Provisional translation of “L’enseignement des Mathématiques”



Translation of the first chapter by Jean Piaget of Beth, Choquet, Dieudonné, Lichnerowicz, Gattegno e Piaget (1955) “L’enseignement des Mathématiques”


The present a provisional translation and commentary of the first chapter of the book “L’enseignement des mathematiques.” whose title we translate as “The mathematical structures and the operational structures of intelligence.” This book is made of six chapters written by the following authors: Beth, Choquet, Dieudonne, Lichnerowicz, Gattegno e Piaget and published in 1955 and contains 173 pages. This book has never been translated into English. The only available translation in a Spanish done by Adolfo Maillo and Alberto Aizpun and published in 1968.

The book was reviewed in English by Goldstein (1957).


Part I

Whether The article that makes up this chapter is the summary of a conference presentation at the Colloquium of the Rochette near Melun in 1952. This colloquium devoted among others to the study of the mathematical and psychological structures was started by a presentation of J. Dieudonne on the first of these two subjects. The essay that followed answered the one by Dieudonne from a psychological point of view. Thus it refers frequently to the theses of that author not summarized in the present volume, but in accordance with those of the Bourbaki in general. (Compare with Bourbaki L’architecture des mathematiques, in Les grandes courants de la pensee mathematique, edited by F. Le Lionnais). Translator note: In the original book the previous footnote is a title footnote one has the practical point of view of a teacher charged with the teaching of the mathematical verities or the theoretical point of view of an epistemologist who is reflecting on the nature of mathematical entities, the central problem seems to be between these knowing whether the mathematical connections originate from the activities of the intellect or this intellect discovers them as an an external and pre-existing reality. Or this problem as ancient as western philosophy may today be posed in terms of psychology, more specifically, child psychology. It is among others to the study of mental development to show us whether the play of actions of the subject, then the operations of thinking, are sufficient to explain the construction of mathematical entities, or if they are discovered from the outside as are physical entities with their objective properties, and also certain ideal entities that are constituted out of language syntagma are. These are imposed on the individual by a social group of which he or she is a member (and one knows enough that the comparison between logic-mathematical entities and linguist links is supported by many logicians whether they are conventionalists or platonists).

But, if the methods to approach this eternal problem may be renewed by invoking the genetic psychology Translator note: It is not clear to me whether this term refers to “behavioral genetics” or “genetic epistemology.” The second discipline was created by Piaget himself. It is the study of the origins (genesis) of knowledge. His claim is that the method through which knowledge is obtained is a factor in the validity of that knowledge., the terms of the problem itself were recently renewed by the perspectives opened, thanks to the Bourbaki Translator note: Bourbaki is a pen-name for a group of mathematicians, mostly French, who starting in 1935 wrote a series of books on advanced mathematics., on the architecture of mathematics and by the fundamental role that these works attribute to the notion of “structure.”

Mathematicians have searched for e long time at the basis of mathematics for simple entities, which were imagined to be more or less of an atomistic nature. These were the whole numbers, which Kronecker attributed to God Himself unlike all other mathematical entities that originated by out of human activities. Those were the point, the line etc. where their composition generated space. But, they themselves were always given entities that the mind was called both to contemplate and manipulate, since the reflection had not yet become aware of the operations where it superposed them on the simple entities, just as a the tools that a brick layer uses to cement the prior given materials for the building of a wall or house.

But if the foundations consist of “structures” and if the construction proceeds from them going from simple to complex and from the general to the specific, the perspectives are different. A structure such as, for example, a “group” is an operating system systeme operatoire: the question is thus to know whether to elements of very different nature to whom we apply the structure existed previously. That is to say, they have a significance that is sufficiently independent from the structure, or whether on the contrary it is the action of the structure – an action not specified at the beginning, because the order in which we become aware is the opposite of the order of the origin – which gives to the elements their essential properties. More precisely, the psychological problem (that is the only one we will discuss) is to establish whether the entities that provide elements to the structures are the product of operations that create them or whether they pre-exist the operations that are later applied to them.

Now, the modifications that the idea of structures causes in the play of definitions and proofs are significant in this respect. Instead of defining the elements in isolation, by convention or by construction, the structural definition consists in in characterizing them by operational relations that they support among themselves as function of the system. And the structural definition of an element will be the demonstration of the necessity of this element, because it has been placed as belonging to a system where its parts are interdependent. This way a principle of totality is given from the start, and this totality is by necessity of operational nature. Even if in a system of pure relations such as structures of order, if the product of two relations is still an operation, it is that the relations are coordinated among themselves by operations of the logic of relations.

Not less revealing are the transformations that are introduced thanks to the notion of structure in the “architecture” of mathematics, which amounts to saying that within the order of construction or of relation of the innumerable classes that it is possible to distinguish in these abstract entities. It is possible to say in this respect that the introduction of the structures represents an progress that is analogous to what comparative anatomy has realized in biology, in that it replaces a classification based on the internal and genetic connections to a classification that is satisfied by the external characters and in their static discontinuity. Starting from a few fundamental structures, the following step consists in differentiating them, from general to particular, and to combine them with each other, from the simple to the complex: from from a hierarchy that replaces the previous domains that are juxtaposed a series of planes superposed according those two modes of generation. What follows is a new principle of totality that subordinates the elements or classes of elements to the dynamism of a proper construction.

Furthermore, we note the great interest, by the psychology of mathematical thinking, of the modality of the discovery of structures – and here we return to our original alternatives of the continuity between the work of intelligence and the mathematical construction or the exteriority of ideal entities that the mind perceives as external. Initially, the examination of the steps of the mathematicians take to obtain the fundamental structures seem to favor the second of these theses: far from deducing them at once, it starts with analogies that are discovered afterwards within forms of reasoning that play in domains without apparent relationship, later in some sort of induction, as one proceeds with experimental data, and reconstructs common mechanisms to evince the most general laws of the examined structure. It is then that the axiomatization intervenes, then the use, that is the application of these general laws to particular theories by progressive differentiation. Moreover, the shift from primary structures to secondary structures if performed by the combination of several structures: here another time this combination is not a deduction, because we have to apply new axioms to each new structure to be able to integrate new elements.

But this reasoning, somewhat inductive of the discovery of structures is on the contrary very revealing of the relations that support structures with the different elements that they organize. If historically these elements appear to be previously given to the structure that is discovered, and if this way the latter has essentially the role of a reflexive instrument that is destined to release their most general characteristics, we should not forget that, psychologically, the order in which we become aware of them is the opposite of the order of their origin: that which is the first in the order of the construction appears to the the last to be reflexively analyzed, because the subject becomes aware of the results of the mental construction before reaching its inner mechanisms For example the late introduction, with Cantor, of the operations of the one-to-one and reciprocal correspondences when on the contrary it is one of the operations of generation of the whole numbers with children and primitive cultures..Far from being a decisive argument in favor of the indipendence of “structures” in relation to the work of the intelligence, their late and quasi inductive discovery makes us on the contrary suspect their primitive and generative character. But if what is fundamental appears at the end of the analysis, the opposite is not necessarily true and the problem to show the possible connections between the mother structures of the edifice of mathematics and the operational structures that the study of mental development allows us to consider as formative of the logic-mathematical construction is thus still unresolved. That is what we try to examine now in the field of psychogenesis.

The three fundamental structures on which the edifice of mathematics rests, according to Bourbaki, are the algebraic structures, where the prototype is the “group”, the structures of order, where a variety commonly used today (with extremes of others in certain cases) is the “network” Translators note: lattice?, and the topological structures. This number is not exhaustive, and the development of mathematics may increase it. But with today’s level of knowledge these three structures are the only ones that are irreducible among themselves and thus play the role of mother structures.