ora-0062

4.2 Three meanings of categorical truth

The phrase “categorical truth” has three progressively stronger readings.

  1. Representational adequacy: the objects, transformations, and invariances posited by a core system admit a faithful categorical model.

  2. Explanatory adequacy: universal properties and functorial constraints compress regularities that would otherwise require unrelated rules for different tasks or modalities.

  3. Developmental adequacy: using the doctrine changes what can be identified from the presentations available to a learner, and its permitted repairs match the distinction between assimilation and accommodation.

The first reading is necessary but weak. The second says why category theory is useful. The third is the ORACLE claim: the prior contributes to learnability rather than merely redescribing a learned result.

Definition 4.1 Categorical core-knowledge claim

A categorical core-knowledge claim is a tuple

\[ (\mathbb T_i,\mathfrak H_i,P_i,\mathcal Q_i,\mathcal R_i) \]

consisting of a fragment sketch, a hypothesis class, an evidence observer, a query language, and an admissible repair class. It is empirically contentful only if replacing \(\mathbb T_i\) by a weaker doctrine changes at least one observable identification, invariance, composition, or repair prediction.

This formulation separates mathematical validity from empirical truth. A theorem may correctly describe models of \(\mathbb T_i\) even if evidence later shows that infants do not employ the predicted distinction. Conversely, an observed competence does not select a unique categorical explanation without comparison to weaker or competing doctrines.