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1. mathematical models [Joe B & Michael] Of course, mathematical models work in the natural sciences, and from this many in the human sciences reckon that mathematical models should work too. Piaget argued repeatedly that [i] mathematics is a normative science and that [ii] logic is too, from which he drew the conclusion that [iii] "what works" in science is a normative model and there are at least two varieties, either valid. Actually, there are others too, such as law and?So [iv] let's check out a normative model under [ii]. Why? See 2. 2. logical necessity?the phenomenon of normativity in Piaget's project [Joe B] Piaget repeatedly identified the formation of logical necessity as his central problem. Anyone who doubts this is welcome to check Cambridge Companion to Piaget [CCP] p.69. The reason why this is a problem - well revisit Plato's Republic, Hume's Treatise, Kant's 1st Critique, or Wittgenstein's Tractatus?. Psychology has addressed a minor part of this problem by investigating the frequency of different types of such reasoning in different populations. But the major problem is its formation - how does anyone develop an understanding of logical necessity from the factual world in which we live [Tractatus: the world is the totality of facts, not thing]? Under [iv], Piaget's models were logical and normative and that is why he made such claims as "A subject is always "norm-laden" [Un sujet est toujours “normé”]". (p. 113) Smith, L. (2006). Norms and normative facts in human development.?In L. Smith & J. Vonèche. (Eds.). Norms in human development. [pp.103-37]. Cambridge: Cambridge University Press. For more on this, see CCP p 75. I don't see a complete answer to Piaget's problem in Piaget; nor anywhere else. But I do see contributions towards an answer - see Piaget's?papers on Necessity and Reason; there's a bit more from me in CCP ch. 3 Piaget, J. (1986). Essay on necessity. Human Development, 29, 301-14. Piaget, J. (2006). Reason. New Ideas in Psychology, 24, 1-29. 3. equilibration as a normative mechanism?[Joe M] Here is Piaget on this: "[During my education and training at school and university in Neuchâtel with Reymond] I had arrived at two ideas central for my point of view and which, moreover, I have never given up. [One is that] all knowledge is always assimilation of a datum external to the subject's structure?The second is that the normative factors of thought correspond biologically to a necessity of equilibrium of self-regulation: thus logic would in the subject correspond to a process of equilibrium" (1972, p.8). The successful outcome of this process is access to logical necessity, a normative notion par excellence. Or again: "An intelligent act can only be characterised as logical…from the moment that certain norms impress a given structure and rules of equilibrium upon this matter. Logic is not co-extensive with intelligence, but consists in the set of control rules used by intelligence in its self-direction." (1932, p. 405). 4. Stages?[Joe M] I agree with what you say - fine. Except for this. Stages are not the basic unit of analysis in P's constructivism:?"Constructions are without end, i.e. there is neither a first nor last level?[Construction sans fin...régression sans fin ]" (CCP, p. 73). Piaget repeatedly noted that his stages were stages of intelligence; he did not offer stages of perception, affectivity, behaviour in lower biological species. Nor were the mechanisms of stage-transition themselves stages - how could a state be a process? The best way to interpret this is not to use "stage" as the unit of analysis, nor to banish stages; but rather to see the latter as a special case of the former. The unit of analysis that I take from Piaget is: framework [cadre] - see chap, 3 in CCP;?see also this nice quote from Piaget's Abstraction book, p. 320 that is explicit: "from the beginning and even among our youngest subjects, a physical fact is recorded only with in a logical-mathematical; framework, however elementary it may be" (1977/2001, p. 320). The structure fitting a stage is a framework; but so too are any of the more primtive levels of action and thought in construction that is bi-directionally "sans fin". 4. Regression?[Joe M] There are two reasons why I interpret Piaget's account of human construction to include the possibility of regression, assuming that the unit of analysis is a framework. One is that he said things like this about development: "With development, the more elementary structures become incorporated into higher level structures" (1971, p. 2) => If there is development, structural integration occurs. Note the "if. And he also said things like this: DE is the study of the “mechanisms in the growth of the different fields of knowledge. The distinctive characteristic?of this discipline would therefore consist in analysing, in all domains relevant to the origin [genèse] or elaboration of fields of scientific knowledge, the transition from states of lesser knowledge to more elaborated states?1 Footnote 1 "Of course, the study of the growth of the fields of knowledge comprises, as (more or less exceptional) special cases, set-backs [reculs] or absence of modifications. The decadence of Greek science from the Alexandrian period and that of Arab science, for example, leads to very interesting problems, corresponding to questions of regression in individual development (aphasia for language and verbal thought, etc.)" (1957, p. 8 - my translation.) As an example of regression in science, how about the notion of the ether in 19th century physics. As an example of regression in psychology, how about MIC in the book on formal operations: he predicts a large plank of wood will sink; tests this and finds it floats so "he leans on the plank with all his strength to keep it under water" (1958, p. 22). Freud referred to reality-denial; MIC went further in trying to make reality do what he believed it should do, i.e. he had representational abilities but was reverting to using sensori-motor abilities.