Note that, according to the reconstructionist approach, there should be an isomorphism between the representation and the entity it stands for: ``representation of something is an image, model, or reproduction of that thing'' [Suppes et al., 1994]. This amounts to the Aristotelian idea of representation by resemblance, which happens to have been discredited so thoroughly as to become a rare point of consensus in the philosophy of mind [Cummins, 1989]. Barring resemblance, or ``first-order structural isomorphism'' [Shepard, 1968] between the object and the entity that stands for it internally, what relationship can qualify as representation of something by something else? Shepard (1968) suggests ``second-order'' isomorphism: ``... the isomorphism should be sought --- not in the first-order relation between (a) an individual object, and (b) its corresponding internal representation --- but in the second-order relation between (a) the relations among alternative external objects, and (b) the relations among their corresponding internal representations. Thus, although the internal representation for a square need not itself be square, it should (whatever it is) at least have a closer functional relation to the internal representation for a rectangle than to that, say, for a green flash or the taste of a persimmon,'' (Shepard and Chipman, 1970, p.2).
[IMAGE ]
Figure 6: Clustering by similarity, and its representation that
fulfills the requirement of second-order isomorphism (in this case,
isomorphism between distance ranks in the represented and the
representing spaces).