ora-0065

4.5 Six systems, six categorical burdens

The declarations of Chapter 2 can now be read as research hypotheses rather than finished models. Table 4.1 records the categorical content and the corresponding burden for each system.

Core system	Candidate categorical content	What would make it non-vacuous?
Objects	Relative category of persisting states; paths, boundaries, contact, and exclusion	Identity judgments should be invariant under declared occlusions and viewpoint changes, but not under transformations that violate cohesion, continuity, or contact constraints.
Number	Ordered commutative magnitude object with resolution-dependent indistinguishability	Approximate comparisons should respect order and combination coherently across presentations while remaining distinct from exact natural-number recursion.
Space	Groupoid of viewpoints acting through a geometric observer	Route composition and reorientation should be predicted from transformations among views rather than memorized separately for each view.
Form	Shape functor separated from object identity	Shape equivalence should transport across identity-preserving viewpoint changes and fail in controlled ways under genuine deformation.
Agents	Behavioral coalgebras together with goal and efficiency comparisons	Self-propelled trajectories should support counterfactual predictions that motion-only coalgebras or object-path models do not determine.
Social beings and language	Indexed social categories linked to an admissible subcategory of language coalgebras	Communication should update shared context coherently, while formally possible but humanly impossible grammars remain outside the learnable class.

Table 4.1 The six Spelke systems become categorical hypotheses only when their formal declarations impose discriminating observational and developmental burdens.

The table also exposes where the current abstractions are weakest. Objects and space already have well-developed notions of invariance, paths, and gluing. Approximate number requires a principled account of psychophysical resolution rather than a convenient enriched order. Agency requires more than behavioral coalgebra: intention and efficiency must yield predictions not already present in trajectory fitting. Social cognition cannot be reduced to a poset if reciprocal communication, attention, and belief change are essential.