Corey J. Maley
Department of Philosophy
Cognition, Agency, and Intelligence Center
Purdue University
coreymaley.net
The thesis of this paper is simple: structural representation and analog representation turn out to be two names for the same kind of representation. This may seem implausible, given that typical examples of structural representation (e.g., maps) are much more complex than typical cases of analog representation (e.g., liquid thermometers). Here I will outline what it takes to extend the simple account of analog representation to cover more complex cases of analog representation, then show that the result is also an account of structural representation.
First, what is structural representation? It is purportedly a unique type of representation, often deployed in neuroscience and cognitive science. A number of theorists have offered accounts and critiques (e.g., (Facchin 2024; Artiga 2023; Nirshberg and Shapiro 2021; Shea 2014; Swoyer 1991)). Unlike propositional or other symbolic representation, the format of structural representations contains information that, in some special sense, reflects information about the target or referent of the representation. More specifically, as the name suggests, the structure of the representation itself somehow mirrors the structure of what is represented. One simple example is the topographic organization of visual cortex area V1: neurons near one another represent areas of the visual field that are near one another in the same way that areas of a map that are near one another represent locations that are near one another. “Near one another” here means something rather specific: given a neuron N that represents an area A of the visual field, the neurons to the left, right, above, and below N will represent the areas to the left, right, above, and below A, respectively, just as in an ordinary map.
Some of my earlier work has focused on analog representation, also a purportedly unique type of representation (Maley 2011, 2023b, 2023a). Typical examples of analog representation include things like analog clocks or liquid thermometers. In this type of representation, some physical magnitude increases (or decreases) as the thing being represented increases (or decreases). As temperature increases, the height of the liquid in the thermometer literally increases. This is unlike a digital thermometer, where, as the temperature increases, a sequence of symbols changes from one pattern to another. A height of 74 cm (representing, say, 74 degrees F) is literally taller than a height of 73 cm; but the digit sequence “74” is simply a different sequence than “73.” This kind of account has been called the mirroring account of analog representation: the structure of the representation “mirrors” the structure of what is being represented.
These examples of analog representation are what we might call one-dimensional: there is one dimension along which the representation varies, reflecting the single dimension along which the variable of interest varies. Other examples of analog representation have multiple dimensions: photographs, 3-D models, maps, and animations are all examples. But what exactly makes these examples analog?
Here’s the short version of my attempt to answer that question: analog representation can be extended to multiple dimensions, where different dimensions function as something like dependent and independent variables. Take a greyscale photograph as an example; call the scene depicted in the photograph the image (we will assume this image is two-dimensional: the image of a photograph is the two-dimensional projection of a three-dimensional scene). The individual points of the photograph correspond to points of the image, where the grey value of the point on a photograph represents the luminosity of the corresponding point in the image. We have two independent dimensions of variation: the vertical dimension and the horizontal dimension (or the x-axis and the y-axis, if you like). The dependent dimension is the grey level at each point.[1]
Interestingly, extending analog representation into multiple dimensions also makes clear precisely what kinds of mirroring relations constitute these representations. Consider a single point in the photograph. The grey level of this point increases/decreases as the luminosity of the point it represents increases/decreases. But with respect to what does this increase/decrease happen? When we asked the similar question about the thermometer, the answer was time: the liquid height increases/decreases as temperature changes with respect to time. But nothing changes in a photograph: after all, it is a static representation. For the photograph, the answer is that the grey value changes with respect to changes in the two independent dimensions. Pick a point P on the photograph, which represents point M on the image; as you examine points to the left of P, you are examining points that represent points to the left of M. In other words, as you go left of P, the variation in the grey level reflects variation in the luminosity of points to the left of M.
In the one-dimensional case of the thermometer, the mirroring involved is between temperature and height: variation in the height of the liquid reflects variation in the temperature. More generally, variation in the magnitude doing the representing reflects variation in the magnitude being represented. But note that in the photograph, we have multiple mirroring relations among different magnitudes. First, we have the relation between the grey level of a pixel and the luminosity of the point it represents. But we also have the two directional dimensions of variation: in short, the locations of the points in the photograph (i.e., the two-dimensional grid of points that constitute the photograph) represent the locations of points in the image (another two-dimensional grid).
There is more to say about the right way to make “mirroring” precise, and about what kinds of magnitudes allow for different kinds of analog representations (i.e., whether analog representations can have “parts,” and thus be subject to the “parts principle” or not). In the paper, I attempt to do just this: characterizing mirroring in terms of homomorphisms and couching the uniqueness of analog/structural representation in terms of magnitudes, which come with clear ordering relations—forming the basis for non-arbitrary structure preservation—for free. For now, however, note that this spelling out of analog representation turns out to also be a characterization of structural representation, without remainder on either side.
I think this is interesting in and of itself: analog representation and structural representation come from different disciplines; the fact that they converge on the same characterization suggests that there is something robust about this type of representation. Additionally, however, it adds some theoretical clarity to how we understand this type of representation. In some respects, we know more about analog representation than we do structural, while in other respects, the converse is true. If they really are just the same type of representation, then what is known about one applies to the other. Or, if that turns out not to be the case, perhaps illustrating why certain examples are, say, structural but not analog (or vice versa) in the context of this account will make more precise exactly how they do differ.
References
Artiga, Marc. 2023. “Understanding Structural Representations.” The British Journal for the Philosophy of Science. doi:10.1086/728714.
Facchin, Marco. 2024. “Maps, Simulations, Spaces and Dynamics: On Distinguishing Types of Structural Representations.” Erkenntnis 90:2743–64. doi:10.1007/s10670-024-00831-6.
Maley, Corey J. 2011. “Analog and Digital, Continuous and Discrete.” Philosophical Studies 155(1):117–31. doi:10.1007/s11098-010-9562-8.
Maley, Corey J. 2023a. “Icons, Magnitudes, and Their Parts.” Crítica. Revista Hispanoamericana de Filosofía 55(163):129–54. doi:10.22201/iifs.18704905e.2023.1411.
Maley, Corey J. 2023b. “The Analog Alternative.” in Mind Design III, edited by J. Haugeland, C. F. Craver, and C. Klein. MIT Press.
Nirshberg, Gregory, and Lawrence A. Shapiro. 2021. “Structural and Indicator Representations: A Difference in Degree, Not Kind.” Synthese 198:7647–64. doi:10.1007/s11229-020-02537-y.
Shea, Nicholas. 2014. “Exploitable Isomorphism and Structural Representation.” Proceedings of the Aristotelian Society 114(2pt2):123–44. doi:10.1111/j.1467-9264.2014.00367.x.
Swoyer, Chris. 1991. “Structural Representation and Surrogative Reasoning.” Synthese 87(3):449–508. doi:10.1007/BF00499820.
[1] (On my account of analog representation, it does not matter whether the variation happens continuously or in discrete steps along different dimensions. An analog clock does not cease to be analog when it ticks in discrete steps, and a photograph still represents what it does qua analog representation whether it is pixelated or continuous, and whether it has only 16 grey levels or continuously-many. Back to multidimensional representation.)