For the whole difficulty of [natural] philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces.
Newton Principia, “Preface”
Abstract
The analysis of theory-confirmation generally takes the deductive form: show that a theory in conjunction with physical data and auxiliary hypotheses yield a prediction about phenomena; verify the prediction; provide a quantitative measure of the degree of theory-confirmation this yields. The issue of confirmation for an entire framework (e.g., Newtonian mechanics en bloc, as opposed, say, to Newton’s theory of gravitation) either does not arise, or is dismissed in so far as frameworks are thought not to be the kind of thing that admits scientific confirmation. I argue that there is another form of scientific reasoning that has not received philosophical attention, what I call Newtonian abduction, that does provide confirmation for frameworks as a whole, and does so in two novel ways. (In particular, Newtonian abduction is not inference to the best explanation, but rather is closer to Peirce’s original idea of abduction.) I further argue that Newtonian abduction is at least as important a form of reasoning in science as standard deductive and inductive forms. The form is beautifully summed up by Maxwell (Proc Camb Philos Soc II:292–294, 1876): “The true method of physical reasoning is to begin with the phenomena and to deduce the forces from them by a direct application of the equations of motion.”
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Notes
Those who advocate some form of holism about knowledge will also hold that frameworks admit of confirmation. Quine, the canonical example of a holist, thought that frameworks admit empirical confirmation because he thought everything does, even pure mathematics. My reasons for claiming that frameworks admit confirmation differ from those of the holist: as will become clear in the paper’s argument, I claim they do only in so far as the representations they allow us to construct make substantive, direct contact with the physical systems they purport to treat. That is certainly not true of pure mathematics qua mathematics. It is also the case that some advocates of inference to the best explanation (IBE), such as Lipton (2004), claim that it allows for confirmation of frameworks. See footnote 42 below for a comparison of the ways that advocates of IBE claim that it supports confirmation of frameworks to the ways I argue Newtonian abduction does.
In order to try to avoid misunderstanding, I emphasize that I am not claiming that all forms of reasoning treated by systems, such as the Bayesian one, in which one tries to model confirmataory relations, are subsumed by the schema that I claim represents the “standard form of confirmation”. Philosophers have given strong arguments that the Bayesian framework, for example, can model the confirmatory relevance of forms of reasoning such as statistical inference, unification, explanation, and partial entailment. I still think, nonetheless, that the—admittedly gross—simplification of using the logical form of HD as the comparative foil for the structure of confirmation in Newtonian abduction—as I will do throughout the paper—is justified for my purposes, since all the other forms of reasoning that, e.g., Bayesianism treats (unification, explanation, partial entailment, and so on) have the same difficulty capturing the kinds of confirmation that Newtonian abduction endows frameworks with, what I call below structural and modal confirmation.
I will speak of “equations of motion” in this paper, but it should be understood that what I say applies also to field equations, and indeed to any mathematical relations and physical principles a framework or theory posits as governing and constraining the behavior of the types of physical system it treats. I will discuss examples of more general such types of relation and principle at the end of Sect. 3.
It seems to be the case that many if not most contemporary philosophers believe that Peirce’s notion of abduction is in fact IBE. That is wrong. An attentive reading of Peirce shows that his notion of abduction was not equivalent to contemporary conceptions of IBE, and that neither did he ever champion any such idea. Detailed exegesis to demonstrate this is beyond the scope of this paper. I will only invite the reader to read Peirce (1878a, 1903, 1955) with an unbiased eye, and note that nowhere does Peirce use the “goodness” of the hypothesis as grounds for formulating it based on the evidence. He certainly expected that one would, in the natural course of events, evaluate the goodness of the hypothesis once one had produced it by abduction, but the goodness of the hypothesis plays no role for him in its production by abductive reasoning. As I will argue, the same holds true for Newtonian abduction.
I discuss almost all the issues here in greater detail and to greater depth in Curiel (2019).
My conception of a framework is in many ways similar to, and indeed inspired by, Carnap’s conception of a linguistic framework (Carnap 1956), particularly in the way that a framework in both senses serves to define a fixed sense of physical possibility relevant to the kinds of system the framework treats. Carnap’s conception is too broad and vague, however, to do the work I require of it. Stein (1992) provides an insightful and illuminating, albeit brief, discussion of the differences between a Carnapian framework in the original sense and a framework in the sense of a structure in theoretical physics of the type I am sketching here, though he restricts attention to the strictly mathematical parts of one, as I do not. Lakatos (1970) has some affinity with the gist of this view, in particular his notion of the “hard core” of “research programs”, though, again, the differences in detail outweigh the similarities. There is perhaps more affinity with the “research traditions” of Laudan (1977), in so far as different ones can share and swap important methodological and theoretical principles, as can happen with frameworks in my sense. A discussion of these comparisons is beyond the scope of this paper.
It is also characteristic of an appropriately unified kind of physical system, one treated by a theory in my sense, that there exist a set of scales at each of which all quantities the theory attributes to the kind of system simultaneously lose definition—the breakdown scales—as does not happen with the entirety of the family of all types of physical system treated by a framework. In other words, every theory has a single, unified regime of applicability, bounded on all sides by scales characterized by the values of different combinations of its physical quantities. For classical fluids, for example, the definitions of their pressure, fluid flow, viscosity, and all the rest break down at spatial and temporal scales a few orders of magnitude greater than those of the mean free-path of the fluid’s constituent particles. There is no a priori reason why the definitions of all the different physical quantities represented by the theory should fail at the same characteristic scales, even though, in fact, those of all known theories do, not only for classical fluids but for all physical theories we have. This seems, indeed, to be one of the markers of a physical theory, the existence of a set of characteristic scales for its physical quantities, at each of which all the theory’s physical quantities simultaneously lose definition. This is a fact that deserves philosophical investigation.
See Landau and Lifschitz (1975, ch. v, §49) for an exposition of the physics of Navier–Stokes theory, and Curiel (2017c) for a deeper and more extensive discussion of Navier–Stokes theory with regard to the issues I discuss here. There is a subtlety I am glossing over: Navier–Stokes theory requires partial-differential equations for its formulation, which were not a part of the original Newtonian framework. Strictly speaking, therefore, Navier–Stokes theory is formulated in an appropriate extension of the original Newtonian framework. How frameworks can be extended in such ways is a fascinating problem, but one I cannot discuss here.
I argue elsewhere for the claim that general relativity is a framework in this sense, and not a theory. Nothing in the paper hinges on the claim, so if you object to it, let it go.
One can as well consider mixed systems, with, say, a fixed value for mass but indeterminate value for Hooke’s constant. These raise interesting questions, but they are beside the point here.
I have deliberately taken my terminology from biological taxonomy, inspired by the remark of Peirce (1878b, p. 143):
Now, the naturalists are the great builders of conceptions; there is no other branch of science where so much of this work is done as in theirs; and we must, to great measure, take them for our teachers in this important part of logic.
There is much of insightful relevance in the lead-up to this remark, about how one individuates and characterizes genera and species of physical systems in my sense, which it would be illuminating to discuss, but it would take us too far afield. I am tempted to describe structure at the level of a framework as phylar, and to call the family of all types of physical system treated by a framework a phylum—and so all physical systems would fall under the kingdom of physics—but I suspect it would just be distracting to the reader.
This idea bears obvious and interesting comparison with the distinction between data and phenomena as drawn by Bogen and Woodward (1988). In so far as I understand their distinction, their idea of data more or less corresponds with my idea of “experimentally or observationally gathered results”, but their notion of phenomena, in so far as it seems to try to capture something like general patterns in the world, does not neatly square with my conception of a concrete model, which is the result of appropriately transforming the results of a single experiment (or family of related experiments).
I discuss many of them in detail in Curiel (2019).
There are other types of Newtonian abduction, involving for example the derivation of specific equations of motion from generic equations of motion and concrete models. One can, for instance, determine the ratio of Hooke’s constant to the mass of a spring from a concrete model in conjunction with the generic equations of motion for a simple harmonic oscillator. This is a simple-minded example of what Harper (2011) and Smith (2014) call theory-mediated measurement. More substantive examples are the determination of the relative masses of the planets in Newton’s derivation of universal gravity and the determination of \(\omega \) in Brans-Dicke theory by Shapiro time-delay (Harper 2011). In these cases, the logical form of the reasoning is the same as in Newtonian abduction. The difference is in the contents of the terms in the formula: the antecedent is the theory (e.g., Newtonian gravitational theory), not the framework; the lefthand side of the biconditional is still the concrete model; but the righthand side are the values of the parameters being determined. I discuss this further in Sect. 4. One can also consider a weaker form of Newtonian abduction: not a biconditional with a single theory (a single set of generic equations of motion), but rather a biconditional with a family of related theories. Later in this section, we will see an example of this in the discussion of Newton’s framework for the investigation of light and color.
I do not claim that all non-abductive forms of deductive reasoning and other similar forms of inference in science are HD; because details of difference in their form do not matter for my purposes, I ignore them.
The Area Law states that a planet in its orbit around the sun sweeps out equal areas in equal times, the area swept out being that of the region through which the line from the planet to the sun moves. (In modern terms, this is equivalent to the conservation of angular momentum.) The Harmonic Law states that, given the elliptical form of the orbits, the ratio of the square of the orbital period to the cube of the semi-major axis is the same for all planets.
The analysis of his arguments up to this step captures the heart of what is known in the literature as Newton’s “deduction from the phenomena” (Harper 1990; Worrall 2000). It is in this sense that the standard account of deduction from the phenomena captures part of Newtonian abduction. As subsequent discussion will make clear, however, the standard account does not capture the full logical structure of Newtonian abduction, and thus cannot support the confirmatory weight I attribute to it.
There are several subtleties of the derivation I gloss over, such as his use of Corollaries vi and vii to Proposition iv in Book i, which are propositions for concentric circles, not ellipses as he knew the planetary orbits to be. The use of those propositions was justified because, when a planet in an elliptical orbit is at a distance equal to the semi-major axis (i.e., when it is 90 or 180 degrees from aphelion), then its centripetal acceleration exactly equals that of a planet in a uniform motion circular orbit of that radius and having the same period, and that suffices for the use of the corollaries. See Stein (1990) and Stein (1994, p. 639ff.) for more detailed discussion of the logical structure of Newton’s reasoning, and exposition and explanation of those sorts of subtleties.
It is important to note that, strictly speaking, Newton’s initial “deduction” does not work, because, as Newton well knew, the concrete models he relied on did not exactly, only approximately, instantiate the Keplerian relations. Here is where what Harper (2011) calls Newton’s method of successive approximations comes into play. In effect, Newton decided to assert the validity of the simple abductive proposition and then push it as far as he could by successive refinements of the approximative models, each one markedly improving on the previous. (See Stein 1967, pp. 177–180 for a concise and lucid discussion of Newton’s guiding methodological principle here and the way he applied it.) It is worth remarking as well that HD cannot accommodate this method, as it can do nothing but strike the hypothesis from the record of viable candidates at the first glimpse of such inconsistency with the data.
It would be an interesting project to characterize the necessary and sufficient conditions on the structures (topological, algebraic, geometric, etc.) a framework imposes on its family of theories required for the framework to support Newtonian abduction.
On this view, the entirety of a framework is a dynamic entity, as are individual theories, evolving over time as new theoretical and experimental techniques and practices are developed and accepted, and so theories themselves will be as well. I think this is the right way to think about these matters for many if not most purposes in those parts of philosophy of science studying scientific theories. The contemporary practice of treating theories as static, fixed entities, especially in work of a more technical and formal character, can lead to serious philosophical error. An adequate semantics of a theory, for instance, should reflect and accommodate its dynamic nature.
That Newton uses the term ‘general induction’ in Rule iv has no bearing on my arguments. The word ‘abduction’ did not exist then, and, in any event, as is made clear by Newton’s gloss on the rule following its statement, Newton is using ‘general induction’ to refer, among other things, to the pattern of reasoning he employs in deriving his theory of universal gravity, which I claim is in fact Newtonian abduction.
More precisely, he used Proposition lxvi and its corollaries (Book i), which are themselves derived from the Precession Theorem. The content of the Precession Theorem itself carries the burden of the argument.
In fact, this is not true for Mercury, as Le Verrier well knew. There was an extra 39 arcseconds (\(39''\)) per century of precession that his calculations could not account for. He labored for the next 14 years to produce a mechanism to explain the discrepancy, even postulating hitherto unobserved celestial bodies and other such ad hoc devices, but nothing worked (Le Verrier 1859). Indeed, by the end of the 19th Century the inexplicability of the aberrant precession was such a great embarrassment that many eminent physicists had already concluded that Newtonian gravitational theory could not be fundamentally correct, even before the development of special relativity (a historical fact that seems to be not so well known as it ought), based entirely on abductive use of the Precession Theorem. See Newcomb (1895a, b, 1905) for an extended discussion and summation of the experimental knowledge of the aberrant precession at that time, when the anomalous amount of Mercury’s precession was finally fixed at \(43''\) per century, and see Freundlich (1915) for an exhaustive argument that Newtonian gravitational theory could not account for it. To get a sense of how small the angle \(43''\) is, imagine the appearance of the diameter of a penny from a distance of about 30 miles. The apparent length of its diameter on the eye, projected back to the penny, subtends an angle of that size. It is a testament to the profound confirmatory entrenchment of Newtonian gravitational theory in particular at the time, and Newtonian mechanics in general, that a discrepancy of this infinitesimal angle per century in a planetary orbit caused such consternation in and provoked such labor from the leading lights of the scientific community for more than 70 years. Of course, we now know that the error arises from general relativistic effects, and cannot be accommodated by Newtonian gravitational theory.
Indeed, Newton had even more instances of such reasoning than only that based on the Precession Theorem. According to Corollary vii to Proposition iv, Book i, the Harmonic Law also provides support for the derivation of such subjunctive conditionals. Over and above the biconditional between the Harmonic Law and the inverse-square form of the force law, the corollary shows that the periods of the planetary orbits are proportional to a power greater than the Harmonic Law’s 3/2 power of the semi-major axes if and only if the centripetal forces fall off more rapidly than the inverse square law, and contrarily that the periods are proportional to a power less than 3/2 of the semi-major axes if and only if the centripetal forces fall off more slowly than the inverse squared law. See Harper (1999, 2011, pp. 114–120) for discussion.
A good example of this from contemporary physics is the parametrized post-Einsteinian framework of Yunes and Pretorius (2009), which manifestly supports subjunctive reasoning of this form, with the explicit intent of sharpening the recent observation of gravitational waves by LIGO (Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration) 2016) as tests of general relativity, by providing a framework within which one can probe for deviations from general relativity’s predictions in a parametrized, controllable form: the dynamics of the observed coalescence of the binary black hole system deviates from general relativity’s models in this way, by this parametrized amount, if and only if the observed gravitational waveform exhibits this quantifiable feature.
For an exposition of Newton’s framework for light and color, and the investigations and abductive reasoning that led him to it, see Curiel (2001) (though I do not refer to the form of reasoning as Newtonian abduction in that paper), and for a more extensive and deeper discussion of those investigations, with a direct bearing on the relevant issues, see Stein (Unpublished(a), Unpublished(b)).
Due to limitations of space, I cannot give a detailed argument that Maxwell’s reasoning for his equations governing the electromagnetic field is abductive. I will note here only that, in Maxwell (1856), his derivation of the equations, including the necessity of the novel term representing the so-called displacement current, at bottom takes the form of a biconditional between, on the one hand, the electromagnetic phenomena observed and regimented by, inter alia, Œrsted, Ampère, and (most of all) Faraday, and, on the other hand, what we now call Maxwell’s equations, where the biconditional is implied by a Newtonian framework comprising the theory of a particular kind of Newtonian fluid. His final, complete derivation in Maxwell (1864) also has this logical form, though the biconditional’s antecedent is a Newtonian framework comprising a completely abstract theoretical representation of a medium whose dynamical behavior is governed by a Newtonian form of elasticity. In Maxwell (1891), he abductively derives the equations in the framework of Lagrangian mechanics. I emphasize that these claims are crude and naive in the extreme, requiring detailed historical and technical exegesis for their complete elucidation and defense. Nonetheless, I also claim that they capture the heart of the matter. In the same vein, it is worth considering the argument of Hertz (1884) concerning Helmholtz’s formulation of electromagnetism that he intended to be neutral between Maxwell’s theory and Weber’s action-at-a-distance theory (in other words, Helmholtz’s formulation of a framework subsuming both). Hertz argued, in effect abductively, that Helmholtz’s framework is in fact inconsistent with Maxwell’s theory, in so far as it cannot represent the existence of free electromagnetic radiation, which itself abductively favors Maxwell’s theory. Again, a detailed defense of this claim is beyond the scope of this paper.
It is a little delicate to explain the way in which a family of spacetime models such as the FLRW or Schwarzschild ones is relevantly like a set of generic equations of motion, but one can do this, and when one reconstructs the reasoning involved, say, in proving Birkhoff’s Theorem (which implies the uniqueness of Schwarzschild spacetime given the assumed symmetries), it is indeed abductive in form. If you object to the claim that general relativity is a framework, then think of this as the abduction of specific equations of motion from concrete models and generic equations of motion (general relativity considered as a theory, not a framework).
For a graphic illustration of the method, see Curiel (2014, §4), in particular the discussion of how one can read the generalized forces off from the form of the second-order vector fields on the tangent bundle of the configuration space for a given genus of physical system, those second-order vector fields representing the allowed dynamical evolutions of the system, i.e., its individual models.
One can therefore understand Newtonian abduction as providing grounds for the position of the “sophisticated instrumentalist” as characterized by Stein (1989). (I thank Tom Pashby for pointing this out to me.) This does not, however, imply that Newtonian abduction by itself militates in favor of instrumentalism, nor realism either for that matter.
It is important to note that no concrete model on its own has any of these structures; only rich enough families of them exhibit the structures as relations among them.
After I finished this manuscript, I discovered that Kuipers (2001, pp. 208–209) uses ‘structural confirmation’ to refer to confirmation derived from instantiation of another relation, that of partial entailment, i.e., the probabilistic degree to which \(A \vee B\) entails A. The two should not be conflated.
Of course, everything said of structural confirmation of a framework holds as well, mutatis mutandis, for a theory. Theories also are amenable to structural confirmation, in the same way, by abductively showing that the structures intrinsic to the theory are appropriate and adequate for representing and reasoning about the concrete models of the different species of physical system the theory purports to treat.
Indeed, structural evidence of the kind relevant to abduction in general differs in important ways from “raven counting” evidence and from HD predictive evidence, primarily because of the non-trivial mathematical relations among theories one must account for in formulating and analyzing the evidence. A general discussion is beyond the scope of this paper.
To get a sense of the difficulty and depth of the technical problems one faces, consider the family of all possible “theories” one can construct in the Newtonian framework, i.e., all (say) twice-differentiable functions (force laws) on Newtonian spacetime. This can not even be turned into a topological manifold, since the different force laws can depend on any number of different variables (position, velocity, fluid flow, shear-stress, and so on), and so are functions with infinitely many different possible domains. Restrict attention, therefore, to the family of all possible theories for a fixed set of physical quantities (say, those appearing in Navier–Stokes theory). This, then, forms an infinite dimensional vector space, modeled on the field of real numbers, which can be turned into a Banach space by choice of a norm; there is, however, no obvious, unique, physically significant choice of norm, though many are available (e.g., the \(\sup \)-norm, and so on). There is as well no obvious, unique, physically significant topology to put on the space, though many topologies are available (compact-open, Whitney, one based on the \(\sup \)-norm, and so on). One now wants to put something like a probability measure on it that respects the topology, in the sense that small perturbations don’t drastically change the measure of open sets, so one wants a Borel measure that is, if not invariant under linear translations, then uniformly bounded in some way by the size of the translation. There is, strictly speaking, no probability measure on such an infinite-dimensional space, and no translation-invariant Borel measure on it at all (Curiel 2017b). It is an open—and difficult—question whether one can construct such a measure that is uniformly bounded under translations. Even if one were to construct such a measure, it would necessarily assign infinite size to the space as a whole, and so one would need some sort of regularization scheme to extract meaningful probabilities for non-trivial open sets. See Curiel (2017b) for a discussion of how to try to do this in the context of defining physically significant probability measures in cosmology, and the many further problems that arise.
One may be able to get the technical machinery under control if one can “localize” the family of laws to an appropriately restricted subfamily. Consider the case of Newton’s and Le Verrier’s use of the Precession Theorem to support the inverse-square form of Newton’s gravitational law. The “state space” of laws now consists of functions of the form \(\displaystyle \frac{1}{r^{2 + \epsilon }}\), for small \(\epsilon \ge 0\) and, say, \(\le .5\). This naturally forms a compact 1-dimensional manifold with boundary, with a natural, physically significant metric function on it (the absolute difference in \(\epsilon \)). There is thus an obvious, physically significant topology to put on the space, based on the metric. The absolute difference in \(\epsilon \) gives the space a locally affine structure: if \(\displaystyle \frac{1}{r^{2 + \epsilon _1}}\) and \(\displaystyle \frac{1}{r^{2 + \epsilon _2}}\) are in the space (\(\epsilon _1 < \epsilon _2\)), then so is \(\frac{1}{r^{2 + \epsilon '}}\) for all \(\epsilon ' = \lambda \epsilon _1 + (1 - \lambda ) \epsilon _2\), for \(\lambda \in [0, \, 1]\). One now wants to put a probability measure on it that respects the topology and the affine structure, in the sense that small perturbations don’t drastically change the measure of open sets. One therefore wants a Borel measure that is invariant under affine translations, or at least uniformly bounded in some way by the size of the translation. There are many such measures.
I do not claim that standard forms of HD reasoning, and standard accounts of confirmation based on them, cannot be modified or extended to cover the form of reasoning I discuss here. I claim only that no account I know of in its present form has the capacity to do so. It would be an interesting project to attempt to modify or extend extant accounts to try to do so.
I am glossing over a subtlety here. The concrete models one compares to the individual models of each framework will not always be the same, but will rather sometimes have to be constructed from the raw data in different ways using the different theoretical concepts and structures of the different frameworks, so as to fit into the different abductive propositions of each framework. That is not in general a problem. In such cases, the individual models of one can be “translated” into those of the other in a way that preserves enough physical significance for one to have confidence that the individual models in the different theories represent the same dynamical behaviors of the same physical systems. Such translations often take the form of limiting or approximative constructions. For example, Malament (1986), in effect, shows how to do this for general relativity and Newtonian gravitational theory. If one cannot construct such translations, then one has no reason to believe that the two theories are representing the same behaviors of the same physical systems. Nonetheless, it is sometimes the case that the same concrete models can in fact be identified with individual models of different frameworks, as in the case of Keplerized planetary orbits and individual models in general relativity (Schwarzschild spacetimes with test particles traversing geodesics in a 3\(+\)1 representation of spacetime) and Newtonian mechanics (two-body solutions of Newtonian gravitational theory).
I believe that Newton’s infamous proclamation, “Hypotheses non fingo,” is best understood in the context of his characteristically abductive form of reasoning: whatever is not derived abductively from the phenomena is a hypothesis, and those he avoids as much as possible, precisely because, being merely postulated and then tested by weaker forms of reasoning (such as HD or induction), they cannot accrue the same degree of certainty as propositions derived by abduction.
Works such as Lipton (2004, ch. 7) and Henderson (2014) are not relevant to my arguments, as their discussions of abduction treat it as IBE. Lipton does talk about the capacity for IBE to confirm entire frameworks (e.g., on p. 60, in his discussion of Newtonian mechanics and special relativity). His reasons for saying this, however, are based on theoretical—non-empirical—virtues, such as unification, scope and explanatory power, that he claims accrue to frameworks based on their role in IBE. As should be clear, my arguments that Newtonian abduction can confirm frameworks derive directly from the logical form of the reasoning itself in its immediate contact with empirical evidence.
I want to emphasize again that none of this has anything to do with any issue pertaining to realism and anti-realism. There is no claim made or needed that the structure manifest in the phenomena, i.e., the structured data, is “really” part of the furniture of the world, in some deep metaphysical sense. That it is manifest in the phenomena, in the sense that one can identify the structured data the phenomena yields with individual models, suffices for the soundness of the evidential and confirmatory relations at issue. Those relations are agnostic about realism, as, again, any good confirmatory relation should be.
Stein (1967) provides a compelling confutation of this claim, by showing that, in Newton’s development of his abstract framework of dynamics, his metaphysics of space, time and motion, and his theory of universal gravitation, much conceptual and theoretical clarification—learning—was required before the empirical data could be properly comprehended at all, much less used as the basis for the construction of a fully fledged theory.
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I thank Bill Harper for many enjoyable and edifying discussions about Newton, scientific reasoning, and evidence, and for having written such a wonderful book (Harper 2011). I also thank him for detailed comments on an earlier draft of the paper, including catching an error in §3 in my discussion of Hall, Brown, and the precessions of Mercury and the Moon. I thank Tom Pashby for comments on an earlier draft, including interesting suggestions for elaboration in future work. I thank an anonymous referee for detailed criticisms and questions. I am grateful to Howard Stein as well, as always, for many illuminating and pleasurable conversations on all of these matters. Work for this paper was funded by Grant CU 338/1-1 from the Deutsche Forschungsgemeinschaft.
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Curiel, E. Framework confirmation by Newtonian abduction. Synthese 198 (Suppl 16), 3813–3851 (2021). https://doi.org/10.1007/s11229-019-02400-9
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DOI: https://doi.org/10.1007/s11229-019-02400-9