A comprehensive reference to category theory for students and researchers in mathematics, computer science, logic, cognitive science, linguistics, and philosophy. Useful for self-study and as a course text, the book includes all basic definitions and theorems, as well as numerous examples and exercises.

According to 'Ontic Structural Realism' (OSR), physical objects—qua metaphysical entities—should be reconceptualised, or, more strongly, eliminated in favour of the relevant structures. In this paper I shall attempt to articulate the relationship between these putative objects and structures in terms of certain accounts of metaphysical dependence currently available. This will allow me to articulate the differences between the different forms of OSR and to argue in favour of the 'eliminativist' version. A useful context is provided by Floridi's account of the (...) relationship between 'ontic' and 'epistemic' structural realisms and I shall conclude with some brief remarks on possible extensions of OSR into other scientific domains. (shrink)

Structural realist interpretations of generally relativistic spacetimes have recently come to enjoy a remarkable degree of popularity among philosophers. I present a challenge to these structuralist interpretations that arises from considering cosmological models in general relativity. As a consequence of their high degree of spacetime symmetry, these models resist a structuralist interpretation. I then evaluate the various strategies available to the structuralist to react to this challenge. †To contact the author, please write to: Department of Philosophy, 9500 Gilman Drive, 0119, (...) University of California, San Diego, La Jolla, CA 92093‐0119; e‐mail: [email protected]. (shrink)

Some mathematicians and philosophers contend that set theory plays a foundational role in mathematics. However, the development of category theory during the second half of the twentieth century has encouraged the view that this theory can provide a structuralist alternative to set-theoretical foundations. Against this tendency, criticisms have been made that category theory depends on set-theoretical notions and, because of this, category theory fails to show that set-theoretical foundations are dispensable. The goal of this paper is to show that these (...) criticisms are misguided by arguing that category theory is entirely autonomous from set theory. (shrink)

In his paper ``What is Structural Realism?'' James Ladyman drew a distinction between epistemological structural realism and metaphysical (or ontic) structural realism. He also drew a suggestive analogy between the perennial debate between substantivalist and relationalist interpretations of spacetime on the one hand, and the debate about whether quantum mechanics treats identical particles as individuals or as `non-individuals' on the other. In both cases, Ladyman's suggestion is that an ontic structural realist interpretation of the physics might be just what is (...) needed to overcome the stalemate. The main thesis of this paper is that, whatever the interpretative difficulties of generally covariant spacetime physics are, they do not support or suggest structural realism. In particular, I hope to show that there is in fact no analogy that supports a similar interpretation of the metaphysics of spacetime points and of quantum particles. (shrink)

The paper defends two claims; Viewed from the perspective of the substantivalism/relationism debate, structural spacetime realism is a form of relationism; However, if we managed to reinforce Rynasiewicz’s point that the general theory of relativity makes the substantivalism/relationism dispute “outdated”, the re-elaboration of Stein ’s 1967 version of structural spacetime realism to be proposed here proves to be a good, antimetaphysical solution to the problem of the ontological status of spacetime.

We outline Ladyman's 'metaphysical' or 'ontic' form of structuralrealism and defend it against various objections. Cao, in particular, has questioned theview of ontology presupposed by this approach and we argue that by reconceptualisingobjects in structural terms it offers the best hope for the realist in thecontext of modern physics.

This paper sets out a moderate version of metaphysical structural realism that stands in contrast to both the epistemic structural realism of Worrall and the—radical—ontic structural realism of French and Ladyman. According to moderate structural realism, objects and relations (structure) are on the same ontological footing, with the objects being characterized only by the relations in which they stand. We show how this position fares well as regards philosophical arguments, avoiding the objections against the other two versions of structural realism. (...) In particular, we set out how this position can be applied to space-time, providing for a convincing understanding of space-time points in the standard tensor formulation of general relativity as well as in the fibre bundle formulation. (shrink)

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. (shrink)

Focusing on spacetime singularities, Earman engages with a host of foundational issues at the intersection of science and philosophy, ranging from the big bang to the possibility of time travel.

The ‘ontic’ form of structural realism , roughly speaking, admits a complete elimination of the objects in the discourse of scientific theories, leaving us with structures only. As put by the defenders of such a claim, the idea is that all there is are structures and, if the relevant structures are to be set-theoretical constructs , as it has also been claimed, then the relations which appear in such structures should be taken to be ‘relations without the relata’. As far (...) as we know, there is not a definition of structure in standard mathematics which fits their intuitions, and even category theory seems not to correspond adequately the OSR claims. Since OSR is also linked with the semantic approach to theories, the structures to be dealt with are taken to be set-theoretical constructs. But these are ‘relational’ structures where the involved relations are built from basic objects , and so no complete elimination of the relata is possible, although it would be adequate for characterizing OSR. In this paper we present a definition of a kind of relation that does not depend on the particular objects being related in the sense that the ‘relation’ continues to hold even if the relata are exchanged by suitable objects. Although there is not a ‘complete’ elimination of relata, our definition might be viewed as an alternative way of finding adequate mathematical ‘set-theoretical’ frameworks for describing at least some of the intuitions regarding OSR within a ‘set-theoretical’ framework. (shrink)

The debate between substantivalists and relationists about spacetime was given a new lease of life approximately twenty years ago, when John Earman and John Norton published an argument for the conclusion that, in the light of general relativity, substantivalism is untenable. Responses to Earman and Norton’s argument generated a proliferation of ‘substantivalisms’, and a debate between them that was, to the ears of at least some, distinctively metaphysical in character.

Indeed, this is the first serious book-length study of the subject by a philosopher of science.

Preanalytically, we are all scientific realists. But both philosophers and scientists become uncomfortable when forced into analysis. In the case of scientists, this discomfort often arises from practical difficulties in setting out a carefully described set of objects which adequately account for the phenomena with which they are concerned. This paper offers a set of representative examples of these difficulties for contemporary physicists. These examples challenge the traditional realist vision of mature scientific activity as struggling toward an ontologically well-defined world (...) picture. They challenge antirealist alternatives as well. (shrink)

According to ‘Ontic Structural Realism’ (OSR), physical objects—qua metaphysical entities—should be reconceptualised, or, more strongly, eliminated in favour of the relevant structures. In this paper I shall attempt to articulate the relationship between these putative objects and structures in terms of certain accounts of metaphysical dependence currently available. This will allow me to articulate the differences between the different forms of OSR and to argue in favour of the ‘eliminativist’ version. A useful context is provided by Floridi’s account of the (...) relationship between ‘ontic’ and ‘epistemic’ structural realisms and I shall conclude with some brief remarks on possible extensions of OSR into other scientific domains. (shrink)