lin-0263

22.2.2 Appearance, reality, and theoretical invention

McCarthy argued that learning cannot be reduced to classifying appearances [ McCarthy , 2007 ] . Appearances are partial and dependent on the circumstances of observation, whereas the objects and mechanisms invoked by an explanation are intended to remain coherent across views, times, and actions. In categorical language, an observation functor

\[ O:\mathcal R\longrightarrow \mathcal A \]

maps candidate realities to appearances. An observed \(a\in \mathcal A\) generally determines a fiber of compatible realities, not a unique inverse image. If the probes available to the learner do not separate that fiber, reality is not identifiable from passive appearance.

The harder case is not selection among already represented latent states but invention of a new theoretical sort. Ancient atomism proposed that visible matter was generated from invisible constituents. Dalton’s atomic theory turned that ontological proposal into a constrained explanatory system: elements, atoms, molecules, fixed combinations, and quantitative relations could jointly account for stable regularities in chemical appearance. The theoretical vocabulary was not copied from sense data. It was introduced to make a larger family of observations compose.

This is a declaration-level repair. A persistent obstruction in the old sketch motivates an augmentation

\[ \mathbb S \longrightarrow \mathbb S^{+} = \mathbb S+ \{ \text{new sorts, generators, and laws}\} . \]

The expansion creates a new forward model from theoretical reality to appearance. Its value lies not merely in fitting the observations that motivated it, but in transporting earlier successes and generating consequences that can be tested independently.

Intervention sharpens the test. Distinct candidate realities may agree on passive appearances yet respond differently when experimental conditions are changed. McCarthy’s inverse problem and Pearl’s intervention problem are therefore successive stages rather than competing diagnoses:

\[ \begin{aligned} \text{appearance} & \longrightarrow \text{candidate reality},\\ \text{candidate reality} & \longrightarrow \text{interventional consequence},\\ \text{interventional consequence} & \longrightarrow \text{admission or rejection}. \end{aligned} \]

LINCS adds one further possibility: when every candidate inside the current language fails, the repair may change the language itself.

Stage	Model-level workflow	Theory-level workflow
Declare	Fix \(\mathbb S\) and a candidate model \(D\)	Fix the current theory and the language of permitted extensions
Differentiate	Measure how an obstruction moves under model variation	Measure which generators, axioms, or universal properties sustain the failure
Quotient	Remove decision-null or presentation-null model directions	Identify presentation-equivalent extensions and mere definitional changes
Localize	Assign the obstruction to components, contexts, or overlaps	Assign explanatory pressure to missing sorts, arrows, equations, cones, or cocones
Repair	Change the realized maps or representations	Propose \(\iota :\mathbb S\to \mathbb S^{+}\) and compatible expanded models
Admit	Test the repaired model on held-out probes	Test conservativity, old successes, independent evidence, and novel predictions

Table 22.4 The six-stage workflow lifted from model repair to theory augmentation.

At the theory level, differentiation need not mean that sketches form a smooth parameter space. It means constructing a typed sensitivity analysis: which declared relation produces the persistent obstruction, which added generator could mediate it, and which new universal property would make the candidate explanation testable? Localization therefore returns not only a faulty component but an explanatory gap in the current presentation.

The resulting search cannot range indiscriminately over all theories. Accessible-category methods suggest one disciplined possibility: bound the arity and size of candidate generators, construct larger candidates from presentable pieces, and preserve filtered-colimit structure where the semantics requires it [ Makkai and Paré , 1989 , Adámek and Rosický , 1994 ] . This does not supply an algorithm for scientific discovery. It supplies a categorical search language in which proposed extensions and their model categories can be compared.

Admission contract

An augmented sketch must do more than absorb the anomaly that proposed it. Admission requires an explicit restriction functor, recovery or principled rejection of successful old models, and consequences tested on evidence not used to construct the augmentation. Novel prediction is stronger evidence than retrospective fit.

In this form, an abductive leap is neither an unexplained jump nor a scalar loss reduction. It is a proposed expansion of the theory’s generators and axioms, together with transport obligations and empirical or deductive tests. The examples of quantization, spacetime geometry, natural selection, and latent causal structure can all be read this way, although their appropriate sketch languages will be very different.