From Piaget, Structuralism, Routledge and Kegan Paul, 1971 (p. 6-10).
That wholeness is a defining mark of structures almost goes without saying, since all structuralists - mathematicians, liguists, psychologists, or what have you - are at one in recognizing as fundamental the contrast between structures and aggregates, the former being wholes, the latter composites formed of elements that are independent of the complexes into which they enter. To insist on this distinction is not to deny that structures have elements, but the elements of a structure are subordinated to laws, and it is in terms of these laws that the structure qua whole or system is defined. Moreover, the laws governing a structure's composition are not reducible to cumulative one-by-one association of its elements: they confer on the whole as such over-all properties distinct from the properties of its elements. Here is an example of what we have in mind: the integers do not exist in isolation from one another, nor were they discovered one by one in some accidental sequence and then, finally, united into a whole. They do not come upon the scene except as ordered, and this order of the integers is associated with structural properties (of groups, fields, rings, and the like), which are quite different from the properties of number individuals, each of which is even or odd, prime or non-prime, and so on.
The idea of wholeness does, however, raise a good many problems, of which we shall take up just the two principal ones, the first bearing on its nature, the other on its mode of formation (or preformation).
It would be a mistake
to think that, in all domains, the epistemological alternatives reduce
to just two options: either admit wholes defined in terms of their structural
laws, or allow only for atomistic compounding of prior elements. No
matter what area of science we subject to scrutiny, whether we consider
the perceptual structures of the Gestalt psychologists or the social
wholes (classes or entire societies) of sociologists and anthropologists,
we find that not one but two alternatives to atomism have
made their way in the history of ideas, only one of which appears to us in tune with the spirit of modern structuralism.
The first consists in simply reversing the sequence that appeared natural to those who wanted to proceed from the simple to the complex (from sense impressions to perceptual complexes, from individuals to social groups, and so forth). The whole which this sort of critic of atomism posits at the outset is viewed as the outcome of some sort of emergence, vaguely conceived as a law of nature and not further analyzed. Thus, when Comte proposed to explain men in terms of humanity, not humanity in terms of men, or when Durkheim thought of the social whole as emerging from the union of individuals in much the same way as molecules are formed by the union of atoms, or when the Gestalt psychologists believed they could discern immediate wholes in primary perception comparable to the field effects that figure in electromagnetism, they did indeed remind us that a whole is not the same as a simple juxtaposition of previously available elements, and for this they deserve our gratitude; but by viewing the whole as prior to its elements or contemporaneous with their "contact," they simplified the problem to such an extent as to risk bypassing all the central questions—questions about the nature of a whole's laws of composition.
Over and beyond the schemes of atomist association on the one hand and emergent totalities on the other, there is, however, a third, that of operational structuralism. It adopts from the start a relational perspective, according to which it is neither the elements nor a whole introduction and location of problems that comes about in a manner one knows not how, but the relations among elements that count. In other words, the logical procedures or natural processes by which the whole is formed are primary, not the whole, which is consequent on the system's laws of composition, or the elements.
But at this point a second, and much more serious, problem springs up, the really central problem of structuralism: Have these composite wholes always been composed? How can this be? Did not someone compound them? Or were they initially (and are they still) in process of composition? To put the question in a different way: Do structures call for formation, or is only some soil of eternal preformation compatible with them?
Structuralism, it seems, must choose between structureless genesis on the one hand and ungenerated wholes or forms on the other; the former would make it revert to that atomistic association to which empiricism has accustomed us; the latter constantly threaten to make it lapse into a theory of Husserlian essences, Platonic forms, Kantian a priori forms of synthesis. Unless, of course, there is a way of passing between the horns of this dilemma.
As is to be expected, it is on this problem that opinion is most divided, some going so far as to contend that the problem of the genesis of structures cannot so much as be formulated because structure is of its very nature non-lemporal (as if this were not in its own way a solution of the problem, namely, the choice of a preformational view of the origin of structures).
Actually, the problem we now are discussing arises with the notion of wholeness itself. It can be narrowed down once we take the second characteristic of structures, namely, their being systems of transformations rather than static forms, seriously.