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 authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of (...) the other. I suggest that ‘i’ functions like a parameter in natural deduction systems. I gave an early version of this paper at a workshop on structuralism in mathematics and science, held in the Autumn of 2006, at Bristol University. Thanks to the organizers, particularly Hannes Leitgeb, James Ladyman, and Øystein Linnebo, to my commentator Richard Pettigrew, and to the audience there. The paper also benefited considerably from a preliminary session at the Arché Research Centre at the University of St Andrews. I am indebted to my colleagues Craige Roberts, for help with the linguistics literature, and Ben Caplan and Gabriel Uzquiano, for help with the metaphysics. Thanks also to Hannes Leitgeb and Jeffrey Ketland for reading an earlier version of the manuscript and making helpful suggestions. I also benefited from conversations with Richard Heck, John Mayberry, Kevin Scharp, and Jason Stanley. CiteULike Connotea Del.icio.us What's this? (shrink)

Øystein Linnebo and Richard Pettigrew have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They argue that their theory of abstract structures proves a consistent version of the structuralist thesis that positions in abstract structures only have structural properties. They do this by defining a subset of the properties of positions in structures, so-called fundamental properties, and argue that all fundamental properties of positions are structural. In this paper, we argue that the structuralist thesis, even (...) when restricted to fundamental properties, does not follow from the theory of structures that Linnebo and Pettigrew have developed. To make their account work, we propose a formal framework in terms of Kripke models that makes structural abstraction precise. The formal framework allows us to articulate a revised definition of fundamental properties, understood as intensional properties. Based on this revised definition, we show that the restricted version of the structuralist thesis holds. (shrink)

We demonstrate that the quantum-mechanical description of composite physical systems of an arbitrary number of similar fermions in all their admissible states, mixed or pure, for all finite-dimensional Hilbert spaces, is not in conflict with Leibniz's Principle of the Identity of Indiscernibles (PII). We discern the fermions by means of physically meaningful, permutation-invariant categorical relations, i.e. relations independent of the quantum-mechanical probabilities. If, indeed, probabilistic relations are permitted as well, we argue that similar bosons can also be discerned in all (...) their admissible states; but their categorical discernibility turns out to be a state-dependent matter. In all demonstrated cases of discernibility, the fermions and the bosons are discerned (i) with only minimal assumptions on the interpretation of quantum mechanics; (ii) without appealing to metaphysical notions, such as Scotusian haecceitas, Lockean substrata, Postian transcendental individuality or Adamsian primitive thisness; and (iii) without revising the general framework of classical elementary predicate logic and standard set theory, thus without revising standard mathematics. This confutes: (a) the currently dominant view that, provided (i) and (ii), the quantum-mechanical description of such composite physical systems always conflicts with PII; and (b) that if PII can be saved at all, the only way to do it is by adopting one or other of the thick metaphysical notions mentioned above. Among the most general and influential arguments for the currently dominant view are those due to Schrodinger, Margenau, Cortes, Dalla Chiara, Di Francia, Redhead, French, Teller, Butterfield, Giuntini, Mittelstaedt, Castellani, Krause and Huggett. We review them succinctly and critically as well as related arguments by van Fraassen and Massimi. (shrink)

Particle indistinguishability has always been considered a purely quantum mechanical concept. In parallel, indistinguishable particles have been thought to be entities that are not properly speaking objects at all. I argue, to the contrary, that the concept can equally be applied to classical particles, and that in either case particles may (with certain exceptions) be counted as objects even though they are indistinguishable. The exceptions are elementary bosons (for example photons).

In this paper I introduce a novel strategy to deal with the indiscernibility problem for ante rem structuralism. The ante rem structuralist takes the ontology of mathematics to consist of abstract systems of pure relata. Many of such systems are totally symmetrical, in the sense that all of their elements are relationally indiscernible, so the ante rem structuralist seems committed to positing indiscernible yet distinct relata. If she decides to identify them, she falls into mathematical inconsistency while, accepting their distinctness, (...) she finds herself unable to account for it. I show that the ante rem structuralist has in fact the resources to account for the distinctness of indiscernibles and that these resources come from the very symmetry properties of the mathematical objects that seem to pose problems for her. (shrink)

Towards Non-Being develops an account of the semantics of intentional predicates and operators. The account appeals to objects, both existent and non-existent, and worlds, both possible and impossible. This paper formulates replies to a number of the more interesting objections to the semantics that have been proposed since the book was published.

In some sense, both ontological and epistemological problems related to individuation have been the focal issues in the philosophy of mathematics ever since Frege. However, such an interest becomes manifest in the rise of structuralism as one of the most promising positions in recent philosophy of mathematics. The most recent controversy between Keränen and Shapiro seems to be the culmination of this phenomenon. Rather than taking sides, in this paper, I propose to critically examine some common assumptions shared by both (...) parties. In particular, I shall focus on their assumptions on haecceity as an individual essence, haecceity as a property, the classification of properties, and thereby the search for the principle of individuation in terms of properties. I shall argue that all these assumptions are mistaken and ungrounded from Scotus’ point of view. Further, I will fathom what consequences would follow, if we reject each of these assumptions. (shrink)

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 criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the main questions and issues that (...) have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts. (shrink)

Saunders' recent arguments in favour of the weak discernibility of (certain) quantum particles seem to be grounded in the 'generalist' view that science only provides general descriptions of the worlIn this paper, I introduce the ‘generalist’ perspective and consider its possible justification and philosophical basis; and then look at the notion of weak discernibility. I expand on the criticisms formulated by Hawley (2006) and Dieks and Veerstegh (2008) and explain what I take to be the basic problem: that the properties (...) invoked by Saunders cannot be pointed to as ‘individuators’ of otherwise indiscernible (and thus numerically identical) entities because their ontological status remains underdetermined by the evidence and the established interpretation of the theory. In addition to to this, I suggest that Saunders does not deal adequately with bosons, and cannot do so exactly because he subscribes to PII and the generalist picture. The last part of the paper contains a critical examination of the claim (or at least implicit assumption) that the generalist picture should be regarded as obviously compelling by the modern-day empiricist. (shrink)

Recent work in the philosophy of mathematics has suggested that mathematical structuralism is not committed to a strong form of the Identity of Indiscernibles (II). José Bermúdez demurs, and argues that a strong form of II can be warranted on structuralist grounds by countenancing identity properties, or haecceities, as legitimately structural. Typically, structuralists dismiss such properties as obviously non-structural. I will argue to the contrary that haecceities can be viewed as structural but that this concession does not warrant Bermúdez’s version (...) of II but, rather, another easily falsified version. I close with some reflections on reference vis-à-vis structurally indiscernible objects. (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)

Bundle Theory is the view that every concrete particular object is solely constituted by its universals. This theory is often criticized for not accommodating the possibility of symmetrical universes, such as one that contains two indiscernible spheres two meters from each other in otherwise empty space. One bundle theoretic solution to this criticism holds that the fact that the spheres stand in a weakly discerning—i.e., irreflexive and symmetric—relation, such as being two meters from, is sufficient for the numerical diversity of (...) the spheres. For this solution to be effective, however, it should be established that weak discernibility not only necessitates but also explains numerical diversity. In this paper, I argue that the fact that two objects have a certain distance between them does explain why they are non-identical. I also argue that the worry that the weak discernibility approach has some circularity problems is not well-founded. (shrink)

Bundle Theory is the view that every concrete particular object is solely constituted by its universals. This theory is often criticized for not accommodating the possibility of symmetrical universes, such as one that contains two indiscernible spheres two meters from each other in otherwise empty space. One bundle theoretic solution to this criticism holds that the fact that the spheres stand in a weakly discerning—i.e., irreflexive and symmetric—relation, such as being two meters from, is sufficient for the numerical diversity of (...) the spheres. For this solution to be effective, however, it should be established that weak discernibility not only necessitates but also explains numerical diversity. In this paper, I argue that the fact that two objects have a certain distance between them does explain why they are non-identical. I also argue that the worry that the weak discernibility approach has some circularity problems is not well-founded. (shrink)

Abstractvan Fraassen (The empirical stance, 2002) contrasts the empirical stance with the materialist stance. The way he describes them makes both of them attractive, and while opposed they have something in common for both stances are scientific approaches to philosophy. The difference between them reflects their differing conceptions of science itself. Empiricists emphasise fallibilism, verifiability and falsifiability, and also to some extent scepticism and tolerance of novel hypotheses. Materialists regard the theoretical picture of the world as matter in motion as (...) a true and explanatory account and insist on not taking ‘spooky’ entities or processes seriously as potential explanations of phenomena that so far lie outside the scope of successful science. The history of science shows us that both stances have been instrumental in the achievement of progress at various times. It is therefore plausible for a naturalist to suggest that science depends for its success on the dialectic between empiricism and materialism. A truly naturalist approach to philosophy ought then to synthesise them. Call the synthesized empiricist and materialist stances ‘the scientistic stance’. This paper elaborates and defends it. (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)

Tim Räz has presented what he takes to be a new objection to Stewart Shapiro's ante rem structuralism. Räz claims that ARS conflicts with mathematical practice. I will explain why this is similar to an old problem, posed originally by John Burgess in 1999 and Jukka Keränen in 2001, and show that Shapiro can use the solution to the original problem in Räz's case. Additionally, I will suggest that Räz's proposed treatment of the situation does not provide an argument for (...) the in re over the ante rem approach. (shrink)

The notion of strict identity is sometimes given an explicit second-order definition: objects with all the same properties are identical. Here, a somewhat different problem is raised: Under what conditions is the identity relation on the domain of a structure first-order definable? A structure may have objects that are distinct, but indiscernible by the strongest means of discerning them given the language (the indiscernibility formula). Here a number of results concerning the indiscernibility formula, and the definability of identity, are collected (...) and a number of applications discussed. (shrink)

There has been considerable discussion in the literature of one kind of identity problem that mathematical structuralism faces: the automorphism problem, in which the structure is unable to individuate the mathematical entities in its domain. Shapiro (Philos Math 16(3):285–309, 2008) has partly responded to these concerns. But I argue here that the theory faces an even more serious kind of identity problem, which the theory can’t overcome staying within its remit. I give two examples to make the point.

Simon Saunders argues that, although distinct objects must be discernible, they need only be weakly discernible (Saunders 2003, 2006a). I will argue that this combination of views is unmotivated: if there can be objects which differ only weakly, there can be objects which don’t differ at all.

A prominent version of mathematical structuralism holds that mathematical objects are at bottom nothing but "positions in structures," purely relational entities without any sort of nature independent of the structure to which they belong. Such an ontology is often presented as a response to Benacerraf's "multiple reductions" problem, or motivated on hermeneutic grounds, as a faithful representation of the discourse and practice of mathematics. In this paper I argue that there are serious difficulties with this kind of view: its proponents (...) rely on a distinction between "essential" and "nonessential" features of mathematical objects, and there's no good way to articulate this distinction which is compatible with basic structuralist commitments. But all is not lost. For I further argue that the insights motivating structuralism (or at least those worth preserving) can be preserved without formulating the view in ontologically committal terms. (shrink)

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. (shrink)

Everything you always wanted to know about structural realism but were afraid to ask Content Type Journal Article Pages 227-276 DOI 10.1007/s13194-011-0025-7 Authors Roman Frigg, Department of Philosophy, Logic and Scientific Method, London School of Economics and Political Science, Houghton Street, London, WC2A 2AE UK Ioannis Votsis, Philosophisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, Geb. 23.21/04.86, 40225 Düsseldorf, Germany Journal European Journal for Philosophy of Science Online ISSN 1879-4920 Print ISSN 1879-4912 Journal Volume Volume 1 Journal Issue Volume 1, Number 2.

ABSTRACT This paper reveals David Hilbert’s position in the philosophy of mathematics, circa 1900, to be a form of non-eliminative structuralism, predating his formalism. I argue that Hilbert withstands the pressing objections put to him by Frege in the course of the Frege-Hilbert controversy in virtue of this early structuralist approach. To demonstrate that this historical position deserves contemporary attention I show that Hilbertian structuralism avoids a recent wave of objections against non-eliminative structuralists to the effect that they cannot distinguish (...) between structurally identical but importantly distinct mathematical objects, such as the complex roots of $-1$. (shrink)

Recently, several authors have claimed to have found graph-theoretic counterexamples to the Principle of the Identity of Indiscernibles. In this paper, I argue that their counterexamples presuppose a certain view of what unlabeled graphs are, and that this view is optional at best.

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.

Using the Hilbert-Bernays account as a spring-board, we first define four ways in which two objects can be discerned from one another, using the non-logical vocabulary of the language concerned. Because of our use of the Hilbert-Bernays account, these definitions are in terms of the syntax of the language. But we also relate our definitions to the idea of permutations on the domain of quantification, and their being symmetries. These relations turn out to be subtle---some natural conjectures about them are (...) false. We will see in particular that the idea of symmetry meshes with a species of indiscernibility that we will call `absolute indiscernibility'. We then report all the logical implications between our four kinds of discernibility. We use these four kinds as a resource for stating four metaphysical theses about identity. Three of these theses articulate two traditional philosophical themes: viz. the principle of the identity of indiscernibles, and haecceitism. The fourth is recent. Its most notable feature is that it makes diversity weaker than what we will call individuality : two objects can be distinct but not individuals. For this reason, it has been advocated both for quantum particles and for spacetime points. Finally, we locate this fourth metaphysical thesis in a broader position, which we call structuralism. We conclude with a discussion of the semantics suitable for a structuralist, with particular reference to physical theories as well as elementary model theory. (shrink)

Keränen (2001) raises an argument against realistic (ante rem) structuralism: where a mathematical structure has a non-trivial automorphism, distinct indiscernible positions within the structure cannot be shown to be non-identical using only the properties and relations of that structure. Ladyman (2005) responds by allowing our identity criterion to include 'irreflexive two-place relations'. I note that this does not solve the problem for structures with indistinguishable positions, i.e. positions that have all the same properties as each other and exactly the same (...) relations to all objects (including themselves). I conclude that realistic structuralists must compromise and treat some structures eliminativistically. (shrink)

Noneism a is form of Meinongianism, proposed by Richard Routley and developed and improved by Graham Priest in his widely discussed book Towards Non-Being. Priest's noneism is based upon the double move of building a worlds semantics including impossible worlds, besides possible ones, and admitting a new comprehension principle for objects, differerent from the ones proposed in other kinds of neo-Meinongian theories, such as Parsons' and Zalta's. The new principle has no restrictions on the sets of properties that can deliver (...) objects, but parameterizes the having of properties by objects to worlds. Modality is therefore explicitly built in - so the approach can be conveniently labeled as "modal noneism". In this paper, I put modal noneism to work by testing it against classical issues in modal logic and semantics. It turns out that - perhaps surprisingly - the theory performs well in problems of transworld identity, which are frequently considered to be the difficult ones in the literature; faces a limitation, albeit not a severe one, when one comes to transworld individuation, which is often taken as an easy issue, if not a pseudo-problem; and may stumble upon a real trouble when dealing with what I shall call 'extensionally indiscernible entities' - particular nonexistent objects modal noneism is committed to. (shrink)

The existence of structures with non-trivial authomorphisms (such as the automorphism of the field of complex numbers onto itself that swaps the two roots of – 1) has been held by Burgess and others to pose a serious difficulty for mathematical structuralism. This paper proposes a model-theoretic solution to the problem. It suggests that mathematical structuralists identify the “position” of an n-tuple in a mathematical structure with the type of that n-tuple in the expansion of the structure that has a (...) name for every element in it. (shrink)

This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up naturalisation of Badiou’s philosophical approach (...) to mathematics and a top-down naturalisation. Articulating my particular understanding of what realism and naturalism should commit us to, I propose a creative fusion of Badiou’s attention to metamathematical results with a structural-informational metaphysics, proposing a ‘matherialism’ uniting the more daring speculative insights of the former with the naturalist and empiricist commitments motivating the latter. (shrink)

There are mathematical structures with elements that cannot be distinguished by the properties they have within that structure. For instance within the field of complex numbers the two square roots of −1, i and −i, have the same algebraic properties in that field. So how do we distinguish between them? Imbedding the complex numbers in a bigger structure, the quaternions, allows us to algebraically tell them apart. But a similar problem appears for this larger structure. There seems to be always (...) a background and a context that we rely upon. Thus mathematicians naturally make use of Kantian intuition and references fixed by names and denotations. I argue that such features cannot be avoided. (shrink)

Discutindo acerca das centenas de detecções de planetas extrassolares, que supostamente aconteceram desde 1989 e que ele considera (incorretamente) como instâncias de observações, Peter Kosso disse, justamente, que segundo os parâmetros de Bas van Fraassen esses objetos celestes seriam observáveis. Ora, tais astros poderiam sem dúvida ser observados diretamente (sem a necessidade de instrumentos), nas condições apropriadas. Mas, acrescenta Kosso, “esse tipo de epistemologia externalista, que permite que a justificação se baseie em informação que não temos a disposição (nós não (...) estamos em condição de ver planetas extrassolares a olho nu [atualmente]), não ajuda a decidir quais particulares afirmações científicas garantem a crença” (Kosso 2006, 225, nota 1). Van Fraassen provavelmente incorre em uma petição de princípio não muito diferente daquelas que ele muitas vezes desmascarou em textos de autores realistas, quando apela para o fato de o objeto detectado ser observável, para atribuir o estatuto de ‘observação’ a uma certa detecção. O ponto é, mais uma vez, entender o que vale como observação, particularmente quando isso envolve o uso de algum instrumento. Endossar aquilo que Otávio Bueno chama de ‘padrão internalista’ parece garantir solidez à qualificação de determinadas ações como observações. Uma das principais consequências dessa adoção é que tal critério, não operando nenhuma distinção relevante entre detecções ‘diretas’ e detecções mediadas por instrumentos, permite que a linha que separa observáveis e inobserváveis seja traçada de maneira diferente de como van Fraassen acredita que deveria ser, abarcando, entre os observáveis, mais (tipos de) entidades do que esse parece estar disposto a admitir. (shrink)

In a recent paper, Sun Demirli (2010) proposes an allegedly new way of conceiving of individuation in the context of the bundle theory of object constitution. He suggests that allowing for distance relations to individuate objects solves the problems with worlds containing indiscernible objects that would otherwise affect the theory. The aim of the present paper is i) To show that Demirli’s proposal falls short of achieving this goal and ii) To carry out a more general critical assessment of the (...) issue by appraising the costs and benefits of Demirli’s view as well as of existing alternatives. (shrink)