ora-0035

2.1.2 The generative-theory refinement

The fixed core sketch and the theory learned from experience must be distinguished. The former constrains what kinds of worlds are initially considered possible. The latter is a compact, revisable account of how the particular world is generated. A child playing with construction pieces may first identify configurations and transformations, then discover that joining, juxtaposition, and separation generate a much larger family of constructions subject to reusable equations.

Definition 2.2 Generatively presented world

A generatively presented world consists of

\[ (\mathcal C,\mathbb S,\mathfrak D,\mathbb A,G), \qquad \mathbb A=\operatorname {Th}_{\mathfrak D}(\mathbb S), \qquad G:\mathbb A\longrightarrow \mathcal C, \]

where \(\mathcal C\) is a world category, \(\mathbb S\) is a sketch of generators and relations, \(\mathfrak D\) is a preservation doctrine, \(\mathbb A\) is the theory presented by that sketch, and \(G\) is a structure-preserving interpretation.

ORACLE may therefore seek either \(\mathcal C_\star \) alone or the stronger generative target

\[ (\mathcal C_\star ,\mathbb S_\star ,\mathfrak D_\star , \mathbb A_\star ,G_\star ). \]

The stronger target explains indefinitely many composites using a finite presentation. It also creates new ambiguities: different signatures can present equivalent theories, and different theories can have equivalent model categories on the semantic worlds currently available to the learner. Literal recovery of a privileged generating list is therefore not the invariant objective. The learner must declare whether it seeks equivalence of completed theories, equivalence of their model semantics, or only factorization of a specified query doctrine.

This refinement sharpens Piagetian accommodation. Adding a newly encountered composite inside the old theory is assimilation. Adding a generator, imposing an equation, restricting an operation’s domain, or changing from a Cartesian to a resource-sensitive monoidal doctrine changes the theory by which experience is organized. Chapter 6 makes this distinction algorithmic.