NOTE-142: Wallace — Stating Structural Realism

Contribution

The paper gives structural realism a statement that its metaphysical critics can engage with (Dorr, Hawley, Sider). It does so by replacing the object/relation framing with a distinction between two views of theories, language-first and math-first. It then shows that the move to math-first theories plus representation-as-success is by itself enough to realise Worrall’s and Ladyman’s goals. It defines two precise positions, math-first ESR (math-first physics with language-first metaphysics) and math-first OSR (math-first metaphysics: a “Model of the World”, not a “Book of the World”), and works out their consequences for AdS/CFT and for higher-level ontology.

Key insight

The standard examples of underdetermination and discontinuity across theory change are differences between predicate precisifications of mathematically equivalent (or instantiating) theories. Curvature versus force in gravity, Newton versus Einstein, and AdS versus CFT are all examples. At the level of mathematics these differences are invisible. So realism about the mathematics, understood as successful representation, is structural realism, and the remaining question, whether an object/property/relation description is fundamental or derivative, matters only at the unknown fundamental level.

Assumptions

• Math-first view of theories: a theory is a structured collection of mathematical models, whose theory/world relation is representation, like a map. The math-first view is argued to fit physics since the late nineteenth century better than the language-first view. Wallace defends it only indirectly, through the examples in §§3–5, and says so (pp. 6–7).
• Theoretical equivalence as mathematical equivalence, taken case by case. Set-theoretic isomorphism is too strict and categorical equivalence apparently too permissive (p. 11). The notion is admitted to be undeveloped (§13).
• Reduction as instantiation, approximate and domain-restricted (§5; Wallace 2012 ch. 2).
• Genuine strong underdetermination is between mathematically equivalent theories. Given the theory-ladenness of observation, empirical equivalence can hardly be shown otherwise (p. 23; cf. Norton 2008). The quantum measurement problem is set aside as too contested.
• Scope: physics only. The special sciences are touched only in a closing “hunch” (p. 38).
• Coherence of math-first metaphysics is assumed from §11 on, not established (p. 28).

Key results

• §3, empirical contact: “approximately true” is either undefinable or a disguise for “approximately”-statements inside the mathematics. Domain restriction is automatic on the math-first view.
• §4, equivalence: Euclidean space (coordinatised, affine or manifold presentations), the position-representation wavefunction (N-place on E³, 1-place on configuration space, or a function on N-tuples), Newtonian mechanics (trajectories or occupied curves), scalar fields (a function on M, or a pair of functions from a manifold) and AdS/CFT are each one theory on the math-first view, yet have no plausible language-first intertranslation. “Mathematical equivalences between theories mix up ontological categories” (p. 14).
• §5, reduction: language-first reduction needs holistic translation, mereology or instrumentalism (gases, hadrons as quarks, phonons). Math-first instantiation handles phonons (a symmetry-preserving Hilbert-space isomorphism under a state restriction) and fluids (autonomous low Fourier modes) directly.
• §8, math-first scientific realism: “Our successful scientific theories succeed, at least approximately, in representing the systems which they are used to model, including features of those systems that are unobservable”. This is already structural, with no further selective scepticism, so there is “no meaningful distinction to draw between ‘standard’ and ‘structural’ realism” (p. 24).
• §9, math-first ESR: a completed ToE underdetermines the metaphysicians’ Fundamental Theory, which leaves two options: Kantian humility, going further than Langton or Russellian monism, or a metaphysical methodology that can no longer claim continuity with physics.
• §10, math-first OSR: metaphysics is itself given mathematically. It realises French & Ladyman’s complaint that logic and set theory are ill-suited to structuralism, and Quine’s revisability of logic.
• §11, ontology as derivative (after Saunders 2003): existence claims are what the best predicate precisification entails. Precisification is the most fundamental level at which we can speak of what exists, but not the most fundamental level. On OSR, ties can make ontology indeterminate. For AdS/CFT, “what is the dimension of spacetime at the most fundamental level” has no answer (Fig. 2). On math-first ESR one side is correct (Fig. 1; Butterfield 2021).
• §12, higher-level ontology: on OSR, the high-level ontology is grounded in the high-level theory, which is instantiated by the low-level theory, and that relation constitutes the real pattern (Fig. 3). Ontology is scale- and domain-relative; unification is at the level of mathematics. On math-first ESR, Fig. 4 (direct grounding, primitive ontology) loses contact with high-level science, and Fig. 5 (separate ESR at each level) is bizarrely disunified. That leaves Fig. 6/7: every level but the fundamental is treated OSR-style.
• §13: “structural realism” names two projects. Math-first realism fulfils the first (Worrall, French, Ladyman). The second, priority theses and anti-individualism within the object/relation framework (Dasgupta; Esfeld et al.), “has grown distant” from those motivations.

Claims

id	claim	strength	support
C1	Contemporary physics is better read math-first than language-first.	weak–moderate	indirect support from examples, §§3–5; explicitly not defended directly
C2	Math-first theoretical equivalence is much coarser than any language-first notion, and mixes ontological categories.	moderate	five worked examples, §4; no formal definition
C3	Serious strong underdetermination arises only between mathematically equivalent theories.	moderate	argument from theory-ladenness, p. 23; the measurement problem is excluded
C4	Math-first scientific realism blunts the underdetermination and theory-change objections, and is already structural realism.	moderate	argument, §8, resting on C2, C3 and reduction-as-instantiation
C5	Even a completed physics underdetermines language-first fundamental metaphysics, weakening its claimed continuity with science.	moderate	argument, §9
C6	On math-first OSR, choosing a predicate precisification can use anthropocentric criteria legitimately, and may be indeterminate (AdS/CFT).	moderate	argument, §11
C7	Real patterns can be analysed as instantiation plus best precisification.	proposal	definition, p. 33, with no worked test
C8	Math-first ESR and OSR agree on all non-fundamental ontology.	moderate	elimination of the Fig. 4 and 5 alternatives, §12
C9	Math-first metaphysics is coherent.	assumption	conceded to be “significantly less well understood” than the language-first notions, §10

Concepts

• Language-first / math-first view of theories: roughly the syntactic and semantic views, renamed to cancel formalisation connotations and to allow hybrids.
• Predicate precisification: a presentation of a mathematically given theory in terms of objects, properties and relations.
• Math-first scientific realism: realism as approximate representational success, including of unobservable features.
• Math-first ESR: math-first physics plus language-first metaphysics (a Book of the World, grounding the physics).
• Math-first OSR: math-first physics plus math-first metaphysics (a Model of the World); ontology is derivative.
• Instantiation: reduction as a substructure of low-level models realising high-level structure. For state spaces, a map that commutes with the dynamics and preserves commonly interpreted structure.
• Real pattern (as defined here): X is real patterns in Y iff T_Y instantiates T_X and X, Y are the best precisifications of T_X, T_Y.

Connections

The paper builds on and restates Ladyman (1998), French & Ladyman (2003), French & Saatsi (2006), Saunders (2003, 2016) and Ladyman & Ross (2007), and answers Sider (2011, 2020), Dorr and Hawley. The SEP entry (LIT-045) is the map of the “second conception” literature that Wallace diagnoses; the paper cites its 2020 edition for the continuing centrality of “relations without relata”. Ladyman & Lorenzetti’s effective OSR (LIT-184) should be read with §12. Both make ontology scale-relative through real patterns, and Wallace’s instantiation-plus-precisification definition is a candidate precisification of that notion. Chakravartty (LIT-196, read as x6) targets exactly the priority-thesis OSR that Wallace sets aside. On Wallace’s framing, Chakravartty’s dilemma is internal to the language-first “second conception” and does not touch math-first OSR, which makes no claim that relations are prior to relata. Morganti’s coherentism (LIT-173) and Floridi’s informational SR (LIT-152) likewise sit within the second conception. French (LIT-170, read as x7) borrows Wallace’s earlier conventional-QFT stance, and his representation-spanning structure is a case where math-first equivalence becomes contentious: unitary inequivalence.

Bearing on metaphysics: the paper defends math-first OSR as a metaphysics. The most fundamental description of the world is mathematical; objects, properties and relations are derivative and exist only as the best precisification. It also defends a metametaphysical thesis: language-first fundamental metaphysics is underdetermined by completed physics and lacks methodological continuity with it. Grounding, levels and emergence (real patterns) are central. The tag is justified.

Bearing on the record

It is a strong candidate source for any THEORY in this record about theoretical equivalence, reduction as instantiation, or scale-relative ontology. Its real-pattern definition gives a checkable form to claims about emergence and levels. It carries nothing for ML practice. The analogy between “equivalent formulations” and ML reparametrisations is not drawn in the paper and should not be imported. Nothing belongs in the Anthology of the SOTA.

Limitations

• By the author’s own account (§§1, 13) it states rather than defends: “the views I describe are inevitably only first drafts”.
• It leans on a notion of mathematical equivalence that is not developed. “We seem to know it when we see it” (p. 38). Contested cases (gauge-related descriptions, unitarily inequivalent representations, dualities not yet proven) are exactly where the view needs it to be sharp.
• The coherence of math-first metaphysics is assumed for §§11–12, and Sider’s demand for a framework “adequate to the foundations of mathematics and science” is answered only by turning it back on predicate logic.
• AdS/CFT is used as a paradigm of equivalence, though Wallace notes that it is neither sharply stated nor proven, and that the bulk may be emergent rather than equivalent (fn. 10).
• It covers physics only; its extension to the special sciences is a declared “hunch”.

Open questions

• What precise notion of mathematical equivalence serves the view: categorical equivalence restricted how, or something else? And does it classify disputed cases (gauge, inequivalent representations) the way physicists do?
• Is the real-pattern definition extensionally right? It could be tested on phonons, fluids and quasi-particles.
• Can math-first metaphysics answer Sider’s adequacy demand directly, for instance by showing that category-theoretic or model-theoretic frameworks can carry fundamental metaphysics?

Corrections to the seeded skim

• The dossier says that on the math-first view reduction is “an asymmetric version of equivalence, with no need for bridge laws” (§5). Wallace says reduction is an asymmetric version of equivalence “in some sense on either view” (p. 15). On the math-first view specifically, reduction is instantiation: a substructure of the low-level models realises the structure of the high-level models. For state-space instantiation, that means a map from the low-level to the high-level state space that commutes with the dynamics and preserves commonly interpreted structure (p. 16). What math-first reduction lacks is the language-first need for bridge laws and for mereology (p. 17).
• The dossier puts “§12 and Figures 1–7 (pp. 31–37)” together. Figures 1–2 (AdS/CFT under math-first ESR and under math-first OSR) belong to §11 “Ontology as derivative” (pp. 30–32). Figures 3–7 (higher-level ontology) belong to §12.
• The dossier leaves out §9’s central argument (pp. 24–27). On math-first ESR, even a completed ToE underdetermines the metaphysicians’ Fundamental Theory, because many predicate precisifications fit it. So metaphysics’ claimed methodological continuity with science fails. Physicists’ appeals to simplicity are either “inference to the only explanation that is any good” or ways of choosing between empirically different theories while data are pending, and metaphysical choices are neither.
• The dossier leaves out the paper’s explicit analysis of real patterns (§12, p. 33): “X consists of real patterns in Y iff there are theories T_X and T_Y such that T_Y instantiates T_X and X, Y are respectively the best predicate precisifications of T_X, T_Y”. The pragmatic element sits in the precisification; the instantiation relation has none. This is the paper’s most reusable definition.
• The dossier suggests checking whether the precisification criteria (intelligibility, explanatory power, simplicity) “smuggle back a language-first standard”. Wallace addresses this (p. 30). On OSR the criteria do not need to be truth-tracking, only good ways of describing the mathematics, so even “flatly anthropocentric” criteria are defensible, as in Strawsonian descriptive metaphysics. On ESR the same criteria would have to track the truth, which he says is unmotivated.
• The dossier says the ESR/OSR choice “concerns only unknown fundamental physics, because non-fundamental physics comes out the same”. This holds on Wallace’s argument that math-first ESR has only one viable account of higher-level ontology (Fig. 6), after rejecting grounding high-level ontology directly in low-level ontology (Fig. 4, which he associates with primitive ontology) and a disunified high-level ESR (Fig. 5). It is not a stipulation.
• metaphysics tag: justified. philosophy-of-science rightly leads.