It is a common assumption about models that they are representations. This the starting point for several debates about scientific models. A number of papers discuss the issue of what it might mean for non-linguistic representations like set theoretic structures or isomor- phisms to represent a phenomenon: see Frigg (2006), Giere (2004), Su´arez (2003). Argu- ments for different views of how models represent (whether by similarity or isomorphism) also abound: seeFrench (2003),Giere(2004),Teller(2001),van Fraassen(1980). Furthermore, it is commonly assumed that being representations is essential to models playing the roles they do in science. Bokulich (2011) evaluates three accounts of what makes a model explanatory, all of which assume that models are representations or descriptions of phenomena.
In this sense, models are still considered second class scientific tools; in order to be useful they must approximate, by accurately representing, one of the first class scientific tools like a theory or a real-world system. In this section I first review some of the arguments for models being essentially representational, then explore the possibility of non-representational models. I argue that in some cases, it is not in virtue of being representations, but rather in virtue of being appropriate stand-ins or exemplars that models make themselves useful.
5.3.1 Representation in Explanation
I’ll call models that act primarily as representations, representational models. In some cases, representational models can be used to explain. Teachers use globes to explain geography to their students, and it is the fact that the globe represents the relevant geographical features that makes this possible. That the globe accurately represents the shapes and locations of countries and bodies of water is what makes it useful for explaining why Syria is not Iran’s route to the sea, for example. Here the globe is a model of the earth, and its explanatory power comes from its being a representation of the target system.
Another teacher might use a globe and a piece of string to explain the unintuitive fact that the shortest flight path from Pittsburgh to Paris goes roughly over Gander, Newfoundland, despite it being further north than either the start or end point. In this case the globe needs to represent geometrical features rather than geography: the relative positions of the three cities, and the latitude and longitude lines are the essential features for this explanation. If these features of the system are accurately represented, then once again the globe can be used as a representational model, this time to explain why shortest flight paths between distant cities often curve significantly north or southwards.
Roughly this manner of models being useful in explanation in virtue of being represen- tations is overwhelmingly emphasized in discussions of model explanations.
5.3.2 Beyond Representation
There is, however, a growing chorus of voices questioning the hegemony of representation in modeling, both within the general philosophy of science literature, and in cognitive science.
One problem is that there is no clear consensus about what a representation is. Some of the competing notions include similarity accounts, isomorphism, and homomorphism. There are known problems with each of these in terms of whether they include too much or too little, and whether they can do the work representations are supposed to do, of standing in for their targets and grounding inferences back to those targets.
Knuuttila (2011) argues for what she calls an “artefactual approach” motivated by the conviction that seeing models as representations is too limiting. She denies that the rep- resentational relationship is “privileged.” Instead, she emphasizes how models function as “external tools for thinking, the construction and manipulation of which are crucial to their epistemic functioning” (Knuuttila 2011, 263). In Knuuttila and Boon (2011) this criticism of representational accounts of modeling is continued, and the artefactual approach explored further.
Ramsey(1997) argues that appeals to representation in discussions of connectionist mod- els are largely confused. In Ramsey (2007) he expands on this critique to consider how rep- resentation is appealed to in computational theories of cognition more generally. He argues that the internal states and structures invoked as representations in connectionist modeling and cognitive neuroscience do not really serve as representations in those models at all. Both what he calls the receptor notion and tacit notions of representation are rejected as not being capable of doing the jobs representations are expected to do in these models, and as adding “nothing of explanatory significance” (Ramsey 2007, 186). He claims that if connectionist models provide an accurate picture of how the brain gives rise to cognitive capacities, “then those capacities are not driven by representational structures” (Ramsey 2007, 187).
If models do not operate primarily as representations, it makes sense to look beyond representations, and explore other, non-representational options for understanding models. My view of how models work is compatible with Knuuttila and Boon’s idea of models as concrete objects that can be manipulated to generate knowledge. Their focus on models being constructed and on the role of agents in interacting with models is a bit different than mine, as they are interested in answering mainly epistemological questions. I’m mainly concerned with the ontic status of the models, and the methodologies used in modeling. Another source of inspiration for my non-representational view of models is Morgan(2002),
in which she distinguishes between models that are representations and models that are representatives (although Morgan defines models as representations).
The proposal I’ll articulate in more detail in later sections takes the type of model I’m interested in to be physical stand-ins for their targets. These stand-ins allow for something like surrogative explanation, as described by Swoyer (1991). Swoyer’s surrogative explana- tions are enabled by what he calls structural representations. The line between this notion of representation and the models I’m referring to as non-representational may be a fine one. The point I want to emphasize is that in drawing inferences from the model to the target, the fact that the model represents the target (if it does) plays no significant role, so calling these representations does no useful work (for my purposes). There are certainly other functions for models in which representation does play a role.
Returning to the example of explaining why shortest flight paths often curve north or southwards, a prop like a globe is certainly a useful pedagogical tool, but there are other ways of explaining the same phenomenon where it is less clear what explanatory role representation is playing.
Shortest flight paths can also be explained using equations for great circle distances and geodesics. These equations can be thought of as representations, certainly, if you let variables represent features of the system like the coordinates of each city. It is not obvious that the variables representing coordinates is where the explanatory power comes from, however. The equations on their own, without any variable assignments, are provably true. They guarantee that for any pair of non-antipodal points on a sphere, there is a unique great circle, and the shortest distance between those points is the shorter arc into which the points divide the great circle. The form of the equation for that arc shows whether it curves north, south, or neither.
None of this depends on the variables representing anything in particular. You can do the proof very well with the variables uninstantiated. In applying the proof to a particular case, there are two options available. In the representational option, you might substitute points within Paris and Pittsburgh for the variables, then calculate the equation for the flight route, and check the form of that equation.
fact, applicable to anything that meets the assumptions required for the proof to work. Namely, Paris and Pittsburgh are on a surface close to a sphere, and are not antipodal, so they meet the assumptions. From this you can conclude that there is a unique great circle passing through them, the shorter arc of which curves northward. This can be proven without instantiating any variables. This second use of the proof depends on facts about the locations of the cities on an almost-sphere and the truth of the equations about great circles and geodesics, not on instantiating the variables such that they represent the locations of the cities.
In this sort of demonstration, a general fact is proven, of the form, ∀x(P (x) → C(x)), where P (x) means a set of premises is true of x, and C(x) means the conclusion is true of x. From this general fact, if the premises are true of the target system t, then by universal instantiation and modus ponens, the conclusion is true of the target too. This does not require the target to be represented by the model (in this case the equation). The explanation just requires a target system to meet certain criteria, and for a generalization to be true of things meeting those criteria.
In this example, I would not say that the equations are physical stand-ins for the target system. The point is to show the form the inferences take in this sort of explanation. This same inferential structure is how I think experiments on models can be applied to their target systems.