ifc-0053
3.8 Assimilation and accommodation in differential form
Piaget’s distinction between assimilation and accommodation can now be made more precise. Let \(X\in \mathcal M\) denote the maintained schema or theory state. An infinitesimal variation at a generalized point \(x:S\to X\) is a map
When the application supplies an integration semantics, such a variation may generate a path or finite update \(X\rightsquigarrow X'\). A bare tangent category does not guarantee that every tangent vector integrates, so the update must remain part of the application-level admission contract.
A proto-accommodative probe is a section of the interpreted side bundle \(B_{\mathrm{acc}}\to M\) that tests a local change in the organization, observer language, or coupling of a maintained theory. It remains a probe inside the current double repair package; it is not yet a theory extension.
An assimilatory update is an admitted change \(X\rightsquigarrow X'\) whose variation, observations, and repair are expressible using the maintained ambient category, Weil action \(F\), interpretation, and observer family. It changes the state described by the theory without changing what counts as an admissible local variation.
Assimilation is therefore not synonymous with a small numerical step. A long trajectory, a composite of many probes, or a large parameter update may still be assimilatory if it remains inside the geometry already declared. What matters is whether the current conceptual coordinates and their transition rules suffice to describe the change.
Pressure for finite theory extension begins when they do not. A persistent defect may show that a direction is absent, that locally admissible variations fail to close, that an observer conflates distinct mechanisms, or that the present coordinates do not support the invariant required by the evidence. The response can occur at several levels:
move to a new base representation or stratum still available in \(\mathcal M\);
enlarge the observer family or the admitted probe language;
replace the tangent action \(F\) by a richer action \(F^+\); or
propose a change of ambient theory through candidate differential extension data \((J,\bar J,\chi ):(\mathcal M,F)\to (\mathcal M^+,F^+)\).
An admitted differential accommodation is a finite, typed change to the maintained representation or its differential semantics whose coherent finite realization, semantic transport, and new consequences have passed a registered test independent of proposal construction. Such a change may have been proposed because no admitted assimilatory repair accounted for a persistent obstruction. In its strongest form it consists of a differential theory extension \((J,\bar J,\chi )\), together with a repaired state \(r:\bar JX\to X^+\) in the extended universe and its admission record.
This definition distinguishes a local proto-accommodative probe from accommodation of the declaration. It also makes accommodation relative to a declared resolution. A change that requires a new object at one level may be an ordinary motion inside a richer theory at another. The audit record must therefore say which of \(X\), the observers, the probe doctrine, \(F\), or \(\mathcal M\) changed.
Piaget’s conservation task illustrates the distinction geometrically [ Piaget and Inhelder , 1974 ] . Pouring water changes visible height and container shape. An appearance-based schema follows those directions but does not yet represent conserved quantity. Accommodation introduces an invariant observer \(q\) for which the pouring direction \(v\) is null:
even though other observers vary. The child has not merely fitted a better coefficient; the representation now quotients a family of appearances by a quantity-preserving transformation.
Object permanence gives the complementary case. Occlusion removes the object from the immediate visual observer without removing it from the maintained world model [ Piaget , 1954 ] . Accommodation introduces a persistent state component whose identity is transported across visible and occluded contexts. Drescher’s Schema Mechanism provides an early computational analogue: updates of predictive schemas use existing items, while the construction of a synthetic item enlarges the available representational vocabulary [ Drescher , 1986 , 1991 ] .
. Attempt assimilation and proto-accommodation before a finite theory change. Use \(\Theta _{\mathrm{int}}\), or the classical observer \(\Omega _{\mathrm{cre}}\), to test whether their local orders are compatible. Accommodate the declaration only when a persistent, independently checked obstruction shows that the maintained geometry is inadequate; then record exactly which part of that geometry changed.