Arbitrary objects, species structures: metaphysical mathematical structuralism

Non-eliminative structuralism, for example that in the version of S. Shapiro, faces the problem of the so-called incompleteness of mathematical objects and the problem of permutation. The article analyzes the concept of generic mathematical structuralism developed by L. Horsten, which claims to solve these problems while adhering to non-eliminativist, realist positions. The paper presents the key characteristics of the conception of specific mathematical structuralism based on the concept of an arbitrary object offered by K. Fine and the idea of generic structures. According to this conception, each arbitrary object is associated with a domain of individual objects - its values. Thus, with each arbitrary number a domain of individual numbers is associated; with every arbitrary person - a domain of individual people. An arbitrary object has properties that are common to the individual objects of the associated domain. Generic structuralism treats mathematical structures as generic structures, while mathematical objects - as arbitrary. Generic structures themselves are defined by the relation of instantiation - the relation of being in a state. As a version of non-eliminative structuralism, the concept of generic structuralism avoids difficulties encountered by other formulations of this position. Another interesting feature of the concept is the shift of attention from ontological to metaphysical problems, which played a secondary role in the debate about mathematical structuralism. In this regard, we consider the problem of the independence and definiteness of arbitrary objects, which was already pointed out by K. Fine, to be one of the important problems. As applied to Fine’s concept of arbitrary objects, interesting results have already been obtained by means of independent-friendly logic. The application of its conceptual means, in our opinion, will make it possible to obtain metaphysical results important for the structuralist philosophy of mathematics. The advantages of the analyzed concept are identified in the paper, and the directions of its further development are outlined.