An intelligent system is able to gain information about its environment, reason rationally about it, and then act appropriately. In order to achieve this, it must be able to model its environment. It must represent it somehow. Thus representation is one of the foundations of intelligence. Unfortunately our understanding of the nature and process of representation is relatively immature compared to what we think we understand about other aspects of intelligence. In philosophy and in science, whenever we realise that so much theoretical weight is resting on so feeble a foundation, we know it is time to focus our attention at that foundation.

When analysing the nature of representations it is both useful and tempting to refer to their interpretation by cognitive systems such as humans (see, for example, Peirce's triadic analysis of representation, as presented by Von Ekhardt, 1993, Ch.4, pp.145-159). However, any explanatory account of representation which appeals to mental representational faculties runs the risk of being circular if it is then used to account for mental representation. An approach that is sure to avoid any such circularity is one which relies on no abstract appeal to the mental. This brief analysis of the role of resemblance in representation attempts such an approach. No appeal to interpretation is made. Such a move is necessary at some point in any naturalistic account since, at bottom, there can be no triadic analysis of representation. It requires a purely dyadic grounding relation between representational vehicle and content which does all of the work in explaining not only representation (in all its forms), but also misrepresentation. If successful, the resulting theory will apply equally to all representation, including the mental. It must prove wrong Crane's (1995, p.13) view that mental representation is the most fundamental kind of representation.

The choice of this approach, with one neat stroke, besides eliminating the risk of circularity in attempting to explain mental content, reduces the need for a lengthy discussion of the three aspects of a triadic analysis. What remains here is a discussion of the problems of representation and content determination, followed by a look at higher-order resemblance relations and how plausible they might be as the basis for addressing such problems.

How can one part of the world be about another? This is the problem of representation. What is required to address this problem is a theory of meaningfulness.

What is it in virtue of which that Figure 1 is taken to represent a particular set of stars? It might be argued that there is a degree of convention involved, the arrangement is very familiar to those in this part of the world (and the caption provides a clue to others). But assuming you had no prior encounter with this symbol (and we ignore the caption), as long as you were able to see the southern night sky it would be possible for you to identify the figure with the constellation Crux (The Southern Cross).

The topmost star-shape in Figure 1 represents the most northerly star of Crux, not by virtue of any properties or information embodied in itself, but by its spatial relation to the other star-shapes. The spatial relations between shapes are in proportion to the relative angular displacements of the five stars as they currently appear from Earth. The figure represents the constellation in virtue of the correspondence between those relations.

A depiction of Crux more often uses stars of differing sizes to represent differences in the apparent magnitude (brightness) of the stars as seen from Earth. Sometimes star-shapes with differing numbers of points are used with the numbers of points signifying features of something else. In those cases, the significance of a star-shape having, say, seven points is entirely a matter of convention. Regardless of the shapes or colours of its constituent elements, the figure gains most of its representational power through the spatial relations between them. Each of the dots in Figure 2 has difficulty in representing anything in itself, but as part of a collection of dots on the page, it suddenly means something. So meaning can be attributed to representations on a grander scale: a representational system identifies with another system by virtue of a higher-level resemblance, and parts of the representational system can bear meaning without having to carry information intrinsically. Within the appropriate context, a dot of ink can represent a particular star.

Given that something is a representation, the content problem is to answer the following question:

A. how does that representation have the particular content (meaning, referent) that it does?

Such "how" questions are generally poorly posed - from it we can infer two quite different and more specific questions, namely:

A1. by what process does a representation come to have a particular meaning (or, what is the origin of representational content?); and

A2. what is it about a representation in virtue of which it bears a particular content? (or, what is the nature of a vehicle that has representational content?)

A plausible answer to the second question might tell us everything about the representation relation yet reveal nothing as to the origin of such an entity. For example, the simple notion that representation is by virtue of resemblance says nothing about how such a state of affair might come to be in the world. By contrast, a naturalistic theory which provides a plausible account of the origin of representation (answering the first question) would also imply its nature.

What is required to address the problem of representational content is a theory of meaning. Returning to the example of the stars depicting a constellation, it is evident that it may be futile to focus on individual representations. Instead, the problem of representational content should be about how, and in what sense, a representational system can resemble another system in such a way that parts of the first system can correspond to (and mirror) parts of the second. Outlined below is an information theory of meaning, based on higher-order resemblance at a systemic level.

Resemblance is relation between two systems. My photocopy of this page (the original) bears a high degree of resemblance to the original due to the fact the large number of its properties are identical to the corresponding properties of the original. This systemic relation between corresponding properties is a first-order isomorphism between the two systems. The same kind of isomorphism is what makes a photo of a scene resemble that scene. Each element of the photo represents a corresponding element of the scene. The isomorphism is first-order because the corresponding properties between the two entities are of corresponding kinds: colours correspond to colours, lengths and shapes correspond to lengths and shapes (subject to perspective), and so on. The properties of the elements of a system may be thought of as the set of unary relations on elements of the system. Higher-order relations, such as those describing the arrangement of a collection of elements also participate in the resemblance relation: the spatial relations between elements of the photo correspond to spatial relations between elements of the scene. The piano keyboard provides another example: the vertical separation of written musical notes along one line of sheet-music corresponds to the horizontal separation between the corresponding keys on the keyboard.

The undeveloped film, however, only caries patterns of chemical depletion on its surface, yet it still manages to resemble the scene in some sense. Because levels of brightness and colour in the scene are represented by various levels of chemical depletion in the undeveloped film, the resemblance relation is no longer one of first-order isomorphism. Instead it may be described as a second-order isomorphism, in this case an analog isomorphism (Palmer, 1978, pp.295-297). As with first-order isomorphism, when a system resembles another by virtue of an analog isomorphism, parts of the first system resemble corresponding parts of the second system. However, in a second-order isomorphism the corresponding properties need not be of the same kind, as in the size of a star-shape corresponding to the relative brightness of a star. In the case of second-order isomorphism, the relations need not be of the same kind. Returning to the piano keyboard example: the relative proportional frequency difference between the notes played on the piano corresponds with the horizontal difference in position of the corresponding keys on the keyboard.

Resemblance relates pairs of systems whose corresponding properties exhibit a constant relation. It is a purely descriptive concept which offers no explanation for what it describes. Hence although it may possibly serve as a descriptive basis for the nature of representation (answering A2), it can never, in itself, provide explanation as to its origin (A1).

What is required for a complete explanation is a theory about how one system can come to be isomorphic to another. It seems plausible to suppose there is some kind of influence of one system on the other: a causal link between the systems. In the case of a live motion video image, there is a continuous and reliable causal stream of influence between the scene being captured by the camera and its depiction on a television screen. It is by virtue of this continuous causal link that the image on the screen comes to resemble the scene. Hence we can only understand "how" (A1) the image represents the scene by understanding the nature of the continuous causal link from the scene to the image.

Actually, the image on screen never represents the scene as it is now, since the causal chain of influence between the two inevitably involves a time delay. It appears that any isomorphism carried by causal influence between two spatially separated isomorphic dynamical systems must exhibit some such temporal lag. This, then, provides one explanation for a kind of misrepresentation. So everyone who still has my old telephone number next to my name in their address books, are not really misrepresenting my number, but instead are representing it properly, and just suffering from the effect of a temporal lag in a causal link.

However, dynamical systems do not have to be continuously linked by such causal chains of influence to remain isomorphic. For example, a cheap replica of a wristwatch can maintain almost the same time as the original. The only causal link in this case ends once the replication is complete. Allowing for a predictable, gradually increasing time lag between the two systems, their time displays remain isomorphic (in this case first-order). In addition to the correspondences between constituent elements, their properties and relations, the watches also share corresponding sequences of processes and events (changes in properties and relations over time). However, the internal mechanisms need not be isomorphic: the original might use an entirely mechanical timer whilst the replica might rely on the electronic resonance frequency of quartz crystal semiconductor. If corresponding parts of two systems have equivalent functions within their respective systems, then there is a functional isomorphism between the two systems. As with analog isomorphism, understanding the resemblance alone is not sufficient for a complete understanding of how (A1) one system represents the other. The causal link also needs to be understood.

An example of a very peculiar way of representing a horse is provided by nature in its DNA sequence. What might be said about the correspondence relation between the notional system of all horse DNA codes and the system of all horses? Evidently it is not a relation of first-order isomorphism, nor structural isomorphism, although there obviously does exist an isomorphic relation that maps between the two. Even functional isomorphism doesn't appear to be the appropriate mapping. Such equivalence might be termed operational isomorphism since particular operations defined over the systems need to be available to render explicit certain information embodied in the systems (Palmer, 1978, pp.264-266) before any apparent resemblance can emerge. Sometimes these operations embody information that is crucial to the correspondence, in which case the representation alone cannot be considered to represent without them. For example, consider the following strange-looking sentence:

Only after you are informed to ignore all words except those in bold or italic typeface, does the system above come to represent a determinate state of affairs. Given a different operation, the same system might represent something different, or even become more meaningless. Hence the representational power here is shared between the system and the operations defined on it.

Like many theories of representation, the resemblance theory has problems accounting for misrepresentation. However, it does provide an answer in terms of the systemic view of resemblance: if representation is due to correspondences between systems, then misrepresentation may occur where the two systems are isomorphic on a larger scale, but certain constituent elements miss-out on corresponding appropriately.

Combining the resemblance theory with a causal explanation allows for further understanding of misrepresentation in terms of failures in the causal link: breaks; temporal lags; distortions; poor fidelity; and so on. The causal link establishes the representational relationship, and its shortcomings are what gives rise to misrepresentation.

Another problem with a resemblance theory is that it is a symmetrical relation whereas representation is generally considered to by asymmetrical (Crane, 1995, p.16). Although a horse and its photo seem to equally resemble each other, the horse is not generally taken to be a representation of its photo. But perhaps resemblance is similarly asymmetrical: we rarely say that the horse resembles its photo. The photo is only one aspect of the appearance of the horse taken from a particular angle at a particular distance, at a particular moment, in a particular setting, under particular lighting conditions. It represents a subset of all the properties of the horse. The horse also existed before its photo, and it was the appearance of the horse that helped cause the resulting photo. Perhaps then it is these other kinds of asymmetries which carry over into the representation relation. Resemblance does not have to account for it.

Theories of second-order resemblance provide for powerful, descriptive analysis of the correspondences between representational systems and the systems they represent. They are powerful in the sense that they reduce the constraints on what kinds of physical representational systems might conceivably represent other kinds of systems. In particular, they open the door of plausibility for the brain to represent the world. However, by themselves, they offer little explanation for the process by which such representations emerge. For this reason they must be combined with a causal theory or representation.

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