Ian G. Barbour is Professor of Science, Technology, and Society at Carleton College, Northefiled, Minnesota. He is the author of Myths, Models and Paradigms (a National Book Award), Issues in Science and Religion, and Science and Secularity, all published by HarperSanFrancisco. Published by Harper & Row, New York, Hagerstown, San Francisco, London, 1976. This material was prepared for Religion Online by Ted and Winnie Brock.

There are in science a number of different kinds of model which serve a diversity of functions.í They are used, that is, for very diverse purposes. First there are experimental models which can actually be constructed and used in the laboratory. These include replicas or ëscale modelsí representing spatial relationships, and ëworking modelsí representing temporal sequences. From a wind-tunnel model of a proposed airplane design, the lifting force of a particular wing structure can be estimated. In an ëanalogue modelí, certain features of one system are simulated by the behaviour of another system in a different medium -- for instance, a hydraulic flow model of an economic system, or an electrical circuit model of an acoustic system. Such models are used to solve practical problems when it is difficult to experiment on the primary system, or when the relevant mathematical equations are unknown or too complex to solve. In these cases one physical system is actually built to serve as a model of another physical system.²

Second, at the opposite extreme, there are logical models. The logician or the pure mathematician starts from the axioms and theorems of a formal deductive system. A logical model is a particular set of entities which satisfy these axioms and theorems. For example, a set of points and lines in geometry is a logical model for Euclidís formal axioms. The mathematician uses it to illustrate the abstract system and to give a possible interpretation of it. Note that here he is dealing entirely in the realm of ideas; neither the formal system nor the model of it are physical systems.³

Third, mathematical models are between these two extremes. They are symbolic representations of quantitative variables in physical or social systems. Examples might be: equations proposed to express the relation between supply and demand in economics, or the growth of a population in time. A mathematical model may in turn be physically represented by the electrical circuits of a computer; computer models of economic, political, military and transportation systems are widely used today. At the moment, the point to note is that a mathematical model resembles the primary system only in formal structure; there are no material or physical similarities. It is a symbolic representation of particular aspects of a physical system, and its chief use is to predict the behaviour of the latter.

My main concern in this chapter is a fourth kind, theoretical models. These are imaginative mental constructs invented to account for observed phenomena. Such a model is usually an imagined mechanism or process, which is postulated by analogy with familiar mechanisms or processes. I will maintain that its chief use is to help one understand the world, not simply to make predictions. But I will also claim that it is not a literal picture of the world. Like a mathematical model, it is a symbolic representation of a physical system, but it differs in its intent to represent the underlying structure of the world. It is used to develop a theory which in some sense explains the phenomena. And its origination seems to require a special kind of creative imagination. In the subsequent chapter theoretical models will be compared with models in religion.

A theoretical model, then, is an imagined mechanism or process, postulated by analogy with familiar mechanisms or processes and used to construct a theory to correlate a set of observations. I will call the source of the analogy ëthe familiar systemí, where ëfamiliarí means better understood rather than everyday. The model drawn from the familiar system suggests a theory. It also suggests possible relationships between some of the terms of the theory and some observation terms; these correlations linking theory and observation are called ërules of correspondenceí. A theoretical model, in short, is used to generate a theory to explain the behaviour of an observable system. The relation of theory and observation is examined in Chapter 6 below. In the present chapter attention is directed to the distinctive role of models in the generation of theories.

Let me give an illustration from physics since it is the scientific field I know best: the ëbilliard-ball modelí of a gas. Consider a box full of a gas, such as air, and imagine that the gas is composed of very tiny elastic spheres bouncing around. If one assumes that the mechanical behaviour of the hypothetical spheres is similar to the familiar behaviour of colliding billiard balls, a theory can be developed (the Kinetic Theory of Gases). The theory involves equations interrelating the mass (m), velocity (p), energy and momentum of the hypothetical spheres. Of course none of these theoretical properties can be observed. But the model also intimates that some theoretical terms might be related to observable properties of the gas (for example, the momentum change of the ëparticlesí colliding with the containing wall might be identified with the pressure of the gas). With these assumptions one can derive several of the well-known experimental Gas Laws -- Boyleís Law, for instance, which states that if the volume (V) of a gas is reduced by 50% (by compressing the air in a bicycle pump, for example) then the pressure (P) of the gas will double.

The model thus leads to a theory, and the theory accounts for patterns in experimental observations. These ralationships are portrayed schematically in the diagram.

The double arrows signify the formal deduction of experimental laws from the theory together with rules of correspondence. Three features of the billiard-ball model, and others like it, should be noted:

1. Models are analogical. Similarities with a familiar situation are posited in some respects (the positive analogy), and differences are posited in other respects (the negative analogy). Thus our hypothetical tiny elastic spheres were assumed to have mass and velocity, as billiard balls do, but not colour. Notice that the analogies postulated may be physical (e.g., elasticity and mass), and not simply formal as in the case of logical or mathematical models. In the origination of a novel theory the scientist may propose a model incorporating analogies drawn from several familiar situations, together with radically new assumptions. In the diagram all the lines going into the model are shown dotted because its origins lie in an act of creative imagination and not in purely logical inference. In general, we would have to show additional dotted arrows coming in from other familiar situations at the left. In imagining a model there is implicit or explicit reference to what is familiar and previously intelligible, but there is also considerable novelty and freedom. One can assign to it whatever properties one thinks might contribute fruitfully to the theory.

The history of science provides many examples of this combination of analogy and innovation in the creation of models which were useful in generating theories.⁴ The ëBohr modelí of the atom, in which ëplanetaryí electrons revolve in orbits around a central nucleus, resembles the solar system in certain of its dynamical properties; but the key assumption of quantum jumps between orbits had no classical parallel at all. Again, the model of vibrating oscillators was prominent in the development of a theory of the specific heat of metals. Among more recent examples is the ëliquid drop modelí of the nucleus. Somewhat different in character, but equally crucial in the origination and interpretation of a theory, is the model of an ideal heat engine in the field of thermodynamics. In each case the model aided the formulation of the equations of the theory and also suggested rules of correspondence between certain theoretical terms and observable variables.

2. Models contribute to the extension of theories. The use of a model may encourage the postulation of new rules of correspondence and the application of a theory to new kinds of phenomena. Thus the equations of the Kinetic Theory were applied to new experimental domains (including gas diffusion, viscosity and heat conduction) which involved types of observation very different from those of the gas laws. A model may also be crucial in the modification of the theory itself It was the model, not the formalism of the theory, which led to the hypothesis of particles having a finite size and attracting each other; when the theory was thus amended, van der Waalís equations for gases under high pressure could be derived. The revised model (elastic spheres with attractive forces) departs from the simple billiard-ball model, yet it would never have occurred to anyone without the latter. As Mary Hesse points out, clues for the modification of a theory often arise in exploring the ëneutral analogyí -- that is, the features of the familiar situation whose inclusion in the model has neither been explicitly affirmed nor denied.⁵ She argues that because of its suggestiveness and open-endedness, a model is a continuing source of plausible hypotheses:

3. A model is intelligible as a unit. It provides a mental picture whose unity can be more readily understood than that of a set of abstract equations. A model is grasped as a whole; it gives in vivid form a summary of complex relationships. It is said to offer ëepistemological immediacyí or ëdirect presentation of meaningí. Because of its vividness and intelligibility it is frequently used for teaching purposes to help a student understand a theory. But even at the critical stages of scientific discovery itself, scientists report that visual imagery often predominates over verbal or mathematical thinking, according to several studies.⁷ Images are creative expressions of the human imagination in the sciences as in the humanities. There are of course, no rules for creativity; but it has been pointed out that analogies, models and metaphors are common in the search for new kinds of connection and new ways of looking at phenomena.⁸ Campbell suggests that models also provide a distinctive form of intellectual satisfaction which the scientist values.⁹

Several words of caution are needed, however. The ëintuitive intelligibilityí of a model is no guarantee at all concerning its validity; deductions from the theory to which the model leads must be carefully tested against the data and, more often than not, the proposed model must be amended or discarded. Models are not advanced as guaranteed truths; they are used to generate plausible hypotheses to Investigate. They are a source of promising theories to test. Again, a model need not be picturable, though it must be conceivable, in both science and religion. Visualizable features may be selectively suppressed, as when we imagine colourless elastic spheres. In quantum physics mechanical models are given up and there are severe limitations on the use of visualizable models. In a later chapter I will maintain, nevertheless, that even in quantum physics there are models with the three characteristics I have mentioned -- models which are analogical, extensible and intelligible as units.

What is the relation between theoretical models and the world? There have been four alternative views of the status of models, and each has been closely associated with a particular view of the status of theories:

1. Naive realism. With a few exceptions, most scientists until the present century assumed that scientific theories were accurate descriptions of ëthe world as it is in itselfí. The entities postulated in theories were believed to exist, even if they were not directly observable. Theoretical terms were said to denote real things of the same kind as physical objects in the perceived world. Theoretical statements were understood as true or false propositions about actual entities (atoms, molecules, genes, etc.). The main difficulty with naive realism is that we have no access to ëthe world in itselfí, especially in the sub-microscopic domain; there is no way to compare a theory directly with ërealityí. Moreover, theoretical concepts are not given to us by nature; they are mental constructs, and often are only very remotely connected with observations. The history of science does not show the kind of simple convergence and cumulation which naive realism would lead one to expect; instead, there have been radical conceptual changes and paradigm shifts, as we will see in Chapter 6 below.

Corresponding to a naively realistic view of theories is a literalistic view of models. Models were taken as replicas of the world, ëpictures of realityí. Lord Kelvin said in 1884: ëI never satisfy myself until I can make a mechanical model of a thing. If I can make a mechanical model I can understand it.í¹⁰ But such literalism always runs the risk that one will push an analogy too far and neglect important differences between the new situation and the familiar analogue. Thus the analogy between light waves and sound waves, which was so fruitful at one stage in the history of science, led to the erroneous assumption that light, like sound, must be transmitted through a medium (the hypothetical ëaetherí). The nineteenth-century predilection for picturable mechanical models has been thoroughly undermined by quantum physics which has shown that the atomic world is very unlike the world of familiar objects.

2. Positivism. To the early positivists, a theory is a summary of data, a formula for giving a resumé of experience. Theoretical concepts are merely convenient categories for classifying observations. In British thought there has been a strong empiricist tradition, going back to Bacon, Hume, and Mill, which has emphasized the observational side of science. When physicists in the early twentieth century used concepts further and further removed from observations, positivist philosophers of science, such as Bridgman and Carnap, looked on these abstract concepts as purely mathematical symbols for correlating observations. They wanted to accept only theoretical terms which could be ëoperationally definedí in observational terms; they claimed that all theoretical statements should be exhaustively translatable into observation statements."

However, positivists were unable to carry out in practice their programme for translating theoretical into observational statements. It was realized also that scientific progress would be hindered if their programme could be achieved, since the extensibility of a theory arises from its application to new situations. A theory may be relevant to an indefinite number of new kinds of observation. We will also see in Chapter 6 that the scientist never has the bare data, uninterpreted by theory, which positivists sought; there is no neutral observation-language, since ëall data are theory-ladení.

Positivists have usually dismissed models and held that theories can be inferred directly from observations by a process of inductive generalization. (In the diagram above, they want to keep only the right column, and both the model of ëtiny elastic spheresí and the analogy with billiard-balls would be omitted; the double arrow of inference would have to point upward to represent the induction of theory from data, rather than downward to represent the deduction of expected observations from theory.) I have argued, on the contrary, that models often play an essential part in the origins and continued development of scientific theories. Theories are the product of creative imagination, often mediated through models, and not the result of simply generalizing from the data.

3. Instrumentalism. Instrumentalists agree with positivists that theories are not representations of the world. They hold that theories should not be judged by truth or falsity, but by their usefulness as calculating devices for correlating observations and making predictions. Toulmin calls theories ëtechniques for making inferencesí, whereby experimental predictions can be made from initial observations.¹² Theories are also organizing guides for directing research and practical tools for technical control. Unlike positivists, however, instrumentalists acknowledge that theories are the product of manís creative imagination. They maintain, moreover, that theoretical terms cannot be exhaustively translated into equivalent observation terms. Theoretical terms are not eliminable, and the most powerful concepts may have no direct correspondence to observation terms.

The corresponding instrumentalist view of models can be called fictionalism. It is said that models, too, are neither true nor false, but only more or less useful mental devices. They are regarded as temporary psychological aids in setting up theoretical equations; having served their purpose, they should be discarded. (One would start, as in the diagram above, with a model leading to a theory, and the deductive arrow would point downward from theory to data; but once one had the theory, the model would be erased as superfluous.) Models are ëdisreputable understudies for mathematical formulasí, or in Duhemís words, ëprops for feeble mindsí.¹³ Even the more cautious instrumentalists, such as Richard Braithwaite, consider models to be dispensable; they are only ëa convenient way of thinking about the structure of the theoryí.¹⁴ Braithwaite urges us to avoid all reference to such unobservable entities as elastic particles. I will examine this view in the following section.

4. Critical realism. Like the naive realist (and unlike the instrumentalist), the critical realist takes theories to be representations of the world. He holds that valid theories are true as well as useful. To him, science is discovery and exploration as well as construction and invention. The scientist, he insists, seeks to understand and not just to predict or control. Unlike the naive realist, however, the critical realist (along with the instrumentalist) recognizes the importance of human imagination in the formation of theories. He acknowledges the incomplete and selective character of scientific theories. Theories, in short, are abstract symbol systems which inadequately represent particular aspects of the world for specific purposes. The critical realist thus tries to acknowledge both the creativity of manís mind and the existence of patterns in events not created by manís mind. Descriptions of nature are human constructions but nature is such as to bear description in some ways and not others. No theory is an exact account of the world, but some theories agree with observations better than others because the world has an objective form of its own.¹⁵

I will be defending a critical realism concerning theoretical models, a position between literalism at the one extreme and fictionalism at the other. Let us grant that a model is a mental construct and not a picture of reality. It is an attempt to represent symbolically, for restricted purposes, aspects of a world whose structure is not accessible to us. No direct comparison of model and world is possible. But let us preserve the scientistís realistic intent in his use of theoretical models. The extension of theories seems to require that models and the questions they suggest be assigned a more important status than instrumentalism allows. The scientist today usually takes his models seriously but not literally. Models are limited and inadequate ways of imagining what is not observable. They remain hypothetical; gases behave as if they were composed of tiny elastic spheres. The ëas ifí reflects both a partial resemblance and a tentative commitment.¹⁶ Leonard Nash puts it thus:

Of the four positions outlined above, naive realism and positivism have few defenders today. Instrumentalism has many adherents and merits detailed discussion. It is also of interest here because it closely parallels the claim that models in religion are ëuseful fictionsí. This phrase, once more, is not meant to imply reference to what is known to be false. It is not like a ëfictitious nameí, which is a kind of deliberate deception. Nor is it like a ëlegal fictioní (e.g., that a corporation is a person), which a court treats as if it were true, though it is known to be false. A ëuseful fictioní is a mental construct used instrumentally for particular purposes but not assumed to be either true or false. In this section I will try to analyse the instrumentalist position with a minimum of technical terminology, but the reader who finds the argument difficult to follow could proceed to the following section.

To Braithwaite, a scientific model is a temporary psychological aid in the formation of a theory. The model is dispensable once the theory has been elaborated; it is a ëheuristic deviceí and not in any sense a representation of reality.¹⁸ Braithwaite holds that the ideal scientific theory has only two components: an abstract calculus (that is, a set of axioms and derived equations whose terms are uninterpreted mathematical symbols) and a set of rules of correspondence relating some of these abstract symbols to observation terms. The postulates of the theory may originally have been embedded in a model, but they should be separated from it and stated as formal axioms. The theoretical terms of the calculus obtain their meaning indirectly from the observation terms and not from the model. There are, in this view, two interpretations of the abstract calculus of the Kinetic Theory: the initial interpretation in terms of imaginary elastic spheres, which can be ignored once the theory is worked out, and the subsequent interpretation in terms of observable pressures and volumes, which remain scientifically significant.

Now it seems to me that the instrumentalist account can be criticized in regard to each of the three characteristics of models mentioned earlier. First, by stressing mathematical isomorphism it gives prominence to formal analogies and neglects substantive analogies. But there have been many historical cases in which rules of correspondence were suggested by analogies between observations. Thus parallels between the brightness of light and the loudness of sound, and between the colour of light and the pitch of sound, gave the clues for applying a wave theory to light when a wave theory of sound was already familiar.¹⁹ As Achinstein points out, physical similarities in some features of a pair of situations provide grounds for the plausibility of investigating possible similarities in other features.²⁰ More typically, however, the substantive analogy is not observed but postulated, as when the physical properties of inertia and elasticity were attributed to the unobservable gas particles. Several recent articles have asserted that models have implicit substructures and complex associations lacking in the equations of the theory but essential to the continued growth of science.²¹

Second, having eliminated models, the instrumentalist seems unable to provide adequately for the extension of theories. Neither an uninterpreted formalism nor previous rules of correspondence give any clue as to possible new rules of correspondence or extensions to new types of observation. Moreover, a theory can be applied to a new domain only ig contrary to the instrumentalist thesis, its theoretical terms do preserve their meaning when new correspondence rules are formulated. The meaning of the theoretical terms comes from the model, not from the observation terms. It was precisely the concepts of mass and velocity, occurring originally in the theory of mechanics and attributed to the hypothetical gas particles, which suggested possible correlations with very different observation terms in the study of viscosity, or in Einsteinís explanation of the ëBrownian movementí. The concept of particle velocity also suggested further novel experiments, such as those with molecular beams. What occurred was not a change of meaning but a new way of testing relations among terms with unchanged meanings -- velocity, in this case. Or take the concepts of mass: ëwe can talk about the mass of a billiard ball, or the mass of a gas particle, or the mass of the moon, precisely because the concept of mass is not uniquely tied to any particular type of observation.²²

Third, the instrumentalist tends to neglect the importance of models because he is not concerned about the process of discovery. He pictures scientific theories as completed formal systems, and considers models to be of merely historical or psychological rather than logical interest. This seems a rather static view of science -- a logicianís ideal, perhaps, but one which can say nothing about the way theories originate or the way science actually develops. Scientists themselves seem to have little interest in setting up formalized axiomatic systems; there are in fact few, if any, clear examples of theories which have been completely axiomatized. There is no reason to think that scientific progress would be furthered if its concepts were replaced by bare uninterpreted symbols.

Furthermore, most instrumentalists hold that the goal of science is prediction -- which is achieved by equations (interpreted calculi) rather than by models. Their claim is that explanation is equivalent to

prediction; to explain an event, they say, is to subsume it under a law, which is equivalent to showing that the event could have been predicted from knowledge of the law and the boundary conditions.²³ This thesis has come under considerable recent criticism.²⁴ Toulmin has departed from his earlier instrumentalism; he now holds that theories and models have explanatory force because of the intelligibility and generality of their ideas; they yield a type of understanding which even the most accurate prediction-formula lacks.²⁵ He cites the fact that the Babylonians could predict eclipses with precision from time-series tables but could offer no reasons for their occurrence. This is too complex an issue to discuss here, but I would submit that if understanding rather than prediction is the goal of science, models cannot be replaced by predictive mathematical formalisms.

In addition, models contribute to the unity of knowledge. The presence of analogies in the structures of two or more theories promotes systematic integration and the linking of widely divergent domains. Nagel writes:

Nagel grants to models a continuing and irreplaceable role in the coherent extension and unification of scientific explanations. He does not treat them as temporary expedients which should be eliminated as soon as possible, but accords them an enduring and significant role in scientific thought.

Now a reformulated instrumentalism which acknowledges an enduring and significant role for models and which remains open and non-committal concerning their ontological status, is very close to critical realism. However, I would argue that critical realism provides a logical justification, lacking in instrumentalism, for the continuing role of models. Moreover critical realism is consonant with the scientistís quest for coherence. Instrumentalism can offer no objection to the employment of two contradictory theories or two inconsistent models if both are useful. Yet scientists do seek coherence, even when they are aware that they have not achieved it. Often they use a plurality of models, but they do not see them as unrelated to each other, and new discoveries have arisen from attempts to resolve conflicting theories. Even the use of complementary models in quantum physics (Chapter 5) does not negate this quest for coherence.

Finally, I will suggest in Section 5 below that scientists do actually view some kinds of models as making a tentative ontological claim of the sort which critical realism defends. Scientists are motivated by the desire to know and understand, and not simply to predict and control. They consider theories and models as making tentative truth-claims, beyond their usefulness as tools for classifying phenomena. In particular, they hold that there are entities in the world something like those described in the model; they believe there is some isomorphism between the model and the real structures of the world.

Scientific models seem far removed from literary metaphors. Yet there are some interesting parallels which warrant brief comment. In the previous chapter, I stated that a metaphor proposes analogies between the familiar context of a word and a new context into which it is introduced. There is a tension between affirmation and denial; in other words, both positive and negative analogy are present. For metaphors, as for models, it is the neutral analogy which invites exploration, and which prevents reduction to a set of equivalent literal statements. Metaphors were seen to be irreducible because they are open-ended.

I cited Max Blackís view of metaphor as the selective transfer of some of the familiar associations of a word; certain features of the new situation are emphasized and others ignored. The sentence ëMan is a wolfí leads us to construe man as wolf-like; metaphor, in general, encourages us ëto construe one situation in terms of anotherí. Black goes on to propose that scientific models are systematically-developed metaphors. A model suggests new ways of looking at a problematical situation by transferring some of the features of another situation which is better understood. ëIt may help us to notice what would otherwise be overlooked and to shift the relative emphasis attached to details -- in short, to see new connections.^í27 Black stresses the role of imagination in both the sciences and the humanities. Hesse follows Black and speaks of theoretical explanation in science as ëmetaphoric re-descriptioní. She notes that neither metaphor nor model is private or merely subjective in its use, since in both cases the ideas and implications associated with the familiar domain are shared by a community of language users:

Donald Schon has also given a protracted comparison of models and metaphors. He holds that both offer programmes for exploring new situations. Neither models nor metaphors subsume analogous situations under general concepts already formulated; instead, they both intimate a similarity not yet fully conceptualized. One is asked, as it were, to find features of the old in the new; one is offered new ways of looking at a phenomenon.²⁹ Harré has pointed out that many scientific terms are themselves metaphorical and carry an important component of meaning from their original context. Electrical ëcurrentí, for example, is not simply defined by ammeter readings but carries an implicit reference to the current in a river. Such ëpicture-carrying expressionsí, he says, are essential for the growth of science, and without them ëthe theory would lead nowhereí:

Even an analogy which was not essential to the formulation of a theory can influence its future development -- as, for example, when molecular biologists speak of the genetic ëcodeí of DNA molecules in terms of ëlettersí, ëwordsí, ësentencesí, and ëpunctuationí.

We will note later that in science there is no sharp line separating theoretical language from observational language; the distinction is relative, shifting, and context-dependent. All observation-reports are theory-laden; the theoretical framework influences what is taken to be ëdataí. There is a close parallel in the interaction of metaphorical language and literal language; there is no sharp line between the two, but only a distinction which is relative, shifting, and contextdependent.³¹ ëMan is a wolfí invites reflection not only on wolf-life characteristics of man, but also on man-like characteristics of the wolf, which is seen thereafter as more human. Again, a term initially introduced metaphorically (e.g. ëfoothillí or ëskyscraperí) may come to be used as a standard word and the original analogy is forgotten. Metaphors, like models, influence the supposedly literal reporting of facts, and they extend language by the creation of new meanings.

I do not, of course, intend to equate metaphors and models. A metaphor evokes many types of personal experience, including emotional and valuational responses. A scientific model, on the other hand, is systematically developed, and the positive and negative analogy are specified, even though the neutral analogy remains open for further exploration. Above all, scientific models lead to theories which can be tested experimentally (Chapter 6 below). Nevertheless there are enough similarities between metaphors and models to illustrate the importance of analogical imagination in very diverse fields of human thought. Although metaphors are not literally true, they do, in Wheelwrightís words, ësay something, however tentatively and obliquely, about the nature of what isí. They can help to illustrate for us the range of alternatives which lie between literalism and fictionalism.

I will conclude by noting again the variety of types of model in science and trying to specify those to which the position of critical realism might be applicable. Recall that in broadest terms the function of models is the ordering of experience and that within science this may involve a wide diversity of specific types of activity. I mentioned experimental models, such as wind-tunnel models of proposed airplane designs, which are used to obtain approximate solutions to practical problems when it is difficult to experiment on the primary system. Mathematical models, such as the equations for the growth of a population of insects, are used to make quantitative predictions of particular variables. Computer models can be used to carry out calculations with many variables among which specifiable relationships are assumed, and can thereby simulate the behaviour of quantifiable aspects of such complex systems as urban transportation networks. These ësimulation modelsí are prominent in the new fields of ëoperations researchí and ësystems analysisí. They are heuristic aids in problem-solving.

Sometimes the same mathematical model is applicable to two different kinds of physical system, and one system is said to be a model of the other. These can be called ëformal analoguesí, since there are two quite distinct physical domains which could never be confused with each other. For instance, the same second-order differential equation is applicable to the vibratory oscillations of a pendulum, an electric circuit, and a violin string, but neither the observable variables nor the theoretical concepts have anything else in common. Mathieuís equation occurs in analysis of the motion of an elliptical membrane and of the equilibrium of an acrobat -- but there the resemblance ends. Any analogy is purely formal; there are no parallels beyond this mathematical isomorphism. The only point of the comparison is to make use of familiar mathematical procedures in computation.

Some models embodying formal analogies were indeed introduced as deliberate fictions. Maxwell, for instance, showed that the equations of an electric field would be the same as those for the flow of ëan imaginary incompressible fluidí; the purpose of invoking the latter was ëto make the mathematical theorems more intelligible to certain mindsí.³² At least in his early work he seems to have regarded the incompressible fluid and the electric field as analogues whose only resemblance is mathematical isomorphism. Even when there is no explicit disclaimer of the status of the model, the scientist usually knows when he is introducing it as an imaginary construct which is not intended to represent the world. I would want to distinguish these cases from theoretical models which the scientist usually views more realistically.

The character of a scientistís commitment to a theoretical model may vary widely in the course of its history. When it is first introduced, it may be used very tentatively for very limited purposes, correlating a narrow range of phenomena. But the scientist tries to develop a consistent model covering as many aspects of the phenomena as possible. As the scope and reliability of theories to which the model leads increases, his confidence in it also increases. In the process, the model may be altered considerably; we saw that the model of gas particles came to include features such as mutual attraction which are not found at all in billiard balls. The model becomes more complex, and draws from many other analogies besides the initial one. The original model may still be employed as a useful approximation, but it is then recognized as a deliberate simplification.

To complete the earlier example, we should recall that the ëelastic spheresí were identified with the hypothetical molecules which Gay-Lussac had posited from experiments on the combining volumes of chemically active gases. The search for such models linking two sciences having quite different observation terms would not be encouraged by the fictionalist position. In the present century, models of molecules have not been abandoned but have undergone further modification under the impact of quantum physics, as we shall see later. Many analogies besides those with billiard balls have contributed to the more recent models of a gas particle. The quantum physicist Max Born has written: ëAll great discoveries in experimental physics have been due to the intuition of men who made free use of models which were for them not products of the imagination but representatives of real things.^í33

But there is a wide variety even among theoretical models. Some, such as the ëdouble-helixí model of the DNA molecule, are closer to observations and can be taken more literally. Yet even in these cases one must remember that only certain aspects of the world are brought into prominence by the model, while other aspects are neglected (e.g., the model represents spatial relationships among the DNA components but not the character of the bonds between them). Other models, such as the abstract psi-functions of quantum physics, seem to invite a fictionalist interpretation. Yet even in that case there is a referential intent and a necessity of experimentation which are not present in pure mathematics. Most physicists hold that electrons exist, even though they are not waves or particles. Perhaps some biologists verge on literalism and some physicists verge on fictionalism, but the majority of practicing scientists are probably closer to the intermediate position which I have called critical realism.

Critical realism recognizes that models are selective; they allow us to deal with only restricted aspects of events. Entities in the world are assumed to be two stages removed from the familiar systems on which the model is based: (1) gas molecules are not the ëtiny elastic spheresí of the model (if we are not naive realists), and (2) ëtiny elastic spheresí are not billiard balls (if we have kept negative analogy in mind). The critical realist makes only a tentative commitment to the existence of entities something like those portrayed in the model. He says that gas molecules exist, and are in some ways like tiny elastic spheres -- or, he would now say, like the wave and particle models of quantum physics.

Let me summarize the main themes of this chapter. First, models have a variety of uses in science. They serve diverse functions, same practical and some theoretical. Second, theoretical models are novel mental constructions. They originate in a combination of analogy to the familiar and creative imagination in inventing the new. They are open-ended, extensible, and suggestive of new hypotheses. Third, such models are taken seriously but not literally. They are neither pictures of reality nor useful fictions; they are partial and inadequate ways of imagining what is not observable.