20.3 Why two interacting directions matter

Piaget’s distinction between assimilation and accommodation supplies the book’s cognitive intuition. Assimilation improves a model’s fit or competence within its current organization. Accommodation changes that organization when continued assimilation is no longer adequate. Neither direction alone is creative in the sense developed here. Unconstrained accommodation merely changes the rules whenever they become inconvenient; assimilation alone can become increasingly competent inside a theory whose vocabulary is wrong.

The computational hierarchy is more explicit than this psychological analogy. Proto-accommodation probes or proposes a changed organization; accommodation names the finite package change after independent admission; and theory extension is the special case that changes the presentation or generative language.

The double structure records their interaction. Order can matter: adapting a skill and then revising its interface need not agree with revising the interface and then adapting the skill. The mixed bracket makes that order-sensitivity observable, while the quotient removes changes already available along either side separately. Weil probes then ask whether the residual is stable under a controlled family of infinitesimal tests. The result is not a mystical detector of creativity. It is a mechanism for asking a precise question: does the interaction reveal a direction missing from the current theory?

This also explains the book’s use of the word infinitesimal. A finite conceptual change may be proposed because many local comparisons point toward the same obstruction. The infinitesimal layer supplies evidence and localization; a finite package comparison records the proposal, with a sketch extension when the language changes. DIAL therefore does not assert that every breakthrough is a tiny parameter update. It asks whether a discontinuous revision can be made accountable to a structured family of local failures.