Online research in philosophy

It is generally assumed that rigidity plays a key role in explaining the necessary a posteriori status of identity statements, both between proper names and between natural kind terms. However, while the notion of rigid designation is well defined for singular terms, there is no generally accepted definition of what it is for a general term to be rigid. In this paper I argue that the most common view, according to which rigid general terms are the ones which designate the same kind in all possible worlds, fails to deliver a posteriori necessities. I also present an alternative view, on which the work of explaining a posteriori necessities is not done by rigidity, but by a related metasemantic notion, which I call actuality - dependence.

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.

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.

This paper explores the consequences of the two most prominent forms of contemporary structural realism for the notion of objecthood. Epistemic structuralists hold that we can know structural aspects of reality, but nothing about the natures of unobservable relata whose relations define structures. Ontic structuralists hold that we can know structural aspects of reality, and that there is nothing else to know—objects are useful heuristic posits, but are ultimately ontologically dispensable. I argue that structuralism does not succeed in ridding a structuralist ontology of objects.

It is generally assumed that everything that can be said about dependence with the notion of strong global supervenience can also be said with the notion of strong supervenience. It is argued here, however, that strong global supervenience has a metaphysically distinctive role to play. It is shown that when the relevant sets include relations , strong global supervenience and strong supervenience are distinct. It is then concluded that there are claims about dependence of relations that can be made with the global notion of strong supervenience but not with the “local” (individual) one.

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.

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be algebraic structuralists all the way down.

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 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.

The main contribution of this paper is a novel account of ontological dependence. While dependence is often explained in terms of modality and existence, there are relations of dependence that slip through the mesh of such an account. Starting from an idea proposed by Jonathan Lowe, the article develops an account of ontological dependence based on a notion of explanation; on its basis, certain relations of dependence can be established that cannot be accounted by the modal-existential account. Dependence is only one of two main topics of this paper, for it is approached via a discussion of the category of substance. On a traditional view, substances can be characterised as independent entities. Before the background of a modal-existential account of dependence, this idea appears problematic. The proposed notion of explanatory dependence is shown to vindicate the traditional approach to substance.