Robin Findlay Hendry,
Department
of Philosophy
University of Durham
50, Old Elvet
Durham DH1 3HN
In the philosophy of science, model building has been the subject of much discussion of late, enjoying the status of 'primary phenomenon' of applied physical theorising. There is broad agreement that models are extralinguistic mediators between theories and phenomena, but competing philosophical views of theories offer differing views of how theories and models are related. Any such view must be able to provide a plausible account of how theories and models are used in the context of explanation and prediction. It is the purpose of this paper to present a number of objections to one influential account of this relationship, the semantic view of theories. In Section 2, I will present an alternative, supported by some briefly-summarised examples of historical and contemporary theory-construction from physics and chemistry.
On what Suppe has called the 'received view' (see Suppe [1977]), a theory is construed as a system of axioms couched in a theoretical language. Hence theories are primarily linguistic structures. Uses of a theory to explain and predict are reconstructed as deductions from its central axioms. To the extent that such extralinguistic items as models are accorded any role in the construction or application of a theory, the authors of the received view presented them as heuristic devices. Following Hesse and Campbell, models were conceived of as emerging from analogical connections between different domains: in a famous example, a model based on an analogical connection with billiard balls might play a central role in the construction of equations governing the behaviour of gas molecules. In the classic statement of the received view, Nagel was willing to accord models of this kind central, and possibly even indispensable, heuristic and pragmatic roles in the context of theory construction. But models were conceived of as playing no part in individuating the content of completed theories, and therefore their explanatory and predictive power ([1979], pp.107-17). The shortcomings of the received view are very well known: problems with the central devices to which its standard formulations appealed-the distinction between observational and theoretical vocabularies, partial interpretation and correspondence rules-gave rise to intractable difficulties in giving an account of the meaning of theoretical terms, and therefore in the identification of theoretical content (see Suppe [1977], Chapter IV). Secondly, the received view obscured the fact that theories do not make contact with phenomena directly, but rather theoretical models are brought into contact with models of the data (see Suppes [1969] and Suppe [1977]). Critics of the received view traced its central defects to its identification of theories with linguistic structures.
On the semantic conception of theories, to present a theory is to present a family of models, with satisfaction of some set of equations being the criterion of family membership. Models are abstract extralinguistic structures that are not merely contingently associated with theories, but rather constitute the very stuff of which theories are made. A theory may well be associated with a particular linguistic formulation (that may take the form of a set of equations), but this should not be mistaken for the theory itself. Now it must be admitted that the semantic view allows for more perspicuous accounts of both the internal structure of theories (see for instance Giere [1988] and Lloyd [1988]), and of the 'external' relationship between theory and phenomena (see especially van Fraassen [1980]).
However, in a joint paper (see Hendry and Psillos, [forthcoming], Section 3), I have argued that in making the relation between model and equation a semantic one, the semantic view opens itself to two objections: (i) firstly, models are no more required to play the role of primary bearers of linguistic meaning than were propositions before them; (ii) secondly, in putting models in this position, the semantic view obscures the fact that models themselves are representational devices. Even if models (as abstract mathematical structures) do play an essential role in the individuation of a theory's content, it is a (category) mistake to infer that the models could constitute that content. The content of a physical theory is what it has to say about real-worldly physical systems. We use equations to say these things, and the semantic view is right in stressing models as a way of showing how equations convey their message. But the models themselves are a medium, not the message.
Turning to the 'external' relations, the semantic view is, according to its proponents, close to scientific practice because a theory is applied by constructing a model of the target system. However, Psillos and I have argued that even under the semantic view, theories must be construed as having linguistic components (see Hendry and Psillos, [forthcoming], Section 3). In addition, I have argued elsewhere that two different notions of 'model' are at work in the semantic views' account of the relationship between theory and data. On the one hand there is the kind of model that is familiar from mathematical logic: a structure that satisfies some set of axioms. On the other hand, there is the kind of model that represents some particular real item, much as a model of the Eiffel Tower does (see Hendry [1997]). Within the semantic view of theories, standard accounts of model construction, and of empirical adequacy, trade on the identification of the two different kinds of model: we apply a theory by selecting one of its models to represent some real system. This does not do justice to practice, however, for the scientist's model of a real system will often fail to satisfy the equations of the theories of which it is an application: nomologically disuniform models are common, models that fall under no one set of theoretical laws.
In conclusion, the semantic view presents theories as families of models, but has a monistic account of models themselves: models are the kinds of things that stand in logical relationships like satisfaction to such linguistic items as equations. As a result, it is difficult to see how, in the absence of a straightforward conflation of the relations of satisfaction and representation, models could stand in the kinds of representative relations to real-worldly systems that they must do, if a plausible account of theoretical representation is to be given.
The semantic view of theories presents the relationship between theories and models as a constitutive one, in the sense that, under the semantic view, theories are constituted by families of models. In a joint paper, I have argued that the relationship between theories and models is, instead, a historically contingent one, and also a complex one. Theories are historically contingent complexes of different representational media (see Hendry and Psillos, [forthcoming], Section 4), of which models are but one component. It is also easily forgotten that mathematics itself is a representational medium, albeit a linguistic one. Causal and theoretical claims can also be made using non-linguistic items like graphs and (real) models: some historical theories, like molecular structure in the nineteenth century, have grown up entirely around such representations. For the sake of continuity with this homely sense of 'model', models are conceived of primarily as representational devices, but models that are abstract objects may stand in logical relations (like that of satisfaction) to linguistic items. Which particular models are associated with a given theory is contingent, in the sense that they might have been different (see examples (i) and (ii), below). In addition, the relationship between theory and (representational) model is a complex one. The semantic view is right to contend that 'theoretical models' (i.e. models that by definition satisfy quantum mechanics) are indeed what a theory like quantum mechanics brings to the task of representing real systems. But theoretical models do not represent anything on their own, or do so only very rarely. They can, however, play a part in the construction of models that do represent real systems (see example (iii)).
(i) Bohr's Atom
In developing the first successful quantum-theoretic atomic model, Bohr imported equations that had previously been associated with classical-mechanical treatments of the solar system. A well-known mechanism for this kind of transfer, proposed by Mary Hesse, is investigated: the metaphorical redescription of the target system (in this case, the hydrogen atom and ionised helium), in the theoretical terms of the 'source' (i.e. solar) system, licenses the transfer of mathematical structure associated with the source system to the target system. It is argued that this mechanism accounts for key features of Bohr's progress, for one key difficulty experienced by Bohr in giving a mechanism for the production of atomic spectra within his model, and also for Bohr's own worries about the coherence and consistency of the model. The central feature of Hesse's mechanism of the transfer of mathematics, however, is that the representational power of imported equations is a function of their prior historical associations, in this case with the classical model of the solar system.
(ii) Wave mechanics and matrix mechanics
In 1926, Schrödinger, Eckart and Pauli produced proofs of an intimate theoretical relationship between the matrix mechanics of Heisenberg, Born and Jordan, and Schrödinger's own wave mechanics. This relationship has often been read as equivalence: here were two theories that in some sense 'said the same thing'. But this reading is plausible only on an impoverished view of theories. The equations associated with the two theories were at best shown by Schrödinger to be intertranslatable, but physical theories are not just sets of equations, so intertranslatability is not equivalence: Schrödinger's interpretation in particular suffused his presentation and development of wave mechanics. In wave mechanics he had consciously imported a second-order differential equation that in classical mechanics had been used to describe wave processes. This is no mere historical point: arguing from his intended wave interpretation, Schrödinger seems to have thought that wave mechanics and matrix mechanics might have become mathematically non-equivalent from the point of view of their future development. Hence the contingent historical associations of the equations with which Schrödinger formulated wave mechanics are required to distinguish it from its rival, matrix mechanics.
(iii) Quantum Chemistry
The relationship between a theory and the model that is the product of its application within a particular domain need not be a hierarchical one. Within quantum chemistry, the models of molecules within which quantum mechanics is applied cannot plausibly be presented as models of quantum mechanics, because they are nomologically disuniform. Take, for instance carbon dioxide (CO2), whose spectrum is very well understood: here is how one textbook of spectroscopy describes CO2:
The CO2 molecule is linear and contains three atoms; therefore it has four fundamental vibrations ... The symmetrical stretching vibration is inactive in the infrared since it produces no change in the dipole moment of the molecule. The bending vibrations ... are equivalent, and are the resolved components of bending motion oriented at any angle to the internuclear axis; they have the same frequency and are said to be doubly degenerate (Silverstein, Bassler and Morrill [1981], p.96).
The next step is to apply quantum mechanics. There are models in quantum mechanics for simple rotating bodies, and for simple oscillators: they are usually to be found in the chapter of the textbook on quantum mechanics after the chapter in which the Schrödinger equation was introduced. With some adjustments, the quantum-mechanical rigid rotator and harmonic oscillator allow us to quantise the rotational and vibrational motions that background chemical theory tells us the carbon dioxide molecule must exhibit. This provides the energy levels: differences between these energy levels correspond to spectral lines (in the infrared region in the case of CO2's vibrational modes). These explanations-and the models that ground them-do not seem to fit the accounts of application that are offered within the semantic view: motley collections of theory-fragments from different areas (classical molecular structure, idealised quantum-mechanical systems, statistical mechanics) pull together in the explanatory process. No-one possesses the Schrödinger equations for whole molecules that would be satisfied by this molecular model. If satisfaction of a Schrödinger equation be the criterion for a model being an application of quantum mechanics (as it would seem to be on the semantic conception), then the above model of carbon dioxide does not constitute an application of quantum mechanics. A more sophisticated account of theory application is required. (For an extended examination of models in quantum chemistry, see Hendry [1998].)
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