Trope sheaves. A topological ontology of tropes

Abstract

In this paper I want to show that topology has a bearing on the theory of tropes. More precisely, I propose a topological ontology of tropes. This is to be understood as follows: trope ontology is a „one-category”-ontology countenancing only one kind of basic entities, to wit, tropes.1 Hence, individuals, properties, relations, etc. are to be constructed from tropes. However, the world can’t be considered as a mere set of tropes. Tropes do not come as a heap, so to speak, rather, the set of tropes has to have some structure. In this paper I want to deal with the problem of what kinds of structures are necessary to make trope theory work. The example of geometry may explain how the role of structure is to be understood here: The starting point of geometry as the theory of space is not an unstructured set of space-points, from a geometrical point of view space has to be conceived of as a structured set of space points. Similarly, trope theory is to be considered as a theory of structured sets of tropes. This proposal is, of course, not new. From the very beginning of trope theory, philosophers have realized that tropes as „the very alphabet of being” (Williams (1953:7)) do not suffice. We need a kind of syntax, at least. Thus, the set of tropes has to be endowed with some further structure. Williams and others proposed two „complementary” relations I want to call in the following the relation of compresence and the relation of resemblance. Often, it has not been quite clear what requirements those relations have to satisfy. The example of geometry should warn us that such an approach is bound not to be overly successful. Ponder- ing on what tropes „really are” and remaining vague about their structural relations corresponds to a conception of geometry that concentrates on what points and lines „really are” while ignoring their structural relationships. This amounts to a rather outdated style of geometric theorizing. I contend that an analogous assertion holds for trope theory. For this reason, I propose to pursue trope theory along similar lines as Hilbert did for geometry, i.e. considering trope theory as a structural theory of tropes. Fine recent examples of this line of research can be found in Bacon (1987), Bacon (1988) and Fuhrmann (1991). Although the spirit of my approach resembles theirs, I chose a different line of investigation: I intend to deal with the hitherto unnoticed topological aspects of trope structures. More precisely, I want to show that the appropriate formal framework for trope theory is provided by themathematical theory of sheaves (cf. Tennison (1975), MacLane/Moerdijk (1992)). The outline of this paper is as follows: In the next section we recall and formalize the basic concepts of trope theory. As starting point I take the concept of a trope space, i.e. a set of tropes endowed with the relations of compresence and of resemblance. Traditionally, both of these relations are taken as equivalence relations. In this paper I want to show that a more convenient choice for resemblance is to conceptualize it topologically. As a first step of this task, I show that the resemblance relation need not be assumed to be an equivalence relation. Rather, it is sufficient to consider it as a similarity relation. This leads to the replacement of trope spaces by generalized trope spaces. (Generalized) trope spaces give rise to bundles in the sense of mathematical theory of bundles (cf. Goldblatt (1978), MacLane/Moerdijk (1992)).5 In section 3 the topological properties of trope bundles are elucidated. We get that trope bundles give rise to sheaves in the sense of mathematical theory of sheaves (cf. Goldblatt (1977), Tenni- son (1975), MacLane/Moerdijk 1993). In section 4 we introduce continuous sections of trope sheaves as surrogates of universal properties. The relation between section properties and individual properties is investigated. As it turns out, it is just the mathematically well-known relation of a sheaf space to its presheaf of sections. We show that if there exist universals the section properties can be identified with them. In section 5 we close with some remarks on what is to be understood by a general topological trope ontology.