Contribution
The paper imports Kuhn’s (1970) notion of a “generalization-sketch” into the dispute over the Price equation. It argues that critics and users are both right. The critics are right that the equation is not a model and predicts nothing on its own. The users are right that it is fruitful, because its role is that of a schema from which specific models are derived. Luque supports this with a derivation (§3), a survey of results the equation recovers and of its extensions across fields (§4, Tables 2–3), an invariance argument (§5), two worked case studies (§6.1–6.2), and replies to the tautology and predictive-power objections (§7).
Key insight
Price’s equation stands to evolutionary biology as f = ma stands to mechanics. It is a law-sketch that is “empty in [its] predictive content” (p. 31) until a model supplies the entities, the characters, the time step and the causes. Its value lies in unification and in heuristic guidance, not in prediction. Demanding dynamic sufficiency or predictions from it is a category mistake.
Assumptions
Premises and authorities the argument rests on:
- Derivation (Frank’s standard form, §3): the equation assumes only a mapping from ancestors i (frequency q_i, character z_i, fitness w_i) to descendants (q′_i = q_i w_i / w̄, z′_i). The one restriction stated is a closed population in which every entity at t + 1 descends from, or is itself, an entity present at t, which excludes migration (p. 8). The equation holds over any time interval.
- Kuhn’s account of generalization-sketches (Kuhn 1970, p. 465; Kuhn 2000) and of exemplars as the way scientists learn to specialise the sketch.
- Unification as explanatory value (Kitcher 1993), and “buy generality at the expense of predictive power” (Birch 2014, p. 400).
- Sober’s (1984) distinction between consequence laws and source laws.
- Cartwright’s (1999) “fundamentalism” and Rice & Papadopoulos’s (2009) “axiomatic theories”, taken as descriptions of the style of theorizing the equation embodies.
- Ramsey–Quine theoretical holism (via Sklar 2013): theoretical terms get their meaning from their role in a theory. Used in §7.1 to say that both F = ma and the Price equation acquire empirical meaning only through interpretation.
- A premise from the philosophy of mathematics: mathematical theorems are not tautologies, which he supports with Gödel’s incompleteness theorems and Putnam (1975) (p. 28, fn 17).
Key results
What it argues:
- Eq. (1), p. 7: Δz̄ = Cov(w, z) + E(wΔz). Eq. (2): Δz̄ = Cov(ω, z) + E_ω(Δz), with relative fitness ω. Eq. (3), continuous time (Page & Nowak 2002). Eq. (4): Δz̄ = Cov(ω, z′) + E(Δz), Okasha’s preferred causal partition. Eq. (5): Δz̄ = Cov(ω, z) + Cov(ω, Δz) + E(Δz), after Godfrey-Smith 2007, for when selection and transmission interact non-additively. All are collected in Table 1.
- Unification (§4): the diallelic viability-selection result Δp = p(w*_A − w̄)/w̄ is re-derived from the covariance term (pp. 10–11). So are Robertson’s secondary theorem, Δz̄ = Cov_A(w, z), and the “classical interpretation” of Fisher’s fundamental theorem, Δw̄ = Var_A(w). Footnote 7 notes that the exact theorem concerns only the partial change caused by selection. The paper also lists applications in selection theory, stochastic evolution, ecology, epidemiology, non-genetic inheritance and culture, and mentions economics and cosmology. Tables 2 and 3 collect about 25 derived identities and extensions.
- Invariance (§5): following Frank (2012a), selection can be written as Δ_s z̄ = β_zw F(Δq̃), with F the Fisher information in the frequency fluctuations. Infinitely many combinations of frequency change and character value give the same total selection. Against an objection he credits to Jun Otsuka, Luque argues that invariances always stem from a model’s mathematical form, as Galilean invariance does for Newtonian models, and that they can still have empirical consequences (pp. 15–16).
- Generalization-sketch (§6): the equation provides a common language (covariances, expectations, regressions). It works as a consequence law and a “promise”, and needs exemplars to learn. Hamilton’s 1979 course notes are offered as evidence that it was taught that way (pp. 22–23).
- Case studies: Day & Bonduriansky (2011) derive eqs. (6a–b), overlapping-generation versions that separate selection, “reproductive transmission” and “survival transmission”, and then eqs. (7a–b) under a first-order fitness approximation. These are specialised into models of environmental noise, maternal effects, indirect genetic effects and transgenerational epigenetic inheritance (§6.1). Rice’s stochastic Price equation shows that all moments of the fitness distribution, not just the mean, contribute to directional evolution; the second and third terms pull the population toward minimum variance and maximum skewness in fitness (§6.2).
- Tautology (§7.1): the equation is “not a tautology” in the logician’s sense (p. 28). The critics’ real complaint, that it follows from notational definitions, has the same shape as the 18th–20th century dispute over whether F = ma is a definition (d’Alembert, Mach, Poincaré, Wilczek). Both acquire meaning only by interpretation within a theory. The biological interpretation, “change over time, ancestor/descendant relations, and phenotype” (Rice 2004), is empirically grounded (p. 30).
- Predictive power (§7.2): dynamic sufficiency belongs to particular models, not to the equation, and van Veelen et al. (2012) concede as much. On van Veelen’s claim that the equation cannot answer Type 1 (top-down) questions, Luque replies that predictive power is not the only explanatory value and that generalization-sketches are predictively empty by nature (pp. 31–32).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | The Price equation follows from a mapping between ancestor and descendant sets with no simplifying assumption, and is restricted only by population closure (no migration) | strong | standard derivation, §3 (pp. 6–8); scope note citing Kerr & Godfrey-Smith 2009 |
| C2 | Classical results (diallelic viability selection, Robertson’s secondary theorem, Fisher’s fundamental theorem in its “classical interpretation”) can be derived from it | strong for the diallelic case; moderate for the others | derivation of the diallelic case (pp. 10–11); the Robertson and Fisher results are stated, with derivations deferred to Frank 2012a and Walsh & Lynch 2013 |
| C3 | It unifies work across selection theory, stochastic evolution, ecology, epidemiology, non-genetic and cultural inheritance | moderate | literature survey, §4 and Tables 2–3; the survey is illustrative and, by the author’s note (fn 8), not exhaustive |
| C4 | It has invariance “like other scientific principles” | weak | rests entirely on Frank’s (2012a) information-geometric identity; the analogy to physical symmetries is argued informally (§5) |
| C5 | It is a Kuhnian generalization-sketch (a schema, not a model) | moderate | conceptual argument by analogy with f = ma, plus two case studies in which researchers derived specialised models from it (§6) |
| C6 | It is not a tautology | weak | the argument that mathematical truths are not logical tautologies relies on Gödel/Putnam (§7.1, fn 17). It answers the word rather than the critics’ point, which Luque himself restates as “follows from notational definitions”. |
| C7 | The demand for predictive power or dynamic sufficiency is misplaced, because sketches are predictively empty and unification is an explanatory value in its own right | moderate | informal argument by analogy with Newton’s second law and Kitcher-style unification (§7.2) |
| C8 | Researchers “should adopt the Price equation … and this use will be rewarded” | weak | an assertion of the prescriptive aim (pp. 2, 23); no comparative evidence that Price-based modelling outperforms direct modelling |
Concepts
- generalization-sketch — after Kuhn (1970): a schematic law whose symbolic form changes from one application to the next, such as f = ma becoming mg = m d²s/dt² for free fall. Kuhn also called it a “law-sketch”; structuralists call it a “guiding principle” (fn 1).
- abstractness / unification / invariance — the three features Luque says the Price equation shares with other scientific principles. Invariance here means Frank’s invariance of total selection across combinations of frequency change and character value.
- consequence law vs source law — Sober (1984): a consequence law says how forces, once present, change a system; a source law says what produces the forces.
- fundamentalist / axiomatic theorizing — starting from postulates believed true and deriving a system’s mathematical rules, as opposed to idealised models built on simplifying assumptions (Cartwright 1999; Rice & Papadopoulos 2009).
- dynamic sufficiency — a model’s capacity to be iterated to predict any future state (Lewontin 1974). Luque, following Frank and Gardner, calls it a property of models, not of the equation.
- Type 1 / Type 2 questions — van Veelen (2005): top-down (from assumptions, deduce implications) vs bottom-up (from data, find the model that fits).
Connections
The paper builds on Frank’s (1995, 2012a, 2012b) and Rice’s (2004, 2008) pro-Price programme, on Gardner’s “meta-model” reading (2008), and on Okasha’s remark that the equation is “a schema that may be used to understand all other evolutionary models” (2010, p. 426). It argues against van Veelen (2005), van Veelen et al. (2012) and Nowak & Highfield (2011). Birch (2014) appears only for the two versions of Hamilton’s rule and the generality-for-predictive-power trade.
In the record, it pairs with the units-of-selection material: LIT-160 (Units and Levels of Selection, read) and LIT-201 (Alternative formulations of multilevel selection, still a seed). There the multilevel Price partition (Table 2, “multi-level Price equation (Okasha 2006)”) underwrites decomposing selection by level. Luque touches multilevel selection only in the survey (p. 12).
philosophy-of-science: the paper addresses the epistemic and semantic status of a fundamental equation. Is it an empirical law, a definition or a schema? Does unification count as explanation without prediction? How does a mathematical identity acquire empirical content? Luque’s positions are: a schema (a Kuhnian generalization-sketch and a Soberian consequence law); unification is an explanatory value in its own right (Kitcher); and empirical content comes from interpretation within a theory (Ramsey–Quine holism). The tag is justified and should lead.
Bearing on the record
It supplies a standard reference for the “just an identity” objection to general equations. The move it makes, granting that a general identity is predictively empty and locating its value in unification and model-construction, is the reusable content, and it transfers to any record discussion of fundamental equations in a science. No THEORY document in this record is sourced to it.
It carries nothing for ML practice. The Price equation has been used to analyse learning dynamics elsewhere, but this paper does not mention machine learning, and no instruction for the Anthology of the SOTA follows from it.
Limitations
- The tautology argument (§7.1) is the weakest part. Establishing that theorems are not logical tautologies does not answer the critics’ substantive point that the equation adds no empirical content, and Luque concedes that point in §7.2 anyway. The abstract’s “some recent disputes about it could be diverted” is fair; any stronger reading, that the disputes are resolved, is not supported.
- The invariance section borrows its only substantive result from Frank (2012a) and gives no derivation. The claim that this invariance is of a kind with physical symmetries rests on analogy.
- The prescriptive claim (researchers should adopt the equation) is asserted with no comparison against other routes to the same models. Luque concedes that his case-study authors “found previous results”, such as Lande 2007 and Gillespie 1974/1977, that were already derivable by other means (p. 27).
- Descriptive coverage stops at about 2015, and the survey is explicitly non-exhaustive (fn 8).
- The version read is the PhilSci manuscript, not the typeset article. Wording and pagination may differ.
Open questions
- Does the generalization-sketch reading survive the known non-uniqueness of Price partitions, such as eq. (2) vs (4) vs (5), and the dependence on how ancestor–descendant maps are drawn? Luque notes the partitions (§3) but not the problem this poses for the invariance claim. A treatment showing which partitions a given causal model licenses would settle it. Otsuka (2015), cited in passing, points that way.
- Is there a principled difference between the Price equation and f = ma, given that force has independent source laws (gravitation, Coulomb) while selection “forces” arguably do not? The paper gestures at Sober’s source/consequence distinction but does not say what the biological source laws are.
Corrections to the seeded skim
- The skim’s section map is wrong. The paper runs: §2 population genetics and diffusion theory; §3 derivation of the Price equation and its equivalent forms (Table 1); §4 abstractness and unification; §5 invariance (Frank’s information-theoretic reading of the covariance term); §6 the Price equation as a generalization-sketch, with case studies §6.1 genetic and non-genetic inheritance (Day & Bonduriansky 2011) and §6.2 stochastic evolution (Rice 2008; Rice & Papadopoulos 2009; Engen & Saether 2014); §7 critical views, split into §7.1 tautologies and mathematics and §7.2 predictive power; §8 conclusion. Kuhn’s generalization-sketch belongs to §6, not §4.
- Hamilton’s rule is not a test case, and there is no Hamilton’s-rule section. Birch’s two versions (a special version tied to simplifying assumptions, and a general version derived from the Price equation) appear only in footnote 13, p. 22. The “almost never holds” vs “always holds” framing is not in the text. The summary’s “defuses disputes such as the one over Hamilton’s rule” is therefore wrong. The disputes the paper addresses are van Veelen’s and Nowak’s: the charge that the equation is a tautology or mere identity, and the charges of dynamic insufficiency and lack of predictive power for “Type 1” (top-down) questions.
- The structuralist “theory-net” is not developed. Structuralists appear only in footnote 1, where Díez & Lorenzano’s term “guiding principle” is given as a synonym.
- The paper does not “vindicate the forces analogy in evolutionary theory against its critics”. It never engages the forces-analogy debate. The Newton parallel serves three narrower purposes: generalization-sketch status (§6), the history of disputes over whether F = ma is a definition (§7.1), and Sober’s (1984) source-law vs consequence-law distinction, under which the Price equation is a consequence law (p. 21). The “fundamentalist”/axiomatic approach is in §6 (Cartwright 1999; Rice & Papadopoulos 2009) and in the conclusion, not in a separate §§6–7.
- The skim missed that §7.1 argues the Price equation is not a tautology in the strict sense, on the grounds that mathematical truths are not logical tautologies (it cites Gödel and Putnam). It also missed that the paper states a prescriptive aim: researchers “should adopt the Price equation” (pp. 2, 23).
- Access has changed since the skim: the PhilSci-Archive PDF is now directly retrievable, and it is 36 pp., not the 22 pp. that the dossier reported for the SciSpace mirror copy.
- The dossier’s open question about invariance: the paper’s invariance is Frank’s (2012a) invariance of total selection across combinations of frequency changes and character values, which Luque treats as a property of the model’s mathematical form (pp. 15–16). He does not discuss how that depends on the choice of population or ancestor–descendant map. The only scope restriction he states is a closed population with no migration (p. 8), which he says Kerr & Godfrey-Smith (2009) lifted.
- philosophy-of-science tag: justified. The work is about the epistemic status of a scientific principle: law or definition, model or schema, unification as explanation, and predictive power.
- primary topic: philosophy-of-science (unchanged). natural-sciences is justified (evolutionary biology). mathematics is defensible under its “mathematical frameworks applied to other fields” clause but is the weakest of the three tags.