STUDIES IN GENETIC EPISTEMOLOGY XXIII EPISTEMOLOGY AND PSYCHOLOGY OF FUNCTIONS SYNTHESE LIBRARY

MONOGRAPHS ON EPISTEMOLOGY,

LOGIC, METHODOLOGY, PHILOSOPHY OF SCIENCE,

SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE,

AND ON THE MATHEMATICAL METHODS OF

SOCIAL AND BEHAVIORAL SCIENCES

Managing Editor:

JAAKKO HINTIKKA, Academy of Finland and Stanford University

Editors:

ROBER T S. COHEN, Boston University

DONALD DAVIDSON, University of Chicago

GABRIEL NUCHELMANS, University of Leyden

WESLEY C. SALMON, University of Arizona

VOLUME 83 STUDIES IN GENETIC EPISTEMOLOGY Published under the direction of JEAN PIAGET Professor of the Faculte des Sciences at Geneva

XXIII

EPISTEMOLOGY AND PSYCHOLOGY OF FUNCTIONS

By

JEAN PIAGET, JEAN-BLAISE GRIZE, ALINA SZEMINSKA, AND VINH BANG

With the Collaboration of Catherine Fot, Marianne Meylan-Backs, Francine Orsini Andrula Papert-Christophides, Elsa Schmid-Kitzikis and Hermine Sinclair

D. REIDEL PUBLISHING COMPANY

DORDRECHT-HOLLAND/BOSTON-U .S.A. Libnuy of Congress Cataloging in Publication Data

Epistemology and psychology of functions.

(Studies in genetic epistemology; 23) (Synthese library; v. 83) Translation of Epistemologie et psychologie de la fonction. Includes bibliographical references and indexes. I. Cognition in children. 2. Knowledge, Theory of. 3. Functions. I. Piaget, Jean, 1896- II. Series: Etudes d'epistemologie genetique ; 23. BF723.C5E613 155.4'13 77-6792

ISBN-13: 978-90-277-1242-4 e-ISBN-13: 978-94-010-9321-7 DOl: 10.1007/978-94-010-9321-7

ETUDES D'EPlSTEMOLOGIE GENETIQUE X Xlii EPlSTEMOLOGIE ET PSYCHOLOGIE DE LA FONCTION

First published by Presses Universitaires de France, 1968 Translated from the French by F. Xavier Castellanos and Vivian D. Anderson

Published by D. Reidel Publishing Company, P.O. Box 17, Dordrecht, Holland

Sold and distributed in the U.S.A., Canada, and Mexico by D. Reidel Publishing Company, Inc. Lincoln Building, 160 Old Derby Street, Hingham, Mass. 02043, U.S.A.

All Rights Reserved Copyright © 1977 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover 1st edition 1977 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any informational storage and retrieval system. without written permission from the copyright owner PREFACE

Years ago, prompted by Grize, Apostel and Papert, we undertook the study of functions, but until now we did not properly understand the relations between functions and operations, and their increasing interactions at the level of 'constituted functions'. By contrast, certain recent studies on 'constitutive functions', or preoperatory functional schemes, have convinced us of the existence of a sort of logic of functions (springing from the schemes of actions) which is prior to the logic of operations (drawn from the general and reversible coordinations between actions). This preoperatory 'logic' accounts for the very general, and until now unexplained, primacy of order relations between 4 and 7 years of age, which is natural since functions are ordered dependences and result from oriented 'applications'. And while this 'logic' ends up in a positive manner in formalizable structures, it has gaps or limitations. Psychologically, we are interested in understanding the system• atic errors due to this primacy of order, such ·as the undifferentiation of 'longer' and 'farther', or the non-conservations caused by ordinal estimations (of levels, etc.), as opposed to extensive or metric evaluations. In a sense which is psychologically very real, this preoperatory logic of constitutive functions represents only the first half of operatory logic, if this can be said, and it is reversibility which allows the construction of the other half by completing the initial one-way structures. Furthermore, with respect to schemes of action, as opposed to general and operatory coordinations, functions constitute the common source of oper• ations and of causality and it is also in regard to this second point of view that the studies contained in this volume present certain new data. The first part of this work, authored by J. Piaget in collaboration with C. Fot, M. Meylan-Backs, F. Orsini, A. Papert-Christophides, E. Schmid-Kitzikis, H. Sinclair and A. Szeminska, bears above all on constitutive or preoperatory functions and on their gradual transformations into constituted functions, linked to operations. 641 subjects from 3 to 12-13 years of age were tested on this point. Part II is from the pen of Vinh Bang and relates 5 experiments which he conducted over a period of years on the quantification of constituted func- VI PREFACE tions, and specially on proportionality (based on 353 subjects from 6 to 14 years of age). Part III comprises two theoretical studies. In the first, J .-B. Grize reviews the history of the logical structure of functions, including the different levels considered in the preceding chapters. In the second, J. Piaget draws the 'General Conclusions' from these studies. INTRODUCTION

Reading a book by Piaget is like entering a system. To a large extent he and his collaborators (but particularly he) are builders of an impressively struc• tured whole - an experimentally based and controlled set of truth judge• ments about knowing and knowledge. It seems almost irrelevant to choose just a single book out of his enormous oeuvre. Always one meets a number of fundamental concepts which have arisen through many decades of experi• mental work with hundreds of collaborators, and from very productive thinking. Without doubt, for more than forty years Piaget has been (and still is) the spearhead and the creating and integrative power behind several generations of famous co-workers in 'the circle of Geneva'. It is impossible to mention all these women and men who were and are explicitly and implicitly present at the fabrication of so many writings. In the highly interesting autobiography of his scientific development, Piaget himself thankfully acknowledges the help of psychologists Barbel Inhelder, Alina Szeminska, Vinh Bang, Bresson, Greco, and Fraisse, the psycholinguist Sinclair, the mathematicians Henriques and Beth, the physicists Garcia and Halbwachs, the cyberneticians Papert and Cellerier, the methodologist in biology Nowinski, the logicians Apostel, Grize, Wermus, "and so many more". And here we are at the heart of the Centre International d'Epistemologie Genetique, where the studies in genetic epistemology originate and come about under the direction of Piaget. This particular book from 1968 was the twenty-third to be published since 1956. With the permanent aid of the Rockefeller Foundation since the beginning, and with that of the Fond National Suisse de Recherches Scientifiques since 1964, a production of approximately two volumes per year up till now (1977) has been maintained. I hardly need to say that this all surpasses the frontiers of psychology in the usual sense. Piaget was and his successors are directing the Faculte de psychologie et sciences de l'education of the University of Geneva. From about 1920, as chef des travaux at the Institut Jean-Jacques Rousseau under his predecessor Claparecte, Piaget began his very original studies in experi• mental child psychology and especially the cognitive development from VIII INTRODUCTION

childhood to adolescence, in which observations of his own two children played a major role. But significantly (I will come back to the underlying meaning of this) at an earlier date he started as a young man to study 'natural sciences' and wrote a doctoral thesis about the varieties of snail species living under different ecological conditions in Swiss lakes at different altitudes. At the same time he was attracted to and studied traditional philosophy (which, in the course of his scientific development, he surpassed - see his very thoughtful book Sagesse et illusions de la philosophie, 1966). This led in his early years, at the Universities of Geneva and Neuchatel, to teaching not only child psychology but also the philosophy of science and the development of ideas as reflected in the history of science. Even courses in human sociology for students (remember the studies on biological ecology) were amongst his tasks. With these and other starting points in his scientific career, stemming from the 'irrational' roots of his personality (with vital early questions about the essence of life and truth and securities), it is understandable that his very productive experimental and synthetic mind engendered a sizable problem convergency rather early. He wished to discover a kind of 'psychological embryology' of intelligence (using biological preformation) in which the relations between the acting and thinking subject and the objects of his experience, and between the subject and other subjects, had to be seen as a special case of the relation between the biological 'organism' and its surroundings. The difference being in the first case exchange of informajion and in the second case exchange of material. Piaget occupied himself temporarily with the development of social cooperation between individuals and groups, with social exchange of satis• factions, with valuation, with motivational aspects, with the growth of normative thinking in morals and justice, but his main line was cognitive development in a more strictly mathematical, geometrical and physical sense and its continuation in the development and the stages of intellectual thinking in science, and the epistemological reflection of science on its own foundations. To, for example, the Anglo-Saxon reader, this combination of psychology and epistemology may raise doubts. Is this not mixing up too much experimental psychological knowledge with 'philosophy'? Has psychology not to deal experimentally with thinking and has it not to consider the way in which the human mind or the subject constructs images of the objects in the outer world which are, after inner organisation (in which language plays an important role), a true description of reality expressible in words? Piaget and INTRODUCTION IX his collaborators have always fought against this kind of reductionism in explaining intelligence and intelligent behaviour. See for example 'The gaps of empiricism' presented in 1968 at the Alpbach Symposium on 'Beyond Reductionism'. Reductionism because action and thinking rather than percep• tion are the main sources of children's knowledge of the world around them and, according to Piaget also as an ultimate result, the adult's knowledge of the principles of logic. And (to pursue the line of some of his critics further) isn't epistemology a branch of philosophical thinking? Piaget objects strongly to this. He points out that the time of the epistemology of the old masters in philosophy, Plato, Descartes, Leibniz and Kant, is over. There has come instead a modern scientific epistemology, brought about by the reflections of the great scientific minds of modern times. Through their scientific work they came to questions about the foundations of their knowledge and about methods of purchasing truth. This epistemology is far removed from metaphysics and other philosophical fields and strongly connected to the magnificent develop• ment of modern (mathematical) logic. There are many famous names in this connection in mathematics and physics - to mention only one in molecular biology, Jacques Monod. Piaget had an interesting written discussion with Monod on development and evolution and the role of chance, necessity and active self-construction of the living system as a result of perturbation or promotion by its environment and based on regulative and assimilative biological capacities. Piaget defended himself very early against what he called 'the demon in philosophy' by devoting himself to painstaking experimental work and 'the study of truth' although combined with powerful reasoning. He was always interested in the unobservable to which the 'facts' refer and lead us. In accordance with this and his biological background, already in his early writings he conceived in all domains of life (organic, mental, social) organised or structured wholes and never elements functioning in isolation. Elementary realities we observe are always, in our way of understanding, dependent on bigger totalities which give these elementary realities their meaning. And on all levels there is the problem of the relations between the parts and totality. For example the problem of the species in biology raises the question whether this is 'a reality' or a function of our wider concept in which it is a part. Remember Piaget's studies on varieties of snail species under different ecological circumstances or different totalities of life. This could be con• sidered an analogy of the question of the existence or non-existence of so-called 'mathematical entities' of which we know now that they become x INTRODUCTION

meaningless if we develop mathematical thinking into wider, more powerful mathematical structures. And it is again comparable with the historical question in philosophy on 'realism' and 'nominalism'. And so Piaget turned to our elementary structures of thinking, especially knowing, beginning with and frequently coming back to the study of the very early 'knowledge' of young and very young children. He discovered the 'logic of the child'. Almost from the beginning he started not to be interested in the 'errors' and 'failures' in children's thinking, but in the reason why they must fail in their responses around the problems we 'adults in thinking' put to them. He and his early collaborators developed a fine 'clinical' way of interviewing and interrogating them on their own level, and following them in their stage of logic to higher stages (organisations, systems, structures of thinking, he says) to adolescent and further to adult scientific stages of thinking. Even in the official sciences there is not one but different methods with different stages of thinking behind them. Our knowledge is neither predetermined in the internal structures of our mind nor in the pre-existent properties of things around us. These properties of 'objects' become only gradually known by means of structures of knowledge which are gradually constructed in the developing mind under the influence of experience by action. And here we are in genetic epistemology. Piaget discovered psychology as an experimental field between biology and the old philosophical analysis of knowledge. He was always in search of experimental knowledge about our growing knowledge. The American Psychological Association, when addressing themselves to Piaget at the occasion of some celebration, said that he had approached philosophical questions in an empirical way and that these studies have led to psychology as a 'by-product'. Yes and no. But this formulation is not precise enough. There is no question about main and secondary activities. It is in essence an experimental and analysing approach to the problem of knowing and the immediate reflection on the developing mental structures behind our growing knowledge. It is better to put it like Piaget himself did: these judgements about knowing trespass upon and go beyond the boundaries of experimental psychology with regard to their meaning, but not with regard to their verification. After having studied the adult structures of thinking and what he called the mental operations in it, which develop themselves out of regulative activities. to ever more complex, finally totally reversible structures of thinking, Piaget turned himself to the most primitive ones, which he found and called 'conservation' or 'constancy' of the 'object' under the many and different situations of perception and use. INTRODUCTION XI

The laws that rule this mental development to mental operations in structures of thinking have, according to Piaget, without doubt correspon• dence with the laws of structuration of the nervous system, which can already partly be expressed in the formalised language of qualitative mathematical structures (groups, lattices, etc.). See for this his important book Biologie et connaissance. Piaget puts it as follows: the correspondence of the formal (mathematical, logical) deductive structures to the psychological stages of structuration of mental activity in reality, and with autoregulative, cybernetic models with their realisations in biology, make it understandable that operative thinking (developed out of action of the thinking subject in reality) is the bridge between structured organic life and structured logico-mathe• matical 'realities'. Piaget frequently states that no psychological study of our growing cognitive functions is conceivable without the aid of logical or mathematical models, which represent our highest adult forms of cognitive thinking. He states further that no psychological study of our cognitive functions is possible without constant epistemological analysis of what we know. In the yearly international and interdisciplinary 'plenary' sessions of the Centre International d'Epistemologie Genetique, where the results of very specialised studies are discussed, there is a search for common language, common methods and common conclusions in, for example, highly formal• ised deduction and experimental verification. In this forum the people mentioned at the beginning of this preface are present, together with important guests from abroad, representing the different diSciplines appro• priate to the study under discussion. The experiments are analysed as well as the theory that lies behind them. Different interpretations of the various disciplines with regard to the same experiments are welcomed as a main road to higher reciprocal understanding and higher forms of knowing, and also as a starting point for further questions and further experiments. The specific importance of this book lies in the fact that it raised the question and opened discussion on something that had been dealt with only slightly in the previous years of study on the genesis, stages, and role of mental operations in cognitive thinking. Formerly, up to the age of 6 the child was considered to have a sensory-motor intelligence, but till about that age no operational intelligence, be it only mental operations in the sense of handling the representations of concrete things. During that period there is already an intelligent behaviour, even before the appearance of language, but only in a sensory-motor sense, in dealing with concrete things. During that period there are 'schemes of actions' with coordination and organisation of adaptive action. But this book presents a much more positive view of this XII INTRODUCTION period from, say, 3% to 6. Here again it is with the aid of mathematical, formalised models of adult thinking that the origins of this kind of thinking are studied in the psychological reality of the child and a beginning of understanding is founded on the relations between thinking in functions, which is essentially thinking in dependencies, and operational intelligence, with reversibilities and the possibility of transformations within a structured whole. The function is considered as a precursor of the mental operation. Also the origin of functions is discussed, whether in the physical world around the child or in his actions, his own handling of objects. The book speaks abundantly for itself. Finally, coming back once again to the wholeness of all the works in and about the 'circle of Geneva' around Piaget, I may mention here the existence of a Catalogue des Archives Jean Piaget of the University of Geneva under direction of Professor Barbel Inhelder. This bibliography is divided into three parts, one of which contains all the works of Piaget from 1907 till now, with 1500 titles. The second part contains about 800 index cards dealing with all the works of immediate collaborators. The third part contains the so-called secondary literature engendered by Piagetian thinking with about 1750 index cards. The publisher is G. K. Hall and Co. Publications, Boston, U.S.A. Besides being a model of modern scientific world citizenship, with a never-ending inquisitive, experimental attitude, Piaget is for many a kind of ideal of the old European mind. His unbreakable tie with nature, to which belong periodic retreats and walks in the Swiss mountains, is combined with his powerful rational, integrative thinking and reasoning, far above the fortuitous situation; his healthy taste for good companionship and good life is combined with charming simple style and manners and with an outside soberness; and even his very human slyness. These are all aspects of a kind of Olympianship.

A. SUNIER TABLE OF CONTENTS

Preface v

Introduction vii

PART I I FROM CONSTITUTIVE FUNCTIONS TO CONSTITUTED FUNCTIONS

1. The Coordination of Pairs 3 2. From Constitutive Functions to Equivalence Classes 15 3. From Regularities to Proportionalities 30 4. An Example of Causal and Spatial Functions 49 5. From Coproperties to Covariations: The Equalization and Estimation of Inequalities 65 6. The Composition of Differences: Unequal Partitions 76 7. An Example of the Composition of the Variations of Variations 84

PART III THE QUANTIFICATION OF CONSTITUTED FUNCTIONS

8. The Functional Relation between the Increase and the Decrease of Both Sides of a Rectangle Having a Constant Perimeter - The Transformations of the Perimeter of a Square 103 9. Serial Regularities and Proportions 109 10. The Relation between the Size of a Wheel and the Distance Travelled 114 11. The Establishment of a Functional Relation among Several Variables: Distance Travelled, Wheel Size and Rotational Frequency 121 12. The Inverse Proportional Relationship between Weight Wand Distance D (Arm of a Lever) in the Equilibrium of a Balance 126 13. Conclusion of Chapters 8 to 12: The General Evolution of Behaviors 132 XIV TABLE OF CONTENTS

PART III / THEORETICAL PROBLEMS

14. Analyses to Aid in the Epistemological Study of the Notion of Function 141 15. General Conclusions 167 Index 197