Online research in philosophy

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.

The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts without caution, as the use of anachronistic notation tends to impede, rather than enhance, our ability to recognize the emergent nature of mathematical objects.

[First paragraph] For a long time philosophers thought material objects were unproblematic. Or nearly so. There may have been a problem about what a material object is: a substance, a bundle of tropes, a compound of substratum and universals, a collection of sense-data, or what have you. But once that was settled there were supposed to be no further metaphysical problems about material objects. This illusion has now largely been dispelled. No one can get a Ph.D. in philosophy nowadays without encountering the puzzles of the ship of Theseus, the statue and the lump, the cat and its tail complement', amoebic fission, and others. These problems are especially pressing on the assumption that we ourselves are material objects.

I argue (1) that metaphysical views of material objects should be understood as 'packages', rather than individual claims, where the other parts of the package include how the theory addresses 'recalcitant data' (such as - the denier of artifacts has to account, somehow, for the seeming truth of 'there are three pencils on my table'), and (2) that when the packages meet certain general desiderata - which all of the currently competing views *can* meet - there is nothing in the world that could make one of the theories true as opposed to any of the others.

In this paper, I distinguish scientific models in three kinds on the basis of their ontological status—material models, mathematical models and fictional models, and develop and defend an account of fictional models as fictional objects—i.e. abstract objects that stand for possible concrete objects.

It is sometimes claimed that ordinary objects, such as mountains and chairs, are not material in their own right, but only in virtue of the fact that they are constituted by matter. As Fine puts it, they are “onlyderivatively material” (2003, 211). In this paper I argue that invoking “constitution” to account for the materiality of things that are not material in their own right explains nothing and renders the admission that these objects are indeed material completely mysterious. Although there may be metaphysical contexts in which mysterianism can be accepted with equanimity, I further argue, the question of the materiality of quotidian objects is not one of them.

In this paper I propose a position in the ontology of mathematics which is inspired mainly by a case study in the mathematical discipline if-theory. The main theses of this position are that mathematical objects are introduced by mathematicians and that after mathematical objects have been introduced, they exist as objectively accessible abstract objects.

The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. Mathematical platonists claim that at least some of the objects which are the subject matter of pure mathematics (e.g. numbers, sets, groups) actually exist. Furthermore, they claim that these objects differ radically from the concrete objects (trees, cats, stars, molecules) which inhabit the material world. We take the standard platonistic position to include the claim that platonic objects lack spatio-temporal location and causal powers. Many (perhaps most) mathematical platonists subscribe to this view.1 But some who call themselves (or might be called) mathematical platonists..

This paper attempts to motivate skepticism about the reality of mathematical objects. The aim of the paper is not to provide a general critique of mathematical realism, but to demonstrate the insufficiency of the arguments advanced by Michael Resnik. I argue that Resnik’s use of the concept of immanent truth is inconsistent with the treatment of mathematical objects as ontologically and epistemically continuous with the objects posited by the natural sciences. In addition, Resnik’s structuralist program, and his denial of relational properties, is incompatible with a realist metaphysics about mathematical objects.

In Mathematical Thought and Its Objects, Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a “nature” than that confers on them.