Contribution
From the pages seen, the book does two things. It proposes one definition of structure that covers its uses across the sciences. And it argues, domain by domain (mathematics and logic, physics and biology, psychology), that structures are not static forms. They are systems of transformations, and they are constructed. Ch. I says the synthesis is possible only by separating two problems “always conjoined in fact” but logically distinct. The first is the “affirmative ideal” that goes with the idea of structure, and the page breaks off there; the second was on an unserved page.
Key insight
A structure is a set of elements under laws of transformation, and it regulates itself. What it is can be read off what it does to itself: “all known structures—from mathematical groups to kinship systems—are, without exception, systems of transformation. But transformation need not be a temporal process: 1
- 1 ‘make’ 2” (p. 11).
Assumptions
- Wholes are not aggregates (p. 7): “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”. The laws “confer on the whole as such over-all properties distinct from the properties of its elements”. The integers are the example: they “do not come upon the scene except as ordered”.
- The choice is not wholes or atoms (p. 7). Piaget denies that the epistemological alternatives reduce to wholes defined by laws or atomistic compounding. The third option he argues for was on unserved pages.
Key results
From the pages seen, in order:
- Self-regulation, two grades (p. 15). In logic and mathematics, “an operation is a ‘perfect’ regulation”: an operational system “excludes errors before they are made, because every operation has its inverse in the system”. But “an immense class of structures… whose transformations unfold in time: linguistic structures, sociological structures, psychological structures” are governed by “regulations” that “are not in the strict sense ‘operations,’ because they are not entirely reversible”.
- Why the group concept succeeds (p. 19). Groups are “a kind of prototype of structures in general”. The concept is obtained by “reflective abstraction”, which derives properties “not… from things but from our ways of acting on things”. It rests on coordinations such as the possibility of a “return to the starting point” (the inverse) and of reaching one goal by different routes.
- Bourbaki’s mother structures (pp. 23, 27). Group structure “is quite independent of the intrinsic nature of its elements”. Psychogenetically, topological structures come before metric and projective ones, so “psychogenesis inverts the historical development of geometry but matches the Bourbaki ‘genealogy’”. MacLane and Eilenberg’s categories are mentioned as structuralism “in process of transformation”, with an emphasis on morphisms.
- Logic as general algebra (p. 31). Boolean algebra can be read as a logic of classes, a propositional logic or a two-valued arithmetic. The INRC group (identity, negation, reciprocal, correlative) is a Klein four-group acting on propositional operations.
- The limits of formalization are movable (p. 35). Gödel shows limits that “are not laid down once and for all, like the walls of China”. This is because “there is no ‘form as such’ or ‘content as such’”: each element is form to what it subsumes and content for a higher form.
- Physical structures (pp. 39, 43). Explanation places the real “in the context of possible transformations”, as in least-action paths and probabilistic explanations of entropy. “There are physical structures which, though independent of us, correspond to our operational structures”.
- Organisms (pp. 47, 51). Homeostasis makes the organism “a self-regulating system”. Learning depends on “antecedent structures”, which “need not be innate, nor are they necessarily immutable”. Against Foucault’s man as “a kind of rupture in the order of things”, that rupture is “the product of a great upheaval… which is constituted by life as a whole”.
- Against Gestalt (pp. 55, 59). Gestalt structures appeal to those who want structures “‘pure,’ unpolluted by history or genesis, functionless and detached from the subject”. But logico-mathematical structures obey laws of wholeness without being Gestalten, since “their laws of composition are strictly additive”. And the subject “performs in, sometimes even composes,” what it perceives.
- Sensori-motor origins (p. 63). The functional factors are assimilation and accommodation. The structural elements are order relations, subordination of schemes and correspondences. Coordination yields structures with “a modicum of ‘reversibility’”, first of all the “practical” group of displacements.
- The subject (p. 69). “To be real, a structure must, in the literal sense, be governed from within”. So “it is for our purposes sufficient to define this subject as the center of functional activity”. The page ends as Piaget asks what role is left for the subject if each structure regulates itself.
Concepts
- structure: a whole whose elements are subordinated to laws of composition, which is a system of transformations and regulates itself. The three defining ideas, from the contents and pp. 7–15, are wholeness, transformation and self-regulation. Piaget’s summary definition of ch. I, §1 was on unserved pages and is not quoted here.
- operation / regulation (p. 15): an operation is a fully reversible transformation, a “perfect” regulation; a regulation in the cybernetic sense governs transformations in time without full reversibility.
- reflective abstraction (p. 19): abstraction from the coordination of one’s own actions, not from properties of objects.
- INRC group (p. 31): identity, negation, reciprocal and correlative, acting on binary propositional operations.
Connections
The French original is LIT-802, unread. Saussure is cited in the translator’s note on p. 11 for “synchronic equilibration”, and the record holds the Course as LIT-774. The translator adds bracketed notes “[Trans.]” that supply English references: Saussure’s Course in Baskin’s translation, Bally, Köhler, Bourbaki, and Piaget’s own Origins of Intelligence.
Bearing on the record
- The p. 15 distinction bears on the record’s interest in which transformations of meaning are invertible. Logico-mathematical structures are groups, closed under inverses. Linguistic and social structures transform in time under regulations that are “not entirely reversible”. That is Piaget’s own line between group-like and merely self-regulating transformation systems, stated in 1968.
- p. 11 states that “all known structures… are… systems of transformation”, where the transformation need not be temporal: a definition of structure through its transformations.
- No THEORY is filed. A skim of a quarter of four chapters cannot ground a claim about what is true, and the pages are definitional.
- No instruction for machine-learning practice.
Limitations
- About 17 of some 140 text pages, one in four, chosen by the preview, not by the reader. The arguments of each chapter were seen only in fragments.
- None of the chapters on linguistics, the social sciences (where the publisher says Lévi-Strauss, Foucault, Parsons and Chomsky are assessed) or philosophy was seen, nor the conclusion.
- OCR of page images; symbols on p. 31 are garbled in places.
Open questions
- How does Piaget define the “affirmative ideal” of structuralism on p. 3, and what is the second problem he separates from it?
- In the linguistics and social-science chapters, does he count linguistic structures as only partly reversible, as p. 15 implies, and how does he treat Chomsky’s transformations against that?