The basic relations and functions that mathematicians use to identify mathematical objects fail to settle whether mathematical objects of one kind are identical to or distinct from objects of an apparently different kind, and what, if any, intrinsic properties mathematical objects possess. According to one influential interpretation of mathematical discourse, this is because the objects under study are themselves incomplete; they are positions or akin to positions in patterns or structures. Two versions of this idea are examined. It is argued (...) that the evidence adduced in favor of the incompleteness of mathematical objects underdetermines whether it is the objects themselves or our knowledge of them that is incomplete. Also, holding that mathematical objects are incomplete conflicts with the practice of mathematics. The objection that structuralism is committed to the identity of indiscernibles is evaluated and it is also argued that the identification of objects with positions is metaphysically suspect. (shrink)

In "Mathematical Truth", Paul Benacerraf articulated an epistemological problem for mathematical realism. His formulation of the problem relied on a causal theory of knowledge which is now widely rejected. But it is generally agreed that Benacerraf was onto a genuine problem for mathematical realism nevertheless. Hartry Field describes it as the problem of explaining the reliability of our mathematical beliefs, realistically construed. In this paper, I argue that the Benacerraf Problem cannot be made out. There simply is no intelligible problem (...) that satisfies all of the constraints which have been placed on the Benacerraf Problem. The point generalizes to all arguments with the structure of the Benacerraf Problem aimed at realism about a domain meeting certain conditions. Such arguments include so-called "Evolutionary Debunking Arguments" aimed at moral realism. I conclude with some suggestions about the relationship between the Benacerraf Problem and the Gettier Problem. (shrink)

Wilfrid Hodges; VIII*—Truth in a Structure, Proceedings of the Aristotelian Society, Volume 86, Issue 1, 1 June 1986, Pages 135–152, https://doi.org/10.1093/ari.

Das Ziel: Konstitutionssystem der Begriffe Das Ziel der vorliegenden Untersuchungen ist die Aufstellung eines erkenntnismäßig-logischen Systems der ...

Available for the first time in 20 years, here is the Rudolf Carnap's famous principle of tolerance by which everyone is free to mix and match the rules of ...

The no-miracles argument for realism and the pessimistic meta-induction for anti-realism pull in opposite directions. Structural Realism---the position that the mathematical structure of mature science reflects reality---relieves this tension.

The semantic, or model-theoretic, approach to theories has recently come under criticism on two fronts: (i) it is claimed that it cannot account for the wide diversity of models employed in scientific practice—a claim which has led some to propose a “deflationary” account of models; (ii) it is further contended that the sense of “model” used by the approach differs from that given in model theory. Our aim in the present work is to articulate a possible response to these claims, (...) drawing on recent developments within the semantic approach itself. Thus, the first is answered by utilizing the notion of a “partial structure”, first introduced in this context by da Costa and French in 1990. The second claim is undermined by consideration of van Fraassen's understanding of “model” which corresponds well with that evinced by modem mathematicians. This latter discussion, in particular, has an impact on the continuing debate regarding the relative merits of the semantic and syntactic views and the developments presented here can be taken to provide further support to the former. (shrink)

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...) is about things that really exist. (shrink)

This major publication is a history of the semantic tradition in philosophy from the early nineteenth century through its incarnation in the work of the Vienna Circle, the group of logical positivists that emerged in the years 1925-1935 in Vienna who were characterised by a strong commitment to empiricism, a high regard for science, and a conviction that modern logic is the primary tool of analytic philosophy. In the first part of the book, Alberto Coffa traces the roots of logical (...) positivism in a semantic tradition that arose in opposition to Kant's theory that a priori knowledge is based on pure intuition and the constitutive powers of the mind. In Part II, Coffa chronicles the development of this tradition by members and associates of the Vienna Circle. Much of Coffa's analysis draws on the unpublished notes and correspondence of many philosophers. The book, however, is not merely a history of the semantic tradition from Kant 'to the Vienna Station'. Coffa also critically reassesses the role of semantic notions in understanding the ground of a priori knowledge and its relation to empirical knowledge and questions the turn the tradition has taken since Vienna. (shrink)

In discussions about whether the Principle of the Identity of Indiscernibles is compatible with structuralist ontologies of mathematics, it is usually assumed that individual objects are subject to criteria of identity which somehow account for the identity of the individuals. Much of this debate concerns structures that admit of non-trivial automorphisms. We consider cases from graph theory that violate even weak formulations of PII. We argue that (i) the identity or difference of places in a structure is not to be (...) accounted for by anything other than the structure itself and that (ii) mathematical practice provides evidence for this view. We want to thank Leon Horsten, Jeff Ketland, Øystein Linnebo, John Mayberry, Richard Pettigrew, and Philip Welch for valuable comments on drafts of this paper. We are especially grateful to Fraser MacBride for correcting our interpretation of two of his papers and for other helpful comments. CiteULike Connotea Del.icio.us What's this? (shrink)

This paper examines in detail two paths that lead to Structural Realism (SR), viz. a substantive philosophical position which asserts that only the structure of the world is knowable. The upward path is any attempt to begin with empiricist premises and reach a sustainable realist position. (It has been advocated by Russell, Weyl, and Maxwell among others.) The downward path is any attempt to start from realist premises and construct a weaker realist position. (It has been recently advocated by Worrall, (...) French, and Ladyman.) This paper unravels and criticizes the metaphysical presuppositions of both paths to SR. It questions its very possibility as a substantive-and viable-realist thesis. (shrink)

Recent discussions of the nature of representation in science have tended to import pre-established decompositions from analyses of representation in the arts, language, cognition and so forth. Which of these analyses one favours will depend on how one conceives of theories in the first place. If one thinks of them in terms of an axiomatised set of logico-linguistic statements, then one might be naturally drawn to accounts of linguistic representation in which notions of denotation, for example, feature prominently. If, on (...) the other hand, one conceives of theories in non-linguistic terms, as in the model-theoretic approach, then one might look to analyses of representation in the arts where notions of resemblance tend to be brought to the fore. Thus van Fraassen, for example, has imported such an analysis into his discussion of representation in science and argued that an appropriate account of resemblance can be given in terms of the set-theoretic relation of isomorphism. This has been strongly criticised by Suarez, who argues that just as isomorphism cannot capture representation in art, so it is inappropriate in the scientific context as well. Similarly Hughes draws on Goodman`s rejection of resemblance in art in favour of denotation and, rather confusingly perhaps, favours the latter whilst also maintaining the model-theoretic view of theories. In this paper, I shall examine the debate in terms of four claims: 1. Isomorphism is not sufficient for representation; 2. Isomorphism is not necessary for representation; 3. Models represent but theories do not; 4. Models denote and do not resemble. Each of these claims will be questioned and I will conclude by suggesting that, through appropriate modifications, a form of isomorphism can serve to underpin representation in both the arts and science. (shrink)

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 is structured around the three elements of the title. Section 2 claims that (a) structures need objects and (b) scientific structuralism should focus on in re structures. Therefore, pure structuralism is undermined. Section 3 discusses whether the world has `excess structure' over the structure of appearances. The main point is that the claim that only structure can be known is false. Finally, Section 4 argues directly against ontic structural realism that it lacks the resources to accommodate causation within (...) its structuralist slogan. (shrink)

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. (shrink)

Although much of its history has been neglected or misunderstood, a structuralist 'tendency' has re-emerged within the philosophy of science. Broadly speaking, it consists of two fundamental strands: on the one hand, there is the identification of structural commonalities between theories; on the other, there is the metaphysical decomposition of objects in structural terms. Both have been pressed into service for the realist cause: the former has been identified primarily with Worrall's 'epistemic' structural realism; the latter with Ladyman's 'ontic' form. (...) And both raise important issues of general interest within the philosophy of science and metaphysics, respectively. The former invites questions regarding the identification and appropriate representation of these commonalities; the latter touches on different views regarding the nature of objects, the constitutive role of properties and the seat of causal powers. Both strands have recently come under critical fire. It is my intention to present a unified account of the 'structuralist tendency' which emphasizes the dual roles of structure as representational and constitutive, and to indicate how the more acute critical remarks can be dealt with. (shrink)

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)

James Ladyman has recently proposed a view according to which all that exists on the level of microphysics are structures "all the way down". By means of a comparative reading of structuralism in philosophy of mathematics as proposed by Stewart Shapiro, I shall present what I believe structures could not be. I shall argue that, if Ladyman is indeed proposing something as strong as suggested here, then he is committed to solving problems that proponents of structuralism in philosophy of mathematics (...) such as Shapiro are trying to solve. Attempting to do so, however, brings out a tacit tension in Ladyman's position. I shall argue that the upshot of this is that the ontological import that Ladyman attributes to structures is rather epistemological import properly understood. (shrink)

Hilary Putnam, who may have been the first philosopher to advance the notion that the computer is an apt model for the mind, takes a radically new view of his...

In “Mathematical Truth,” Paul Benacerraf presented an epistemological problem for mathematical realism.

This book argues that the only kind of metaphysics that can contribute to objective knowledge is one based specifically on contemporary science as it really is, and not on philosophers' a priori intuitions, common sense, or simplifications of science. In addition to showing how recent metaphysics has drifted away from connection with all other serious scholarly inquiry as a result of not heeding this restriction, this book demonstrates how to build a metaphysics compatible with current fundamental physics, which, when combined (...) with metaphysics of the special sciences, can be used to unify physics with the other sciences without reducing these sciences to physics itself. Taking science metaphysically seriously, this book argues, means that metaphysicians must abandon the picture of the world as composed of self-subsistent individual objects, and the paradigm of causation as the collision of such objects. The text assesses the role of information theory and complex systems theory in attempts to explain the relationship between the special sciences and physics, treading a middle road between the grand synthesis of thermodynamics and information, and eliminativism about information. The consequences of the books' metaphysical theory for central issues in the philosophy of science are explored, including the implications for the realism versus empiricism debate, the role of causation in scientific explanations, the nature of causation and laws, the status of abstract and virtual objects, and the objective reality of natural kinds. (shrink)

In his Analysis of Matter, Russell advocated an epistemic form of Epistemic Structural Realism which held that our knowledge of the external world is structural. This approach has been criticised by M.H.A. Newman, 137–148, 1928) and Stathis Psillos, S13–S24, 2001). The paper aims to reconstruct Russell’s version of ESR, and defend its experimental and philosophical plausibility. The basic assumption is that without getting a viable experimental handle on the linkage between the structure of perceptions and causal structure of the stimuli, (...) the so called upward path to SR couldn’t be defended conclusively. In this vein, the paper explores the resources of the recent experimental theories of perception. These help establishing a structuralist account of the perception, and provide an account for the causal linkage between the structure of perceptions and the structure of the stimuli. I argue that it is possible to settle for objectivity of this linkage according to the criteria of comprehensiveness, simplicity, and above all rationality, in a way that meets the standards of the scientific realists. (shrink)

The identity and diversity of individual objects may be grounded or ungrounded, and intrinsic or contextual. Intrinsic individuation can be grounded in haecceities, or absolute discernibility. Contextual individuation can be grounded in relations, but this is compatible with absolute, relative or weak discernibility. Contextual individuation is compatible with the denial of haecceitism, and this is more harmonious with science. Structuralism implies contextual individuation. In mathematics contextual individuation is in general primitive. In physics contextual individuation may be grounded in relations via (...) weak discernibility. (shrink)

Wigner famously referred to the `unreasonable effectiveness' of mathematics in its application to science. Using Wigner's own application of group theory to nuclear physics, I hope to indicate that this effectiveness can be seen to be not so unreasonable if attention is paid to the various idealising moves undertaken. The overall framework for analysing this relationship between mathematics and physics is that of da Costa's partial structures programme.

In this essay I shall focus on the adoption of the Semantic Approach by structural realists, including myself, who have done so on the grounds that it wears its structuralist sympathies on its sleeve. Despite this, the SA has been identified as standing in tension with the ontological commitments of the so-called ’ontic’ form of this view and so I shall explore that tension before discussing the usefulness of the SA in framing scientific representation and concluding with a discussion of (...) the implications of the ontological status of theories and models themselves. (shrink)

Halvorson argues that the semantic view of theories leads to absurdities. Glymour shows how to inoculate the semantic view against Halvorson's criticisms, namely by making it into a syntactic view of theories. I argue that this modified semantic-syntactic view cannot do the philosophical work that the original "language-free" semantic view was supposed to do.

Bertrand Russell famously argued that causation is not part of the fundamental physical description of the world, describing the notion of cause as “a relic of a bygone age”. This paper assesses one of Russell’s arguments for this conclusion: the ‘Directionality Argument’, which holds that the time symmetry of fundamental physics is inconsistent with the time asymmetry of causation. We claim that the coherence and success of the Directionality Argument crucially depends on the proper interpretation of the ‘ time symmetry’ (...) of fundamental physics as it appears in the argument, and offer two alternative interpretations. We argue that: if ‘ time symmetry’ is understood as the time -reversal invariance of physical theories, then the crucial premise of the Directionality Argument should be rejected; and if ‘ time symmetry’ is understood as the temporally bidirectional nomic dependence relations of physical laws, then the crucial premise of the Directionality Argument is far more plausible. We defend the second reading as continuous with Russell’s writings, and consider the consequences of the bidirectionality of nomic dependence relations in physics for the metaphysics of causation. (shrink)

The paper compares ontic structural realism in quantum physics with ontic structural realism about space–time. We contend that both quantum theory and general relativity theory support a common, contentful metaphysics of ontic structural realism. After recalling the main claim of ontic structural realism and its physical support, we point out that both in the domain of quantum theory and in the domain of general relativity theory, there are objects whose essential ways of being are certain relations so that these objects (...) do not possess an intrinsic identity. Nonetheless, the qualitative, physical nature of these relations is in the quantum case (entanglement) fundamentally different from the classical, metrical relations treated in general relativity theory. (shrink)

In a recent paper, Suárez and Cartwright return to the example of London and London's construction of a model for superconductivity and raise a number of concerns against the account of this construction presented in French and Ladyman and elsewhere. In this discussion note, we examine the challenge they raised and offer our responses.