DEVELOPMENTAL EPISTEMOLOGY1 Jean Piaget
INTRODUCTION
Classic theories of knowledge initially were concerned with the question, 'How is knowledge possible?' This question was quickly differentiated differentiate d into a multiplicity of problems bearing on the nature of, and preliminary conditions for, logico-mathematical knowledge, experimental knowledge of the empirical kind, and soon. But the common postulate of the various traditional epistemologies is that knowledge is a fact and not a process, and that, even if our various forms of knowledge arc continually incomplete and our various sciences still imperfect, that which is acquired is acquired and can therefore be studied in isolation: hence the posing in absolute terms of problems such as, 'What is knowledge?' or, 'How are the various kinds of knowledge possible?' The reasons for such an attitude, which immediately placed itself sub specie aeternitatis, are to be found in the individual doctrines of the great philosophers who laid the foundations of the theory of knowledge - doctrines such as the transcendent realism of Plato, the Aristotelian belief in immanent but also al so permanent forms, the innate ideas of Cartesian thought or the preestablished harmony of Leibniz, the Kantian a priori limits, or even the postulate of Hegel who, while discovering growth and history in the social products of humanity, would have had them reduced to the total deducibility of a dialectic of concepts. In addition, this attitude to knowledge was influenced not only by these various doctrines, but also by the fact that scientific thought itself has long claimed to have achieved a body of definitive truths, albeit incomplete, and thus permitted the question of what is knowledge to be asked once and for all. Mathematicians, while holding different opinions regarding the nature of mathematical 'entities' remained, until not so very long ago, impervious to ideas of revision and reflexive reorganization; logic has long been considered complete, and the re-examination of the limits of its powers had to wait for Gödel’s theorems; physics, after the triumphs of Newton, believed right up to the beginning of this century in the absolute nature of a significant number of principles; even sciences as young as sociology or psychology, if unable to pride themselves on a solid basis of knowledge, did at least not hesitate (until relatively recently) to attribute to human beings, and thus to the thinking subjects under study, cither an immutable 'natural logic', as Comte would have it (in spite of his law of the three stages and his
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Jean Piaget. Psychology and Epistemology: Towards a Theory of Knowledge. pp. 1—16.
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insistence on the processes of reasoning which were both common and constant factors in these), or instruments of knowledge which were invariant. Thus, under the converging influence of a series of factors, knowledge is currently coming to be considered more and more as a process rather than as a state. The reason for this derives partly from the epistemology of the philosophies of science: Cournot's probabilism and his comparative studies of various kinds of ideas arc already heralding a revision of this kind; works of historical criticism, by throwing light on the contrasts between the various kinds of scientific thought, have notably encouraged this evolution; and the work of Brunschvicg, for example, marks an important turning point in the direction of a doctrine of knowledge as a developmental process. Among the neo-Kantians pronouncements of this kind may be found in the writings of Natorp: '. . . like Kant, we start with the actual existence of knowledge and seek the basis from there. But what is this existence since, as we know, knowledge is constantly evolving? Progression, method is everything ... in consequence, the existence of knowledge cannot be comprehended except as a " fieri". This " fieri" alone is the fact. Any entity (or object) which knowledge attempts to crystallize must dissolve again in the current of development. It is in the last phase of this development, and in this alone, that we have the right to say: "this is (a fact)". What we can and must seek, then, is the law underlying this process.' Then again, Kuhn's work on 'scientific revolutions' is well known. But if the epistemologists have been able to arrive at such straightforward pronouncements, it is because the whole of the development of contemporary science has led them in that direction, and this applies to the realm of deduction as well as experiment. In comparing, for example, the works of present-day logicians with the proofs which satisfied those who are already being described as 'the great forefathers', such as Whitehead and Russell, we cannot help but be struck by the astonishing transformation of ideas that has taken place, as well as by the actual rigour of the arguments employed. The current works of mathematicians who by 'reflection and abstraction' derive new procedures from those already known, or new structures from comparison of previous structures, have led to the enrichment of the most fundamental ideas, not contradicting them as much as organizing them in an unexpected manner. In physics it is coming to be accepted that all of the original principles have altered both in form and content, to the extent that even the best-established laws begin to relate only to a certain level, and change in significance as they change their situation in the body of the system. In biology where the same degree of exactitude has not been attained and where immense problems still remain to be solved, the changes in perspective are also impressive. In addition, it must be remembered that, in the very execution of these changes (which are not entirely devoid of the occasional crisis and which necessarily involve a constant labour of reflexive reorganization), the epistemology of scientific thought has become to an increasing extent the province of scientists themselves The 'fundamental' problems arc thus increasingly incorporated into the system of each of the sciences under consideration, in physics as in mathematics or logic. EPISTEMOLOGY AND PSYCHOLOGY
This fundamental transformation of the knowledge-state into the knowledge-process leads us to restate in a new form the question of the relationships between epistemology and the development or even the psychological formation of ideas and operations. Throughout the history of classical epistemologies, the empiricist traditions alone have had recourse to psychology, for reasons which are not difficult to understand, although these explain neither 2/10
the scant concern for psychological verification in other schools nor the all too summary type of psychology with which empiricism itself has been content. These reasons are, naturally, that if we wish to give an account of the body of knowledge through 'experience' alone, we can justify a proposition of this kind only by attempting to analyse the nature of experience, and we reach the point of having to refer to perceptions, associations and habits, which are psychological processes. But since empiricist, sensualist and other philosophies existed long before experimental psychology, most people were content with these commonsense ideas and with speculative description; this prevented them from seeing that experience is always a process of assimilation I to existing structures, and from giving themselves over to systematic study of the ipse intellectus. As for the Platonic, rationalist or apriorist epistemologies, each believed that it had found some fundamental means of knowledge which was unknown, superior, or prior to experience. However, following an oversight which can doubtless be explained afresh by tendencies to speculate and by disdain for effective verification, these doctrines, while careful to characterize the properties which they attributed to this means of knowledge (the reminiscence of Ideas, the universal power of Reason or the prior and necessary nature of the a priori forms), failed to verify that it was in fact at the subject's disposal. There is, then, whether one wishes it or not, a question of fact. In the case of Platonic reminiscence or of universal Reason, this question is relatively straightforward: it is evident that, before conferring 'faculties' of this sort upon 'all' normal human beings, it would be advisable to examine them and this examination soon displays the difficulties of the hypothesis. In the case of the a priori forms, factual analysis is a more delicate matter, since it is not enough to analyse the consciousness of one's subjects without considering also their previous circumstances; and according to the hypothesis, the psychologist who wished to study them would make use of them himself, as the antecedent conditions of his research. However, history in all its multifarious dimensions is still with us (the history of sciences, sociogenesis and psychogenesis), and if the hypothesis is correct, it must be verified not by the subjects' introspection but by examining the results of their intellectual labours. This examination then clearly demonstrates that it is essential to separate antecedent from necessary conditions, for if all knowledge and in particular all experience imply antecedent conditions, these do not immediately present any logical or intrusive necessity; and if several forms of knowledge result in necessity, then this necessity is to be found at the end and certainly not at the beginning. In short, all epistemologies, even those which arc anti-empiricist, raise questions of fact and thus implicitly adopt psychological positions which, however, lack effective verification, even though this is indispensable as far as sound method is concerned. If, then, our present proposition already holds true as far as static epistemologies are concerned, it is a fortiori even more so in the case of dynamic theories of knowledge. Indeed, if all knowledge is continually in course of development and consists in passing from a state of lesser knowledge to one which is more complete and effective, then it is clearly a matter of understanding this development and analysing it as accurately as possible. This process of growth does not take place haphazardly but forms a developmental sequence, and since in no cognitive sphere does there exist an absolute starting-point of development, this itself must be studied in the socalled formative stages. Since these still form a sequence of development proceeding from antecedent conditions (whether known or unknown) there is a danger of infinite regression (in other words, of an appeal to biology): however, since the problem is that of finding the law underlying processes, and since the final stages (final for the moment, that is) are as important 3/10
in this respect as those known first, the sector of development being considered can offer at least partial solutions, provided that collaboration between historico-critical analysis and psycho-developmental analysis can be secured. The immediate aim of developmental epistemology is, perhaps, to treat psychology seriously and to subject to scrutiny all questions of fact which each epistemology necessarily raises, but at the same time replacing the speculative or implicit psychology with which many have been content by controlled analysis (in the scientific sense of what is known as a control). Let it be said once again, that although this constraint should always have been adhered to, it has currently become increasingly pressing. It is indeed striking to observe that the most spectacular transformations of ideas or structures in the evolution of contemporary science correspond, when the psychogenesis of these same ideas or structures is investigated, to circumstances or characteristics which take into account the possibility of their subsequent transformations. We shall see some examples of this in connection with the revision of the idea of absolute time, since from the outset duration is thought of in relation to speed; or in the evolution of geometry, since from the very beginning topological intuitions precede all metric ones. METHODS
Epistemology is the theory of valid knowledge, and, even if this knowledge is always a process rather than a state, this process is in essence the change from a lesser to a greater validity. It follows from this that epistemology is by its very nature an interdisciplinary subject, since a process of this kind raises questions both of fact and of validity. If it were a question of validity alone, epistemology would merge with logic: the problem here is not purely formal, but reverts to determining how knowledge comes to terms with the real world, and therefore what relationships obtain between subject and object. If epistemology were only a question of facts, it would be reduced to a psychology of cognitive functions, which would not be able to resolve questions of validity. The first rule of developmental epistemology is consequently a rule of collaboration: its problem is to investigate the growth of understanding, and one must therefore engage the co-operation, in each individual issue, of psychologists who are studying development as such, of logicians who formalize the stages or states of temporary equilibrium in this development, and of specialists in the science relevant to the area under discussion; to these must of course be added mathematicians to ensure some liaison between logic and the field in question; and of cyberneticians to ensure liaison between psychology and logic. It is therefore only through the functioning of this collaboration that the requirements of fact and validity can be equally respected. In order to understand the meaning of this collaboration, however, something should be borne in mind which is far too often overlooked: namely that although psychology is not competent to lay down norms of validity, it studies subjects who themselves, at all ages (from earliest childhood to adulthood and right up to several levels of scientific thought), provide norms of this kind. For example, a five- to six-year-old child is still ignorant of transitivity and will refuse to draw the conclusion that A < C if he has seen that A < B and B < C but has not seen A and C together. Similarly, if a given quantity A of liquid is poured from a wide shallow vessel into a tall narrow one, where it takes the form A', the child will refuse to admit that quantity A is conserved in A' although he will accept that 'the same water1 is involved. He therefore recognizes qualitative identity, but rejects conservation of quantity. At the age of seven or eight, on the contrary, he will consider both transitivity and conservation of quantity
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to be simultaneously necessary. The subject as such (i.e., independently of the psychologist) therefore recognizes norms. Several problems arise from this: 1 How docs the individual arrive at his own norms of this kind? This is in essence a psychological question, independent of any competence (which psychology does not in any case possess) with respect to the evaluation of the cognitive significance of these norms; it is, for example, the business of the psychologist to determine whether these norms have simply been transmitted to the child through the adult, (which is not the case), whether they depend on a single experience (which is in fact not at all sufficient), whether they result from language and constructions which are simple semiotic or symbolic forms as well as syntactic and semantic (which is in turn inadequate), or whether they constitute the product of a structuring procedure which is in part endogenous and proceeds by adjustments and progressive self-regulatory procedures (which this time is the case). 2 There is next the problem of (he validity of these norms: it is the business of the logician to formalize the structures appropriate to these successive stages, the structures with 'pre-operations' (without the properties of commutativity, transitivity, or conservation, but with qualitative identities and with equally qualitative directed functions, which in both cases correspond to 'categories' in MacLane's sense, but extremely elementary and trivial ones) or structures with operators (with 'group' or 'groupoid' properties). It is therefore for the logician to determine the value of these norms and the properties of epistemic progress or regression shown by the cognitive developments which the psychologist studies. 3 Finally, there is the question of the interest or the absence of significance of the result obtained by the field of science under consideration. In this respect we shall always remember Einstein's pleasure at Princeton when we related to him the facts of non-conservation of the amount of liquid following transfer from one container to another in children aged four to six years, and how suggestive he found the characteristically late appearance of the concept of conservation of quantity. And if in fact the most elementary and apparently the most selfevident ideas require a long and difficult elaboration, it becomes easier to understand why the establishment of the experimental sciences has been systematically retarded in comparison with that of the logico-mathematical disciplines. NUMBER AND SPACE
With these few pointers in mind let us now attempt to give some examples of results obtained, beginning with the difficult problem of attempts to reduce the concept of number to logic. It is known, for example, that Whitehead and Russell sought to make the ordinals equivalent to classes of classes by a one-to-one correspondence, whereas Poincare considered that the concept of number relies upon an irreducible intuition of n + 1. Since that time Gödel’s theorems have in some sense shown Poincare to be right concerning the difficulties of rcductionism in general, but psychologically the idea of n + 1 is not basic and only takes place in an operational form (together with conservation of number when the arrangement of elements is modified) at about seven to eight years of age, and then in connection with the structuring of classes and asymmetric relationships. It is therefore necessary to look for a solution which bypasses both the reduction of the Principia and the proposition of a fundamental axiom for the concept of natural number. In fact, between the ages of four and seven we can observe the construction of three correlated systems of operation. In the first place the child becomes capable of serialization, that is, a 5/10
transitive sequencing of order relations: A before B, B before C, etc. Secondly, he constructs classifications or 'groupings' of classes, whose simplest form consists in combining the distinct classes A and A' in B, then B and the distinct class B' in C, then C and C in D, and so on. Now let us assume that he is disregarding qualities, that is to say that A, A', W are considered as equivalent and indistinguishable with respect to their qualities (which is indeed the case where identical tokens or counters arc concerned). In this case we would have A + A = B etc, and consequently A + A = A. There is only one way of avoiding this tautology (which is, in fact, to disregard one element or to count the same one twice, and so on), namely, to distinguish A, A , B by their order of enumeration. Effectively this order differentiates them, even if their qualities are disregarded, for we are in fact dealing with a 'synthetic' order, meaning that if the terms are permuted the same order is still found (a first, a second and so on, according to whether the first has no predecessor, the second has only one, and so on). The concept of number thus emerges as the union of the concepts of inclusion of classes and of serial order, that is to say as a new combination, but one based on purely logical properties.
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As for the one-to-one correspondence between classes postulated by the Principia, there is here a kind of vicious circle, for in this context there exist two quite distinct kinds of operations: either a qualified correspondence (one object corresponding to another of the same quality, such as a square and a square, a circle and a circle, and so on), or a generalized correspondence which disregards qualities. But in this latter case the individual object becomes an arithmetical unit and ceases to be merely logical (qualified unique class): to render two classes equivalent by a 'generalized' correspondence thus amounts to introducing the concept of number into class implicitly, in order to extract it explicitly afterwards! Moreover, Whitehead and Russell were also well advised in appealing to the concept of order, since, in attempting to avoid the tautology 1 + 1 = 1 and arrive at the iteration 1 + 1 = 2, they came lo distinguish 0 + 1 from 1 + 0. In saying that the concept of number is the synthesis of inclusion and order relationships, we are simply summarizing what every axiomatization is bound to state in one form or another. This has several implications relating to the specificity of recursive reasoning, and some remarkably early examples arc to be found in children at a level which is already preoperational. As for the problems of space, we have been able to emphasize the essentially operational nature of the formation of this idea, which is not based in the least upon experience through perception, in spite of the attempts of Enriquez to reduce different geometric forms to separate sensory categories. Here the question arises of establishing whether spatial operations, in the course of spontaneous intellectual development (independent of schooling), take place in conformity with historical order (Euclidian metric, then protective intuitions and finally discovery of topological relations); or whether they follow an order of formation conforming more to the theoretical order (topological intuitions at the beginning, followed by the construction, in parallel, of projective space and of a metric which may be Euclidian). Now if we consider separately perceptual and sensory-motor space (which takes shape from the earliest months of existence) and conceptual or operational space, we find in the two fields (albeit with chronological displacement) the same evolutionary principle. Initially topological relationships of neighbourhood, continuity, closure and position in relation to boundaries predominate, and only then do we find simultaneous and correlated construction of Euclidian and projective relationships, leading finally to a co-ordination of viewpoints regarding these latter relationships together with metric references (measures with two or three dimensions 6/10
and natural co-ordinates) as regards the former. It should be noted in particular that for a long time ordinal evaluations prevail over metric consideration: of two straight rods whose equal length has been verified through congruence, the one which is shortly afterwards displaced and projects beyond the other a Mule way is estimated as being 'longer' because it goes* further*; and it is easy to verify that this is not simply a case of semantic misunderstanding because the two projections (the higher rod projecting in front and the lower projecting behind) are not judged to be equal. TIME AND VELOCITY
A further example of contact between problems of developmental psychology and the epistemology of contemporary science is that of relationships between time and velocity. Indeed, it is known that there has always existed a kind of vicious circle involving these two ideas: velocity is defined by reference to time, but durations cannot be measured except by reference to velocities. The epistemological relationship between these two concepts therefore presents a problem. On the other hand, in classical or Newtonian mechanics, time and space arc both absolutes corresponding to simple intuitions (Newton's Sensorium Dei), while velocity is merely a relationship between them. In relativistic mechanics, by contrast, velocity becomes an absolute and time (like space) is relative to it. What then is the view of developmental psychology? Observation shows, in fact, that there exists a basic intuition of speed, independent of any idea of duration and resulting from this primal concept of order which we have just been discussing with respect to space: namely the intuition of kinematic overtaking. If moving object A is behind B at the instant Tl and passes in front of moving object B at the instant T2, it is judged to be faster, and this holds for all ages: nothing intervenes here except temporal order (behind and in front of), and there is no consideration of duration or of space traversed. Speed is therefore initially independent of durations. On the other hand, at all ages concepts of duration presuppose a component of speed (or when speed is not taken into account there is error in estimating duration): if the moving objects A and B leave the same point simultaneously and in the same direction, young subjects will say that they leave at the same time but that they do not stop at the same instant, although they acknowledge that when one stops the other ceases to go. Even when this simultaneity of stopping, denied until about six years of two, is recognized, the subject continues not to believe in the equality of these synchronous durations, and this persists until about eight years of age. Simultaneities and durations are thus subordinated to kinematic effects, and one could give plenty of other examples of this and of the belief in the equivalence 'faster — more time' which is so frequently found before age seven, and which can be explained by a kind of equation: faster = further = more time. In short, the very development of these ideas of speed and time makes it quite clear that there is nothing inevitable about the intuition of universal and absolute time and that, being the product of a certain level of detailed understanding, it should given place to analyses based upon more accurate approximations. OBJECT PERMANENCE, IDENTITY AND CONSERVATION
A further example of unexpected contact between recent scientific history and developmental psychology is given by the concept of object permanence. This idea of permanence, which at the beginning of this century seemed to be both self-evident and necessary, has (as is well 7/10
known) been cast into doubt by contemporary atomic physics, for which an object exists qua object (as opposed to its wave representation) only to extent that it can be localized. It would therefore be interesting to try to establish how the concept of the object is formed, since it is no longer clothed with the same appearance of necessity which its previous history seemed to confer upon him. Analysis of the first year of mental development shows that this idea of object permanence does not correspond to anything innate: the original universe is, during the first months of existence, an objectless universe, composed of perceptual tableaux which appear and disappear by process of resorption, and an object is not sought when it is hidden by a screen the baby, for example, withdraws his hand if he is about to grasp the object and the latter is covered with a handkerchief. At a later stage the child begins to look for the object, lifting the handkerchief at A where it has just been covered; and when the object is displaced to a position beneath B (for example td the right, whereas A was on his left), the child, although he has seen, the object being placed at B, often looks for it at A when it disappear again; that is, he looks in the place where his action has been successful on an earlier occasion. He therefore does not take account of successive displacements of the object which he has nevertheless observed and followed attentively. It is not until towards the end of the first year that he looks for the object unhesitatingly in the place where it last vanished: object permanence is thus closely linked with localization in space, and, as we see, this latter itself depends upon the construction of the 'displacement group' which Poincare rightly placed at the origin of the development of the concept of sensory motor space. Poincare alone saw in this group an a priori form of our activity and thought, because he considered as a primary basic endowment the ability to distinguish changes in position (which car be corrected for by a compensatory movement of one's own body, and changes in state: and in so far as there are no permanent objects everything is in terms of changes in state. The displacement group therefore becomes necessary through the progressive organization of actions, but it is not necessary as a preliminary and therefore does not constitute an a priori method. On the other hand, it is understandable why the object itself, whose permanence depends on the extent m which localization is possible, can lose this permanence in areas when localization is absent. Object permanence, together with that of one's own body image (itself attained through observation of the bodies of others, which are precisely the first objects to become permanent), constitutes the first of the concepts which may be described as 'qualitative identity' in the pre-operational development of the individual. There are many experiments relating to this issue which can be carried out on children of two to three and seven to eight years, asking, for example, whether water which changes its form when its container is changed is still' the same water'; whether a wire which is bent into a straight line or into! curve is still 'the same wire'; whether a' dendrite' which the child sees changing in a few minutes from seed crystal to tree-like form when placed in a liquid is still the same 'dendrite'; or again, in an experiment on perception of apparent (stroboscopic) movement, where a circle appears to change into a square or a triangle, asking whether it is the same object which changes shape or whether there are two objects and no change takes place; and so on. Two clear results have | thus been obtained. The first is that the range of identity increases with age: thus in the case of the 'dendrite' (studied by Voyat) young subjects say that the 'dendrite' as it grows is not 'the same dendrite' since it passes from the class of 'small things' to that of 'medium-size things' or 'large things'; and so on: at seven to eight, on the other hand, it is the same. The second finding is that the early development of concepts of identity occurs well before conservation of quantity: the' water' that is poured is' the same water' even though there may now be a little more if the level is higher, and so on. 8/10
The fact that the concept of identity occurs before conservation of quantity is interesting from the epistemological point of view. Before establishing an operation properly so called (the 'identity operation' of a group, or the addition of a 'null element') identity has only qualitative significance and is obtained by a simple dissociation of constant qualities (the same material, the same colour and so on) and of variable qualities (form, and so on): it does not therefore presuppose any operational structure for its formation and appearance at the same time as unidirectional functions (applications). For example, if a wire is bent into a right angle the child understands from the age of four or five that segment B is augmented as a function of the diminution of segment A and he will say that it is 'the same wire', even though he believes that the total length A + B is modified as the wire is bent. On the other hand, conservation of this total length, or of the amount of a decanted liquid, is not acquired until about seven to sight years because it presupposes operations of quantifying (compensations between the dimension which increases and the dimension which decreases, and so on): the concept of quantity therefore presupposes an act of construction and is not given by simple perceptual verification as are the concepts of quality. At the pre-operational level, the only possible quantification remains of an everyday kind: longer' for example, means 'going further'; hence the non-conservation of decanted liquids, because their quantity is estimated simply by he order of the levels (going 'higher', and so on) without taking account of other dimensions. Conservation therefore does not derive from the concept of identity, as Bruner holds and as Meyerson believed: it presupposes an operational mixture of transformations, which embeds the concept of identity in a wider framework of reversibility (possibility of inverse operations) and of quantitative compensations, together with syntheses which incorporate the concept of number and of measure (discussed in the section on number and space, p. 8). Considerable research has been done on the nature of these concepts of conservation, all of which points towards this interpretation in terms of operational epistemology. CHANCE
A few words remain to be said concerning a concept which is fundamental from the epistemological point of view and whose origin at first sight appears quite different from that of the previous concepts: namely the concept of chance, which was defined by Cournot as an interaction of independent causal sequences, and which thus corresponds to what is generally designated by the term 'entropy'. Entropy is irreversible and increases with an ever-decreasing probability of return to the initial state. We may therefore ask whether at the pre-operational levels, namely before the age of seven to eight, where the child has not yet succeeded in manipulating inverse or reciprocal operations and hence reversibility, he will derive from this very fact some intuition of irreversibility and will thus reach an immediate intuition of probabilistic entropy. To answer this question, we should distinguish two levels: one of actions and one of ideas. On the level of action, it is evident that the child learns, at a very early stage, to take chance fluctuations into consideration, for example to anticipate that a falling object may land either side up, and to evaluate that he will have more difficult} crossing a road if it is full of cars than if there are only a few. But it is another matter to understand the concept of chance as such, in so far as the ideas of interaction or of entropy are concerned, and to distinguish it from the idea of the purely arbitrary or from a system which is planned but unpredictable. Together with Inhelder, we have therefore devoted ourselves to a series of experiments dealing with extremely simple penny-tossing situations, with elementary statistics 9/10
distributions and especially with increasing entropy: for example a box containing ten white beads and ten black beads is rocked from side to side several times in succession to see whether at each successive movement the beads mix together more instead of each returning to its respective side, the black on the left and the white on the right. Two clear findings emerge from experiments of this kind. The first is that until about seven to eight years there is no explicit concept of chance: in principle the whole sequence could be predicted from the behaviour of individual objects, and if the beads mix together, contrary to prediction, they will soon end up by 'unmixing themselves and returning to their original order (often following a kind of 'general post' which will take all the white ones over to the black side, and vice versa). The second conclusion, and this is the crucial point, is that irreversibility is not understood except by reference to deducible reversibility to which it is opposed: in other words, the subject has to construct models of reversible operations in order to understand the existence of processes which do not fit this model and which are not deducible. After which the process comes to terms with chance and attains a calculation of probabilities, but one dealing with ensembles (large groups) and not with individual cases. In short, the evolution of the concept of chance is itself subordinated to the construction of operational models. CONCLUSIONS
These few examples, chosen from many possible ones, show the potential fruitfulness of a method which attempts to arrive at an understanding of mechanisms of knowledge through studying their origins and development. If, as we were suggesting at the beginning of this discussion, knowledge is a continuous process and cannot be crystallized in any one of its momentary states, it is clear that research of this kind is indispensable, as the history of science or of ideas is 'inevitably still full of gaps. A number of deep-rooted prejudices must be overcome, when one is working in the areas of logical epistemology, mathematics or physics, in order to understand that a useful 'liaison can exist with a discipline as limited and to all appearances as insubstantial as child psychology or developmental psychology. But in fact an increasing number of specialists have become interested in our International Centre for Developmental Epistemology and have 'collaborated with us in publications. Twenty-two volumes have already appeared in our collection of Etudes d'epistemologie genetique (Presses Universitaires de France, Paris) and four are still in preparation They deal with the formation of logical structures of the concepts, of number, space, functions, etc.; with interpretations of experiments and with the logical concepts of various kinds of learning, with concepts of order, velocity and time, with relationships between cybernetics and epistemology, and so on. We are currently deep m the difficult study of causality. Each year's work is discussed in a final, symposium, and eminent specialists participating in these gatherings have included W. V. Quine, E. W. Beth, F. Gonseth, T. S. Kuhn, M. Bunge, D. Bohm, W. McCulloch and B. Kedroff.
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