An illustration of the principle of representation by second-order isomorphism appears in Figure 6. Let us now identify mathematical conditions under which isomorphism between distal and proximal relations gives rise to representation which is, in a sense, true to the original. Let [IMAGE ] be the set of distal objects, and [IMAGE ] --- the set of internal tokens that participate in the process of representation. Let [IMAGE ] and [IMAGE ] be functions defined, respectively, over sets of distal and proximal entities (no restriction needs to be placed at this stage on the number of arguments of f,g). According to the requirement of second-order isomorphism, [IMAGE ] is isomorphic to [IMAGE ] under a mapping M if [IMAGE ], where the relation [IMAGE ] is that of set isomorphism, defined over [IMAGE ]. To constrain the choice of M, we have to be more specific in defining the functions f,g. In the context of shape perception, it is natural to consider for this purpose similarity (actually, two similarity functions must be defined, one for the external objects and one for the internal tokens standing for those objects). Intuitively, we would like the representation to capture the similarity relationships within and between natural kinds [Quine, 1969]. More specifically, similarity is seen to be relevant both to recognition, in which case resemblance between the viewed shape and some previously seen ones is to be assessed, and to categorization, where similarities between a shape and a number of shape classes are compared [Nosofsky, 1988,Goldstone, 1994]. For a pair of internal representations, similarity is not directly observable, but can still be defined operationally as the degree to which an optimal stimulus for one token activates the other one [Shepard and Chipman, 1970,Edelman, 1995c].
As noted in [Nicod, 1930,Quine, 1973], similarity should be construed as a triadic relation: knowing that A is more similar to B than to C is much more informative than merely knowing that A is similar to B and B to C. Furthermore, we may assume that similarity is defined qualitatively and not quantitatively, that is, the range of f,g is [IMAGE ]. This assumption limits neither categorization (because knowledge of similarity for every triplet of points belonging to a metric space defines unequivocally the clustering of the points), nor recognition (because the location of each point can be recovered from triadic similarities using nonmetric multidimensional scaling). We thus obtain [IMAGE ], and, analogously, [IMAGE ]. The condition on M then becomes: [IMAGE ]. In other words, the mapping M is required to preserve similarity ranks.