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5.9 Developmental towers of theory

Piaget’s stages of cognitive development suggest a provocative question: could development itself be represented as a succession of increasingly articulated theories? There is no established theorem identifying sensorimotor, preoperational, concrete-operational, and formal-operational reasoning with standard categorical doctrines. Nor do the doctrines in Table 5.1 form a single hierarchy: a PROP and a Lawvere theory make different structural commitments, and a topos is not simply a more mature finite-limit theory. Any correspondence must therefore remain explicitly heuristic [ Piaget , 1952 , 1985 ] .

The formal core of the analogy is instead a developmental tower of presentations

\[ \mathbb S_0 \xrightarrow {\iota _0} \mathbb S_1 \xrightarrow {\iota _1} \mathbb S_2 \xrightarrow {\iota _2}\cdots . \]

At stage \(n\), the presentation \(\mathbb S_n\) determines what can be represented, composed, observed, and asked. Assimilation changes or explores a model

\[ M\in \operatorname {Mod}(\mathbb S_n,\mathcal C) \]

while the presentation remains fixed. A finite extension proposal introduces \(\mathbb S_{n+1}\), adding or reorganizing sorts, operations, equations, observers, admissible domains, or the preservation doctrine itself. It becomes an accommodation of the maintained theory only after transport and admission.

Each stage map has the restriction semantics developed in Section 5.6. Its fibers distinguish an incompatible change from an essentially definitional elaboration and from a genuinely underdetermined expansion. The developmental tower therefore adds no new kind of model comparison; it applies the same comparison repeatedly while making the history of admitted presentations explicit.

Boundary: A lossy functor. It is useful to call the proposed correspondence from Piagetian development to categorical theory change a lossy functor. The phrase is intentionally metaphorical: no source category of cognitive stages and no structure-preserving functor have yet been defined. The mapping forgets psychological, embodied, temporal, and social dimensions while retaining one abstract pattern—alternation between change inside a representational scheme and change of the scheme itself.

Piagetian capacity	Tentative categorical reading	Principal loss in the analogy
Sensorimotor organization	Models with actions and state transitions before an explicit symbolic presentation has been extracted.	Embodied perception and action are not themselves algebraic theories.
Preoperational representation	A signature or free sketch of objects and operations with few registered equations or invariants.	Symbol use, language, and perspective taking exceed a bare signature.
Concrete operations	Equational or finite-product structure supporting composition, reversibility, and invariants such as conservation.	Concrete-operational competence is task- and context-sensitive, not merely equational closure.
Formal operations	Hypothetical reasoning across models and controlled extensions of a theory, including comparison of assumptions and consequences.	Meta-level theory variation does not capture the full psychology of abstract reasoning.

Table 5.2 A deliberately lossy correspondence between Piagetian capacities and categorical theory construction. The rows are explanatory analogies, not identifications of psychological stages with categorical doctrines.

Two safeguards are essential. First, developmental stages should be attached to semantic capacity, not textual presentation. Presentations related by an appropriate Morita-style equivalence should represent the same stage for the purpose of this abstraction. Second, the developmental relation is at most a partial order. Two theories may support different, incomparable capacities; one need not be globally more advanced than the other.

The proposal therefore does not classify human development. It extracts a formal pattern useful for synthetic creativity: equilibration alternates between fitting experience within a current model category and extending the theory that determines that category. The restriction functor and its fibers make the continuity, incompatibility, and underdetermination of each proposed accommodation auditable.