Morphisms and Categories: Comparing and Transforming

245 Pages·1992·4.857 MB·English

by  Jean Piaget, Terrance Brown (editor)

MORPHISMS AND CATEGORIES
Comparing and Transforming
STATEMENT OF SPONSORSHIP
The Jean Piaget Society and the Fondation Archives Jean Piaget encourage
translations of important works not yet translated, support retranslations of
inadequately translated texts, foster consistent translation of technical terms,
and provide translators with expert consultation. Their goal is to promote
easier access to and better understanding of Piaget's ideas by English-speaking
scholars. This translation of Jean Piaget's Morphismes et Categories: Com
parer et Transformer reflects the efforts of these scholarly organizations.
MORPHISMS AND CATEGORIES
Comparing and Transforming
JEAN PIAGET
Gil Henriques Edgar Ascher
Translated and Edited by Terrance Brown
Preface by Seymour Papert
1m
LAWRENCE ERLBAUM ASSOCIATES, PUBLISHERS
1992 Hillsdale, New Jersey Hove and London
Copyright c 1992, by Lawrence Erlbaum Associastes, Inc.
All rights reserved. No part of the book may be reproduced in
any form, by photostat, microform, retrieval system, or any other
means, without the prior written permission of the publisher.
Lawrence Er1baum Associates, Inc., Publishers
365 Broadway
Hillsdale, New Jersey 07642
Library of Congress Cataloging-in-Publication Data
Piaget, Jean, 1896-
[Morphismes et categories. English)
Morphisms and categories : comparing and transforming I by Jean
Piaget, Gil Henriques, Edgar Ascher : translated and edited by
Terrance Brown : preface by Seymour Papert.
p. em.
Translation of : Morphismes et categories.
Includes bibliographical references and index.
ISBN 0-8058-0300-9
l. Genetic epistemology. 2. Categories (Mathematics)
-Psychological aspects. I. Henriques, Gil. II. Ascher, Edgar.
III. Brown, Terrance. IV. Title.
BF723.C5P5213 1992
155.4'13-dc20 91-19072
CIP
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Table of Contents
About the Translation
by Terrance Brown vii
Preface
by Seymour Papert ix
Introduction xvii
Chapter 1 Rotations and Circumductions
with Cl. Monnier and j. Vauclair 1
Chapter 2 The Composition of Two Cyclic Successions
with D. Voelin-Liambey and
I. Berthoud-Papandropoulou 15
Chapter 3 The Rotation of Cubes
with A. Moreau 31
Chapter 4 Compositions and Conservations of Lengths
with I. Fluckiger and M. Fluckiger 43
Chapter 5 The Composition of Differences
with E. Marti and E. Mayer 59
Chapter 6 The Sections of a Parallelepiped and a Cube
with H. Kilcher and J. P. Bronckart 77
Chapter 7 Correspondences of Kinships
with C. Bruhlhart and E. Marbach 91
Chapter 8 A Special Case of Inferential Symmetry:
Reading a Road Map Upside Down
with A. Karmiloff-Smith 111
v
Vi TABLE OF CONTENTS
Chapter 9 Conflicts Among Symmetries
with A. Karmiloff-Smith 123
Chapter 10 Correspondences and Causality
with Cl. Voelin and E. Rappe-du-Cher 137
Chapter 11 Equilibrium of Moments in a System
of Coaxial Disks
with F. Kubli 153
Chapter 12 Comparison of Two Machines and Their Regulators
with A. Blanchet and E. Valladao-Ackermann 167
Chapter 13 Morphisms and Transformations in the
Construction of Invariants
by Gil Henriques 183
Chapter 14 The Theory of Categories and Genetic Epistemology
by Edgar Ascher 207
Chapter 15 General Conclusions 215
Author Index 227
Subject Index 229
About the Translation
Barbel Inhelder informs me that the investigations published in this volume
were carried out at the International Center for Genetic Epistemology during
1973 and 1974. The work itself was finished by Piaget, Ascher, and Hen
riques in 1975, but was not published at that time. It forms a natural sequel
to Recherches sur les Correspondances published in 1980. That volume,
number XXXVII, was the last of the Etudes d'Epistemo/ogie et de Psycho
logie Genetiques. It remains untranslated.
The present translation began before Morphismes et Categories was
published in French (1990) and was translated from the manuscript itself.
Because the French edition appeared before the translation was completed,
it was possible to identify, correct, and incorporate important changes from
that edition into this publication.
As in my other translations, I have strived for fidelity and clarity
throughout this project. The first is a matter of understanding, a formi
dable challenge in parts of this work; the second is a matter of editing. By
the latter, I do not mean that I have deliberately omitted, added, or
rearranged ideas or data. I mean that I have tried to present what Piaget
expressed in endless, complex sentences in a simpler and more natural way.
If this translation proves at all successful, it will in large part be due to the
encouragement of Barbel Inhelder and Jacques Montangero, to the patience
of Judi Amsel of Lawrence Erlbaum Associates, and to the wonderful spirit
of cooperation shown by Piaget's co-authors and collaborators. My burden
was eased substantially by having several of the scholars who actually
carried out the experiments or who wrote chapters in the original version
(several of whom are members of the Translation Advisory Committee of
the Jean Piaget Society) check various parts of this translation. Not only
have they corrected errors and suggested improvements, but in one case,
they have even added new material. In particular, I am grateful to Edgar
Ascher for carefully examining key theoretical sections in addition to his
own chapter. I am also grateful to Gil Henriques for vetting the chapter he
contributed, to Annette Karmiloff-Smith and Edith Valladao-Ackermann
for reviewing chapters and reporting on their experiments, and to Angela
Cornu-Wells for her work on the impossible chapter I.
-Terrance Brown
vii
Preface
Studying this volume has been a very special experience for me, and I think
it will also be for all those who take seriously Piaget's commitment to
finding continuities between the psychogenesis of children's thinking and
the historical development of ideas. This volume contains a particularly
clear statement of his mature position on continuity, with biology as well as
with the history of ideas; it also offers a rich description of one of the most
interesting attempts to use a very sophisticated form of the continuity
hypothesis as a guide to experimentation. Let me quickly note, because such
issues sometimes make for hard reading, that this book, like much of
Piaget's writing, and like all the best literature, can be read profitably on
several different levels.
Who has not picked up a new volume by Piaget and made a first pass at
it by savoring the poetry of the dialogues with children, skimming lightly
over the intervening prose discussion of abstruse issues? Doing so does not
necessarily mean missing the essential: To a surprising extent Piaget's
theoretical positions are brilliantly embodied in these concrete instantia
tions. Many Piaget-watchers who never go back for a second pass still take
profound lessons from the masterly choices of situations and interchanges.
The series of experimental studies described in the following pages can be
recommended as an exceptionally rich source of pleasure, information, and
ideas for those who prefer this style of reading as well as for those who want
to follow him through every turn of thought.
This series of studies is also outstanding in the extent to which it allows
the intellectual personalities of the individual experimental collaborators to
shine through Piaget's integration of their work as part of his larger theo
retical perspective. Although I do not know all the collaborators well enough
to comment on each individually, I am struck by the consistency over many
of the books from Geneva of such features as the crisply logical style of
Berthoud-Papandropolou's experimental studies, the real-world rootedness
of Karmilofrs, and the fascination with mechanism shown in Blanchet and
Ackermann-Valladiio's. The role of the collaborators, as individuals, as
more than just names listed as footnotes to each chapter, had received very
insufficient attention in the discussion of Piaget's methodology.
ix
X PAPERT
I see this as absolutely capital. I hope that one day a historian of Piaget's
work will capture the importance of the cycle in which a very general idea,
such as causality or, in this case, categories, put out by Piaget, is picked up,
Rorschach-like in different ways, by the various members of the team of
collaborators who feed back to Piaget what they, and the children they
worked with, made of the idea: It is hard to imagine a situation richer in the
interplay of assimilations and accommodations than the genese of a study
such as the present book. But without waiting for this historian, the alerted
reader can piece together the traces of the process sufficiently to read the
book as more than the homogeneous intellectual product of a single
brilliant mastermind.
My own experience with this book was not without some pain. The pain
of being reminded the "le patron, is no longer with us was the more acute
because this volume is so pervaded by aspects of his thinking that are least
represented in contemporary Piagetian discussion, the ones most likely to
fall by the wayside as Piaget the epistemologist is himself appropriated by
psychologists and educators. But besides this pain of loss, I also experienced
an intellectual pain of struggling to find meaning in texts that go dizzyingly
in and out of focus. Do I really know what makes a transformation
"morphismique, and can I really see Piaget's use of quite deep and quite
technical mathematical ideas as more than superficial metaphor? I am not
alone in being sometimes beset by such doubts; indeed, I know more than
one among those who were closest and are most loyal to Piaget, and most
grateful for what they have learned from him, who have long ago decided
(at least in private) to treat this side of Piaget as the kind of obsessional
quirk of mind one tolerantly humors in friends and family. But if you
recognize yourself in this description, I urge you to try one last time to go
with the flow of Piaget's thought: In addition to its important new
theoretical advance, this volume offers an excellent exercise ground for
coming to grips with the side of Piaget, in his view a vitally important one,
that so many have found hardest to appropriate.
Buried deeply in the concluding section of the book, Piaget offers one of
the clearest statements of the centrality to his work of the search for
connection with what is most fundamental in science and mathematics: "A
genetic epistemology only has meaning on two conditions. One is if it
demonstrates a continuity between 'natural' thought and scientific thought.
The other, just as essential, is if it explains natural thought in terms of its
biological formation by reattaching it to the organic processes of life
itself."1
1See chapter 15, p. 215. Note the use of the word continuite, a much more subtle statement
of a principle in question than any allusion to the hackneyed recapitulation of phylogeny by
ontogeny.