The Problem of Scientific Modeling by danman3459

The Problem of Scientific Modeling for Structural Realism and the Semantic View of Theories∗ Daniel J Singer singerd@sas.upenn.edu

Many philosophers of science endorse a structural realist view in the debate over scientific realism because it avoids both the no miracles argument and the pessimistic induction argument. While the position may succeed in those respects, I argue here that structural realism, along with a semantic view of theories, is incompatible with a satisfactory account of the practice of scientific modeling. In sections 1 and 2, I lay out the structural realist position, the semantic view of theories, and demonstrate the appeal of the semantic view to the structural realist. In section 3, I provide a basic picture of scientific modeling and frame the theory-data relationship question, the question of how theories connect up with phenomena. In section 4, I briefly discuss three formal approaches to the theory-data relationship question and show why none of them is satisfactory. Then, in section 5, I show that to account for scientific practice, any proper characterization of scientists’ models must contain intensional content that the semantic view is missing when employed by the structural realist, so the semantic view of theories is faulty for the very reason that makes it so appealing to the structural realist. Finally, in section 6, I show that the structural realist, in order to provide a sufficient account of modeling in science, must give up on the semantic view or adjust it in ways antithetical to the original aim of the semantic view.

The Structural Realist Picture

It seems natural to hold the na¨ıve scientific realist position—that things in the world are as true scientific theories describe them. But, the pessimistic induction argument claims that the realist is not justified in believing in the approximate truth of currently accepted theories by induction on the large number of historical examples of disproved scientific theories.1 ∗ Many

thanks to Prof. Michael Weisberg, Prof. Elisabeth Camp, Prof. Scott Weinstein, Prof. Adrienne Martin, and Kristin Williams for helpful comments on earlier drafts of this work. 1 Laudan (1981, p. 33) argues that such a list of examples “could be extended ad nauseam.”

Daniel J Singer

The Problem of Scientific Modeling

Take, for example, the theory of dephlogisticated air: In 1667, Johann Joachim Becher posited a new element, phlogiston, that is released by objects into the air when they burn. Because air affects the length of time for which objects burn, air was taken to have a specific capacity for phlogiston, and similarly, oxygen was taken to be dephlogisticated air. Later, Lavoisier showed that there was no such thing as phlogiston or dephlogisticated air, and so the theory (and the phlogiston that it posited) was rejected by the scientific community (Morris, 2003). The pessimistic induction argument suggests that because almost all past theories had a fate similar to the phlogiston theory’s fate, current theories are likely to suffer the same demise. Worrall (1989) contends that the argument from pessimistic induction misses something: there are important aspects of a theory which are not lost when the theory is discredited. He cites the transition from Fresnel’s theory of optics to Maxwell’s theory of the electromagnetic field: There was an important element of continuity in the shift from Fresnel to Maxwell – and this was much more than a simple question of carrying over the successful empirical content into the new theory. At the same time it was rather less than a carrying over of the full theoretical content or full theoretical mechanisms (even in “approximate” form) . . . . There was continuity or accumulation in the shift, but the continuity is one of form or structure, not of content. (Worrall, 1989, p. 117) Worrall holds that while the ontologies or entities posited by successive theories may conflict (as in the conflict between theories of dephlogisticated air and of oxygen), the relations between the entities, or the structure of the theories, persist at least approximately. So, Worrall suggests relaxing the realist position slightly to avoid the pessimistic induction argument by limiting realist beliefs to just the structure posited by theories. On the other side of the realism debate, the no miracles argument argues for realism by claiming that anti-realist philosophies of science, without positing the approximate truth of scientific theories, cannot explain the success and progress of science. The structuralist realist position can avoid can avoid the No Miracles Argument because the structural realist still holds that certain parts of theories refer in a realist way, and under structural realism, the endeavor of science can be viewed as a discovery of the true structure of the world. Ladyman (1998) pointed out that the simple structural realism originally presented by Worrall is ambiguous as to whether it is an epistemological or metaphysical position. The epistemological position on structural realism is that it is a constraint on our knowledge of 2

Daniel J Singer

The Problem of Scientific Modeling

the world: an epistemological structural realist view holds that all that we can know about a phenomenon is the structure, not the nature of the entities in question. Alternatively, the metaphysical position on structural realism is that it is a constraint on the ability of scientific theories to refer: Da Costa and French explain that “On this view, it is not merely that all that we know, but all that there is, is structure” (2003, p. 189). Ladyman further argues that the epistemological structural realist view “gains no advantage over traditional scientific realism if it is understood as merely an epistemological refinement of it” (1998, p. 411). As a way of sidestepping this debate, I will suggest that while there are varying versions of the structural realist view, an aspect that is shared by both the epistemological and metaphysical camps is the claim that scientific theories are about structure. So, I’ll take structural realism to be the family of views that hold that scientific theories are about structure, and my arguments in the following sections will be targeted against the claim.

The Semantic View of Theories

In order to make sense of the structural realist’s claim about the structure of theories, we have to establish what is meant by “the structure of a theory.” The traditional “received” view of scientific theories identifies a theory with “axiomatic calculi in which theoretical terms are given a partial observation interpretation by means of correspondence rules” (Suppe, 1977, p. 38). On this construction, which was mostly due to the logical positivists, we essentially associate theories with rules or descriptions of the target phenomenon. If structural realism were to employ the “received” view of theories, the “received” view must lend itself to a way of identifying the structure of a theory. One way to do this would be to divide the terms in the language into those terms that refer to structure and those that do not. This was attempted by Maxwell, who applied Ramsification to the propositions of a theory, replacing the non-structural, directly referring terms with descriptive ones (Maxwell, 1970a, 1970b). This process removes constants from the formal characterization of the theory and replaces them with higher-order existentially quantified variables. What remains is a formal theory in which the direct reference to terms is removed. This approach fails in many ways due to the fact that identifying a theory with a Ramsey sentence entails that two theories are equivalent if they have the same empirical consequences, which is clearly in contention with Worrall’s conception of the structural realist position (Ladyman, 1998, p. 412). Instead, as Ladyman suggests, the structural realist should accept an alternative: the semantic view of theories. The main claim of the semantic view of theories is that a theory can be identified with 3

Daniel J Singer

The Problem of Scientific Modeling

a class of models. Advocated by Suppes (1960a), the semantic view associates a theory not with syntactic axioms but rather with the semantic objects, models, that represent the target phenomenon as conceived of by the theory. Suppes is operating under the assumption that “the meaning of the concept of model is the same in mathematics and the empirical sciences,” so to specify a model is to specify an extra-linguistic entity that can be picked out by a variety of linguistic formulations. The semantic view appeals to the structural realist because, with the semantic view, the structural realist could represent the structure of a theory in a model, such as a set-theoretic model, without concern for specifying the structure in an axiomatic system.2 Since the structural realist wants to employ the semantic view to associate a theory directly with models of the structure of the theory, it will be useful to specify more precisely the notion of a model. There are two main views on the reified nature of a model in the semantic account. One view, recently advocated by Godfrey-Smith, thinks of models as imaginary structures that would be concrete if they were real (Godfrey-Smith, 2006). On this view, a model of a three-sexed population is actually an imaginary population with three sexes. This view is appealing because it seems to correctly account for the ways in which scientists talk about their models: a scientist might say “this is a model in which there is a three-sexed population,” and in doing so, he seems to be talking about something akin to fictional worlds. The other, more popular, camp thinks of models as abstract mathematical objects. This view originates in Suppes (1960a) where he claims that every case of the use of the word “model” could be accounted for in a Tarskian set-theoretic way. A model of this type contains a domain of objects and sets which extensionally define relations on that domain. An alternative picture to this one, primarily developed by Van Fraassen (van Fraassen, 1970), which still takes models to be abstract mathematical objects, is the state-space view in which a model is a state-space with transition rules or trajectories through the state-space. Godfrey-Smith’s account takes models to be similar to fictional worlds: a model is an imaginary conception of a way the phenomenon could actually exist. So, since the semantic view takes a theory to be a set of models, on this view, the theory of spring displacement over time would be the set of all ways that an imaginary spring could be displaced over time. The semantic view with Godfrey-Smith’s models equates a theory to all of the ways the phenomenon could concretely exist. However, the structural realist is interested in the semantic approach because of its emphasis on the abstract structure that models could 2 For now, I am glossing over how the structural realist specifies such structure-espousing models, but I will return to that shortly.

Daniel J Singer

The Problem of Scientific Modeling

represent, and associating a theory of springs with all possible springs does not appear to be any more structure-centric than associating the theory with the set of propositions that describe all possible springs, as in the “received” view. So, for structural realists, the use of Godfrey-Smith’s type of models undermines the appeal of the semantic approach compared to the “received” view. In order to make the semantic view useful to the structural realist, models must be stipulated in a structure-centric manner. The most familiar way to do this is with the Tarskian set-theoretic approach favored by Suppes. Here the structure of the theory can be associated with the relations defined explicitly in the model. Alternatively, the state-space view, which takes a model to be a set of states along with transition rules, is the most prominent contender. Here, the theory structure can be associated with the transition rules or some manipulation of them. For the purposes of this essay, although I occasionally make reference to models as state-spaces or Hilbert spaces with transition rules, my arguments are generalizable to any abstract-model representation. It is necessary only that theory models be structure-centric abstract semantic objects to avoid the structural realist’s objection to Godfrey-Smith’s view. I will use “theory model” to mean an abstract structure-centric model of a theory, and I will take an abstract-model semantic view to be the view that a scientific theory is a collection of theory models.

Modeling with the Semantic Approach

As we saw above, the structural realist requires the use of abstract models to exploit the semantic view of theories, and I will show that the abstract-model semantic view is incompatible with an adequate account of the practice of modeling in science. In this and the next sections, I will describe some past approaches to dealing with modeling and resemblance in the abstract-model semantic view, but I will show that none of these accounts is sufficient. Then in Section 5, I will provide an argument that no such account can be given. In the end, since scientific modeling plays such an central role in scientific practice, any acceptable complete philosophy of science should admit of an account of scientific modeling, so I will conclude that the structural realist must either forgo the abstract-model semantic view or adjust the semantic view to include a type of model interpretation that is counter to the original thesis of the semantic view. But first, let’s look at the role of models in science. Much of science is consumed with the use of scientific models. Take, for example, the Lotka-Volterra model of predator-prey interaction, the Bohr model of the atom, or equilibrium models in economics. Generally, scientific modeling has the following form: there 5

Daniel J Singer

The Problem of Scientific Modeling

is some target phenomenon, which can be either concrete or imaginary, and there is some collection of data that is extracted from the phenomenon, such as the displacement over time of a spring with a weight on the end. Modeling is the creation of a model, which has some intended relation to the phenomenon. When the intended relation obtains, the model applies to the target. For example, when a model applies to a phenomenon, the model may explain the phenomenon, such as a model explaining the causes of some aspect of the phenomenon, or the model may account for or predict the data, such as the simple harmonic oscillator (SHO) model of a spring that attempts to describe the displacement over time of a spring as a sine curve. Many models fall into the latter role while completely avoiding the former. For example, one may think that the San Fransisco Bay model, a physical scale model of the bay, is no more explanatory than the bay itself since it attempts to be just like the bay in every way except size. Conversely, the double-helix model of DNA does not try to predict data about DNA but rather gives an explanatory account of the phenomenon.3 Suppes suggests the most important role that models play is that of being the intermediate link between theories and phenomena (1960b), and da Costa and French follow up on this line recently (2003). On these views, there are two main types of models: (1) data models act as representations of physical target systems by representing data or other aspects of the phenomenon, and (2) theory models act as representations of theories, in that they are objects which make theories true (or, in the semantic view, they are just part of the theory). A theory successfully accounts for a phenomenon, in these views, when there is a theory model T of the theory and a data model D of the phenomenon such that T applies to D, in the sense discussed above. The important question that I will focus on in this essay is the question of what is the “applies to” relationship, which I will call the theory-data relationship question: Using an abstract-model semantic conception of theories, what must be true so that a theory model for some theory applies to a data model for some phenomenon? By exploring this question, I will argue that abstract-model semantic approach, as used by the structural realist, cannot give a sufficient answer to this question that accurately accounts for scientists’ models.4 3 For

a more detailed account of scientific modeling, see Weisberg (2007). (1962, p. 262), Achinstein (1968, p. 209-225), and Downes (1992)all have presented different arguments attempting to show that the semantic view of theories cannot adequately account for modeling in scientific practice. French and Ladyman argue that their arguments can be summarized into two principle complaints: (1) the types of models, such as physical models, that the semantic approach is not diverse enough to capture them formally, and (2) that it is “implausible to require that the formal structure of a theory and model be identical.” (French & Ladyman, 1999, p. 106-107) Here, I present a new argument against the structural realist account with an abstract-model semantic view. 4 Black

Daniel J Singer

The Problem of Scientific Modeling

The Theory-Data Relationship

Early incarnations of the semantic view took the proper theory-data relationship to be isomorphism: “Theories, embodied as models, were taken to apply to the world by the relation of isomorphism between models and (parts of) the world” (Teller, 2001, p. 394).5 Reactions against this type of formal account of the relationship spawned Giere’s and Teller’s claim that “models correspond to the world not by a relation of isomorphism but by a looser relation of similarity” (Teller, 2001, p. 395). There has been a lot of discussion of the notion of similarity, but the following brief argument suggests that the structural realist does not need such a non-formal relationship, although this corollary is not essential to my overall conclusion: The main claim of the structural realist is that a successful theory model captures the structure in the world. Since we are not committed to any specific characterization of models (such as models as set-theoretic objects), the semantic view could formulate a characterization of models that is capable of representing the world’s structure. Surely this is possible (in principle) since, if the structure actually exists in some phenomenon in the world, a data model of that phenomenon can be created by simply abstracting the concrete structure in the world, and a theory model can be created using the same characterization as the data model. Then the constructed theory model, like the data model, is capable of representing the structure as simply an abstraction of the structure in the world. So, there should be some way of directly correlating the parts of the successful theory model as such and the structure in the world (or data model). This is exactly what the types of formal approaches, such as isomorphism, do. So, intuitively, some sort of morphism, a map which is relation-preserving, should suffice to capture the theory-data relationship, but my final argument does not require this assumption. I will devote this section to briefly covering three past attempts to answer the theorydata relationship question and responses to them and, in the next section, I show why no relationship, neither formal nor informal, in the structural realist and abstract-model semantic view can satisfactorily account for the use of models in science.

4.1

The Isomorphism Attempt

Given a theory model and data model in the semantic view, one way to characterize the modeling relationship is by isomorphism: This view says that the theory model applies to 5 To

remain consistent with the picture of modeling presented, here I take Teller’s use of “world” to be a data model of some part of the world rather than the actually phenomenon.

Daniel J Singer

The Problem of Scientific Modeling

the data model just in case the two are isomorphic. Versions of this view have championed by Suppe (1977) and Van Fraassen (1980).6 The clear advantage of this sort of approach is that it provides a rigorous notion of accuracy that seems to fit the intuitions about what an ideal scientific theory should do. In the limiting case, we would want our theories to directly correspond to the phenomenon in the way isomorphism requires: We expect, for example, that a successful theory of the displacement of a spring over time in the semantic view should have a theory model that maps directly to the data model of the actual spring. The ability to create this type of map is one motivation behind the identification of theories with models in the semantic view. Despite its intuitive appeal, the isomorphism approach will have at least two problems in dealing with data noise and deliberate idealizations, due to the fact that isomorphism requires strict identity of structure. We take the simple harmonic oscillator theory of springs to be a theory of springs; but no actual spring is a simple harmonic oscillator, and even if it were, a data model of it would not be perfectly harmonic due to measuring errors and imperfect experimental conditions. Therefore, no isomorphism will obtain between a simple harmonic oscillator theory model and the data model of the actual spring. Another problem with the isomorphism approach is that if we take an isomorphism to be only a relation-preserving, bijective map (e.g. the standard definition of isomorphism), we run into issues regarding the proper definition of models, and we find that the relationship obtains in situations where we wouldn’t expect it to. Take, for example, the state-space approach: if we take a state-space model to simply be a set of states with curves defined on the states by transition rules, then a model of a single parameter over time where that parameter increases monotonically will be isomorphic to any other state-space model of the same cardinality of a single parameter over time where the parameter increases monotonically, regardless of whether the models are intended to represent the parameter increasing linearly or exponentially. This is because the transition rules of each model will pick out a curve in the space of states such that for each state there is at most one time value, and since the states do not contain any semantic information about what state of the physical system they represent, from the perspective of the abstract models, the transition rules pick out isomorphic curves. Surely this falls outside of the isomorphism’s advocates’ expectations. They would like that, for example, two states in the state-space approach are mapped to each other iff they represent the same state of the physical system. Unfortunately, that can6 Van Fraassen’s view presented in The Scientific Image is actually one of embedability which can be loosely characterized by isomorphism of a substructure of the theory model to the data model (van Fraassen, 1980, p. 64).

Daniel J Singer

The Problem of Scientific Modeling

not occur in this simple isomorphism approach. In the domain of a Tarskian model, every element is simply a mathematical object since models, in the abstract-model semantic view, are not about any physical system, and every element looks just like every other element.7 This is the fact that powers my main argument in section 5. Another strong objection against the isomorphism approach is that it requires that there be a bijection between the data model and the theory model. While a relation of bijection may satisfy our intuitions of what what a scientific theory should strive towards, it does little to help answer our question of how we should relate actual data models to actual theory models. Given that at any point in time, we will have only finitely many pieces of data, and assuming that in most cases a theory model will posit infinitely many experimental results, it is in practice impossible to use the isomorphism approach to the model-data relationship. So, the next approach relaxes the bijection requirement to try to save a morphism approach.

4.2

The Homomorphism Attempt

To resolve the bijection problem of the isomorphism approach, we might follow Lloyd’s observation that “in practice, the relationship between theoretical and empirical model is typically weaker than isomorphism, usually a homomorphism, or sometimes an even weaker type of morphism” (Lloyd, 1994, p. 14 fn. 2). By dropping the bijection condition from the isomorphism approach, we can seemingly maintain the rigour benefits of the isomorphism approach but without requiring that the map be bijective. Because homomorphisms don’t need to map onto their range but must be functions of their domain, it will be useful to consider two types of homomorphisms which may exist between data models and theory models: those which map from the data to the theory models, which I will call dhomomorphisms, and those which map from the theory models to the data, which I will call t-homomorphisms.8 D-homomorphisms offer a natural means of thinking about a homomorphic relationship between theory and data since we often think of an actual set of data as some subset of the information contained in an ideal theory model. But, due to their strictness of relationpreserving, they are still subject to the problems of noise, idealization, and obtaining in unexpected cases: It may seem that because this approach does not require the morphism to map onto the entire theory model, that we could allow for idealization by relaxing a 7 The

problem could potentially be resolved by putting a metric on the states of the state-space or by some other construction, but the point is that, strictly, the use of isomorphism would require these considerations. 8 Here I will also take both types of homomorphisms to be injections since a non-injective homomorphism ”loses” cardinality information in the connection, which would be unusable for the scientist.

Daniel J Singer

The Problem of Scientific Modeling

theory model of an idealized theory so that it maps from more possible sets of data. For example, in the state-space approach, if we think of the original (idealized) theory model as a state-space model with a sine curve, it seems that we could allow for idealization by also including in the curve a “halo” of the states near states in the curve. This way, the data needs only to be “close” to the theory for the d-homomorphism to attain. This approach is useful and accurate when it comes to approximation,9 where a model is taken to approximate another just in case there is an appropriate closeness of fit between them; but in the case of idealization, that is the inclusion of intentionally false assumptions in the model which may cause large deviations from the data, this approach fails since idealization cannot be fully accounted for as approximation (Cartwright, 1989). Finally, the use of d-homomorphisms as a theory-data relationship brings up a concern not present in the isomorphism account. With the state-space approach to models, the intuitive appeal behind using the bijective map of the isomorphism approach is that it would require a strict structural equivalence between the theory model and the data model: An isomorphism would require every state accounted for in the theory to be in the data and every state accounted for in the data to be in the theory. With d-homomorphisms we have dropped the requirement that every state accounted for in the theory be accounted for in the data. So, notice that if we take d-homomorphisms as the proper “applies to” relationship, we fail to require that theory models be parsimonious, that is the theory may contain unnecessary parts that would not be checked by the data. Consequently, we could develop a trivial theory such that a d-homomorphism would always obtain between any data and the theory. So, data models, with d-homomorphisms, do not put the significant check on the theory that would assure that the theory is true specifically for the data and does not just trivially obtain. This is not a show-stopping problem for the approach though, as it does not appear to be a problem to have theories that are trivially true. While it is no advancement for science to consider the theory that states that everything equals itself, for example, our conception of science should be able to formally account for the testing of such a hypothesis against some physical target system. Secondly, we may find that Occam’s razor is enforced upon theories and theory makers via some other mechanism, maybe in the theory-creating or experiment-design processes, both of which are beyond the scope of this discussion. On the other hand, T-homomorphisms, homomorphisms from the theory model to the 9 In fact, not-with-standing the failure of the semantic view of theories, this approach can be developed into a full theory of approximation where “approximation” is used in a sense such that a theory model approximates some data model just in case there is a limit to the error of the data in the theory.

Daniel J Singer

The Problem of Scientific Modeling

data model, are strikingly incapable of serving as a characterization of the modeling relationship: As suggested above, our expectation of the data, given some theory model, is that the data should be some subcollection of the states in the relations posited by the theory model, because theory models should have a general applicability to many targets and unknown data rather than just telling us about the states of the few target systems that compose the data. For a t-homomorphism to obtain though, every element of structure posited in the theory must obtain in the data. Hence, while the technical requirement of a bijection is alleviated with the use of t-homomorphisms, the problem of bijection remains. A bigger problem for t-homomorphisms is afoot, though: consider a theory model with only one element in its domain and one unary relation that contains that element. This theory is t-homomorphic to any data model that has any non-empty unary relation. Here we see that with t-homomorphisms, the theory need only express some part of the data for the relation to obtain, so essentially, t-homomorphisms require only that theories do not say anything false about the data. That relationship is clearly not a sufficient characterization of the â&#x20AC;&#x153;applies toâ&#x20AC;? relationship, so this is a deadly failure for the t-homomorphism method. The failure of the t-homomorphism approach highlights a very natural idea that underlies a correct account of the theory-data relationship: we expect theories at least to account for the data. This is the same as saying that, in the ideal case, a d-homomorphism obtains. The failure of the t-homomorphism method is due to its failure to account for this obvious premise. A fortiori, we expect the theory to say more about the target than the data alone do: we expect that with ideal theories and data, not only should a d-homomorphism obtain, but that it should not be surjective, i.e. the data does not map onto the whole theory, so that it says more about the target than the data model does. Even though both homomorphism approaches avoid the bijection objection of the isomorphism approach (at least nominally), and the d-homomorphism approach yields an appealing account of proper relationship, ultimately, both of these accounts fall prey to many of the same objections made against the isomorphism account and some new ones as well.

4.3

The Partial Isomorphism Attempt

In this section, I briefly introduce a different tack on the problem of specifying the correct relationship between theory and data in which a special notion of partial isomorphism is used. Here I will drop the assumption that models are generic abstract structure-centric models and instead consider the new formulation of set-theoretic models proposed by da Costa and French.

Daniel J Singer

The Problem of Scientific Modeling

In Science and Partial Truth: A Unitary Approach to Models and Scientific Reasoning (2003), da Costa and French loosen the traditional Tarskian dichotomy, that for each n-ary relation in a model each n-tuple is either in or out of the relation, by allowing that a tuple can have three values with respect to a relation R of the model: it can be in R, out of R, or indeterminate with respect to R. A partial structure is then defined, like the Tarskian notion of a model, as a tuple whose first member is the domain of the structure but whose following members are 3-tuples of sets of tuples hR1 , R2 , R3 i such that an element of the domain is in R1 if it is in the relation, in R2 if it is not in the relation, and in R3 if it is “left open” with respect to the relation. Using partial models, da Costa and French argue that we can produce a sufficient answer to the theory-data relationship question. They hold that the relationship obtains when there is a partial isomorphism between two partial structures: A is partially isomorphic to A0 when a partial substructure of A is isomorphic to a partial substructure of A0 . The notion of a partial structure (or substructure) is so conceived that a total structure (or substructure) constitutes a particular case of a partial structure (or substructure). In other words, we can say that, with regard to a partial isomorphism, certain of the R – some subfamily – stand in a one-to-one correspondence to certain of the R0 . (da Costa & French, 2003, p. 49, variables changed) The idea here is that a theory model applies to a data model just in case there is some subset of the (partial) relations of the models such that if we ignore that subset, the two models are isomorphic.10 In other words, two models are partially isomorphic in case they are isomorphic with respect to some of their relations. This type of formulation, according to the authors, allows for correct characterizations of theories and theory change as it allows for target system states whose status is unknown with regard to the theory. Certainly, this is intuitively a good way to allow for dynamical theory evolution as it would allow that a new theory with more tuples decided (i.e. more tuples not in R3 ) to be considered an extension of an old theory in a very natural way. Da Costa and French also believe that this formulation allows for the appropriate “looseness of fit” to account for idealization and approximation (da Costa & French, 2003, p. 104) and hence may be a viable candidate for the theory-data relationship. Like the other morphism approaches, the partial isomorphism approach is insufficient 10 I take this interpretation of their approach mostly from the last line of their definition, which seems to imply

that there should be a bijective morphism that respects some of the relations in each model. This assumes that by “one-to-one correspondence,” the authors means a one-to-one morphism since “one-to-one correspondence” is standardly interpreted as a bijective map (Barile, 2002).

Daniel J Singer

The Problem of Scientific Modeling

to account for idealization and abstraction, and in addition, this account potentially suffers from a problem of triviality. Pincock (2005) shows that “The partial structures program is forced . . . into thinking that all idealizations are treatable as a series of approximations” (2005, p. 1255). Idealizations, such as the inclusion of intentionally false presuppositions in theories, cannot be accommodated in the partial structures approach because the approach is forced to think of idealizations as approximations for many of the same reasons as the other morphism approaches. The approach also cannot differentiate “well motivated from poorly motivated idealizations when the two idealizations agree on the models or isomorphisms at issue” (Pincock, 2005, p. 1257). This aspect of this approach is similar to how the homomorphism approach fails to account for idealization. While the partial structure characterization of models appears to be a step in the right direction because of its ability to represent dynamical theory change, the use of da Costa’s and French’s notion to characterize the “applies to” relationship as a mangled form of isomorphism is still subject to many of the objections against the other morphism approaches. Even though the partial isomorphism account pursues a good goal of producing a system that allows for the flexibility in the modeling relationship, it fails not only because it, like the other morphism approaches, cannot give a sufficient account of idealization, but also because, as I show in the next section, it lacks the semantic content necessary to make this approach compatible with an adequate account of modeling in practice.

A Case for No Acceptable Relationship

So far, I have presented a range of intuitive possibilities for the theory-data relationship and established that they all face significant obstacles including being too rigid or being reduced to triviality. Yet, the intuition that there may be some formal relationship that can rigorously capture the relationship still stands. In this section, I present a fundamental objection to the project of providing such a formal relationship: I claim that for the structural realist, the semantic view with any species of abstract structures as models is incompatible with a satisfactory account of scientific modeling. The argument gets its force from the fact that abstract, mathematical structures fail to have the intensional power essential to modeling in the natural sciences. Recall that using the semantic view of theories allows the structural realist to specify structure-centric models—models that have structure just like the structure of the things they model. Further, isomorphism is traditionally taken to be a type of formal mapping that exists between two models just in case they are structurally identical. Therefore, 13

Daniel J Singer

The Problem of Scientific Modeling

on the structural realist’s account, two models should be isomorphic if they espouse the same structure (since they would be structurally identical). In standard mathematical logic though, two models are isomorphic just in case the models share a signature (in some given language) and there is a bijection such that a tuple of elements is in a relation Ri of the first model just in case its image is in Ri of the second model. Notice that in the formal definition of isomorphism, two models can have the same structure, but if they are formulated with different signatures, they are not isomorphic: Consider a simple case where one model is identical to another except that one of the models has an additional constant symbol in its signature. Then, by the formal definition, these models cannot be isomorphic. But clearly, the structural realist would want to consider these to be the same model since they share the same structure. So, we have two separate notions of isomorphism: two models are formally isomorphic just in case they satisfy the formal definition, and they are structure-preservingly isomorphic just in case they have the same structure in the structural realist’s formulation of structure-centric models. Essentially, two models are structure-preservingly isomorphic if they are formally isomorphic under some renaming of the structural elements of the model and excluding constant symbols. Particularly for the structural realist, isomorphism is supposed to guarantee identity of structure, so formal isomorphism is too strict as it would not allow that two models with identical structure are isomorphic due of the language of their formulation.11 Therefore, unless otherwise specified, I will use “isomorphism” to mean structurepreserving isomorphism. With this in mind, let us turn to the main argument, which is most easily presented with an example: Let’s say there is a theory about hormone levels in newborn females. For simplicity, assume we are using Tarskian set-theoretic representations of models and that there is a single model A, which is the theory. The theory is about two hormones called “foo” and “bar.” The theory says that foo varies between 10 units and 12 units according to the equation d = sint + 11, where d is the units and t is the time since birth. Similarly, bar varies between 0 and 2 units according to d = sint + 1. So, the domain D of the model A will be the space ℜ2 , where the first dimension represents time and the second represents 11 Swanson

(1966, p. 298) seams to imply the distinction between the two types of isomorphism. N.B. Swanson does not make this point explicitly. Instead, Swanson’s goal is to provide a “classical characterisation of models.” In doing so, he suggests that two models are isomorphic essentially if the relations of one model line up with the relations of the other via a bijective map. We can make this more precise by saying that two models (with finitely-many relations) are isomorphic (to the philosopher of science) if there is a bijective map G from the signature of the first model to the signature of the second such that there is a bijective map between the two models such that a tuple of the first model is in a relation Ri of the first model just in case its image is in the relation R j = G(Ri ) in the second model.

Daniel J Singer

The Problem of Scientific Modeling

units of hormones. Then there will be a predicate F, which picks out the curve for foo in the domain, such that some element (x, y) ∈ D is in the relation F just in case y = sin x + 11. Likewise, there will be another relation B for bar such that some element (x, y) ∈ D is in the relation B just in case y = sin x + 1. Then the model A is hD, F, Bi12 and can be thought of as a graph of the hormone functions in a Cartesian plane (i.e. as two sine curves, one above the other). Assume that the theory accurately describes these hormones. Let’s also say that there is another theory about hormone levels in newborn males. Assume that the male theory also has one set-theoretic model B. The male theory says that newborn males, like females, have two hormones that vary sinusoidally, foo and bar. But in this case, bar varies between 10 units and 12 units according to the equation d = sint + 11, where d is the units and t is the time since birth, and foo varies between 0 and 2 units according to d = sint + 1. So, B will have the same domain D as above, and two relations F and B such that the extension of F in B is the extension of B in A and similarly for B in B. So, B will have the same signature as A, hD, F, Bi. Using structure-preserving isomorphism, A and B are isomorphic since they are identical in structure up to a renaming of the relations. Also, since the role of the model is to be structure-centric, if two models share the same structure (i.e. are structure-preservingly isomorphic or structurally identical), then the two models are identical. Hence, A and B, to the structural realist, are identical models. Further, the semantic view says that a scientific theory is just a collection of models, so since both the male and female theories have the same collection of models, according to the semantic view, the two theories are identical. But surely, any view that delivers this conclusion is confused! The female theory, if it were used in practice with males, would discord highly with the observed behavior of the hormones in the males, and vice versa. The example suggests that theories in the abstractmodel semantic view differ in two significant ways from theories in scientific practice: (1) in practice, theories are about phenomenological targets, and (2) in practice, models of theories assign different parameters of target systems to the parts of the models. This example can be generalized to support the conclusion that no theory-data relationship, under the general abstract-model semantic view, can give an adequate account of practice: Assuming the semantic view, let’s say D is the class of all theory models and all data models for some abstract structure-centric representational scheme for models. There are models f , f 0 , m, m0 ∈ D such that f corresponds to the theory model for the theory of the female hormones described above, f 0 corresponds to the data of the hormones for the female, and similarly for the models m and m0 . Assume (for reductio) that there were 12 Here

I am assuming that the structure of the space ℜ2 is included in D.

Daniel J Singer

The Problem of Scientific Modeling

some binary relationship R(x, y) for all x, y ∈ D such that R(x, y) just in case x applies to y. For such a relation to be consistent with scientific practice, we would expect R( f , f 0 ) and ¬R( f , m0 ) since the female theory does model the female data and does not model the male data. But, the structural realist recognizes that the structure underlying the female and male data is equivalent, so f 0 = m0 . Hence, R( f , f 0 ) ≡ R( f , m0 ), so R cannot be consistent with scientific practice. One may object that this argument seems to depend on the theory-data relationship being a binary relation between two abstract models: A simple response to my argument might suggest that the relation R also be considered a function of another consideration, such as the scientist’s intensions like his association of the female model with the female and not the male. This turns out not to be a problem for the argument: Assume that R(x, y, i) is a ternary relation where x, y ∈ D and i is the intensions of the scientist. Let f , f 0 , m, m0 ∈ D be the same models as in the previous example. Then say that the scientist gathers some data that show that the hormone foo varies between 10 units and 12 units according to the equation d = sin(t) + 11 and that bar varies between 0 and 2 units according to d = sin(t) + 1. So, then we would like the scientist to conclude that the female theory applies here and the male theory does not. The abstract-model semantic view says that the female theory and male theory are simply identified with the models, namely f and m respectively. But, as we saw above, a structure satisfies the female theory iff it satisfies the male theory, so the theories are co-extensional in D and by the semantic view, they are the same theory. Here, the scientist’s intensions fall out since the scientist is required to conclude that the data model applies to the female theory and the male theory or applies to neither, because they are the same theory. So, even the inclusion of the intensions of the scientist is not sufficient to save this approach. The main problem here is that the abstract-model semantic view takes theories to be something they are not: theories in the real scientific practice have intension; to satisfy the structural realist, they may be about structure, but they are also about physical target systems. The harmonic oscillator model of spring is not the same as a harmonic oscillator model of a RLC circuit even though they are both harmonic oscillators. The structural realist employing the abstract-model semantic view is forced to abstract the notion of theory so far that it is no longer connected to actual target systems. Without including this type of intension in our notion of theory, no philosophy of science will be able to give an adequate account of practice. In the next section, I show that for the structural realist who accepts the semantic view to be able to give an adequate account of scientific modeling, he will either have to reject the abstract-model version of the semantic account or else include in 16

Daniel J Singer

The Problem of Scientific Modeling

his account of models some type of interpretation that allows models to be about physical systems. But, I will show that neither of these approaches can perfectly preserve the goal the structural realist.

Fixing Up Structural Realism

By the above arguments, I have shown that the structural realist picture with an abstractmodel semantic view of theories fails because, in scientific practice, theories are about physical systems whereas in the semantic account, they are not. In this section, I present two ways that the view can be adjusted to accommodate practice. First, I sketch a variation on abstract models, which will allow them to be about the objects of scientific inquiry. Then I show how the use of Godfrey-Smithâ&#x20AC;&#x2122;s type of models may solve the problem as well. Finally, in the last section, I argue that neither solution preserves the original intention of the structural realist view.

6.1

Making Models Extend

The abstract-model semantic view takes a model to be an abstract mathematical model, and a theory is a collection of those models. By the previous argument though, this is insufficient, and one option for fixing the problem is to expand the conception of models to include at least the intended scope and semantic assignment of parameters to targets. So, a model is better thought of as a tuple hM, Ii where M is the model as in the unaltered account, and the semantic assignment I is a non-empty set of tuples ht, ii where t is a target system, which qua Weisberg (2007) is abstracted away just to the parameters and variables of interest, and i is a (partial) map from the model (e.g. the domain of set-theoretic model or the dimensions of the domain of the state-space model) to the parameters and variables of the target system. Take for example the theory about female hormones above: As stipulated, it consists of just one model f . So, in this new conception of models, the theory would be just one model conceived of as h f , {ht1 , i1 i, ht2 , i2 i . . .}i where each t p is phenomenological physical target system to which the model is intended to apply (such as the newborn female Stacy), and each i p is a (partial) map from f to t p . In this sketch, this solution seems to allow only for standard sorts of targets of inquiry, but the approach can easily be expanded to include non-existent and generalized targets. The clear benefit of thinking of scientific models in this way is that it resolves the

Daniel J Singer

The Problem of Scientific Modeling

problem of intension by extensionally defining the scope of a model. For example, if the model of the female hormones is h f , {ht1 , i1 i, ht2 , i2 i . . .}i, then the intension of the model can be read right off of the model itself: each t p is some physical target system to which the model should apply, and each i p specifies the way in which the model hooks up with t p . Secondly, using this definition of models, we can make sense of scientists’ statements such as “It’s a model where population growth is exponential.” In the old system, models do not have ‘population growth;’ they are stuck with abstract relations. In the new system, a model’s population growth can be identified with the parts of the model that map to the target system’s population growth. Further, in this system, a theory’s scope can be simply conceived of as the class of target systems associated with its models, i.e. the union of the sets of t p for each model in the theory. This method of defining models allows two models to share the abstract mathematical part of the model without being identical, as the models could differ in their intensions. Here, it seems more plausible that an acceptable theory-data relationship could be found since an “applies to” relation that respects modeling practice could avoid the problems presented above by being a function of the new intensional parts of the models. On this approach, a formal theory-data relation could conclude that the female theory applies to the female and not the male by saying that the relation obtains only if the theory model and data model are about the same physical system, that there is a non-empty intersection between the intensional parts of the models. One added benefit of this approach is its ability to differentiate and generalize models: Pretheoretically, in the example above, three scientific theories naturally fall out: There is one theory of the hormones of each gender, and then there is a generalized theory of newborn hormones. This is analogous to how the SHO theory is a generalized theory of both the more specific theory of spring displacement over time and the more specific theory of RLC circuits. In this approach, all of those theories are naturally definable and differentiable: the female and male are definable as above, and the generalized theory consists of the combination of the two (which still has only the one mathematical model). Another benefit of this solution, that the next proposed solution does not have, is that it preserves the use of abstract models and hence would allow a reformulated type of structural realism to use abstract models to represent structure.

Daniel J Singer

6.2

The Problem of Scientific Modeling

Making Models Intend

Recall that so far in this discussion, I have taken the semantic view of theories to be a view that employs abstract mathematical models, such as set-theoretic models or state-space models, to define a theory. As we’ve seen, Godfrey-Smith advocates a different approach to models, where models are considered to be imaginary objects that would be concrete if they were real. Godfrey-Smith gives the example of a populations model: I take at face value the fact that modelers often take themselves to be describing imaginary biological populations, imaginary neural networks, or imaginary economies. An imaginary population is something that, if it was real, would be a concrete flesh-and-blood population, not a mathematical object. . . . On the view I am developing, the model systems of science often work similarly to these familiar fictions. (Godfrey-Smith, 2006, p. 735) Weisberg (2007) calls this type of model a concrete model. Using concrete models fixes the problem of the abstract-model semantic view in the obvious way: concrete models come with intension built-in. In this conception, a model of the female hormone theory would contain an imaginary female with imaginary hormones that vary sinusoidally according to the above description, and a theory-data relationship would, presumably, operate such that a theory model of the female theory would apply only to a data model of female hormones. As discussed above, this view also has the benefit of being able to make sense of scientists’ statements about models, such as “The growth of this model’s population is exponential,” in the obvious way. While both of these solutions suggest that there are ways in which our conception of models can be changed to avoid my main argument, in the next section, I suggest that neither the concrete model type of modeling nor the extensional modification of modeling presented above preserves the original intension of the semantic view, that scientific theories can be identified with a family of models where “the meaning of the concept of model is the same in mathematics and the empirical sciences” (Suppes, 1960a, p. 289).

6.3

Two Steps Forward, One Step Back

While these two solutions seem to alleviate the problem of practice, both have aspects that partially run counter to the original spirit of the structural realist philosophy of science and the semantic view of theories.

Daniel J Singer

6.3.1

The Problem of Scientific Modeling

Against Making Models Intend

Recall that in section 2, I augmented Ladyman’s (1998) argument that the structural realist should employ the semantic view by arguing that the abstract-model semantic view should win out over the concrete-model semantic view. The point of employing the semantic view at all with the structural realist account is that identifying models with theories allows for a more lucid account of the nature of a theory’s structure, which is needed to flesh out the structural realist’s claim. Using concrete models in the structural realism picture brings us back to square one in the quest to elucidate the structure of a theory, as identifying the structure of an imaginary model may be as hard as identifying the structure in the target system itself. Using concrete models might bring the quest back even further than a recourse to the “received” view: If a concrete model is just some fictional or imaginary phenomenon, whereas the “received” view at least has equations and relations to help define theory structure, picking out structure from a concrete model seems no simpler than picking structure straight out of the target system itself, which must be the goal of all of science for the structural realist. Additionally, Weisberg (2007) points out that it is not clear on this account how to establish identity of models. If we take models of population biology to be imaginary populations, when are two imaginary populations the same population? It may seem that two people cannot share an imaginary populations, so do they have separate models? Further, since the model should be “concrete if they were real,” should our models differ if for example, in one model, emus have blue eyes, and in another, they are brown? The abstractmodel approach ran into problems because it conflated non-equivalent models, and while the concrete-model approach seems to solve that problem, its use seems to suggest that it has the opposite problem. Other contemporary semantic view conceptions employ set-theoretical (Suppes, 1970), state-space (van Fraassen, 1980), or partial structure (da Costa & French, 2003) conceptions of models; all of these conceptions view models as mathematical models, whereas the concrete-model approach does not. There are clear benefits to preserving the mathematical portion of the model that the concrete model approach loses, including the ability to employ formal tools of analysis such as morphisms to evaluate the models against other models. So clearly, if the structural realist were to employ concrete models, he would not be fully preserving the original theme of the semantic view that “the meaning of the concept of model is the same in mathematics and the empirical sciences” (Suppes, 1960a, p. 289).

Daniel J Singer

6.3.2

The Problem of Scientific Modeling

Against Making Models Extend

In section 6.1, I suggested an extensional solution to the problem of theory intension for the semantic view, which expands the infrastructure of abstract models to allow them to be extensionally about target systems. We saw there are challenges for the concrete model approach to overcome to be employed by the structural realist, but unlike the concretemodel approach, the extensional solution at least partially saves the semantic view’s claim that “the meaning of the concept of model is the same in mathematics and the empirical sciences” (Suppes, 1960a, p. 289) by adding to the model new infrastructure that allows it to define its target systems extensionally. Any solution must abandon at least some aspect of this claim though, since as my argument above shows, any correct conception of scientific models must include some kind of model intension, and mathematical models do not, under any standard formulation, have intension. That said, the extensional solution at least preserves the idea that there is a strong connection between mathematical and scientific models, a claim that should be acceptable to many of the supporters of the original semantic view. I suggested above that for the structural realist to make use of the semantic view of theories, he must employ a structure-centric conception of abstract models. This assertion assumes that such a formulation is feasible; but for the extensional solution to be implemented, we would actually need such a formulation. In this section, I demonstrate that formulating such a conception based on currently-used conceptions will be difficult and may pose a large a problem for the structural realist using this conception of models. The extensional solution calls for the addition of a semantic assignment I to be associated with each mathematical model in the standard semantic view. Each I contains a partial map i from the entities that compose the model to aspects of the physical target systems. Recall that the main claim of the structural realist is that a scientific theory of a phenomenon is about the structure of that phenomenon. So, if the semantic view, including the extensional solution’s variation, is correct, then for a scientific model to be useful, part of its map i must map to structure in the world, or else the model is not about structure in the world. Unfortunately, in current conceptions of models, it is unclear what part of the model, if any, could be mapped to the structure of a phenomenon in the world. Consider Suppes’ set-theoretical formulation of models and the partial structure formulation of da Costa and French discussed above: Both of these model conceptions take models to be some domain of individuals with relations defined extensionally on that domain. A standard logic-class

Daniel J Singer

The Problem of Scientific Modeling

example of this type of model would be a domain of people with a binary relation that defines lovers on the domain. Such a model consists solely of the domain and relations (which are simply sets of tuples of individuals), so any map i from the model to the structure in the world must be a map from individuals or tuples of individuals in the model, because nothing else exists in the model. Now consider a model of a simple love triangle. Surely, the information about structure that is represented by such a model is contained in the relations between the lovers, not in the lovers themselves. Consequently, if the structural realist were to use this formulation of models, he would conclude that structure is represented as relations in the set-theoretical model, not in the individuals themselves, and that the map i maps from tuples of individuals to the structure in the world. The problem with this analysis becomes apparent when we consider the nature of the connection between the relation in the model and the structure in the world, when they are connected via the map i. The most natural way for the structural realist to characterize this relationship is to claim that the structure exists in the world just as the relation exists in the model: as a relation between relata. However, if he does this, he posits the existence of some individuals in the world that have properties at least described by the structure (conceived as a relation) in which they are relata. But, if the structural realist does this, he cannot preserve his original goal of avoiding the pessimistic induction argument. After all, the original structural realist position evades the argument precisely by not positing specific entities, and as we saw above in the context of Ramsey sentences, reference to entities by description does not suffice to avoid positing entities. So, the structural realist cannot simply think of structure in the world being just like relations in models, and he is forced to say that a map i that associates parts of the model with structure in the world is a map that maps from relations in the model to something very unlike that relation in the world.13 The challenge for the structural realist using the extensional solution to the problem caused by the lack of model intensionality is to figure out how to account for the relationship between relations in abstract models and structure in the world given that the intuitive notion of identifying the nature of the world with the nature of the model fails. When dealing with problems of representation, we would like that the representer, in some sense, â&#x20AC;&#x153;look likeâ&#x20AC;? the thing represented, which would imply the existence of a kind of simple connection between them. But, the structural realist using the extensional solution is presented 13 A similar conclusion can be drawn out from the state-space conception of models by noting that a state in a state-space model is an abstract representation of some parameters of an individual in the target system. So, any map i would have to be a map from states of individuals or tuples thereof.

Daniel J Singer

The Problem of Scientific Modeling

with a much harder problem since he knows that the thing related from (namely, relations in abstract models) is fundamentally different from the thing related to (namely, structure in the world). For this solution to save the structural realist, he must overcome this problem.

Conclusion

Above I offer two solutions, (1) using concrete models and (2) adding infrastructure to the abstract model account, by which the semantic view can be adjusted to avoid the problem of structural realism not admitting a sufficient account of scientific modeling. Both solution fail to completely preserve the original aim of the semantic view since they deny that scientific models can be thought of solely in terms of mathematical models. Even though the concrete model solution solves the problem of theory intension, it is not useful to the structural realist because it fails to elucidate the notion of structure in a model, which is a main draw of the semantic approach to the structural realist. On the other hand, using the infrastructure solution suggests large hurdles for the structural realist to overcome. The arguments here show that the structural realist is in a much less sound position than the current literature, particularly Ladyman (1998), suggests. In the beginning, the structural realist is inclined towards the definition of scientific theories offered by the semantic view because it appears to provide the structural realist with a structure-centric account of theories. The structural realist hoped to employ the view to create theories that avoid positing theoretic ontologies by focusing solely on structure. But, the arguments here exhibit that the semantic view is faulty in precisely that respect: without an ability to distinguish between the theoretical ontologies employed by scientists, the semantic view is forced to equate non-equivalent theories. Any solution to this problem must force theories to be about physical phenomena, which appears to require the theory to be committed to the existence of some kind of entities in the world. So while the two approaches above appear to correct the exhibited problem, it appears that no sufficient solution to the problem will be able to reach the goal that the structural realist has in employing the semantic view, to focus on theory structure and avoid ontology.

References Achinstein, P. (1968). Concepts of science: a philosophical analysis. Baltimore: Johns Hopkins Press. Barile, M. (2002). One-to-one. From MathWorldâ&#x20AC;&#x201C;A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/One-to-One.html. 23

Daniel J Singer

The Problem of Scientific Modeling

Black, M. (1962). Models and metaphors. In M. Black (Ed.), (pp. 219–243). Ithaca, NY: Cornell University Press. Cartwright, N. (1989). Nature’s capacities and their measurement. Oxford: Oxford University Press. da Costa, N. C. A., & French, S. (2003). Science and partial truth. Oxford: Oxford University Press. Downes, S. (1992). The importance of models in theorizing: a deflationary semantic view. PSA, 1, 142–153. French, S., & Ladyman, J. (1999). Reinflating the semantic approach. International Studies in the Philosophy of Science, 13(2), 103–121. Godfrey-Smith, P. (2006). The strategy of model-based science. Biology and Philosophy, 21, 725-740. Ladyman, J. (1998). What is structural realism? Studies in History and Philosophy of Science, 29, 409–24. Laudan, L. (1981). A confutation of convergent realism. Philosophy of Science, 48, 19–49. Lloyd, E. A. (1994). The structure and confirmation of evolutionary theory (Second ed.). Princeton: Princeton University Press. Maxwell, G. (1970a). Minnesota studies in the philosophy of science, iv. In (p. 181-192). Minneapolis: University of Minnesota Press. Maxwell, G. (1970b). The nature and function of scientific theories. In R. Colodny (Ed.), (pp. 3–34). Pittsburgh: University of Pittsburgh Press. Morris, R. (2003). The last sorcerers: The path from alchemy to the periodic table. Washington, D.C.: Joseph Henry Press. Pincock, C. (2005). Overextending partial structures: Idealization and abstraction. Philosophy of Science, 72, 1248–1259. Suppe, F. (Ed.). (1977). The structure of scientific theories. Chicago: University of Illinois Press. Suppes, P. (1960a). A comparison of the meaning and use of models in mathematics and the empirical sciences. Synthese, 12, 287–300. Suppes, P. (1960b). Models of data. In E. Nagel & P. Suppes (Eds.), Logic, methodology and the philosophy of science: Proceedings of the 1960 international congress (p. 251-61). Stanford: Stanford University Press. Suppes, P. (1970). Set theroretical structures in science. Mimeographed lecture notes, University of Stanford. Swanson, J. W. (1966). On models. British Journal of Philosophy of Science, 17(4), 297–311. Teller, P. (2001). Twilight of the perfect model model. Erkenntnis, 55, 393–415. van Fraassen, B. C. (1970). On the extension of beth’s semantics of physical theories. Philosophy of Science, 37(3), 325–339. van Fraassen, B. C. (1980). The scientific image. Oxford: Oxford University Press. Weisberg, M. (2007). Models. (unpublished manuscript) Worrall, J. (1989). Structural realism: The best of both worlds? dialectica, 43, 99–124.