Online research in philosophy

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.

Consider the aims of the following three influential philosophical views. The semantic view of theories aims to supply the proper form and content of scientific theories. Structural realism aspires to delimit the epistemology and ontology of science. Mathematical structuralism seeks to reveal the epistemological and ontological nature of – you guessed it – mathematical objects. Given their divergent aims they may seem like unlikely bedfellows, but the semantic view of theories, structural realism and mathematical structuralism share enough ground to be able to benefit or suffer from some of the same reasons. What unites the three views is the purely structural analysis of their respective subject matter. The semantic view sees theories as nothing more than families of models, i.e. sets of structures. Representation, according to this view, is a matter of establishing mappings between some models of the theory and target domains. Structural realism judges scientific knowledge and perhaps even ontology to be wholly structural. Mathematical structuralism proclaims that the objects of mathematics are specifiable only up to isomorphism.

The view that mathematical objects are indefinite in nature is presented and defended, hi the first section, Field's argument for fictionalism, given in response to Benacerraf's problem of identification, is closely examined, and it is contended that platonists can solve the problem equally well if they take the view that mathematical objects are indefinite. In the second section, two general arguments against the intelligibility of objectual indefiniteness are shown erroneous, hi the final section, the view is compared to mathematical structuralism, and it is shown that a version of structuralism should be understood as embracing the same view.

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.

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.

This paper explores varieties of scientific structuralism. Central to our investigation is the notion of `shared structure'. We begin with a description of mathematical structuralism and use this to point out analogies and disanalogies with scientific structuralism. Our particular focus is the semantic structuralist's attempt to use the notion of shared structure to account for the theory-world connection, this use being crucially important to both the contemporary structural empiricist and realist. We show why minimal scientific structuralism is, at the very least, a powerful methodological standpoint. Our investigation also makes explicit what more must be added to this minimal structuralist position in order to address the theory-world connection, namely, an account of representation.

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.

(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.

Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell’s insights. Problems of absolutism plague some versions, and, interestingly, Russell’s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modal-structuralism and a category theoretic approach as remaining non-absolutist options. It is suggested that these should be combined.

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.