Online research in philosophy

The debate on structuralism in the philosophy of mathematics has brought into focus a question about the status of meta-mathematics. It has been raised by Shapiro ( 2005 ), where he compares the ongoing discussion on structuralism in category theory to the Frege-Hilbert controversy on axiomatic systems. Shapiro outlines an answer according to which meta-mathematics is understood in structural terms and one according to which it is not. He finds both options viable and does not seem to prefer one over the other. The present paper reconsiders the nature of the formulae and symbols meta-mathematics is about and finds that, contrary to Charles Parsons’ influential view, meta-mathematical objects are not “quasi-concrete”. It is argued that, consequently, structuralists should extend their account of mathematics to meta-mathematics.

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.

In modern physics the notion of structure can be treated as an extension of the notion of law of nature. French and Ladyman’s view concerning the ontological priority of structures over objects is confronted with Psillos’ criticism. This kind of view agrees with the paradigmatic case where the structure is an internal symmetry and the instantiations are elementary particles. An ontological model is proposed which demonstrates the relation between structures and their instantiations in this case. This view which may be categorized as “weak ontic structuralism” is compared with Busch’s treatment of ontic structuralism in the philosophy of mathematics.

Two topics figure prominently in recent discussions of mathematical structuralism: challenges to the purported metaphysical insight provided by sui generis structuralism and the significance of category theory for understanding and articulating mathematical structuralism. This article presents an overview of central themes related to these topics.

(DRAFT) We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to recent criticisms, and our view is that the problems raised stem from the lack of proper, mathematics-free theoretical foundations. We attempt to provide such foundations and show that our foundations have consequences, in the form of theorems, that provide answers to the main questions and problems that have arisen in connection with this form of structuralismn. Our solutions to the problems of structuralism are not developed piecemeal but rather justified by reference to a principled position.

In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.

This paper is about structuralism as a form of reconstructing theories, associated with the work Sneed, Balzar and Moulines among others, and not about "structuralism" is any of its other manifold senses. The paper is a reflection in that it looks back on some earlier work of my own on the subject of structuralism and explanation, in which I argued that structuralism and my 'instance view' of explanation go well together, with structuralism providing the means to develop the idea of a theoretical instance. Bartelborth has suggested a view that has some similarity with my early ideas, so I reflect on those as well. I suggest, in opposition to both positions, that a causal account of explanation might also sit well with structuralism. This paper will appear in a special edition of Synthese edited by Moulines and devoted to structuralism themes.

This last of three articles on Structuralism and Post-structuralism attempts to do four things: (1) to summarize the dispute between Structuralism and Post-structuralism about the stability of meaning; (2) to present three criticisms of Derrida’s dissemination; (3) to assess the worth of these criticisms; and (4) to offer some concluding remarks on Structuralism and Post-structuralism.

I address Grosholz's critique of Resnik's mathematical structuralism and suggest that although Resnik's structuralism is not without its difficulties it survives Grosholz's attacks.

Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions.