1 The regulatory dynamics could couple with the leaves of a foliation of the total interactive dynamic space, e.g., a coupling with the parameters of the foliation, if the mathematical conditions of a manifold are satisfied (Kolar, et al, 1993; Marmo, et al, 1985; Tamura, 1992). A discrete dynamics, on the other hand, such as a typical programming language provides, would not in general manifest a dynamical manifold, and, therefore, not a coupling via a foliation. Nevertheless, a discrete dynamics could manifest a meta-dynamics that would regulate -- control -- a system of directly interactive routines. That is, the distinction between an interactive dynamics and a regulatory dynamics could still be made, for example, in terms of a distinction between interactive processing and control flow.
2 A still more sophisticated view would recognize that such a "relaxation" process is more likely to be a process of mutual selection constraints between interactive dynamics and microgenesis. Mutual constraining relationships among endogenously active processes permit greater flexibility than "simple" relaxation processes, and altering parameters of those mutual constraint processes is still another source of flexibility: mutual selection constraints is a more complete use of the power of variation and selection than is relaxation (Bickhard & Campbell, D. T., in preparation).
3 This is an internal development of processes of evolutionary epistemology -- a kind of process that continues to occur externally. Internal variation and selection processes emerge because of the advantages of adaptability that they offer the organism. They are not in any sense a bringing into the organism or impressing into the organism of something from outside. The internal emergence is parallel to, not instructed from, external processes -- except, of course, via the selection effects of external processes. This is in contrast, for example, to the notions of internalization of Piaget and Vygotsky, which involve specific external structures and organizations being brought into the organism, and at least suggest an encoding of those structures and organizations (Bickhard, 1995b).
4 This differentiation, in fact, is the developmental progression in children. "Consider the following example from a child of 4 years and 11 months who was asked to indicate all the ways that a toy car could get from a point A to a point B in a room: Pie (4;11) "Show me all the ways one can go from A to B." Straight ahead. "Can you make another?" No. "Try it." You could put the car in the garage (he repeats the straight path). "But do another one." He describes a slight curved line. "And another." No. "There are only two to do?" Yes. "Why?" Because there's only one car. We set up the post [a post set on the floor in between A and B]. "Now, do it." It's impossible, because there's a post, so we can't go to B, it would make an accident. "Try." He makes a curved path. I got around it. "And another." He repeats the same curved path, but turns back at the post, having bypassed it, instead of going to B. "Another." A curve from A to B, bypassing the post at the right instead of left. That's not the same. "Are there others?" No. "When you go to school, you always take the same way?" No. "And from A to B? Always the same?" Yes. (Piaget, 1987, p. 19) (from Bickhard, 1988, pp. 502-503) Trying to sort out just what is going on in this example is non-trivial (Piaget, 1987). What is clear, however, is that, although notions of possibility, impossibility, and so on are not unknown to Pie, nevertheless he has the field of modality extremely confused and mixed up -- undifferentiated. The course of development involves, among many other things, a progressive differentiation out of such initial beginnings.
5 The notion of "actuality" is itself subject to variation. It acts, in fact, as a kind of dual to the various kinds of necessity: an actuality in this room; an actuality in this physical world; an actual physical possibility; an actuality in a model, or in some space of possible models, and so on. The notion of Truth climbs up these spaces dually to the notion of necessity descending them. Truth and necessity are dual notions, with Truth focusing on exceptionlessness with respect to one space of consideration and necessity focusing on spaces of possible variations in that space of consideration.
6 In particular, with respect to automorphisms of models. Note that automorphisms are not the only kind of possible variation of extensions: restriction of consideration to automorphisms yields a particular, very powerful, notion of the nature of logic (Sher, 1991). That restriction, in turn, follows from the necessary indeterminism of the particulars of representational extensions: representation is fundamentally implicit, not explicit, so set theoretic structural properties, which are preserved by (structurally invariant under; structurally isomorphic with respect to) automorphisms, are the limit of what can be considered about extensions in general. (More will be represented, of course, about [elements of] the extensions of particular representations.)
7 Note that this notion of "formal" is related to, but is not the same as, the formal approach to rationality. In particular, this notion of formal "simply" means in terms of particular identifiable (formal) properties, and has no assumptions or implications of foundationalism.
8 Invariance under automorphisms of models prescinds both from a full metaphysics of possible elements of extensions, and from particularities of systems of representation. Logic abstracts properties that are invariant with respect to structures of extensions, ignoring the particularities of extensional elements and of representational systems (Sher, 1991, 1996a, 1996b). Such structural invariance, however, does depend on, among other things, the notion of and criteria for identity of elements of those models -- criteria of "entityness" and of identity.
9 Invariance is one of the most important kinds of critical principle (Bickhard, 1980a; Hooker, 1992, 1995). Differing forms of invariance can be found with respect to the cognitions of physical objects; conservations, such as of number, mass, or volume; the differing kinds of invariance that generate the various kinds of geometry; and global and gauge invariances in theoretical physics. The invariances discussed with respect to the Theories of Relativity are examples of global invariances. Earlier, the domain of logic was construed as being constituted as a kind of invariance: invariance under isomorphic structural transformations of abstracted representational extensions (Lindenbaum & Tarski, 1934-1935; Mautner, 1946; Mostowski, 1957; Lindström, 1966a, 1966b; Tarski, 1966/1986; see especially Sher, 1991, 1996a, 1996b). Invariance is a -- perhaps the -- primary form of differentiation from, independence of, the processes of an epistemic agent. The invariances of objects yield a stability in time with respect to most ensuing events, which, in turn, makes possible the representation of a relatively stable world transcending the immediate perceptual environment of the organism -- your home remains relatively invariant, and can be represented as such, even when you are away. The invariances of physics prescind from particularities of the situation of observers and the origins, orientations, and time derivatives of measuring frames. Logic is the domain of properties that are invariant under isomorphic structural transformations of representational extensions -- it is the domain generated by prescinding from the particularities of representational agents and their situations. Invariance is the general form of understanding and representing the world as (relatively) independent of the observer; invariance is agent-decentering.