically sound but have such a poor psychological
content that its scientific value would be negligible (compare, for instance,
Stevens 800) . Psychology needs stressing of the formalistic. Yet, it would soon
prove most unfortunate if one should lose sight of the fact that the main purpose
of psychological theories is, after all, to explain reality. In psychology, for a
long time to come, the richness and fruitfulness of theories should not be judged
by their formalistic perfection alone.

Logically it may be conceivable that in representing psychological dynamics
one can use any kind of concepts. (In other words, one may disregard the
possibility that there is something like a "logic of dynamics.") Practically,
however, form and content of a theory are both fundamental and we mean to
stress both aspects equally. For, a system which is inadequate or unclear as to
the content of its dynamical concepts will soon prove to be unsatisfactory
logically. It would lead us too far to discuss from this point of view Hull's
"Miniature Scientific Theoretical System" (350, pp.

CONTENT OF TEE PSYCHOLOGICAL LIFE SPACE 23

ill, but today he seems somewhat better. She and her husband have been
quarreling more and more often recently. They had a quarrel this morn-
ing. Her husband's parents have suggested that she send the child to
them in the country. The woman is undecided what to do about it.

It is obvious how closely life situation and momentary situa-
tion are connected. In this case, the life situation may serve
as a rather remote background of the momentary situation.
Or it can be that the woman was thinking of her child while
she worked, and in this way the life situation often became
part of the immediate situation. But even when she was
busy repairing her broken thread and no longer thought of
the domestic situation, even then the life situation remained at
least indirectly significant. It affected the state of the person
and thereby the reactions within the momentary situation.

The woman sees the momentary environment, the rooms,
the bed, the household routine, in a different light with each
change in the life situation. Objects which were dear to her
before the trouble with her husband might have become dis-
agreeable, others the more precious. The room in which a
child is ill changes its character and changes it once more when
the child recovers. Their past history thus plays a great part
in determining the psychological import of things for the person.

Although the whole life situation always has some influence
on the behavior, the extent to which one must take it explicitly
into account in the representation of the life space is very
different in different cases. A person who is trying to decide
whether or not to get married, whether or not to go into a
certain business, whether or not to begin a lawsuit against an
influential opponent, will in general act in accord with his whole
life situation. Only happy-go-lucky, superficial, or childish
persons act out of a narrow momentary situation in important
questions of life. On the other hand whether a man who is
taking a walk goes along the right or the left side of the road
will be decided by a much less inclusive momentary situation.
It is easy to observe how the structured part of the life space
becomes wider or narrower under the influence of a new
event.

24 TASK OF PSYCHOLOGY

In representing the life space it is necessary to take into
account these differences of extension. The specific problem
with which we have to deal in a given case determines whether
it is the life situation or the momentary situation which comes
more strongly into the foreground. The life situation is on
the whole the more constant. In representing it one can reckon
with larger time units. These differences in rate of change
do not however mean that there is a fundamental difference
between the two. The concepts which are developed in the
following can be equally well applied to life and momentary
situations, i.e., to any possible life space. 1

QUASI-PHYSICAL, QUASI-SOCIAL, AND QUASI-CONCEPTUAL
FACTS WITHIN TEE LITE SPACE

If one uses the dynamical criterion ("what is real is what has
effects ") to determine the existence or nonexistence of a fact
as a part of a psychological life space, one has to include a great
number of facts. For example, one would have to include all
somatic processes which influence the behavior of a person, for
there can be no doubt that in psychology we have to deal with
psychobiological organization and that psychology is only a
part of the general science of life.

The Quasi-physical Facts. This does not mean that we
have to include within the psychological life space the whole
physical world with its "objective" characteristics in terms of
physics* These facts are to be included in the representation
of the psychological life space only to the extent and in the
manner in which they affect the individual in his momentary
state. We express this by calling them quasi-physical facts.

Even when from the standpoint of the physicist the environ-
ment is identical or nearly identical for a child and for an adult,
the psychological situation can be fundamentally different.
The same is true for the environment of men and animals
(6o 7 pp. 322-323) and also for men of different personality.
Further a physically identical environment can be psycho-

1 Examples of constructive representation of the life situation are given by
T. Dernbo and E. Hanfmann (19); K. Lewin (58).

CONTENT OF TEE PSYCHOLOGICAL LIFE SPACE 25

logically different even for the same man in different conditions,
for instance when he is hungry and when he is satiated* 1 This
does not mean that psychology can ignore the postulate of
general validity for scientific statements. It only means that
the situation must be represented in the way in which It is
"real" for the individual in question, that is, as it affects him.
The confusion of "objective" with "physical" and of "logi-
cally general" with "equal for all" has led to grave conceptual
and methodological errors in psychology.

The Quasi-social Facts. A distinction analogous to that
between physical facts in the sense of physics and quasi-
physical facts is that between objective social facts 2 in terms of
sociology and social-psychological facts which have to be taken
into account in representing a certain life space.

When a mother threatens an obstreperous child with the
policeman and the child obeys her because of his fear of the
policeman, then as far as the representation and explanation
of the child's behavior are concerned we are dealing not with
the actual legal or social power of the police over the child, but
rather with the power of the police as the child sees it.

The same applies to the power of Santa Claus, of the father,
in short, to all social relationships. As regards membership in
a group we have to consider more the belief of the person
and the way , it affects him than legally or sociologically
defined criteria of group membership. In representing the
psychological situation we have to include social, like physical
facts, only in so far and only in the manner in which they
influence the person under consideration. On this account, we
shall speak of quasi-social rather than of social facts.

1 AUport (2), p. 178, says: "Speed apparently is another factor that is homo-
geneous only to physics; in our results speed seems to split into three relatively
independent rates of movement. Many of the speed measures correlate more
highly with non-speed measures than with each other. In short, physical
categories of movement are unsuitable models of the psychological study of
expression."

2 It is not necessary at this point to go into the epistemological question of the
"objectivity" and "reality" of social structures as sociology defines them, nor
into the problem whether it is justifiable to compare this reality with the reality
of physics.

26 TASK OF PSYCHOLOGY

The relationship between purely sociological and quasi-
social facts is complicated by the knowledge that even for
sociology as such the opinions of people about the social
relationships in which they live play an important role. This
is true, even with people who consciously have no adequate
realization of their social relationships. Certainly the behavior
of nations would be different if they were fully conscious of
their real social interdependence. Thus the social-psycholog-
ical facts (the quasi-social facts of psychology) have great
significance for sociology itself and the representations of
psychological, especially of social-psychological facts, might be
applied fruitfully in sociology.

The Quasi-conceptual Facts. In addition to the quasi-
physical and the quasi-social facts one has to consider quasi-
conceptual facts as important for the psychological life space.
We do not want to assume here a conceptual realism, but one
may not neglect the following functional equivalence.

A person may be engaged in solving an extensive conceptual,
for instance a mathematical problem. He has to follow definite
steps in bis thinking to determine the suitable mathematical
relationships and to find his way in a system of mathematical
concepts. The structure of the psychological environment in
which the individual moves about, in which he faces difficulties,
and in which he carries out certain tasks is then essentially
determined by the structure of the mathematical field itself.
The analogy to quasi-physical and quasi-social environment
holds in this respect also, that we again have to deal with
"given" facts which can be more or less adequately compre-
hended and according to whose objective structure the indi-
vidual concerned must adapt himself if he wants to attain
certain goals. In this respect the mathematical facts are
sometimes dynamically not less cogent and insurmountable
than certain physical and social facts.

Again there is not usually a complete agreement between the
objective structure of the mathematical field and the momen-
tary psychological field. If there were such complete cor-
respondence, then the solution of mathematical problems would

CONTENT OF TEE PSYCHOLOGICAL LIFE SPACE 27

be child's play. Rather, the psychological field is usually
incomplete in comparison with the structure of the mathe-
matical task, and is inadequate in decisive points. For the
derivation of the actual psychological events and their dynamics
it is not the mathematical system in itself which is important,
but rather the momentary structure of the psychological field
of the individual. Therefore we shall speak of "quasi-con-
ceptual" facts.

Finally, it may be mentioned that the quasi-physical, quasi-
social, and quasi-conceptual facts are not sharply separated
from each other, but that we are dealing throughout with a
unified psychological life space in which these three groups of
facts can be thought of as representing three only roughly
distinguishable classes.

For the child with his animistic and magic thinking these
differences remain fluid; with adults also there are many tran-
sitions and many facts which can be classified in these groups
only with difficulty.

INFLUENCES BY WAY OF PERCEPTION AND "GROSS
SOMATIC" INFLUENCES

The quasi-physical and quasi-social facts in the psycho-
logical life space need not be an adequate representation of the
objective physical and social facts to which they refer. How-
ever, the structure of these psychological facts depends to a
high degree upon the structure of the physical and social facts.
A change in the quasi-physical facts in the life space of the
person is often the result of an objective change in the physical
environment.

One can roughly distinguish two cases in which the life space
is influenced from the outside: (i) The influence can occur by
way of a perceptual process, usually leading to a change of the
cognitive structure (85) of the field with reference to the
object in question. (2) The influence can be a gross somatic
one. A stone may hit a person and cause injury or loss of
consciousness. This stone need not necessarily appear in the
perceptual field of the person.

28 TASK OF PSYCHOLOGY

It may be questionable whether such gross somatic influences
ought to be treated by psychology. Certainly the perception
of a physical object and an injury inflicted by a stone are events
of very different character. But the effect of a perception
also may go beyond a change of the cognitive structure of the
life space. It may, for instance, produce a change of the goal
and lead to a change in the person's direction of action. On
the other hand a gross somatic influence, for example poisoning,
can also involve far-reaching cognitive changes in the life space.
Perceptual processes as well as gross somatic influences can
therefore change the life space in every respect. One must
remember that the transition from the grossly somatic
effects of physical objects to their effects as objects of perception
is not at all an abrupt one.

This is especially clear in regard to the field of action. When,
for instance, a man is moving a heavy piece of lumber and is
pushed to one side or lifted up by it, when he swims, when he
goes up steps or moves in any other way then the changes
effected by physical objects which occur in the environment
and in the state of the person are not the result of perceptual
processes alone. Moreover, we find even within the psychology
of perception a transition and interplay of the two different
kinds of effects. For instance, looking at a too-intense light
may result in an injury to the eye. Similarly atmospheric
conditions affect a person not only through perception but also
in other ways.

In the following, we shall deal with the purely " somatic"
influences only in so far as they are connected with behavior
and perception. But we want to emphasize that this distinc-
tion is not a fundamental one and that it is not possible for
psychology to disregard entirely the gross somatic influences.
Probably there is a certain dynamical similarity between all
these influences of the physical world on the life space in so far
as they force certain changes upon the individual.

Likewise sociological facts need not necessarily influence the
psychological life space of the person by means of perceptual
processes. An arrest, a change of one's legal position by a new

CONTENT OF THE PSYCHOLOGICAL LIFE SPACE 29

law, or the coming of a new employer can act from the outside,
like a flying stone, and alter fundamentally the situation of an
individual. This alteration need not be the result of the
dynamics of the previous psychological situation.

What concepts psychology can use in dealing with such
influences of "nonpsychological" factors is a difficult question
which we shall treat later. In any case the extent of such
influences on perception and action makes it impossible to
disregard them -in psychology. This means that it should be
possible in principle to represent all bodily changes of the person
in the life space. At the same time it confirms our position
that psychology and biology are not essentially separated but
are distinguished only by a difference of interest.

CHAPTER V
CAUSAL INTERCONNECTIONS IN PSYCHOLOGY

THE HISTORICAL AND THE SYSTEMATIC CONCEPT OF
CAUSALITY

One derives psychological events by tracing them back to
the dynamic relation in which they have their source. This
"tracing back" and the concept of causation which is implied
in doing so are understood in very different ways in psychology.

The question "why" can have two very different meanings
in psychology.

1. Why does a given situation S (i.e., a particular person P
in a particular environment E) have the event B and no other
as a result? As stated above this question is answered if we
succeed in discovering the general law B = f(PE) which is
valid for the dynamic structure of the situation in question.
Thus the event is traced back to the dynamic characteristics
of the momentary situation. The "cause" of the event con-
sists in the properties of the momentary life space or of certain
integral parts of it.

2. Why does just such a situation come into being i.e., why
has the life space in a particular case these particular proper-
ties? This question has a specific historical, or as one can say
more accurately a "historical-geographical" meaning (60,
pp. 328-329). It deals with historical developments, with
chains of causes, and with the point of convergence of these
chains. The answer to this question is obtained only by an
analysis of the history of the individual and of his environment.
We shall speak therefore of "historical concepts of causation"
in these cases in contrast to the "systematic concepts of causa-
tion" which were characterized above.

A physical example: I am sitting in the rain under a tree
whose leaves keep me from getting wet. I ask: "Why don't I

CASUAL INTERCONNECTIONS IN PSYCHOLOGY 31

get wet?" It is possible to answer this question by finding
out the direction and velocity of the falling drops, the position
of the leaves, my own position, etc. In short one can represent
the present situation and, by applying the laws of mechanics
or other relevant laws, derive what event must occur in such
a situation. But the answer to this question could also be as
follows: "It is thanks to your grandfather who planted this
tree that you do not get wet. To be sure the soil is not very
good right here, but your grandfather took special care of the
tree during its first years. Yet if the plan for a new state road
had gone through last year the tree would already have been
cut down and you couldn't sit here without getting wet."

The second answer is an example of explanation in terms of
historical causality. Its characteristic feature is that it gives
an account of the course and interweavings of causal chains of
events, events which happened but once and which can be
placed in certain years and certain geographical locations.
The answer in terms of systematic causality on the other hand
refers to types and to laws in which there are no dates. Both
answers are entirely legitimate and important, but neither is a
substitute for the other, although one has bearing on the other.
For psychology both types of answer are important. The
historical concept plays an especially important role in develop-
mental psychology, both of individuals and of groups. It is
also important in psychopathology where it may be essential
to understand the genesis of the illness.

Even in dealing with systematic questions in experimental
psychology certain "historical" facts must be given special
consideration. As the psychology of will and affect has shown
(20, p. 30), the decision of some of the most fundamental prob-
lems of experimental psychology requires the setting up of
unambiguous situations and this is possible as a rule only by
producing a certain historical development.

In spite of the dose connection between the historical and
the systematic concepts of causation in research, one ought to
distinguish the two much more sharply than is done at present.
The confusion of historical and systematic concepts and problems

32 TASK OF PSYCHOLOGY

is an essential characteristic of the pre-Galilean or Aristotelian
thinking 1 of a period of psychology which is now coming to an
end and which has led to momentous errors. It is one of the
reasons for the inconsistencies of the association theory and the
difficulties involved in applying the concept of experience. In
psychoanalysis, whose contribution lies in great part in its
emphasis on the historical aspect of its problems, it has led
to important misinterpretations. Analogous conceptual con-
fusions frequently occurred in experimental child psychology.

Only when we distinguish sharply in all branches of psychol-
ogy between the two concepts of causation and in this respect
also proceed from "Aristotelian" to " Galilean" ways of think-
ing, can we attack dynamic problems.

The following discussion will be based on the systematic con-
cept of causation, for in dynamics one must undoubtedly give
first place to this concept. Even questions of the dynamics
of historical sequences cannot be answered without insight into
the dependence of the single event on the given situation, i.e.,
without determining the equation B = f(PE). This does not
imply a neglect or underestimation of the historical problems
in psychology.

EXISTENCE, TEMPORAL AND CAUSAL RELATIONSHIPS

In deriving an event from the life space, one has to take into
account several principles that are important for research.
They are expressed in part by the formula B = /(PE), but one
must emphasize them especially because present-day psychol-
ogy so often ignores them.

The Principle of "Concreteness." Only what is concrete
can have effects. This proposition may seem obvious. But
one often ignores it in explaining an event by development, by
adaptation, by the Prinzip der wirkenden Seek (12), by an
abstract drive, and in treating these principles as concrete

1 C/. K. Lewin (59). The same confusion of the two kinds of concepts has
been no less of a handicap in other early periodsj e.g., of economics and history of
art.

CAUSAL INTERCONNECTIONS IN PSYCHOLOGY 33

causes. It would lead too far to deal with all the consequences
of such reasoning. These fallacies arise in part from a con-
fusion between the law that governs the effects of certain
concrete events and these events themselves. Effects can be
produced only by what is " concrete," i.e., by something that
has the position of an individual fact which exists at a certain
moment; a fact which makes up a real part of the life space
and which can be given a definite place in the representation of
the psychological situation. All this is not true of " principles.' '

The Relational Character of Causal Facts. It was char-
acteristic of the Aristotelian way of thinking (59 , pp. 28-30) to
derive an event from the nature of a single object, for instance,
from the personality of an individual as such, from an inner
drive, from an emotion. The question whether heredity or
environment plays the greater part also belongs to this kind of
thinking. The transition to the Galilean thinking involved a
recognition of the general validity of the thesis: An event is
always the result of the interaction of several facts.

This principle has a certain connection with a thesis which
gestalt theory has done much to develop in the psychology of
perception^ namely, that the effect of a stimulus depends in
part upon the nature of the surrounding field. This principle
has penetrated more and more into other fields of psychology
during recent years (47; 44).

If one is to derive events from forces, one will have to recog-
nize that a force is always the result of an interaction of several
facts. We shall repeatedly come across this principle and its
implications.

The Principle of "Contemporaneity." The questions of
the temporal relationship of the event and the dynamic con-
ditions which produce it are very important and have a direct
bearing on almost all psychological problems. By referring to
our formula B = /(P-E) we can state these questions more
precisely: What is the temporal relationship of behavior (B)
to the two factors which make up the situation, person (P)
and environment ()? Furthermore, what is the temporal
relationship between the different parts of the life space?

34 TASK OF PSYCHOLOGY

These questions have usually been discussed in such a way
that one asked whether only past or whether future events also
could cause change. Wundt, for instance, believed that the
characteristic of the causa finalis (teleology) lies in the assump-
tion that future events influence present events. In the case
of the ordinary cause (causa efficient) one generally takes for
granted that something past is the cause of present events,
This point of view occurs frequently in philosophical discussions
that are based on physics.

This emphasis on past or future causes plays an important
part, not only in philosophical discussions of psychology, but
also and that is more significant for us in the actual con-
struction of theories regarding concrete problems. The
reference to future events occurs in more or less explicit form, for
instance in the application of the concepts of drive or instinct, in
theories of play, etc. The causal derivation of present events
from past facts plays a great role in the theory of expression,
emotion, and experience. The nature of this procedure is but
poorly concealed if one connects past experiences with the
present by a bridge of memory. Very often the assumption
of such a connection is made merely because one has followed
this principle of explanation, for instance, when one bases the
universality of expression of certain emotions on phylogeneti-
cally identical, formerly useful forms of behavior instead of
deriving the similarity of expression directly from the similarity
of the situation involved.

Though we are justified in setting up "historical" questions
and looking for causal sequences, yet we must be careful to
avoid historical or half-historical answers to " systematic "
questions of causation. It was typical of the Aristotelian way
of thinking not to distinguish sufficiently between historical and
systematic questions. The result was that one took past or
future facts as causes of present events. In opposition to this
assumption we shall here strongly defend the thesis that neither
past nor future psychological facts but only the present situation
can influence present events. This thesis, is a direct conse-
quence of the principle that only what exists concretely can

CAUSAL INTERCONNECTIONS IN PSYCHOLOGY 35

have effects. Since neither the past nor the future exists at
the present moment it cannot have effects at the present. In
representing the life space therefore we take into account only
what is contemporary.

This implies that in answering questions in regard to "sys-
tematic" causation one has, to represent courses of events as
series of situations (momentary sections). If in the following
discussion we occasionally use a diagram of a path to indicate
a movement in a situation, it is only a symbolical representa-
tion and means that the structure of the situation remains
sufficiently constant during the interval of time in question.

We do not have to discuss the problems of contemporaneity at this
point. In order to determine the contemporaneity for different points of
the field with absolute exactness one ought to take into account in psy-
chology, as is done in modern physics, the velocity of the mediating
processes. 1 But we can neglect this question for the present. It is much
more important to stress the necessity of excluding events which roughly
speaking belong to past and future time.

To be precise one will have to treat the single time sections
not as moments without extension but as differential time sec-
tions in order to be able to determine direction and velocity of
changes at given points. The concept of momentary section
is to be understood in this sense in the following discussion.

One could argue that psychological facts are intrinsically of a
historical nature. As a matter of fact the structure of the
person and the psychological characteristics of the environment
at each moment and in each point are in a decisive way depend-
ent upon the previous history, as experimental investigations
show impressively (34). However, this influence of the pre-
vious history is to be thought of as indirect in dynamic psy-
chology: From the point of view of systematic causation, past
events cannot influence present events, fast .events can .only
have a position in the historical causal chains .whose inter-
weavings create the present situation... .This L fact has oftenjiot
been givSineHmigh^ in psychology.

1 A consequence of this is that the field of contemporaneous events must be
defined differently according to the point of reference in each case (74).

36 TASK OF PSYCHOLOGY

Historical processes in psychology as in other sciences are to
be thought of as dialectical. Whether we look at short inter-
vals or at longer stretches of time we find that periods of
apparently continuous transformations are followed by periods
of crisis with sudden changes of structure.

PAST AND FUTURE; THE UNREAL AND THE
INDETERMINATE IN THE LITE SPACE

The relationship of past and future to the momentary life
space involves several conceptually and practically important
questions which physics does not have to consider.

Existence and Temporal Determination of a Psycho-
logical Fact and Its Content. We shall begin with an exam-
ple from StendahTs Rouge et Noir (82, pp. 104-106) :

Julien, the tutor, had resolved to touch the hand of Mme.
de Renal, his pupil's mother:

Julien, his mind intent on his proposed enterprise, could think of noth-
ing to say. The conversation languished.

"I wonder if this is the way I shall behave on the occasion of my first
duel?" the young man asked himself; for he had too much distrust of
himself and others not to be aware of the mental condition he was in.

Any danger would have seemed preferable to him in his mortal agony.
How he prayed that Mme. de Rnal might think of some forgotten domes-
tic duty and return to the house! The restraint that he was obliged
to put on liimself produced an appreciable alteration in his voice; Mme. de
RnaPs voice, too, was beginning to tremble, but Julien did not notice it;
the conflict between duty and timidity was too severe to admit of his
thinking of anything outside himself. The great clock of the chateau
struck the third quarter past nine, and he had not had courage to attempt
anything. Disgusted with his pusillanimity, he said to himself, "When
that dock strikes ten I will do what I have been promising myself all day
to do, or I will go up to my room and put a bullet in my brain."

After a period of suspense and anxiety it seemed a century during
which Julien in his tense emotion thought his reason must desert him, the
dock above his head struck ten. Each stroke reechoed in his bosom as if
the hammer had fallen on his heart.

Finally, . . .

Such an example shows that there are cases in which a future
event has a strong influence on behavior, on the train of thought,

CAUSAL INTERCONNECTIONS IN PSYCHOLOGY 37

on the mood. (^JBut are we really dealing with the future in these
cases? J If a chii3rtries4o get to a box of blocks that is visible
but difficult to reach/jthe goal certainly is psychologically
present. It is obviousT that for Julien the goal, to touch the
hand of Mme. de Renal, is in the same sense part of the present
life space. 1

The following considerations offer a solution of this dilemma.
The goal as a psychological fact undoubtedly lies in the present.
It really exists at the moment and makes up an essential part
of the momentary life space. On the other hand the " content "
of the goal, th&4ouehifitg-of^^ as a physical or social

fact in the future. Indeed it may not occur at all. The
nature of the expectation and the character of what is expected,
in so far as they act as psychological conditions at the moment,
naturally do not depend upon whether or not the event comes
to pass. 2 In either case the person strives toward a goal
which exists psychologically in the present life space.

In the example given above the time relationship is as follows: The
desire to touch the hand of Mme. de Rnal existed already that morning.
If one were to represent the life space for a given moment of the morning
one would have to include the existence of this goal as a psychologically
real fact. The content of the goal is a future event, an action which is to
be carried out by Julien himself in the evening. This time index of the
content naturally does not determine the temporal position of the situa-
tion in which this content is included as a psychological fact. It is a
qualitatively very important characteristic of this fact that the content
refers to an event on the evening of the same day. This gives the goal a
certain degree of nearness.

When the sun sets the intention will still persist and it will have the
same content that it had in the morning. But even if it has the same
content the dynamic characteristics of this intention are essentially
changed, especially because of the greater temporal proximity to the goal.
Not only does the goal attract Julien, but at the same time he is afraid of
it. The situation is now one of grave conflict. Moreover the goal is, in
another sense than in the morning, continually present.

1 One often says in these cases: "The event is present as an image." Such a
characterization is usually misleading, for there is often no real image of the goal.

2 This shows the fallacy of the formulation that in these cases a future event
acts as cause. The "future event" often never becomes a real, existing fact
that can produce effects of its own.

38 TASK OF PSYCHOLOGY

The dynamically important characteristics of the intention change
again when the time of carrying it out is fixed, and its content in this way
acquires a precise time index.

We find a similar difference in time index of psychological
fact and its content in the memory of, the flight from, or
embarrassment over something that happened in the past.
Again, the psychological reality of such feelings as fear, hope,
or doubt does not depend upon whether or not the content of
these feelings exists in a physical or social sense.

We can make the general formulation:

The existence or nonexistence and the time index of a psycho-
logical fact are independent of the existence or nonexistence and
time index of the fact to which its content refers.

Content as a Property. Nevertheless, the content is in no
way irrelevant, but is of greatest importance for psychological
dynamics. Whether, for instance, an actual goal refers to a
present or a future event, whether this event is thought of as
something that definitely exists, or as something that is only
possible or highly improbable all this forms an essential
characteristic of a goal. Differences in time index and in
existential characteristics of the content imply a qualitative
difference in the psychological facts themselves, that is, they
have formally the position of properties of the psychological
facts.

On the other hand, in psychology as in all other sciences the time index
and likewise existence or nonexistence of the facts themselves (for instance
the difference between possible and real) have not the position of prop-
erties. They do not make it possible to distinguish between qualitatively
different types of psychological facts.

It is an important problem how one ought to characterize the
qualitative difference between these facts and whether, for
instance, facts which refer to future events can always be
represented within the present life space in the same way. 1

1 One could think of representing future and past as the margin of the life
space, but on the whole such a representation does not seem permissible to the
author.

CAUSAL INTERCONNECTIONS IN PSYCHOLOGY 39

The meaning of past and future in a life space is different in
different cases. It is an important developmental fact that the
temporal extent of the life space of a child generally increases
with increasing age.

The Indeterminate. Connected with the problem of the
time index and of existence is the difficult question of deter-
minateness and indeterminateness of psychological facts. This
question is very important for the representation of
situations.

In one respect our example of Julien Sorel is not typical.
His goal is unusually definite, clear, and compact. When he
decides to carry out the intended action at exactly ten o'clock
the goal loses its last trace of indeterminateness.

Usually, goals are much less tangible, less clear and definite.
The vocational goal of the fourteen-year-old may be entirely
vague and uncertain. An expectation also can be very indefi-
nite as to its content. One may feel vaguely that something is
going to happen. Then it may become clearer from what
direction the trouble is to be expected. Gradually one becomes
aware of its real nature. In every region of mental life one
finds every possible transition between the greatest indeter-
minateness and full determinateness.

One must emphasize that the degree of determinateness or
indeterminateness of a goal, of an expectation, of a thoughtis in
each case a dynamically important fact, and that any change
of the degree of determination is a real psychological process.

It is typical of the process of orientation in a new environ-
ment that the regions which are at first unclear gradually
become clearer. The degree of clearness is an essential deter-
minant of the cognitive structure of the life space. It is closely
related to the degree to which one can differentiate the life
space into different regions and is therefore of great importance
for learning and insight. It is essential for all situations in
which practical tasks are to be completed. The definiteness
or indefiniteness of a situation plays a great role in the making
of decisions, in the general firmness of an individual's behavior,
in his inclination to enter into a fight (18), or his aptitude

40 TASK OF PSYCHOLOGY

for leadership (38, pp. 44-59). We are therefore dealing with
important characteristics of every situation.

Indeterminateness leads to special difficulties in representing
the life space. How shall one represent something that in
itself is indefinite? How, for instance, can one speak of the
direction toward an indefinite goal? Does not this inde-
terminateness involve something which in the long run makes a
conceptual representation impossible?

G. E. Miiller in his polemic against the concept of an image
of an indefinite gray has rightly emphasized (67, pp. 42 5/.)
that it belongs to the nature of a fact to have no indetermi-
nate properties. In truth the fundamental scientific methods
of proof rest on the proposition that every reality is univocally
determined.

Thus we are faced with the conceptual difficulty that on the
one hand each fact as such can have only definitely determined
properties, while on the other hand there exists the possibility
of an indeterminate goal, thought or image. I see the solution
of this difficulty in that the indeterminateness of mental events
is an indeterminateness of the content and not of the psycho-
logical fact itself. Fear in the face of an uncertainty that
dominates a man at a certain moment is in itself a perfectly
definite psychological fact whose properties it is possible to
describe. One of these uniquely determined properties is the
specific degree of indeterminateness of what the man fears.

Like the time index so the indeterminateness or the particular
degree of detenninateness of the content is to be thought of as a
property of the psychological fact itself. This property is in
each case uniquely determined as are all other properties of the
psychological fact.

How these properties are to be represented in the life space
remains a difficult and important question. We shall discuss
this point later. But one must not evade the issue because of
philosophical prejudices nor be disturbed by them.

CHAPTER VI

THE PSYCHOLOGICAL LIFE SPACE AS SPACE IN THE
SENSE OF MATHEMATICS

EXAMPLES or SPACE-LIKE RELATIONSHIPS IN THE LIFE SPACE

Whenever a life space is to be represented one tries to show
that a man of such and such a nature is in such and such a
situation :

John is on a ski trip in the mountains.

Janek is stable boy for the coachman of Sameschkin.

Hedwig is waiting at the police station for her passport to be
issued.

Henry has been running around for three days as though he
were out of his head. He had been sure that he would get the
job. Now he has lost his chance.

Paul refuses all invitations. He is completely buried in his
work. He sits at his desk hearing and seeing nothing.

Jack is in the dentist's chair. His eyes are closed and he is
trying to forget that the dentist behind him is preparing the
needle for an injection.

Small Joan is standing at a fence dreaming how the first man
came down from heaven in a very, very tiny ball.

All representations of psychological life space are based on the
fundamental conception of a particular person in a particular
environment. Doubts begin to arise when one asks what is to
be treated as person and what as environment, whether for
instance clothes belong to person or environment. But how-
ever the details of the boundary between environment and
person are worked out, the conception of a person in an environ-
ment is one of relative "position." The relationship "person-
environment" and also the concept of "belongingness to"

42 TASK OF PSYCHOLOGY

person or to environment imply categories which in some sense
can be characterized as spacial.

Besides the person there is "in " this life space a great number
of other quasi-social, quasi-physical, and quasi-conceptual facts.
These facts have a certain definite spacial relationship. The
life space is articulated into " regions" that are qualitatively
different from each other and that are separated by more or
less pervious " boundaries." In our example of the factory
worker (see p. 22) the loom belongs definitely to factory and
occupation. The lunch box may be characterized by a con-
tinuous change between home and factory. Many experi-
ments have shown how important it is in which way the
different regions are " connected' 7 and in what way and to
what degree they are " separated.' 3

I am convinced that these concepts which we use for the
representation of psychological facts, like region, spacial rela-
tionship in life space, connectedness and separateness, belong-
ingness, etc., are real spacial concepts in a strict mathematical
sense. It is very important for psychology to use these con-
cepts in a strict and consistent way. Before we discuss the
question of whether these are really spacial concepts we shall
give a few examples of the way they have been used in our
psychological investigations.

Space of Free Movement.

Example i. We start with a simple example. Two six-
year-old boys are sitting in a bathtub, the one very lively,
excited, and overactive, the other quieter. The excited one
(A) jumps around in the tub so much that the other (B) feels
cramped. Finally B draws a line in the water across the middle
of the tub and tells A to stay within his own region. Whereas
in the beginning (Fig. 10) there was a single unarticulated
region of possible movement for A with the result that for the
other child (B) the actual freedom of movement was very
much restricted, now there are adjacent but sharply separated
regions of free movement for A and for B (Fig. 16).

Example 2. An example of a very limited space of free
movement is the life space of a prisoner. His freedom of bodily

LIFE SPACE IN THE SENSE OF MATHEMATICS

movement is limited to the area of his cell. At certain hours
there may be added to this the corridor and perhaps a work-
room and the courtyard. This represents his whole space of
free bodily movement. The region of things outside the prison
that includes his family, his friends, his business is unattainable
to him.

One could raise the objection that the region outside of the
prison is not in every sense out of reach for the prisoner. It is

FIG. i. Space of free movement, (a) Spaces of movement of A and B are
not separated; (b) they are separated; (c) see text, p. 106.

possible for him to have at least occasional "social communica-
tion" by letter and other channels with his friends and family.
Further, he is able to take real action in a social sense in the outer
field, for instance, by making use of his remaining legal rights.

One could ask whether there is any psychological boundary
between the inner region (the prison) and the outer region, since
the prisoner can, at least in thought, concern himself with
objects and affairs outside the prison. This difficulty in
determining the boundary of the space of free movement dis-
appears as soon as one sees that it is only possible to determine a
region univocally if one refers to specific psychological processes.

We started with a consideration of the bodily locomotion of
the prisoner and the region of which we spoke referred only

44 TASK OF PSYCHOLOGY

to this particular kind of movement. For social locomotion
the structure and spacial relationships of regions in the life
space generally have to be characterized in a way different
from that for bodily locomotion. It is again different for
mental locomotion. This difference is a manifestation of a
principle which we have mentioned before and which is very
important. In order to represent the life space in such a way
as to answer dynamic questions it is necessary to use processes
(operations) as the ultimate elements of construction.

It is characteristic of the life space of a prisoner that his space
of free movement has especially rigid and almost impassable
boundaries. It is dangerous and often impossible to attempt
to cross these boundaries. In this connection one can speak of
the solidity of a boundary. The solidity of the boundary
of the prison is different for bodily, for social, and for mental
locomotions.

Example 3. Two facts are especially important in regard to
the space of free movement of a child : the nature and the extent
of (i) what is allowed to him, and (2) what his own abilities
permit him to do.

With reference to (i) : A child (C) may be forbidden to go
on the street alone, to read certain books, to go into the pantry,
to climb a fence, to pick flowers, or to be impolite to guests.
If within the life space of the child one outlines the region of the
forbidden (/) then the remaining space, i.e., the space of what
is allowed, is for some children relatively small (Fig. 20), for
others considerably larger (Fig. 26). This difference is very
important for the behavior and development of the child, espe-
cially for his independence and his personality.

In addition to the extent of these regions the solidity and
definiteness of the boundaries between the forbidden and the
permitted play an important role. An adult when exerting
strong control over a child can establish clear and solid bounda-
ries between the forbidden and the permitted, or he may make
the region of the forbidden very extensive, but without such
solid boundaries. The result is quite different for the child
in these two cases.

LIFE SPACE IN THE SENSE OF MATHEMATICS

In this example the boundaries do not consist of physical
walls as in the example of the prison, 1 but they are usually of
a social nature, Their solidity depends on the character of
the underlying social relationships, i.e., on authority, affection,
fear of punishment, etc.

FIG. 2. Differences in space of free movement, (a) Less able child in a
situation with many taboos; (&) capable child in a situation with few taboos;
C, child; /, forbidden region; i, region inaccessible because of insufficient ability.

With reference to (2) : The freedom of movement is further
limited by the fact that many goals which of themselves are
permitted cannot be reached because of insufficient mental or
physical ability (Fig, 2, regions ). The region of what is
possible is in many respects much smaller for children than for

1 The solidity of the boundaries in the example of the prison is also often
based on social facts, for instance, on the legal prohibition against leaving the
Jaii.

46 TASK OF PSYCHOLOGY

adults. The gradual extension of this region is one of the most
important aspects of child development.

Between the gifted and the less gifted child there are great
differences in this respect. For the mentally retarded child
who continually meets failure and who is stopped by barriers
which other children of his own age are able to pass, the space
of free movement is far more limited (Fig. 2 a) than for the
superior child (Fig. 26). The enlargement of his space of free
movement is further hampered by the fact that through fear
of failure, he holds back from attempting to enter regions
which have heretofore been closed to him. Since his space
of free movement is thus so greatly limited one has to be
especially careful not to restrict it further by unnecessary
prohibition.

Example 4. The space of movement in the social field is
similarly determined by the region of the prohibited and the
region of the not possible. There is a difference in freedom of
movement between different classes and conditions even where
they are legally equal. The rich man generally has much
greater freedom of movement because of his means. He can
take a special train or an airplane in order to reach his destina-
tion quickly. The poor man may have legally exactly the same
rights as the wealthy one, but what is much more important
for him is the fact that his social dependence and the task of
supplying himself with the immediate necessities of lite, such
as his daily food, narrow down his freedom of movement to a
much greater extent. 1 However one must not forget that
we are dealing with freedom of movement in psychological
space and that in certain situations the freedom of movement
can be exceedingly small for any man. In any case the differ-
ence in space of free movement between persons of different
social classes leads to important differences in behavior.

One of the most important goals of domestic and foreign
politics is to change the space of movement of a single person

1 Hans Fallada, in Little Man What Now? (24), describes very convincingly
these dependencies and how they practically destroy a man's freedom of
movement

LIFE SPACE IN THE SENSE OF MATHEMATICS 47

or of a group. At the same time it is one of the essential
means of reaching a political goal. Political struggles as well
as struggles between individuals are nearly always struggles
over the boundary of the space of free movement.

Example 5 (from a motion picture). A mother has taken
a year-old child away from play and wants to feed him on her
lap. He does not want to eat. He is at the moment dominated
by the tendency "away from eating" or " toward play."
The mother holds the child on her lap and prevents the intended
movement "away from eating." She puts her arm around Mm
so that he cannot break away. The mother's interference has

FIG. 3. Topology of an eating situation: a child is prohibited from leaving
for play. C, child; b, barrier (mother's interference); $, region of eating; sp,
spoon; $1, region of play.

in this case the character of a barrier (b, Fig. 3) between the
region of eating (e) and that of play (pf). This barrier at the
same time keeps the child (C) from pulling away from the spoon
(sp) as it is brought near his mouth. The child now begins to
play on the mother's lap. The mother tries to put an end
even to this possibility of action and limits the child's space of
free movement still further. Thereupon the child tries to
widen his region of free movement and begins to struggle with
the mother.

Locomotion; Forces. While locomotion is not the only
change in the psychological life space, it is one of the most
important of them. There are very different kinds of locomo-
tion. The person himself can move about. He can approach
a goal or flee from another person or from an event. This
movement can take place in a direct path or by way of a detour,
The movement can be free and open or secret, reluctant, and
timid* It can take place rapidly or slowly. All this is true

TASK OF PSYCHOLOGY

of bodily movement in quasi-physical or quasi-social fields. At
the same time there is real movement in the psychological
field that one cannot call bodily movement.

Example 6. The vocational goal of a sixteen-year-old boy
(P) is to become a physician (Fig. 4). The "path" to this goal
(G) leads through definite stages: college-entrance examinations
(ce), college (c), medical school (m), internship (i\ establishing
a practice (pr). The boy may have a fairly clear idea of college.
Medical school and the following stages may constitute a more
or less undifferentiated region "beyond" which lies the goal of
being a physician. Of this the boy may have a false but
nevertheless a clear picture.

FIG. 4. Situation of a boy who wants to become a physician. P, person;
G, goal; ce, college entrance examinations; c, college; m, medical school; i,
internship; pr, establishing a practice.

When he passes his college entrance examinations he has
made a "step forward" on the way to his goal. This move-
ment is certainly not a bodily one. Nevertheless it is real
locomotion, a real change of position in the quasi-social (and
as a matter of fact also in the objective social) life space. The
examinations have brought him a step closer to his goal. The
reality of the change in his position becomes clear when one
considers that many things are now within his reach which
were not before. He can go to college or university, his time
is much more within his own control than before. His social
position too is changed: he can play on the college football
team, go to the dances, etc. His examinations therefore had
for him the character of a boundary between two distinct
regions. He had to cross this boundary if he wished to go
from the one region to the other.

Had he failed in his examinations, then he would not have
made this advance toward his goal. But also in that case

LIFE SPACE IN THE SENSE OF MATHEMATICS 49

there would have been a real change in his life situation. The
failure would have changed the barrier between him and the
region of college, which was shortly before in his immediate
neighborhood. The barrier would seem much more solid,
almost impassable. The youth would be thrown back and
possibly would seek an entirely new goal.

To "come closer" to another person through a conversation
is another case of social locomotion which, although it involves
no physical movement, is psychologically real; it is the same
when one withdraws from a person or a group by a particular
action, or when one evades a social difficulty.

There can also be psychologically real locomotion in quasi-
conceptual fields. One begins a mathematical problem. At
first it presents itself as an unclear, unstructured region.
Finally one may be fortunate enough to approach the solution
of the problem by a particular path. This progress in
thinking is a psychologically real event. The same path can
be taken several times and such repetitions are different from
giving up one path and trying another.

By observing the psychological locomotions one finds that
there are not only thing-like objects in the life space, but also
"spaces between them," i.e., regions which are empty or
pervious in the sense that one can move through them or along
them as though they were passageways or roads. The dynamic
characteristics of regions in respect to possible locomotion
especially the resistance they offer are psychologically very
important.

Not only the person himself can move about within the
psychological life space but also many of the quasi-physical,
quasi-social, and quasi-conceptual objects, for instance, other
persons, animals, objects of all kinds. A social group may
move, the power field of a person may enlarge. Whole regions
can gradually or suddenly assume new positions in the life
space. It often happens that someone, without really moving
himself, is carried by a social movement to a certain place, or
that he is pushed farther and farther from his goal against his
own efforts. In another place we will describe a case in which a

50 TASK OF PSYCHOLOGY

failure causes such a "shifting of the ground under the feet."
Further, it is very important whether the life space as a whole
is relatively stable, or whether it is labile and shifting.

It is a characteristic of many psychological locomotions
that they are "directed" toward a certain goal or away from a
certain region. If such a locomotion is stopped by an impass-
able barrier there can exist a tendency or ; as we can also say, a
"force" in the direction toward this goal. In some cases this
direction is clear and definite and can easily be represented.
For instance, in our examples, the boy has a strong urge to
take the examinations; for the child there is a force away from
the lap of the mother; the prisoner has a strong tendency to
leave the prison. In other cases it is more difficult to deter-
mine unambiguously the direction of the psychological forces.
A conflict may drive a person in two or more different directions
at the same time. There may be a general state of tension
although the directions of the tendencies are not clearly evident.

Regions within the Person. The use of space-like con-
cepts has likewise proved unavoidable for the psychological
representation of the person. Dynamically the person appears
as a "stratified" system which has a definite structure and in
which one can distinguish central and peripheral regions. It
has been shown that it is of great significance for problems of
decision and intention, for questions of memory, of psycho-
logical satiation, of substitute satisfaction, and of emotion,
whether the corresponding processes belong more to "periph-
eral" or to "central" regions. Furthermore it is important
whether certain systems and regions within the person are
"connected" or "separated." It is also relevant whether a
system develops "within" this or that region of the person, and
in what way the single systems go together as "parts" of a
larger system. It is important whether a larger system, for
instance the whole person, is strongly or less strongly differ-
entiated into part systems. Changes in the connectedness
of the systems within the person are real and demonstrable.
They have, as experimental investigations have shown (92,
p. 64), very important consequences.

LIFE SPACE IN THE SENSE OF MATHEMATICS 51

Summary. In attempting to characterize the psychological
life space and events that occur in it one finds again and again
relationships of a specific spacial character. 1

This is especially true of the fundamental conception of the
life space which implies a person "in" a surrounding field. It
is true in the same way of the relationship between the parts
of the environment.

The representation of the life space has to indicate the "posi-
tion" of persons and objects in certain "regions." It has to
take into account locomotions of a quasi-physical, quasi-social,
and quasi-conceptual nature; neighborhood relationships of
regions; boundaries; approaches and withdrawals; expansion
and contraction; and finally movements and forces in certain
directions. Whatever the life space "is," whatever the psycho-
logical facts within it "are," and whatever the regions "are"
of which both person and environment consist, certainly one
of the most important relations between the parts of the life
space is that they exist side by side. The spacial character of
these relationships is further strengthened by the fact that we
have to deal with a coexisting manifold. The question is
whether these are genuine spacial relationships.

We have to meet a great number of objections if we answer
this question in the affirmative. These objections have special
weight in that they are partly intrinsically, partly historically
connected with many philosophical problems. Obviously the
quasi-social locomotions, distances, directions, and regions, in
the psychological life space are not defined by the space with
which physics has to deal. But physical space is the only
empirical space about which one is accustomed to speak in a
scientific, not a metaphorical way. The final criterion of the
applicability of spacial concepts in psychology is their fruit-
fulness for research. Nevertheless it remains necessary to

1 That something spacial can appear as the " content " of a perception, of an
image, or of a thought is well known, and space perception is one of the most
highly developed fields of psychology. It is but a step from this to speak in
similar fashion of the perception of a social space, of space of action, or of experi-
ences of spacial structures in general. It is hardly necessary to explain that in
this book we will not discuss the experience of space.

52 TASK OF PSYCHOLOGY

dear up these philosophical objections. A few points may be
taken up briefly.

Certainly the fact that language employs many metaphors
which imply spatial concepts to describe the behavior and
fortune of a person (for instance, "He has come down in the
world") cannot be taken as proof. Although everyday speech
contains important suggestions for the psychologist, such meta-
phors might rather arouse mistrust against the application of
spacial concepts in psychology.

THE MATHEMATICAL CONCEPT OF SPACE

In order to settle the question of whether the facts that
exist psychologically, i.e., those which have psychological
effects, show spacial relationships one has to go back to the
mathematical concept of space. The person who is not a
mathematician as a rule thinks of space as physical space. He
pictures mathematical space as a geometrical structure extend-
ing in every direction without limits, a structure which can be
visualized to a certain degree and within which one can deter-
mine direction and distance exactly by measurement.

Mathematics, in the course of its long history, has developed
a concept of space which is in no way limited to physical space
or to space which can be visualized. It has progressed from
the investigation of three-dimensional to that of w-dimensional
space, from Euclidian space to curved space, from metrical
to non-metrical space.

The nature of the things whose system constitutes a mathe-
matical space is entirely irrelevant for modern mathematics.
It does not matter whether one thinks of them as physical
objects, temperatures, numbers, colors, events, or anything
else. Only certain relationships and the possibility of certain
operations are relevant. It is these which finally define space
(66, pp. is/.)-

As far as mathematics is concerned there is therefore no
fundamental objection to applying the mathematical concept
of space to psychological facts. The crucial point is whether

LIFE SPACE IN THE SENSE OF MATHEMATICS 53

the relationships that characterize space in mathematics can be
applied adequately to psychological facts, and whether one
can coordinate psychological processes univocally to mathe-
matical operations.

In order to avoid misunderstanding in terminology it should be noted
that one does not think of mathematical space as necessarily unlimited in
all directions. One also designates as space a series of numbers, a sphere,
or a finite line of any curvature.

Topological Space. The mathematical concept of space
can be developed from different fundamental relationships
(31, pp. 2iijf.). From the point of view of psychology it is
especially interesting that one can use the part-whole relation-
ship as the basic one. This basic theorem states that for any
two objects, U and F, of a system for which certain conditions
hold, the following relationship shall or shall not be valid: U is
a part of V (that is equivalent to "F includes U"). By
means of certain monotonous series of inclusion one can charac-
terize the concept of a "point" 1 and further the concept of
"surrounding. 77

Space thus defined is called topological space. By this term
is meant that we are dealing with mathematical relationships ;
which can be characterized without measurement. No dis-
tances are defined in topological space. A drop of water and
the earth are, from a topological point of view, fully equivalent.
A cube and a sphere also are not distinguishable (27). Never-
theless these nonmetrical spaces exhibit important character-
istics which are fundamental also for metrical space. There
is a highly developed branch of mathematics which has grown
up around the concept of connectedness. It deals with sepa-
rated and connected spaces, with the different kinds of con-
nectedness, with the relationship of part sets in different
regions, with boundaries, with cuts, etc. Problems of dimen-
sion can also be treated on the basis of topological concepts
without recourse to metrical properties.

1 The term "point" in these cases is not restricted to point in the usual sense.

54 TASK OF PSYCHOLOGY

It is now generally recognized that the whole-part relation-
ship, and the relationships of the parts to each other play a
fundamental role in psychology. This is true for all branches of
psychology. The concept of connectedness, for instance the
distinction between separate and connected regions, the distinc-
tion between different groupings of regions, is as we have shown
above of prime importance for characterizing both the person
and the psychological environment. Changes of connection,
especially the uniting and separating of regions, are just as
important for the psychology of perception as for the psychology
of intention, satisfaction, or friendship. The basic idea of a
person in an environment is in its conceptual content a state-
ment of a certain topological relationship between two
regions.

Certain binary relationships, i.e., relationships between two
points of topological space (31, p. 210), play an essential role
in topology. In this place we must call attention to the
topological concept of "path" that connects two points. How
parts of a space are connected can be determined to a large
extent by the possibility of such paths and the fact that the
path does or does not intersect certain boundaries. We will
see that the concept of path plays a fundamental role in the
constitution of psychological spaces. One can coordinate
certain psychological facts which have the function of a psycho-
logical connection between two psychological " points" to a
"path" which mathematically connects two points. For
Instance, any kind of locomotion of the person in the quasi-
physical, the quasi-social, or the quasi-conceptual field can be
designated as a connecting process which corresponds to a
topological path.

In addition to the locomotions of the person or other parts
of the psychological space we must call attention to that type
of real connection which one can call "dynamic communica-
tion/' The fact that certain regions in the psychological
environment and within the person influence other regions,
both of the environment and of the person, may be taken as a
criterion for connectedness in the topological sense.

LIFE SPACE IN THE SENSE OF MATHEMATICS 55

In later chapters we will show how psychological and mathe-
matical facts can be coordinated in detail. I hope that it is
now clear that from the point of view of mathematics there is no
reason why these concepts should not be applied to psycho-
logical problems and that psychology has already used, and
cannot avoid using, concepts which are of a topological nature.
It will be necessary to clarify these concepts and to apply them
strictly in order to make them fruitful for the whole field of
psychological dynamics.

Metrical Space* One can ask whether it is possible to go
beyond topological determinations and apply metrical con-
cepts to psychological regions. Metrical space is characterized
by the fact that one can coordinate to any two points, x, y,
of the space a distance xy ^ o for which certain axioms hold,
especially the triangle axiom (31, p. 211). One distinguishes
Euclidian and non-Euclidian space according to the particular
kind of metrics employed. Common to both is the fact that
there are differences in magnitude and that one can use
measurement and numbers in characterizing spacial
relationships.

Certainly at the present time there are no metrical determina-
tions available concerning psychological life space. It is clear
that a double distance in physical space generally does not
correspond to a double distance in psychological space. There
is no doubt that very different quasi-physical distances often
correspond to equal physical distances. A special investigation
is necessary to determine whether the triangle axiom whose
applicability is the premise for all measurement of distance
holds also for quasi-physical fields. It might seem to be impos-
sible to measure in the exact meaning of the word quasi-social
or quasi-conceptual distances. The same difficulties appear
with the problem of magnitude of layers or regions of the
person, or the distance between them.

Not only the concept of distance but also that of direction
goes beyond purely topological determinations. Mathemati-
cally there is a dose relationship between measurement of
angles and measurement of distances. As a matter of fact

56 TASK OF PSYCHOLOGY

the determination of directions in the psychological life space
is as difficult as that of distance.

On the other hand, the history of psychology seems to show
that it is impossible to represent psychological causal inter-
connections without making use of the concept of direction.
It is presupposed in the concept of "direction of a movement"
as well as in the concepts of " tendency" or "force." The
concept of "difference of direction" or of "opposing direction"
can hardly be avoided, for instance in the theory of conflict

(60, pp. 338-339; 54).

Neither can we ignore the fact that in certain cases one can
speak definitely about direction, or at least changes in direction.
This is true not only for the quasi-physical, but also for the
quasi-social or quasi-conceptual fields. The same is true of the
determination of distance. Despite all obvious objections
to any exact measurement in the psychological life space it
seems to be possible sometimes to make definite statements
about distance or at least changes of distance. Such a change
of distance takes place, for instance, when one approaches his
goal in the quasi-social space (see Example 6, p. 48). One
can even compare regions within the life space with each other
as to their magnitude, for instance, regions of free movement
(see Examples i to 4, pp. 42-46).

We shall have to discuss the difficult group of metrical prob-
lems after we have dealt with the topological questions in
detail. We begin with topological problems because they are
the more fundamental both from a mathematical and from a
psychological point of view.

THE APPLICATION or THE CONCEPT OF SPACE AND
PHYSICALISM

The objection has been raised against our representation of
the psychological life space by means of topology that it implies
an application of physical concepts to psychology.

In another place we shall have to take up briefly the rela-
tionship between psychology and physics. Here we want to
emphasize only that the application of topological concepts in

LIFE SPACE IN THE SENSE OF MATHEMATICS 57

psychology is entirely independent of this question. The
objection is based on a misunderstanding of the position of
mathematical concepts in the concept-formation of the single
sciences.

The historical development of sciences brought it about that
the most broad and many-sided applications of mathematics
are in physics. The bond between mathematical and physical
thinking has become so close that in many cases one has lost
the feeling for the difference between the two kinds of concepts.
Nevertheless, there can be no doubt that in physics we are
dealing not with pure mathematics but with applied mathe-
matics (7, pp. 34-35). The more recent developments in
physics, especially the theory of relativity, have again brought
to the fore the importance of this fact.

Not so long ago it was thought that even the application of
numbers to psychology was an unjustifiable application of
physical concepts. As a matter of fact number is employed
not only in physics but also in economics and in history. At
present there is no doubt that numbers are mathematical
tools which may be applied in psychology like "logical"
categories. We do not need to discuss the relationship between
logic and mathematics in this place. Just as concepts of logic
can be applied in different empirical sciences so can those of
mathematics. The application of the same numbers does not
imply a carrying over of concepts of economics into history,
or of physical concepts into economics.

Like numbers, topological concepts are mathematical, not
physical in nature. Wherever one has used them, or will use
them in physics or in other sciences, it is a matter of applying
them in different and equally justifiable ways. The same is
true of metrical spaces, i.e., of the concepts of distance and
direction. In these cases also we are dealing with mathemat-
ical-logical concepts. *And modern mathematics does not
cease to insist that these concepts do not imply that the
elements of the systems about whose relationships statements
are made are physical or are in any way determined in regard
to their content.* This is true also of the concept of "directed

58 TASK OF PSYCHOLOGY

magnitude," i.e., vector, which we will apply in representing
certain psychological facts. Like other mathematical concepts
the concept of vector can be used to represent facts of very
different content. It is not to be confused with the concept of
force, certainly not with the concept of force in physics. To
use spatial concepts in psychology means to treat psychology
mathematically, but does not imply physicalism.

CHAPTER VII

PSYCHOLOGICAL SPACE AND PSYCHOLOGICAL
DYNAMICS

PROBLEMS OF PURE MATHEMATICS AND PROBLEMS OF
COORDINATION

The fact that we have to deal with applied mathematics in
the empirical sciences means that the question of which mathe-
matical concepts we have to use for the representation depends
in each case upon the characteristics of the special empirical
facts. We are not dealing with questions of pure mathematics.
Our task is to represent certain empirical data adequately. The
mathematical concepts which are used are very different for
the treatment of different problems even within one science,
and it may be possible that mathematical concepts are adequate
for the whole field of one science which cannot be applied at all
in another. For instance, it may be that to represent the
psychological life space one has to use mathematical space
concepts which are entirely different from those now used to
represent physical space.

The application of mathematics to psychology involves two
kinds of questions. It is important that we be dear in regard
to the fundamental difference between them. According to
Reichenbach (73, p, 5) we can designate the one kind as prob-
lems of coordination, the other we shall call problems of pure
mathematics. 1 The problem of coordination is essentially an
empirical problem in that its task is to demonstrate the nature
of certain empirical facts, in our case facts of psychological
dynamics, and to coordinate to them mathematical concepts

1 Blumberg and Feigl (40, pp. 289^.) survey the philosophical problems
connected with the concept of coordination. They contrast purely formal or
implicit definitions with application^ (coordinating) definitions.

60 TASK OF PSYCHOLOGY

which represent the logical structure of these empirical relation-
ships adequately. Of course a correct coordination presupposes
a sufficient knowledge of these mathematical-logical concepts
to which one coordinates the empirical facts. That requires
an orientation in regard to the purely mathematical problems
involved. In some cases one may be faced with mathematical
questions which mathematics itself has not yet treated.

In actual research, problems of coordination and purely
mathematical problems are so closely connected that it is not
always easy to distinguish between them. Nevertheless, as
the history of physics shows, the greatest possible accuracy in
this respect is important for the progress of science.

THE INSTABILITY OF PSYCHOLOGICAL SITUATIONS

The instability of psychological situations has often been
emphasized. One might easily think that it constitutes a
difficulty for the representation of the psychological life space
by means of mathematical concepts. It is quite true that the
life space or parts of it often suffer strong and sudden changes,
but this is equally true of certain physical situations. For the
application of mathematics it is quite irrelevant whether a
given situation represents a section through an event which
occurs with great rapidity or whether one has to deal with a
relatively constant situation. Besides, even psychological
structures can show a surprising degree of constancy that some-
times continues over a period of years.

NECESSARY CONDITIONS FOR THE APPLICATION OF METRICAL
AND TOPOLOGICAL CONCEPTS TO THE LlFE SPACE

More essential is a second question which is related to the
problem of the indeterminacy of psychological events (see
pp. 39j(F.)' As we have mentioned before, one can sometimes
determine direction and distances within the life space, that is,
one can determine quasi-metrical characteristics which go
beyond purely topological ones. In agreement with mathe-
matical usage one could speak here of a "metrisation"

PSYCHOLOGICAL SPACE AND PSYCHOLOGICAL DYNAMICS 61

(Metrisierung) of psychological life space (86; 16, pp. 14^- X
and thereby mean the actual development of psychological
organizations which allow of metrical predication.

On the other hand, it is possible that there are cases in which
the psychological life space does not even show those charac-
teristics which are a condition for the application of topological
concepts. For instance, one could think that the perceptual
field of the newborn child is still so undeveloped that it does not
yet fully allow the application of the most simple concepts of
connectedness or the concept of "parts/' and that there only
gradually takes place what we could call a "topologisation"
of the perceptual field.

A similar development may sometimes be observed with
quasi-conceptual fields of the adult, for instance, when he
starts to treat a very obscure and complicated problem. It
can happen that in a certain part of the life space there occurs
a further structurization, or a destructurization. An undif-
ferentiated region, that is, a region in which it is impossible to
distinguish certain parts, may become gradually more differ-
entiated, so that one can determine parts and subparts. This
makes it possible to make certain topological statements about
these subparts. On the other hand if a region is completely
undifferentiated the psychological conditions are lacking which
would permit topological determinations of its parts.

To what extent topological concepts can be used in repre-
senting the life space depends upon the real properties of the
case at hand. It seems to me that one of the most important
general characteristics of the psychological life space is that
it is not infinitely structured, but that it is always structured
only to a certain degree. The representation of the life space
will have to take this fact into account. It would be incorrect
to presuppose in the analysis smaller subparts than actually
exist. That the topological treatment of the life space is
limited in this way is a fact which is of fundamental importance
from a dynamic as well as from a mathematical point of view.
For the spaces which are familiar to the mathematician can
be subdivided ad infinitwn. We shall come back to this fact

62 TASK OF PSYCHOLOGY

later. For the present discussion of topological questions it
suffices to say that the life space is to a certain extent struc-
tured. At least there is a certain topological structuring of
the environment in nearly all situations with which psychology
deals, and no doubt there is always some structuring of the
person. This supplies the empirical premise for the application
of the topological concepts to the life space.

SPACE AND DYNAMICS. THE HISTORY OF THE CONCEPT
OF SPACE IN PHYSICS AND PSYCHOLOGY

In using mathematical space concepts in psychology one
will have to base these concepts on psychological dynamics from
the beginning. 1 In this respect the history of psychology is
essentially different from that of physics.

In physics there has never been a question as to whether
spacial concepts should be used for the representation of
structures and events. Physical space is given visually. It
was from the beginning an essentially metrical space in which
distance and direction were relatively easy to determine. One
was able to employ fixed measures which were represented
simply enough by solid bodies.

Therefore in the beginning the properties of physical space
were not made dependent on dynamics, i.e., not upon the laws
of the physical processes. For this reason the problems of
measurement in physical space often seemed like problems of
pure mathematics. Only the most recent development in
physics, especially the theory of relativity, has revealed the
direct connection between properties of physical space, espe-
cially its metrical properties, and questions of physical dynam-
ics. It has shown the importance of the velocity of light and of
other physical facts for measurement. It has further revealed
the connection between determinations of time and of space,
and it has demonstrated that even if we use solid measuring
rods the nature of the physical processes themselves must be
taken into account. The determination of physical space
therefore, in the last analysis, goes back to physical causal

1 The same is true of the use of spacial concepts in sociology.

PSYCHOLOGICAL SPACE AND PSYCHOLOGICAL DYNAMICS 63

chains (Genesereihen) . Which particular mathematical space
is to be used for the representation of physical structures
depends, as modern physics has shown, on the special laws of
physical dynamics (73). Nevertheless, physics was able to
advance very far in investigating the physical world without
having to consider the relationship between the measurement
of physical space and the physical laws.

Psychology, in this regard, is in a different situation. The
relations of "togetherness" in the psychological life space are
given only to a limited extent. As far as the quasi-physical
facts of the environment are concerned the spacial relations
can to a certain degree often be experienced directly. This is
less the case with quasi-social facts and still less so with regions
within the person. We do not have to discuss at this time
whether this difference between physics and psychology is a
fundamental one or only one of degree. In any case psychol-
ogy, in determining the life space metrically or topologically, is
not in possession of tools which in their immediacy and sim-
plicity are comparable to the solid measuring rods of physics.
One can represent the order of togetherness of the facts which
exist for psychology only if one realizes from the beginning
that the determination of spacial relations has to be based
upon psychological processes and depends therefore on the
nature and the laws of psychological dynamics. For this reason
psychology is already being faced with questions which in some
respects are as "general" as those which recent years have
brought to the fore in regard to physical space.

THE FUISTDAMENTAL CONCEPTS or DYNAMICS

That the spacial structure of the life space depends on psycho-
logical dynamics itself shows the importance of the dynamical
concepts. These are such concepts as cause of change, tend-
ency, resistance, solidity, equilibrium, force, tension, etc. At
present it does not seem to be fruitful to discuss the difficult
general problem of which concepts one has to use to represent
the dynamics of a certain science or according to what criteria
one must select them. I shall limit myself to a few remarks

64 TASK OF PSYCHOLOGY

that ought to help forestall the misunderstandings which seem
most likely to occur in regard to these questions.

1. If we want to derive actual psychological events con-
ceptually, we have to try to represent not only the spacial
relationships but also the dynamic ones in a mathematical way.
For instance, we shall often have to substitute the concept of
"force/' which is clearer and which can be represented mathe-
matically, for the more nebulous concept of tendency.

Nevertheless, it is important to keep in mind that the
dynamical concept and its mathematical representation are not
identical. The same mathematical concept can be used for the
representation of different facts. For instance, a vector can rep-
resent not only forces but also movements. It will be necessary
to distinguish between the formal, mathematical properties of
the dynamic concepts and their content, which is determined by
the psychologically real events to which they are coordinated.

2. It must not be assumed here that all empirical sciences
should use the same fundamental concepts for the representa-
tion of dynamics. Especially it must not be assumed that
psychology has to use the same dynamic concepts as physics.
Only an investigation of psychological dynamics itself can
decide which concepts are adequate for it.

3. As far as we can see at the present time psychology will
have to use dynamic concepts that in some respects show a
formal similarity to the concepts of physics, but which are not
identical with them.

For instance, both sciences can use the concept of "equi-
librium." This implies that one also assumes a concept of
force, since there is a dose connection between the concept
of equilibrium and that of force in so far as equilibrium is a
special case of a constellation of forces. As a matter of fact,
as we shall see, the concept of force and a number of related
concepts can be used in psychology in an exact way. Kohler's
argumentation (43) has made it sufficiently dear that the same
concept of " dynamic whole" or "gestalt" can be used in
physics and in psychobiology; and also that the fundamental
gestalt laws are equally valid for both sdences.

PSYCHOLOGICAL SPACE AND PSYCHOLOGICAL DYNAMICS 65

When we use equivalent concepts of this sort we are not
trying to derive psychological from physical concepts. The
content of these dynamic concepts is determined only by
coordinating them with real psychological or real physical
processes. In so far as one coordinates these concepts to
different real processes, in the one case to psychological proc-
esses, in the other to physical ones, these conceptual fields
remain separate from the point of view of theory of science.
When for instance, in the following we speak of forces in con-
nection with the quasi-social or quasi-conceptual facts in the
life space we define the forces as psychobiological and not as
physical. The application of formally equal dynamical con-
cepts in psychology and physics does not mean a regression to
physics any more than the application of the same mathematical
concepts. In both cases we use processes which are different
in content for the definition of dynamical concepts which are
equivalent in form.

Besides, this formal correspondence between the dynamical
concepts of physics and those of psychology seems to hold only
within limits. For instance the laws which pertain to forces
are probably essentially different in form for the two sciences.

4. In spite of this lack of complete correspondence we have
often chosen terms that are also used in physics to designate
our psychological dynamical concepts. The use of identical
terms for the two concepts rather points to their functional
equivalence within the system of dynamic concepts in physics
and psychology, than to a complete identity in a formal mathe-
matical sense. Yet at the same time I shall try to avoid
using identical terms except when the concepts are at least
similar in form. It would have been easier to decide whether
to introduce a completely new terminology into psychology or
whether to make use of existing terms if we could have foreseen
exactly how far the formal similarity between the concepts
would hold. But this we can learn only by an investigation
of psychological dynamics itself. Actually there is probably
not much danger of misunderstanding as long as one holds to
the definitions as we shall give them.

CHAPTER VIII

THE PSYCHOLOGICAL WORLDS AND THE PHYSICAL

WORLD

If one is accustomed to speak of psychological facts as some-
thing essentially nonspacial, one thinks first not of mathematical
but of physical space, i.e., of the togetherness of the facts
that are real in the sense of physics. An essential character-
istic of this physical space is that it is thought of as a single
coherent space which includes the totality of all physical facts
that exist at a certain time 1 (the whole physical world) and
which includes only physical facts. The facts of psychology,
i.e., these facts which psychology must recognize as real, have,
according to the teaching of physics, no place within physical
space. This is equally true of economic or aesthetic facts.

PHYSICAL AND PSYCHOLOGICAL SPACE

The thesis that psychological facts, such as dreams and
wishes, have no place in physical space has long been accepted
as self-evident by the physicist. This thesis is also generally
accepted in psychology and constitutes the main reason for the
assumption that psychological facts are in general of a non-
spacial nature. Nevertheless attempts are continually being
made in psychology to "coordinate" dynamic facts in psychol-
ogy to physical facts.

It would lead too far to enter into the tangle of single hetero-
geneous tendencies of different value that are presented by the
followers and the opponents of this procedure. It seems to me
^that one can solve this complex of related problems only if one
attacks it from the point of view of the theory of science. This
means that this problem must not be separated from the more

1 We can disregard the complications that the theory of relativity introduces
into this concept.

TEE PSYCHOLOGICAL WORLDS AND THE PHYSICAL WORLD 67

general problem of the relation of the different sciences to
each other, i.e., that instead of contrasting "physical" and
"mental" one has to think in terms of the difference between
physics and psychology, or of the difference between physics
and biology (53).

Then two oppositions result: (a) A distinction within
biology between bodily and mental; (Z>) a much more funda-
mental distinction between "object of physics "" and "object of
biology" (including psychology). The main reason for many
confusions is that the term "physiological" is used in two
different ways; sometimes it is used to designate a certain class
of biological events, viz., "bodily" processes in opposition to
psychological ones and sometimes to designate "object of
physics." The difference between the complex of "physical"
questions on the one hand and "psychological-physiological"
problems on the other is based on the difference in the way the
temporally distinguished units of the existential series (Genese-
reihen) (53) are related to each other. From this difference
in the temporal order follows the difference in the spacial order
in physics and psychology.

We must emphasize that not only psychological facts but
also the "bodily "-biological facts do not belong to the physical
space. This makes it clear that in the following Vhen we speak
of psychological regions, forces, or changes, we are not dealing
with figments of the imagination but with facts which have the
same reality and kind of existence as biological facts in general.

THE SINGLENESS or THE PHYSICAL WORLD AND THE
PLURALITY OF THE PSYCHOLOGICAL WORLDS

Despite the changes in the concept of physical space during
the recent development of physics one has always considered
physical space as one single connected space that includes all
existing physical realities. Psychology, on the other hand,
does not deal with one single connected space of psycholog-
ically real facts. Especially when one takes "effects" as the
basis for the construction of psychological spaces, one will
have to think of each individual's life space as an entirely

68 TASK OF PSYCHOLOGY

separate world. A wish which plays an essential part in the
life space of the individual A may possess no psychological
reality for the individual B.

From the point of view of dynamics the life space of each
single individual is a totality which is equivalent to the totality
of the whole physical world.

Naturally, not only another individual B but also thoughts of other
individuals can have effects in the life space of a person A. But also in
this respect the principle which we discussed above in regard to quasi-
physical, quasi-social, and quasi-conceptual facts is valid, namely, that
one has to take into account only those processes which exist psycho-
logically for the person A.

The older psychology of expression upheld the principle that the mental
processes of another person could only be inferred. This principle as such
is certainly untenable. It is probably based in part on a misapplication
of the principle which in itself is correct, that from the point of view of
psychology the life spaces of different persons are to be treated as separate
worlds which can only affect each other indirectly. (See pp. 6ojf.)

In this respect there is a sharp distinction between the con-
cepts of sociology and those of psychology. Sociology is not
obliged to consider only facts which belong to the life space of
one single individual as being in direct dynamic communica-
tion with each other. For instance, the dynamics of a group of
persons can represent a single system for sociology.

In any case psychology does not deal with one single con-
nected space which represents the totality of its world, as does
physics. Instead its subject matter is a plurality of separate
spaces, i.e., a great number of totalities each of which corre-
sponds to a single man or animal.

THE PHYSICAL WORLD AS A DYNAMICALLY CLOSED 1 UNITY

AND THE PSYCHOLOGICAL WORLDS AS DYNAMICALLY

UNCLOSED UNITIES

Physics treats its space as a closed system of causes and effects
in the following sense: All physical changes are the result of con-

1 The term "dynamically closed" should not be identified with the topological
term "closed." It is irrelevant for our problem whether physical space is
"closed 31 in the topological sense of the word (see p. 89).

TEE PSYCHOLOGICAL WORLDS AND THE PHYSICAL WORLD 69

ditions or changes within the same physical space. According
to physics there are no influences on this space from outside.

In psychology also there is a more or less close dynamic
connection between all the facts which belong to the same
psychological space. The psychological events are determined
by the life space according to the formula B = /(S). So far
therefore, as in physics, a change is the result of conditions or
events within the same space.

Nevertheless, the matter is not so simple in psychology.
That becomes clear if one asks, in connection with the question
of "historical causality" (see pp. 30$), how the situation S
came into being. Physics answers this question by proceeding
from a temporally later situation, 2, to a preceding situation,
Si> and shows how 2 came into being through events which
resulted from situation Si. It can be impossible to derive situ-
ation S 2 from the situation Si because the situation Si which
one chose was not sufficiently comprehensive and therefore
certain facts were neglected which in the interval influenced the
system from the outside. However, theoretically, it is always
possible in physics to choose an Si so comprehensive that S2 can
be derived from it. The facts of Si always exist physically
and thereby have a place in physical space, 1 In this sense there
is, in physics, no effect from outside on physical space.

In psychology one tries in similar fashion, to derive situation
S2 from a preceding situation Si. As a matter of fact that is
sometimes possible. A person A may, in a situation Si, have
intended to perform a certain action. The event BI which
results according to psychological laws from the intention and
the other characteristics of Si may make it possible to determine
sufficiently the later situation S 2 . That means that in this
case we have derived the condition of a life space from that
of the same life space at an earlier moment of time. In so far
the analogy to the physical world is complete.

1 We are here dealing with the fact that the physical causal series (Genese-
reihen) can always be traced back in time. That means that there is always an
object Ot~ n which from the point of view of physics is completely "gen-identical"
with the object Ot- Cf. K. Lewin, (53, p. 49).

70 TASK OF PSYCHOLOGY

But it can also happen that during the event BI, which results
from situation Si according to psychological laws, certain
events enter which themselves cannot be deduced from situa-
tion Si. Thus, while a person A is busy writing a letter the
door may open and an outside person enter unexpectedly.
Or, while someone is marking off the place where his house is
to be built in a field, a herd of cattle may pass and destroy the
lines. Or, when a man is building a fence, there is a slide and
the situation is changed entirely. The influence of such events
on the behavior of A may again be determined by psychological
laws. Nevertheless, in these cases we are dealing with an
influence from outside in so far as at least part of the events
which lead up to situation S% cannot be derived from the
situation Si. Such a derivation would not be possible even
if all the psychobiological laws were known.

Here we are faced with one of the principal differences
between physical and psychological space. In physics it is
theoretically always possible to make Si more comprehensive
when it does not suffice for the derivation of 2. In psychology
such an expansion is often not possible. If one tries to make a
corresponding enlargement and for instance includes the
approach of the outside person to the door of the letter writer A
in A's life space at the time i, one makes a mistake. For at
that time the stranger clearly does not belong to the life space
of A because the behavior of A would have been different if
he had known of the approach of the stranger. We cannot
avoid recognizing that there are such influences from outside
on the psychological life space. That means that there are
changes which cannot be derived from the dynamics of the
psychological life space even if one assumes strict determinism
in psychology and if one has a complete knowledge of the
previous situation and of all psychological laws. These changes
can only be thought of as influences on the psychological life
space which are "alien to psychology."

I do not use the expression "alien to psychology" to indicate physio-
logical or other bodily influences which, as we have seen, have to be
included in the system of psychobiological laws. Instead I use it to

THE PSYCHOLOGICAL WORLDS AND THE PHYSICAL WORLD 71

designate such influences on a situation as cannot be derived from the
psychobiological properties of the preceding situation.

Such alien influences occur frequently. The field of percep-
tion and action can be changed for instance by the fact that an
object is suddenly set in motion by physical causes, that another
person encroaches, that the telephone rings, etc. These
influences from outside can have a definitely social character.
The announcement of new regulations for peddlers, about
unemployment relief, about taxes, can entirely change the
field of action of a peddler, of an unemployed person, or of the
tax-payer. As mentioned above such influences can occur by
way of perception or as gross somatic influences. In all these
cases we find essential dynamic changes of the life space of an
individual which do not depend on the psychological dynamics
of the life space itself. The single psychobiological worlds
do not therefore represent dynamically closed regions in the
sense indicated above.

If one follows up this problem one sees that almost all proc-
esses which are based primarily on psychobiological dynamics
depend to a certain degree on alien factors. If someone saws
a board his behavior is determined not only by his goal but also
by the nature of the wood and the properties of his saw. The
same is true when a year-old child puts one block on another
and finds that they do not stick together however hard he
presses them; or when someone tries to influence a political
group, or to solve a mathematical problem, and finds that
things do not go as he had expected. If someone throws a ball
at a mark, if he tries to influence another person through
praise or blame, if he goes along a street, or looks around, in
all these cases, the actual effects of the psychological event
depend also on facts which are alien to psychology.

We have, further, to call attention to the connection between
cognition and these alien factors. Cognition has always been
treated as a specific characteristic of mental life although it
has not at all such outstanding importance for inner-psycho-
logical dynamics. This may be one consequence of the fact
that the psychological worlds are not dosed. Perception and

72 TASK OF PSYCHOLOGY

cognition often affect the life space in such a way that the
structure of certain of its parts corresponds in a high degree
to the objective structure of what is perceived (44). How far
and at what points the two structures correspond in concrete
cases is very important for the success of an action, and is
decisive for the value of a plan. The comprehension of the
intrinsic nature of the alien factors, whether of physical rela-
tionships, of mathematical problems, or of social groups, is
therefore of the greatest importance for every achievement.

It is not always easy to decide, in particular cases, what is to be taken
as alien and what as real psychological influences. A child may want to
get a doth to clean up water that the "naughty" doll has spilled. If he
asks an adult where to find the cloth, then the adult's answer represents
an alien factor, i.e., it cannot be derived from the preceding situation in
the life space of the child. But if the child already knows where the cloth
is, one is inclined to speak of a simple psychological relationship. Further,
certainly the kind and order of visual impressions which the child receives
as he walks through the rooms depend upon the arrangement of physical
objects in the rooms, yet when the child is thoroughly familiar with his
environment one does not think of this fact as constituting a break in the
psychological causal chain. For then the change in the visual impressions
is a result of the behavior of a child. Nevertheless, even in this case the
change in the situation is partly determined by alien factors.

The concept of a world which is dynamically not "closed"
but within which there exists nevertheless a strict determinism
offers some difficulty at first, and one might ask whether it is
at all possible to give a conceptually clear definition of such a
space. Closer examination, however, shows that a mathe-
matical representation of such a world is quite possible.

We have to consider two possibilities: The influence "from
outside" can affect the psychological life space at every point
or only in certain regions.

The mathematical representation of the second case is simple.
It means that one has to distinguish within the life space
"inner points" and "boundary points." The life space there-
fore would be a "limited" and "closed" region, i.e., a region
which includes its own boundary. (These mathematical con-
cepts will be explained in Chap. X.) The boundary points

TEE PSYCHOLOGICAL WORLDS AND THE PHYSICAL WORLD 73

would correspond to those zones of the life space that can be
influenced from the outside. This representation would be
correct if for instance all influences on the life space from the
outside were mediated by the surface of the body of the person.
The boundary of the life space would be a simply connected
region which, represented two-dimensionally, would correspond
to Fig. 5. Only psychological laws would then govern the
inner part of the life space. Alien influences would affect only
these boundary points.

The other possibility, viz., that each point of the life space
can be affected by alien influences, can also be represented

Boundary points
"of +he fife space

Life space of the incfiv/olvoi} P

psychologic* 1

PIG. 5. The life space as a "dynamically not closed" world. P, person; ,

environment.

mathematically. It would mean that all points of the life
space are boundary points. This postulate is fulfilled if one
imbeds the space of psychological facts in a space which has
one dimension more. Then, regardless of the number of
dimensions of the psychological life space, every point becomes
a boundary point in relation to the space of more dimensions. 1
Let us assume that the life space could be represented by a
two-dimensional manifold, for instance by a plane. Then the
events that occurred could be derived according to psycho-
logical laws from the structure and the dynamic properties
of the facts represented in this plane. Nevertheless, each point
of the plane would be a boundary point in regard to a three-
dimensional space. This three-dimensional ' ' hull ' J would make
it possible to represent those physical, social, or other facts

which are alien to psychology and which do not influence the

1 To illustrate: Each inner point of a disk is a boundary point in relation to a
three-dimensional space in which the disk lies; each inner point of a line, that is
of a one-dimensional space, is a boundary point in relation to the plane.

74 TASK OF PSYCHOLOGY

life space at the present moment but which can affect it in the
future. Within this hull would hold not psychological, but
other laws, 1 In this case the life space could again be "limited J '
(as in Fig. 5). But it could also correspond to the unlimited
plane. An analogous possibility of a transition to a hull of
more dimensions exists mathematically also in the case in
which the life space itself represents a manifold of three, four,
or more dimensions. 2

In answering the question how the boundary points in the
life space are distributed one will have to take into account
that not only is the life space influenced from the outside, but
that effects can also work in the opposite direction; that is,
behavior can affect those regions which are not subject to
psychological laws-
One must not conclude from the unclosedness of the psycho-
logical worlds that there is no use in speaking of a psycho-
logical or a psychobiological causal relationship. The task of
dynamical psychology is to derive univocally the behavior of a
given individual from the totality of the psychobiological facts
that exist in the life space at a given moment (B = /OS)). To
this also belong all those facts at the boundary points which
influence the person at the present moment but which them-
selves owe their existence partly to alien events. In so far,
therefore, the task of deriving the behavior B from the totality
of S remains unchanged and has not lost anything of its psycho-
logical character. In so far there is no formal difference
between psychology and physics. The difference consists in
the fact that there are no boundary points in the physical
world which depend on nonphysical factors. 3

1 In constructing this non-psychological hull one would probably therefore have
to go backward in time. In that respect this hull would be formally similar to
other fundamental "genese" relationships in biology, for instance pedigree.
Cf. K. Lewin (53, pp. 83-85, 144).

' 2 In mathematics, too, one distinguishes the questions of the inner structure of
an ^-dimensional space from its relation to its n-plus-one-dimensional hull. C/.
K. Menger (66).

* It is irrelevant for this discussion whether this conception of the physical
world is correct.

THE PSYCHOLOGICAL WORLDS AND THE PHYSICAL WORLD 75

It is obvious that psychology must take into account also
physical and social facts which obey non-psychological laws
and which control the events in the " foreign" hull of the life
space. For these facts determine the boundary points of the
life space and are therefore of great importance for all events
in it. Every act of influencing another person, whether in
laboratory experiment or in everyday life, consists in creating
such a hull, one which affects the boundary points of the life
space and thereby the life space itself in a certain way.

To summarize : psychology has to assign a separate space to
each single individual and his own environment. Each such
space corresponds to the totality of a psychobiological world.
(From the point of view of theory of science it is equivalent to
the whole physical world.) These worlds are "dynamically
not closed"; they have boundaries or each of their points
exhibits boundary properties in relation to certain influences
which are alien to psychology.

CHAPTER IX

MATHEMATICAL REPRESENTATION AND
PSYCHOLOGICAL THEORY

In concluding this general part I want to discuss shortly
several points of our procedure which have been misinterpreted.

CONCEPT, PICTURE, AND SYMBOL

Our aim in representing a psychological situation is not to
give a visual picture of it. Illustrations may serve as a pleas-
ant pedagogical device, but this is not of essential importance
for our real task. In recent times one occasionally finds
diagrams used in different ways in psychology.

I select two examples: Charlotte Biihler (12, p. 237) illus-
trates the change in the relation of familiarity to like and
dislike with the diagram which is given in Fig. 6. It is char-

- ... P* ^_ _^--"^1 Unfamiliar

Famtljoir pl e0 isLfre^><^Curios'i^v +

Unfamiliar ple&sure^><Z3joreotorn Fofmilioir

FIG. 6. [From Ch. Biihler (12).]

acteristic of such illustrations that qualitatively determined
classes of objects, events, or behavior are brought into certain
relations to each other. Such, illustrations can of course be
entirely legitimate. But they are graphic illustrations of
abstract relations, not representations of concrete situations.
Further examples of geometrical concepts in psychology that
do not represent spacial relations of the life space are the con-
cept of dimension as used by Boring (5) in referring to dimen-
sions of properties, the concept of ^-manifold as employed by
N. Raschevsky (72) in discussing the biophysics of space and
time, and the concept of vector as used by Thurstone (83) in
connection with factor analysis.

MATHEMATICAL REPRESENTATION

More closely related to ours axe representations of the kind
which Koffka uses to make dear the relation between behavior
and environment. "G is the geographical environment. It
produces BE, the behavioral environment; in this and regulated
by it KB, real behavior takes place, and some parts of it are
revealed in PHB, phenomenal behavior. In some sense BE,
RB and PEE occur within the real organism, RO, but not in
the phenomenal Ego, which belongs to PHB. RO is directly
affected by G and acts back upon it through RB" (47, p. 40)
(see Fig. 7). This illustration is more nearly concerned with
the task of representing the structure of the life space than
that of the previous example. But it is clear that this figure

FIG. 7. [From Koffka (47).]

is meant only as a "picture" of the structure and not as a
mathematical representation in our sense. If this were not
so it would be incorrect to say of the behavioral environment
(BE}: "In this . . . RB, real behavior takes place" and
nevertheless to represent in the figure the behavioral environ-
ment as a one-dimensional limited line which does not enclose
RB. One would have to ask further why behavior, RB, is
represented as a two-dimensional region while the behavioral
environment BE, and the geographical environment G, is
represented as a one-dimensional line.

La the following we are not dealing with the representation
of the dependency of certain classes of events on each other
nor with pedagogical aids to visualization (28, p. 170; 49,
p. 161) but with the conceptual determination of the dynamic
properties of concrete situations. That we generally illustrate
this conceptual representation by a figure i a matter of second-

78 TASK OF PSYCHOLOGY

ary importance. We even have to emphasize that the figures
lead to misunderstanding if a reader interprets them in terms
of the usual metrical geometry instead of topology. The
diagram on the paper is in fact only a picture of certain topo-
logical structures which for their part are to serve as a con-
ceptual representation of psychological facts.

This means that it is not our aim to find arbitrary symbols
for the representation of situations. The mathematical con-
cepts should "picture" the dynamic properties of the situation
only in the sense in which concepts represent facts.

One can distinguish in our figures between the arbitrary symbols (for
instance, that the point of application of a force is indicated by the head
of an arrow; that the strength of a force is indicated by the length of the
arrow; that the strength of a barrier is shown by the thickness of a line)
and the representation of the topological relationships themselves (for
instance, the closedness or not-closedness of a curve).

If one wants to take the term symbol in a broader sense, one
can even consider concepts as symbols for real events. But
that would mean that the use of mathematics in physics is of a
symbolic nature.

The mathematical concepts which we shall use in the follow-
ing for the conceptual representation of psychologically real
facts are to be understood no less strictly than mathematical
representation in physics. Mathematical concepts are dis-
tinguished from other means of representation, such as the
symbols of ordinary speech, in that they belong to a system of
concepts which are related to each other in a univocal way.
The scope and unambiguousness of these relationships are what
makes the coordination of mathematical systems to real facts
so fruitful for investigation and this is no less true for psychology
than for other sciences.

CONCEPT AND MODEL

We have intentionally avoided the use of any model of a
physical or of a rionphysical nature for the explanation of
psychological dynamics.

MATHEMATICAL REPRESENTATION 79

Like an illustration the working out of a model can have a
certain value. On the other hand it can, especially in psychol-
ogy, involve serious dangers: a model usually contains much
that is purely arbitrary. One usually uses it like an illustration
only in so far as the analogy holds, i.e., really only as long as it
is convenient. As soon as consequences ensue which do not
agree with the real facts, one evades the difficulty by asserting
that it is after all only a model or an illustration. One says,
"A comparison is not an equation." How far one uses the
model for explanation and at what point one discards it as no
longer binding is purely arbitrary. In this respect model (7, p.
53) and illustration are sharply distinguished from the mathe-
matical representation which we are trying to attain. If one
decides to represent a real fact by a mathematical concept
then one is forced to acknowledge all the consequences which
are involved in this concept. This certainly makes the task
a difficult one. On the other hand science will obtain the real
benefit of the application of mathematical concepts only if it
uses them in an absolutely binding way.

PHYSIOLOGICAL THEORIES OP PSYCHOLOGICAL PROCESSES

One must welcome every attempt to go beyond vague ideas
to concrete formulations in psychological dynamics. Often
psychologists who take this task seriously have been driven to
attempt to make "physiological" theories. I do not consider
that tendency fortunate. Certainly one cannot object in the
least to applying physiological methods and to including in the
theory material so obtained. This is in line with the general
standpoint of this book and does not need to be especially
emphasized. But the so-called physiological theories which are
based on psychological facts have almost always the character
of a physiological or rather physical model of which the same
is to be said as of other models. With a physiological model
too, the relationships with which one is concerned are not
expressed directly in concepts but only indirectly in illustra-
tions, and these often include superfluous specializations. The

So TASK OF PSYCHOLOGY

task of scientific research is, however, to determine the dynamic
characteristics of the facts themselves.

At present, there can be no more doubt that such a determi-
nation of the dynamic properties of the underlying facts is also
possible by " psychological" technique in the narrower sense,
for instance by psychological experiment. We have to deter-
mine more and more exactly the properties of these dynamic
systems which have full psychobiological reality. It is there-
fore meaningless to duplicate these dynamic systems by coordi-
nating physiological systems to them, since the properties of
the dynamic systems which are known by means of psycho-
logical technique are already the properties of the real psycho-
biological systems themselves. It would not be correct to
apply the concept of isomorphism, which has genuine value in
other connections (45, p. 38), to the relationship between
" psychological" and "physiological" systems. For at the
level of dynamics we are not dealing with a duality but with
identical systems.

Working out a problem in terms of a physiological model
can sometimes have heuristic value. Very often however it
only creates the illusion of a transition to a greater degree of
reality without advancing the conceptual determination of the
dynamic properties. When Kohler wants to characterize
dynamic properties of psychological facts he often does it by
speaking of the properties of the " brain field." For the reasons
given above we do not follow him in this terminology 1 ; but
at the same time we want to call attention to the fact that the
structure of the brain field coincides with the structure of what

1 As a matter of fact it seems to be difficult with such a terminology to avoid
the danger of referring too directly to the most simple fields of physical forces.
Kohler says, "As the distance is enlarged objectively, exactly the same occurs
in the brain field . . . ** and "... physiologically, the increase of distance in
the brain field will correspond exactly to the stress which, as a field of force, was
tending in that same direction" (44, p. 390). To conclude from these proposi-
tions that the strength of a psychological tension is a monotonous function of
the distance between a person and his goal would not agree with the results of
experimental investigations. Besides, the relation between the magnitude of the
tension and the magnitude of the forces, under the conditions in question, is in
physics also not a simple one.

MATHEMATICAL REPRESENTATION Si

we call life space, certainly in its main features. Kohler
distinguishes within the brain field regions which correspond
to the surrounding field and a special region which corresponds
to the person himself. He coordinates to locomotions of the
person changes of the position of this region in the brain field.
Indeed the methods which Kohler uses in determining the
structure and the properties of the brain field agree essentially
with the methods which serve us in determining the structure
and properties of the life space. The agreement is shown
especially in that questions of dynamical connection and of
relations of position play an especially fundamental role. If
one tries to make an explanation in terms of dynamic relations,
one comes to assumptions about the structure and the proper-
ties of the basic facts that seem to be equivalent, at least in
their mathematical form, regardless of the terminology in which
they are expressed. One must always go back to these invari-
ant relations in the last analysis if one wants to derive the
behavior of a person conceptually. It seems to us therefore
that the principal task of any dynamic psychology is to investi-
gate and represent directly these relations.

REPRESENTATION AND EXPLANATION

The objection has often been raised against our representa-
tions of situations that it is self-evident that one can derive
from the represented situations the events one wants to explain.
It is said that our representations do not explain, that they
only describe.

If one wants to understand the meaning of this objection,
one must remember that one has often explained psychological
processes by deriving them from entities "behind " the proc-
esses. (The concept of instinct has often been used in such a
way.) The derivation of events which we give here implies
only one kind of tracing back, namely, the progress from
phenomenal to conditional-genetic characteristics of the objects
and events themselves (60, pp. 318-321). This however seems
to me to be not a defect but rather one of the most important
positive characteristics of the new way of building theories.

82 TASK OF PSYCHOLOGY

The task of dynamic psychology is to find the psycho-
logical laws and to represent the situation in such a way that
the actual events can be derived from it in a conceptually
univocal manner. If the objection is raised that it is self-
evident that the events follow from these representations of
situations, one has to answer that this is exactly our purpose.
What we are trying to do is to represent situations in such a way
that the events follow from them "self-evidently," namely as
purely logical consequences.

If one wants to call this "description/ 5 it is not worth while
to quibble over words. But if one considers conceptual deriva-
tion and the transition from phenomenal to dynamic facts as
the characteristics of an explanation, then what we have here
is in fact explanation. And we would like to add that this
is the only kind of explanation which psychology can acknowl-
edge after the transition from Aristotelian to Galilean ways of
thinking.

These somewhat subjective objections to the new way of
thinking show again how similar, from the point of view of
theory of science, the present state of psychology is to the state
of physics at the time of Galileo and Newton. In that period
of physics one can observe a very similar change in the meaning
of explanation, namely, a repudiation of theories of the older
style which do not seek an explanation in the relationship of
the dynamic facts themselves but in entities which are "behind
these facts" (15; 60).

We hardly need mention that theories and working hypothe-
ses are not lacking in the new procedure. The dynamic struc-
ture of a situation is not an immediately given fact. As we
have said the complete representation of even one given situa-
tion would presuppose the solution of all psychological problems
and the knowledge of all psychological laws. For scientific
research the difficulties begin as soon as one tries to represent a
"given " situation. A complete representation of one situation
would mean that the whole task of psychology is completed.
The representation can be made only step by step and its prog-
ress must be parallel to the investigation of the dynamic laws.

MATHEMATICAL REPRESENTATION 83

The representation of a situation implies no less theory
the laws which it presupposes. It is important that the investi-
gator be fully aware of this. At the new stage of development
the representation of facts is so closely interwoven with the
formulation of laws that one can say, in regard to dynamic
facts, "A correct representation of what 'is' is at the same time
an explanation of what happens. "

We do not want, in this place, to continue the discussion of
methodological questions nor to treat further the properties of
the life space as a whole. For one comes to the answers to
these fundamental questions only as one progresses with the
special problems of psychology. These special problems and
the value and usefulness of mathematical means of representa-
tion are to a certain extent independent of the way in which
one solves the general problems.

PART TWO
TOPOLOGICAL PSYCHOLOGY

A. CONCEPTS OF TOPOLOGY WHICH ARE
FUNDAMENTAL FOR PSYCHOLOGY

There are two groups of concepts which are fundamental
for the representation of psychological situations. They are
intimately connected and make up the framework of the whole
system.

1. Formal mathematical concepts (for example boundary,
region, connection, vector),

2. Psychological-content concepts (for example solidity of
boundaries, fluidity of materials, strength of psychological
forces).

The formal mathematical concepts shall serve as guides in
our presentation. To be sure, one can apply them in psychol-
ogy only if one coordinates them correctly to psychological
contents that are finally defined by observable processes.

Since the necessary mathematical sources are not always
easily available for the psychologist, it might be worth while
to begin with a short survey of these concepts. We have tried
to present these mathematical concepts in a simplified form
which is adapted to the needs of psychology. 1

According to a mathematical point of view, I have arranged
our questions in two groups :

a. Topological problems.

6. Vector problems.

In general one may say that the topological tools allow us to
determine which events are possible in a given life space and
which are not possible. Vector concepts are necessary to
determine further which of the possible events will actually
occur in a given case. Accordingly within general problems
of psychological dynamics we can roughly distinguish between

1 It is not our task to derive these concepts mathematically from each other.
We only want to make them intelligible to psychologists.

86 TOPOLOGICAL PSYCHOLOGY

problems of topological psychology (presented as Part Two of

this book) and of vector psychology.

One may further enumerate the following groups :
. Problems of dimension. (These problems will be treated
with the topological problems.)

d. Inducing fields.

e. Tensions. (Topics d and e will be treated in connection
with vector psychology.)

CHAPTER X

CONCEPTS OF TOPOLOGY FUNDAMENTAL FOR
PSYCHOLOGY

The determination of topological relationships is the funda-
mental task in all psychological problems. Changes of connec-
tion are the most important changes both in the psychological
environment and in the structure of the person. At the same
time the topological relationships are fundamental for the
mathematical side of our problem.

Topology, as the most general science of spatial relations,
can be based on the relationship between "part" and "whole"
or in other words on the concepts of "being-
included-in." 1 Closely related to these con-
cepts is that of the "surrounding" of a
"point."

If A is a part of B one can write A < B
(A is a part of 5) and B > A (B includes A).
By A+B, the "sum" of A and B, one
understands the totality of all points which
are included in either A or B. By the "intersection " of A and
B (the part which is common to A and B} is meant the totality
of points which are part of both A and B. One expresses
intersection as A * B. In Fig. 8 the sum of the rectangle A and
the disk B is a rectangle with a half disk. The intersection
is the half disk which lies within the rectangle.

If A is part of B then A + B = B and A B = A, Further,
for every part of a space the equation A+A = A = A*A
holds. Two regions A and B are called "foreign" if they have
no common parts, i.e., if the intersection A B equals o.

1 For this and the following definitions cf. K. Mengei (66, p. 17) ; F. Hausdorff
(31, p. 244); W. SierpMski (79); 0. Veblen (87).

88 TOPOLOGICAL PSYCHOLOGY

THE CONCEPT OF REGION. THE CONNECTED REGION

We shall start with the topological concept of "region." A
region is called "connected" if every point of it can be connected
with every other point of it by a path which lies entirely within
the region. The region represented in Fig. 9 is therefore
connected. The region in Fig. 10 consisting of B and C is not
a connected one. The region consisting of the points in Fig. n
is also not a connected one. Thus the concept of region does
not imply that its parts must be connected.

The region B in Fig. 10 is in itself a connected one Just as
the region C is. From a topological point of view there is no
difference between the regions A, 5, and C. One can use as

FIG. 9. Connected region. FIG. 10. JB + C is a FIG. n. Not con-
not connected region. nected region.

criterion of the topological equivalence the following fact:
by a process of continuous transformation it is possible to
convert any one of these regions into any other without chang-
ing the connections within the region, i.e., by stretching or
bending without tearing. Topologically there is no difference
between a circle, an ellipse, a regular or irregular polygon
with any number of sides, and the figure A (cj. Fig. 9). Like-
wise, there is no difference between a sphere, a cube, a cylinder,
and a cone.

Differences in size are also disregarded in topology. There
is no topological difference between a drop of water and a
sphere the size of the sun. One cannot say that the distance
between points i and 2 in Fig. 9 is less than the distance
between points i and 3. The impossibility of determining
size refers not only to distances but also to angles. Neverthe-
less, a number of important mathematical statements can be

CONCEPTS OF TOPOLOGY FUNDAMENTAL FOR PSYCHOLOGY 89

made about spacial relationships of such topologicaily defined
entities.

CLOSED AND OPEN REGIONS

Topology distinguishes between "open" and "closed"
regions. Examples of closed regions of two dimensions are a
disk with its boundary, or the regions A, B, and C in Figs. 9
and 10, if one includes the contour as part of the region. Exam-
ples of open regions are the inner region of a disk, the regions
Ay 5, and C without their boundaries, or an unlimited plane.

An open region is usually characterized as a region for each
point of which there is a surrounding that lies entirely within
the region.

For a dosed region this characteristic does not hold: each
surrounding of a boundary point contains points which do not
belong to the region. This is true however small a surrounding
one may choose for the boundary point. A closed region is
therefore characterized as a region which includes its boundary
points.

LOOTED AND UNLIMITED 'REGIONS

The distinction between "open" and "closed" regions is
not to be confused with that between "limited" and "unlim-
ited" regions. An example of an unlimited two-dimensional
region is a plane; of a limited two-dimensional region a triangle.
The limited regions within a plane can be visualized by the
fact that one can draw a dosed curve in which they lie. An
open region can nevertheless be limited (examples: the inner
regions of the Figs. 4, B, and C). The plane is an open,
unlimited region,

SIMPLY AND MULTIPLY CONNECTED REGIONS

The connected regions A, B } and C in Figs. 9 and 10 and
likewise region D in Fig. 12 are simply connected. In order
to understand what this means we must first consider another
concept. A path which connects two boundary points of a
region and which, aside from these boundary points, lies wholly

go TOPOLOGICAL PSYCHOLOGY

in the inner part of the region is called a "cut." A simply
connected region is defined by the fact that any cut destroys
its connectedness. For instance, the cut c divides the con-
nected closed region D of Fig. 12 into two parts Di and D%
such that it is impossible to connect a point of DI with a point
of Z>2 by a path without crossing c or some other part of the
boundary of Di or J9 2 .

A cut does not necessarily have this effect in every connected
region. For instance, the cut c in region E of Fig. 12 does not
destroy its connectedness. This region has the character of a
ring and its boundary consists of two closed curves m and n.
In spite of the cut c every point of the region E can be con-
nected with every other point by a path which lies wholly
within E. But if one adds to the first cut a second one, E is
D

FIG. 12. D is a simply connected region. E is twofold connected. F is fivefold

connected.

no longer connected. Such a region is therefore called "two-
fold connected."

Region F in Fig. 12 is "fivefold connected." It is possible to
make four but no more than four cuts in such a way that F
remains connected, for instance, the cuts ci, c^ c^ and c.

The reader may get an idea of the binding nature of these topological
properties by trying out the different series of cuts that can be made in F
without destroying its connectedness.

JORDAN CURVE, BOUNDARY, PATH

It is necessary to discuss briefly the mathematical charac-
teristics of the simply connected dosed regions. Such regions,
in a certain sense, represent the simplest case for dynamic
problems in psychology. We shall use them as our most
important elements of construction both for the problems of
the psychological environment and for those of the person.

CONCEPTS OF TOPOLOGY FUNDAMENTAL FOR PSYCHOLOGY gi

The boundary of a simply connected limited region has the
character of a Jordan curve. A Jordan curve is defined as a
topological, i.e., as a one to one continuous, image of a circle.
It is therefore a closed curve which does not intersect itself.
Aside from this the shape of the curve is irrelevant. The
borders of the regions A (Fig. 9), B 7 C (Fig.
10), D (Fig. 12) are Jordan curves. On
the other hand, the boundary of the region
E, which consists of two separate curves FlG

(m and n), is not a Jordan curve. Like-
wise the boundaries of region F (Fig. 12) and of Fig. 13 are
not Jordan curves.

The Jordan curve has several characteristics which make it
of fundamental importance for our representations. It divides
the plane into two regions, an inner and an outer one, in the
following sense (cf. Fig. 14). Each point of the outer region O,
for instance i, can be connected with
any other point of this region, for
instance 2, by a path (m) which does
not intersect the Jordan curve. Like-
wise one can connect every point of
the inner region /, for instance 3, with
FIG. ij^-jordan curve every other point of the inner region,

determining an inner region f instance 4, by a path (0) which doCS
(J) and an outer region (0). ti J r \ s

not intersect the Jordan curve. On the

other hand each path (r) which connects a point (5) of the
inner region with a point (6) of the outer region intersects the
Jordan curve. The Jordan curve is in this well-defined sense
the boundary between the inner and the outer regions.

By path one understands the connection between two points
by a Jordan arc, i.e., by a part of a Jordan curve. A path is
therefore a curve which does not intersect itself.

FOREIGN REGIONS

Some characteristics of boundary and connectedness which
are essential for psychology become still dearer if one starts
from the concept of "being foreign." As we have explained,

92 TOPOLOGICAL PSYCHOLOGY

two regions A and B are called foreign if they have no common
part, or more correctly expressed, if the intersection of A and B
is empty (A B = o). It is important for psychology that it is
possible to distinguish between different cases in this respect.

The open regions B and C of Fig. 10 are foreign and likewise
the open regions D and E of Fig. 15. But while the boundaries
(&) of B and C are foreign (b B ' 'be = o) , there
are boundary points of D which at the same
time are boundary points of E, that is, the
boundaries of D and E have an intersection
which is not empty (&D 6^ ?^ o). Any
FIG. 15. Foreign re- boundary serves both to separate and con-

gions with not foreign ect t Q reg i ons . X&S double function
boundaries. &

is important for psychology.

The difference between the relation of the regions in Fig. 10
and those in Fig. 15 can further be expressed as follows: The
sum B + C of the closed regions B and C is not connected. The
sum of the closed regions Z> and E is connected. We can now
give one more definition of the concept "connected": A region
(which is not empty) is called connected if it cannot be divided
into two foreign (dosed and not empty) parts (66, p. 197).

This concludes our preliminary presentation of mathematical-
topological concepts. We shall not go further into the many,
and often very complicated possibilities of topological rela-
tionships. The treatment of the psychological examples will
give ample occasion to make the mathematical concepts
familiar.

B. TOPOLOGY OF THE PSYCHOLOGICAL
ENVIRONMENT

We shall start our discussion of topological problems in
psychology by considering the psychological environment. In
order to simplify the presentation we shall occasionally use
concepts such as that of force which will not be defined until
we discuss vector psychology.

CHAPTER XI

PSYCHOLOGICAL REGIONS, LOCOMOTIONS, AND
COMMUNICATIONS

COORDINATING DEFINITIONS

One should choose coordinating definitions in such a way
that they hold without exception and are univocaL As far as
possible we shall try to use reversible coordinations.

The Psychological Region. Definition: To each part of
the life space a region is to be coordinated.

Thus we have to represent as a region (i) everything in
which an object of the life space, for instance a person, has its
place; in which it moves; through which it carries out locomo-
tions; (2) everything in which one can distinguish several
positions or parts at the same time, or which is part of a more
inclusive whole.

This definition implies that the person itself has to be repre-
sented as a region in the life space, further that the life space as
a whole is a region.

The reverse of the definition of a psychological region also
holds: everything that is shown as a region in representing a
situation must be a part of the life space.

94 TOPOLOGICAL PSYCHOLOGY

In determining whether we are dealing with one or with
several psychological regions one can build on either of two
facts: (i) one can characterize a region by its qualitative
properties and can find out its relations of position by deter-
mining which regions are contained in others (have the relation
to each other of part and whole, X C7), how they overlap
(X Y 76 o or = o), which regions have common boundaries (i),
and which do not (bx b Y ** o or = o). (2) One can build on
psychological processes which connect different points (part
regions) in the life space, for instance on locomotions. The
locomotions cross or do not cross certain boundaries or other
regions. This characteristic makes it possible, on the basis
of the coordination of locomotion and path (about which see
below), to make topological statements about the regions to
which the points (part regions) belong.

The following are examples of qualitative characterizations
of regions: ground which is easy or difficult to walk on ; region of
a forest; a region within which one may be seen from a certain
point; a region of a certain color; the sphere of influence of a
person; a social group; an occupation; a region in which certain
actions are permitted. To determine the position of such
regions, for instance the sphere of influence of a person, one can
ascertain whether it overlaps the spheres of influence of other
persons or groups and if so which; with which spheres of influ-
ence it has or has not points of contact (common boundaries).

In determining the position of regions one can use any bodily,
quasi-social, or quasi-conceptual locomotion. We have already
given sufficient examples of these locomotions. Locomotion
makes it possible to make statements about the position not
only of the region of departure and of destination but also of
those regions which the locomotion crosses.

In psychological investigations one will have to use qualita-
tive properties as well as locomotions in determining psycho-
logical regions and their positions. Sometimes the one,
sometimes the other gives better results. On the whole the
more reliable method of determining the topological char-
acteristics of regions seems to be that of referring to locomotions.

PSYCHOLOGICAL REGIONS, LOCOMOTIONS 95

The determination of a region, for instance by certain quali-
tative characteristics, does not in itself imply whether or not
this region is a connected one. For instance, the region which
corresponds to the property of a person or the region which
corresponds to a certain social group has to be represented
sometimes as a connected, sometimes as a non-connected
region according to the actual distribution of the property or
of the members of the group. Also in this point therefore the
psychological concept of region agrees with that of mathematics
(see p. 88).

Psychological Locomotion. Definition: A path is to be
coordinated mathematically to each psychological locomotion.

As stated above, one understands by path a part of a Jordan curve,
i.e., a curve which does not intersect itself. On the other hand psycho-
logical locomotion can, at least in a certain sense, pass the same place
twice. In these cases locomotion would have to be represented by a curve
which intersects itself. However, we shall generally speak of paths, since
this is very unlikely to lead to error in the practical application of our
concepts. (Besides, mathematics itself sometimes uses the concept of
path in this more general sense.) It should be emphasized once more that
in the following discussion we mean by locomotion not only quasi-physical
but also quasi-social or quasi-conceptual locomotion.

The question arises whether this coordinating definition can
be reversed. Such a reversal would take the following form:
To each path in the life space corresponds a locomotion.
However, there are cases in which one can connect mathe-
matically points in two different regions of the life space, but
when the corresponding locomotion can actually not be carried
out. For instance, in our example the prisoner cannot carry
out bodily locomotion from the region within the prison to the
region outside. Nevertheless, in this case other objects in the
life space of the prisoner can carry out such a locomotion and
he himself can move in his thoughts from one region to the
other. But it is at least conceivable that there can be regions
in the life space into which even a conceptual locomotion
cannot be carried out.

In spite of this difficulty it is possible to reverse the definition.
We have already mentioned that it is sometimes easy, some-

96 TOPOLOGICAL PSYCHOLOGY

times difficult to carry out a locomotion. A locomotion which
cannot be carried out can therefore be thought of as an extreme
case of difficult locomotion. Thus the concept of an impossible
locomotion is entirely legitimate and even unavoidable. In
other words one has to distinguish between the applicability
of the concept of locomotion and its factual possibility. For
our purposes we can state the definition in its reverse form as
follows: To each path in the psychological life space corresponds
a locomotion which can or cannot be carried out.

Analogous definitions are common in mathematics. Thus one usually
defines function as follows: a is called a function of b if the value of a varies,
or does not vary, with the value of b.

On the basis of these coordinating definitions of the psycho-
logical region and the psychological locomotion one. can repre-
sent mathematically the topological relationships of an unlim-
ited number of different situations. It is the task of the single
psychological investigations to carry out these constructions.
Here we only want to demonstrate by means of simple examples
the general method of making such constructions.

BEING INSIDE OR OUTSIDE OF A REGION

We have explained that the different social positions differ
significantly in space of free movement that is available to the
person. A change in a person's social position can often be
represented as a locomotion from one region to another. In
general the fact that a person (P) is in a particular region (R) 1
at a given moment (P < R) is of decisive importance for his
behavior. We shall explain that more exactly by means of a
few examples.

An investigation has been made of the effect of social pressure
on the behavior of children during meals. 2 It shows that one
of the most important means by which the adult induces the
child to eat an undesired food is tp bring Mm into the "eating

1 The region R may be defined in such a way as to include the objects located
in it.

* This investigation (18) has been carried out in the nursery school of the
College of Home Economics of Cornell University,

PSYCHOLOGICAL REGIONS, LOCOMOTIONS

situation." If a particular kind of food is not desired, the
otherwise unified action of eating usually breaks up into a
series of separate steps such as: putting the hand on the table
(h) ; taking the spoon (sp) ; putting the food on the spoon (f) ;
bringing the spoon halfway to the mouth (to); bringing it to
the mouth (f); taking the food into the mouth (i); chewing

eotrincf

<gK

- -*

FIG. 16. Situation of child facing disliked food, (a) before entering the
region of real eating; (6) after entering the region of real eating. In this and the
later figures the following symbols are used:
< force: the direction of the arrow represents the direction of the force; its

length, the strength of the force; its point the point of application of the force,
< locomotion : the point of the arrow designates the place of termination

of the locomotion, the other end of the dotted line designates the point of

departure,
-f- positive valence.
negative valence.

(cA); swallowing (sw). These steps correspond topologically
to a series of regions (cf. Fig. 160). The procedure of the adult
is sometimes to bring the child (C) step by step through these
regions closer to the region of the "real eating" (chewing and
swallowing). In doing so he usually meets with increasing
resistance in accordance with the fact that with approach to
the undesired action the repulsive forces (represented as arrows
in Fig. 16) increase. However, as soon as the food is once
in the mouth it is often not spit out, even when the adult has
fed the child against its will. Instead the child goes on to
chewing and swaEowing the food.

98 TOPOLOGICAL PSYCHOLOGY

One can show in detail that this change of behavior is brought
about essentially by the fact that as the child enters the region
of "real eating" his position and the direction of the field
forces are entirely changed. When the child is in one of the
preceding regions, for instance, when he holds the spoon half-
way to his mouth, then a region of greater unpleasantness into
which the adult tries to push him, still lies ahead. The adult
therefore may have to exert great pressure to induce the child
to make a locomotion into the disagreeable region. When the
child is once within this region of real eating then the region
which lies ahead of him is a more pleasant one of relative
freedom (Fig. i6J). The child therefore often prefers a loco-
motion in this direction to spitting out, which is a locomotion
in the direction of a disagreeable fight with the adult.

I cannot discuss further the various and often very compli-
cated details of this situation. But I would like to point out
that a similar technique is often used to force an adult to do
something against his will, for instance in a political struggle.
A social group may fight bitterly against the attempt to change
its position. If however one succeeds in bringing about a
fait accompli the group will accept its new position without
resistance. This is one reason why the fait accompli is so
dreaded in politics.

Another example: A two-year-old child does not want to
stop playing and go in to wash his hands. The mother who
knows that the child especially likes to wet the wash cloth in
the basin for himself asks, "Do you want to wet the doth or
should I do it?" The child wants to do it and so he lets
himself be washed without further trouble. The question
has sufficed to transfer the child from the play situation to
the washing situation and it begins to behave according to the
requirements of the new region.

Again, Werfel (8ya), in his book The Forty Days of Musa
Dagk, describes how a group of Armenian peasants decide to
defend themselves on a mountain against their common
enemies. But they cannot come to an agreement about how
to settle the property rights in their mountain camp. The

PSYCHOLOGICAL REGIONS, LOCOMOTIONS 99

priest finds a way to delay the discussion of the question. Later
when they are once on the mountain the question of property
adjusts itself according to the situation in a way which had
been strongly rejected while they were still outside of the
situation.

The importance of the act of decision for behavior lies
mainly in the fact that by it the person changes his position.

These examples may suffice to make it dear how important
it is whether one stands within or outside of a situation;
expressed in mathematical terms whether one stands within
(P < R) or outside of a certain region (P R = o).

Why the region in which a person stands is so important for
his behavior becomes intelligible when one realizes how great
the change is which is brought about by a transition from one
region into another, even within an otherwise unchanged life
space. As a rule all relations of neighborhood are changed
by such a transition: Regions which were before adjacent to the
region of the person are no longer so and vice versa. Different
locomotions are now possible and impossible. Even when the
same regions are still attainable the course of the path which
one must follow to reach them has changed because of the new
point of departure. This usually means a change in direction
and distance of other regions of the life space and thereby a
change in the direction and magnitude of the forces which affect
the person. But above all, it is important that the region
itself in which he is located has a different character,

In short, the dynamic condition of a person depends in almost
every respect directly on his position in a certain region.
Methodologically therefore in almost every psychological
problem one should give first place to the question of the region
in which a person is at a certain moment, or what change of
position is just occurring,

THE INNER STRUCTURE OF A PSYCHOLOGICAL REGION

The topological properties of a region can be of very different
kinds. In the following we shall discuss several pertinent

loo TOPOLOGICAL PSYCHOLOGY

questions in connection with simple examples of quasi-physical
regions which we have already treated.

Determination of the Connectedness of Regions. A
simple example of the determination of the topological structure
of a region by reference to possible locomotions is that of the
"space of free movement." We have characterized the space
of free movement as the totality of regions to which the person
in question has access from his present position. Within this
region of free movement the person can carry out locomotions
from each point to every other without leaving the region,
i.e., without having to break through its boundary. On the
basis of the coordination of psychological locomotion and
mathematical path one can therefore designate the space of

FIG. 17. Minority group, (a) Unscattered; (&) scattered. A, The minority
group; J?, the majority group.

free movement of a person as a connected region. (The reader
will remember that a connected region is defined by the fact
that any two of its points can be connected by a path which
lies wholly within the region.)

In determining the connectedness of a region in psychology
it is not possible to observe the locomotions between all the
parts of the region. It is sufficient to find out whether the
essential parts can be connected by locomotions whose paths
do not leave the region as a whole.

Nonconnected Regions* For the behavior of a person who
belongs to a minority group of a country the distribution of his
group within the country is of great importance. The minority
may be scattered, or it may live as a closed group. In the
latter case the minority group (A) corresponds to a connected
limited region within an enclosing region (5) which is to be
coordinated to the other parts of the population of the country

PSYCHOLOGICAL REGIONS, LOCOMOTIONS 101

(Fig. 170). If the group is scattered it corresponds topologi-
caily to a nonconnected region (Fig. ijb). (A 1 + A* + A* +
... is a not connected region.)

In so far as we are dealing with the geographical distribution
one can demonstrate that the parts of the group are separated
topologically simply enough by the fact that one cannot carry
out a bodily locomotion within the region of the group from
one of its parts to any other part. But it is also possible to
determine the structure of such a group in regard to its social
relationships. A group which is characterized by race or social
position may live in a city without much social contact with
the rest of the population. (One may think of the Negroes
in New York; the nobility in a medieval Italian city, or any
other exclusive social group.) This group would have to be
represented as a connected region in so far as its social life is
concerned if all of its parts can come into social contact with
each other without the mediation of other circles which do not
belong to this group. When social or business intercourse has
to be carried out through members of other groups then the
group has to be represented as regards its social or business life
as a nonconnected region. One can see how it is possible in
this way to determine in detail which parts of the region are
connected and which are separated* Naturally the topological
structure of the subparts can be determined in the same way.

One realizes how important the dynamic consequences of the
connectedness or nonconnectedness of a group are if one con-
siders how the behavior of its members is affected by the break-
ing up of a connected group. The breaking up of the Ghettos
and the scattering of the Jews into smaller groups had as its
result changes of behavior which are caused to a large extent
by the change of the topological structure of the group. For a
separation into nonconnected parts has dynamically the result
of a weakening of the inner connectedness and the degree of
mutual dependency (58, p. 182). It means further that, other
tMngs being equal, the surfaces of contact between this region
and other regions are enlarged; also the magnitude of the forces
which affect the group as such and its single members often

102

TOPOLOGICAL PSYCHOLOGY

are essentially influenced by the change of the connectedness
of the group.

One must note in this example that it is possible to determine
the topological properties of the social structure of a group on
the basis of its social intercourse. It may seem doubtful
whether we are really dealing with locomotions in this case;
that is, whether in social intercourse the one person or the one
group actually passes from its own region to the region of the
other person or other group. There is no question that inter-
course brings about a connection between the two regions.
But this connection often has the character of a "communica-
tion" rather than that of a locomotion; one region shifts toward

FIG. 1 8. (a) Locomotion from A to J5; (ft) communication between A and B.

the other until a contact is made or one part of the region is
so moved forward that it makes a bridge to the other region.
Such a representation seems especially suitable if we are dealing
not with intercourse between single persons but between groups.

Figures i8a and 186 show the difference between the two
kinds of connection. In one case a one-dimensional path (w)
leads from a point (i) within the region A to a point (2) of
the region B. In the other case a two-dimensional arm (part
region) A l reaches out from A to the region B so that it touches
region B or partly overlaps it. (A + B is a connected region.)
In the second case we shall speak of "communication" (see
p. 126) . Locomotions and communications are in some respects
of equal value for the determination of topological relation.
This point will come up again in our discussions.

As we have mentioned above very different structures result
if one takes as the basis for the determination of a region
different kinds of locomotion or communication: the

PSYCHOLOGICAL REGIONS, LOCOMOTIONS 103

graphical structure of a social group can be very different from
its social or vocational structure.

Multiply Connected Regions. One can ask whether a
given space of free movement is a simply or a multiply con-
nected region.

Let us take as an example the space of free movement which
was discussed on page 45. It consists of the regions of what
one "can" do and what one "is allowed" to do. Such a space
of free movement is, as a rule, a multiply connected region.
The space of free movement is probably always surrounded by
regions which the person is not able to enter. This would
mean topologically that the space of free movement is a limited
region (see p. 89). For it would lie entirely within a ring
of inaccessible regions.

But it does not usually happen that all regions of the for-
bidden and the impossible belong to this surrounding ring and
make up with it one connected region. In most cases there
will be within the region of the allowed certain islands of the
forbidden, i.e., regions of the forbidden which are not connected
with each other.

The nonconnectedness of these islands can be proved by
reference to locomotions. For instance the regions which are
characterized by such prohibitions as "You must not cross
the road alone," "You must not copy in school," "You must
not be impolite to a certain person," may usually be uncon-
nected, especially when the one prohibition issues from the
parents, the second from the teacher, and the third from a good
friend. The transition from one of the forbidden regions to
another forbidden region will usually be possible only by
passing through the region of the allowed. This would prove
that the two forbidden regions are not connected. At the
same time it means that the space of free movement is in these
cases multiply connected.

In some cases several regions of the forbidden are connected.
This can have the dynamic effect that if once the boundary
of the forbidden is crossed the person can pass from one part of
this region to another with relative ease.

104 TOPOLOGICAL PSYCHOLOGY

In such a situation other factors, especially the eventual weakening of
the social fields which induce the prohibition, can play a role. This
weakening is not caused by the topological connectedness of the different
regions of the forbidden.

As a rule it is dynamically of no great importance to deter-
mine exactly how many times a multiply connected region is
connected. It remains, however, important whether a space
of free movement is frequently interrupted by islands of the
forbidden or whether a space of equal extent is relatively free
from such islands. A child may have a region of his own (a
playroom, or a playground) in which he can follow his
inclinations undisturbed. This means that this field is
distinctly free from regions of the forbidden. If the child
has to play in rooms which are also used by adults, he con-
tinually runs up against more or less extensive regions of the
forbidden.

Limited and Closed Regions. The topological structure
of the situation in respect to bodily locomotions is especially
simple in our example of the bathtub (see p. 42). Let us dis-
cuss the situation of the child A.

In the beginning A considers the whole bathtub as his space
of free movement; since there is water in the tub, that is a
material which does not hinder the locomotions in question (we
shall limit ourselves to locomotions of the whole body). The
space of free movement can be characterized mathematically
in a very simple way: it is a connected limited region. The
connectedness results from the possibility of locomotions from
each point to every other on paths which lie wholly within the
region. The fact that the region is limited is in this case mani-
fested very simply by the surrounding rim of the bathtub.

But this does not tell us whether this space is a dosed or an
open region. Mathematically, as we have seen, it is character-
istic of boundary points that they have no surrounding which
lies wholly within this region. If one includes the boundary
points in the space of free movement one thereby characterizes
this space as a closed region. If one does not, the space of free
movement would be an open region.

PSYCHOLOGICAL REGIONS, LOCOMOTIONS 105

From a psychological point of view one can say: If one con-
siders the edge of the tub as the real boundary of the space, then
one is inclined not to include the boundary as part of the space,
for naturally the child can only move about in the inner part of
this region. This would mean that the space of free movement
is defined as an open region. But, on the other hand, one could
consider the points of the inner surface of the real edge of the
tub, or perhaps the boundary line of the water at the edge of the
tub, as the boundary of the space of free movement. Then one
could not object to including the boundary in the space of free
movement. In this case the space of free movement would be
characterized as a closed region. We see that, from the point
of view of psychology, it is irrelevant whether this space of free
movement be defined as an open or as a dosed region.

The question of whether a limited psychological region is to be
characterized as an open one or as a dosed one (that is, whether
or not one should include the mathematical boundary curve in
the region) seems to be a matter of minor importance. One
must not forget that the open limited regions always have a
boundary or a hull (66, p. 29).

Much more important are the dynamic characteristics of the
boundary, for instance, its solidity. From the point of view of
mathematics the edge of the tub is not really a boundary with-
out thickness but is itself a region. We shall return to this
question when we discuss boundaries.

In addition to the limitation of the space of free movement in
our bathtub example the kind of connectedness is of factual
importance. In the beginning the space has the character so
it seems at first of a simply connected region. This follows
mathematically from the fact that the boundary of the region
is a Jordan curve. But this characterization of the space of free
movement of A is not entirely correct. There is an "island"
within this space, namely the boy B. The behavior of A would
be quite different if B were not there. We do not want to dis-
cuss here the important difference between a permeable body of
water and such a "thing" as a person. But it may be well to
make it dear that one can indude the second child in a matte-

io6 TOPOLOGICAL PSYCHOLOGY

matically consistent description without becoming psychologi-
cally unsound. B is a part of the life space of A and is therefore
according to our definition (p. 93) himself a region. The body
of B has the character of a thing (cf. p. 115), that is, a region
which A cannot enter or cross. It is therefore not a part of A J s
space of free movement. The presence of B in the tub makes
the space of free movement, if one wants to be exact, a doubly
connected region. If B makes a connection between his body
and the edge of the tub by grasping the edge e (Fig. ic) with his
arm B', B produces a cut (see p. 43)* through the space of free
movement, but the space maintains its connectedness. A is
still able to reach all points of the tub by making detours around
B.

As a matter of fact B carries out his intention of limiting ^4 J s
space in another way. He connects two points of the outer
border by a line which he draws across the water with his finger,
according to Fig. ib (see p. 43). The originally connected
space is divided by this cut into two regions. Only one part,
the one in which A is, retains the character of a space of free
movement, although it is now more restricted. The other part
becomes a "power field" of B(Ps) which A may not enter.
The division made by the cut means at the same time a certain
clarification of the situation (PB*PA = o). A is now sole
master within his own region while, in the original region, his
freedom of movement was hampered at least in the neighbor-
hood of B (P B -Pi 5*0).

It is further important that A's space of free movement
remains adjacent to that of J5. If one wants to express this
fact in exact mathematical terms one can say: the topological
intersection of the boundaries of the two spaces of free move-
ment (u and &*) is not empty (b b s 7* o). This is an example
of a determination of the adjacency of two regions without
reference to locomotion.

One might raise the question whether the body of A himself
should not be treated as an object in his own space of free move-

1 He makes a connection between different boundary points of the space
of free movement by means of a path which lies within this space.

PSYCHOLOGICAL REGIONS, LOCOMOTIONS 107

ment. This would mean that this space is at first a threefold,
later a twofold connected region. In principle there is no
objection to such a representation, and there are cases in which
it is even advisable from a psychological point o| view. On the
whole it will be better to represent the own person as point or
region which moves about within the space of free movement
but which is not a foreign region in it. We shall later return
to the question of the representation of the person in the life
space.

In regard to each region which has the character of a thing,
one can ask whether it makes up a part of the space of free
movement itself or whether it lies within this space without
belonging to it. Psychologically this question is only meaning-
ful when we are dealing with sufficiently large or otherwise
significant objects that may for instance be obstacles to certain
locomotions. The answer to this question will depend upon the
special character of the situation and the special kind of
locomotion.

REPRESENTATION AS PATH OR AS MORE THAN
ONE-DIMENSIONAL REGION

It is sometimes doubtful whether one should represent a
certain fact in the life space as a path or as a region. We have
already met this question: Social intercourse can be a loco-
motion which must be represented as path or it can be a com-
munication which has to be represented as region (or part
region).

Mathematically the concept of region also includes one-
dimensional manifolds (see p. 88). A path therefore can
be understood as a region no less than those parts of the
life space which we have represented by two-dimensional mani-
folds. The question how many dimensions the life space has
will be discussed later (see p. 193). In any case there is a great
difference between a two- or more-dimensional region in which
or through which a path may take its course and these paths
themselves. The very fact that they differ in number of
dimensions is important. It is of still greater significance

lo8 TOPOLOGICAL PSYCHOLOGY

psychologically that (in our case) paths are coordinated to
locomotions, i.e., to processes in time, but psychological regions
to the areas in which these processes occur. It is therefore an
important question whether one should represent a certain
psychological fact as a path or as a two- or more-dimensional
region.

In the example of the child who had to eat something which
he did not like we have represented the actions which lead up to
eating and eating itself as regions through which the child had
to move. We have further spoken of a person's occupation as a
region. In experimental work too we have found that it is
sometimes useful to represent activities as psychological regions
in the life space.

It may seem surprising that an action can be characterized
psychologically as a region, and moreover as a region of the
psychological environment rather than of the person. We do
not claim that actions have always to be characterized as
regions. Generally one thinks of an action as an event, a
process in time. This process can have the character of a
quasi-physical, quasi-social, or a quasi-conceptual locomotion
and is then to be represented as a path.

However, it seems advisable, in certain cases, to represent
actions as regions. For instance a child may be busy playing
with its dolls. The mother calls it to eat its supper or to go to
bed. The difficulties which are typical in such cases are con-
nected with the fact that the child has to leave the region of a
certain occupation and has to enter the quite different region of
eating or sleeping. Or again, if a man leaves his business to go
on a vacation, or if he changes from one occupation to another,
we are dealing with a locomotion across a boundary. The
actions of playing and eating have, at least at this moment, the
character of regions in which the child is located, out of which
or into which he has to go ; that is the character of regions in the
environment.

The regions of action correspond to other regions in the life
space also in that one can move about in, them, that they can
be parts of more inclusive regions, and that they can contain

PSYCHOLOGICAL REGIONS, LOCOMOTIONS 109

part regions. A child who has to solve a problem in arithmetic
for his school work may move about within the region of the
problem until he has found the solution. He may then leave
the region and go to another task in arithmetic until he has
finished that one as well. From there he may go on to other
school work, for instance the preparation of a French exercise.
The appearance of a playmate may cause the child to leave the
region of school work before he has finished and to go and play
with his friends. In this example the arithmetic problem is
part of the whole region of school work and includes part
regions which correspond to the single problems. The single
problem can include part regions which correspond to the single
mathematical operations.

The locomotions within such a region can have a purposive
character. In the case of the arithmetic problem the solution
of the task is the goal which controls the locomotion. In other
cases the process within the region of activity may have less
the character of striving toward a goal than that of staying
or moving about within a region. This is true of such activities
as dancing, the infant's thumb sucking, the child's play with
dolls. One's regular work also can be purposive to different
degrees. For the unskilled laborer work has more the char-
acter of a field within which he remains than it has for the
skilled workman (50).

In many cases it may be doubtful whether one moves at all
while one is performing an action in a given region of activity.
To sit quietly and muse may be considered as staying within the
same region and even at the same point. At least one is not
engaged in any swift locomotion.

The temporal process of locomotion, independent of its
velocity, is a change in the position of a person whose course can
be represented by a path. This path as we have mentioned
before (see p. 95) must not be thought of as representing a part
of the life space as it exists at a given moment but rather a
change of position within a field which otherwise remains
sufficiently constant. Such a representation is essentially an
abbreviated representation of a sequence of situations and is

no TOPOLOGICAL PSYCHOLOGY

therefore possible only within periods of time in which such a
constancy exists. 1 The actions on the other hand which are to
be represented as regions in the life space are coexisting mani-
folds which possess a certain structure and a certain degree of
differentiation.

REPRESENTATION AS POINT OR AS MORE THAN
ONE-DIMENSIONAL REGION

The relation between paths and regions becomes clearer if we
consider the relation between points and regions in the life space.
It is sometimes doubtful whether we should represent a psycho-
logical fact as a point or as a more-dimensional region. (Mathe-
matically a point is a o-dimensional region.)

To a locomotion we have coordinated a path, i.e., a connec-
tion between two points. Psychologically also it seems entirely
correct to say that a locomotion leads from a beginning to an
end point. This end point is often a goal toward which a
person strives. As a matter of fact one can represent goals as
points. On closer consideration however one finds that a
psychological goal is always a region. This is true of the apple
toward which the child strives 2 as well as of the occupational
aim of the youth who wishes to become a physician. Goals are
not points but regions into which a person would like to enter
or in relation to which he wishes to have a certain position.

Also if one tries to characterize the intermediate points of
such a path, one is often faced with facts that one cannot
represent as points, but only as regions. One can represent the
process of working an arithmetic problem in which one has to
perform first a multiplication, then an addition, and finally a
division as a locomotion to which one coordinates a path from a

1 We usually represent a locomotion in this case as a broken line which starts
at the moving region and whose end point is characterized by an arrow. The
moving region is shown in the position at the beginning of the locomotion.
Such a representation makes it possible to distinguish between force and
movement.

2 It is usually not correct to designate the material object itself as the goal.
The goal is usually an action or a state, for instance the eating ot an apple or the
possessing of an object.

PSYCHOLOGICAL REGIONS, LOCOMOTIONS

III

beginning point (i) by way of certain intermediate points
(2, 3, and 4) to an end point (5) (Fig. 190). This implies that
one has to represent multiplication as a region of action (M u)
which the path crosses and not as a point. The progress from
multiplication to addition (Ad) and division (Di) means that a
person has passed from one region to another. The question
arises whether one can determine exactly the points of the path
within these regions. These points are at first only character-
ized by the fact that they lie in such and such a region. We can
determine their position more exactly only if we succeed in

FIG. 19. The maximum exactness with which the position of a point can be
determined depends upon the degree of differentiation of the region in which it
lies, (a) Relatively undifferentiated regions; (&) relatively differentiated regions.
Mu, multiplication; Ad, addition; Di, division; i, 2, 3, 4, 5, different points.

characterizing subregions within the larger regions. It may be
possible to divide the region which corresponds to a complicated
division into subregions according to the part operations
involved. The topological relationships of these part regions
are determined to a certain degree by the sequence of these
operations. Then one can say that the path which is to be
coordinated to the action passes through the subregions in such
and such a sequence (cf. Fig. 196).

It is important for the determination of the life space that
the position of a point cannot be characterized except in terms
of its position in such and such a region. The exactness of tiiis
determination depends, in the single case, upon the extent to
which one can distinguish subregions within the region in
question. 1

1 The same is true of the determination of real points in physics.

112 . TOPOLOGICAL PSYCHOLOGY

With this fact is connected a second: if it is true that one can
determine the position of a point only in so far as one can divide
the regions in which it lies into part regions, it does not make
any difference whether one speaks of the beginning point and
end point of a locomotion or of its beginning region and end
region. The point becomes thereby, for our consideration,
equivalent to the region in which it lies and which can be no
further differentiated. In other words: in psychology the
point is equivalent to a region which is not structured into parts.

This means for our representation two things: first, if a
psychological fact is represented by a point, one is justified in
considering this point, provided one is to examine it more
closely, as a more-dimensional region; second, one can some-
times represent unstructured regions in the life space as points. 1

In some cases one can make use of these facts in the repre-
sentation of the person. We have seen that it is always neces-
sary to represent both person and environment in the life space.
We also have mentioned that one has to distinguish within the
person certain strata and regions. The person himself is
therefore a more than zero-dimensional region. For certain
problems however one can represent the person in a first
approximation as a point for the following reason : the person is
a strongly unified whole. When we are not dealing with the
dynamic differentiation of the person into part regions we can
consider the person as a single system and can represent it as an
undifferentiated region or as a point in the sense explained
above. One can use such a representation especially in those
cases that concern the locomotion of the person as a whole.
The same representation is valuable in treating forces which
affect the locomotion of the person. To be sure it is admissible
only when the point of application of the forces within the
person is not important; this means again when the person in
this respect can be thought of as an undifferentiated whole. 2

1 Also from a purely mathematical point of view a point is an unstructured
region.

2 In these cases we usually represent the person in our diagrams as a limited,
simply connected, and undifferentiated region.

PSYCHOLOGICAL REGIONS, LOCOMOTIONS 113

For more exact representation one will always have to show the
person as a differentiated region and the specific point of
application of the force.

LOCOMOTIONS OF A SURROUNDING FIELD

Occasionally there are definite changes of position of the
person in the surrounding field when the person does not carry
out any active locomotion himself.

Some of these cases are very simple. A man may have been
swept along by external social changes. The social position
of a person P may change greatly as a result of gain or loss in the
influence of his family or of the business concern with which he
is identified. History shows how closely the rise or fall of
single persons is bound up with the fortunes of whole groups.
A change of a person P in his environment E often appears as
active locomotion of P when in reality this change is not a result
of a movement of P in relation to his immediate social environ-
ment, i.e., the group G. In reality it may have been brought
about by a movement of the group G in relation to the whole
field.

It is easy to characterize social locomotions of this kind.
They correspond to the change of place of a person who is in a
moving train. Such cases offer no special conceptual diffi-
culties. One only has to take into account the relative char-
acter of all movement.

More surprising than this kind of locomotion are other cases.
We can offer an example from our moving-picture material. A
two-year-old child C who still has trouble in walking up- or
downstairs without support wants to place his ball on the land-
ing. In order to do so he has to go up three steps. Topo-
logicaUy we could represent the initial stage of the situation as
follows (cf. Fig. 200). Between the goal G and the child C
there is a barrier which consists of the following zones: climbing
the first step (ci), climbing the second step (cz), the third step
(^3), and finally going beyond the edge of the landing which is
still a danger zone (dz) from which the ball may roll back.
Let us assume that the child has already picked up the ball (Bo).

TOPOLOGICAL PSYCHOLOGY

(The child C and ball may be represented as regions which have
a partially common boundary.)

The child succeeds in bringing the ball up all three steps to
the danger zone dz (Fig. 206). Then he drops the ball and it
rolls down again. Thereby the following situation comes into
being (Fig. soc). The goal is suddenly moved to a distance.
There are now more regions between the child and his goal than
in the original situation. In order to reach his goal the child

(a)

olz

Be*

(b)

0,3

0.,

P l

\*Q

(c)

FIG. 20. " Ground moves under one's feet/' (a) Child starts to climb the
steps carrying a ball; (6) the goal is nearly reached; (c) the ball has rolled down.
C, Child; Ba t ball; G, goal, cl, Climbing up the first step; 02, the second step;
c3, the third step, dx, Climbing down the first step; d2, the second step; ^3, the
third step, pi, Packing up the ball; dz, danger zone.

must now go through the following regions: He must climb
down the third step (^3), the second step (^2), the first step (Ji),
and pick up (pi) the ball. Then he must again climb up the
first step (<PI), the second step (c2\ and the third step (03) with
the ball in his hands, and he must take the ball beyond the
danger zone (dz) on the landing.

Without doubt there occurs a significant change in spacial
relations of C and G. Since all locomotions can be determined
only relatively there is no reason for not speaking of locomotion
in this case. The causes of this locomotion are essentially
different from those of the active movement of the child between

PSYCHOLOGICAL REGIONS, LOCOMOTIONS 1x5

the first and the second stages. Then it was the ball which
separated itself from the child and carried out the locomotion to
pi. At the same time however the spacial relationship of C
and G underwent a marked change as a result of the locomotion
of the ball. Since C did not bring about this change of position
by active movement on his own part, and furthermore since he
was not passively moved by another person, he might have the
feeling that the "ground moved under his feet." Such an
event may well be characterized as a locomotion of the sur-
rounding field in relation to the person. Locomotions of this
kind are often accompanied by other changes in the structure
of the field.

As we have mentioned above there seems to be no doubt that
in psychology as in other sciences locomotion can be determined
only relatively, i.e., as a change of position of one region in rela-
tion to others. One can therefore raise the question whether it
means anything to distinguish the movement of the person
from an opposed movement of the surrounding field. The time
has not yet come when psychology can enter into a discussion
of this problem which corresponds to the question regarding the
relativity of movement in physics.

THING AND MEDIUM

Psychological regions of the environment can show very
different dynamical properties. They can offer either great or
slight resistance to locomotion; they can attract, can be neutral,
or can repulse; they can represent living beings or objects; they
can exhibit any degree of fluidity or elasticity; they can react
differently to different influences. We shall discuss a few of
these properties later. At this place we shall only emphasize
a certain kinematic difference.

If one represents an activity as a region one represents it at
the same time as a "medium" (32). A goal toward which one
moves, on the other hand, has not the character of a medium
but of a "thing." The person who mpves also is such a thing,
as is likewise another person in the life space.

Tl6 TOPOLOGICAL PSYCHOLOGY

From a kinematic point of view one may call a region a
medium (in opposition to a thing) if movements can be carried
out in or through it (cf. 32),

There is no question about the thing character of a great
number of quasi-physical objects like a ball, a table, a tool, etc.
The medium character of certain other regions is also clear, for
instance the air for the flyer. In other cases of quasi-physical
regions the character is not so unambiguous. It can change
with changes in the situation. A hut in the mountain has the
character of a thing as long as one is trying to reach it from a
distance. As soon as one goes in, it serves as a region in which
one can move about. Similarly, a bucket may at first impress
a child with its object character. Later, in the course of a game
with water it may become a region in which different kinds of
locomotions can be carried out. The fact that it is not a loco-
motion of the whole person but only a locomotion of a hand is
irrelevant for our consideration.

This example may show that, as far as the medium character
of a region is concerned, one must think not only in terms of
locomotion of the whole person but also of any other regions.
It can well be that what is a thing for the locomotions of the
whole person is a medium for the locomotions of other objects.
For instance a toy bank is a medium in regard to the coins that
are in it.

A somewhat different example of the fact that an object can
be at one time a thing and at another a medium is the difference
in the psychological import of an object for a child and for an
adult. A barrel may be a thing for an adult while a child can
move about in it as in a medium. This is one reason why the
same outer world may have different meaning for child and
adult.

In regions other than the quasi-physical it also often happens
that a thing changes into a medium or the reverse. The home
work that a pupil has to hand in within three weeks, and an
operation which a person must undergo at the end of six
months have the character of a thing for the person concerned.
The same holds for many events or actions which are tempo-

PSYCHOLOGICAL REGIONS, LOCOMOTIONS 117

rally distant and which are imdifferentiated regions in the life
space. Gradually as the time for the operation or the final
date for handing in the home work conies nearer it becomes
something that one must "go through. " We may say that
there is a clearer differentiation of its regions into subregions in
which one can move about, and that the medium character
finally becomes entirely evident as one enters the region.

One thinks generally of a medium as a region which offers no
resistance to a movement, whereas a thing seems something
dynamically compact and solid. However, one must realize
that regions may offer all possible degrees of resistance. There
are regions which can be crossed but which still act as obstacles
to movement. For bodily locomotion for instance, a thick
underbrush is a medium which offers definite "friction/* This
friction can increase until it is impossible to advance farther.
Then one is no longer dealing with a space of free movement
but with a boundary of this space. This example shows clearly
that there are all possible transitions between the dynamical
properties of thing and medium.

CHAPTER XII
BOUNDARIES OF PSYCHOLOGICAL REGIONS

DEFINITION AND DETERMINATION OF PSYCHOLOGICAL
BOUNDARIES

In making use of the mathematical definition of a boundary
one can give the following definition:

Definition: We shall designate as the boundary of a psycho-
logical region those points of a region for which there is no sur-
rounding that lies entirely within the region.

One can determine existence and position of a certain bound-
ary in the psychological life space in different ways according to
the nature of the case under consideration. It is possible that
one can survey, for instance in the quasi-physical field, a
number of regions simultaneously and determine the boundaries
between them without difficulty.

For dynamical problems psychological locomotion plays
again an important role. However, it would not be correct to
determine, as perhaps seems most simple, the boundary points
as those points "beyond which one cannot go" without leaving
the region. For such a determination would presuppose the
concept of direction, which is not admissible in topology.

It would be less objectionable to designate as the boundary
points those points of a psychological region which one can not
encircle without leaving the region. In the case of the space of
free movement one could for instance say: Its boundary points
are those points which the person can touch but not encircle.
But this determination also is not unassailable. For we find
cases in which one can encircle boundary points of a region
without leaving it; namely, when we are dealing with boundary
points toward a second region which lies like an island entirely
within the first region. If one can exclude this possibility in
a given case the procedure is correct.

118

BOUNDARIES OF PSYCHOLOGICAL REGIONS 119

In carrying out a locomotion the experience of crossing a
boundary is often a clear one. This is for instance the case
when one climbs over a fence or enters a strange house for the
first time; or, to use an example of a quasi-social locomotion, if
one is admitted to membership in a dub by some special cere-
mony. Thereby the position of the boundary is quite accu-
rately determined. However, there are cases in which one can
establish with certainty that the locomotion has proceeded
from one region into another one, although the crossing of the
boundary does not become evident as a special event during the
locomotion. For instance, one can gradually pass from one
social circle into another. A path may lead from the moun-
tains into lower hills and on to a plain, or from a great city
through more and more open suburbs into the country and it
may be impossible to describe definite boundaries between these
regions. The same is true for all gradual transitions between
two regions. For instance, it can happen in conversation that
one is not even aware of a "gradual transition." That the
person has passed the boundary can then be inferred only
indirectly from the fact that he is in another region. In these
cases it even remains doubtful how many boundaries and inter-
mediate regions the locomotion has crossed.

SHARPNESS OF A BOUNDARY; BOUNDARY ZONES

Even when the transition occurs gradually one can make
statements about the position of the boundaries. One can in
such cases think of the boundary as a boundary zone, that is,
not as a one-dimensional but as a more-dimensional region.

The position of this intermediate region is determined above
all by the fact that it intersects the path which corresponds
to the locomotion concerned between its beginning point and
end point. It is often possible to determine the position of the
intermediate zone still more exactly if one can locate all points
of the path which definitely lie in the beginning region or end
region. By such a convergent approach one can often deter-
mine the position and width of the boundary zone with a high

120 TOPOLOGICAL PSYCHOLOGY

degree of accuracy. According to the width 1 of this zone of
gradual transition we shall talk of more or less sharp boundaries.
We have already pointed out that on closer examination we find
that all real boundaries of psychological regions are not curves
or surfaces without thickness but that they themselves are
more-dimensional regions. However, there are considerable
differences in this respect.

In the example of the bathtub the edge is so thin that it has
psychologically not the character of an enclosing region but

PIG. si. Boundary zone between two regions. /, inner region; O, outer region;
BZ, boundary zone.

that of a boundary without thickness. To some degree the
same may be true of the walls of the prison. But if there is a
ditch or a barbed wire entanglement behind the wall, the
boundary assumes much more definitely the character of a zone.
The same is true if the prison wall is protected by machine guns.
Then in making his escape the prisoner must also pass the
danger zone of the machine guns. In such cases the inner
region (7) is separated by a boundary zone (BZ) from the
outer region (0) (cf. Fig. 21). Instead of two regions (7 and O)
and a boundary (7?) one can then speak of three regions (7, 0,

1 The concept of width, goes beyond topology. However sometimes one can
make use of the fact that one boundary zone lies entirely within another. The
transition to a greater exactness of determination of boundaries has meaning also
from a purely topological point of view because it can be based directly on the
relation of *' bemg-contained-in."

BOUNDARIES OF PSYCHOLOGICAL REGIONS 121

and BZ} which have a definite topological relation. Each of
these regions can of course have part regions.

Definition: We call a boundary zone between two regions
(m and n) that region (BZ) which is foreign to m and n and
which has to be crossed by a locomotion from one of them to
the other (m 9 BZ = o; n BZ = o; m + # + BZ is a connected
region).

In the example of the prison the boundary zone is a connected
region which separates as a whole an inner and an outer region
and whose border consists of two separate Jordan curves.

If the width of a boundary zone is psychologically irrelevant,
one can represent it as a one-dimensional boundary. Such a
representation is permissible as a first approximation even when
we are really dealing with a boundary zone. The relation
between boundaries and boundary zones is similar to that
between points and more-dimensional regions. As we have seen
one can sometimes use a point to represent undifferentiated
regions. It is dear that in a similar way a boundary can stand
for a boundary zone which is not differentiated in depth. One
can always proceed later to a more exact representation by
means of a boundary zone.

It often happens that in the course of events the character of a
boundary changes, for instance when the person concerned
approaches the boundary or begins to think about it. Then
boundaries which have at first the character of a one-dimen-
sional curve may later differentiate themselves into boundary
zones. The reverse may also happen. We found a similar
dependency on psychological distance when we discussed thing
and medium.

Finally, it can happen that regions assume the character of a
boundary zone because a path which connects two other
regions has to cross it. An example: a child who is eating his
dinner must prepare his lessons before he can play. In this
case the lessons assume the character of a boundary zone.

We mentioned that one can think of unsharp boundaries as
boundary zones whose width corresponds to the degree of
unsharpness. Sharp psychological boundaries correspond best

122 TOPOLOGICAL PSYCHOLOGY

to mathematical boundaries. On the other hand not every
boundary with pronounced depth implies an unsharp transition.
An example from social psychology may serve as a demon-
stration. While the boundary between different economic
classes is in general relatively unsharp and is characterized by a
gradual transition, the boundary of some social groups such as
an exclusive club is sharply defined. This means that for every
person it is clearly determined whether or not he belongs to the
group. Nevertheless the boundary of such a group can have
the character of a boundary zone. In order to join the dub
for example it may be necessary to have one's name put on a
waiting list in advance. Sometimes several such stages are
prescribed. Therefore the existence of a boundary zone does
not necessarily lessen the sharpness of the boundary, for the
boundary zone itself may be a region which is clearly structured
and sharply defined as to its boundaries.

The degree of sharpness of the boundary seems to be of great
importance for the inner structure of a social group, especially
for its homogeneity and for the processes within it. An
example is the difference between social life in the United States
and in Germany. It seems to me to be one of the most impor-
tant characteristics of the social structure of the United States
that many socially relevant regions are more sharply bounded
than in Germany. This can be observed in small things of the
daily life, as well as in the political, vocational, and social
structure of the country (for instance, in the importance which
one attaches to punctuality, or the exactness with which a
schedule is made out; in occupational specialization; in the
sharpness with which governmental responsibilities are defined
(55)). Pedagogically it is of great importance whether the
regions of play, eating, sleeping, and working in the life space
of a child are dearly and sharply separated or whether there
exist broad regions of unclear transitions. The same is true,
as we have mentioned, of the regions of the permitted and the
forbidden, of freedom and coerdon. Undear zones of unsharp
transitions lead more often to tension and conflicts. When
dealing with this question also one will have to take into account

BOUNDARIES OF PSYCHOLOGICAL REGIONS 123

the special characteristics of the situation. Incidentally in
pedagogical matters too, especially as regards what is forbidden
and permitted, the boundaries of the regions seem to be com-
paratively more sharply defined in America.

One must not confuse sharpness of boundaries with their
solidity (see p. 124). The life space of the child for instance
corresponds, as we shall see, to a relatively fluid medium.
Nevertheless it seems to show an especially strong tendency to
sharp boundaries. It may perhaps be a general rule that
unsharp boundaries are more apt to be found in a relatively solid
than in a relatively fluid medium. Certainly individuals differ
in respect to the sharpness of the boundaries within the life
space and in the tendency to avoid unsharp boundaries.

DYNAMIC PROPERTIES OF PSYCHOLOGICAL BOUNDARIES

Boundaries as well as boundary zones can have very different
dynamic properties.

Barriers. Dynamically the principal difference between the
prison wall and the edge of the bathtub is that the one is much
more easily crossed than the other. In general the resistance
which a boundary offers to being crossed is very important.
This resistance can have all values between almost zero and
infinity. This is true for unsharp as well as for sharp bound-
aries. If the prison is surrounded by an open field, there is a
gradual transition between the danger zone in which an escaping
prisoner could be reached by a machine gun and more distant
regions which are less dangerous. In other cases, for instance
if there is a dense wood just beyond the open field, the region of
greatest risk is sharply bounded. In both cases the boundaries
of this region need not offer any special resistance to the bodily
locomotion of the prisoner. It may be possible to go through
the gate that separates street and garden without noticeable
resistance. In general there is no difficulty in crossing the
sharply defined boundary between sidewalk and street pave-
ment. Also the experience of a transition is not necessarily
limited to boundaries which are difficult to cross but can also
occur with easily passable boundaries, especially when the

124 TOPOLOGICAL PSYCHOLOGY

boundary is sharp and the two regions sufficiently different in
quality.

At the other end of the scale of difficulties are the impassable
boundaries. For the man who cannot swim, a river may be
impassable. Each insurmountable object is an example of an
impassable boundary for quasi-physical locomotion. The
physical properties of impassable boundaries can be of very
different nature. Spring floods or an unusually swift current
may make a river impassable for the swimmer; the speed of a
railroad train means a strong boundary against any attempt
to board or leave the train while it is in motion.

Like the boundaries of the quasi-physical regions those of the
quasi-social regions can be different in regard to their dynamical
properties. It may be very difficult for an outsider to gain
access to a certain social group. However, in social regions
too, the boundary does not necessarily imply difficulty for
locomotion. Many dubs represent well-defined groups,
although entrance into them may offer no real difficulties.
Also the boundary of a crowd, for instance of spectators or of
people who stage a demonstration, may easily be crossed. The
strength of the boundary can suddenly increase. If, for
instance, during a demonstration the crowd is attacked from all
sides, it may become difficult to separate oneself from it.
. Definition: We shall call boundaries (boundary zones) which
offer resistance to psychological locomotion "barriers/' We
shall speak of barriers of different strength according to the
degree of their resistance.

We shall continue to use the concept of boundary in a purely topo-
logical sense. The term "psychologically real" boundary therefore does
not imply defined dynamic properties.

Our examples have shown that a barrier can offer different
kinds of resistance to locomotions. It can have very different
degrees of "solidity," different degrees of "rigidity" or "elas-
ticity" (see p. 159). It can oppose a locomotion at a certain
point like a fence. (This is true of the edge of the river for the
person who cannot swim.) Or it can have the character of a

BOUNDARIES OF PSYCHOLOGICAL REGIONS 125

boundary zone, which offers resistance although it does not
make further progress impossible. (This is true for instance
of a piece of land which is difficult to cross.) In this case one
can speak of "friction." Finally the barrier can have the
character of a more or less permeable membrane.

In discussing the space of free movement we have already
mentioned that the boundary of a psychological region can be
easily passable for one kind of locomotion and impassable for
another. The strength of a barrier is therefore always to be
defined in relation to a certain kind of locomotion. It is not
only different for quasi-physical, quasi-social, and quasi-con-
ceptual locomotion, but also for different kinds of quasi-
physical locomotion (swimming, driving, looking). The follow-
ing example from Mrs. Lindbergh's North to the Orient (61,
pp. 220-221) gives a very impressive description of such a
difference. Colonel Lindbergh and two physicians were leaving
a crowd of starving Chinese in a flood area to whom they had
tried to take medical supplies: "Looking down on the spot they
had just left, the men in the plane were acutely conscious of the
miracle of their escape. A moment before they had been down
in that crowd of starving people, some of whom might live until
spring; many would die before the waters receded. Now,
headed for Nanking, safety, food, and shelter were as assured to
the fliers as in their own homes. Separated from those desper-
ate people below only by a few seconds in time, only by a few
hundred feet in distance, they were yet irretrievably removed in
some fourth dimension. The two worlds were separated by a
gulf which, although not wide, was deep, perilous, and unbridge-
able. At least it was unbridgeable to the owners of the sam-
pans* The fliers had crossed over from one world to another as
easily, as swiftly, as one crosses from the world of nightmare to
the world of reality in the flash of waking.

"They had a gun; they had a plane powerful as any genii
to be summoned from a magic lamp. And yet, magic rests on a
knife-edge a lamp, a tinder-box, an 'open sesame.' It is a
hair-bridge between captivity and escape; safety and danger;
life and death* The pull of a trigger, the press of a switch

126 TOPOLOGICAL PSYCHOLOGY

without these, the three magicians flying back to Nanking
would have been simply three people in a starving, dying, and
devastated land."

The difficulties of crossing the boundary are not always the
same in entering a region and in leaving it. Thus we have to
recognize that the dynamic characteristics of a boundary may be
different for locomotion in different directions.

A boundary does not necessarily have the same strength at
all points. It often has parts which one can pass easily and
others which offer great difficulty. The fact that different
sectors of a boundary can have different dynamical properties
is important, for instance, for the problem of detour.

The boundaries of social regions too do not usually have the
same solidity at all points. Success or failure in entering social
groups often depends upon whether one finds the correct
approach. The successful impostor is especially clever in
choosing his points of approach.

It may be mentioned that even a single person can be thought
of as a social region in the sense discussed above. Boundaries
of different degrees of strength correspond dynamically to the
different degrees of accessibility which distinguish individuals
from each other (55). Again the boundary of a person is not
equally strong at all points. In trying to make a contact it is
important to find the right approach.

Boundaries Which Affect Communication, In determin-
ing boundaries of psychological regions and their dynamical
properties we cannot limit ourselves to the consideration of
psychological locomotion. As we have already said, com-
munications are of no less importance for quasi-physical than
for quasi-social and quasi-conceptual fields.

Definition: By degree of communication of a region a with a
region b we understand the degree of the influence of the state of
a on 5.

In so far as communications are concerned we designate
psychological boundaries in general as dynamic "walls." We
speak of the strength of a wall in the sense that a high degree of
communication corresponds to a weak wall. For the concept

BOUNDARIES OF PSYCHOLOGICAL REGIONS 127

of barrier the difficulty of a locomotion across a boundary is
relevant; here we are dealing with boundaries which affect the
influence exerted by the state of one region on the state of
another.

In treating the topology of the person we shall have oppor-
tunity to discuss more in detail the problems of communication
of regions. However, it must be emphasized that these prob-
lems have bearing on the psychological environment as well.
For instance, as we have mentioned, the degree of communi-
cation between different social groups is of essential
importance.

The fact that a region a is in communication with a region b
does not, according to our definition, imply that 6 is in equally
close communication with a. As we have seen, the strength of
the resistance which a barrier offers to locomotion can vary
according to the direction of the locomotion. Likewise the
strength of a dynamic wall can have a different value for proc-
esses of communication from a to 6 and from 6 to a.

An example is the communication between two persons when
one looks at another. If a mother looks her child in the eye
when she is trying to induce him to carry out a certain action
or to emphasize a command, the looking is certainly a real
process which can have a great influence on the course of events.
One could think of representing "looking at" as a locomotion.
It certainly is a kind of intercourse. However, the objection
can be made that it is not the whole person A which carries out
the locomotion to 5. Yet the "looking at" brings A into
contact with B. "Looking at" in this respect corresponds for
instance to a touching of B by stretching out the hand. As a
matter of fact "looking at" can be a direct substitute for the
touching of the child by the mother. "Looking at" therefore
would have to be represented as a reaching out of an "arm";
or topologically as locomotion of a part A r of A in such a way
that the part touches B without separating itself from the main
region of A. (A + A' + B is a connected region.) (Fig. 220,.
This representation agrees with that of social intercourse,
Fig. i&c, which we have discussed on page 102.)

128 TOPOLOGICAL PSYCHOLOGY

The mother establishes the contact to influence her child,
that is, to change the state of the child in a certain way. The
"looking at" is therefore an establishment of contact for the
purpose of a communication in the defined sense. This com-
munication occurs only if the mother succeeds in catching the
child's eye. Mother and child must look at each other. The
child often tries to avoid the influence by dodging the mother's
glance. He avoids getting into communication with the
mother. Sometimes he may look at the mother in an impudent
way. In these cases the child is closed to the influence of the

PIG. 22. Communication of A with B by "looking at.** (a) Represented as
"arm" of A^ (b) represented as separated region of A\ (c) represented as power
field of A. A, person looking at Bj A\ region corresponding to "looking at."

mother's glance in spite of the fact that he looks at her; there
exists an inner wall which more or less blocks the influence of
the mother. The child wants to preserve his own state and
even to influence the state of the mother. The opposite case is
realized when the child looks at the mother ready to carry out
her least wish.

A similar situation exists if one looks at a work of art in an
uncritical manner and gives himself entirely to it. In this case
the onlooker actively establishes a communication by "looking
at" and then assumes an attitude of complete receptivity; that
is, he tries to make the dynamical walls between himself and
the work of art as weak as possible and to allow the influence
to proceed toward his own person. If the glance only plays
across a number of objects, the communication is usually very
weak.

BOUNDARIES OF PSYCHOLOGICAL REGIONS 129

Thus in all these cases "looking at" establishes a contact.
But the direction in which the influence occurs (whether mainly
from A to B or from B to A) and the degree of communication are
very different. It depends on the state of the person who looks,
on the state of that on which he looks, and on the kind of looking.
The degree of communication always depends on the properties
of the communicating regions and the kind of communication.

Another fact of general methodological importance may be
pointed out in connection with this example. One could
object to the representation which we have given on the
grounds that the bridge between A and B which is established
by looking has not the character of a continuously solid part
of A as is the case when A reaches out his hand toward B. It
should therefore not be permissible to represent "looking at"
as a reaching out of an arm. Without doubt there are essential
differences between these two cases, and one could think of
representing "looking at" according to Fig. 2 si as a touching of
B in which there is no continuous connection between A and B.
A would then throw his glance to B like a ball (-40- TJlis bal1
however has not the character of a solid body but rather that of
a force. Indeed direction and kind of looking are directly
related to what one can call the power field, the sphere of
influence of a person and what one can represent psychologically
as a field of forces, Wiehe (91) found that these fields of forces
are in general stronger and reach farther in front, in the direc-
tion of the person's glance, than behind him. The dynamic
nature of the arm which reaches from A to B can probably best
be thought of as that of a field of forces (Fig. 22*;) ; "looking at "
can be considered as a change of the position and intensity of
this field of forces. However, the topological correctness of our
representation is not thereby impaired. There is no reason
why one should not also treat fields of forces as regions and
represent their relations of position, in a first approximation, by
topological means. Certainly the dynamic nature of such
regions needs a characterization which goes beyond topology.
This, however, as we have said repeatedly, is true of all psycho-
logical regions.

130 TOPOLOGICAL PSYCHOLOGY

The question whether "looking at" is to be represented as an
arm A f according to Fig, 220 or as a region A f which is separated
from A according to Fig. 226 also must be asked if one con-
siders "looking at" as a field of forces. There may be cases in
which Fig. 226 is preferable. In general however Fig. 22^,
which corresponds to Fig. 22^, may be more correct. For it is
possible to impair the communication that exists between the
mother and child who are looking each other in the eye by
interference at any point of the immediate region between A
and B or to block it entirely by erecting an opaque wall between
them.

Boundary Zones Which Can Be Passed Only with Diffi-
culty. As we have mentioned, a psychological barrier need not
have the character of a thing but may be like a boundary zone
that can be crossed only with difficulty. Since one usually
understands by the term "barrier" a solid thing-like object this
group of psychologically real boundaries may be treated
specifically in this place.

We have mentioned already, that the resistance of a boundary
zone is different according to the kind of locomotion concerned.
The same stormy lake which is impassable for a swimmer and
which a sailboat can cross only with difficulty may offer easy
passage to a rugged steamer. If one wants to overcome a
barrier, one usually does it not by continuing the original kind of
locomotion with increased efforts but by choosing another kind
of locomotion against which the barrier is weaker. The
problem which the barrier sets is essentially one of finding the
most suitable kind of locomotion or communication. This is
true of social communication, for instance of trying to get a
message to a political prisoner in spite of barriers. The use of
tools also is closely connected with this question.

We shall return later to the relation between solid barriers
and boundary zones which can be passed only with difficulty.

Zones of Undetermined Quality. So far we have treated
barriers and boundary zones whose qualitative characteristics
were, at least to some extent, determined. However, there are
rather frequently cases in which the boundary zone not only

BOUNDARIES OF PSYCHOLOGICAL REGIONS 131

contains undetermined sectors but where the approach to a
point is impossible because the intervening region cannot be
determined or where it is, so to speak, psychologically "empty."
Unsolved mathematical problems often offer just this kind of
difficulty. One knows that it is possible to solve the problem,
but one cannot see any approach to the solution. We can say
about the situation only that there is within the life space a
region G which is separated from the person P and which corre-
sponds to the solution of the problem (Fig. 230). But in this
case the intervening region Z7 between P and G does not consist
of empty space in the sense of a medium which can be crossed
easily. It is rather a region whose quality cannot be deter-
mined sufficiently and which therefore cannot be crossed.

If one represents such a situation more exactly, then one has
to say: there is for P the region of the mathematical task A
which, in so far as it is a problem, can be sufficiently char-
acterized (Fig. 236). For P only the starting point (sp} within
A) which corresponds to the way in which the question was set,
is accessible. Only this region sp therefore is part of P's space
of free movement (C, D, E, . . . ). (The boundary of this
space of free movement may correspond to the line 5.) There is
further a region G within A which corresponds to the solution.
Sometimes, to be sure, it is not even certain that such a region G
exists at all. In any case no pathway is visible from sp to G
because the quality, of the intervening zone cannot be
determined.

The situation often develops in such a way that the person
succeeds in finding a region r (Fig. 23^) which is connected with
G and which he hopes to reach more easily from sp. Little by
little there can appear a larger group of such regions connected
with G (r\ s, t } D}. At the same time one usually tries to find
more regions connected with sp (c, d, e) in such a way that one
can hope finally to build a bridge from the starting region sp
to the solution G } i.e., a series of regions the topological sum of
which sp + c + d + e+ * + v + t + s + Gisa connected
region. When one has to reach a certain region by moving
through such an unstructured zone it often remains uncertain,

132

TOPOLOGICAL PSYCHOLOGY

as long as one has not really completed the bridge, whether the
regions s } t, v and c } d, e, which are developed at first, will or
will not serve as an approach to the solution.

This type of barrier is not limited to conceptual locomotion in
mathematical tasks. Similar situations can often be observed
in connection with bodily locomotions. For instance one may

FIG. 23. Boundary zone of undetermined quality, (a) A mathematical task
involving an undetermined boundary zone; (&) situation in the beginning stage;
(c} attempts to bridge the gap by proceeding from both ends. A, Region cor-
responding to the mathematical task; P, person; G, goal (solution of mathematical
task) ; t7, undetermined region between person and goal; sp, region corresponding
to starting point; c t d, e, r, s, A 9 , regions, determined in character, which are
intended to bridge the gap between s$ and G\ V Z>, E, F, part of P's space of free
movement; 6, boundary of JP's space of free movement.

want to go from the railway station in a strange city to a certain
house without having a map or without the possibility of asking
for information. Or, one wants to find a person whose house
and name one does not know. In such cases there exist the
well-defined barriers of the kind we have discussed in an earlier
chapter, e.g., the obstacle of physical distance. But besides
difficulties from such barriers the principal difficulty lies in the
fact that one does not know whether any given movement brings
him closer to his goal or takes him farther from it. A character-
istic property of barriers of this type is that they depend

BOUNDARIES OF PSYCHOLOGICAL REGIONS 133

directly on one's knowledge or rather on one's ignorance of the
situation. According to Tolman's concept of cognitive struc-
ture one can say that the difficulty in these cases consists in the
fact that the field is unstructured with reference to cognition.
If, for instance, the stranger finds a map of the city the difficulty
is removed.

Maze experiments with rats offer an example of the over-
coming of such difficulties of locomotion on the basis of cognitive

FIG. 24. Maze learning. Connectedness or not connectedness with respect to
the region containing food is the aspect according to which the field is structured.
st, Starting point; I, position of rat; a, b, c, regions adjacent to i; F, region con-
taining food; /, food.

structuring of the field. A rat may have found the food for the
first time in a new maze. It is brought back to the starting
point. It then "knows " that it is possible to get from its place
to the food, but it does not yet know the path. The main task
of orientation in such a maze is the following: the rat may have
run from the starting point st (Fig. 24) to the first branching
(i). It is then faced with two possibilities, to enter region b
or to enter region c. The fundamental difference between b
and c in this case is that region c is a part of a connected region
F which contains the food/(/< F;JF> C), whereas region b
(seen from station i) is not connected with the region of the
food (F b = o). The rat "knows" the path as soon as it is
able to decide at each branching (2, 3, 4, 5) which of the adjacent

134 TOPOLOGICAL PSYCHOLOGY

regions "leads to" the food and which does not. In other
words, the rat is able to find its way as soon as it knows the
topological relationships of the regions well enough to make
the correct decision at each critical point. The analogy to the
mathematical task is obvious. In such maze experiments
the first structuring into an articulated series of regions often
occurs near the food (regions g, h, 5, i}. In these cases the
rat learns the maze from the goal backward.

If one puts a rat into the maze without food, he gets a chance
to " orient" himself, which means that what is first an unstruc-
tured field becomes structured. Insofar as this process of
structurization tends to be complete, the animal will know at
any point the relation to the adjacent regions and perhaps also
to the more distant ones.

In the case in which the animal learns from the beginning
to go after the food, there, too, is a process of orientation and
structurization but the structurization is a very special one
with a " start" and "end." In the non-reward orientation
case the structure will be much more variable and will permit
of more aspects (Auffassungen) .

What happens at the moment one puts food in the maze after giving
the animal the orientation period is, to my mind, so far as cognitive
processes are concerned, this: The field undergoes a restructuring so that
the one aspect which enclosed the start-end relation will become dominant.
One would assume, also, the following conclusions: First, after an optimum
time has been given to the animal for the orientation period, an increase
in the orientation time should be of no further help to the " latent learning"
(85). Second, there should be cases in which the second learning should
occur through one repetition. I assume thereby that the restructuring
of an already structured field can be done with rats by one sudden act.
This statement has certainly some limitations and will not hold for very
complicated mazes or for unintelligent rats. Third, it may be possible
to create mazes so that the "natural" aspect resulting from the first
structuring would be of a type which would be difficult to restructure
quickly. I don't know how difficult this restructuring could be made,
but I think there should be appreciable differences for different set-ups.

In locomotions in the quasi-social field also a person may come
to regions which he cannot cross because they do not have

BOUNDARIES OF PSYCHOLOGICAL REGIONS 135

sufficient cognitive structuring. It is only in rare cases that the
path which leads to the social goal is clearly known in advance.
For instance the occupational goals of a young man are often
of such a kind that the region between his present position and
the goal is not at all, or only vaguely, structured with reference
to cognition.

The difficulties which result from the absence of cognitive
determination of the boundary zone are very common, They
are essentially different from other types of barriers and from
everything that one has been accustomed to think of as a
barrier if one is guided by the idea of a physical barrier. But
these zones actually hinder locomotion and must therefore be
called barriers. They are comparable to barriers of the
highest degree of solidity in so far as locomotions through such
zones usually are impossible as long as their cognitive structure
is not sufficiently known. On the other hand this impossibility
is not a result of solidity of the barrier or of friction within the
boundary zone such as we have discussed above. The impassa-
bility of such a zone depends in a special sense on the " knowl-
edge " of the person. The friction of a cognitively well-defined
boundary zone is not removed by the fact that the person
concerned knows of it. The real locomotion through the region
remains difficult in spite of this knowledge. On the other hand
the difficulty of locomotion which results from a cognitive
indetermination of a zone is actually removed by a recognition
of its properties. Therefore we find in these cases a peculiar
relationship between the knowledge of a zone and the possibility
or impossibility of locomotion through it.

Even with qualitatively well-characterized barriers of given
solidity or given friction there is a relationship to cognitive
factors. A change in the knowledge of the boundary zone, or in
other words a change of its cognitive structure, may show the
existence of a part of the boundary zone which can be crossed
more easily than the parts which were known originally.
Furthermore, a more exact knowledge of boundary zones often
allows one to find a different kind of locomotion by means of
which one can cross the zone without meeting great resistance.

CHAPTER XIII

THE RELATIVE POSITION OF TWO REGIONS

The concepts of psychological region, of psychological
boundary, of locomotion, and of communication allow us to
represent an infinite number of different structures of the
psychological life space. To make these representations is a
task of empirical psychology. Here it must suffice to discuss
a few simple cases.

FOREIGN REGIONS

The relative position of two psychological regions is topo-
logically especially easy to represent when one simply con-
nected region is enclosed by another. An elementary example
is the space of free movement in a prison. If we disregard the
differentiation within the prison, one can say: The walls of the
prison, like a Jordan curve, separate an inner, connected,
limited region (the space of free movement) from an outer
region (the region of "freedom"). In the construction of every
prison use is made of the fundamental mathematical fact that
each path from a point of the inner region to a point of the
outer region must intersect the Jordan curve: aside from the
dynamic property of the walls (their solidity), it is above all
their topological properties, namely, their arrangement as a
closed curve, which makes the escape of a prisoner impossible.

The following fact too is directly connected with the topo-
logical relationships. The cell C in which the prisoner may be
located represents in itself a connected limited region which is
bounded by a Jordan curve. 1 In order to gain freedom he
prisoner must not only surmount the boundaries of this region
but also the outer walls of the prison. How many successive

1 As noted above we refer, for the sake of simplicity, only to the two-
dimensional in this discussion.

136

THE RELATIVE POSITION OF TWO REGIONS

137

obstacles the prisoner has to surmount in making his escape
depends essentially on topological relationships, namely, on
how many regions (C, RI, R^ Rz, - ) lie one within the other
(C < Ri < Rz < * ' ' ) in such a way that their boundaries
(bx) have no common parts (be * &v * a " * * = o. Fig. 253).
If the prisoner's cell lies on the outer wall (w) of the prison

PIG. 25. Topology of the prison, (a) Series of walls without common parts; (&)
walls with common parts. P, person; Pr, prison; w, wall.

(Fig. 256), that is, if the boundaries of the cell (i c ) and of the
whole prison (bp r ) have common parts (bp r b c ?* o), the
prisoner would have to cross only one boundary.

If two foreign psychological regions do not correspond to an
inner and an outer region which are separated by a Jordan
curve, it is psychologically important to determine whether or
not they have a common boundary (Figs. 10 and 15). We
have already explained how this can
be determined (see p. 94).

OVERLAPPING REGIONS ; THE RELATIVE
WEIGHT OP SITUATIONS

PlG. 26. Boundary zone

The Cases in which tWO regions an< l the overlapping of regions.
, , , ,, , 2, common part of A and B.

overlap, in part or wholly, play an

important role in psychology. One can often consider the
boundary zone z between two regions A and B as an area in
which the two regions A and B overlap (Fig. 26). This is true
for instance of boundary regions between two occupations or
two branches of science. The boundary zone becomes thereby
an intersection of the regions A and B both of which are defined
as including 0. (z = A B.)

138 TOPOLOGICAL PSYCHOLOGY

A psychologically important application of the concept of
overlapping is the overlapping of two situations. A child may
be eating and at the same time listening to the song of a bird.
The listening can be the major and the eating the minor activity,
or the reverse. Between the two extremes many transitions
are possible. Such cases in which one is involved to different
degrees in two different activities are of common occurrence.
But they offer considerable difficulty for description as well as
for treatment of their dynamic facts. One can meet some of

these difficulties in the follow-
ing way.

One can say in such cases
that the person P is in two
regions at the same time.
Each of these regions is usu-
ally well structured and has
the character of a situation.
One can therefore speak of an
overlapping of two situations.

FIG. 27. Overlapping situations. These situations (Si and 2)

*"* * foreign regions, but
have a common intersection.
This is proved by the fact that the person P is at the same
time in both regions (Si > P;5 2 > P; therefore 5i'5 2 5*0)
(Fig. 27). The psychological structure and content of the two
partly or wholly overlapping situations can be very different.

If two or more situations overlap in such a way, each situ-
ation possesses at a particular moment a certain "relative
weight " (importance, potency) for the person. The change
of this relative weight is a dynamically important process.
The forces which result from a situation seem other things
being equal to increase and decrease with the relative weight
of this situation. The change of the relative weight of a situ-
ation is one of the principal ways of influencing other persons
and is often used for pedagogical purposes (18).

It is sometimes possible without special difficulty to dis-
criminate with sufficient exactness between several degrees of

TEE RELATIVE POSITION OF TWO REGIONS 139

relative weight and to characterize the state of the life space
by the quotient of the relative weights of two or more situations.

DIFFICULTIES IN REPRESENTING THE RELATIVE POSITION OF

Two REGIONS

As long as one limits oneself to two regions it is easy to deter-
mine their relative position and the properties of their boun-
daries. But when the two regions lie within a group of other
regions the problem of characteriz-
ing their relations becomes a more
difficult one.

In Fig. 28 for instance one can
easily enough say that the regions
A and B are foreign to each other;
furthermore that they do not have
common boundaries. But one can-
not make this statement with assur- FIG. 28. Different

.,. , j . ,, , ties of viewing the boundary

ance if one includes in the concept zon esof Aand#. (For example.
of boundary that of boundary the region 4 : + 1 2 + 15 + 20 +

J . J 21 + 14 might be considered as

zone. One can consider the regions the boundary zone ^ of A\ and

12, 4, 14, , *, I 9 , 16, IS, II, 10,

6 5 as part regions of a boundary of 5; in this case the two

, , . . T . .. , , ary zones would have no common

zone b A which surrounds A; further par t; ZM-&B * o. if one con-
fix rpcnrmq T9 Tf IT TO 16 IO si <* ers > however, the more inclu-

tne regions 5, 12, 15, 11, 19, 10, 10, siveregionl +3+4+5+6 +
I 7? 9; 7? 2 > 6, can be considered as 10 + n -t- 12 + 15 4- 19 H- *o -h

P i t i T 21 4- 22 -f- 13 4- 14 as &j. and

parts of a boundary zone b B around region II+6+2 + 7+9 + I7
B. A and B would then be two + * 6 + ioas& St then^-&* -

. , , , 6 4- 10 4- ii.)

regions whose boundary zones nave

common parts (b A b B = 5 + 12 + 15 + n + 6 + 10 + 19 +
1 6). On the other hand, one could treat only the region
4 + 14 + 21 + 20 + 15 + 12 as the boundary zone of A and
consider the topological sum of the regions 2, 7, 9, 17, 16, 10, 6,
as the boundary zone of B. In this case the boundary zones
of A and B would have no common part.

Topologically there is no reason to consider the indicated
regions as a boundary zone between A and B. One could for
instance instead think of the topological sum of the regions
16 and 20 as an arm which brings B into communication with A,

140 TOPOLOGICAL PSYCHOLOGY

Obviously the psychological justification of these different
interpretations depends on the dynamic character of the
regions concerned. But the very fact that there are so many
possibilities of interpretation occasionally leads to confusion
in psychological research and it often requires considerable
experience to know where one is dealing with intrinsic differ-
ences and where the difference is only one of expression. It
seems to me therefore advisable to take up a few examples in
order to discuss these difficulties. In doing so we shall only
use concepts which we already have explained. But we shall
employ them, as in practical experimental work, according to
the requirements of the concrete problem, not in systematic
order. We shall introduce these discussions in the form of a
problem. In this way the relations between the different
topological concepts and between the topological and dynamic
concepts will become dearer.

The Two Principal Possibilities for the Representation of
the Inaccessibility of a Point. Let us represent the follow-
ing situation: "A person has a certain goal, but at the moment
it is difficult or impossible for him to reach it." (Vector psychol-
ogy has to discuss the forces which may be involved in such a
case (54, p. 253). Here we shall consider only the topological
aspects of the problem.)

The cases in which an obstacle makes it difficult to reach a
goal are frequent, and of many different kinds. The goal may
be a certain job and the obstacle may be the fact that there is a
second applicant who has more influential connections. In
other cases the applicant's own incompetency, the fact that he
does not yet hold certain degrees, or that he is disqualified on
account of his citizenship or religion may constitute the obstacle.
The goal may be going to a dance, marriage, a business trans-
action, or picking a flower. The obstacle may be a prohibition
which rests on a law or perhaps on the authority of a person, or
it may be social convention by which the person feels himself
bound.

However different the cases are in detail they have at least
the one fact in common, that two separate points or regions,

THE RELATIVE POSITION OF TWO REGIONS 141

the person P and the goal G, are distinguishable and that a
barrier B makes locomotion from P to G difficult or impossible.
As the most simple example, we can think of the case in which a
physical barrier blocks the approach to the goal.

One could attempt to represent such a situation by the
diagram of Figure 293. A line B representing the barrier lies
between P and G. But such a representation does not express
the fundamental fact that there is no usable path between P
and G. The representation offered in Fig. 2ga leaves open the

^___ w

(<0 (&) 00

FIG. 29. Topologically inadequate representations of the inaccessibility
of a goal. The representations (fl) f (&), and (c) are not different topologically.
P, person; G, goal; J? t barrier; 10, Wi 9 wi, ws, paths.

possibility of many such paths from P to G, for instance Wi, iv$,
wz. From a topological point of view such a representation
means that P and G are points of one connected region (see
p. 88).

This fact is in no way changed if one gives the barrier the
form of that in Fig. 296 where G is "almost entirely" enclosed
by B. In this case too P and G lie in one connected region:
there is a path w from PtoG which does not cross B.

One has to be dear about the fact that topologically the
representation in Fig. 296 is in no way better than that in Fig.
290. Both representations are topologically equivalent even
to Fig. 29*; where B does not lie "between" P and G in the
sense of ordinary geometry.

If one wants to represent the "unattainability" of G in a
way which is topologically adequate, one has to be sure that P
and G do not belong to one connected region. This means that
one must represent the barrier as a Jordan curve which divides

142 TOPOLOGICAL PSYCHOLOGY

the whole field in such a way that P and G belong to two different
regions.

We can do this in two and only two ways: either the goal G
lies in the inner region J and the person P in the outer region O
(G < 7; P < 0) (Fig. 30*2) or the person lies in the inner region
and the goal in the outer (Fig. 310) (G <0\P < 7).

The psychological differences between these two represen-
tations become clear when one characterizes more closely the
different regions of the life space which belong to the inner and
the outer field. If the person is in the outer region then
relatively few regions (#, 5, c,G,Fig. 306) are unattainable to him.
In the remaining space he can move about freely. If, on the

6 +

(3oa) (3ia)

FIGS. 3O# and 310. The two fundamental possibilities of representing inac-
cessibility: (3 Off) G lies in the inner region, P in the outer region; (31 a) P lies in
the inner, G in the outer region. B, barrier represented as Jordan curve; G, goal;
P t person.

other hand, the person is within the barrier (Fig. 31 J), then
the space of free movement is limited to a narrow region and
everything else (a, J, c, . . . , g) is unattainable. In this case,
the situation therefore has to a much higher degree the char-
acter of a "restraining situation" like the situation of the
prison.

As an example one can use the difference between a situation
in which a command is supported by means of promise of reward
and a situation in which the command is supported by threat
of punishment. In order to make a threat of punishment
effective one has to create a restraining situation, even when the
command refers to a definite task. The space of free move-
ment of the person has to be limited to a sufficiently small
region. Otherwise the person will escape at the sides (.52,
pp. Qfijf.). In the case of reward the space of free movement
of the person can remain unlimited. Only the access to the

TEE RELATIVE POSITION OF TWO REGIONS 143

region of the reward is limited; i,e. f one cannot get the reward
without first passing the region of the task. A restraining
situation, or in other words, the presence of an outer barrier
which makes the situation inescapable plays an important role
for certain emotional processes, for instance anger (20) and

PIG. 306. Elaboration of Pig. 300. Region a + b + c + G is inaccessible for P

PIG. 3i&. Elaboration of Pig. 31 a. Region a

inaccessible for P.

despair (52, p. 195; 19). In regard to the resulting forces too
there are important differences between the cases in which the
person is within the Jordan curve and those in which he is out-
side of it.

There are, in a certain sense, transitions between the two
cases. Their dynamic difference rests in part on the different
extent of the space of free movement. If the person is within
the barrier, the space of free movement can be enlarged by

144 TOPOLOGICAL PSYCHOLOGY

including further regions in the inner field, so that the character
of a restraining situation gradually becomes weaker (52, p. 128).
On the other hand, if the person is outside the barrier, the
enlargement of the unattainable regions within the barrier, or
the appearance of further islands of the unattainable in the life
space can more and more limit the space of free movement.

The size of a space of free movement is not a topological con-
cept. Topologically one can say only what part regions belong
to it. Nevertheless one can determine in this way an extension
over new regions or a restriction of the space of free movement.
A limited space of movement sometimes seems to grow psycho-
logically when the region differentiates itself into a number of
subregions. Even an objectively small region can, in this way,
have the significance of a relatively wide field for the person
concerned.

Occasionally outer barriers have at the same time the char-
acter of a protection against influences from the outside. For
the gangster the prison may serve as a not unwelcome protec-
tion against attack by his rivals. Often a prisoner of many
years feels unprotected against the dangers of the outer world
as soon as he leaves the prison. It is then difficult to persuade
him to leave his home or even to expose himself to the gaze of
other people from whom he was safe while he was in prison.

In our graphic representation, the degree of solidity of a barrier is
generally designated by the thickness of the line.

Topological and Dynamical Aspects of the Represen-
tation of Limitations. One can raise the question whether it
is possible to represent by topological means the fact that a
goal is attainable but only with difficulty. It may seem pos-
sible to express the difficulty by leaving only a small gap in the
Jordan curve and making it the smaller the greater the degree
of difficulty. However one must not forget that there are no
size differences in topology. As we have mentioned there is no
difference between Figs. 296 and 290. The degrees of difficulty
between the impassable barrier on the one hand and the
boundary which offers no resistance on the other hand cannot

TEE RELATIVE POSITION OF TWO REGIONS 145

be characterized topologically but only dynamically. When the
goal is attainable, but more or less difficult to reach, one still
has to represent the barrier as a dosed curve or a ring-like
boundary zone. Only in this way can we express topologically
the fact that there is a barrier " between " P and G. To the
different degrees of difficulty correspond dynamic differences of
the boundaries, for instance different degrees of solidity.

Discrete Paths and Their Totality. One could try to
represent the inaccessibility of a goal in the following way,

I \ ^

PIG. 32. Attempt to represent inaccessibility by discrete blocked paths. G,
goal; P, person; WT., a>z, w^ r paths; t% 9 ci, cz, blocked points.

without using a Jordan curve. One could start with certain
paths wi } wa, w$ between person P and goal G and could repre-
sent the impassability of the different paths by cuts Ci, C2, 3,
which signify the impassable points in each of these paths
(Fig. 32), restricting the representation of the life space to a
one-dimensional space (see p. 193).

By such a representation the obstacle assumes the character
of an unconnected set of discrete points. This may be ade-
quate in a case in which we are dealing with a definite number
of separate paths to the goal, as in a maze experiment. In
other cases it is not correct 3 for (i) such a representation implies
the assumption that it is impossible to go "along" the obstacle
(and this doubtless is possible at times), and (2) it does not show
that there are other paths open between P and G besides those

146 TOPOLOGICAL PSYCHOLOGY

which are especially indicated. The Jordan curve takes into
account the totality of possible paths, but the representation of
Fig. 32 does not do so.

As we have mentioned before it is very important in repre-
senting a situation to take this totality of all possible events into
account. If the representation of a psychological life space
is to be more than an illustration without compulsory
consequences, the coordinations between psychological and
mathematical facts have to be strictly maintained. Each
representation in which P and G belong to a connected
region, as in Fig. 32 (considered as a more-than-one-dimensional
space) would show positively that there are still further paths
7 between P and G. The representation would

therefore be false in an important point.

Figure 32 is not sufficient even for the repre-
sentation of the paths of a maze. The fact
that there are no other possible paths is not
expressed. One must represent a maze topo-
t o u r' p r 3 o t> i e m logically as a branching of regions with barriers

tfon^ ^ chad- on botl1 sides to P revent escape (Fig. 24).
B, 'u-s'haped Such a representation would be correct even
bench; r, toy. Qr ^ e i eva t ec j maze where the paths are

not enclosed by physical walls but where the rat is unable to
leave the paths.

Homogeneous and Differentiated Barriers; Approach
and Withdrawal. A one-year-old child C stands behind a
U-shaped bench B (Fig. 33). He wants to get the toy T on the
other side of the bench but has not yet a sufficiently broad
survey of the situation to carry out the necessary detour.
In this case also there is a barrier for bodily locomotions which
makes it impossible for C to reach T. Therefore according to
our definitions C and T, as far as this locomotion is concerned,
do not belong to a connected region. The barrier has to be
represented in this case also as a Jordan curve (Fig. 300 or 31 a)
in spite of the fact that it is not closed physically. Vector
psychology has to take up in detail the conceptually difficult
problem of detour (see 54; 60).

THE RELATIVE POSITION OF TWO REGIONS 147

We already have mentioned that the single parts of a barrier
can correspond to different degrees of difficulty. We shall
now discuss briefly an example, taken from our films, of a
barrier which is physically homogeneous, psychologically
inhomogeneous.

A toy T stands within a circular iron barrier 7. A one-and
one-half-year-old child C who is outside of the barrier wants to
get the toy (Fig. 340 is a diagram of the physical relationships).
Besides, the mother M is in the room. After a series of futile
attempts to climb over the barrier the child runs to the mother
for help. This turning toward the mother need not have the

() (b)

FIG. 34. A young child wishes to reach a toy which lies inside a circular
barrier, (a) Physical situation; (&) psychological situation. C, Child; T, toy;
/, barrier; M, mother; G, goal; wi, ws t paths.

character of a turning away from the toy. But it can have the
meaning of an "indirect " turning toward the toy (54, p. 253).
The child may have realized suddenly that not only the iron
barrier but also the mother stands between him and the goal.
In this case there occurred a restructuring of the psychological
situation which is indicated in Fig. 346: while the barrier
between C and the goal G at first had to be characterized as
relatively homogeneous it consists now of at least two parts
(sectors) one of which corresponds to the iron barrier 7, the
other to the mother M. Path Wi corresponds to reaching the
toy by crossing the physical barrier, path Wz to getting the toy
through the help of the mother.

Therefore even when we are dealing with quasi-physical
fields the representation of the psychological field will have
to follow exactly the relations of connectedness which are
defined by the psychological functions.

148 TOPOLOGICAL PSYCHOLOGY

Barriers and Adits. A person is given the following prob-

8 X 3 X 74

lem r; He begins by canceling the two into the eight

2x3

and the three into the three* Let us represent the moment
before he performs these operations. One again can start
from the fact that there is a goal G, namely, the solution of the
problem, and that the person P is separated from this goal by
a barrier B which is not very strong (Fig. 300).

However, if one looks more closely it may seem doubtful
whether it is admissible to speak here of a "barrier." The
development of the whole situation may be as follows: the
person finds himself faced with a problem. The solution of
the problem Pr is then a region outside of which P is located
(Fig. 350). This region is not an entirely homogeneous field
but shows from the beginning or very soon a certain structure.
Although the goal G, the solution, is not yet fully evident the
path to the goal becomes dear. The person realizes that one
must first cancel two into eight and three into three and then
multiply four by seventy-four. The path to the goal is there-
fore characterized as a series of operations, namely, division Di,
division D*, multiplication M (Fig. 356).

We meet here again the fact which we have mentioned before
(see p. 107), namely, that the concepts of action and of path
shift between a one-dimensional locomotion and a more-
dimensional region. The single steps of locomotion correspond
to part regions of the task. The performance of the operations
would correspond to a locomotion wi of P through these regions
toG.

However, the representation in Fig. 356 is not yet satisfactory
if one considers the necessity of taking into account all the
consequences of the representation. Figure 356 leaves open
the possibility that P reaches the goal G directly by way of the
path w% without crossing regions Z>i, Dz, and M. This would
imply that P can arrive at the solution without really carrying
out the operations. Such a process is psychologically not with-
out meaning. The person may for instance know the result
because he has made the same operations before, or because

THE RELATIVE POSITION OF TWO REGIONS 149

somebody has told him the answer. If, however, as we shall
assume in our case, this is not true, the representation in
Fig. 356 is not sufficient.

There are two different ways of meeting this inadequacy of
representation. One can (i) use the concept of dynamic
solidity of a boundary, or (2) use purely topological means.

Ad i : Since it is a question of excluding certain locomotions,
which the representation, used so far, still allows, one might
represent as passable only those boundaries of the regions
concerned which correspond to the path w\. We can represent
the fact that there is only one possible path to G by giving the
character of an impassable barrier B to the outer boundary
of the whole region Pr (Fig. 35^) with the exception of the
boundary of DI. Thereby one succeeds in showing that person
P can reach goal G only by passing regions D x , D 2 , and M.

Ad 2: If one does not use impassable boundaries, one can
limit the possibilities to this one kind of pathway by surround-
ing the region of the goal G by a series of concentric ring-like
regions which correspond to the operations Af, D 2 , and DI
(Fig* 35^)- In this case again P's only passageway to G is
DI, Dz, and M.

The question arises in what respect the two representations
agree and in what respect they are different. The most impor-
tant topological difference between the two representations
consists in the fact that G has in the second case (Fig. 35^)
only one adjoining region, namely M, whereas in the first case
(Fig. 35c) there are parts of the boundary of G which are not
at the same time parts of the boundary of M. Similarly
region M has in Fig. 35^ only regions G and Z>2 as neighbors;
in Fig. 3sc on the other hand there is still a further region.
This holds also for D 2 and DI. If one represents the topology
of the situation by Fig. 356, one can achieve limitation to one
approach only by representing part of the boundary between
the part regions of the problem and the surrounding regions
as an impassable barrier. Figure 35^ does not have to use
the concept of the solidity of a barrier because in this repre-
sentation there are no boundaries which make it possible to

ISO TOPOLOGICAL PSYCHOLOGY

enter "from the side." The two representations are therefore
actually somewhat different, and it is a question which is the
correct one.

One could object to the representation in Fig. 35*; on the
grounds that in reality one cannot observe a solid barrier B
between the regions G, M , D% and a general outside region. It
is not such a barrier but the inner logic of the mathematical
problem which makes the paths w$, and w$ (Fig. 356) impass-
able. This "logical impossibility " is doubtless represented
more adequately by the topology of Fig. 35^ than by the barrier
B in Fig. 35*;, which seems somewhat arbitrary.

The representation by Fig. 35^ may be adequate for the
case in which the person concerned thinks of the sequence of
the three operations DI, D%, and M (canceling two into eight,
three into three, and multiplying by seventy-four) as the only
possible path to G. As a rule the person knows that there are
also other paths to G, at least the long one of performing the
several operations in the order in which the problem is stated
8 . 3 . 74 -T- 2 -5- 3 (corresponding to the regions M $, M&, D 3 ,
D 4 ). This means that the representation of the situation by
means of a barrier (Fig. 35^) is correct in so far as M is not the
only region which adjoins G. Besides M there is at least
region D as a possible neighboring region of G. Before the
person decided to use path Di 9 D$, M there may have been
a situation when he was wavering between it and the other
possibility. If one would base the representation of such
a situation on the principles underlying Fig. 35^, it would lead
to Fig. 350: two possible adits (Adi, and Ad%) which correspond
to different operations lead from P to G. To exclude the possi-
bility of a "direct " approach to G one would have to coordinate
an outer barrier B to each of these adits.

If there are still more adits (Ad*, Ad^) which the person can
see, one would have to indude in the representation further
regions and combinations of regions which are connected with
G (Fig. 35/). It is not necessary that each of these approaches
is dearly structured in advance. It can be that a path is
visible as a whole, but unstructured (Ad 4).

THE RELATIVE POSITION OF

FIG. 35. Two fundamental ways of representing a situation, when a goal can
be reached only by certain approaches. This limitation of accessibility can be
represented either by purely topological means or with the help of dynamical
concepts, (a) Indicates the undifferentiated; (b) the differentiated situation in
the beginning stage of a mathematical task without representation of the lim-
ited accessibility. The representation of this limitation by purely topological
means is indicated in (d) and elaborated in (#) ; the representation with the help
of dynamical concepts is indicated in (c} and elaborated in (e), (/) and (**) (A)
and (j) show the relations between the two representations: in (fc) the approaches
are viewed as a boundary zone between P and G' f (/) identified the impassable
barrier B in Figs, (c), (0), (/), (*) with certain unstructured sectors U inserted in
(g). P, person; Pr, mathematical problem; (?, goal (solution of problem); M*
Mi, Mz, different multiplications; Di, Dz, D^Dt, different divisions; B, impassable
barrier; Adi, Adz, Ad^, Ad*, different adits; U, qualitatively undetermined regions.

152 TOPOLOGICAL PSYCHOLOGY

One can try to represent the situation which is shown In
Figs. 350 and 35/ in analogy to Fig. $$d by purely topological
means. One has then to distinguish different sectors within
the ring region which surrounds G (Fig. 35$). Figures $$f
and 35g, which represent the same fact, in the one case by means
of impassable barriers, in the other by purely topological
means, are obviously more similar than the corresponding
Figs. 356 and 35^. Nevertheless, the main difference is
maintained: In Fig. 35^ no further regions adjoining G are
possible except the adits which are represented as sectors; in
Fig. 35/ this possibility exists. Besides, there are impassable
barriers in Fig. 357 but not in Fig. 35^. To be sure one cannot
say that there is in Fig. 35^ no barrier at all between P and G:
P has to overcome the difficulties of the operation if he wants
to get to G. Therefore, in this case also there exists a barrier;
but it has the character of a boundary zone which is not
impassable and which consists of the part regions of the prob-
lem itself (cf. p. 130). When we think of the adits as one
region Adi + Ad^ + Adz + Ad* (Fig. 35^), then we see that
we have again the above mentioned (Fig. 300) typical con-
stellation in which a person P wants to attain a goal G. Con-
sidered in this way Fig. 35^ shows a barrier between P and G
which is structured in sectors and also in its depth.

In Fig. 35/ also one can conceive the region D\ + D z + M ,
as part of a boundary between G and P which has the
character of a passable barrier like that of Fig. 35^. But
besides this boundary zone there are other parts of the boundary
of G which have the character of impassable barriers.

This becomes still dearer in the following representation:
We have seen that the psychologically real barrier generally
has a certain depth. We know that size differences are topo-
logically irrelevant From that it is dear that one does not
change Fig. 35/ topologically if one presents it in the form of
Fig. 352. Thereby the two ways of representation (Fig. 35^
and 35^) become still more similar. The topological difference
consists only in that there are in Fig. 35^" besides the four path-
ways (Ad i, 2, 3, 4) still further sectors within the boundary

THE RELATIVE POSITION OF TWO REGIONS 153

zone which axe missing in Fig. 35^. (These sectors are indi-
cated in Fig. 351 by black.) One can make the repre-
sentations completely analogous if one inserts more sectors
(7) between the four adits in Fig. 35^ (cf. Fig. 357)- T^ 6
introduction of such sectors is justified if one considers that the
person is often uncertain whether there are still other adits
to G, and what characteristics they have. Figure 35; is an
adequate representation of the situation when P is not certain
that there are four and only four paths to G. (If these inserted
sectors contract to zero, one gets Fig. 35^ again.)

What do these further sectors (U) of the boundary zone imply
for the possibility of a locomotion from P to G? The main
characteristic is that they are entirely unstructured. They can
be characterized qualitatively only as "possibly existing adits
of some kind." We are therefore dealing with those psycho-
logically unqualified regions which we already have treated
as a special kind of psychological boundary zone (pp. 130$".).
As long as it is impossible for P to determine their quality
they cannot be used as paths to G. The U zone therefore
represents an impassable barrier for P. In this sense Fig. 357
actually corresponds to Fig. 35^. Thereby the relationship
between the two methods of representation with which we
began, that of the dynamic barrier on the one hand and of the
topological structure on the other hand, becomes evident.

This example may have clarified once more the previously
discussed connections between a boundary and a boundary
zone, between a boundary and a barrier, and between the
dynamically different kinds of barrier. Furthermore it brings
out the connection between boundary and path.

It is not a peculiarity of this case that the regions M, D\,
D<L appear on the one hand as a pathway from P to G, and on
the other hand as a barrier between P and G. Rather, it is a
general property of every boundary that it connects and at the
same time separates two regions. This is especially obvious
if we are dealing not with one-dimensional boundaries but with
boundary zones. The fundamental fact is that there are three
regions, A, B, and C, and that the path leads from a point

154 TOPOWGICAL PSYCHOLOGY

in A to a point in C by way of B. It depends upon one's point
of view and also upon the ease witli which B can be crossed
whether one prefers to treat B as a boundary zone between
A and C or to treat it as a pathway from A to C. In reality
both points of view are always possible and have to be taken
into consideration.

CHAPTER XIV
STRUCTURAL CHANGES

DIFFERENTIATION, INTEGRATION, AND RESTRUCTURING

A region which is at first homogeneous can become articu-
lated into a number of part regions. Such differentiations are
among the most frequent and important psychological proc-
esses. There hardly exists a psychological problem in which
they are not involved in one way or another. Topologically
one can think of them as a breaking up of regions into subregions
which can easily be treated mathematically. Naturally there
exists an unlimited number of possible variations in the kind,
speed, sequence, and degree of differentiation into part regions.

These differentiations can be closely connected with cognitive
processes (85, p. 440), for instance with experience or with an
act of insight (46). Differentiation may result from other
causes too. The development of the life space from infancy
to adulthood can be characterized to a large extent as a process
of differentiation (48; 23, p. 263; 81, pp. 129-137 and 162).

Perhaps not less often than a differentiation one can observe
the mathematically opposite process of a dedifferentiation or
integration. Such a unification can be observed for instance
in certain emotional situations (20, p. 118). In these cases
it is usually the effect of strong tensions. In other cases such
a unification of systems, which at first are separated, is produced
by intellectual processes.

Finally, we find a group of changes of the surrounding field
which one cannot think of as differentiation or integration.
The number of part regions of a whole may remain the same,
although their relative position is changed. In such a case we
shall speak of "restructuring." Very often a loss or increase of
differentiation may accompany a restructuring.

155

156 TOPOLOGICAL PSYCHOLOGY

Naturally, an endless number of such changes of structure is
possible. 1 Their special characteristics have to be determined
in each single case. Like other courses of events one will
have to represent differentiations, integrations, or changes of
structure as a series of situations which correspond to the
different cross sections of time,

CHANGES OF STRUCTURE AND LOCOMOTION

The change of the structure of a region has usually a different
character from that of a locomotion. Nevertheless, there is a
dose connection between the two processes. Topologically a
locomotion of a person P from a region A to a region B always
implies a restructuring of the whole field: as a result of the
locomotion the region that corresponds to the person P becomes
part of a different region (P < B, instead of P < A).

The same is true of the locomotion of other persons or objects
in the life space. Each such change of position implies a more
or less important change of the structure of the environment.
This is especially dear in cases in which the environment moves
although the person does not actively contribute to the move-
ment (cf. p. 114).

The fact that a communication is brought about between two
regions by the reaching out of an "arm" (p. 102) can be thought
of as a kind of transition between locomotions and other changes
of structure. Such a locomotion of an "arm" can change the
structure of the life space to a considerable degree although the
regions involved do not themselves show a marked movement.

CHANGES OF MAGNITUDE AND OF DISTANCE

In the case of a differentiation of a whole region, it might
seem possible to speak of the formation of smaller part regions;
in the case of an integration, of the formation of larger regions.
This however can be done only under special conditions. As

1 S. Fajans (23), pp. 24ojf., and Sliosberg (81), pp. 129-137, describe struc-
tural changes resulting from difficulties in reaching a goal and in the situation of
embarrassment. K. Lewin, (52) pp. 114-1^0, discusses changes in a situation of
reward and punishment; T. Dembo and E. Hanfmann (19) compare situations
of patients in a mental hospital.

STRUCTURAL CHANCES 157

we have mentioned, the topological characteristics are inde-
pendent of quantitative determinations. One cannot expect
therefore to express changes of magnitude or distance by means
of topological concepts.

In certain cases, however, it is possible in psychology to make
statements about size or rather changes of size on the basis of
topological determination. Let us consider an example: it is
not possible topologically to say that region A in Fig. $6a is
larger than region B. However if B is entirely included in A

FIG. 36. Topological conditions (a) unfavorable and (6) favorable to the
comparison of size in the life space. Regions A and B may be psychologically
comparable as to size if B < A and A = B 4- N; N j& o.

(Fig. 366) and if A has part regions other than -B, we can
sometimes say psychologically that A is larger than B.

In this case as well one cannot speak of size without going beyond
topological concepts. Topology can only determine the relation of
"being-contained-in" or of "part-whole." From the point of view of
the theory of sets there is as a rule no size difference between the whole
region and one of its parts: their points can be coordinated in one to one
correspondence.

A simple example of the shrinking of a region is the change
in the space of free movement of the child A in the bathtub
example. After the boy B has established a boundary for the
movements of A across the middle of the bathtub (Fig. ib)
A's space of free movement is definitely smaller than it was
in the situation represented in Fig. la. Parts which previously
belonged to the space are excluded and no new regions are
added.

Another typical example of the shrinking of the space of free
movement is that which occurs for a first child when a second

158 TOPOLOGICAL PSYCHOLOGY

child is born. The first child may, for instance, have to share
his room, his table, his toys with the new one. The mother
can no longer give so much time to the first child and the child
must be considerate of the smaller one in many ways. Each
new prohibition limits the space of free movement. On the
other hand the lifting of a prohibition or the acquisition of a
new skill results in an extension of the space.

Such statements about the size of the whole space of free
movement can be made only if, aside from the addition or loss
of certain part regions, there are no significant changes in the
size of other parts. After a marked reduction of the space of
free movement there can appear adaptations which psycho-
logically compensate, at least in part, for the shrinking. The
remaining regions can be differentiated into subregions in such
a way that it becomes doubtful whether, from a psychological
point of view, the life space has become really smaller than it
was before.

Nevertheless, at least at the moment in which sufficiently
large part regions are taken away or added, there is a marked
increase or decrease in the size of the whole space of free move-
ment. The other part regions usually remain sufficiently
unchanged at least for a short time. Therefore the velocity
of the change is often of great importance for its effect on
behavior.

Like size relationships psychological relationships of distance
can be represented by topological means only when the regions
which are coordinated to the distance m are entirely part of
those regions which determine distance n. For instance, in
the example of the child who wanted to carry his ball up the
steps (see p. 114) one is justified in saying that the distance
between goal G and child C has decreased between the first and
second stages (Fig. 2oa and V) and that it has increased between
the second and third stages (Fig. zob and c). In the first
case there occurs no change except that certain part regions
of the intervening zones drop out. In the other case new
regions are added without important change in the properties
of the part regions.

STRUCTURAL CHANGES 159

In dealing with dynamical problems it is often a question
not of comparing any given situations with each other, but
rather of determining changes of situation. Therefore one can
more often make statements about size on the basis of topo-
logical concepts than one would expect in view of the rather
special or specific conditions to which such statements are
limited.

DYNAMIC CONDITIONS OF STRUCTURAL CHANGES;
FLUIDITY, ELASTICITY, PLASTICITY

Changes of the structure of the life space often have impor-
tant dynamic consequences and depend directly on dynamic
factors, especially on the distribution and magnitude of forces.
The treatment of these questions therefore presupposes the
concepts of vector psychology. At this point however we
shall briefly indicate several differences in state of regions which
are important for structural changes.

In discussing thing and medium (see p. 115) we have treated
dynamical properties of regions which are important for loco-
motion. Furthermore we have attributed different degrees of
solidity to the boundaries and we have spoken of boundary
zones which offer different degrees of friction. This charac-
terization was based upon the resistance which these boundaries
or zones offer to locomotion. One has to ask a corresponding
question in regard to the ease with which the structure of the
field can be changed.

We can speak of different degrees of fluidity of the situation.

Definition: A situation is the more fluid the smaller the forces
which are necessary, other conditions being equal, to produce a
given change in the situation.

The greater fluidity can be a general characteristic of the
situation in the sense that it is more fluid in regard to all sorts
of influences. It would then have equal effect for instance in
regard to the release of tension systems and in regard to changes
of topological structure. Generally the fluidity of the situation
is different for different kinds of influences. Resistance to
locomotion can therefore be treated as a special kind of fluidity.

160 TOPOWGICAL PSYCHOLOGY

The degree of fluidity of a situation plays an important role
in all processes and is one of the fundamental dynamic prop-
erties of a situation. Regions of greater irreality (see p. 196)
generally correspond to more fluid media than regions of a
higher degree of reality. Within the level of reality also
different regions seem to possess very different degrees of
fluidity. For pedagogical reasons one often tries to give a
relatively great stability to certain regions within the life space
of the child (regions which are connected with the routine of
dressing, eating, or sleeping) ; there are other regions in which
the child ought to feel really free and which one tries to keep
relatively fluid. The general degree of fluidity of the environ-
ment is pedagogically very important. The degree of social
stability of a group to which a person belongs is usually different
at different times. An inflation, for instance, can lead to a
great fluidity of the social field.

The fluidity of the environment is closely connected with the
state of the person. Fatigue seems to produce an instability
not only of the person but also of the psychological environment.
The frequently noted affectivity of the small child when he
first awakens from sleep may be directly connected with the
instability of the surrounding field at this moment. In general,
situations are most fluid in statu nascendi. There seems to
exist a dose connection between the cognitive uncertainty of
the structure of a situation and its general degree of fluidity.
The solidity of a region generally increases as it remains
constant over a longer period of time. There are however
exceptions. Sliosberg has shown that it is necessary to dis-
tinguish between different degrees of fluidity for regions which
correspond to certain tasks and play materials (81, pp. 148-149
and 176-177). Playful actions and play situations generally
have a more fluid character than serious ones. Frank (250,
p. 293) has shown that the level of aspiration is more easily
raised in a playlike situation.

There are two ways in which one can represent different
degrees of fluidity: (i) by attributing to different regions as
wholes the qualitative characteristic of greater or less change-

STRUCTURAL CHANGES 161

ability; (2) by ascribing to the regions boundaries of different
degrees of solidity. In the latter case the average fluidity
of the whole region depends on the solidity of the inner frame-
work which is characterized by the boundaries of the part
regions. The main difference between the two ways of repre-
sentation is the following: in the first case the single part region
(a or ) is considered as dynamically homogeneous; in the other
case one distinguishes between the solidity of the border and
that of the inner part of the single subregions. If the solidity
of the region a is greater than that of the region b it would mean
in the first representation that locomotions within the region a
are more difficult than those within the region b. According
to the second representation locomotions within a may be
carried out as easily as locomotions within b.

The concrete situation therefore determines which of the two
representations is more adequate in a given case. There is
no doubt that a weakening or loss of the boundaries between
the different regions of the environment can lead to a marked
fluidity of the whole field. This can be observed in the social
field in revolutionary times when the barriers between groups
or barriers established by prohibitions break down; or when
a child who has been brought up in strict obedience is suddenly
placed in a field in which barriers of prohibition are not clearly
evident. The individual differences in such cases show that
in addition to the solidity of special boundaries one always
has to deal with the general stability of the particular life space.
Therefore one must always consider the characteristics both
of the boundaries and of the regions themselves or, as we shall
call it, their "material."

The definition of fluidity which we have given leaves open
the question of "elasticity."

Definition: By elasticity we understand the tendency of a
changed region to return to its original state. It is obvious
that the degree of elasticity can be different in regions of the
same fluidity.

Both elasticity and fluidity are involved in "plasticity." By
plasticity one should understand the ease with which a relatively

1 62 TOPOLOGICAL PSYCHOLOGY

lasting and stable change can be made in the structure of a
region. A too low as well as a too high degree of fluidity is
unfavorable for plasticity. As a matter of fact not only very
rigid persons but also certain psychopathic types which are of
especially high fluidity are difficult to mold; also a too high
degree of elasticity of a region can be disadvantageous to its
plasticity. The plasticity of the environment depends in part
on the state of the person. Katz (41, p. 127) mentioned the
relation between plasticity and need. Certainly the plasticity
of the life space changes in the course of the development of
the person (see p. 190).

CHAPTER XV

THE LIFE SPACE AS FINITELY STRUCTURED SPACE

We shall now conclude our considerations of the topology
of the environment. But before we discuss the topology of the
person we must point out a fundamental fact which concerns
the whole life space.

In treating mathematical spaces it is assumed that space
is infinitely divisible. It is possible to break up each region 1
into part regions and therefore to distinguish part regions of
part regions ad infinitum. This property of space is also
presupposed by topology. One has to ask whether the psy-
chological life space has these characteristics. In the course
of our considerations we have twice met facts which are related
to this problem namely, in discussing how accurately one
can determine points in the life space (see p. in) and in
treating the psychologically unqualified regions (see p. 130).

As we have seen one can determine the relative position of
a point in the life space only by reference to the region in
which it lies. The accuracy of this determination depends
therefore on how far one can proceed in dividing regions into
subregions. Few if any psychological regions can be divided
into smaller subparts ad infinitum. In the example of the
child who had to eat something that he did not want (see p. 97)
one can distinguish the regions: putting the hand on the table;
taking the spoon; bringing it part way to the mouth; etc.
But it is not admissible to distinguish for instance within
the region "taking the spoon" as special regions "moving the
hand forward a hundredth of a millimeter," "moving the hand
forward a second hundredth of a millimeter," etc. In other
words it is usually possible to divide psychological whole
regions into part regions but this can not be carried on ad

1 More exactly: each more-than-o-dimenslonal region.

164 TOPOLOGICAL PSYCHOLOGY

infinitum. An analysis which attempts to do this is not more
accurate; it is psychologically wrong.

How far one may proceed with the division depends on the
case at hand. If a child likes to eat a certain kind of food
and conveying a bite to his mouth is one single process, then
the distinction between the regions "bringing half way to the
mouth" and "bringing to the mouth " may be an inadmissible
analysis of psychologically unstructured unities. The extent
to which regions can be divided therefore varies. It depends
on the momentary condition of the region concerned. (We
have given examples of changes in degree of structuration in
discussing the boundary zone (see p. 121), further in the
chapter on thing and medium.) But it always has a certain
value for a certain region of the life space at a given moment;
in other words it has an objective character.

A consideration of the unqualified zones which can act as
barriers leads to the same result. These zones are usually
in themselves unstructured (see examples, p. 131), With
increasing orientation they may gradually differentiate them-
selves; but at a given moment there are always certain part
regions within which it is psychologically impossible to dis-
tinguish further parts.

We shall call a region which cannot be divided into dis-
tinguishable part regions "unstructured"; a region whose part
regions can be divided indefinitely into further part regions
"infinitely structured"; a region in which division into part
regions is possible, but cannot be carried out indefinitely,
"finitely structured." 1 In this sense we shall speak of unstruc-
tured, of finitely and infinitely structured spaces. The psy-
chological life space is thus a finitely structured space.

It is not possible to discuss the question here whether
physical space is infinitely or only finitely structured. Heisen-
berg's principle of indeterminacy which suggests itself at this
point does not imply that physical space is finitely structured.

1 This term is less likely to lead to misunderstanding than the term "struk-
turiert" (structured) which I have used in "Der Richtungsbegriff in der
Psychologic" (54).

THE LIFE SPACE AS FINITELY STRUCTURED SPACE 165

For In regard to fields which are not in motion there are,
according to this principle, no absolute limits for the accuracy
of the determination of position. At least there is a difference
in degree of structu redness between the physical space and
the life space. While physical space is everywhere structured
to the microscopic level the psychological life space often
contains macroscopic regions which are unstructured. Besides,
the limits of structuredness of the life space vary to a very
high degree.

Riemann (76) mentions in his well-known treatise Ueber die
Hypotheses, welcke der Geometric zu Grunde liegen that it is not
necessary logically that spaces should be infinitely divisible.
As far as I know, mathematics has not yet followed up this
suggestion; it would certainly be of greatest interest for psy-
chology. However, topology allows a representation of the
psychological life space which is sufficiently adequate for most
problems.

C. TOPOLOGY OF THE PERSON

CHAPTER XVI

THE PERSON AS A DIFFERENTIATED REGION IN THE

LIFE SPACE

Up to this point we have represented the person as a con-
nected and undifferentiated region or point within the life
space. But such a representation can serve only as a first
approximation. In reality the person is not an entirely
homogeneous unity but a highly differentiated object (670).

That psychology has to distinguish parts within the region
which represents the person can be demonstrated mathe-
matically by pointing out two facts. As we have seen it is
not necessary that the whole person participate in the bodily
locomotion. Sometimes only one hand reaches toward the
goal, or only the eyes are directed at an object. It may even
happen that different parts of the body are carrying out
different activities at the same time. This means mathe-
matically that one has to distinguish parts within the region
which represents the person.

For psychology a second fact is still more important. As we
have seen the behavior depends on the state of the environment
and that of the person: B = f(PE). In this equation P and
E are not independent variables. The structure of the environ-
ment and the constellation of forces in it vary with the desires
and needs, or in general with the state of the person. It is
possible to determine in detail the dependency of certain facts
in the environment (e.g., the decrease of field forces, change of
valences) on the state of certain needs (e.g., the extent to which
they are satiated). Thereby it becomes evident that a change
of a certain need, for instance its satiation, does not change all

166

THE PERSON AS A DIFFERENTIATED REGION 167

needs in the same direction and to the same extent. This
makes it necessary to distinguish within the person a multi-
tude of different regions whose changes of state are to a certain
extent independent of each other.

These considerations meet an objection which is sometimes
raised against our representations. It is said that it suffices
for the derivation of behavior to represent either environment
or person. In reality however it is impossible to derive the
psychological processes in the life space without including
changes both of person and of environment in the representation.
(All so-called physiological theories which do not contain a rep-
resentation of the environment are for this reason inadequate.)

One will ask for criteria on the basis of which one can deter-
mine what is to be represented as a region of the environment
and what as a region of the person. In answering this question
it could be pointed out that the "self" is experienced as a
region within the whole field (44, pp. SIQ/-)- Tkk criterion is
however not sufficient. We have seen that the goals and
concepts which popular psychology has often attributed to the
inner person as a rule have to be represented as part of the
environment. From a dynamic point of view the following
facts may be considered: one can treat everything as environ-
ment In which, toward which, or away from which the person
as a whole can perform locomotion.

One will have to treat the question whether a psychological
region belongs to person or to environment with the same
topological methods by means of which one determines other
positions in the life space. These determinations depend on
the concrete facts of the individual case. Therefore for different
life spaces there may be considerable differences in the struc-
ture and boundaries of the person. However, the agreement
is great enough to allow us to make several general statements.

CHAPTER XVII

FUNDAMENTAL CONCEPTS AND COORDINATING

DEFINITIONS FOR THE REPRESENTATION

OF THE PERSON

COORDINATING DEFINITIONS FOR ENVIRONMENT AND FOR

PERSON

The mathematical concepts for the representation of the
psychological person are the same as those for the environment.
'The coordinating definitions also can remain unchanged.
Nevertheless, the determination of the structure of the person
has a different character from that of the environment. This
results from the fact that in constructing the person we cannot
use locomotion as a basic dynamic operation. For, from a psy-
chological point of view the person himself cannot in general
be considered as a medium within which an object carries
out locomotions from one part region to another. In determin-
ing boundaries and connections between part regions within
the person one must rather consider a general dynamic rela-
tionship, namely, the "degree of dynamic dependency" of
one region upon another.

Dynamic Dependency. It can easily be shown that the
different parts of the person differ in the^clegree to which they
are related to each other. It may be that the fulfillment of a
wish changes the whole person, for instance his behavior in
business as well as his behavior toward his family and toward his
friends. In other cases most of the personal regions may
remain almost entirely uninfluenced by the fulfillment of a
wish. There are great differences in the extent of the personal
regions which are essentially influenced by events in one region.

The dynamic interdependency of two regions implies that the
state of the one is influenced by the state of the other. It

168

THE REPRESENTATION OF THE PERSON 169

coincides therefore with the concept of dynamic communica-
tion which we have already defined (see p. 126). It is char-
acteristic for the determination of the topology of the person
that it has to be based almost exclusively on communications
and degrees of communication.

We begin by defining the concept of region of influence by the
following coordinating definition.

Definition: a and b are parts of a connected 1 region (region
of influence) if a change of state of a results in a change of
i state of b.

This definition obviously does not allow the determination
of the boundary of the part regions (a, 5, . . .) and their
relative positions. But it does allow the determination of the
including whole region and its boundaries. Let a, J, c, J ? e, f,
g, ... be regions within the person. One can then determine
which other part regions belong to the same whole region as a
if one finds out whether or not a certain change of a changes the
state of b, c, d, . . . If, for instance, the state of b, d> e would
change with that of a, but the state of c,J, g would not change,
then a, b, d, e would be part of one region A (A > a* A > b;
A > e; A > d); c, /, g on the other hand would be parts of
regions foreign to A (A c = o; A / = o; A g = o). A
change of c may show further that c and/ are parts of the same
region B to which g does not belong (B > c;B > /; B g = o).
A consteUation which could correspond to Fig. 37 would thereby
be ascertained. (The fact that regions A, B, C are foreign to
each other does not tell whether or not they have common
boundaries.)

It is dear that these determinations of regions would give
different results if one started with changes of different kinds.
We met a similar dependency with all determinations of region.
Furthermore, we arrive at different wholes according to the
part regions with which we begin. This is only an expression
of the fact that the degree of communication between two

1 In the long run it may prove to be more fruitful to use the concept of "region 7 '
in this definition instead of that of "connected region" and to determine the
connection by an investigation of the "paths of influence" (see p. 172).

170

TOPOLOGICAL PSYCHOLOGY

regions a and b can be different in the direction a to b
from that in the direction b to a,

The fact that the wholes are different for different kinds of
changes of state makes it possible to proceed step by step in the
determination of the topological structure of regions: one can
produce a second change of state of a which is different from the
first. Then it is usually a different group of regions which is
involved in the change: it may contain the region d (Fig. 38);
further the regions c 9 h, I, which were not affected the first

PIG. 37. -Determining "regions of
influence" and their limits by means of
the dynamical interdependence of their
parts. For instance, a change of the
state of a influences b, d, e but not c t f, g.
A > a +*> -f d + e\B ><?+/; C> g;
A - B C = o.

FIG. 38. Regions determined by a
kind of change of state, which is
different from the kind determining
Fig. 37. In this case a change in the
state of a does not influence e, b, but
d,h,l,c. D>a+d+h+l+c;
D b o; D e o.

time. Regions e and b may show no considerable change.
The region D which is thus defined by the new change of state
has a part in common with A and B. Thereby a certain
relation between the regions A and B is determined. Further
it is now possible to distinguish within the region A two sub-
parts (i and 2) of which the one contains a and d, the other
e and b. For the second kind of influence the communication
between the two subparts (i and 2) is weaker than between
a and d. This determines a psychological boundary within
A which has the character of a dynamic wall in the sense defined
(see p. 126). Similarly region B is divided into two parts,
region D into three parts by the overlapping of A, B, and D.

THE REPRESENTATION OF THE PERSON 171

The following procedure is especially valuable. One can
vary the degree of the change of state and keep both the
nature of the change and the region a in which it starts constant.
On the whole one can assume that the stronger the change of
state of a the more inclusive the affected region. In this way
one comes to a series of regions which lie entirely within each
other (M, N, 0; Fig. 39) and which correspond to the different
degrees of intensity of the change in the initial region a.

This principle can be applied frequently, although not with-
out exception. It can happen that the kind of process suddenly
changes if a stimulus is increased
beyond a certain point. Such an
increase also can have the effect that
the regions become dynamically
closed against each other. There-
fore it is possible that under certain
circumstances a smaller region of
influence corresponds to a stronger

^ ^ FIG. 39. Different degrees

change of the initial region, of change in the state of a
Whether one is dealing with such S^'S^tafaSS?

an exception can be determined by sli gkt change of a; N, region

. . affected by a greater change of

means Of the principles Which We a; 0, region affected by a stai

have given. However, the assump- f ^ c ^ e j n > 1 ^ e stateof
tion holds for most processes and

initial regions as long as the influence is not too greatly
increased. Extremely strong influences regardless of their kind
usually affect the state of the whole person.

These methods by which one can determine topological
relations on the basis of dynamic dependence make use of the
concept of "being-contained-in" or "part-whole/' Besides
one can make determinations whose form suggests the concept
of locomotion and of path. Two regions usually can influence
each other in different ways within an area of communicating
regions. (Regions A and B in Fig. 28, for example, could
communicate by way of region 20 + 1 6 ; or through 15 + 11 + 6;
or through 14 + 3 + 1 + 2, etc.) The way in which a cer-
tain influence is mediated between two regions depends on the

172

TOPOLOGICAL PSYCHOLOGY

nature of the regions concerned and the special structure and
properties of the possible intermediate regions. One often can
distinguish between different " paths of influence" and it is
possible sometimes to determine the one which is actually used.
We have mentioned for instance that a person can be influenced
by "looking at 37 only when his eyes can be caught. One can
say that many influences by which the environment affects the
inner-personal regions occur by way of perception. Whether
or not one can induce a person to perform a certain action
depends not only on which inner regions are touched but also
on the sequence in which they are touched.

It may be clear without further explanation that one can use
determinations of such paths of influence in making topological
statements about the relative position of regions and their
boundaries or boundary zones. We are then dealing not with
movements of regions but with "movements" of changes of
state. In these cases impassable boundary zones correspond
dynamically to walls rather than to barriers (see p. 126).

Boundaries and Boundary Zones. The representation of
the inner structure of the person, in so far as it is based on
dynamic dependencies, uses therefore as constructive elements
regions which correspond to dynamic unities of the highest
degree. They are regions whose parts are so closely connected
that each change of one part results in a change of all other
parts. These regions are therefore'dynamic unities' or gestalten
in the sense in which Kohler (43) uses the word.

The unity of these regions always is determined in relation
to a special kind of change. In regard to other kinds of
change the parts of these regions may be dynamically separated.
But this does not affect their dependency in relation to the
first kind of change. The unity of the regions is therefore, in
spite of its relativity, objective in character.

The dynamic dependency or independency of regions rests
(i) on the qualitative properties of the regions concerned, (2)
on the properties of their boundaries and boundary zones.

Ad i : It is possible that two regions, a and J, are in immediate
contact with each other and not separated by any dynamic

THE REPRESENTATION OF THE PERSON 173

walls. Nevertheless their qualitative properties can be such
that a change in the state of a has no apparent influence on the
state of Z>.

Ad 2: The qualitative properties of two regions could be
such that an interdependency of their states would exist if
they were not separated by certain boundaries or boundary
zones.

Therefore the boundaries within the person, like those within
the environment, rest in part on a qualitative difference
between neighboring regions, in part on the properties of the
boundaries and boundary zones themselves. Boundaries
within the person also exhibit different degrees of dynamic
penetrability; to these correspond different degrees of inter-
dependency of regions or if one thinks of the interdependent
regions as parts of a whole region different degrees of dynamic
unity.

The mathematical concepts allow us to distinguish only
connected and not-connected regions. In 'terms of topology
there are no transitional cases. Dynamically, however, there
are doubtless transitions between completely dependent and
completely independent regions. We have already met an
analogous difficulty in discussing the mathematical concept
"boundary" and the dynamical concept "barrier" in regard
to the psychological environment (see p. 144). In determining
the structure of the person also we have to use certain dynamic
characterizations of regions and boundary zones which go
beyond the mathematical concepts.

Remarks about Strong and Weak Gestalten; Gestalten
with Different Degrees of Dynamic Unity, We are
obviously dealing here with the same considerations which
have been fundamental for the concept of dynamic gestalten.
One might be tempted to use the concept of weak and strong
gestalt in Kohler's sense to characterize the different degrees
of dynamic connectedness. However, Kohler, as he has told
me, feels that these terms should be used in their original sense
according to which one has to speak of a strong gestalt if the
diange of one part of the gestalt involves changes in the form

174 TOPOLOGICAL PSYCHOLOGY

(changes of spacial distribution) of the other parts (43, p. 161).
One speaks of a weak gestalt if no changes of form occur even
in those cases in which the interdependency of the parts is
very great. These terms therefore do not refer to different
degrees of dependency. In order to express the degree of
dependency of the parts of a dynamic whole we will speak of
gestalten of greater or less unity. This distinction is not based
upon a logical dichotomy as is the distinction between strong
and weak gestalten. Instead it signifies a continuous series
with the "and-sum" (Wertheimer), that is, a group of dynami-
cally disconnected regions, at the one extreme and at the other
a gestalt of the highest degree of interdependency of parts.

DYNAMIC PROPERTIES OF PERSONAL REGIONS

The determination of the topology and dynamics of the inner
personal regions has on the whole a more indirect and abstract
character than the determination of the structure of the environ-
ment; probably because dynamic interdependence and not
locomotion is the main determinant of the topology of the person.

In so far as we consider the position of the parts of the person
we shall speak of regions. When we are dealing with their
state we shall speak of systems, especially if we have to deal
with state of tension.

There is a great variety of influences which can be used to
determine the mutual dependency and position of part regions
of the person, for instance fatigue and psychological satiation.
The process of satiation of a certain action depends, as Karsten
(40, pp. 197-227) has shown, on the relationship of the corre-
sponding region within the person to its neighboring regions.
Conversely one can therefore determine, on the basis of obser-
vations of satiation, relationships of neighborhood and of
connectedness within the person. Of the many different kinds
of influences we shall discuss only the change of state of
tension.

Tension. To discuss the nature of tension in detail one has
to consider vector problems. In this place we can give only
a general characterization. Tension is a state of a region.

THE REPRESENTATION OF THE PERSON 175

Strictly speaking one can determine only differences of tension;
a difference in tension tends to produce changes in the direction
of a leveling of tension. Tension is therefore a state of a
region relative to that of another region and it involves certain
forces at the boundaries of the region.

Experimental investigations of different kinds show that
certain characteristics of the environment, especially the
presence of a goal or the tendency to a locomotion, are con-
nected with a state of tension in the person. The carrying
out of a locomotion or the reaching of a goal can at the same
time mean the release of a tension. The experiments show
further that although to a certain extent this change involves
the person as a whole, one can satisfy or leave unsatisfied
different needs more or less independently. Therefore one
has to coordinate to these needs states in different part regions
of the person. Thus within the person we can speak of differ-
ent systems whose degree of tension can change relatively
independently.

Groups of Tension Systems. To determine the way in
which different tension systems are connected one can make use
of their mutual dependency, applying the method that we have
discussed above. A common, relatively simple relation between
two tension systems is given if one is a part of another. That
holds, for instance, if the one system corresponds to a subgoal
of a more inclusive goal. The tension of the part system will
then usually cease in case the more inclusive system loses its
tension. 1

The problem of substitute satisfaction offers another case of
connection between two systems. An action b has a dynamic
substitute value (56; 62, pp. 226^.; 64, pp. 31-32) for the action
^ when the tension in the system which corresponds to a is
released as soon as the tension system corresponding to b is
released. That means that the two systems a and b must be
sufficiently connected parts of one larger system (Fig. 400);
they must not be dynamically independent systems (Fig. 40^).

1 Ovsiankina (68), p. 351. The part- whole relation of inner-personal regions
plays an important part in papers by Schwarz (77, 78) and D. K. Adams, (i).

176 TOPOLOGICAL PSYCHOLOGY

It has been shown that the substitute value of b depends upon
the degree to which a and b are unified (62, pp. 243^.).

There is a peculiar relation between substitute action and the
use of tools or preparatory actions (56). This relation becomes
understandable if one remembers that in both cases the problem
of relative dependency of two systems is involved: in case of
substitution the relation between two part systems si and s<>
within one whole system 5 (S > $1 + $a), in the second case
the relation between a part system $i to its whole s(S > $i).
It is clear that these relations are not entirely different.

As Zeigarnik (92) and Bierenbaum (4, pp. i34jf.) have
demonstrated it is possible to produce experimentally relatively

() (6)

FIG. 40. Conditions under which one activity has substitute value for
another, (a) System a related to the original task, and system b related to the
substitute task are parts of one connected system; (6) the systems a, and 6 are
dynamically separated.

complicated groups of systems within the person and to deter-
mine to a considerable extent the structure of these systems,
especially the kind and degree of connection within the group.
For instance it is possible to produce systems corresponding to
a series of tasks in such a way that in one case a closely con-
nected group and in another case (with the same tasks) rela-
tively isolated systems arise. The position of a single tension
system within or outside a group of systems also can be con-
trolled experimentally. Bierenbaum was able to trace in detail
the process of unification of systems which were originally
separate.

A number of quite different investigations of satiation
(40, pp. 20I/.), tension (92), forgetting (4, 25, 75), and sub-
stitution (62, pp. 232^.), have agreed in showing that as a
rule psychologically adjacent regions within the person cor-
respond to actions or tasks which are related as to their content,
although this is a principle which must be applied cautiously.

THE REPRESENTATION OF THE PERSON 177

STRUCTURE OF THE PERSON

Inner-personal Regions and the Motor-perceptual
Region* If, on the basis of these considerations, we try to
determine the structure of the person as a whole, we come to the
following interpretation. The person is to be represented as a
connected region which is separated from the environment by
a Jordan curve. Within this region there are part regions.
One can begin by distinguishing as such parts the "inner-
personal" regions (/) from the motor and perceptual region
(M). The motor and perceptual
region has the position of a boundary-
zone between the inner-personal
regions and the environment (E; Fig.
41). Two groups of facts stand in
favor of such a representation.

1. Needs or other states of the Pla ^H^^iogy of the
inner-personal regions can influence person, u motor-perceptual

. _ , , region; 7, inner-personal re-

the environment only by way of a bod- g ion; p, peripheral parts of i;
ily expression or a bodily action, that w a1 f rt8 o Z: E '
is, by way of a region which one can

call the motor region. Koffka (47, p. 342) uses the term "the
executive" for this region. According to the previous dis-
cussions of "paths of influence" (see p. 172) we have therefore
to represent the motor region as a boundary zone between the
inner-personal regions and the environment. One of the most
important processes in this motor region is speech. It plays a
great role in the communication between the person and his
social environment. The use of gestures, "smiling at" and
"looking at" belong here. The position of the motor region
as an intermediate between the environment and the inner-
personal regions holds for purposeful actions as well as for
undirected affective discharges of tension, i.e., for all changes of
the environment E resulting from the state of the inner-
personal region.

2. We find an intermediate region again when we consider
the influence in the opposite direction, namely psychological

178 TOPOLOCJCAL PSYCHOLOGY

changes of the inner-personal region resulting from changes
of the environment. This intermediate region corresponds to
the perceptual system in the broadest sense of the word, that
is, to hearing, seeing, etc. It is identical in part with the motor
region. The eye for instance can both express and perceive.
Other parts, like the ear, serve with man to transmit events only
in one direction, from outside in. In any case the boundary
zone between the inner regions and the environment includes
both motor and perceptual systems.

It is to a certain degree arbitrary where one draws the
boundary between the motor-perceptual system and the inner
regions, whether for instance one considers the understanding
of speech as an event within the boundary zone or within the
inner-personal systems. The essential task is to determine the
relative position of the regions in question, and the degree of
communication between them and their neighboring regions.
The same is true of the boundaries between the motor region
and the environment. Both determinations depend upon the
nature of the person and also upon the momentary state of
the life space. During a medical examination the boundaries
of the body are at the same time the boundaries between the
person and the environment. But usually the clothing has to be
counted as part of the person. The outer boundary of a
child may be different when he is in contact with his mother
and when he is with a stranger. In cases of embarrassment,
for instance when one is suddenly exposed to critical glances
of a stranger, the clothing and the whole appearance is often
strongly emphasized and stands out as a special zone within
the boundary region of the person. Under certain circum-
stances regions which are usually hidden can lie open or can be
easily discerned through the surface layers.

With the motor-perceptual region, as with every boundary
zone, one must consider its dynamic properties. There are,
as we mentioned before, great individual differences which
depend upon age and personality.

The difference between motor and inner regions is certainly
not only a difference of position but is also a difference of f unc-

THE REPRESENTATION OF THE PERSON 179

tion within the whole person. In a certain sense the motor
systems have the position of "tools" of inner systems.

As an example of a disagreement between certain functional relations
and the topology of the quasi-physical field we shall consider the relation-
ship between mother and infant in the life space of the mother. It
seems to me characteristic of this relationship that the mother picks
the child up, lays him down, washes him, etc., without asking him. She
uses direct bodily force in a dictatorial manner. Thus, the mother con-
trols the infant by her will in a way which is only slightly different from
the way in which she controls her own body.

On the other hand the actions of the mother are wholly at the service
of the infant. She tries to act entirely according to its needs, i.e., the
needs of the child control the actions of the mother. Functionally there-
fore the needs of the child, as the mother understands them, have for her
the position of an inner system, that is, of a system which directs the
motor region M of the mother. It becomes clear that such a representa-
tion is meaningful when one considers that the birth of the child does not
complete his psychological separation from the mother. The psycho-
logical separation in the sense of the freeing of the child from the mother
is usually completed only much later. At the same time however the
child has to some extent the position of an object in the environment or
one may say it has the position of a part of her own body, but a part
which is in direct contact with the rest only at certain times, for instance
when the child is being fed. The antagonism between functional depend-
ency and bodily separation leads to typical inner conflicts of the mother.

Within the motor-perceptual region one can distinguish again
between more " peripheral " and more " central ' ' regions. Such
distinctions play an important role in the theory of perception
(47; 460; 6oa). In so far as action is concerned the motor
region seems to possess a relatively high unity: it is difficult
to carry out four or five unrelated activities at the same time.
It seems that the motor system can be connected dynamically
with only one inner region or one relatively unified group of
such regions at a given moment. If the motor system were
to be guided by all the needs of a person at the same time, his
behavior would become chaotic. The muscular tonus in one
part of the motor region is closely connected with that in the
others (2, 290). The technique for psychodiagnostics, as used
by Luria (63), seems to be based essentially on creating a close

I So TOPQLOGICAL PSYCHOLOGY

communication between certain inner-personal regions and a
certain part of the motor region.

Central and Peripheral Inner-personal Strata. Within
the inner regions of the person one can distinguish between
more central (c) and more peripheral strata (p 9 Fig. 41). The
necessity for such a distinction showed itself in Karsten's
experiments (40, pp. 236-237) on psychological satiation:
actions which belong to more central strata are ceteris paribus
more quickly satiated. It is of great general importance
whether a psychological process belongs to more central or to
more peripheral strata. Dembo's experimental investigations
(20, pp. loijf.) on anger have shown the significance of this
factor for emotions. If only peripheral strata of the person
are touched, manifestations of anger occur more easily. The
outbreaks of anger are then more superficial. If more central
strata are involved an open outbreak of affect is more rare.
Indeed the boundary zone between the central strata (p,
Fig. 41) and the environment (JS) is stronger than the boundary
zone between the peripheral strata (p) and the environment.
Besides, the central regions may be surrounded by a specific
functional wall (B c , Fig. 420)- The peripheral strata come
more easily into connection with the motor region to which they
lie closer. Therefore expression usually occurs more readily
when events of more peripheral strata are concerned. One
speaks about personal matters only under special circumstances.

This is not only because the more central strata have on the
whole less direct access to the motor region. Events in
the opposite direction also, that is, from the environment to
the inner regions of the person, usually reach the more central
regions less easily. In conversation the way to the peripheral
regions of the person is ahnost always open. But it is difficult
to touch the real core of the person*

The relationship between the peripheral or the central posi-
tion of an inner-personal region on the one hand and the degree
of its accessibility and its ease of expression on the other hand,
is not an entirely fixed one. It depends upon the momentary
state of the person and upon the characteristics of the situation.

THE REPRESENTATION OF TEE PERSON 181

With some persons it seems to be easy to touch certain central
places and to injure them like an "open wound." Some of
these central regions seem to be always ready to communicate
with the motor systems. Not less important than the topo-
logical position of the systems therefore are their dynamic
properties and the dynamic properties of their boundaries.
These are usually quite different for the different regions within
one stratum and may change for the whole stratum. An
example of a relatively simple change of dynamic relations

(<*) (&)

FIG. 42. Relations between various strata of the person under different
circumstances, (a) The person in an easy situation: the peripheral parts p of the
inner-personal region 7, are easily accessible from outside J5; the more central
parts c are less accessible; the inner-personal region 7 influences the motor region
M relatively freely. (&) The person under stress, in state of self-control: the
peripheral parts p of the inner-personal region I are less accessible than in (a) ;
peripheral and central parts (c and ) are more closely connected; communication,
between 7 and M is less free, (c} The person under very high tension: unification
(primitivation, "regression") of the inner-personal region I. M, motor-percep-
tual region; 7, inner-personal region; p, peripheral parts of 7; c, central parts of
7; E, environment; B c , dynamic wall between c and p' t B Vr dynamic wall between
7 and M .

between the different strata is the transition from a state of
superficial anger to a state of profound anger. When the
person is in a quiet mood the boundary (JB P ) between the
peripheral strata and the motor region is dynamically relatively
weak, but the boundary (J? c ) between the peripheral and
central regions of the person dynamically strong (Fig. 420). If
a situation of higher affective tension arises, the person usually
replies with greater "self-control." To such self-control
corresponds a greater separation of the peripheral strata from
the motor region. At the same time the inner regions become
relatively more unified (Fig. 426). Dembo (20) has shown
that if the affective tension is increased the resulting unifica-
tions can reduce the person to a more primitive level (Fig. 42*7)*

182 TOPOLOGICAL PSYCHOLOGY

If the tension in the inner regions is still further increased it
can break through to the motor region.

The separation of the inner regions from each other and their
connection with the motor region can undergo changes of very
different kinds. In the state of joy the inner-personal regions
seem to be relatively unified and especially little separated
from the motor zone. Joy expresses itself easily. Here again
we find important individual differences.

Dynamically a more central position and greater tension
of the inner systems are in many respects equivalent.

Individual Differences in the Structure of the Person.
At the present time we do not know very much about the kind
of connection between the different systems and strata of the
person. Yet the structure of the person shows considerable
individual differences,

The Degree of Differentiation of the Person. One of the most
important dynamic differences between child and adult is that
the person of the child is less differentiated into part regions.
The growth of the psychological environment and of the person
of the child does not mean simply quantitative increase in
size, but it is at the same time essentially a process of differ-
entiation (So; 48; 30, p. 8; 51, p. 206; 71, pp. 199-200; 36; 37)
and, to some extent, of integration.

We cannot at this point discuss in detail the psychological
problems of the processes of differentiation and integration
but we have to ask what conceptual means are available for a
comparison between the degrees of differentiation of different
persons. One could say that the statement "the person A is
more differentiated than B" is inadmissible for the following
reason: it is a thesis about the number of part regions of a
whole region, namely, the person, and it seems doubtful
whether there is any value in comparing the number of part
regions of two persons.

We distinguished between two possibilities of determining
regions: the characterization by certain qualitative properties
on the one hand and by locomotions or communications on the
other (p. 94). If one uses the qualitative characterization, one

TEE REPRESENTATION OF THE PERSON 183

can speak of different regions whenever one can make quali-
tative distinctions between regions. It is dear that this is an
entirely relative standard, for what seems to be a homogeneous
region at a superficial examination may show qualitative
differentiations when one looks more closely. The number of
distinguishable regions would thereby be made dependent on
the degree of accuracy of the examination. This seems to make
an objective comparison of the number of part regions impos-
sible. For instance in both child and adult the first analysis
shows the same number of regions, namely, a central and
a peripheral inner-personal stratum and the motor stratum.

If one determines regions by means of the concept of dynamic
communication, their unity is determined by their dynamic
wholeness. But in this way also we achieve only a relative
determination of the units of regions since we find different
degrees of wholeness. It is for instance possible to consider
child and adult each as one single dynamic region.

Another way of approaching this problem is suggested by
our discussion of finitely and infinitely structured spaces. It
might be possible to designate as "smallest regions" those
regions within the person which at a given moment can no
longer be broken up into psychologically meaningful part
regions. As a matter of fact the assumption that such objec-
tively not-further-structured dynamic unities are the structural
elements of the person seems to be justified. Unfortunately
at the present time a comparison between the degree of differ-
entiation of different persons is not possible in this way.

Nevertheless, the dynamic connection can be used in deter-
mining the degree of differentiation of the person. Even if we
can designate child and adult each as one single dynamic
region, still the degree of wholeness of this system is greater
with the child than with the adult: a change of one part of the
system in the child usually influences all other parts to a much
greater extent than in the adult. 1

1 E. Dufiy (21, 22), for example, found that in adults the muscular tensions tend
to be more differentiated among particular groups of muscles than in children.
We cannot discuss special problems at this point, especially not the difference

1 84 TOPOLOGICAL PSYCHOLOGY

This degree of dynamic unity of the whole person can be
taken under certain conditions as a criterion of the degree of
differentiation into part regions, that is, if one takes as basic
unities regions which show the same degree of dynamic separa-
tion from the neighboring regions.

The following consideration shows that we are justified in
using the degree of unity of the whole region as a criterion of
the degree of its differentiation into regions of a certain degree

A 3

PIG, 43, Two systems ot the same structure and the same degree of differentia-
tion but of different degrees of dynamic unity. The system B as a whole shows
a less degree of unity than A. The parts I, 2, 3, 4, 5, of the system A are less
separated than the parts x, 2, 3, 4, 5, of the system J5. A given change of one of
these parts (e.g., 5) will therefore affect the other parts (i, 2, 3, 4) to a lesser
degree in B than in A.

of separation. If within two whole regions neighboring part
regions depend on each other to an equal degree, that whole
region will in general be more strongly unified which contains
fewer part regions. The reverse is also true: whole regions
of the same degree of differentiation show a stronger dynamic
unity if their part regions are less separated from each other.
A and B may be two whole systems which are differentiated to an
equal degree, i.e. } they contain the same number of part systems
(A - IA + ZA + $A + 4* + $A]B = IB + 2* + 3* + 4* + SB)
(Fig. 43) . To simplify the example we shall assume that the part
regions are in themselves homogeneous and of the same quality in
A and in B. The structure shall also be of the same nature
in A and B. The only difference shall be that the part regions
of A are separated from each other by less strong walls than the
part regions of B. In this case B is dynamically a whole of a
lower degree of unity than A, in spite of the fact that it has the
same degree of differentiation.

in the kind of interdependency between certain systems in ttie child and in the
adult, which may be as important as the mere difference in "the degree of
differentiation.

THE REPRESENTATION OF THE PERSON

18$

The statement that a more differentiated system shows a
lower degree of unity, is not valid without exception, even if
we presuppose the same degree of dependency of neighboring
regions on each other. As a third factor we have to mention
the kind of structure of the whole, that is, the special arrange-
ment of the part regions. The degree of unity of the whole
region can vary even when the number of part regions and the
degree of separation of neighboring part regions are the same.
This is shown for instance by a comparison of systems A, B,
and C in Fig. 44. While there are in A and B part regions

FIG. 44. The dependency of the degree of unity of a system on its structure.
The degree of differentiation (number of part systems) of the system A , B, and C
and the degree of separation of neighboring part-systems are the same. Never-
theless the degree of unity of the whole system is diff erent for A , B and C.

which can influence each other only through three separating
walls, in system C we find never more than one wall between
the part regions, in spite of the fact that the number of part
regions is in all three cases the same.

It would be a difficult and important task of a "general
gestalt theory" 1 to investigate how the degree of unity of a
whole depends on its structure. The following proposition
seems to contain a fundamental principle: the dynamic unity
of a whole depends not only on the relation of the parts of the
whole to each other but no less on the relation of th.e whole
to its environment. As a rule a greater separation from the
environment increases the inner unity of the whole.

In determining the degree of dynamic unity of the whole
person one must therefore always take into account (i) the

1 K6hler's investigations (43) are an approach to this problem. Cf. also
Rashevsky (72).

186 TOPOLOGICAL PSYCHOLOGY

degree of differentiation, (2) the degree of dynamic separation
of neighboring part regions, (3) the special structure. If we
assume that the structure is roughly the same in different
persons, we have still to consider factors i and 2. It is how-
ever possible to separate these two factors if one succeeds in
comparing the dynamic strength of the walls between the
single part systems of the person concerned.

Such a comparison can sometimes be made. Investigations
of satiation and substitution have shown that the psychological
systems of certain feeble-minded persons are separated from
each other by relatively strong and rigid walls. On the other
hand it is characteristic of the sensitive problem child that
there is only a small degree of separation of the part regions
from each other. One can express this also in the following
way: the person of the problem child corresponds to a more
fluid, the person of the feeble-minded child to a more rigid
material (52, pp. 209-210). The normal child is intermediate
in this respect. The lesser dynamic separation of the sys-
tems in the problem child involves a closer connection between
his central and peripheral strata. Therefore the condition
of the central regions more readily expresses itself, for instance
in a stormy affective outbreak. At the same time these expres-
sions show a superficial character. The central strata of these
children lie, from the point of view of dynamics, less deep; even
if the degree of differentiation were the same as that of normal
children, the boundary zone which lies between the central
strata and the environment would sheath the central strata
less than with the normal. Actually the degree of differentia-
tion of the feeble-minded as well as of the problem child is less
than that of a normal child of the same age. This shows itself
in their infantilism and in the primitiveness of their behavior.

An observation of the process of differentiation in the develop-
ment of the individual child shows that the older child is in
general a more differentiated system. But the difference
between adult and child, between children of different ages, and
between the adult and the aged is certainly not only one of
differentiation of the whole system. At the same time there

THE REPRESENTATION OF TEE PERSON 187

is at least also a difference in the fluidity of the whole person ;
further a difference in the kind of structure (52, pp. 207$".).

The degree of primitiveness of behavior seems to be a good symptom
of the degree of differentiation of a person. Also the achievement of a
person in an intelligence test seems to depend above all on the degree of
differentiation of the person, or at least on the differentiation of certain
part regions.

The influence of malaria treatment in cases of general paralysis as
shown by Galant-Rattner (29), is an example of change in the state of
the person.

The Kind of Structure and the Function of the Part Regions.
Individuals differ not only in the degree of differentiation of the
whole person but also in the way in which the different part
regions are arranged, in the strength of the connections between
the different individual part regions, and the function which
they have in the life of the person. We shall call these char-
acteristics the "kind of structure of the person."

Within the same person the different part regions are not
differentiated to the same degree. This is obvious for the
different regions which are connected with knowledge and
skills. Individuals also differ markedly as to which part
regions are highly and which are poorly differentiated. We
find similar differences in kind and degree of differentiation
in non-intellectual regions, for instance in those inner-personal
regions which are connected with family, friendships, or occupa-
tion. With the feeble-minded certain "irreal" regions (see
p. 196) which are important for phantasy seem to be relatively
little differentiated (51). Or again if one speaks of a "harmoni-
ous character" it is meant that the different part regions of
the person are relatively well balanced.

The functional significance of such regions can be very
different. The region related to occupation for instance can
play either a fundamental or a more secondary role in the
structure of the person. It can have its source in very different
needs. The significance which a certain activity has for a
person and the satisfaction which it offers him depend on the
functional significance of this region in the life of the person.

l88 TOPOLOGICAL PSYCHOLOGY

The degree of dynamical connectedness of the different parts
of the person can be nearly equal within the whole region of the
person, or certain regions can separate themselves to an espe-
cially high degree from the others and develop relatively
independently. This can be observed in the normal person and
it seems to be important for certain mental diseases.

In those cases in which Freud speaks of "complex" and
McDougall of "dissociation" (650) there is probably also a con-
siderable degree of isolation.

A person's structure is often relatively constant over a long
period of time. Nevertheless a great change in the environ-
ment, falling in love, being "converted," or some other decisive
event can bring about a far-reaching change of structure which
may be either temporary or lasting.

The question of the structure of the person has special bearing
on developmental psychology. For the structure of an indi-
vidual at any given time is a product of his development. The
differentiation of the whole person, connectedness, relative
differentiation, and function of the single part regions seem to
undergo typical changes during individual development (26).
The investigation of these processes about which we know very
little will be possible only along with the investigation of the
general laws and only if one carefully considers the different
conceptual problems which we have discussed.

The Connection between Dynamic and Topological
Factors. In concluding our discussions of the topology of the
person we shall treat somewhat more extensively an example
which throws light upon the connection between the dynamic
and the topological properties of personal regions. This
dependency becomes especially dear if one considers the momen-
tary changes of personal structure.

We have already mentioned that according to the results
obtained by Dembo a state of great inner tension brings about
a dynamic unification of the person. This greater unity
rests at least in part on the fact that the degree of separateness
of the part systems is a relative value and has to be related to
the magnitude of the relevant forces. Dynamic walls which

THE REPRESENTATION OF THE PERSON 189

may suffice to separate part systems to a high degree, if the
forces in question are small, can become relatively unimportant
in the face of the strong forces which arise during a state of
high tension. For this reason a great increase in inner tension
means ipso facto a dedifferentiation of the person.

Certain findings of Coghill (17) in regard to the development
of behavior of embryos may be explained in a similar way.
These experiments show that behavior must not be character-
ized as a gradual genesis of more inclusive reactions out of
what are at first isolated reflexes. Rather the first responses
to stimuli are undifferentiated reactions of the whole organism.
Only gradually more differentiated responses of single parts
of the whole organism come into being. The development
of embryonic behavior like the psychological development
from childhood to maturity (see p. 203) can show besides
differentiation some integration (13; 14; 71, p. 200).

The differentiation occurring in development does not
entirely destroy the original unity of the fundamental dynamic
systems. This is shown by the fact that the embryo, having
attained a certain degree of differentiation, drops back to a more
primitive level when the connection with the blood stream of
the mother is severed or the fetus becomes fatigued; 1 that is,
it again responds to a stimulus with a more undifferentiated
whole reaction. We find here therefore a temporary "regres-
sion " like that which occurs when a child is sick or in the tension
of anger. In explaining this fact one usually points to the
proposition that differentiation of reactions results from an
inhibition (69, p. 43; 3, pp. 418, 442) of the original process by
newly developed centers. In dynamic terms this means that
the differentiated reaction is caused by an opposition of forces,
It may however be more simple to bring these processes directly
into connection with the fundamental fact that the embryo
gradually differentiates itself into regions which dynamically
are relatively separated.

1 ANGULO y GONZLES, A. W. (3), p. 420. The experiment was made with
embryos removed, at different ages, from the uterus in such a way that the con-
nection with the blood stream of the mother was kept intact.

190 TOPOLOGICAL PSYCHOLOGY

That the embryo responds with undifferentiated whole
reactions implies that certain inner systems or the motor strata 1
are unified to such a high degree that a stimulus produces a
reaction of the whole body, a (Fig. 450) may be the system
whose lack of differentiation is responsible for the reaction of
the whole body to the stimulus concerned, (a may be identical
with the whole organism or with one part of it.) The later
differentiation of the reaction may rest on the fact that a has
differentiated itself into the part regions i, 2, 3, 4, 5 (Fig. 456).

(a) (6) (c)

FIG. 45. Regression and the structure of the person, (a) Relatively undiffer-
entiated state of development (the whole systems contain relatively few part-
systems) ; the regions a, b, c, d are separated. (&) Later state of development of
the same individual: regions a, b, , d, show further differentiation into sub-
regions; the newly developed part regions I, 2, 3, 4, 5 of region a are separated
by weaker walls than the older regions a, b, c t d\ under the stress of relatively
strong forces these stronger walls (corresponding to the older primitive structure)
become again the main determinants and result in "regressive" ("infantile")
behavior (similar to that in the stage (a)); this holds also when the person
becomes very fluid, (c) Differentiation with change of the basic structure: cer-
tain newly developed walls between part regions have become stronger than the
older partitions; in this case a primitivation would lead to a behavior somewhat
different from that which occurs in a case of primitivation of (&) .

The separation of biologically "young" parts from each other
is usually less rigid than that of older parts. The dynamic
walls between the regions i, 2, 3, 4, 5 are therefore to be repre-
sented for this time as less rigid than the walls between a as a
whole and its neighboring regions 6, c, d. From this assump-
tion, which biologically is at least very probable, there follows:

i. If the intensity of the stimulus is increased, the rigidity
of the new walls will not suffice to keep the part regions as rela-
tively independent systems in the face of the stronger influence.
(This follows from the general relation between the degree of
separation and the magnitude of the effective forces.) There-
fore again there will be a reaction of the whole body. As a

1 Either of these assumptions would suffice.

THE REPRESENTATION OF THE PERSON 191

matter of fact one can attain a pronounced reaction of the
whole organism even with adults if the stimulus which is chosen
is strong enough. It is characteristic of responses to very
strong stimulation, for instance to shock, as well as for the
responses of undifferentiated organisms, that they are relatively
independent of the point of application of the stimulus (3,
p. 434). This results directly from the fact that the strength
of the walls is only relatively determined.

2. A similar effect must occur if one, instead of increasing
the magnitude of the forces, brings about a greater fluidity of
the systems concerned. Each change of state of the organism
in the direction of greater fluidity or lability will probably affect
the young walls more strongly than the older, already more
rigid ones. Psychological experiments with adults indicate
that such an increase in the fluidity of the whole system occurs
during fatigue (92, pp. 65-70). What changes of state bring
about a dedifferentiation of the systems depends on the dynamic
properties of the factors which separate the part systems and on
the kind of influences in the face of which the separation should
be effective. It is however at least not improbable that, in a
fetus too, fatigue leads to an increase in the lability of the whole
organism, and therefore weakens the relatively new differen-
tations to a higher degree than the older ones.

3 . The hypothesis just presented allows us to determine under
what conditions a dedifferentiation will not lead to behavior
which corresponds to an earlier state of development. We
have assumed that this correspondence of behavior rests on the
fact that the later developed part systems are separated from
each other by more plastic walls than the older ones. But
it can happen that the younger walls become in the course of
time more rigid than those which were developed earlier. This
changes the structure of the larger systems and the relationship
of the single part systems to them. If at this moment a situ-
ation occurs which produces a temporary dedifferentiation of
the person, the behavior will become again more primitive,
but it will not be the same as the behavior at the earlier stages
of development. For the remaining units of system will differ

192 TOPOLOGICAL PSYCHOLOGY

essentially from the original ones (Fig. 45^). As a matter of
fact we find, in spite of all similarities, considerable differences
between the behavior of an infantile adult and that of a child.
This difference is probably not caused merely by a difference in
the properties of the material of the system but also by a
difference in their structure.

The dynamic structure of the organism can be altered not
only by changes in the relative strength of the walls by which
the different part systems are separated from each other; under
certain circumstances existing walls may actually be weakened
or destroyed.

INTEGRATION AND DEDEBTERENTIATION

Besides the process of differentiation there seems to occur a
process of "integration" during development (McDougall,

65*).

This process counteracts the process of differentiation insofar
as it creates a greater degree of interdependence of the different
systems of the person and in this way makes for a higher degree
of unity of the person as a whole. Mathematically this inte-
gration could be viewed as a reversal of the differentiation so
that there would be no difference between the dedifferentiation
occurring under the specific (more or less " abnormal' ') con-
ditions discussed above the normal process of developmental
integration.

Dynamically speaking, however, these processes seem to be
definitely different in character. The integration of the person
during development seems not to be a strict reversal of the
differentiation and also not a simple restructuring of the inner
personal system. It is rather a process by which a certain
system (or group of systems) becomes "dominant" insofar as
it is able to impose certain patterns of action and to build up
certain quasi needs, I am inclined to venture the theory that
we have to deal here with the relation of an inducing field to
an induced field (Lewin, 52, p. 97). In other words, the position
of the dominant system is dynamically equivalent to that of
Che center of a social power field. The same type of dynamic

THE REPRESENTATION OF THE PERSON 1 9 20

interdependence Is probably characteristic for the relation
between the inner psychological region and the motor region or
between a person and a tool. It differs from that inter-
dependence of neighboring tension systems which we discussed
previously.

These problems can be considered in 'detail only in vector
psychology. However, it might be well to limit the term
" integration " in psychology to those cases where the unification
of differentiated regions is due to the establishing of the hier-
archic relation between an inducing field and an induced field.
A unification due to destruction (or weakening) of the dynamic
walls that separate systems might be called "dedifferentiation"
as opposed to "integration."

D. DIMENSIONS OF THE LIFE SPACE

CHAPTER XVIII

THE DIMENSIONS OF THE LIFE SPACE

In our figures we have used so far a plane, that is a two-
dimensional space, for the representation of psychological fields.
One can raise the question: is it correct to use such a manifold
for this representation ? In other words, how many dimensions
has the life space?

MATHEMATICS OF DIMENSIONS

Mathematics has only within recent years found a way to
treat problems of dimension satisfactorily. Differences in
dimension are not differences in size or in potency of the space. 1
One can coordinate one to one the set of points of a line to the
points of a limited two-dimensional region or of a three-
dimensional body. In considering how many dimensions one
ought to attribute to the life space, one therefore does not have
to take into account the purely quantitative question of the
space "available" in the representation.

Mathematics shows that dimension is a property of the
"inner structure" of the space, a property which is closely con-
nected with topological characteristics. It is characteristic of
a two-dimensional space, for instance a plane, that within it
there is no possibility of connecting each of five or more points
with each other such that the connecting lines do not intersect.
Further, as we have said, in a two-dimensional space it is
impossible to connect a point within a circular area with a point
outside of it, without intersecting the boundary of the area.

1 For the following discussions cf. K. Menger, (66), pp. if. In this book one
finds also remarks about the historical background, pp. 83^.

194 TOPOLOGICAL PSYCHOLOGY

In a three-dimensional space such a path is possible. These
facts show clearly how important the number of dimensions of
psychological life space is, for instance, for the question of
what locomotions are possible.

The determination of the number of dimensions in modern
theory of dimension rests essentially on a general relation
between a spacial object and its boundary. The boundaries of
a three-dimensional body for instance are surfaces, that Is, they
are themselves two-dimensional. A surface is limited by lines,
that is, a two-dimensional space is limited by one which is one-
dimensional. The one-dimensional finite line is limited by
points which themselves have the dimension zero. In general
one can say: the boundary of an w-dimensional space is (n i)-
dimensional (66, p. 80; 79, pp. 207-208). It will be dear,
without going into the mathematics of these problems in detail,
that on the basis of this fact a procedure of recursion is
possible. One can begin with an empty set as a ( i)-dimen-
sional space and from it proceed to the higher dimensions.

From the point of view of mathematics there is no reason to
limit the number of dimensions to three. The progression to
spaces of n dimensions is possible without difficulty. Mathe-
matics deals also with spaces whose number of dimensions is
different at different points. It might seem enticing to psy-
chology to make free use of the possibilities which the intro-
duction of a more complicated space or of a space of many
dimensions would offer. In the beginning however it is impor-
tant for reasons of discipline and economy in making theories
to introduce no more dimensions than are absolutely necessary.
Therefore our problem has to be formulated as follows: what
is the minimum number of dimensions that is required to repre-
sent the life space ?

It may seem advisable to distinguish our problem from another in
which it is also possible to speak in a mathematical sense of dimensions.
One can think of the different properties of an object or a system as
variables and can represent them by means of a system of coordinates
which has as many dimensions as there are distinguishable properties.
Physics speaks in this sense of a phase space. (In the same way time can

THE DIMENSIONS OF THE LIFE SPACE 195

be represented as a fourth dimension.) Similarly Boring speaks of
dimensions of the stimulus (6) in psychology. It should be clear that the
question of the number of dimensions of the life space is entirely different
from the question of the number of distinguishable "properties" of
psychological objects or events.

THE DIMENSIONS OF THE PSYCHOLOGICAL ENVIRONMENT

Reality. In the beginning of our experimental work two-
dimensional spaces sufficed for the representation of situation.
For some time however we have been using three-dimensional
spaces.

In determining the number of dimensions of the life space it is
useful to refer to the concept of locomotion and the above
mentioned relations between the dimension of a region and its
boundary. That the number of dimensions of a life space is
greater than zero results from the fact that movement is
impossible in spaces of zero dimension. The fact of locomotion
means that the life space is at least one-dimensional.

The boundaries of a one-dimensional space are themselves of
zero dimension; they correspond to points. Is the point an
adequate representation of the boundaries of psychological
regions? To represent the life space as one-dimensional, that
is, as a group of lines, seems at first thought to contradict the
facts. Nevertheless, it is not easy to reject such a represen-
tation in a mathematically incontestable way. For one-
dimensional spaces can consist of very complicated networks.
For instance, one can define one-dimensional spaces in such a
way that it is possible to distinguish at each point any number
of directions. Nevertheless, I believe that the nature of the
psychological processes which actually occur makes it impossible
to represent the life space as one-dimensional. It is often
possible to go "along a boundary " in the psychological field.
Furthermore, one can often distinguish different sectors as
parts of a connected boundary. It is, as we have mentioned,
important for the behavior of a person that the different parts
of a boundary can differ in their strength. These facts show
that there are boundaries which do not consist of one point,
but which consist at least of lines, that is, of one-dimensional

ig6 TOPOLOGICAL PSYCHOLOGY

manifolds. And a boundary which has at least one dimension
implies that we are dealing with a space which is at least
two-dimensional.

It is more difficult to answer the question whether one should
use $till more dimensions. When we are dealing with quasi-
physical fields, it may be necessary to represent the psycho-
logical field as three-dimensional space corresponding to the
three-dimensionality of physical space. To circumvent, e.g.,
a two-dimensional barrier, certain locomotions are possible
between the points of a three-dimensional physical region that
would not be possible in a two-dimensional space. From the
properties of these locomotions one may be able to demonstrate
that the quasi-physical field, like the physical space, has also
three dimensions.

Degrees of Irreality. A two-dimensional representation of
the quasi-social and quasi-physical fields has sufficed for most of
the psychological problems we have treated so far. It is how-
ever necessary to make the transition to a further dimension
when we have to distinguish between different a degrees of
reality. 7 ' A daydream, a vague hope, has in general less reality
than an action; an action sometimes has more reality than
speech; a perception more than an image; a faraway "ideal
goal " is less real than a "real goal " that determines one's imme-
diate action (34; 52, p. 250). Action itself can be of very
different degrees of reality. Processes which concern strong
needs of the person and in which he has to surmount strong
physical or social barriers have usually a high degree of
reality. Among the quasi-conceptual processes one can for
instance distinguish carefully planned consideration of the ways
and means which lead to a certain goal from free play of
phantasy, which is more irreal (64; 20).

It is important to distinguish the concept of degree of reality in psy-
chology from the epistemological concept of reality. For the epistem-
ological concept there are no different degrees of reality. An intermediate
between existence and non-existence is not possible. The psychologically
"irreal" objects are real for psychology in an epistemological sense; that
is they exist, they have effects. It must further be emphasized that the

THE DIMENSIONS OF THE LIFE SPACE 197

physical reality of the object concerned is not decisive for the degree of
psychological reality. For the primitive or the child living in a "magic
world" those facts which are believed are real (64).

Experiments have shown that the degree of reality is a very
important dynamic property of almost all psychological objects
and processes. This has come out especially in experiments
about the level of aspiration (34), the origin and effects of sub-
stitute action (64, 81), the formation and change of goals (34),
emotional processes (20) 7 memory (u) 3 and play (81).

We have found it necessary to use a special dimension for the
representation of degrees of reality. If one represents the
totality of real regions as two-dimensional space, for instance as
a plane, one has to coordinate to the different degrees of reality
different planes which lie one above the other. In demonstrat-
ing mathematically the necessity of such a representation, one
will have to consider that locomotions are possible within the
same degree of reality and between regions of different degrees
of reality. The totality of what belongs to the same degree of
reality, for instance the world of wishes of a person at a certain
moment, is itself a region. In it one can distinguish part
regions. We can speak of movements in the irreal strata as in
the stratum of reality. Therefore it is possible to determine the
topology of these regions. The topological structure of a
stratum of irreality can be similar to that of the stratum of
reality. Under certain circumstances however its structure is
typically different from the structure of the plane of reality,
especially if the situation in the plane of reality is very dis-
agreeable for the person (52, pp. 146/0

In following the principle of using no more dimensions than
necessary to represent the life space, one will have to ask
whether or not it is possible to represent the irreal levels without
introducing a new dimension. A closer examination shows
that such a representation is not permissible.

Two methods which are somewhat alike suggest themselves if one tries
to represent regions of a lesser degree of reality without using a further
dimension. One could attempt to indicate levels of successively decreas-
ing reality as a chain of regions in such a way that to each region corre-

1 98

TOPOLOCICAL PSYCHOLOGY

sponds a lesser degree of reality than to the one preceding it (Fig. 460).
Or one might represent them as a series of ring-like regions each of which
is entirely enclosed by the succeeding one (Fig. 466).

However, such a method would not permit an adequate representation
of certain locomotions. This becomes clear if one distinguishes between
the single part regions within the same level of reality, a 1 , b\ c l } d\ e l ,
. . . may be part regions of the degree of reality R l j a 2 , 5 2 , 2 , c? 2 , e 2 , . . .

FIG, 46. Attempt to represent regions of different degrees of reality -without
introducing an additional dimension. The different degrees of reality are repre-
sented in (a) as a chain of regions, in (6) as a series of ring-like regions Ri, R* t R* r
R< (Ri > a^R^> b\ Ri > cS Ri > #; R* > a, R J > H R* > c*, R* > d>; 1^ > a,
^ 3 > b*, R* > c 8 , R 3 > rf 3 ) ; the regions d l , &+ d z correspond with respect to their
contents.

may be part regions of the degree of reality J? 2 ; a 3 , & 3 , c z f . . . may be
part regions of the degree of reality R*. The difference in the degree of
reality may not be very great and the structure of the different levels of
reality may be roughly similar. Furthermore, regions d 1 in R 1 , cP in -R 2 ,
and d z in J? s , may belong together psychologically as far as their content is
concerned.

The two-dimensional representation has the disadvantage that a move-
ment within the same level of reality, for instance from b 1 to a 1 , almost

TEE DIMENSIONS OF THE LIFE SPACE igq

always implies an approach to or a withdrawal from regions of a higher or
lower degree of reality. Furthermore, there is the difficulty that if one
represents a locomotion from one region into another region of correspond-
ing content but lower degree of reality, it must cross other regions of the
same degree of reality first. For instance one has to cross the regions
a 1 , & 2 , d*j a?, c 3 , in order to go from d l to d 2 and d\ Within the actual
psychological life space too one cannot move directly from every region
of one degree of reality to every region of another degree of reality. But
certainly there occur psychological locomotions which pass a whole series
of different degrees of reality within a region of the same content. If one
represents the levels of different degrees of reality as a two-dimensional
space these locomotions cannot he represented as continuous paths with-
out falsifying the relations of connectedness of regions within the different
levels. This shows that one has to introduce a new dimension for the
representation of differences in degree of reality if one wants to represent
such locomotions adequately.

Incidentally these considerations also make clear a fundamental mathe-
matical characteristic of dimension. Points or other part regions of
spaces of different numbers of dimension can be coordinated in one-to-one
correspondence only when one destroys the topological relations: the
two-dimensional representation (Fig. 466) destroys the actual existing
connections between d 1 , d* } d*.

Since spaces of no less than two dimensions correspond to
regions of the same degree of reality the life space has to be
represented by at least three dimensions (Fig. 47) if one takes
into account the different degrees of reality. We coordinate
to the different degrees of reality planes or levels which lie one
above the other. In our diagrams we shall represent levels of
higher degree of irreality (/) by planes which lie higher. One
could of course also do the reverse.

One of the most important dynamic differences between
different degrees of reality is the greater fluidity of the more
irreal levels (/). This greater fluidity shows itself in different
facts: (i) Barriers in the environment offer relatively little
resistance (one can do what one wants to do in irreality (20,
PP- 3^jf-)) ! ( 2 ) the boundaries of environmental regions can be
shifted more easily and are less definitely determined (81,
p. 149) ; this holds also for quasi-conceptual regions of different
degrees of reality; (3) a diffuse discharge of an inner-personal
tension system occurs more quickly (n, p. 2); this corre-

200

TOPOLOGICAL PSYCHOLOGY

sponds to weaker walls of the Inner systems; (4) the boundaries
between person and environment are less clear and the struc-
ture of the environment depends to a greater extent on the
needs of the person (52, p. 146).

PROBLEMS OF DIMENSIONS OF THE PERSON

The question arises how many dimensions has the region
which represents the person within the life space. It would be

FlG. 47. "Representation of the different degrees of reality by an additional
dimension of the life space. R, more real level; I, more irreal level; P, person.
In a level of greater reality the barriers are stronger and the person P is more
clearly separated from his environment.

wrong to represent the person only in the plane of reality. For
he can act also within the more irreal levels. Furthermore,
there seem to exist inner-personal systems which one has to
coordinate to irreal levels.

This problem raises questions to which it is hardly possible to
give definite and detailed answers at the present time. One
certainly has to represent the transition from sober reasoning
to phantastic attempts to solve a problem as a transition to
processes within irreal levels. But are we dealing in these

THE DIMENSIONS OF THE LIFE SPACE 201

cases with locomotions of the person as a whole? If the person
were at each moment within only one level of reality, one would
have to represent him as two-dimensional. However, such a
thesis is open to doubt since the person, even during the flight
into irreality, stays within the level of reality in part at least,
namely, as a bodily being. This is proved by the fact that he
can be influenced by processes within this level even if he is
somewhere else in his phantasy. There may be cases in which
the possibility of influence is so weakened that one has to repre-
sent the person as two dynamically relatively separated regions
which belong to different levels of reality. Such a condition
may sometimes be realized in schizophrenia.

One might also think of representing the person as a region
which belongs at the same time to more than one level of reality.
In this case the person like the environment would have to be
represented as a region of at least three dimensions. It may
sometimes reach farther, sometimes less far into the different
planes of reality and the "center of gravity" of the person may
belong to different levels at different moments.

Finally the transition to levels of lesser degree of reality
could in most cases be thought of as a locomotion of parts of
the person. Such a locomotion of a part region of the person
must not necessarily lead to a breaking up of the unity of the
person. Assuredly we find such movements within the level
of reality.

On the whole it seems to me that the psychological facts
speak for attributing to the person the same number of dimen-
sions as to the life space as a whole and for representing the
person in all the levels of reality which the life space concerned
possesses. It often is possible to represent the transition of the
person to other levels of reality as a change in the relative
weight (see p. 137) of the different levels of reality. It is, for
instance, possible to treat the different levels of reality, Ri, R%,
Rz . . . , in which the person P is at the same time, as an over-
lapping situation (seep. 138, P < Ri,P < R^P < R$, . . , ).
The behavior of a person as a whole is influenced at a given
time by the various levels of reality to a different degree. (The

202 TOPOLOGICAL PSYCHOLOGY

relative weight WRI of Ri may be 70 per cent, of Wm, 5 P er cent >
of W&, 10 per cent, etc.) These relative weights of the various
levels differ under different circumstances and for different
persons. A "flight into the irreality" would mean that the
relative weight of certain levels of higher reality (e.g., WRI} is
diminished and that the weight of certain more irreal levels
of the life space (e.g., W s ^) that may have been zero or only
slightly greater than zero has increased considerably (e.g.,
W R4l = 40 per cent instead of W* 4 = 5 per cent as at first).
Such representation of a "flight into irreality" avoids many
difficulties which a representation by locomotion of the person
from one level of reality to another naturally shows. It has
also the advantage of emphasizing a change in the importance
of the various regions within the person, which is characteristic
of such transitions. Certainly such change is not only struc-
tural but functional as well.

It is relatively easy to distinguish between person and environment
within a level of high reality. (We shall at this place disregard the
question of the accurateness of the determination of boundaries.) In any
case it is not difficult to show that there are places in the environment
which the person does not occupy. In levels of great irreality however
it is often uncertain whether one should designate a certain region as part
of the person or as part of the environment.

The stratification of person and environment into levels of
different degrees of reality must not be confused with the dis-
tinction between central and peripheral strata within the person.
For within the same level of reality one will have to discriminate
between more central and more peripheral inner regions. On
the whole, processes within the more irreal planes seem to have
a doser dynamic relation to the core of the person and to his
central needs. It is a special problem whether one has to
represent the motor stratum only within the level of reality.

THE DIFFERENTIATION OP THE LlFE SPACE IN THE DIMENSION
REALITY-IRREALITY

It is not only when phantasy is especially active that the life
space possesses a stratification into regions of different degrees

THE DIMENSIONS OF THE LIFE SPACE 203

of reality. This is always the case. The degree of this strati-
fication however depends upon the situation as a whole (81,
p. 150). In some cases the transition of the person to levels
of a different degree of reality is an extension or contraction of
the life space (that is, of person and environment) in the reality-
irreality dimension rather than a locomotion within an other-
wise constant field. Moreover 3 the relative weight of levels
of different degrees of reality can change.

It seems to be very different with different persons and at
different moments how far the life space is extended in the
direction toward irreality. One may ask whether there exist
intrinsic limits of the life space in this direction, that is, whether
there exists a plane of greatest irreality. It would hardly be
possible at the present time to justify such an assumption,

There is much that speaks for the assumption of a level of
greatest reality. One could attempt to establish the thesis of
the existence of such a plane of te full" reality in each life space
by calling attention to a special relation of the motor stratum
to reality (see p. 177). In addition we might be tempted to
connect conceptually the full reality with the objective physical
or social influences from the "outside " on the life space (see
p. 72) . Reality is essentially characterized, as far as dynamics
is concerned, by its independence of the will of the person.
Nevertheless, there are facts which run counter to the suppo-
sition of an absolute level of full reality. Someone may believe
that he lives in full reality until events of a still harder reality
teach hjm better.

The facts which speak for a relativity of the plane of highest
reality become still more significant if one thinks of the develop-
ment of the life space from childhood to maturity. We have
already mentioned that this development is to be characterized
by an extension and especially by a differentiation of person and
environment. Such a gradual differentiation of the life space
can also be observed in the dimension of differences of reality.
A great number of psychological facts lend support to the
assumptions: (i) that there are only slight differences in degree
of reality within the life space of the small child (Fig, 480); (2)

204

TOPOLOGICAL PSYCHOLOGY

that the levels within the life space of the small child correspond
on the whole to an intermediate degree of reality (80; 70; 42,
p. 706 ; 52, p. 104; 48, pp. 3<56/ .; 39) (tf* Fig. 48a and i). That
means that the levels of reality of the small child are to be
characterized as relatively irreal in comparison with those of
the adult, and that the levels of irreality of the child are com-
paratively real (8 1, p. 162). It seems therefore advisable not
to introduce the concept of a plane of "absolute reality.' 1 We

"V"-

PIG, 48. Life space of a child (a) and of an adult (6). The life space of the
adult is more highly differentiated in the dimension, reality-irreality. The range
of reality and irreality layers in the life space of the child corresponds to an inter-
mediate range of layers in the adult's life space.

have already mentioned that for different individuals and for
the same person at different moments of time the degree of
differentiation of the life space into levels of different degrees of
reality varies. Further investigations are necessary before
one can answer these problems definitely. But the concepts
and methods discussed will, I think, suffice to meet any con-
ceptual difficulties in representing the levels of irreality within
the topology of the life space.

The concepts which we have developed in this book concern
the whole psychological life space, that is, person and environ-

THE DIMENSIONS OF TEE LIFE SPACE 205

ment. They allow treatment of all problems of position and
connections of the life space and its parts. They are applicable
to quasi-physical as well as to quasi-social and quasi-conceptual
facts. By means of these concepts one can represent the struc-
tural changes of person and environment and all kinds of
locomotion. To a certain extent one can also deal with those
problems of psychology that are dynamic in a narrower sense of
the word, for instance the friction of a region, the solidity of a
barrier with respect to locomotions, the degree of separation
of regions with respect to communications of different kinds,
and the degree of wholeness (dynamic gestalt) of systems of the
environment and the person. Finally one can treat certain
problems of tension and changes of state of regions, for instance
the liquidity or solidity of a region. ,

But these dynamic questions are connected in one way or
another with problems which go beyond topological psychology.
Topological psychology can represent only the framework of
events which are (C possible " in a life space. In order to deter-
mine which events actually occur and what conflicts underlie
them one has to take into account directed magnitudes, especi-
ally the concept of psychological force and the concept of field
of force. The relevant concepts and coordinating definitions
are to be treated as part of vector psychology.

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Spurenfeld, PsychoL Forsch., 1933, i8 r 299-342.

76. RTEMANN, GEORG: Ueber die Eypothesen, welche der Geometric zu Grunde

liegen (3rd ed.), BerHn: J. Springer, 1923, 48 pp.

77. SCHWARTZ, GEORG: Ueber Ruckfalligkeit bei Umgew3hnung, Teil I. Ruck-

fall tendenz und Verwechslungsgefahr, PsychoL Forsch.^ 1927, 9, 86-158.

78. SCHWARZ, GEORG: Ueber RUckfalligkeit bei Umgewohnung, Teil II. Ueber

Handlungsganzheiten und ihre Bedeutung fiir die RiickfalUgkeit, PsychoL
Forsch.., 1933, 16, 143-190.

79. SIERPINSKI, WACLAW: Introduction to general topology, Toronto: Univ. of

Toronto Press, 1934, 238 pp.

BIBLIOGRAPHY 21 1

So. STERN, WILLIAM: Psychology of early childhood up to the sixth year of age,
supplemented by extracts from the unpublished diaries of Clara Stern, trans-
lated from 3d ed., New York: Holt, 1924, 557 pp,

Boa. STEVENS, S. S.: The operational definition of psychological concepts,
Psychol. Rev., 1935, 42, 517-527.

81. SLIOSBERG, SARAH: Zur Dynamik des Ersatzes in Spiel- und Ernstsituationen,

Psychol. Forsch., 1934, 19, 122-181.

82. STENDHAL, (BEYLE, M. H.): Red and Black, translated from the French by

E. P. Robins, New York: 1898.

83. THURSTONE, L. L.: Vectors of mind multiple-factor analysis for the isolation of

primary traits, Chicago: Univ. of Chicago Press, 1935, 266 pp.

84. TOLMAN, EDWARD CHASE: Psychology versus immediate erperience, Phil, of

Science, 1935, 2, 356-380.

85. TOLMAN, EDWARD CHASE : Purposive behavior in animals and men, New York :

Century, 1932, 463 pp.

86. URYSOHN, PAUL: Zum Metrisationsproblem, Math. Ann., 1925, 94, 309-315.

87. VEBLEN, OSWALD: Analysis situs (2nd ed.), New York: American Mathe-

matical Society, 1931, 194 pp.

870. WERFEL, FRANZ: The forty days of Musa dagh, New York: Viking Press,
1934, 824 pp.

88. WERTHEIMER, MAX: Ueber Gestalttheorie. Symposion, 1927, I. (Also

separately published as " Sonderdrucke des Symposion," Heft I.)

89. WERTHEIMER, MAX: Untersuchungen zur Lehre von der Gestalt I, Psychol.

Forsch., 1922, i, 47-65.

90. WERTHEIMER, MAX: Untersuchungen zur Lehre von der Gestalt II, Psychol.

Forsch., 1923, 4, 301-350.

91. WIEHE, F. : Die Grenzen des Ichs (in preparation) .

92. ZEIGARNIK, BLUMA: Ueber das Behalten von erledi^ten und unerledigten

Handlungen, Psychol. Forsch., 1927, 9, 1-85.

GLOSSARY

The concepts contained in this glossary are grouped as follows: (I) meth-
odological concepts, i.e., concepts related to epistemology, logic, theory of
science, and methodology; (II) concepts of topological psychology. This group
contains (A) mathematical concepts; (B) psychological concepts. In fhfc latter
subgroup are placed (i) concepts by which directly observable facts are coordi-
nated to topological concepts; (2) dynamic concepts. This classification is not
intended to be strict, but may aid in clarifying the logical position of the various
concepts.

I. METHODOLOGICAL CONCEPTS

Approximation, method of : This method determines first the structure of the
life space as a whole and proceeds gradually by determining more and more
specific properties to greater exactness-
Cause: One has to distinguish historical and systematical causation, (a)
Systematical causation: An event is considered as a function of the total
situation at a particular time. The cause of an event is always the inter-
relation between several facts, (b) Historical causation: An answer is given
to the question why an individual situation at a certain historical time and
at a given geographical position has these particular properties.

Concepts: One can distinguish (a) mathematical concepts like boundary,
region, vector; (&) dynamic (conditional-genetic) concepts like force, ten-
sion, resistance, fluidity.

Concrete : That which has the position of an individual fact which exists at a
certain moment. In psychology the concrete can be represented as a part
of the life space or as a property of such a part.

Construct: A dynamic fact which is determined indirectly as an "intervening
concept" by way of "operational definition." A construct expresses a
dynamic interrelation and permits, in connection with laws, the making of
statements about what is possible and what is not possible.

Definition, coordinating: Certain observable facts are correlated to certain
mathematical concepts. A coordinating definition should be univocal and
if possible reversible.

Dimension of properties : One can think of the different properties of an object
as variables and can represent them by means of a system of coordinates
which have as many dimensions as there are distinguishable properties.
Such a system of coordinates is sometimes called phase space.

Dynamic : Facts or concepts which refer to conditions of change, especially to
forces, are called dynamic. Dynamic facts can be determined indirectly
only (see Construct).

Existence: That which has effects. The existence and the time index of a
psychological fact are independent of the existence and the time indices of
the facts to which its content refers.

213

214 GLOSSARY

Explanation: Representation of a concrete situation in such a way that the
actual events can be derived from it with the help of general laws.

Historic process (dialectic) : Actual order of occurrences in the given world.
Generally periods of apparently continuous transformation are followed by
periods of crisis involving sudden changes of structure.

Influences "from outside** : Influences on a life space which cannot be derived
by psychobiological laws from the psychobiological properties of the pre-
ceding situation.

Law, empirical : A law defines the functional relationship between various facts.
These facts are conceived as types, i.e., historical time indices do not enter a
law. A psychological law can be expressed by an equation, e.g., of the form
B /(). The laws serve as principles according to which the actual
events may be derived from the dynamic factors of the situation.

Phenomenal fact: Fact which can be observed directly.

Prediction : Only if, in addition to the law, the special nature of the particular
situation is known is prediction possible.

Reality: The epistemological concept of reality (see Existence) is to be dis-
tinguished from the psychological concept of "degree of reality,'* which
refers to certain strata of the life space.

Representation: Conceptual image (in the mathematical sense) of facts.
Scientific representation presupposes (i) concepts which permit logically
strict derivations and (2) coordinating definitions between concepts and
observable data.

Space, mathematical : see Mathematical Concepts.

Space, phase : see Dimension of properties.

Space, physical: Space which includes the totality of all physical facts that
exist at a certain time (the whole physical world).

Space, psychological : see Life space, in Psychological Concepts.

World, physical : Totality of more or less interdependent physical facts. All
physical changes are the result of conditions or changes within one connected
physical space. According to physics there are no influences upon physical
objects "from outside" this space.

World, psychological: Totality of more or less interdependent psychological
facts. There exist a plurality of psychological worlds, corresponding to a
plurality of not connected psychological spaces (life spaces) . These worlds
are influenced "from outside."

n. CONCEPTS OP TOPOLOGICAL PSYCHOLOGY

A. Mathematical Concepts

Boundary point: Any surrounding of a boundary point of a region contains
points which do not belong to that region.

Cut: A path which connects two boundary points of a region and which, aside
from these boundary points, lies wholly within the region.

Dimension: The boundary of an ^-dimensional space is (n i) -dimen-
sional. Points of spaces of different numbers of dimensions can be coordi-
nated to each other in one-to-one correspondence only when their
topological relations are destroyed.

GLOSSARY 21 S

Intersection : The intersection or common part of regions A and B (A 2?) is

the totality of points which are part of A and of B.
Jordan curve : A topological image of a circle.
Path ; Connection between two points by a Jordan arc.
Region, closed : A region which includes its boundary points.
Region, connected : A region every point of which can be connected with every

other point by a path which lies entirely within the region.
Region, foreign: A is foreign to B if the intersection of A and B is empty

(A-B =o).
Region, open : A region for every point of which there is a surrounding that

lies entirely within the region.
Region, simply connected : A region whose connectedness is destroyed by one

cut.
Space, mathematical: Does not refer to any particular entities but only to

relations. For these relations certain axioms hold.
Space, metrical: To any two points of such space a distance is coordinated

for which certain axioms hold, especially the triangle axiom.
Space (region), structured finitely: A space (region) which can be divided

into distinguishable part regions, but which is not infinitely structured.
Space (region), structured infinitely: A space (region) whose part regions can

be divided infinitely into further parts.
Space (region), unstructured: A space (region) which cannot be divided into

distinguishable parts.

Space, topological : Nonmetrical space, for which certain axioms hold. Part-
whole relation and connectedness are among its basic concepts.
Sum, topological : The sum of regions A and B (A -f B) is the totality of the

points which are included in either A or B.

Topological image : A one-to-one correspondent continuous image,
Topologically equivalent (isomorphous) : A region is topologically equivalent

to another if it is possible to convert the first into the second by a process of

continuous transformation without changing the connection within the

region, i.e. t by stretching or bending without tearing.

B. Psychological Concepts
i. CONCEPTS MAINLY CONCERNED WITH DIRECTLY OBSERVABLE FACTS

Action, representation of : Under some circumstances action is represented as
a region; under others it is represented as a path.

Adit : A region perceived as permitting locomotion to a certain region.

Ann : A part of a region which, without separating from the region, makes a
locomotion to or communicates with another region.

Behavior : By behavior we mean any change in the life space, which is subject
to psychological laws (see Influences "from outside"). Behavior (B) at a .
given time is a function of the life space (L) at that time. B = /().

Boundary, of a psychological region : Those points of a region for which there
is no surrounding that lies entirely within the region. The presence of a
boundary within the environment or person can be determined by means
of locomotions or communications. A boundary of a psychological region
is not necessarily an obstacle to locomotion or communication.

216 GLOSSARY

Boundary, sharp: Psychologically one can distinguish sharp and unsharp
boundaries. In the case of a sharp boundary it can be determined for every
point of the life space whether or not it belongs to the region in question.

Boundary zone (between region M and N) : That region (BZ) which is foreign

to M and N and which has to be crossed by a locomotion from M to N.
* Differentiation : The degree of differentiation refers to the number of subparts
within a region. Under certain circumstances the degree of dynamical
unity can be used as inverse criterion for the degree of differentiation.

Distance: Although " distance" is not a topological concept, distances in the
life space can be compared by topological means if the path corresponding
to one distance is a part of the path corresponding to the other distance.

Environment: Everything in which, toward which, or away from which the
person can perform locomotions is part of the environment.

Field : Space, conceived as having a certain characteristic at every point.
> Integration: The process by which the number of subparts of a region is
reduced.

Life space: Totality of facts which determine the behavior (J5) of an individual
at a certain moment. The life space (L) represents the totality of possible
events. The life space includes the person (P) and the environment ().
B f(L) f(P,E) . It can be represented by a finitely structured space.

Life space, foreign hull of : Facts which are not subject to psychological laws
but which influence the state of the life space.

Locomotion : Change of position. Locomotion can be regarded as a change of
structure : the moving region becomes a part of another region. Locomotion
can be represented by a path which can or cannot be carried out. This
path characterizes a change of position within a field which otherwise
remains sufficiently constant. One can distinguish quasi-physical, quasi-
social, and quasi-conceptual locomotions.

Person: The person is represented as a differentiated region of the life space;
however in the first approximation he can be represented as an undifferenti-
ated region or a point.

Person, parts of: (i) Motor-perceptual stratum (region); (2) inner-personal
stratum (region): (a) peripheral regions, (b) central regions. The motor-
perceptual stratum has the position of a "Boundary zone between the inner-
personal regions and the environment.

Point: From a psychological point of view the main property of a point is its
character as an unstructured region.

Position, determination of : The position of a point in the life space is character-
ized by the region which includes it. The exactness of the determination
depends upon the extent to which one can distinguish subregions within the
region in question.

Region, connected psychological : A and B are parts of a connected psycho-
logical region (i) if a locomotion from A to B is possible without leaving the
region; (2) if a change of state of A leads to a change of state of JS.

Region, determination of: (i) A psychological region can be determined by its
qualitative properties and by the topological relations of the region or of its
boundary to other regions or their boundaries; (2) by psychological processes
which connect different points, especially by locomotions or communications.

GLOSSARY 217

Region, psychological : Part of the life space. Everything that is represented

as a region in characterizing a psychological situation must be a part of the

life space. A region is not necessarily a connected one.
Region of undetermined quality : A region whose cognitive structure or quality

is not sufficiently determined for the individual. In some respects, it has

dynamically the character of a barrier.
Regions, neighboring: Regions which have a common boundary (boundary

zone) .

Regions, overlapping : Regions which have a common part.
Restructuring : Change of the relative position of part regions without change

of their number.
Situation: Life space or part of it conceived in terms of its content (meaning).

The life space may consist of one situation or of two or more overlapping

situations. The term situation refers either to the general life situation

or the momentary situation.
Situation, overlapping: Two or more situations which exist simultaneously

and which have a common part. The person is generally located within

this common part.
Space of free movement: Regions accessible to the person from his present

position. The space of free movement is usually a multiply connected

region. Its limits are determined mainly by (i) what is forbidden to a

person, (2) what is beyond his abilities.
Structure of a region : Refers to (i) degree of differentiation of the region, (2)

arrangement of its part regions, (3) degree of connection between its part

regions.

2. DYNAMIC CONCEPTS

Accessibility : The ease with which a region can be reached by locomotion or
communication. Degree of accessibility can be represented by barriers or
walls. The accessibility of a person can be represented by boundaries of
different strengths between the environment and certain inner regions of the
person.

Barrier: A boundary (boundary zone) which offers resistance to locomotion.
The degree of this resistance can be different (i) for different kinds of loco-
motion, (2) for locomotion in different directions, (3) at different points of
the barrier.

Barrier, impassable : A boundary (boundary zone) which is impassable for the
locomotion in question.

Barrier, inhomogeneous : A barrier which offers different resistance at different
points.

Barrier, outer : A barrier encircling the person.

Communication: Two regions are in communication if a change of the state
of one region changes the state of the other. The degree of communication
corresponds to the degree of dynamical dependence. The degree of com-
munication depends upon (i) the kind of communicating processes, (2) the
properties of the communicating regions, and (3) the boundary (boundary
zone) between them. The degree of communication from A to B need
not be the same as that from B to A. Communication can be represented
by a boundary zone (wall) ; a low degree of communication corresponds to a
strong wall.

2l8 GLOSSARY

Elasticity : The tendency of a changed region to return to its original state.
The degree of elasticity can differ for regions of the same fluidity.

Equilibrium: A constellation of forces such that the forces at a point are
opposite in direction and equal in strength.

Fluidity: The smaller the forces necessary (other conditions being equal) to
produce a certain change, the more fluid a thing or medium is. The fluidity
of the same region can be different for different kinds of influences.

Force: Cause of change; a basic concept of vector psychology. Properties
of a force are: strength, direction, and point of application. Strength and
direction can be represented by a vector.

Friction: The resistance which a passable region shows to locomotion.

Gestalt : A system whose parts are dynamically connected in such a way that
a change of one part results in a change of all other parts. This unity may
differ for different kinds of changes.

Irreality: see Reality

Material properties : Those factors determining the state of a region which are
conceived as properties of the region itself.

Medium : A region in or through which locomotions can be carried out.

Need : A need corresponds to a tension system of the inner-personal region.

Plasticity: The plasticity of a region corresponds to the ease of producing a
relatively lasting and stable change in its structure.

Power field : The sphere of influence of a person. It can be represented as a
field of inducing forces.

Reality, degree of: A property of psychological facts. Differences in degree
of reality can be coordinated to a special dimension of the life space. The
more irreal levels show a greater fluidity. The structure of a more real
level depends less upon the will of the person. The degree to which the
life space is structured in the dimension reality-irreality depends upon the
specific character, e.g., age, of the person and the momentary situation.

Regression : Corresponds to a decrease in the differentiation of the person.

Rigidity: Boundaries (barriers, walls) are the more rigid the greater the forces
necessary to overcome them. Rigidity of a region can differ for different
types of processes.

Solidity: see Rigidity.

Structure, cognitive : Structure of the life space corresponding to the knowledge
of the person.

Substitute value: An action b has a substitute value for the action a if the
tension of the system corresponding to a is released when the tension of the
system corresponding to b is released.

System: A region considered in regard to its state, especially to its state of
tension.

Tension: A state of a region relative to surrounding regions. It involves
forces at the boundary of the region which tends to produce changes such
that differences of tension are diminished.

Thing : A region in or through which locomotions cannot be carried out.

Valence : A valence corresponds to a field of forces whose structure is that of a
central field. One can distinguish positive and negative valences.

Wall : Boundary (boundary zone) considered as to its influence on communica-
tion. A stronger wall corresponds to a smaller degree of communication.

Weight, relative : Relative importance of one of two or more overlapping situ-
ations (regions) within a life space.

AUTHOR INDEX

Adams, D. K., 175, 207
Allport, Gordon W., 25, 179, 207
Angulo y Gonzales, 189, 191, 207
Aristotle, 8, 9, 10, 82

Birenbaum, Gitta, 176, 207
Blumberg, A. E., 59, 207
Boring, E. G., 76, 195, 207
Bridgman, P. W., 17, 21, 57, 79, 207
Brown, J. F., 8, 21, 197, 199, 207
Biihler, Charlotte, 32, 76, 207

Carmichael, Leonard, 189, 207
Carnap, Rudolf, 61, 207
Cassierer, Ernst, 82, 207
Coghill, G. E., 189, 207

Dembo, Tamara, 15, 31, 39, 96, 138,
143, iSS, 156, 180, 181, 188, 196,
197, 199, 207, 208

Dostoievsky, F., 13

Duffy, Elizabeth, 183, 208

Fajans, Sara, 156, 208

Fallada, Hans, 46, 208

Feigl, Herbert, 19, 59, 207, 208

Forrer, Sarah, 176, 208

Frank, J. D., 39, 9$, 138, 160, 207,

208
Frank, L. K., 188, 208

Franklin, Philip, 53, 208

Freud, Sigmund, 3, 5, 77, 188, 208

Galant-Rattner, R., 187, 208
Galileo, L. J. A., 8, 9, 10, n, 82
Goldstein, Kurt, 179, 208
Gesell, Arnold, 182, 208

Hanfmann, Eugenia, 24, 143, 156, 208
Hausdorff, Felix, 53, 54, 55, 87, 208
Heider, Fritz, 115, 116, 208
Homburger, August, 12, 208
Hoppe, Ferdinand, 35, 136, 147, 208
Hull, C. L., 22, 208

Irwin, O. C., 182, 208

Jack, Lois M., 40, 208
Jaensch, E. R., 204, 209

Karsten, Anitra, 174, 176, 180, 209
Katz, David, 162, 209
Kltiver, Heinrich, 204, 209
Koffka, Kurt, 12, 20, 33, 77, 155, 177,

179, 182, 204, 209
KQhler, Wolfgang, 12, 33, 64, 72, 80,

81, 155, 167, 172, I73i 174, 179,

185, 209
Kttnkel, Fritz, 77, 209

319

AUTHOR INDEX

Lau, Ernst, 109, 209

Lewin, Kurt, 67, 69, 74, 79, 81, 101,

122, 126, 140, 142, 143, 144, 146,

147, 156, 164, I75 176, 179, 182,
186, 187, 196, 197, 200, 204, 209
Lindbergh, Anne S., 125, 210
Lissner, Kate, 175, 176, 210
Luria, Alexsande R., 179, 210

McDougall, William, 188, 210
Mahler, Wera, 175, 196, 197, 210
Menger, Karl, 52, 74, 87, 105, 193, 194,

210

Mueller, G. E., 40, 210
Murphy, G., 166, 210

Newton, I., 6, 82

Ovsiankina, Maria, 175, 210
P

Pavlov, I. P,, 189, 210
Piaget, Jean, 204, 210
Pratt, Karl C., 182, 189, 210

Restorff, Hedwig von, 176, 210
Riemann, Georg, 165, 210

Sakuma, Kanae, 179, 210
Schwarz, Georg, 175, 210
Sierpinski, Waclaw, 87, 194, 210
Sliosberg, Sarah, 155, 156, 160, 197,

199, 203, 204, 211
Stendhal (Beyle, M. H.), 36, 211
Stern, William, 182, 204, 211
Stevens, S. S., 22, 211

Thurstone, L. L., 76, 211
Tolman, Edward Chase, n, 21, 27,
i33> 134, iSS, 211

Urysohn, Paul, 61, 211

V
Vernon, Philip E., 25, 207

Waring, E. B., 39, 96, 138, 207
Werfel, Franz, 38, 211
Wertheimer, Max, 13, 174, 211
Wiehe, F., 129, 211
Wundt, W., 34

Rashevsky, N., 76, 185, 210
Reichenbach, Hans, 35, 58, 63, 210

Zeigarnik, Bluma, 50, 176, 191, 211

SUBJECT INDEX

The letter G is used to indicate page numbers found in the Glossary
A B

Ability, 44, 45

(See also Space of free movement)
Abstraction, 10, 16-17
Accessibility, 126, 180, Gai7
Action, 148, G2i5

as a path, 108

as a region, 180
Activity, as a medium, 115

minor, 138
Adit, G2i5

and barriers, 148-154
Adult, 155, 183, 186, 204

infantile, 192

space of free movement of, 46
Affectivity, 160
Age, 178, 186
Airplane, 125
Analogy, 79
Analysis, 17, 164
Anger, 15, 143, 180, 181
Angle, 88

Appearance, and reality, 18, 22
Approach, 146-148

direct, 150

Approximation, method of successive,
6, 16-17, G2i 3

and person, G2i6
Arm, 102, 127, 156, G2i$
Aspects, 134

Aspiration, level of, 15, 197
Assumptions, minimum of, 6
Atmosphere, social 19
Authority, 45
Axiom, triangle,

Barrier, 6, 123, 130, 141, 145, 190,

and adits, 148-154

and cognition, 135,

differentiated, 146148

homogeneous, 146148,

impassable, 49, G2i7

outer, 143, 144, G2i7

physically homogeneous, psycho-

logically inhomogeneous, 147
(See also Rigidity)
Behavior, G2i5

as function of situation, n, 12, 74
Being contained in, 87, 120, 157
Belonging to, 41
"Between," 141
Biology, and physics, 67
Body, 106, 178, 190
Boundary, 42, 44, 49-91, 118136, 194

and barrier, 153, G2i7

and boundary zone, 121, 172173

common, 137, 139

and communication, 126-130

fluidity, 161

and locomotion, 94, 119

passability of, 123, 124, 149

and path, 153

between peripheral and central
regions, 181

of the person, 167, 200

psychological, 118, Gsi5

sharp, 119-123, G2i6

social, 45, 122, 124, 126
Boundary point, 89
Boundary zone, 114, 119-123, 130,
137, 151, 153; G2i6, G2i7

and boundaries, 172

221

222

SUBJECT INDEX

Boundary zone, and pathway, 154
Bridge, 129, 132

Case, pure, 10
Catalogue, of types, 15
Causa, efficiens, 34

finalis, 34
Causality, 30-40

historical, 9, 30-32, 69, 213

in psychology, 30-40

systematic, 30-32, 0213
Cause, 10, n, 0213

and force, GsiS

and historical origin, 9

physical, 71

relational character of, 33
Causes, chains of, 30

concept of, in different epochs, 10
Central, 180-182
Chain, causal, 62-63
Change, 99, 169, 171

bodily, 29

cause of, GaiS

of connectedness, 50

of distance, 156-159

of magnitude, 156-159

of position, 48, 113, Gsi6

of state, 172, 0217-218

of structure, 156, 159-162, G2i6

velocity of, 158

(See also Tension)
Child, 27, 179, 182, 183, 192, 197

first, 158

gifted, 46

life space of, 204

mentally retarded, 46

space of free movement of, 44, 45
Class, social, 46
Classification, 6, 1,0, 17
Clearness, degree of, 39
Closed curve, 91
Closed region, 89, G2i$
Clothing, 178
Club, 119, 122

Cognition, 71, 133* *35
Cognitive structure, 39, 132-133, 150
Communication, 54, 93-117? 126-130,
169, Gai7

and arm, 102,

and boundary,

social, 43

(See also Accessibility)
Complex, 188
Concept, 13, 85, G2i3

conditional-genetic, 21

constructive, 6

dynamic, 10, 63-65, 205,

historical, 30

intervening, 21,

mathematical, 76, 78, 85, 6213,
G2 14-2 15

methodological, 0213-214

and model, 78-79

psychological, 0215-218

and symbol, 76-78

systematic, 30

topological, 60-62, Gai5
Concreteness, 32-33, 0213
Condition, 68-69
Conflict, 122, 179

Connected region, 54, 90, 100, 101,
174, 0215

and learning, 133
Conscious, 26
Construct, 16, 0213
Contact, points of, 94

social, 101

Contemporaneity, 35
Content, as property, 38-39
Convention, social, 140
Conversation, 49, 180
Conversion, 188
Coordination, 59-60

of psychological to physical facts, 66

reversible, 93

Correspondence, one-to-one, 91, 199
Crossing, a boundary, 126
Crisis, G2i4
Cut, 90, 106, G2I4

SUBJECT INDEX

223

Danger zone, 114

Daydream, 18, 196

Decision, 39, 99, 134

Dedifferentiation, 189, 191

Deduction, 16

Definition, coordinating, 93-96, 168-

193, 213

genetic, 10

operational, 6, 0213

reversible, 6213
Dependency, degrees of, 174

dynamic, 168-172, 0217
Depth of boundary, 122, 152
Derivation, 6, 14, 16, 74, 81, 82
Description, 13, 82

and constructive representation, n
Despair, 143
Destructurization, 61
Determination, conceptual, 77

degree of, 39

exactness of, in, G2i6

indirect, 0213

temporal, 36-38
Determinism, 70
Detour, 146
Development, 46, 162, 188, 189

historical, 31

of the life space, 203-204
Diagram, 78

Differentiation, in, 121, 155-156, 187,
G2i6

degree of, 182-187, G2i6

in the dimension reality-irreality,
202-205

and unity, 185

(See also Regression)
Difficulty, degrees of, 144
Dimension, 53, 73, 107, 1^3-195, 0214

of the life space, 193-205, G2i8

of the person, 200-202

of properties, 76, G2I3

of reality-irreality, 197, G2i8

of stimuli, 195

(See also Irreality)
Direction, 50, 55, 99, 195

offeree, G2i8

Discharge, diffuse, 199

Dissociation, 188

Distance, 88, 99, 156-159, Gai6

quasi-conceptual, 55

quasi-physical, 55

quasi-social, 55

Distribution, geographical, 101
Divisibility of regions, 164, 0215
Dream, 66

Dynamic (see Fact; Concept; Factor)
Dynamically closed, 68
Dynamics, and space, 59, 62-63

Eating, 47
Economics, 32
Ego, phenomenal, 77
Elasticity, 124, 161-162,
Embarrassment, 178
Embryo, 189

Emotion, 34, 155, *8o, 197
Environment, 6, 23, 162, 167, 168-174,
G2i6

behavioral, 77

coordinating definitions for, 168-174

dimensions of, 195-200

geographical, 77

and life space, G2i6

and need, 200

and person, 11-12

physical, 18

psychological, 18

social, 1 8
Epoch, Aristotelian, 9, 10, 32

Galilean, 9, 10
Equilibrium, 63, 64, G2i8

(See also Forces)
Equivalence, functional, 65
Event, 156, G2I3

as function of total situation, 6213

future, 37

Exactness, of determination, in, 6216
Executive, 177
Existence, 35, 32-36, 36-38, 67, 0213

and experience, 18-19

and quality, 38
Experience, 155, 34

direct, 20

224

SUBJECT INDEX

Experience, and existence, 18-19

past, 34

Experiment, 10, 75
Explanation, 21, 34, Si, 6214

and description, 7
Expression, 34, 68, 180

superficial, 186

Extension, of the space of free move-
ment, 144, 158
Eyes, 1 66

Factor, dynamic, 188-192

nonpsychological, 29, 71

topological, 188-192
Fact, collecting of, 4

conditional-genetic, 19

direct observable, 0213, 0215-217

dynamic, 82, 0213

economic, 66

empirical, 59

geographical, 31

as part of the life space, 0213

phenomenal, 6214

psychobiological, 65

quasi-conceptual, 26-27

quasi-physical, 24-25

quasi-social, 25-26

social, 25
Fait, accompli, 98
Fatigue, 160, 174, 189
Fear, 40

Feeble-minded, 186
Field, G 2 i6

of action, 28

brain, 80

offerees, 129, G2i8

inducing, 86

mathematical, 26

perceptual, 61

psychological, 26

quasi-physical, 147
Fight, 38, 39

Flight, into irreality, 201, 202
Fluidity, 150-162, 190, 199, G2i8
Forbidden region, 44, 103, 6217

(See also Space of free movement)

Force, 47-50, 63, 64, 78, 97, 129, 175,

188, 189, 191, 205, 0213, G2i8

(See also Equilibrium; Fluidity;

Friction; Valence)
Foreign, regions, 136

hull of life space, 75, G2i6
Forgetting, 176
Form, 173

Friction, 117, 125, 135, G2i8
Friendships, 187
Future, 34, 35
and past, 36
Function, of region, 187-188

Genesereihe, 67, 69

Germany, social structure of, 122

Gestalt, 33, 6 4, i7 2 , 185, 205, G2i8

strong, 173-174

weak, 173-174
Gestures, 177
Ghetto, 101
Glance, 129
Goal, 37, 48, 114, 140, 175, iQ7

clear, 39

ideal, 196

and object, no

real, 196

as region, no

as thing, 115
Gravity, center of, 201
Ground, moving, 115
Group, scattered, 100

social, 49, 94

of systems, 176

Hearing, 178
Help, 147 ,
Heredity, 33
Hypotheses, working, 82

Ideal, 1 8, 196
Illustration, 146
graphic, 76

SUBJECT INDEX

225

Impossible events, 14

logical, 150

Inaccessibility, 140-144
Indeterminate, 36, 39-40
Individual, 68, 75, 6213

case, as an infinite task, 17

differences, 178, 182-188
Infancy, 155
Infantilism, 186, 190
Influence, 127

direction of, 129

gross somatic, 27-29, 71

from outside the life space, 70, 203,
0214

by perception, 27-29
Inhibition, 189

Inner-personal stratum, 172, 177, 200,
G2i6

central, 180, G2i6

peripheral, 180, G2i6
Inner region, 91
Inside, a region, 96-99
Insight, 155

Instability, of psychological situa-
tions, 60

Integration, 155-156, 189, G2i6
Intelligence, 187
Intensity, 171
Intercourse, 127

social, 101, 102
Intersection, 87, G2i$

of boundaries, 106

empty, 92

Inventory, of behavior, 16
Irreality, 160, G2i8

degrees of, 196-200

dynamic properties of, 199

(See also Reality; Dimension, of

the life space)
Island, 105, 1 1 8

of the forbidden, 103
Isomorphism, 80, G2i$

Jordan curve, 90-91, 105, 136, 141,

146, 177, G2is
Joy, 182

Kinematic, 115

Lability, 191

Law, 1 6, 30, 69, 1 88,

application of, 1 1

and construct, G2I3

and individual case, 8-9

psychobiological, G2I4

and rule, 10

and variables, n
Lawfulness, 10
Leadership, 40
Learning, backward, 134

maze, 133

Level of aspiration, 160
Life space, 11-12, 14-17, 27, 28, 33,
202-205, G2i4, G2i6

of animals, 24

boundary of, 73, G2i6

and brain field, 81

of the child, 123-203

contemporary, 35

content of, 18-29

determination of, in

of different persons, 68

differentiation of, 202-205

dimensions of, 193-205

extension of, 24

and fact, 6213

as finitely structured space, 163-
165, G2i6

form, hull of, 75, 6216

and mathematical space, 59

of men, 24

and physical facts, 24

representation of, 12-13, G2i6

as space, 41-58

as totality of possible events, 16
Limitations, representation of, 144-145
Locomotion, 47, 50, 93-117, 148, l66 >
167, 194, 205, G2i6

of arm, 102,

bodily, 43, 113

and boundary, 118,

226

SUBJECT INDEX

Locomotion, determined relatively, 114

and dimension, 195

directed, 50

of a field, 113-115

and fluidity, 159

of a group, 113

and irreality, 201

mental, 44

and path, 95, 108-109

psychological, 95-96

quasi-conceptual, 4 2-50, 1 10-1 1 8,
132, G2i6

quasi-physical, 42-50, 113, G2i6

quasi-social, 42-50, G2i6

and relative weight, 202

representation of, no

social, 44, 49

and structure, 156

(See also Accessibility; Medium)
Looking, 128

Looking at, 28, 127, 128, 177
Love, 188

Macroscopic structure, 165
Magnitude, 156-159
Malaria, 187
Material, 161

fluid, 186

rigid, 1 86

(See also Elasticity; Plasticity)
Mathematics, applied, 57

and physics, 57

pure, 59-60
Maturity, 189
Maze, 133, 145

elevated, 146

Means, topological, 149, 152
Measurement and laws, 63
Medium, 168

for child, 116

fluid, 123

and resistance, 117

and thing, 115
Member of a group, 101
Membership, 119
Membrane, permeable, 125

Memory, 197
Mental fact, 67
Method, 13

constructive, 16

in psychology, 9
Methodology, 6213
Metrisation, 60
Microscopic, structure, 165
Minority group, 100
Model, and concept, 78-79
Mother, 179, 189
Motor-perceptual stratum, 177-180,

G2i6

Motor stratum, 190
Movement, relativity of, 115

social, 49
Multiplication, as a region, in

Need, 179
Neighbor, 149
Neighborhood, 174
Newborn, 61
Newton, 82
Number, 57

Object, and process, 16
Objective, and physical, 25
Obstacle, 140
Occupation, 94, 108, 187
Occurrence and quality, 9
Old biological part, 190
Operation, mathematical, 53, 148
Organism, 191
Orientation, 39, 133, 164
Origin, historical and cause, 9
Outside, a region, 96-99

Part, 53, 87, 93, 169
common, 87, G2I5
of the person, 168, 201
of a system, 50
young, 190

(See also Differentiation; Integra-
tion)

SUBJECT INDEX

227

Part- whole-relation, 157, 171

and distance, G2i6
Past, 34, 35

and future, 36-40
Path, 49, 54, 89, 90, 91, 95, 148, 6215

as change of position, 109

and dimension, 199

of influence, 172, 177

as sequence of situations, 109

totality of, 145-146
Peasants, 98
Pedigree, 74
Perception, 27-29, 172, 179

spacial, 51

Peripheral region, 180-182
Period (see Epoch)
Permitted region, 122
Person, 6, 112, G2i6

boundary of, 167

dimensions of, 200-202

coordinating definitions for, 168-174

and environment, 1112, 41, 202,
G2i6

and life space, 166-168, G2i6

parts of, 1 66, G2i6

as region, 93, 106

differentiated region, 166-168

as thing, 104

topology of, 166-193

as a whole, 112
Phantasy, 187, 196, 201, 202
Physical, 147

Physicalism, and space, 56-58
Physics, 20

Physiological theories, 67, 167
Plane, 193
Plasticity, 161, 162
Play, 160, 197

as a region, 109
Playroom, 103
Plurality of psychological worlds,

67-68
Point, 53, no, in, 166, G2i6

of application, 113

boundary, 72

discrete, 145

inner, 72

and region, 110-113

Poisoning, 28

Position, 41, 51, 93, 94, 99, 163, 205,
G2i6

of a boundary zone, 119

geographical, G2I3

and region, in

relative, 136-154

social, 48
Possible events, 14

determination of, G2i3
Postulates, 22
Power field, 49, 106, 129
Prediction, 13, 6214
Pressure, social, 96
Primitivation, 181, 190
Primitiveness, 186, 197
Principles, of concreteness, 32-33

of contemporaneity, 33-36

Heisenberg's, 164
Prison, 120, 123, 144
Prisoner, 42
Problem, of coordination, 60

mathematical, 60

solution of, 132, 148-154
Problem child, 186
Process, dialectical, 36, 0214

physical, 62
Prohibition, 104, 158
Property, 195

dimension of, 194, G2i3

qualitative, 94
Protection, 144
Psychoanalysis, 32
Psychodiagnostic, 179
Psychology, animal, 4

and biology, 29

child, 4

developmental, 31

history of, 62

of perception, 3

present state of, 3-7

social, 4, 26

topological, 86, 205, Gai3, G2i4~2i8

vector, 86, 205
Psychopathic, 162
Psychopathology, 4, 31
Punishment, 142

228

SUBJECT INDEX

Quantitative determinations, 156
R

Real, psychologically, 124
Reality, 19, 160, 195-196, G2iS

degrees of, 196

epistemological concept of, 196,

level of greatest, 203
(See also Irreality)
Receptivity, 128
Region, 42, 43, 88-89

and action, 6215

central, 50, 179, 0215

dosed, 89, 104, 189, 0215

connected, 88, 89, 95, 104, 141, 146,

I( $9> 173, 215, G2i6
multiply, 89-90, 103-104
simply, 80-90, 105, 136, 0215

determination of, G2i6

forbidden, 44, 103

foreign, 87, 91-92, 136-137, 6215

of influence, 169

inner, 91

inner-personal, 177-180, G2i6

intermediate, 119

large, 156

limited, 89, 104-107

as medium, 116

more-dimensional, 120

motor-perceptual, 177-180, G2i6

neighboring, 172, G2i7

open, 89, 105, G2i5

outer, 91, 142

overlapping, 137-139, G2I7, G2i8

peripheral, 50, 179, 180, G2i6

permitted, 44

within person, 50

personal, 174-177

position of, 94

psychological, 93-117, G2i6

qualitative properties of, 173

qualitatively undetermined, 130-
135, 141, G2i7

small, 156

Region, structured, finitely, 164,

structured, infinitely, 164^215

staying in a, 109

undifferentiated, 61, in, 121

unlimited, 89

unstructured, 112, 164, G2I5
Regression, 181, 189, 190, G2i8
Relation, binary, 54

causal, 32-36

part-whole, 53-54, 157, 171

spacial, 41-52

temporal, 32-36

univocal, 78
Release, of tension, 175
Representation,

of action,

constructive, 6-n

of degrees of reality, 197, 200

of dynamical concept, 64

of events, 35

and explanation, 81-83

of facts and theory, 83

of the future, 38

incomplete, 17

of indeterminateness, 40

of the life space, 12-13

of limitations, 144

of limited accessibility, 151

logical impossibility, 150

mathematical, 76-83

by means of topological or dynami
cal concepts, 149-154

as path, 107-110, G2i$

of the person, 168-193

of single case, 8

of situation, 8-14, 76
Resistance, 49, 63, 97, 159, 6217

of a boundary, 123

to locomotion, 124, G2i8
Restriction, of the space of free move
ment, 144

Restructuring, 134, 155-156,
Revolution, 161
Reward, 142
Rigidity, 124, 190, G2i8

of person, 162

(See also Barrier; Wall)
Rod, measuring, 163

SUBJECT INDEX

229

Rule, 10

and exception, 10

Satiation, 166, 174, 176, 180, 186
Schizophrenia, 201
Sciences, empirical, 64

universal, 21
Sector, 131, 147, 195

of a boundary, 126
Seeing, 178
Self, 167
Selfcontrol, 181
Separation and unity, 185
Series of situations, 156

and locomotion, 108-109
Sharpness, of a boundary, 119

and solidity, 123
Shrinking of a region, 157
Similarity of actions, 176
Simplification, right, 17

wrong, 17

Singleness of the physical world, 67-68
Situation, 16, 137, 0217

concrete, and law, n

eating, 96

fluidity of, 160

life, 22-24, 0217
as background, 23

momentary, 22-24

overlapping, 0217

relative weight of, i37-*39> 218

restraining, 144

as the total of possibilities, 15, 0213
Size, 88, 157

decrease, 158

increase, 158
Smiling at, 177
Social environment, 177

looking at, 177
Sociology, 26, 62
Solidity, 63, 105, 124, 135, 149* 218

of boundary, 44

and sharpness, 123

(See also Rigidity)
Somatic, 27-29

Space, 193
divisible, 165
and dynamics, 59-65
Euclidian, 55
of free movement, 42, 103, 104, 136,

142, 0217

the history of the concept of, 62-63
mathematical, 41-58, 163, 0214
metrical, 53, 55-56, 0215
n-dimensional, 74, 194, G2i4
non-Euclidian, 55
nonmetrical, 53
one-dimensional, 194, 195
phase, 194, 0213

physical, 51, 52, 66-67, 164, 6214
and physicalism, 56-58
psychological 55, 66-67
structured finitely, 163-165, 183,

0215
structured infinitely, 61, 164, 183,

topological, 53-55, 0215

two-dimensional, 193

unstructured, 164, 6215
Space of free movement, 4247

as connected region, 100

as limited region, 103

shrinking, 157
Speed as boundary, 124
Spheres of influence, 94
Stability, 160

of a social group, 160
Starting point, 131
State, 174

of the person, 166

of a region, 127
Statistics, 12
Status nascendi, 160
Steps, series of separate, 97
Stimulus, 190
Stratum, 180-182, 186 .

central, 186, 202

inner-personal, 180-182, G2i6

of irreality, 197

motor-perceptual, 177-180

peripheral, 180-182, 202

of reality, 197

230

SUBJECT INDEX

Strength, of a barrier, 1 24

and kind of locomotion, r 25

of a wall, 127
Structure, 155-162, 184, 205, 0217

cognitive, 29, 133, 150, G 218

determination of, 170

of a group, 101

inner, 74, 99-107, 122, 193

kind of, 185, 187-188

and locomotion, 156

of the person, 177-192

of a region, 99

vocational, 103

(See also Region, structured)
Structurization, 61, 134
Struggles, political, 47, 98
Substitute, 15, 175, 176, 186, 197
Substitute value, 175, G2i8
Sum, topological, 87, 139, 0215
Surmnative, 174
Surface, of the body, 73
Surrounding, 53, 87, 89, 6214
Symbol, arbitrary, 78

and concept, 76-78
System, 174, G2i8

of behavior, 15-16

comprehensive, 70

of concepts, 5, 6, 9

of deduction, 16

differentiated, 186

motor, 178, 179

perceptual, 178

physiological, 80

psychological, 80

and region, 174

speculative, 5
(See also Tension system)

Task, mathematical, 132

of psychology, 15
Tendency, 55, 64
Tension, 63, 86, 122, 155, 159, 173,

176, 182, 188, 189, Gsi3, Gai8
muscular, 183

(See also Substitute value)
Tension system, 175, 177, 199, GaiS

Theory, in different epochs, 9

empirical, 4

formalistic, 22

physiological, 79-8 r

psychological, 76-83

of relativity, 66

of science, 65, 66

speculative, 4
Thing, G2i8

and medium, 115-117
Thinking, animistic, 27

magic, 27
Time, 69

historical, G2I3

and space, 62

Time index of facts, G2 13-214
Tools, 176, 179
Topological and dynamic factors, 188-

192

Topological image, G2i$
Topological relation and dimension,

199
Topological sum, 7, 8, 139, G2i$

(See also Factors)
Topologically equivalent, 141,
Topologisation, 61
Topology, 87-92, 113
Totality, 68

of physical facts, 66,

of possible cases, 16

of possible events, 14-17, 146

of psychological facts, 6214

(See also Life space)
Touching, 127

Transformation, continuous, 88,
Transition, 99, 123

gradual, 119

unclear, 122
Turning, away, 147

toward, 147
Type, and historical time indices,

Uncertainty, cognitive, 160
Unconscious, 20
Understanding of speech, 178

SUBJECT INDEX

23*

Undetermined zones, 130-135, 151,

United States, social structure of, 122
Unity, degrees of, 173-174, G2i6

dynamic, 172, 184
Unreal, the, 36

and indeterminate, 36-40
Unstructured region, 131, 134, 153,
0215, G2i6

Valence, 166, G2i8
(See also Force)
Validity, general, 25
Vector, 58, 76, G2i3, G2i8
Vector psychology, 159, G2i8

Wall, 126, 128, 170, 173, 190, G2i8
strength of, 186
(See also Rigidity)

Weight, relative, 137-139, 201, 202,

G2i8
(See also Situation, overlapping;

Region, overlapping)
Whole, 87, 169
dynamic, 64
and method of approximation,

Why (see Causality)

Width, of boundary zone, 120

Will, 203

Withdrawal, 146-148

Work, as region, 109

World, dynamically closed, 68^75

dynamically not closed, 68, 75

magic, 197

physical, G2I4

and life space, 24, 66-75

psychological, G2i4
Wound, 181

124619