12 Chekhov’s Gun
If there is a gun on the wall in the first act, it must be fired by the third act.
—Anton Chekhov
Science is rife with models. Some, such as the hydraulic model of San Francisco Bay, are material. Others, such as the virial equation, are abstract. Like epistemically effective literary fictions and thought experiments, epistemically effective models are felicitous falsehoods. How, given their manifest inaccuracies, can they contribute to the understanding of their targets? A model’s mode of reference is complex. Building on a paper of R. I. G. Hughes (1996), I argue that a model represents its target as having certain features. It does so by exemplifying those features and imputing them to the target. Effective models afford an understanding of their targets because their simplifications, idealizations, elaborations, and distortions make salient important features of the targets.
Representation
Science, we are told, is (or at least aspires to be) a mirror of nature, while art imitates life. If so, both disciplines produce, or hope to produce, representations that reflect the way some part of the mind-independent world is. Scientific representations are supposed to be complete, accurate, precise, and distortion-free. Although artistic representations are granted more leeway, they too are supposed to resemble their subjects. Underlying these clichés is the widespread conviction that representations are intensional surrogates for, or replicas of, their objects. If so, a representation should resemble its referent as closely as possible.
This stereotype, I have urged, is false and misleading. It engenders unnecessary problems in both the philosophy of science and the philosophy of art. It makes a mystery of the effectiveness of sketches, caricatures, scientific models, and representations with fictional subjects. Indeed, the stereotype strongly suggests that there is something intellectually suspect about such representations. Caricatures exaggerate and distort. Sketches simplify. Models may do all three. Many pictures and models flagrantly fail to match their referents. Representations with fictional subjects have no hope of matching, since they have no referents to match. The same subject, real or fictive, can be represented by multiple, seemingly incongruous representations. These would be embarrassing admissions if representations were supposed to accurately reflect the facts.
Mimetic accounts of representation fail to do justice to our representational practices. Many seemingly powerful and effective representations turn out on a mimetic account to be at best flawed, at worst unintelligible. Nor is it clear why we should want to replicate reality. As Rebecca West said, “A copy of the universe is not what is required of art; one of the damn things is ample” (1928, 131).1 To replicate reality would simply be to reproduce the blooming, buzzing confusion that confronts us. What is the value in that? Our goal should be to make sense of things—to structure, synthesize, organize, and orient ourselves toward things in ways that serve our ends.
The problem lies in the metaphor of the mirror and the ideal of replication. Neither art nor science is, can be, or ought to be, a mirror of nature. Epistemically effective representations in both disciplines embody and convey an understanding of their subjects. Since understanding is not mirroring, failures of mirroring need not be failures of understanding. Once we recognize the ways science affords understanding, we see that the features that look like flaws under the mirroring account are actually virtues. A first step is to devise an account of scientific representations that shows how they figure in or contribute to understanding. It will turn out that an adequate account of scientific representation also affords insight into representation in the arts.
The term ‘representation’ is irritatingly imprecise. Pictures represent their subjects; graphs represent the data; politicians represent their constituents; representative samples represent whatever they are samples of. Thus far I have relied on context to fix the reference of different tokens of the term. In this chapter further regimentation is called for. We can begin to regiment by restricting attention to cases where representation is a matter of denotation. Pictures, equations, graphs, charts, and maps represent their subjects by denoting them. They are representations of the things they denote.2 It is in this sense that scientific models represent their target systems: they denote them. But, as Bertrand Russell (1968) insists, not all denoting symbols have denotata. A picture that portrays a griffin, a map that marks the route to Mordor, a chart that records the heights of Hobbits, and a graph that plots caloric emissions are all representations, though they do not represent anything. To be a representation, a symbol need not itself denote, but it must be the sort of symbol that denotes. Griffin pictures are representations, then, because they are animal pictures, and some animal pictures denote animals. Middle Earth maps are representations because they are maps, and some maps denote real locations. Hobbit height charts are representations because they are charts, and some charts denote magnitudes of actual entities. Caloric emission graphs are representations because they are graphs, and some graphs denote real processes. So whether a symbol is a representation is a question of what kind of symbol it is.
Following Goodman, let us distinguish between representations of p and p-representations. If s is a representation of p, then p exists and s represents p. But s may be a p-representation even if there is no such thing as p (Goodman, 1968). Thus, there are griffin-pictures even though there are no griffins to depict. There are caloric-descriptions even though there is no caloric to describe. There are also mixed cases. The class of dog-representations includes both factual and fictional representations. Factual dog-representations are representations of actual dogs; fictional dog-representations lack denotata. Literary fictions and thought experiments are p-representations that do not denote p.
Denoting symbols with null denotation may seem problematic. Occasionally it is said that in the absence of griffins, there is no basis for classifying some pictures as griffin pictures and refusing to so classify others. Such an objection supposes that the only basis for classifying representational contents is by appeal to an antecedent classification of their referents. This is just false. We readily classify pictures as landscapes without any acquaintance with the real estate—if any—that they represent. I suggest that each class of p-representations constitutes a small genre, a genre composed of all and only representations with a common ostensible subject matter. There is a genre of griffin-representations and a genre of ideal-gas-representations. We learn to classify representations as belonging to such genres as we study those representations and the fields of inquiry that devise and deploy them. This is no more mysterious than learning to recognize pictures as landscapes without comparing them to the terrain they ostensibly depict.
Some representations denote their ostensible objects. Others do not. Among those that do not, some—such as caloric-representations—simply fail to denote. They purport to denote something, but there is no such thing. They are therefore defective. Others, such as ideal-gas-representations, are fictive. Since they do not purport to denote any real object, their failure to denote is no defect (see Frigg, 2010). We know perfectly well that there is no such animal as a griffin, no such person as Othello, no such gas as the ideal gas. Nonetheless, we can provide detailed representations as if of each of them, argue about their characteristics, be right or wrong about what we say respecting them, and, I have argued, advance understanding by means of them.
Representation-As
Symbol x is, or is not, a representation of y depending on what x denotes. And x is, or is not, a z-representation depending on its genre. This enables us to form a more complex mode of representation in which x represents y as z. In such a representation, symbol x is a z-representation that as such denotes y. Caricature is a familiar case of representation-as. Winston Churchill is represented as a bulldog; Teddy Roosevelt is represented as a bull moose.
Hughes (1996) argues that representation-as is central to the way that models function in science. A diagram represents a spring as a harmonic oscillator; the Hardy—Weinberg model represents a species as mating randomly; an ideal gas model represents molecules as displaying no mutual attraction. A model, Hughes says, is a theoretical construct that involves three components: denotation, demonstration, and interpretation. Denotation is the relation of elements of the model to elements of its target. Demonstration is the operation of the model according to its own ‘internal dynamic’, which enables us to use the model to derive conclusions. Interpretation is a matter of reading the conclusions derived from the model back into the target. For example, a model—such as an equation—denotes a pendulum, an actual swinging bob. By plugging in values and performing a calculation, we demonstrate the relation between two mathematical variables. We then interpret the values of the variables as length and period, and impute them to the material pendulum.
Denotation is straightforward. It is the semantic relation of a name to the thing it names, of a picture to the thing it depicts, of a predicate to the things it is true of. But more needs to be said about demonstration and interpretation. A demonstration can be mathematical or material. “If we want to apply the wave theory of light to two-slit interference, for example, we can do one of two things. We can either model the phenomenon mathematically, or we can get out a ripple tank and model light as a wave motion in the literal sense. In either case we will find that the distance between interference fringes varies with the separation of the sources, and also with the frequency of the waves” (Hughes, 1996, S322). Mathematics does not typically strike us as dynamic. Equations sit quietly on the page. In saying that an equation has an internal dynamic, Hughes means that we reason with it, use it as an inference ticket, draw consequences from it. What is dynamic is the drawing out of consequences via inference. A major insight of his analysis is that models are not static representations. They are tools. They enable us to figure things out.
Hughes is right as far as he goes, but he stops too soon. The problem is that there are always more consequences to be drawn from any demonstration. We need to know what direction to take in our reasoning and where to stop. That is, we need to know which of the inferences that we might legitimately make should be read back onto the target. Again exemplification comes to the rescue. The demonstration, whether in the ripple tank or on paper, has irrelevant features—ones that ought to be ignored. If we continued the demonstration endlessly, the number of irrelevant features would proliferate. But being irrelevant, they are not exemplified. They should be set aside. Interpretation involves ascertaining which features the demonstration exemplifies, hence which ones we should project onto the target. This involves understanding how, in what respects, and with what degree of precision the model represents. Models are symbols. To interpret them correctly requires appreciating which of their elements represent and what those elements refer to.
Representation-of—that is, denotation—can be achieved by fiat. We simply stipulate: let x represent y and x thereby becomes a representation of y. This is what we do in baptizing an individual or a kind. It is also what we do in ad hoc illustrations, as, for example, when I say (with appropriate accompanying gestures), ‘If that chair is Widener Library, and that desk is University Hall, then that window is Emerson Hall’ in helping someone to visualize the layout of Harvard Yard. Like a chair or a desk, a p-representation can by arbitrary stipulation denote any object. We might, for example, point to a tree-picture and stipulate that it denotes the philosophy department. But our arbitrary stipulation does not bring it about that the tree-representation represents the philosophy department as a tree.
Does representation-as then require similarity? In that case, what blocks seemingly groundless and arbitrary cases of representation-as is the need for resemblance between the representation and the referent. Only a department that resembles a tree could then be represented as a tree. But similarity does not establish a referential relationship (Goodman, 1968; Suárez, 2003). Representation is an asymmetrical relation; similarity is symmetrical. Representation is irreflexive; similarity is reflexive. Perhaps this shows only that similarity is not sufficient for representation-as. Something else determines direction. Once the direction is established, it is the similarity between symbol and referent that brings it about that the referent is represented as whatever it is represented as.
The problem is this: Via stipulation, we have seen, pretty much anything can represent pretty much anything else. So nothing beyond stipulation is required to bring it about that one thing represents another. But similarity is ubiquitous. This is the insight of nominalism. For any x and any y, x is somehow similar to y. Thus if all that is required for representation-as is denotation plus similarity, then for any x that represents y, x represents y as x. Every case of representation turns out to be a case of representation-as. In one way or another, the philosophy department is similar to a tree-picture, but it is still hard to see how that fact, combined with the stipulation that a tree-picture represents the department, could make it the case that the department is represented as a tree-picture, much less as a tree. Suppose we add that the similarity must obtain between the content of the p-representation and the denotation. Then for any x-representation and any y, if the x-representation denotes y, the x-representation represents y as x. In that case, a tree that represented the philosophy department would not represent it as a tree. But a tree-picture that represented the philosophy department would represent it as a tree.
The trouble is that contentful representations, as well as chairs and desks, can be used in ad hoc illustrations like the one I gave earlier. If the portrait of the dean on the wall represents Widener Library, and the graph on the blackboard represents University Hall, then the map in the corner represents Emerson Hall. This does not make the dean’s portrait represent Widener Library as the dean. Evidently, it takes more than being represented by a tree-picture to be represented as a tree. Some philosophy departments can be represented as trees. But to bring about such representation-as is not to arbitrarily stipulate that a tree picture shall denote the department, even if we add a vague intimation that somehow or other the department is similar to a tree. The question is, what is effected by such a representation?
To explicate representation-as, Hughes (1996) discusses Sir Joshua Reynolds’s painting, Mrs. Siddons as the Tragic Muse. The painting denotes its subject and represents her as the tragic muse. How does it do so? It establishes Mrs. Siddons as its denotation. It might represent Mrs. Siddons, a person familiar to its original audience, in a style that that audience knows how to interpret. Then, without further cues, they could recognize that the picture is a picture of her. But the painted figure need not bear any particular resemblance to Mrs. Siddons. We readily take her as the subject even though we have no basis for comparison. (Indeed, we even take Picasso’s word about the identities of the referents of his cubist portraits, even though the figures in them do not look like anyone on earth.) Captioning the picture as a portrait of Mrs. Siddons suffices to fix the reference. So a painting can be connected to its denotation by stipulation. The painting is a tragic-muse-picture. It is not a picture of the tragic muse, there being no such thing as the tragic muse. But it belongs to the same restricted genre as other tragic-muse-representations. To recognize it as a tragic-muse-picture is to recognize it as an instance of that genre. Similarly in scientific cases. A spring is represented as a harmonic oscillator just in case a harmonic-oscillator-representation as such denotes the spring. The harmonic-oscillator-representation involves idealization. So it is not strictly a representation of a harmonic oscillator, any more than the Reynolds is a picture of the tragic muse.
In both cases, a representation that does not denote its ostensible subject is used to denote another subject. Since denotation can be effected by stipulation, there is no difficulty in seeing how this can be done. The difficulty comes in seeing why it is worth doing. What is gained by representing Mrs. Siddons as the tragic muse, or a spring as a harmonic oscillator, or in general by representing an existing object as something that does not in fact exist? The quick answer is that the representation affords epistemic access to features of the object that are otherwise difficult or impossible to discern. They exemplify those features.
Nor does this occur only when a subject is represented as something fictional. Legend has it that when people objected that his portrait of Gertrude Stein did not look like her, Picasso replied, “No matter. It will” (Goodman, 1968, 33). Although this remark might be dismissed as yet another expression of Picasso’s arrogance, Robert Schwartz (1985) argues that we should take it seriously. Through his portrait, Schwartz maintains, Picasso gave Stein a new look. In so doing, he not only reconfigured Stein, he changed the world. Schwartz’s position can be elaborated to show how scientific models do the same. The first step is to explain how a portrait can literally give someone a new look and what follows from it.
Schwartz maintains that through his portrait, Picasso brought people to see Stein differently. It did so, I suggest, by exemplifying features of Stein that were normally overlooked. Picasso did not portray her as a dumpy, middle aged, Midwestern woman, but as magisterial, imposing, someone to be reckoned with. Through his portrait, he made these features of Stein—being magisterial, being imposing, being someone (or something) to be reckoned with—salient. We see Stein differently as a result. If we had previously overlooked the fact that she had those features, then not only our perception of her but also our conception of her is reconfigured once we come to recognize them.
Moreover, Schwartz insists, we gain more than just a new way of looking at and thinking of Stein. We also gain new criteria for what it takes to look like Stein and what it takes to be like Stein. These criteria afford bases for likening. People, or indeed things, that we would never previously have counted as looking like Stein are now recognized as doing so, for they share the features that her portrait exemplifies. They therefore have something significant in common not just with Stein but also with each other. They belong to an extension that we are now in a position to call the Stein-lookalikes. If, via the picture, we see in Stein herself features we had overlooked, then we can see others as like her in ways that go beyond her appearance—as being imposing, magisterial, someone to be reckoned with, for example. We then can recognize others as Stein-alikes on the basis of something other than their appearance.
At this point, metaphysicians might object that the extension consisting of all and only what we are now in a position to call the Stein-lookalikes and the extension of what we are now in a position to call the Stein-alikes existed all along. Picasso did not bring the extensions into existence. He did not create them. This is true. Extensions, marked or unmarked, are equally real; none is brought into existence by baptizing, exemplifying, or construing. But most extensions strike us as motley collections of apparently disparate objects. Their members have nothing notable in common. The extension consisting of the Gobi Desert, an apple pie, and a baseball in Detroit have something in common with one another: they are all members of that extension. This is true but trivial. There’s little reason to care about it. From our antecedent perspective, the extensions we now call the Stein-lookalikes and the Stein-alikes seemed arbitrary collections of unrelated things. What Picasso brings out is that these particular collections are by no means arbitrary—their members have something significant in common. They share the constellation of features that the portrait exemplifies.
Still, one might balk, at least about the Stein-alikes. Arguably we can concede that the portrait instantiates, and therefore is capable of exemplifying, features like looking magisterial. But a portrait itself can’t be magisterial. The claim that it exemplifies being magisterial is blocked from the outset, since exemplification requires instantiation. To handle this objection, we need to appeal to representation-as. The portrait represents Gertrude Stein as magisterial insofar as it denotes her, belongs to the genre of magisterial-figure-portrayals, and belongs to that genre because of the way it portrays her—namely as having certain discernible features that its subject purportedly shares with other members of that genre. (‘Purportedly’ because it is possible to misrepresent someone as magisterial.)
Modeling
Models, like fictions, can simplify, omitting confounding factors that would impede epistemic access to the properties of interest. They can abstract, paring away unnecessary and potentially confusing details. They can distort or exaggerate, highlighting significant aspects of the items they focus on. They can augment, introducing additional elements that focus attention on properties of interest. They can insulate, screening off effects that would otherwise dominate. They do these things in order to exemplify.
A familiar worry, first encountered in the discussion of Galileo’s thought experiment, recurs here. The same solution holds. Many scientific models, such as equations and diagrams, are incapable of instantiating the properties they apparently impute to their targets. If they cannot instantiate a range of properties, they cannot exemplify them. Suppose, via an equation, we model a pendulum as a simple harmonic oscillator. Since exemplification requires instantiation, if the model is to represent the pendulum as having a certain mass, the model must have that mass. But, not being a material object, the model has no mass. It cannot exemplify the mass of the pendulum. Indeed, the model does not exemplify mass. Rather, it exemplifies an abstract mathematical property, the magnitude of the pendulum’s mass. Where models are abstract, they exemplify abstract patterns, properties, and/or relations that may be instantiated in physical target systems. It does no harm to say that they exemplify physical magnitudes. But this is to speak loosely. Strictly speaking, they exemplify mathematical (or other abstract) properties that can be instantiated physically.
Even concrete models display the sort of divergence from their targets that we saw above. The San Francisco Bay-Delta model—a massive, hydraulic scale-model of San Francisco Bay and its environs (see Weisberg, 2013)—has no tides. So it cannot exemplify the effect of tides on the water in the bay. But it mimics the tides by gradually pumping salt water into and out of the model’s basin. It thus exemplifies an effect that is instantiated by the ebb and flow of tidal waters in the bay. Speaking loosely, we might say that it exemplifies the effects of the tides. Speaking strictly, we should say that it exemplifies an effect that is instantiated both by flow of water in the model itself and by the tides.
Both literary fictions and scientific models exemplify properties and afford epistemic access to them. By omitting or downplaying the significance of confounding factors (the Napoleonic wars in Pride and Prejudice, intermolecular attraction in the ideal gas, friction in the model pendulum), they constitute a cognitive environment where certain aspects of their subjects stand out. They thereby facilitate recognition of those aspects and appreciation of their significance. They thus give us reason to take those aspects seriously elsewhere.
Of course, this does not justify a straightforward extrapolation to reality. From the fact that Elizabeth Bennet was wrong to distrust Mr. Darcy, we cannot reasonably infer that young women in general are wrong to distrust their suitors, much less that any particular young woman is wrong to distrust any particular suitor. But the fiction exemplifies the grounds for distrust and the reasons those grounds may be misleading. Once we have seen them clearly in a fictional setting, we may be in a better position to recognize them in everyday situations. Nor from the fact that ideal gas molecules exhibit no mutual attraction can we reasonably infer that neither do helium molecules. But the behavior ideal gas molecules exemplify in the model may enable us to recognize such behavior amid the confounding factors that ordinarily obscure what is going on in actual gases. In using a model, we implicate that the features of the phenomena that it brackets are for current purposes negligible. Here, we suggest, we can safely ignore them.
Epistemic Access
Return to Reynolds’s representation of Mrs. Siddons as the tragic muse. The tragic muse is a figure from Greek mythology who is supposed to inspire works of tragedy—works that present a sequence of events leading inexorably from a position of eminence to irrecoverable, unmitigated loss, thereby inspiring pity and terror (Aristotle, 1973). A tragic-muse-representation portrays a figure as capable of inspiring such works, one who exemplifies such features as nobility, seriousness, inevitability, and perhaps a somber dramaticality, along with a capacity to evoke or arouse pity and terror. To represent a person as the tragic muse is to represent her in such a way as to reveal or disclose such characteristics in her or to impute such characteristics to her.
The ideal gas law is an equation ostensibly relating temperature, pressure, and volume in a gas. To satisfy that equation, a gas would have to consist of perfectly elastic spherical particles of negligible volume and exhibiting no mutual attraction. The law thus defines a model that mandates specific values for size, shape, elasticity, and attraction. With these parameters fixed, the interdependence of the values of temperature, pressure, and volume is exemplified. The law and the model it defines are fictions. There is no such gas. Indeed, so far as we can tell, there could be no such gas, since such a gas would violate several well-founded laws of physics. Nevertheless, the model advances our understanding of gas dynamics. It exemplifies a relation that is important but hard to discern in the behavior of actual gases. The model is a representation—a denoting symbol that has an ostensible subject and portrays its ostensible subject in such a way that certain features are exemplified. It represents its target (its denotatum) as exhibiting those features. So to represent helium as an ideal gas is to impute to it features that the ideal gas model exemplifies. By setting the parameters to zero, it construes the actual size, shape, inelasticity, and mutual attraction of the molecules as negligible. Strictly, of course, in helium the values of those parameters are not zero. But if they are negligible, they can safely be ignored. In that case, the imputation allows for a representation that discloses regularities in the behavior of helium that a more faithful representation would obscure. The model then foregrounds the interdependence of temperature, pressure, and volume, making it and its consequences manifest.
Representing a philosophy department as a tree might exemplify the ways the commitments of the various members branch out from a common, solid, rooted tradition, and the ways the work of the graduate students further branch out from the work of their professors. It might intimate that some branches are flourishing while others are stunted growths. It might even suggest the presence of a certain amount of dead wood. Representing the department as a tree, then, affords resources for thinking about it, its members and students, and their relation to the discipline in ways that we otherwise would not.
I said earlier that when x represents y as z, x is a z-representation that as such denotes y. We are now in a position to cash out the ‘as such’. It is because x is a z-representation that x denotes y as it does. x does not merely happen to denote y and happen to be a z-representation. Rather, in being a z-representation, x exemplifies certain properties and imputes those properties or related ones to y. ‘Or related ones’ is crucial. A caricature that exaggerates the size of its subject’s nose need not impute an enormous nose to its subject. By exemplifying the size of the nose, it focuses attention, thereby orienting its audience to the way the subject’s nose dominates his face or, through a chain of reference, the way his nosiness dominates his character. The properties exemplified in the z-representation thus serve as a bridge that connects x to y. This enables x to provide an orientation to its target that affords epistemic access to the properties in question.
There is no guarantee that the target has the features the model exemplifies, any more than there is any guarantee that a subject represented as the tragic muse has the features that a painting representing her as the tragic muse exemplifies. This is a question of fit. A model may fit its target perfectly or loosely or not at all. Like any other case of representation-as, the target may have the features the model exemplifies. Then the function of the model is to display those features and make their significance manifest. We may see the target system in a new and fruitful way by focusing on the features that the model draws attention to. In other cases, the fit is looser. The model does not exactly fit the target. A target that does not instantiate the specific properties its model exemplifies may instantiate more generic properties that subsume the exemplified properties. If gas molecules are roughly spherical, reasonably elastic, and far enough apart, then we may gain insight into their behavior by representing them as perfectly elastic spheres with no mutual attraction. Perhaps we will subsequently have to introduce correction factors to accommodate the divergence from the model; perhaps not. It depends on what degree of precision we want or need.
Sometimes, although the target does not quite instantiate the features exemplified in the model, it is not off by much. Where their divergence is negligible, the models, although not strictly true of the phenomena they denote, are true enough of them. This may be because the models are approximately true, or because they diverge from truth in irrelevant respects, or because the range of cases for which they are not true is a range of cases we do not care about, as for example when the model ‘blows up’ at the limit. Where a model is true enough, we do not go wrong if we think of the phenomena as displaying the features that the model exemplifies. Whether a representation is true enough is a contextual question. A representation that is true enough for some purposes or in some respects is not true enough for or in others. This is no surprise. No one doubts that the accuracy of models is limited. In yet other cases, the model simply does not fit. Then it affords little or no understanding of its target. Not everyone can be informatively represented as the tragic muse. Nor can every object be informatively represented as a perfectly elastic sphere.
Earlier I dismissed resemblance as the vehicle of representation. I argued that exemplification is required instead. But for x to exemplify a property of y, x must share that property with y. So x and y must be alike in respect of that property. It might seem, then, that resemblance in particular respects is what is required to connect a representation with its referent (Giere, 1999). There is a grain of truth here. If exemplification is the vehicle for representation-as, the representation and its object resemble one another in respect of the exemplified properties. But resemblance, even resemblance in a particular, relevant respect, is not enough, as the following tragic example shows.
On January 28, 1986, the space shuttle Challenger exploded because, owing to cold weather, its O-rings failed to effect a seal. The previous day, engineers involved in designing the shuttle had warned NASA about that very danger and faxed data to NASA to support their concern. The printouts contained complex representations conveying vast amounts of information about previous shuttle flights. They included measurements of launch temperatures for previous flights and measurements of six types of O-ring degradation after each flight. Had loss of elasticity been plotted against temperature, the danger would have been clear. The evidence that the O-rings were vulnerable in cold weather was contained in the data. But it was obscured by a mélange of other information that was also included (Tufte, 1997). So although the requisite resemblance between the highly complex, multipurpose model and its target obtained, it was overshadowed in the way that a subtle irregularity in an elaborate tapestry might be. As it was presented, the data instantiated but did not exemplify the correlation between temperature and loss of elasticity. They did not represent the O-rings as increasingly inelastic as the temperature dropped. Because the correlation between O-ring degradation and temperature was not perspicuous, the NASA decision makers did not see it. The launch took place; the shuttle exploded; the astronauts died. When the goal of a representation is to afford understanding, its merely resembling the target in relevant respects is not sufficient. The representation must make the resemblance manifest.
Drawing on Hughes’s account, I have argued that models exemplify features they share with their targets and impute those features to their targets. Where they are successful, they afford epistemic access to aspects of their targets that we might otherwise miss. When an economic model represents a transaction as occurring in a situation where there is perfect information, it is not purporting or pretending that actual financial transactions occur in such contexts. Rather, it factors out what might be seen as interference. By exhibiting what would occur in such a context, it equips us to understand why and to what extent differences in information available to different parties to a transaction matter.
Models are selective. They highlight some features of their targets by marginalizing or downplaying the significance of others. The utility of any particular model is therefore limited. The Hardy—Weinberg model represents a population as consisting of infinitely many members of a species who mate randomly, where that population is not subject to mutation, migration, or genetic drift. It exemplifies the pattern in the redistribution of alleles in the absence of evolutionary pressures. Inasmuch as evolutionary pressures are always present, the model cannot, nor does it pretend to, account for allele distribution generally. It is, however, very useful for some purposes. If population geneticists want to understand how significant an evolutionary factor such as migration is, they need a base rate. They need, that is, to know how alleles would redistribute in its absence. Features that can be neglected in some contexts or for some purposes cannot always be neglected.
By exemplifying a feature, a model affords epistemic access to it, and provides reason to suspect that it is significant. It thereby equips us to see the target differently than we otherwise might. Instead of thinking of a feature as just one of innumerable features of the target, we have reason to take it seriously. Familiar gas models represent gas molecules as lacking attractive force. If the results of our calculations are confirmed when we read them back into the target, we have reason to think that intermolecular forces do not play a significant role in gas dynamics. Knowing as we do that every material object attracts every other one, we do not conclude that there is no attraction. Rather, we conclude that for the sort of understanding we currently seek, at the level of precision we are interested in, intermolecular attraction is negligible. This suggests that we think of the target in terms of the features exemplified in the model. Roughly, it invites us to think of actual gases as ideal gases with distortions, springs as harmonic oscillators with confounding features, and so forth.
Likening
Models liken. If effective, they exemplify features that the members of their target class share. They thereby have the capacity to bring to light significant affinities among members of the class. Let’s look at a couple of cases. Strictly, the equation pV = nRT would hold only of spherical molecules. In effect, the model defines a fictional gas whose molecules are spherical. However, the model is supposed to illuminate the behavior of real gases, whose molecules are lumpy. Moreover, molecules of different gases have different configurations of lumps. On the face of it, the shapes of oxygen molecules and argon molecules do not strike us as particularly similar. But the model intimates that the actual differences in shape can be ignored. Where the model is appropriate, they are negligible. And where the differences are negligible, gas molecules are relevantly alike. By representing actual gas molecules as spheres with distortions, and making the case that the distortions have negligible effects, the model likens actual gas molecules to one another.
A more interesting case is biology’s evolutionary game-theoretic model. It applies to, among other things, predator—prey interactions, mating behavior, fertility and fecundity rates, and altruism. It is a dynamic model, displaying change over time. Even if one is completely convinced that natural selection underwrites all these phenomena, my list seems to consist of wildly disparate items. It is far from obvious that, at any level of abstraction, predator—prey interactions and altruism are the same sort of thing. Off the cuff, they seem virtually antithetical. Although we readily grant that in one way or another, natural selection accounts for all of the items on my list, there seems no a priori reason to suspect that there is a single way that natural selection accounts for all of them. It would not be surprising if each had a separate evolutionary explanation. What the game-theoretic model does is show that at a certain level of abstraction, these phenomena are all alike. They are instances of the same sort of process. The model thus unifies the domain and effects a likening among seemingly disparate phenomena. We can now appreciate why, from an evolutionary perspective, altruistic behavior and predatory behavior are alike.
If we think of the target in terms of the features the model exemplifies, we may recognize and appreciate the significance of affinities we would otherwise overlook. Newton’s law of gravity enables us to recognize that the moon’s staying in orbit, the ebb and flow of the tides, and the apple’s falling from the tree are fundamentally the same sort of thing. The glaring differences between them are to be ignored. In important physical respects, the motions are alike—they satisfy the same equation, and thereby exemplify the same sort of behavior. Perhaps surprisingly, a good way to understand why satellites remain in orbit is to represent them as falling bodies.
Models do not, of course, only omit. If they did, there would be no reason to consider them inaccurate. They could be construed as schematic or partial representations of the phenomena they bear on. But models also distend, distort, exaggerate, and even introduce elements that answer to nothing in the target. These distentions, distortions, exaggerations, and interpolations scaffold representation-as. A distended, distorted, or exaggerated model can exemplify features that obtain in the target but are difficult to discern there. For example, according to Kepler’s laws, the orbit of the Earth is elliptical, with one focus at the sun. The major axis and the minor axis are almost equally long. The shape of the ellipse is very close to that of a circle. But to highlight the fact that the orbit is not a circle, models of Kepler’s laws typically represent the major axis as considerably longer than the minor axis. This would be objectionable if we took such models to exemplify the actual shape of the orbit. But since they exemplify only the orbit’s being elliptical rather than circular and not the precise shape of the ellipse, there is no difficulty. Maxwell’s representation of electromagnetic radiation as involving rotating vortices separated by ‘idle wheels’ denotes that radiation and, through the introduction of fictional elements, makes manifest how the relational structure of the production and transmission of electric and magnetic forces in an electromagnetic medium is like the relational structure of the production and transmission of mechanical forces in a mechanical medium (Nersessian, 2008, 29–48). Through this model, Maxwell effected a likening of electromagnetic and general dynamical processes.
To properly understand a model or other representation-as we must interpret it correctly. Some features of the representation are inert. It makes no difference, for example, what typeface the statement of Kepler’s laws is in or whether the hydraulic model of San Francisco Bay is properly oriented toward the magnetic North Pole. The laws represent the orbit of the Earth, regardless of font, and the model would simulate the effects of the tides equally well if it were rotated 30 degrees. Some features function as scaffolding. They do not themselves exemplify features of the target. But they enable the model to exemplify the features it imputes to the target. The idle wheels in Maxwell’s model and the canine features in Churchill’s caricature scaffold in that they enable the representations to highlight the features they exemplify. They are thus not irrelevant to how the symbol represents, although they do not figure in what it represents. They are not to be read back into the target. Finally, there are the features that the model exemplifies: these are the ones it imputes to its target.
Thus far, I have assumed that a model represents something that is, in Williams’s phrase, there anyway. The target is as it is regardless of how or whether it is represented. If so, the role of the model is to bring to light unanticipated or unappreciated aspects of the phenomena it models. Sometimes this is so. The planets are there anyway and are unaffected by their susceptibility to being modeled as point masses. If all models fit this mold, it would not be unreasonable to balk at the claim that anything properly called ‘worldmaking’ is going on (see Goodman, 1978). But there are cases where the relation between model and target is more dynamic and interactive than this picture suggests. The demarcation of the target is the product of a negotiation between an antecedent partition of the domain and the resources a model can bring to bear.
Hasok Chang’s (2004) account of the tandem development of the thermometer and temperature brings this out. One might have thought that the inventors of the thermometer began with an antecedently recognized magnitude—temperature. Their challenge was to come up with a way to measure it. That is: given temperature, invent the thermometer. But, Chang argues, this is an oversimplification. At the outset, there were phenomenologically distinct experiences as of hotter or colder. Being grounded entirely in sensory experience, judgments about felt differences were subject to unwelcome fluctuations. Two perceivers of the same phenomenon would give different verdicts. The same phenomenon, when judged against different backgrounds, would be felt to be hotter or colder. “Put one hand in a bucket of hot water and the other one in cold water; after a while take them out and put them in a bucket of lukewarm water; one hand feels that the water is cool, the other feels it is warm” (Chang, 2004, 43). Hint that it is a hot coal, and an ice cube dropped down someone’s back will be felt as a burn. Scientists couldn’t rest satisfied with these verdicts. Too much seemed to depend on the conditions of perception and sensibilities of the perceivers. So they sought to create a device that would measure the ‘real’ differences, but bracket those that depended on the varying sensibilities and conditions of perceivers. They couldn’t quite succeed. The best devices they could come up with did not quite reflect the differences in felt hotness or coldness even when obvious biases and idiosyncrasies were controlled for. Initially, of course, this might just be evidence that their devices were inadequate. But eventually, they came to trust the devices. Rather than concluding that thermometers were not really measuring heat, scientists concluded that what perceivers take to be experiences of heat are actually experiences of heat with confounding factors, such as barometric pressure and humidity. They redefined the target of their measurement to legitimize the measuring device.
The thermometer is not just a measuring device, it is a model. It represents changes in the height of a column of mercury as changes in temperature. That is, the thermometer exemplifies magnitudes or changes magnitude and imputes them to the target. So Chang’s discussion shows that models can figure in the constitution of the phenomena they model. The process of inventing the thermometer was in part a process of constituting a specific magnitude to be measured.
Multiple Models
This account enables us to see why different models of the same target serve different purposes. For some purposes, representing light as a wave is valuable; for others, it is preferable to represent light as a particle. For some purposes, representing a customer as an ideally rational agent with full information is appropriate; for others, representing her as an individual with idiosyncratic and mutually inconsistent desires and preferences and with limited information is preferable. Each model exemplifies different features and affords epistemic access to different aspects of the target.
Another question looms. Why, when we refine a model, does its more idealized predecessor endure? The ideal gas pV = nRT was exported from thermodynamics to statistical mechanics. The model has undergone numerous refinements which successively de-idealize its assumptions. The van der Waals equation, for example, accommodates intermolecular forces which the earlier model scrupulously ignores. Further refinements in the equations of state yield increasingly realistic representations. Eventually we get the virial equation,
PV/NkT = 1 + B/V + C/V2 + D/V3 + E/V4 + …,
“which can be rendered arbitrarily precise by extending the equation indefinitely, with each added term being derivable from increasingly detailed assumptions about the intermolecular forces” (Doyle et al., 2015, 5). Armed with the virial equation, why don’t we jettison the predecessors or consign them to the dustbin of the history of science? Boyle, Charles, van der Waal, et al. seem to have been supplanted.
One possible answer is intellectual inertia. We don’t clean our conceptual closets all that often, so remnants of outgrown commitments remain. This, perhaps, is why we remember the old models. But it doesn’t explain why we continue to use them, particularly if we consider them outdated. And it certainly does nothing to justify their continued employment.
Another possibility is that the older models are easier to use. They require fewer and simpler calculations. But the virial equation is not particularly difficult. The math is straightforward. So there is little practical advantage in using the earlier models. And even if there were a practical advantage, it is far from obvious that availing ourselves of that advantage would be epistemically legitimate.
I suggest that it is a mistake to think that the virial equation is to be preferred simply because it can be made arbitrarily precise, and can accommodate increasingly detailed and accurate assumptions. I am not denying that the equation is an impressive accomplishment. Nor am I denying that it is the right model to use in certain circumstances. What I am denying is that what we want of a model is as precise and detailed a representation as we can get. The aim of modeling is to come up with a representation that affords an understanding of the phenomena, not one that replicates the phenomena. That is why we can be entirely justified in deploying an idealized model when a more realistic one is available. The highly idealized pV = nRT exemplifies a relation between pressure, temperature, and volume, and in imputing it to its target implicates that further factors, such as intermolecular forces, are negligible. By omitting the negligible, we put ourselves in a position to appreciate the relations that matter. In contexts where those forces are negligible, we do well to set them aside. If, for example, we want to understand how a pressure cooker works, or how a football might deflate in cold weather, we need no more than pV = nRT. In other contexts, intermolecular forces are nonnegligible. Then we should use a different model—one that accommodates them to the extent that they are nonnegligible. Maybe this means using the van der Waals equation. Maybe it means going all the way to the virial equation.
If I want to make the world safe for the ideal gas model, however, there is a worry that has to be faced. I said earlier that not every feature of a model exemplifies. So a seemingly efficient way of accommodating the cases where intermolecular forces are negligible and the cases where they are only slightly significant would be to use the virial equation and interpret it so that only some of the terms exemplify. That would amount to neglecting the terms whose contribution is negligible. Rather than taking different models to exemplify different features of the target, we would assign different interpretations (in Hughes’s sense of ‘interpretation’) to a single model. This could be done. It might even be a good idea. Still, there is the pesky ideal gas law rattling around in our brains. Why?
According to Chekhov, if a gun is on the wall in the first act of a play, it must be fired by the third act. His edict has an analogue in science. Suppose we are in a context where pV = nRT would be appropriate. Instead, we might use the virial equation,
PV/NkT = 1 + B/V + C/V2 + D/V3 + E/V4 + …,
and ignore the bits we don’t need. But if we don’t need them, why are we given them? What is the point of all the additional bells and whistles? (What is that gun doing on the wall if it’s never going to be fired?) To include multiple terms is to implicate that they have a bearing on the issue. This follows from Grice’s second maxim of quantity: “Do not make your contribution more informative than is required” (1989, 26). In including unnecessary details, we court misunderstanding; we invite the audience to conclude that the details are in fact necessary. Somehow, even in this simple case, audience members may think, we are supposed to recognize the significance of E/V4.
But all models have features that do not represent. Kepler’s laws are in some font or other; the diagram of the harmonic oscillator is some height or other. The hydraulic model of San Francisco Bay is some distance or other from the magnetic North Pole. To treat these features as significant is to misinterpret the model. Similarly, even if Chekhov could have left the gun off the wall, the wall would be some color or other. He could not insist that if the color of the wall performs no function, the wall should have no color, for a transparent wall would itself be deemed significant. The standard accommodation to the inevitable is to make the ineliminable features as unobtrusive as possible. So the wall would be a neutral shade, the font would be a standard font, the diagram would be designed simply to fit neatly on the page, and the model of San Francisco Bay would have no features that even hinted that its distance from the magnetic North Pole mattered. Arguably, then, we could supplant the other models with the virial equation and simply interpret it so that the irrelevant features are, like the color of the wall or the font of the equation, sidelined. This could be done. But it would make the model ambiguous. Sometimes the equation would exemplify the value of E/V4, sometimes it would not.
Science places a premium on intersubjective agreement. Because scientists build on one another’s findings, it should be clear exactly what those findings are and what justifies them. So other things equal, science seeks univocal symbols. Ambiguity is thus, as far as possible, to be avoided. This, I suggest, is why, rather than replacing the more idealized equations of state with the virial equation, statistical mechanics retains them. One way or another, context has to be accommodated. The choice is between admitting multiple models, each suited to a separate range of contexts, and admitting a single model whose interpretation varies with context.
Because models depend on exemplification, they are selective. A model makes some features of its target manifest by overshadowing or ignoring others. So different models of the same target may make different features manifest. Where models are thought of as mirrors, this seems problematic. It is hard to see how the nucleus of an atom could be mirrored without distortion as a liquid drop and as a shell structure.3 Since a single material object cannot be both fluid and rigid, there might seem to be something wrong with our understanding of the domain if both models are admissible. But if what one model highlights is that in some significant respects the nucleus behaves like a liquid drop, and another model highlights that in some other significant respects it behaves as though it has a shell structure, there is in principle no problem. There is no reason why the same thing should not share some significant properties with liquid drops and other significant properties with rigid shells. It may be surprising that the same thing could have both sets of features, but there is no logical or conceptual difficulty. The models afford different perspectives on the same reality. And it is no surprise that different perspectives reveal different aspects of that reality. There is no perfect model (Teller, 2001) for the same reason that there is no perfect perspective. Every perspective, in revealing some things, inevitably obscures others (van Fraassen, 2008).
Conclusion
In chapter 2, I said that models like the ideal gas and the harmonic oscillator are felicitous falsehoods. They afford an understanding of their targets because their inaccuracies do not interfere with, and may even enhance, their epistemic function. We see now how this can be so. Effective models exemplify features they share with their targets and represent their targets as having those features. This makes them epistemically felicitous. But not all models are, even in my extended use of the term, falsehoods. A model house on a building site is a real house. You could live in it. Eventually someone probably will. A model organism is a real, live organism—a mouse or frog or fruit fly. Like models that are felicitous falsehoods, such models exemplify features they share with their targets and represent their targets as having those features. And like all exemplars, they symbolize selectively. The model house shares a vast number of features with houses it serves as a model of. It does not, however, exemplify all of them. Not being made of marshmallow is a feature that the model and its target share; but it is not exemplified by the model or imputed to the target. Drosophila melanogaster is a model organism that exemplifies general genetic features bearing on heritability of traits, susceptibility to mutation and the like; and it represents other organisms as having those features. It does not, however, in its standard uses, exemplify features that are peculiar to invertebrates, flying animals, or insects. The model house exemplifies its floor plan, the placement of windows, the height of the roof, and the like. It typically does not exemplify the colors of the walls in the different rooms, the way it is furnished, and so forth. Features that are not exemplified are representationally inert. Those features, whether shared with the target or not, are negligible. Understanding is not impeded by neglecting them.
Notes
1. A variant of this quote is regularly misattributed to Virginia Woolf. I am grateful to John Kulvicki for giving me the correct wording and the correct reference.
2. This use of ‘denote’ is slightly tendentious, both because denotation is usually restricted to language and because even within language it is usually distinguished from predication. As I use the term, predicates and generic nonverbal representations denote the members of their extensions. See Elgin (1988), 19–35.
3. I am grateful to Roman Frigg for this example.