Source
De Haro & Butterfield (2025), LIT-199, chapter 14, read in full in NOTE-178 from the arXiv manuscript. Page numbers are printed ones.
What was actually shown
Definitions (§14.3). A linkage map link: T_b → T_t from a bottom to a top theory is non-injective and a partial homomorphism; it is reductive (Nagelian) iff surjective. Ontological emergence is a non-commuting diagram, i_b ≠ i_t∘link: the top theory’s interpretation refers to something the bottom’s, carried across, does not. Epistemic emergence is novelty only in the bare theories, as irreducibility or computational intractability.
The incompatibility. A duality is an isomorphism of model roots, so the duals are equinumerous and the map between them is a bijection. That rules out the non-injective linkage that emergence needs, reductive or irreducible. The reading grades this moderate (C7): strictly proven for finite sets, with a physical-grounds argument for the infinite case (fn. 57).
Three ways to combine them (§14.3.4).
- Emergence beside duality: in AdS/CFT the emergence map runs along the renormalization-group (radial) direction, not along the duality map (Fig. 14.8).
- Effective duality: an approximate or limiting duality with no exact isomorphism.
- Epistemic emergence.
A related result. Reduction (formal) and ontological emergence (interpretative) are compatible with each other (C8), shown on Ising spontaneous magnetisation, massless against massive particles, and Seiberg–Witten. Duality, being symmetric, in general gives no priority; reductive emergence gives priority as definitional dependence (§14.4).
What could have come out otherwise. The definitions could have let a bijection carry novel reference. Given them, the finite case is a proof, and the infinite case is where the argument could still fail.
What this does not say
- That dualities indicate emergence or fundamentality. They are “by themselves not indicative of emergence or fundamentality” (§16.2, p. 529).
- That AdS/CFT is an exception to that. It is the exception only to incompatibility with non-epistemic emergence, because an extra dimension appears beside the duality. On a theoretically equivalent reading, Horowitz and Polchinski’s “emergent gravity” “cannot be understood as ontological emergence” (§14.3.3, p. 489).
- That emergence in other senses is excluded. The notion here is Butterfield-style novelty with the robustness condition dropped, a “semantic or interpretative” condition (p. 479). Information-theoretic measures of emergence answer a different question, and the two do not conflict (NOTE-178).
- That the infinite case is proven. It is argued.
- That gauge–gravity duality is exact. It is a conjecture, and the common core there is partial.
Connections
- The record’s other emergence accounts are Rizi (LIT-150) and emergence as conversion of information (LIT-027), and THEORY-222, on which a coarse description can carry more effective information than the micro one. Those are measures over a system’s dynamics. This account is about interpretation maps between theories, so it neither supports nor contradicts them.
- The same book’s account of underdetermination and structural realism is THEORY-690, which uses the same formal/interpretative split.