1. Introduction

Most of us would accept that scientific theories represent, although there has been relatively little consideration of the nature of this representative function. What discussions there have been have tended to import preestablished conceptions from analyses of representation in the arts, language, cognition, and so forth. Of course, which of these analyses one favors will depend on how one conceives of theories in the first place. If one thinks of them in terms of an axiomatized set of logico-linguistic statements, then one might naturally be drawn to accounts of linguistic representation in which notions of denotation, for example, feature prominently. If, on the other hand, one conceives of theories in nonlinguistic terms, as in the model-theoretic approach, then one might look to analyses of representation in the arts where notions of resemblance tend to be brought to the fore.

Van Fraassen, for example, has imported such an analysis into his discussion of representation in science and argued that an appropriate account of resemblance can be given in terms of the set-theoretic relation of isomorphism (Reference van Fraassen1994). This has been strongly criticized by Suarez, who argues that just as isomorphism cannot capture representation in art, so it is inappropriate in the scientific context as well (Reference Suarez1999). Similarly, Hughes draws on Goodman's rejection of resemblance in art in favor of denotation and, rather confusingly perhaps, favors the latter whilst also maintaining the model-theoretic view of theories (Reference Hughes1997). In what follows, I shall examine the debate in terms of three claims:

1. Isomorphism is not sufficient for representation;

2. Isomorphism is not necessary for representation;

3. Models denote and do not resemble.

Each of these claims will be questioned and I will conclude by suggesting that, through appropriate modifications, a form of isomorphism can serve to underpin representation in both the arts and science.

2. Isomorphism is Not Sufficient for Representation

That isomorphism is not sufficient for representation seems to be acknowledged by all participants in the discussion. Consider, for example, some markings left by wind and sea on a stretch of sand: even if these were isomorphic to a face for example, still, it is typically suggested, we would not say that the markings represented a face. The crucial element that is missing in such cases is that of intention: in order for something—a painting, a sculpture, markings in the sand, whatever—to represent, there must be the relevant intention involved (van Fraassen Reference van Fraassen1994; see also Suarez Reference Suarez1999). And—or so it is claimed—the same must hold for representation in science.

But why must it? Let us consider again the face in the sand. Here the issue, I suggest, is not so much the nature of representation but the prior one of whether such a face constitutes art or not. Consider the analogous case of the sea and wind carving the Lorentz transformations into the sand. Are we going to similarly insist that such markings do not represent relativistic phenomena of some sort because the relevant intention is absent? The difference seems to be that in the latter case we do not take the causal provenance of the markings themselves as having any bearing on the constitution of the theory as an object. The theory is “there” in the sand, in a sense, in a way in which the respective work of art is not. And, of course, this is what lies behind those recurrent and amusing stories of works of art being mistaken for heaps of rubbish for example (one of the most recent such examples being Damien Hurst's ad hoc gallery “installation” of the detritus of the artist's studio, which was subsequently swept away and put into bin liners by the gallery's caretaker).

What this relates to is an ontological issue: what is a theory? We should note immediately that a similar question can be asked about certain kinds of art—such as musical compositions for example, and some of the discussions here bear a similarity with those in the philosophy of science—but in the case of paintings, say, or sculpture, an answer is forthcoming comparatively easily. Putting things rather crudely for the moment, we could point to the massive rectangle of canvas on the wall of the Spanish pavilion and say “That is Picasso's Guernica,” or similarly, point to the tank full of formaldehyde and say “That is Hurst's cow and calf.” However, we are typically reluctant to similarly appeal to ostension in picking out the theory of relativity, say. The black marks in my photocopy of Einstein's 1905 paper do not constitute the theory, any more than do the marks in Einstein's original draft or the wind-blown squiggles on the beach. And it is precisely this point that is drawn upon by adherents of the model-theoretic approach—such as van Fraassen—when they urge that theories should be regarded as extra-linguistic entities. But if they are so regarded then the distinction between marks that were intended by Einstein to represent certain phenomena and marks carved into the sand by the wind, appears to evaporate.

My suggestion, then, is that drawing on representation in art in order to power a discussion of representation in science is not entirely straightforward. In particular, whereas intentions are typically drawn upon in art in order to distinguish “artistic” objects from other kinds,Footnote ¹ this is not the case in science. Hence arguments in which resemblance or its formal equivalent is not sufficient to capture the representational relationship in the case of theories, because intention needs to be added to the mix, may not be as powerful as might be first thought.

3. Isomorphism is Not Necessary for Representation

The principle argument for this claim is that there exist examples of works of art which are not, indeed, cannot be, isomorphic to anything because, at one extreme, they enter into a multiplicity of representational relationships and, at the other, they may enter into no such relationship (Suarez Reference Suarez1999). Thus, as an example of the first possibility, consider Picasso's Guernica. In this case—it is claimed—on the one hand it represents the “concrete pain” of the inhabitants of Guernica but on the other it also represents the “more abstract threat” posed by the rise of fascism in Europe. Therefore, “the canvas cannot be placed into a one-to-one correspondence, not to mention an isomorphism, with those things that it represents” (Reference Suarez1999, 78). At the other extreme, the example of Mondrian is given and it is suggested that to focus on representation in such cases is to miss the point of the art: “[t]his type of abstract painting is capable of inducing esthetic, or emotive responses, but it does not do so by ‘representing’ anything” (Reference Suarez1999, 79).Footnote ²

However, such cases do not decisively rule out an account of representation based on isomorphism, as Budd, for example, demonstrates (Budd Reference Budd1993). His approach begins with a crucial distinction between one's “visual world” and one's “visual field”:

The relationship between the two can be understood in terms of abstraction: my visual experience represents the world as a collection of objects “spread out in three-dimensional space” and my visual field is what is left when we abstract the apparent distance of these objects from me. Thus, for example, if my visual world contains a circular object that is titled away from me, then within my visual field this object will appear elliptical. As Budd makes clear, one should not characterize this difference in terms of the visual world and visual field containing different objects, but rather in terms of different accounts of how my visual experience represents the world as being, one complete and one partial. This distinction then allows Budd to firm up the claim that a painting represents by sharing properties with its subject:

This “sharing” of properties can then be captured via the notion of isomorphism, where this is taken to hold between the structure of the surface of the painting and the structure of the relevant visual field. So when you look at a painting, what you see is the structure of the surface of the painting as being isomorphic with the structure of the visual field of the state of affairs that the painting depicts. This yields the following account of pictorial representation:

Putting it in a nutshell, representation consists in the perceived isomorphism of structure (Reference Budd1993, 162; my emphasis).

As an illustration of this approach, Budd considers Holbein's famous picture, The Ambassadors. Here, as is well known, the foreground of the painting includes a depiction of a skull that is represented anamorphically, so that it appears as a skull only if looked at from one side of the picture. From Budd's point of view, the depiction of the skull is isomorphic with the representation in the visual field of a skull in the precise sense that there exists a one-to-one mapping between the points that compose the two, but

Even more significantly, cases of “abstract” art can also be accommodated. First of all, it must be acknowledged that in many cases the visual field representation of the picture surface may be “strikingly unlike” that of the relevant state of affairs. The perceived dissimilarities can take two (obvious) forms: the visual field representation of the picture surface can lack features possessed by the visual field representation of what is depicted, or the former can possess additional features that the latter does not. Of course, as Budd is clear to note, this absence or presence of features does not imply that the spectator sees what is depicted as lacking or possessing the relevant features. The absence of color in a black and white drawing, for example, is not understood by the spectator as indicative of a lack of color in what is being depicted but only as not indicating any color; as Budd puts it, “the object, as depicted, has an indefinite appearance in the dimension of color.” By eschewing color, the artist's intention is simply “to depict only the spatial structure of a state of affairs and the comparative brightness of its parts, perhaps, not the colours of its constituent objects” (Reference Budd1993, 164). Likewise, a spectator viewing a painting or drawing in which stippling or cross-hatching is used to represent variations in tone does not read these features into the objects being depicted. In general, we typically—but not always—abstract from certain details of the picture surface.

Thus, in considering the example of Picasso's Head of a Girl, the loops and spirals of Picasso's drawing correspond to the visual-field representation of a girl's hair only when other features are abstracted away. In general, schematic depictions resemble the visual-field representations of the corresponding objects “only with respect to the structural features they depict” (Reference Budd1993, 164). As Budd emphasizes, the perceived isomorphism refers to structure “at a certain level of detail” (Reference Budd1993, 165) and one can easily see how this account also accommodates Picasso's Guernica. It can be extended to cover Mondrian's work as well, if this is understood as a form of schematic depiction in which what is being depicted is “pure relationship.”Footnote ³

More intriguingly, Budd claims that it can capture depictions of “impossible” or contradictory objects such as these featured in the works of Escher, for example. If we consider the famous case of the three-pronged tuning fork, which has two rectangular bars at one end and three cylindrical bars at the other, then what we have, in effect, are two incompatible isomorphisms:

However, we cannot impose on the lines a single, consistent interpretation and attempts to do so lead to well-known visual discomfort.

What can we conclude from all this? First of all, that accounts of representation in art that invoke isomorphism cannot be so easily dismissed (at least not on the basis of the examples so far presented). Secondly, that one such account can accommodate what might, at first sight, seem to be the problematic cases of representation in “abstract” art, such as Picasso's Guernica, the representation of pure structure, as in Mondrian and the representation of “inconsistent” objects, as in Escher's drawings. As I shall indicate, in capturing such cases, this account offers a nice parallel to a generalization of the model-theoretic account as applied to theories and models in science.

4. Models Denote and Do Not Resemble

Budd's approach can be understood as part of the backlash to Goodman's classic rejection of resemblance-based accounts of representation in favor of one based on denotation (Goodman Reference Goodman1969). Moving back from art into science, Hughes has incorporated Goodman's account into the model-theoretic framework, yielding his own DDI (Denotation, Demonstration, and Interpretation) view (Hughes Reference Hughes1997). A model then “stands for” or refers to a physical system, and elements of the model denote elements of the subject. Furthermore, scientific representations possess an “internal dynamic” that allows us to make predictions of both everyday and, crucially, novel facts. Through this internal dynamic we “demonstrate” the relevant results. And a given model may provide the internal dynamic of another so that we get a hierarchy of models, and of representations, in which a model at one level of the hierarchy is the subject of and represented by a model at a higher level. Finally, these results that are demonstrated within the model have to be interpreted in terms of its subject. It is only after such interpretation that we can see “whether theoretical conclusions correspond to the phenomena, and hence whether the theory is empirically adequate” (Reference Hughes1997, 333).

However, transporting Goodman's approach into the scientific context is not unproblematic. As is well known, the primary basis for his rejection of resemblance is the perceived lack of correspondence between the properties of resemblance and representation: the former is symmetric and reflexive, for example, whereas the latter is not (Goodman Reference Goodman1969, 4). The obvious way to overcome this objection is to insist that we do not simply model a phenomenon, we model it as something. Thus, for example, we model a gas as a system of billiard balls, or we model a superconductor as a giant diamagnet (see French and Ladyman Reference French and Ladyman1997). Furthermore, even if it is granted that when it comes to artistic objects, “almost anything may stand for anything else” (Goodman Reference Goodman1969, 5), it is not clear that this is the case for models. Not anything can serve as a scientific model of a physical system; if the appropriate relationships are not in place between the relevant properties then the “model” will not be deemed scientific.

Hughes's emphasis on denotation also puts him at odds with the model-theoretic approach he espouses. Words are typically held to have denotation and, of course, the analysis of how words represent might well be, and typically is, quite different from the analysis of pictorial representation.Footnote ⁴ Furthermore, denotation itself cannot do all the work, as Hughes acknowledges. One must also include some account of interpretation, but if one is going to do that, why bother with denotation to begin with? Why not return to an account based on isomorphism?Footnote ⁵

More importantly, however, in describing the relationship between theories and models from the perspective of his approach, Hughes draws on an interesting example. Unlike Suarez, he allows for the possibility that both “low-level” models and “high-level” theories represent. In particular he notes that Bohr's famous atomic model can be described as a “local” model and represents a clearly specified type of system (Reference Hughes1997, 330). Quantum mechanics, on the other hand, can be considered a “global” theory and deals with a heterogeneous collection of physical systems. In this case we can still say that there is representation insofar as each individual system in this collection can be represented by a particular model defined in terms of the theory. What a global theory defines is not a particular model but a class of models and in the application of the theory to a particular system, it is a “local” member of this class that represents. According to Hughes, “[t]here is thus no significant difference between the representations supplied by local and global theories” (Reference Hughes1997, 331).

Now, in the case of the Bohr's model of the atom we have a famously inconsistent model and in this context it raises the issue of whether and in what sense one can have inconsistent representations. On the Goodman account, one would have to appeal to some notion of inconsistent denotation, that could perhaps be related to the various forms of paraconsistent logic that are widely available. Alternatively, one could try to accommodate such cases within the model-theoretic approach, although there have been relatively few studies along such lines.

Laymon, for example, notes that one of the advantages of the model theoretic approach is that it

Thus his primary concern is with the way in which theories that are inconsistent with one another are used to account for phenomena, rather than with the explanatory role of internally inconsistent theories themselves. However, he does emphasize the role of idealizations and approximations in generating a certain “looseness of fit” between theories and their computational consequences, sufficient to allow mutually inconsistent theories to be used. Now, idealization and approximation can be accommodated within the model-theoretic approach through the introduction of “partial isomorphisms” as the fundamental relationship between theoretical and data models (see French and Ladyman Reference French and Ladyman1998). In particular, they allow for the way in which two models can “share” parts of their structure.

The technical details have been given elsewhere, but, in summary, are as follows: A partial structure is a set-theoretic construct $\mathsf{a}=\langle D,R_{\mathrm{i}\,},\rangle _{\mathrm{i}\,\prime \in \mathrm{I}\,}$ , where D is a nonempty set and each R_i is a partial relation. A partial relation R_i, i′ ∊ I over D is a relation which is not necessarily defined for all n-tuples of elements of D (see da Costa and French Reference da Costa and French1990, 255). Each partial relation R can be viewed as an ordered triple 〈R ₁,R ₂,R ₃〉, where R ₁, R ₂, and R ₃ are mutually disjoint sets, with $R_{1}\surd R_{2}\surd R_{3}=A^{n}$ , and such that: R ₁ is the set of n-tuples that belong to R, R ₂ is the set of n-tuples that do not belong to R, and R ₃ is the set of n-tuples for which it is not defined whether they belong or not to R. If we have two partial structures

and

(where R _k and R′ _k are partial relations as above, so that $R_{k}=\langle R_{k1},R_{k2},R_{k3}\rangle $ and $R\prime _{k}=\langle R\prime _{k1},R\prime _{k2},R\prime _{k3}\rangle $ ) then a function f from D to D′ is a partial isomorphism between $\textsf{a}$ and $\textsf{a}$′ if 1) f is a bijective and 2) for all x and y in D, R _k1xy iff R′ _k1f(x)f(y) and R _k2xy iff R′ _k2f(x)f(y) (French and Ladyman Reference French and Ladyman1999; Bueno Reference Bueno1997). Of course, if $R_{k3}=R\prime _{k3}=\mbox{\o} $ , so that we no longer have partial structures but “total” ones, then we recover the standard notion of isomorphism (see Bueno Reference Bueno1997).

In terms of this framework, one can then understand how “computational consequences” can be obtained from an “internally” inconsistent theory such as Bohr's (see da Costa and French Reference da Costa and Frenchforthcoming). It is by acknowledging that there is a certain internal “looseness of fit” between the component elements of the model, that inconsistency can be comfortably accommodated. However, my concern here is with the issue of how such a model might be said to represent, for, in a sense, the corresponding “object” or system is an impossible one.

Here I can only sketch a response (some of the details can be found in da Costa and French Reference da Costa and Frenchforthcoming): putting things a little crudely, Bohr's theory contains elements of both quantum and classical physics and if we were to focus on each element to the exclusion of the other, we might be tempted to say the theory represents a quantum or classical system respectively. However, this would be to ignore what effectively “binds” the theory into a whole and allows the two disparate elements to coexist, as it were, and that is Bohr's central notion of a “stationary state.” It is here that the two contradictory elements come together: classical mechanics applies to the dynamics of the electron in the stationary state, while quantum theory comes in when the transition between such states is considered. However, it is important to note that it is not only in the discreteness of the stationary states that we have conflict between quantum and classical physics but also in what has been called “one of the most audacious postulates ever seen in physics” (Pais Reference Pais1991, 147), namely, the assertion that the ground state is stable, so that an electron in such a state will not radiate energy and spiral into the nucleus as determined by classical physics. This is the central inconsistency of the Bohr model and together with their discrete nature it is what makes the stationary states so peculiar.

Of course, as the Bohr model evolved and came to be supplanted, this peculiarity was eventually understood in terms of the new quantum mechanics and as a consequence, the formal inconsistency evaporated (or, better, came to be replaced by “higher-order” or interpretational incongruences that the principle of complementarity was intended to resolve; see da Costa and French Reference da Costa and Frenchforthcoming). At the time the model was proposed, however, the notion of a stationary state was not understood at all, or at best, only partially, and if one were to represent Bohr's model in terms of partial structures, the stationary states would have to be located among the R₃, as relationships which had not yet been established to hold or not. Characterizing the model in this way one can accommodate the partial and conceptually “blurred” nature of the stationary states that allows for the internal “looseness of fit” between the component elements of the model. And this in turn, gives us an idea of how the model can still be said to represent: what it represents is a system that has elements of classical and quantum physics but has at its heart this poorly understood and conceptually indistinct notion.

There is a nice parallel here with Budd's account of the artistic representation of impossible objects: if we focus on, or abstract out, the classical and quantum aspects of Bohr's model, then each can be taken to represent, respectively, classical and quantum objects. It is when these aspects are brought together, and we wonder what the model as a whole represents, that we experience a kind of cognitive discomfort akin to the visual discomfort we encounter when we look at one of Escher's drawings. Of course, in the Bohr case this comfort was eventually assuaged or, perhaps, merely transposed to the level of interpretation, through the introduction of a formally consistent theory but—and this is marks a difference between the scientific and artistic cases—no such developments can, or should, be expected in the case of Escher's work. More generally, Budd's account can be seen as the counterpart, for artistic objects, of the partial-structures approach for scientific ones. Analogously to the latter case, pictorial depictions resemble the visual-field representations of the corresponding objects only with respect to certain features and, as I noted, the perceived isomorphism refers to structure only at a certain level of detail.

5. Conclusion

My aim in this paper has been to suggest that the “isomorphic” account of representation is not ruled out by the kinds of examples that have been deployed against it. First of all, we need to be careful in importing conclusions drawn from examples in art into the domain of science. Secondly, even in the artistic realm a notion of isomorphism has been defended. Furthermore, this notion is interestingly partial: in particular, if we characterize both the structure of the surface of the painting and the structure of the relevant visual field in terms of something akin to partial structures, then Budd's central idea that the visual-field representation of the picture-surface can lack features possessed by the visual-field representation of what is depicted, or the former can possess additional features that the latter does not, can be captured in terms of partial isomorphisms holding between these partial structures. The parallels are perhaps most striking when it comes to the interesting case of inconsistent representations.

Of course, this is not to deny the differences between scientific objects and artistic ones; we could certainly invoke Budd's account to capture the relationships between Picasso's sketches and the finished Guernica (as an exercise parallel to French and Ladyman's partial structures analysis of the construction of the London-London model of superconductivity, for example; see French and Ladyman Reference French and Ladyman1997), but still we might balk at taking Picasso's masterpiece to be diachronically “open” in the way theories are. Nevertheless, there are enough parallel aspects to suggest that what we have here is a common account of representation and one that encourages further exploration.