There is a standard view of relations, held by philosophers and logicians alike, according to which we may meaningfully talk of a relation holding of several objects in a given order. Thus it is supposed that we may meaningfully—indeed, correctly—talk of the relation loves holding of Anthony and Cleopatra or of the relation between holding of New York, Washington, and Boston. But innocuous as this view might appear to be, it cannot be accepted as applying to all relations whatever. For (...) there is an important class of metaphysical and linguistic contexts which call for an alternative conception of relation, one for which the order of the relata plays no role and in which the application of the relation to its relata is achieved by other means. (shrink)

This article is an extended critical study of Kit Fine’s The limits of abstraction, which is a sustained attempt to take the measure of the neo-logicist program in the philosophy and foundations of mathematics, founded on abstraction principles like Hume’s principle. The present article covers the philosophical and technical aspects of Fine’s deep and penetrating study.

I consider different versions of a structuralist view of mathematical objects, according to which characteristic mathematical objects have no more of a 'nature' than is given by the basic relations of a structure in which they reside. My own version of such a view is non-eliminative in the sense that it does not lead to a programme for eliminating reference to mathematical objects. I reply to criticisms of non-eliminative structuralism recently advanced by Keränen and Hellman. In replying to the former, (...) I rely on a distinction between 'basic' and 'constructed' structures. A conclusion is that ideas from the metaphysical tradition can be misleading when applied to the objects of modern mathematics. (shrink)

In the ordinary way of representing relations, the order of the relata plays a structural role, but in the states themselves such an order often does not seem to be intrinsically present. An alternative way to represent relations makes use of positions for the arguments. This is no problem for the love relation, but for relations like the adjacency relation and cyclic relations, different assignments of objects to the positions can give exactly the same states. This is a puzzling situation. (...) The question is what is the internal structure of relations? Is the use of positions still justified, and if so, what is their ontological status? In this paper mathematical models for relations are developed that provide more insight into the structure of relations “out there” in the real world. (shrink)

Kit Fine develops a Fregean theory of abstraction, and suggests that it may yield a new philosophical foundation for mathematics, one that can account for both our reference to various mathematical objects and our knowledge of various mathematical truths. The Limits ofion breaks new ground both technically and philosophically.