European Journal for Philosophy of Science

, Volume 2, Issue 3, pp 249–274

Defending the Semantic View: what it takes

Authors

    • Department of Philosophy LA 101University of Montana
Original paper in Philosophy of Science

DOI: 10.1007/s13194-011-0026-6

Cite this article as:
Le Bihan, S. Euro Jnl Phil Sci (2012) 2: 249. doi:10.1007/s13194-011-0026-6

Abstract

In this paper, a modest version of the Semantic View is motivated as both tenable and potentially fruitful for philosophy of science. An analysis is proposed in which the Semantic View is characterized by three main claims. For each of these claims, a distinction is made between stronger and more modest interpretations. It is argued that the criticisms recently leveled against the Semantic View hold only under the stronger interpretations of these claims. However, if one only commits to the modest interpretation for all the claims, then the view obtained, the Modest Semantic View, is tenable and fruitful for the philosophy of science.

Keywords

ModelsSemantic ViewScientific theoriesScientific modelsStructureStructuralism

1 Introduction

From the seventies to the late eighties, the Semantic View had seemed to become the new orthodoxy in philosophy of science.1 Over the last two decades however, a controversy has arisen over the consistency and usefulness of the Semantic View through an abundant literature, some of which is heavily critical of the program2 and some of which is energetically defending it.3 The main aim of this paper is to articulate a version of the Semantic View that is not touched by the criticisms recently leveled in the literature and, albeit being limited in scope, is also potentially fruitful for the philosophy of science.

As a starting point, let us consider that the Semantic View is constituted by the three following claims:4
  1. 1.

    (Models)—Scientific theories can be presented through the class of scientific models that scientists typically use to represent the domains of their theories;

     
  2. 2.

    (Scientific=Logical)—Scientific models can be construed as logical models;

     
  3. 3.

    (→ Adequacy)—Considering scientific theories from the point of view of their models and construing scientific models as logical models provides the means to give an adequate account of what scientists typically use to represent the world in actual practice.

     

Each of these three claims have been criticized in the literature. They are analyzed along with their criticisms in turn in the following sections. For each claim, a distinction is made between stronger and more modest interpretations. The motivation for the most modest interpretation is given. Further, it is shown that the criticisms recently leveled in the literature rely on the stronger interpretations. It is also argued that both advocates and critics of the Semantic View failed to recognize the distinction between the two interpretations. Once the distinction is recognized, and the stronger interpretations are rejected, the view obtained, which is called here the Modest Semantic View, is tenable.

The Modest Semantic View is defended as a methodological prescription to use model theory as a tool for the rigorous analysis of the structure of what scientists typically use to represent the world in actual practice. Just as any tool, the tool offered, as powerful as it may be, is efficient only within its proper domain of application. We argue that the Semantic View is well equipped for the analysis only of the structural—but not of the functional or pragmatic—features of what scientists typically use to represent the world.

We conclude by emphasizing to what extent the Modest Semantic View can be useful for philosophers of science, but is limited in scope. In particular, an important consequence of our analysis is that the Modest Semantic View cannot help to defend either a realist, or an anti-realist, stance towards science. That said, the advantage given by the latter feature to the Modest Semantic View is that it should prove useful to all sides of the realism debate, including the structural realists, the constructive realists, the constructive empiricists, and the “toolbox instrumentalists”.5 But let us start with an analysis of the first of the three main claims of the view: (Models).

2 (Models): theories, models and scientific practice

In this section, we begin by motivating and presenting (Models) in its modest interpretation. We then turn to the criticisms that have been leveled against (Models). We argue that these criticisms rely on a stronger interpretation of (Models). We conclude that only the Modest Semantic View, explicitly committed to the modest interpretation of (Models), remains tenable.

Central to the Semantic View is the idea that philosophers of science should stop studying scientific theories from the point of view of their axiomatic structure—as the “received view”6 is taken to recommend, and to focus on scientific models instead. One can justify this recommendation in the following way.

First, one urges philosophers of science to stop studying scientific theories from the point of view of axioms because such a method results in philosophers of science neglecting a large portion of well-recognized scientific practice. Granted, for those theories that are axiomatizable, working on possible axiomatizations can yield important insights into the structure of these theories. That said, the domain of philosophy of science cannot be restricted to axiomatizable theories, simply because most of the scientific theories that are used in scientific practice are not axiomatizable in any interesting sense.7 Neuroscientific, paleontological, or biological theories, for example, cannot be formulated through nice sets of axioms. However, such theories are well-accepted and widely used in actual scientific practice. As such, they are part of the proper domain of philosophy of science. That they are theories that are not fruitfully axiomatizable should not lead one to conclude that these theories are not scientific. Instead, it should be taken as a proof that the received view is lacking.

As a remedy to the problem above, it is proposed to focus on scientific models. An important motivation for this is the factual observation that scientists typically use scientific models to represent the world.8 In Newtonian physics for example, one can use the two gravitational body system with point masses to represent the Earth orbiting the Sun, and a block moving down an inclined plane to represent two kids sledding in the winter. Note that scientific models are not only used for mature physical theories. Models are used throughout the scientific realm, including for example the computational model of the brain in neuroscience, the Lotka–Volterra models of predator–prey interactions, or the well-known double helix model of DNA. In short, the claim is that, by studying scientific models, one in fact studies what scientists typically use to represent the world in actual practice. This is one of the main motivations for (Models), the first of the three claims which together constitute the Semantic View:

(Models)—Scientific theories can be presented through the class of scientific models that scientists typically use to represent the domains of their theories;

In its most modest interpretation, (Models) is based on the following partial, descriptive and rather uncontroversial statement about scientific practice: an important part of what scientists do with scientific theories is to use models to represent the world. The idea is then that focusing on these models provides a good standpoint for analyzing one typical way in which these theories are used in actual practice. A good way to begin such an analysis is thus to obtain the class of scientific models that are typically used to represent a given domain directly from the scientists working in the field.9 In its most modest interpretation, (Models) is then the methodological prescription to take these models as the object of study:

(Models)Modest—Scientific theories and scientific practice can be partially characterized by the scientific models that scientists typically use to represent the world.

It is crucial that the description of theories by (Models) is partial, and that it gives a methodological prescription for the analysis—rather than a complete definition—of scientific theories. By contrast, critics of the Semantic View attack (Models) under a much stronger interpretation, one according to which (Models) provides a complete view of scientific theories.10

Let us take two examples. First, let us consider Margaret Morrison, who is arguably one of the leading figures in the recent literature against the Semantic View. In her (2007), she criticizes the Semantic View for failing to give an adequate account of the role theories play in science along the following lines (Morrison 2007, p. 202):

If a theory is just a family of models (emphasis added), then what does it mean to say that the model/ structure is a realization of the theory?

The model is not a realization of the theory because there is no theory, strictly speaking, for it to be a realization of. In other words, the semantic view has effectively dispensed with theories altogether by redefining (emphasis added) them in terms of models.

As the quote shows, one of the main premises of Morrison’s argument is that, within the semantic view, “the theory is simply defined or identified to its models” (p. 201). Tim Maudlin (2007), our second example, seems to make a similar assumption when addressing van Fraassen’s objection to the nomological theory of explanation—i.e that laws are not primitives but derived from symmetries. Maudlin notices that van Fraassen relies on the Semantic View to bypass laws when accounting for scientific theories. Maudlin then investigates whether the Semantic View could provide the right kind of tool for a non-nomological theory of scientific explanation. According to Maudlin, one option would be to say that (Maudlin 2007, p. 38):

A theory explains a fact in the actual world only if that fact obtains in all (or an appropriately defined most) of its models.

Maudlin however concludes that this will not do, with the following argument (p. 39):

Consider the theory that results if we begin with our present theories of physics, chemistry, etc. but exclude all the models which do not contain living beings. This is a perfectly good class of models, and hence a theory (emphasis added). In this theory, the existence of life in the universe is a ‘law’. Hence the existence of life in the universe is explained as the consequence of a ‘law’...And clearly the same trick can be played for any known fact: simply restrict the class of models from the old theory to those in which the fact holds, and thereby produce a theory that explains it. This does not describe actual practice.

Maudlin’s main assumption in this argument is clear: “we must take seriously the idea that the theory is just the class of models” (p. 39).

Both Maudlin and Morrison thus base their criticisms of the Semantic View (as failing to give an account either of scientific explanation or of the cognitive role of theories) on a strong interpretation of (Models), which takes (Models) as including a complete view of scientific theories, i.e. the view that scientific theories are simply sets of models. Is such a strong interpretation fair?

While one could argue that the modest interpretation is the interpretation that most fits the spirit of the view, one must admit that advocates of the Semantic View are not explicit about whether or not they take (Models) as including a complete view of scientific theories. Let us take a couple of examples. Van Fraassen can be read as favoring the modest interpretation of (Models) in his (1980, p.64):

To present a theory is to specify a family of structures, its models...

To see this, one should pay attention to the fact that he does not claim that theories are either defined by, identifiable with, even less reducible to, sets of models. All that he claims is that theories can be presented through their sets of models. That said, the ambiguity reappears in his (2008, p. 309):

On the semantic view, a theory offers us a large range of models—in fact, while a theory may have many different formulations, its sets of models is what is important.

Similarly, while one could argue that for Giere, a theory does not reduce to a set of models, given that it also includes as an essential part a “theoretical hypothesis” by which the models of the theory are related to the world (1988, p. 65), it is also clear that Giere does not recognize the importance of rejecting the stronger interpretation of (Models) as is shown in his (1988, p. 48):

This makes possible to identify a theory [...] with the set of models that would be picked out by all the different possible formulations.

The claim is not that advocates of the Semantic View adopt the stronger interpretation of (Models), but that they failed to make the distinction between the two interpretations explicit, and to recognize the crucial importance of rejecting the stronger one. In doing so, their views fall prey to the criticisms of the kind leveled by, among others, Morrison or Maudlin. The first element of the Modest Semantic View which is here advocated is to explicitly reject the stronger interpretation of (Models) in favor of the more modest one.11

Within the Modest Semantic View, (Models) is not interpreted as providing an exhaustive account of scientific theories. The Modest Semantic View commits only to the partial description of scientific practice that an important part of the use of scientific theories in actual practice and across almost all domains of science consists in using scientific models to represent the world. (Models) consists then in the methodological prescription to consider scientific theories from the point of view of these models. Only the modest interpretation of (Models), and hence the Modest Semantic View, remain untouched by criticisms of the type of Morrison and Maudlin above.12

3 (Scientific=Logical), the structural point of view

In this section, we begin by motivating and explaining (Scientific=Logical) in its most modest interpretation. We then show that the criticisms leveled against (Scientific=Logical) rely on a stronger interpretation. Here as in the case of (Models), it is noted that most advocates of the Semantic View have missed the importance of the distinction. In the light of our analysis, it appears that it is crucial to make the distinction between the two interpretations explicit and to reject the stronger one. Only the Modest Semantic View, which is explicitly committed to the most modest interpretation of (Scientific=Logical) is tenable.

In the preceding section, we emphasized that, within the Semantic View, one considers a scientific theory from the point of view of the models that scientists typically use to represent the domain of the theory in actual practice. Now that the semanticist has these models in hand, what does she propose to do with them? Here enters another crucial aspect of the Semantic View, i.e. the commitment toward formal methods. A fundamental intuition behind the Semantic View is that formal methods are fruitful for the systematic treatment of issues in philosophy of science and scientific methodology.

Note that it is an intuition that the advocates of the Semantic View share with the advocates of the received view. We take it that the Semantic View does not prescribe a change in the kind of analysis that is deemed useful in philosophy of science, but a change in tools for this analysis. The received view was perceived to be restricted to reconstruct scientific theories using only first order logic plus identity (see Suppe 1977a, pp. 16, 50), and it was considered that this was just too poor a tool to give an account of the complexities of scientific theories.13 The upshot is that the received view could not consider anything but oversimplified theories. Adding the axioms of (a fairly uncontroversial version of) set theory is just what is needed to be able to give an account of most important scientific results, because it includes most of mathematics.14 In short, model theory is advocated as the correct formal framework for the systematic treatment of issues in the philosophy of science.

Central to model theory is the notion of logical model, that is, the Tarskian notion of model (see Löwenheim (1879), Tarski (1933), or Gamut (1990) for a rigorous presentation). A logical model is a set-theoretical structure that gives an interpretation to the possible sets of sentences in which a theory is formulated so that the sentences are true on the interpretation.15 For example, consider Euclidean geometry, which can be formulated as a set of axioms. In these axioms, the terms ‘point’ and ‘line’ appear. A triangle on a plane is a structure which provides an interpretation of such terms. Within this structure, all the axioms of Euclidean geometry are true.16 Hence, the triangle on a plane is a model of Euclidean geometry. By contrast, a triangle on the surface of a sphere is not a model of Euclidean geometry, because it is a structure which gives an interpretation to the axioms of Euclidean geometry such that not all of these axioms are true within this interpretation. In short, logical models are logical structures which are essentially characterized by their truth-making function. Within the Semantic View, it is thus considered that the notion of logical model, just as the other resources of model theory, can be helpful for the systematic analysis of scientific models.

Such a commitment to formal methods in general, and to model theory in particular, is, as we see it, the main motivation for (Scientific=Logical), which complements (Models) in the following way. The Semantic view appears as the methodological prescription to consider scientific theories from the point of view of scientific models, and to use model theory, including the notion of logical models, as tools for the formal analysis of these scientific models. This is essentially what we take the second claim of the Semantic View to be about:

(Scientific=Logical)—Scientific models can be construed as logical models.

In its most modest interpretation, (Scientific=Logical) tells us the following. Once the semanticist has gotten directly from scientists the class of models that these scientists typically use to represent a given domain, then she can construe these models as abstract structures.17 Such an abstract structure can be analyzed with the formal tools of model theory. In particular, scientific models can be construed as logical models.

Now of course, given that logical models are essentially characterized by their truth-making function, it must be shown that scientific models are true of a theory. The most modest interpretation of (Scientific=Logical) takes it that the sense in which scientific models are truth makers of a corresponding theory is almost trivial: a model, is, by definition, true of some theory. As a consequence, the various models that are used to represent a domain of interest are logical models of various theories. There is typically a large variety of models which are used to represent a given domain, from the models of data and of experiments to the most theoretical models. Clearly, not all such models are consistent with one another. This is because these models include various kinds of approximations. Some of the approximating methods used result in equations that are incompatible with the original ones. Hence all the models used for a domain cannot be true of the same, high-level theory. That said, it is the case that the various models used are true of various theories. Note that this involves neither that all these theories be used in common scientific practice, nor even that they be fully and explicitly formulated. They might not all correspond to what one would call a substantial scientific theory. But that’s not what is required by the most modest interpretation of (Scientific=Logical). Instead, what is required is that any of various models used to represent a given domain can be construed as an abstract structure that can be analyzed from a model-theoretic perspective.

There is no need to look for exotic theories in order to illustrate this point: a well-behaved theory like Newton’s mechanics will do. Take Newton’s laws of motion plus the law of gravitation as our high level theory. Now, consider for example the model of the block going down an inclined plane. Such a model is incompatible with our high level theory: contrary to what Newton’s law of gravitation prescribes, the gravitational force is here taken to be constant. So, the model of the block moving down an inclined plane is not true of the high level theory. That said, it is true of a lower-level theory, say the g-constant theory. At lower levels, other models are used, and the series continues down to the models of data and experiments.

According to the modest interpretation of (Scientific=Logical), the variety of models used to represent a given domain thus constitute a whole hierarchy of models standing between the phenomena and the high level theory. To this hierarchy of models, by virtue of the definition of models, corresponds a hierarchy of theories (see Suppes 1969, p. 34 [1962] for a similar idea of a hierarchy of models). The upshot is then that, under its most modest interpretation, (Scientific=Logical) becomes:

(Scientific=Logical)Modest—The whole hierarchy of scientific models that scientists typically use to represent a certain domain can be formally studied as abstract structures from a model-theoretic point of view.

This is not, however, how (Scientific=Logical) is understood by the critics of the Semantic View. On this matter, such critics generally follow Redhead’s pioneering paper (1980).

In this paper, Redhead identifies an important class of models which he calls ‘impoverishment models’. What characterizes impoverishment models is that they contradict the high-level theory. They arise from the simplification of the equations of the theory considered in order to get the solutions more tractable. Such models are disconnected from the theory in so far as the theory does not provide the means to mathematically justify the approximations. Moreover, the simplified equations generally have quite different solutions than the original equations. Accordingly, such models simply cannot be true of the high-level theory.

One of the most recent examples of a criticism along Redhead’s lines is Margaret Morrison (1999):

...the classification of theoretical models suggested by the semantic view does not seem rich enough to capture the many ways in which models are constructed and function. The semantic view characterizes theoretical models as ‘models of theory’—there is a basic theoretical structure that does more than simply constrain the acceptable models, it provides the fundamental building blocks from which the model is constructed. ...However there is reasonable evidence to suggest that this represents only a very limited picture of model building in physics; many models are constructed in a rather piecemeal way making use of different theoretical concepts in nothing like a systematic process. (p. 43)

The models themselves are not strictly ‘theoretical’ in the sense of being derived from a coherent theory, some make use of a variety of theoretical and empirical assumptions. (p. 44)

In the above quotes, Morrison criticizes the Semantic View for not providing an adequate account of model construction. According to her, the semanticists take it that models which are used to represent a given domain are derived from the high level theory of the domain through a systematic (formal) process. In this case, all models used to represent a given domain have to be truth-makers of the high-level theory. That Morrison bases her argument on such an interpretation of (Scientific=Logical) is clear in the following quote:

According to the semantic view, a theory is simply a family of models .... All [models used to represent mechanical systems] obey the fundamental laws of mechanics and in that sense can be seen to be instantiations or applications of these laws in conjunction with specific conditions relevant to the system, which together define the model. (p. 41)

Such an interpretation of (Scientific=Logical) is much stronger than the interpretation we proposed. Within Morrison’s interpretation, (Scientific=Logical) is not seen as the rather modest claim that any scientific model, when seen as an abstract structure, can be construed as a logical model, that is, a model which is true, by definition, of a theory, and whose structure can be studied with the tools of model theory. Instead, it is interpreted as the claim that all the models used for a particular domain are true of, and systematically derived from, the high level theory of the domain.

There is no doubt that the literature from the last two decades on model construction has been of great importance for philosophy of science. It is not our intention to deny that good progress has been achieved in our understanding of some neglected aspects of actual scientific practice. For example, one could argue that various works in this field allow us to get a clearer picture of the kinds of approximation which are used in model building.18 Our contention is that, however, none of this work constitutes a basis for fair criticisms against the Semantic View as we construe it. This is in part because, as we said above, the Modest Semantic View does not commit to the stronger interpretation of (Scientific=Logical).

One has to grant that most advocates of the Semantic view do not clearly reject the naive view that all models used to represent a given domain are true of the high level theory of the domain. Only Suppes seems to be explicitly advocating a more complex view of how theories and models relate to each other and to the world (see for example Suppes 1969, p. 34 [1962]).19 By contrast, the following quote, for example, seem to suggest that van Fraassen adopts the naive picture described above (van Fraassen 1980, p. 64):

To present a theory is to specify a family of structures, its models; and secondly, to specify certain parts of those models (the empirical substructures) as candidates for the direct representation of observable phenomena.

Likewise, Giere endorses the naive view in his (1988, pp. 82–83) and his (1999, pp. 92–93).

Because the stronger interpretation is untenable, if one wants to adopt the Semantic View, then one must commit only to the Modest Semantic View, which explicitly rejects the stronger interpretation of (Scientific=Logical) in favor of the most modest one. (Scientific=Logical) is then interpreted as the methodological prescription to use the notion of logical model and the resources of model theory as tools for the formal analysis of the whole hierarchy of scientific models that scientists use to represent a certain domain. It remains to see what can be achieved with these tools. This is the burden of the next section.

4 (→ Adequacy): structure, function, and pragmatics

Up to now, the Modest Semantic View is the methodological prescription to:
  1. 1.

    focus on the scientific models that scientists typically use to represent the world when studying scientific theories;

     
  2. 2.

    study this variety of models as abstract structures from a model-theoretic perspective.

     
What remains to be explained is what is to be accomplished with such methodological prescriptions. This is the question that (→ Adequacy) is meant to answer:

(→ Adequacy)—Considering scientific theories from the point of view of their models and construing scientific models as logical models provides the means to give an adequate account of what scientists typically use to represent the world in actual practice.

The various ways in which (→ Adequacy) can be interpreted correspond to the various levels of adequacy that one can achieve when giving an account of scientific models. The difficulty of course is to distinguish clearly between levels of adequacy and to spell out precisely what form of adequacy the Modest Semantic View can hope to achieve.

What is needed is a characterization of what counts as ‘an adequate account of scientific models’, where scientific models are taken to be ‘what scientists typically use to represent the world’. In order to do so, a detour from the domain of scientific theories and scientific models should prove useful. Let us try to determine first what counts as ‘an adequate account of sailboats’, where sailboats are taken to be ‘what sailors typically use to sail’. As we shall see, at least three different kinds of account of sailboats can be distinguished. To these three kinds of account will correspond three different forms of adequacy. This analysis will finally be applied to the case of scientific models.

First, one could consider a sailboat in terms of its structure. This would consist in considering the blueprint of the sailboat, and describe what its various parts are, and how they relate to one another. On top of this, one could compare various sailboats from the point of view of their structure. Among interesting structural features of a sailboat is, for example, the ratio of length and maximal width. It should be emphasized that one can give an account of the structure of a sailboat independently of the fact that they are used to sail. The structure of sailboats can be investigated without any mention of these elements which are external to sailboats but are important for the use of sailboats for sailing—water, currents and wind. By no means is it claimed here that such an account would be complete. However, it can be said to be adequate, even if in a narrow sense. An account is indeed structural-adequate if and only if the description given of the structure of a given sailboat corresponds to the actual structure of this sailboat.

Arguably, one would require that a complete account of a sailboat contain at least, in addition to the account of the structure itself, an account of how the sailboat’s structure helps serve its function. In such an account, the ratio of length and maximal width would not be studied solely as a structural part, but rather in terms of the balance between speed and stability. One cannot give such an account without mentioning external elements like water, currents and wind. One has then to study the relationships between the internal structure of sailboats and these external elements such that it be explained that sailboats can be used for sailing. An account is thus functional-adequate if and only if the relationships between the internal structure of sailboats and the external elements of the use of sailboats explain how sailboats can fit their function. Notice that one can give an adequate functional account of a sailboat without deciding on which sailboat is preferable for actual sailing, that is to say, without deciding on the pragmatics of sailboats. That said, a functional account, even if it does not account for the pragmatics of sailboats and sailing, can still be adequate, qua functional account.

To give a pragmatic account of sailboats consists in giving an account of the types of choices that sailors face and make in considering the specific goals and means they have for sailing. Goals and means vary of course, so do the choices. Typically, someone interested in racing does not make the same choices as someone interested in family vacations: more or less emphasis is put on speed and comfort, respectively. Of course, the cost of the sailboat also enters into account. A pragmatic account of sailboats thus takes into account, besides the structure and the function of sailboats in general, the specific goals and means that we have when we actually choose a sailboat to sail. More precisely, a pragmatic account is expected to account for the choices made in explaining how these choices result from how the structural features of the boat, along with how these features result in specific functions, relate to the specific goals and means that sailors have. An account is thus pragmatic-adequate if and only if the relationships between structure, function, goals and means are so described that the choices made are explained.

We have found that there are at least three types of account of sailboats, to which correspond three forms of adequacy—structural, functional and pragmatic. Let us emphasize again that the lower-level accounts can be given independently of the higher levels: structural features can be studied independently of functional or pragmatic considerations, and functional features can be studied independently of pragmatic considerations. That said, this is not the case when going down the scale: functional features crucially depend on structural features, and pragmatic choices depend on both functional and structural features. To put it simply, the way sailboats are used depends on the internal make up of sailboats, and our choices regarding sailboats depend on both the internal make up of sailboats and how this affects their possible usage. Now, let us see how our analysis of the kinds of account and corresponding forms of adequacy one can give of sailboats applies to scientific models.

Following the analysis above, one can distinguish three ways in which to account for scientific models and three corresponding forms of adequacy. A given scientific model can be accounted for in terms of its structural features, which can further be compared with the structure of other models, independently of their function. An account is structural-adequate if and only if the structure described is indeed the structure that the model of interest possesses. One can then study how this structure relates to typical functions of scientific models. Given that an important function of scientific models is to represent the world, a functional-adequate account of scientific models should explain how the internal make up of scientific models makes it possible to use them to represent the world. In other words, an account of scientific models is functional-adequate only if it answers the question of representation. Finally, one can study what choices scientists face and make when actually constructing and using a scientific model. An account of scientific models is then pragmatic-adequate if and only if the structure of the models along with how this structure makes the model serve specific functions are related to the goals and means of scientists such that typical choices that these scientists make are explained. Typically, scientists have to make choices concerning the relative emphasis to put on either the accuracy of the models and the tractability of the solutions. Cost, or epistemic accessibility (which depends on what kinds of instrument they have) are other important and highly pragmatic considerations. So, an account of scientific models can be pragmatic-adequate only if it answers the difficult question of the appropriateness of approximations.

With the above analysis of the three types of account of corresponding three levels of adequacy in hand, we can now deploy three interpretations of (→ Adequacy). Stronger interpretations of (→ Adequacy) consists in the claims that studying scientific theories from the point of view of scientific models, and studying scientific models from a model-theoretic point of view, is sufficient to give an functional- and/or a pragmatic-adequate account of scientific models. By contrast, the Modest Semantic View commits only to the claim that studying scientific theories from the point of view of scientific models, and studying scientific models from a model-theoretic point of view, is sufficient to give a structural-adequate account of scientific models. This is because model theory provides the right kind of formal tools for such an account. In particular, it makes the notion of structure of models and of structural relationships between models precise.

Such a precise and well-defined notion is useful in addressing both issues of internal structure of theories and interrelations between theories. For example, van Fraassen (1991) uses the method presented in his (1989), to examine the controversies that have plagued the philosophy of quantum mechanics since the inception of the theory. He offers a formal presentation of the issue of whether or not quantum phenomena can be described by deterministic causal models. Lloyd (1994) [1988] has shown how the semantic conception is an interesting framework in which to study population genetics. Within the framework of the Semantic View, Lloyd offers a categorization of the various mathematical models in population genetics, leads a discussion of units of selection based on this analysis, and offers a systematic analysis of types of confirmation based on the investigation of the relationships between data models and theoretical models. Focusing on the structures of these models, she is, for example, able to distinguish between various kinds of group selection, and to argue against genic selectionism.20 In his (2002), Suppes shows how proving representation theorems with model theory is useful to address the question of reductionism. To prove a representation theorem is to show that there is a specific class S of models such that every model of a theory T is isomorphic to a member of S. In other words, all the models of the theory are “represented” by a model in S. This is arguably an interesting way to formalize the possible reduction of a theory T to the theory T′ associated with the restricted class S of models. To the extent that the structuralist program shares with the Semantic View the idea that taking a model-theoretic perspective on theories is an efficient way to study the internal structure of theories and intertheoretical relations, one can mention some of their achievements to illustrate our point. Besides the paradigmatic cases of reconstruction of theories exposed in Balzer et al. (2000), structuralists have been discussing classic issues such as the issues of reduction (see Balzer et al. (1987, chap. VI), and, more recently, Bickle (2002) or Niebergall (2002), among others), of holism (Gähde 2002), and of incommensurability (see Balzer et al. 1987, chap. VI.7, and, most recently, Caamaño 2009). These examples show that model theory is an appropriate formal tool for the systematic investigation of the structure of theories and the interrelations between theories, and thus for giving a structural-adequate account of scientific models. This is the minimal sense in which (→ Adequacy) is understood within the Modest Semantic View:

(→ Adequacy)Modest—Studying scientific theories by studying their models and construing scientific models as logical models provides the means to give a structural-adequate account of what scientists typically use to represent the world in actual practice.

This is not, however, how (→ Adequacy) has been interpreted within the criticisms recently leveled against the Semantic View. Critics have been repeatedly pointing out that the Semantic View cannot give the tools for either a functional- nor a pragmatic-adequate account of scientific models. Morgan and Morrison (1999) contains multiple contributions to that effect. In the first chapter, Morgan and Morrison explicitly state that, while the Semantic View provides “some analysis” of “the importance of models in scientific practice”, “there is much more to be said concerning the dynamics involved in model construction, function and use” (p. 10). In the same volume, Suárez argues that “the semantic view lacks the resources to provide us with an understanding of how, in practice, models mediate between theory and the world” (Suárez 1999, p. 172), and Cartwright that the semantic view is not capable of accounting for the production of models as an “incredibly difficult and creative activity” (Cartwright 1999, p. 247). There are many ways to argue that the Semantic View fails to provide either a functional- or a pragmatic-adequate account of scientific models. We will only focus on the function of representation and the pragmatics of approximations, two essential features of the function and pragmatics of scientific models, respectively. To show that the Semantic View fails to account for such features is sufficient to show that it is unable to reach either functional- or pragmatic-adequacy.

The Semantic View does not provide the appropriate tools for providing an account of the representational function of scientific models.21 As noted earlier, it is rather uncontroversial that an essential function of scientific models is to represent the world. Consequently, a functional account of scientific models cannot be adequate unless it at least addresses the issue of representation. Now, what does the representational function of scientific models consist in? A minimal characterization of representation is the following: it is a relation between a representans and a representandum. In our case, the representans is a scientific model, while the representandum is a phenomenon in the world. Now, we have seen that, within the Semantic View, one construes scientific models as abstract formal structures and uses model theory to study their structural properties as well as their structural relationships with other formal structures. The tools that the Semantic View offers, that is, logical models and model theory, are adequate tools only to study formal relationships between formal structures. The problem is of course that the world is not a formal structure. Consequently, the issue of how scientific models represent the world, that is, the question of how abstract formal structures relate to non-abstract, non-formal things such that ‘representation’ occurs simply cannot be addressed with the tools proposed.

Two notes of clarification are in order. First, it is not implied by the argument above that the world has no structure at all. The Semantic View is not committed to any metaphysical claim concerning the fundamental nature of the world. Second, the argument does not imply either that the tools of the Semantic View cannot prove useful for the analysis of the problems of representation, evidence and confirmation, when the analysis is confined to the study of the relationships between scientific models and the data models, in which the world is already considered to be represented by abstract structures. Suppes’s work on evidence, Lloyd’s work on confirmation, the work of the structural realists, and, more recently, van Fraassen’s work on empirical grounding (see van Fraassen 2008), are cases in point.22 That said, no account of how ultimately the data models represent (and misrepresent) the world can be given in model-theoretic terms. Consider, for example, one of the square tiles that are covering a kitchen counter. One might think that such a tile—in the world—can be easily compared to a square—the mathematical structure. We identify the tile as approximately square by noticing that both the tile and the mathematical square share some structural features, e.g. the angular separation between the edges, and it would seem that, even if it might be peculiar, in this case, one can use model theory to do so. But it should be clear that, in fact, we cannot compare the tile itself with the mathematical square. Instead, we measure the tile’s angles, thus building a model of the tile, which we compare with the mathematical square. Formal methods—including model theory—are the right kind of tool for this last task, but it does not offer any way to say whether the procedure that you used to construct your model is a good or a bad one.23 So, the Semantic View does not give the means for an adequate account of the representational function of scientific models. The upshot is: if (→ Adequacy) is taken as the claim that the Semantic View does give the means for such an account, then the Semantic View fails.

The Semantic View does not provide the means to give an adequate pragmatic account of scientific models either. Recall that, in order to give an adequate pragmatic account of scientific models, one has to explain the choices that scientists make over scientific models in relation to their specific goals and means. In other words, one has to understand how certain types of scientific models (characterized by certain structural features and related functions) are chosen depending on certain goals and means. Now, one of the crucial aspects of model construction consists in deciding over which approximations are appropriate. The problem is that there is no formal justification for these approximations. Even less is there any formal justification for which approximation is preferable in regards to the scientists’ means and goals.

Of course, knowledge of the formal structure of the various possible models is an important part of the decision process. Back to sailboats, the choice over which kind of boat is the most appropriate for someone depends on the various possible structures that sailboats can have. Similarly, the choice over which kind of scientific model is the most appropriate depends on the possible internal structure that scientific models can have. Model theory can certainly be used—and fruitfully so—to rigorously analyze, compare, and gauge the structure of approximation relationships between high level and low level theories and models. One should refer here to the results of the structuralist program, in which the approximation relation is formally analyzed in terms of the topological concept of uniformity taken from Bourbaki (1954).24

That said, a structural account is not sufficient. Within an adequate pragmatic account, the various possible structures have to be understood not only in light of how they relate to each other but also in light of how they relate to specific pragmatic considerations.25 In other words, while the Semantic View may offer the tools to analyze the structural relationships between high level models and their lower level approximate versions, or between various approximate models, it cannot account for why such or such approximations is better justified given such or such pragmatic goal. Just as in the case of the question of representation, the Semantic View fails to provide the appropriate tools to account for such relationships simply because these relationships are not formal relationships between formal structures. Consequently, if (→ Adequacy) is taken as the claim that the Semantic View does provide such means, then the view fails.

So, the arguments to the effect that the Semantic View program fails because of its incapacity of giving an account of the function and use of scientific models hold only under strong interpretations of (→ Adequacy). It remains to be seen whether the advocates of the Semantic View committed to such interpretations. Surely enough, the emphasis on the notion of structure is pervasive in the Semantic View literature. That said, it should be recognized that advocates of the Semantic View seem quite ambitious as to what the view can help us accomplish. Arguably, both Suppe’s account in terms of partial-structures in his (1989) and Giere’s account in terms of similarity in his (1988) include the claim that the tools offered by the Semantic View are useful for supporting a realist view. This suggests that they take the Semantic View to be capable of analyzing the theory-world relations, which we have shown is not the case. Van Fraassen also seems quite optimistic as to what the Semantic View has to offer. In his (1980), he hopes to formally characterize empirical adequacy, a theory–world relation, in terms of isomorphism. In his (1989), he hopes that the Semantic View could allow us to give answers to the problems of scientific explanation, confirmation and modality.

None of the advocates of the Semantic View has explicitly recognized the above tripartite distinction between the structural, functional and pragmatic accounts of science, and explicitly rejected the stronger interpretations of (→ Adequacy) that the tools offered by the Semantic View could allow us to provide a functional and/or a pragmatic account of science. Only within the Modest Semantic View is it clearly recognized that studying scientific models from a model-theoretic point of view does not allow one to give either a functional- or a pragmatic-adequate account of scientific models. The Modest Semantic View only commits to the claim that studying scientific models as formal, abstract structures gives the means to give an adequate account of the structure of what scientists typically use to represent the world. Thus, the Modest Semantic View does not fall under the usual criticisms leveled in the literature and offers useful tools for the systematic analysis of the structure of scientific theories.

5 Conclusion: the Modest Semantic View

The Modest Semantic View (MSV) consists in the following claims:
  • (Models)Modest—Scientific theories and scientific practice can be partially characterized by the scientific models that scientists typically use to represent the world;

  • (Scientific=Logical)Modest—The whole hierarchy of scientific models that scientists typically use to represent a certain domain can be formally studied as abstract structures from a model-theoretic point of view;

  • (→ Adequacy)Modest—Studying scientific theories by studying their models and construing scientific models as logical models provides the means to give a structural-adequate account of what scientists typically use to represent the world in actual practice.

The first two claims taken together amount to a methodological prescription: it indicates what tools are proposed within MSV. Holding the first claim implies holding a partial descriptive claim about scientific practice: scientists typically use scientific models to represent the world. Such a claim does not amount to a complete account of either scientific representation, scientific practice, or scientific theories. The second claim is essentially the methodological prescription to use model theory for the formal analysis of scientific models construed as abstract structures. The claim is not that all scientific models used to represent a given domain are truth-makers of a single unified theory of the domain. The naive picture of how theories, models and phenomena relate to each other is replaced by the more realistic picture that a whole hierarchy of sets of models and corresponding theories stand in between high level theories and the data models. The last claim is a claim of adequacy: it is stated that the method proposed allows one to provide a structural-adequate account of what scientists typically use to represent the world. The form of adequacy that MSV can achieve is rather narrow: MSV does not provide the appropriate tools to account either for the function or for the pragmatics of scientific models. Nor is it supposed to.

There is no question that it is essential for scientific models to have a representational function. Nor is there any doubt that pragmatic considerations are crucial elements of the practice of model building and use. If the Semantic View is supposed to be a comprehensive view of science, then it had better say something about the functional features and the pragmatics of scientific models. MSV does not pretend to be a complete account of scientific theories and scientific practice. We agree with Suppes (1994, p. 214) that:

It is a myth engendered by philosophers—even in the past to some extent by myself—that the deductive organization of physics in nice set theoretical form is an achievable goal. A look at the chaos in the current literature in any part of physics is enough to quickly dispel that illusion, this does not mean that set theoretical work cannot be done, it is just that its severe limitations must be recognized.

Our analysis above allows us to clearly identify the severe limitations that have to be recognized: MSV cannot give either a functional or a pragmatic account of scientific models.

That said, on the positive side, just as Suppes remarks: this does not mean that no set theoretical work can be done. And we want to argue that such set-theoretical work can be useful for the philosophy of science. One can give a simple argument for this. As mentioned before, any functional or pragmatic account of scientific models will depend on the structural features of these models. Such structural features we have argued can be adequately analyzed by MSV. Hence, the structural account that MSV gives can serve as a crucial component of any full account of scientific theories and scientific practice. Let us make this point clearer in taking the example of the usefulness of the Semantic View concerning the highly debated question of scientific realism.

It should be clear that an important consequence of our argument is that the Modest Semantic View cannot support either a realist or an anti-realist stance toward science. This should not come as a surprise, given that the various advocates of the Semantic View take various stances toward the realism debate: while van Fraassen defends his famous “constructive empiricism” or, more recently, his “empiricist structuralism”,26 Suppe declares himself a “quasi-realist”, and Giere a “constructive realist”. Finally, Suppes’ position has been characterized as an “operationalism” of a “sophisticated sort” by Moulines and Sneed (1979, pp. 76–77) . One can now understand such a variety in the light of our analysis of the Semantic View.

To pick a side within the realism debate roughly amounts to taking a stance concerning the question of the extent to which scientific theories and/or models are true of the world. This in turn requires providing an account of the relationships between scientific theories and/or models and the world. But we have argued that the Semantic View fails to provide the tools necessary to give such an account. More precisely, model-theory can only be useful in studying the relationships between high-level theories and models and lower level models, including the data models. In order to defend a realist or an anti-realist view, however, one has to say something about how the high level theories and models relate to the world that goes beyond the embedding of data models in theoretical models. While the realist will defend the view that high level theories and models do represent actual features of the world that are not captured by the data models, the anti-realist will defend the opposite view, or at least will suggest that we should remain agnostic about such claims. Clearly, the analysis of the structural relationships between the high level theories and models and the data models is not going to help support such claims. So, the Semantic View does not support either a realist or an anti-realist stance.

That said, the Semantic View can serve as a useful tool on either side of the debate. Let us take for examples two recent and opposite views. On the one hand, let us consider the version of structural realism defended by Bueno, Da Costa, French, and Ladyman, and on the other hand the toolbox view of science defended by Cartwright, Shomar and Suárez.

The idea of structural realism is at least more than a century old, as defended by Poincaré in his (1952) [1905] and (1958) [1906] [1914]. Whether or not one agrees with the claims of structural realism, one must admit that the consideration of the formal definition of structures allow one to make the view both precise and clear. In particular, the account in terms of partial structures as defended by Bueno et al. is arguably one of the best of the realist views of science currently available.27 As French and Ladyman argue in their (1999), the tools provided by the semantic approach are quite versatile and prove useful to account for the various kinds of models used in science—from the iconic models to the more theoretical ones. Structural analysis is also central to their account of how scientific models represent data models, which in turn is essential to their version of structural realism.28 Thus, the Modest Semantic View can be seen as an essential tool for the structural realist.

We would like to argue that it also can be useful to the “toolbox view of science” as defended by Suárez and Cartwright. Consistently with our argument above, we agree with Suárez and Cartwright (2008) that the structural analysis offered by the Modest Semantic View will not allow one to provide a full account the intellectual processes of model building and use within scientific practice (p. 79). That said, we disagree with them when they seem to suggest that a structural analysis “will be so abstract as to be almost empty of content” (p. 79). For example, if their own analysis of the structure of the London Model is correct, it still yields a non-trivial result: that the relationships between the London Model and the old models of superconductivity cannot be captured by any form of isomorphism, whether partial or not. Thus is identified and characterized, even if negatively, a specific type of model building, which they call “piecemeal borrowing” and which they deem essential to scientific practice. Arguably, which kind of relationships holds between these models will be part of any account of the processes of scientific theorizing, and this question is a question which can be answered through set-theoretical work. In this sense, the Modest Semantic View can also be useful to make anti-realist views of science, such as the idea of “the piecemeal borrowing” in the practice of model building, both precise and clear.

The upshot of our analysis is the following. First, the Modest Semantic View does not fall prey to the recent criticisms leveled against the Semantic View in the literature. Second, while being itself limited in scope, the Modest Semantic View can be useful for any complete account of scientific theories. In particular, it can be useful for both sides of the realism debate.

It should have appeared to the reader that there are some significant affinities between the core tenets of the Modest Semantic View and of the structuralist program of Sneed, Moulines, Balzer et al. For reasons unknown to us, the structuralist program has been largely ignored within the discussions on the Semantic View in the English-speaking world. Given the affinities between the structuralists’ views and our own, there is no doubt that a systematic comparison between the two views would be an important and interesting project. One can hope that the Modest Semantic View could serve as a bridge between the two traditions. Unfortunately, such a project could not be accomplished within the confines of this paper, and we have to leave it for the future.

Footnotes
1

Major developments of the Semantic View are in Suppes (1960, 1962, 1967), Suppe (1974, 1977a, 1979, 1989), Giere (1979, 1988, 1999), and van Fraassen (1972, 1980, 1989, 1991, 2008). In their subsequent works, all these authors typically remained in favor of the Semantic View broadly speaking, even if they each developed particular versions of it, and never formed a unified view. A much more unified view has been initiated by Sneed (1971), and developed in Europe and especially in Germany thanks to the work by Wolfgang Stegmüller (1976, 1979a, b, 1986), C. U. Moulines (1975a, b, 1982, 1991, 2002) and Wolfgang Balzer (1978, 1982, 1985), among others. The view associated with these authors has been called the “non-statement view” but it is now known as the “structuralist program”. One classic work associated with the structuralist program is Balzer et al. (1987). Though this article is devoted to the Anglo-Saxon tradition of the Semantic View, we shall point out some obvious affinities and differences between our view and the structuralist program on various occasions in what follows.

 
2

See for example Cartwright (1983, 1989, 1999), Cartwright et al. (1995), Ereshefsky (1991), Downes (1992), Morrison (1999), Suárez (2003), Thomson-Jones (2006), Frigg (2006), Morrison (2007), Suárez and Cartwright (2008), and Krause and Bueno (2008). Even if not explicitly directed against the Semantic View, Redhead (1980) is an important background contribution to the debate.

 
3

Important recent work in which the program of the Semantic View is defended and developed has been undertaken by Bueno et al. in various publications, including: Bueno (1997), French and Ladyman (1999), Bueno et al. (2002) and Da Costa and French (2003).

 
4

What follows is a conceptual reconstruction of the Semantic View. No pretension is made to give a historically adequate account of how the Semantic View developed. Note also that the differences between the set-theoretical approach of Suppes, the state–space approach of van Fraassen and the relational system approach of Suppe are irrelevant here. We agree with Suppe (1989, p. 4) and da Costa and French (2003, p. 23) that van Fraassen’s and Suppe’s approaches can be conceived within Suppes’ set-theoretical framework.

 
5

The “toolbox instrumentalists” conceive of theories as tools for the construction of models through “piecemeal borrowing”. See Cartwright et al. (1995) and Suárez and Cartwright (2008).

 
6

To conceive of scientific theories as set of laws or of axioms was part of what has been labeled the “received view” by Putnam in his (1962). It is not clear that the label refers to a single unified view rather than to a variety of positions taken at different times by various authors (for a presentation of the historical development of the received view, see for example Friedman 1999 or Parrini et al. 2003). That said, the view is generally taken to be as presented in Carnap (1966) and Hempel (1963).

 
7

Suppe (1977a, pp. 62–66) provides an analysis of what counts as a fruitful axiomatization.

 
8

It is well known that there are various kinds of ‘scientific models’. In this paper, no stance is taken as to any particular mode of representation (structural, analogous, iconic etc.) by which scientific models represent the world. For a synthesis of the notions of model in science, see Frigg and Hartman (2006). The relationships between the notions of scientific models (as representing phenomena in the world) and logical models will be discussed in Section 3.

 
9

Or, as it is often easier, from the scientific literature in the field. Note that, in doing so, one avoids the problem of having to define what is the theory before being able to consider the models used by the scientists to represent the world. The class of models that the semanticist sets himself to study is simply the class of models that are used to represent a certain class of phenomena.

 
10

This distinction between the modest and the stronger interpretation of (Models) is related, but not identical to the distinctions between intrinsic and extrinsic characterization of theories by Suppes (1967, pp. 60–62, and 2002), and between representational and constitutional role of models by da Costa and French (2003, p. 34).

 
11

Whether or not this claim holds for structuralist program as well is an interesting question. Clearly, in spirit, and just like the proponents of the Semantic View, the structuralists appear not to adopt the stronger interpretation of (Models), since their “theory-elements”—roughly, elements of what is commonly called a theory—contain, in addition to sets of models, other components, including, among others, the intended domain of application of the theory and the necessary approximations associated with such application. See Schmidt (2008) for a short exposition of the notion of theory-element, and for more details, see Balzer et al. (1987). The question of whether or not the structuralists systematically make the distinction explicit and reject the first interpretation of (Models) requires a detailed investigation that falls beyond the scope of this paper.

 
12

Our modest interpretation of (Models) is similar to Downes’ “Deflationary Semantic View” as described in his (1992), which is constituted by the claim that “model construction is an important part of scientific theorizing”, while rejecting the claim that “all scientific theories are simply families of models” (p. 151). That said, the Modest Semantic View defended here differs from Downes’ version of the Semantic View, because it contains, in addition to the modest interpretation of (Models), the modest interpretations of (Scientific=Logical) and of (→ Adequacy), as described in the following two sections.

 
13

The common claim that the received view demands that scientific theories be axiomatized in first order logic has been recently challenged in the literature (see Lutz 2010).

 
14

On the question of whether the received view and the Semantic View are equivalent due to the Completeness Theorem, see the reviews on van Fraassen (1980) by Friedman (1982) and by Worrall (1984) along with van Fraassen’s answers in van Fraassen (1985, p. 301 sq.) and van Fraassen (1989, p. 211 sq.), which we believe are satisfactory. The discussion is summarized in da Costa and French (2003, pp. 30–31). On the question of wether the Semantic View—using first order model theory—is no better off than the received view—using first order logic plus identity, see French and Ladyman (1999) and references therein.

 
15

Rigorously speaking, the interpretation is a domain along with a function assigning extensions to non-logical terms.

 
16

It should be clear that “truth” is used here in the weak sense of “satisfaction”, or “truth within an interpretation”.

 
17

It is important to note that scientific models do not have to be mathematical structures. It suffices that scientific models possesses a structure (roughly, a domain and relations between the members of this domain). Any structure, whether or not this structure is captured by nice mathematical equations, can be considered from an abstract point of view. This leaves room for an account of structures within the sciences in which nice mathematical equations are not often possible to obtain.

 
18

A distinction between at least between three kinds of approximation arises from the literature: (1) Aristotelian abstractions consist in choosing the relevant parameters and variables of the system at hand—cf. Cartwright (1989, chap. 5); (2) Galilean idealizations consist in neglecting some parameters and variables that are clearly relevant the situation studied—cf. McMullin (1985); and (3) mathematical impoverishments consist in radically modifying the original equations such that solutions be tractable—cf. Redhead (1980).

 
19

This is true at least within the Anglo-Saxon tradition of the Semantic View. The structuralist program also developed, using the notion of a theory–net, a more subtle view of the complexities of the relationships between theories, models, and the data models. In a theory–net, various more “specialized” theories branch from a core theory-element. For more details on such an account, see Balzer et al. (1987, chap. IV).

 
20

It should be noted that Lloyd’s analysis does not include the assumption that population genetics is all there is to evolutionary theory. Rather, she aims at giving an account of the structure of evolutionary theory in a broader sense, including issues about species formation and extinction, group selection, and the tempo and mode of selection (1994, chap. 1, note 4 and chap. 3) [1988].

 
21

Morrisson (1999) is often cited for posing the criticism of the Semantic View along these lines. That said, Cartwright (1983, 1989) has been developing similar views over the past twenty years. More recently, Thomson-Jones (2006) gives an analysis of the relation between logical models and scientific models, the former being characterized by their truth making function, and the latter by their representational function. Brading and Landry (2006) and Frigg (2006) gives also a systematic analysis of the problem. It should be noted that van Fraassen presents the same problem in his (2006), and offers an extended discussion of it in his (2008, chap. 11), where he defends his “empiricist structuralism”.

 
22

This applies to the structuralist treatment of the “empirical claims” associated with a theory as well. See Balzer at al. (1987, chap. II) for more details.

 
23

Note also that model theory is useless when it comes to addressing the question of whether theories represent the features of the world that are not captured by the data models—a question on which the realist and the empiricist depart from one another. More on this point later.

 
24

Technically, they distinguish between four types of approximation (Balzer et al. 1987, chap. VII.1). The one we are interested in here is the first one: “model construction approximation” (Balzer et al. 1987, p. 325).

 
25

It should be noted that Balzer et al. (1987, VII.2.2) manage to formally characterize a set of necessary (but not sufficient) conditions for an approximation to be “admissible”. For example, they formalize the rather intuitive notion that a given admissible approximation should remain so if one changes only “purely theoretical” aspects of the models associated with it. They also clearly recognize, however, that some strong pragmatic components of model construction are not formally explicable.

 
26

The exegesis of van Fraassen’s thought, and in particular, the study of the difference between constructive empiricism and empiricist structuralism falls outside of the scope of this paper.

 
27

See for example Da Costa and French (2003) for a systematic treatment of their view.

 
28

Note that this is not contradictory to our claim that the tools offered by the Semantic View are useless when it comes to solve the fundamental problem of representation. French and Ladyman are well aware of this and commit only to the claim that the Semantic Approach allows us to give an account of the relationships between the various models in the hierarchy of models as described in Section 2 (see note 12, p. 119, in their 1999). This remains true within Ladyman’s most recent ontic structural realism as exposed in Ladyman and Ross (2007). There, Ladyman and Ross explain that formal methods are good for representing and investigating “the relationships between theoretical models and models of the phenomena” (p. 117), but not for explaining how what they call “structures”, which “are to be understood as mathematical models”, can represent “real patterns” in the world (pp. 118–120). This holds true also for the accounts of evidence by Suppes, of confirmation by Lloyd, and of empirical grounding by van Fraassen in his (2008).

 

Acknowledgements

I am extremely grateful to Armond Duwell for extensive discussions on earlier versions of this paper. I also wish to thank Michael Dickson for perceptive comments, as well as Leah McClimans for pressing on the notion of adequacy in the last section. Finally, I wish to express my gratitude to two anonymous referees for pointing out the affinities between the views of the structuralist program and my own.

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© Springer Science + Business Media B.V. 2011