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  1. On Kinds of Indiscernibility in Logic and Metaphysics.Adam Caulton & Jeremy Butterfield - 2012 - British Journal for the Philosophy of Science 63 (1):27-84.
    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 (...)
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  • The Identity Problem for Realist Structuralism.J. Keranen - 2001 - Philosophia Mathematica 9 (3):308--330.
    According to realist structuralism, mathematical objects are places in abstract structures. We argue that in spite of its many attractions, realist structuralism must be rejected. For, first, mathematical structures typically contain intra-structurally indiscernible places. Second, any account of place-identity available to the realist structuralist entails that intra-structurally indiscernible places are identical. Since for her mathematical singular terms denote places in structures, she would have to say, for example, that 1 = − 1 in the group (Z, +). We call this (...)
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  • Discerning elementary particles.F. A. Muller & M. P. Seevinck - 2009 - Philosophy of Science 76 (2):179-200.
    We maximally extend the quantum‐mechanical results of Muller and Saunders ( 2008 ) establishing the ‘weak discernibility’ of an arbitrary number of similar fermions in finite‐dimensional Hilbert spaces. This confutes the currently dominant view that ( A ) the quantum‐mechanical description of similar particles conflicts with Leibniz’s Principle of the Identity of Indiscernibles (PII); and that ( B ) the only way to save PII is by adopting some heavy metaphysical notion such as Scotusian haecceitas or Adamsian primitive thisness. We (...)
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  • Structure and identity.Stewart Shapiro - 2006 - In Fraser MacBride (ed.), Identity and modality. New York: Oxford University Press. pp. 34--69.
    According to ante rem structuralism a branch of mathematics, such as arithmetic, is about a structure, or structures, that exist independent of the mathematician, and independent of any systems that exemplify the structure. A structure is a universal of sorts: structure is to exemplified system as property is to object. So ante rem structuralist is a form of ante rem realism concerning universals. Since the appearance of my Philosophy of mathematics: Structure and ontology, a number of criticisms of the idea (...)
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  • Criteria of identity and structuralist ontology.Hannes Leitgib & James Ladyman - 2008 - Philosophia Mathematica 16 (3):388-396.
    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 (...)
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  • Identity, indiscernibility, and Ante Rem structuralism: The tale of I and –I.Stewart Shapiro - 2008 - Philosophia Mathematica 16 (3):285-309.
    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 (...)
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  • Realistic structuralism's identity crisis: A hybrid solution.Tim Button - 2006 - Analysis 66 (3):216–222.
    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 (...)
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  • (1 other version)The identity of indiscernibles.Max Black - 1952 - Mind 61 (242):153-164.
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  • What numbers could not be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.
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  • Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - Oxford, England: Oxford University Press USA.
    Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.
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  • Philosophy and Model Theory.Tim Button & Sean P. Walsh - 2018 - Oxford, UK: Oxford University Press. Edited by Sean Walsh & Wilfrid Hodges.
    Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions -/- Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in mathematics textbooks: these are aimed squarely at mathematicians; (...)
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  • Grades of Discrimination: Indiscernibility, Symmetry, and Relativity.Tim Button - 2017 - Notre Dame Journal of Formal Logic 58 (4):527-553.
    There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discrimination have been studied in some detail. (...)
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  • (1 other version)Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2002 - Philosophy and Phenomenological Research 65 (2):467-475.
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  • (1 other version)Structuralism.Geoffrey Hellman - 2005 - In Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic. Oxford and New York: Oxford University Press.
    With developments in the 19th and early 20th centuries, structuralist ideas concerning the subject matter of mathematics have become commonplace. Yet fundamental questions concerning structures and relations themselves as well as the scope of structuralist analyses remain to be answered. The distinction between axioms as defining conditions and axioms as assertions is highlighted as is the problem of the indefinite extendability of any putatively all-embracing realm of structures. This chapter systematically compares four main versions: set-theoretic structuralism, a version taking structures (...)
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  • Are quantum particles objects?Simon Saunders - 2006 - Analysis 66 (1):52-63.
    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).
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  • What constitutes the numerical diversity of mathematical objects?F. MacBride - 2006 - Analysis 66 (1):63-69.
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  • Physics and Leibniz's principles.Simon Saunders - 2002 - In Katherine Brading & Elena Castellani (eds.), Symmetries in Physics: Philosophical Reflections. New York: Cambridge University Press. pp. 289--307.
    It is shown that the Hilbert-Bernays-Quine principle of identity of indiscernibles applies uniformly to all the contentious cases of symmetries in physics, including permutation symmetry in classical and quantum mechanics. It follows that there is no special problem with the notion of objecthood in physics. Leibniz's principle of sufficient reason is considered as well; this too applies uniformly. But given the new principle of identity, it no longer implies that space, or atoms, are unreal.
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  • (1 other version)Structuralism.Geoffrey Hellman - manuscript
    With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the set-theoretic foundations of mathematics, and the influential work of the Bourbaki, certain views called “structuralist” have become commonplace. Mathematics is seen as the investigation, by more or less rigorous deductive means, of “abstract structures”, systems of objects fulfilling certain structural relations among themselves and in relation to other systems, without regard to the particular nature of (...)
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  • Cantorian Set Theory and Limitation of Size.Michael Hallett - 1984 - Oxford, England: Clarendon Press.
    This volume presents the philosophical and heuristic framework Cantor developed and explores its lasting effect on modern mathematics. "Establishes a new plateau for historical comprehension of Cantor's monumental contribution to mathematics." --The American Mathematical Monthly.
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  • ontributions to the Founding of the Theory of Transfinite Numbers. [REVIEW]Georg Cantor - 1916 - Ancient Philosophy (Misc) 26:638.
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  • (1 other version)Cantorian Set Theory and Limitation of Size.Michael Hallett - 1990 - Studia Logica 49 (2):283-284.
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  • (1 other version)Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2000 - Philosophical Quarterly 50 (198):120-123.
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  • (1 other version)Cantorian Set Theory and Limitation of Size.Michael Hallett - 1986 - Mind 95 (380):523-528.
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  • On some putative graph-theoretic counterexamples to the Principle of the Identity of Indiscernibles.Rafael De Clercq - 2012 - Synthese 187 (2):661-672.
    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.
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  • Russell's absolutism vs.(?) Structuralism.Geoffrey Hellman - manuscript
    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 (...)
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