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  1. Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - Oxford, England: Oxford University Press.
    Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...)
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  • Realism in Mathematics.Penelope Maddy - 1990 - Oxford, England and New York, NY, USA: Oxford University Prress.
    Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. (...)
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  • Reasoning with Arbitrary Objects.Kit Fine - 1985 - Oxford and New York: Blackwell.
    Contents: Preface VII; Introduction 1; 1. The General Framework 5; 2. Some Standard Systems 61; 3. Systems in General 147; 4. Non-Standard Systems 177; Bibliography 210; General Index 215; Index of Symbols 219-220.
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  • Science Without Numbers: A Defence of Nominalism.Hartry H. Field - 1980 - Princeton, NJ, USA: Princeton University Press.
    Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.
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  • Rigor and Structure.John P. Burgess - 2015 - Oxford, England: Oxford University Press UK.
    While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics (...)
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  • Dedekind's Structuralism: An Interpretation and Partial Defense.Erich H. Reck - 2003 - Synthese 137 (3):369 - 419.
    Various contributors to recent philosophy of mathematics havetaken Richard Dedekind to be the founder of structuralismin mathematics. In this paper I examine whether Dedekind did, in fact, hold structuralist views and, insofar as that is the case, how they relate to the main contemporary variants. In addition, I argue that his writings contain philosophical insights that are worth reexamining and reviving. The discussion focusses on Dedekind''s classic essay Was sind und was sollen die Zahlen?, supplemented by evidence from Stetigkeit und (...)
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  • Platonism and Aristotelianism in Mathematics.Richard Pettigrew - 2008 - Philosophia Mathematica 16 (3):310-332.
    Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at 'face value' is one on which the expressions 'N', '0', '1', '+', and 'x' are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic (...)
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  • How to Defeat Opposition to Moore.Ernest Sosa - 1999 - Philosophical Perspectives 13:137-49.
    What modal relation must a fact bear to a belief in order for this belief to constitute knowledge of that fact? Externalists have proposed various answers, including some that combine externalism with contextualism. We shall find that various forms of externalism share a modal conception of “sensitivity” open to serious objections. Fortunately, the undeniable intuitive attractiveness of this conception can be explained through an easily confused but far preferable notion of “safety.” The denouement of our reflections, finally, will be to (...)
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  • Realism, Mathematics and Modality.Hartry Field - 1988 - Philosophical Topics 16 (1):57-107.
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  • Philosophical Explanations.Robert Nozick - 1981 - Harvard University Press.
    Nozick analyzes fundamental issues, such as the identity of the self, knowledge and skepticism, free will, the foundations of ethics, and the meaning of life.
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  • Philosophical Explanations. [REVIEW]Robert Nozick - 1981 - Ethics 94 (2):326-327.
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  • Realism, Mathematics & Modality.Hartry H. Field - 1989 - Blackwell.
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  • Nominalism, Naturalism, Epistemic Relativism.Gideon Rosen - 2001 - Noûs 35 (s15):69 - 91.
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  • Structuralism as a Philosophy of Mathematical Practice.Jessica Carter - 2008 - Synthese 163 (2):119 - 131.
    This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure and the set. In the (...)
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  • Individuation of Objects – a Problem for Structuralism?Jessica Carter - 2005 - Synthese 143 (3):291 - 307.
    . This paper identifies two aspects of the structuralist position of S. Shapiro which are in conflict with the actual practice of mathematics. The first problem follows from Shapiros identification of isomorphic structures. Here I consider the so called K-group, as defined by A. Grothendieck in algebraic geometry, and a group which is isomorphic to the K-group, and I argue that these are not equal. The second problem concerns Shapiros claim that it is not possible to identify objects in a (...)
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  • Philosophical Explanations. [REVIEW]Robert Nozick - 1981 - Philosophy 58 (223):118-121.
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  • On What There Is.W. V. Quine - 1953 - In From a Logical Point of View. Cambridge, Mass.: Harvard University Press. pp. 1-19.
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  • The Foundations of Mathematics in the Theory of Sets.J. P. Mayberry - 2000 - Cambridge University Press.
    This book will appeal to mathematicians and philosophers interested in the foundations of mathematics.
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  • What is the Benacerraf Problem?Justin Clarke-Doane - 2017 - In Fabrice Pataut (ed.), New Perspectives on the Philosophy of Paul Benacerraf: Truth, Objects, Infinity. Springer Verlag.
    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 (...)
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  • A Single Axiom for Set Theory.David Bennett - 2000 - Notre Dame Journal of Formal Logic 41 (2):152-170.
    Axioms in set theory typically have the form , where is a relation which links with in some way. In this paper we introduce a particular linkage relation and a single axiom based on from which all the axioms of (Zermelo set theory) can be derived as theorems. The single axiom is presented both in informal and formal versions. This calls for some discussion of pertinent features of formal and informal axiomatic method and some discussion of pertinent features of the (...)
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  • Book Review: Stewart Shapiro. Philosophy of Mathematics: Structure and Ontology. [REVIEW]John P. Burgess - 1999 - Notre Dame Journal of Formal Logic 40 (2):283-291.
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  • J.P. Mayberry: The Foundations of Mathematics in the Theory of Sets. [REVIEW]W. W. Tait - 2002 - Bulletin of Symbolic Logic 8 (3):424-426.
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  • Mathematical Truth.Paul Benacerraf - 1973 - Journal of Philosophy 70 (19):661-679.
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  • Philosophical Explanations.Robert Nozick - 1981 - Mind 93 (371):450-455.
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  • What We Talk About When We Talk About Numbers.Richard Pettigrew - 2018 - Annals of Pure and Applied Logic 169 (12):1437-1456.
    In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.
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  • Mathematics Without Numbers: Towards a Modal-Structural Interpretation.Bob Hale & Geoffrey Hellman - 1992 - Philosophical Review 101 (4):919.
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  • Two Types of Abstraction for Structuralism.Øystein Linnebo & Richard Pettigrew - 2014 - Philosophical Quarterly 64 (255):267-283.
    If numbers were identified with any of their standard set-theoretic realizations, then they would have various non-arithmetical properties that mathematicians are reluctant to ascribe to them. Dedekind and later structuralists conclude that we should refrain from ascribing to numbers such ‘foreign’ properties. We first rehearse why it is hard to provide an acceptable formulation of this conclusion. Then we investigate some forms of abstraction meant to purge mathematical objects of all ‘foreign’ properties. One form is inspired by Frege; the other (...)
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  • Category Theory as an Autonomous Foundation.Øystein Linnebo & Richard Pettigrew - 2011 - Philosophia Mathematica 19 (3):227-254.
    Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other (...)
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  • On What There Is.W. V. O. Quine - 1948 - In Robert B. Talisse & Scott F. Aikin (eds.), The Pragmatism Reader: From Peirce Through the Present. Princeton University Press. pp. 221-233.
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  • Structuralism, Invariance, and Univalence.Steve Awodey - 2014 - Philosophia Mathematica 22 (1):1-11.
    The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...)
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  • Structuralism and Representation Theorems.George Weaver - 1998 - Philosophia Mathematica 6 (3):257-271.
    Much of the inspiration for structuralist approaches to mathematics can be found in the late nineteenth- and early twentieth-century program of characterizing various mathematical systems upto isomorphism. From the perspective of this program, differences between isomorphic systems are irrelevant. It is argued that a different view of the import of the differences between isomorphic systems can be obtained from the perspective of contemporary discussions of representation theorems and that from this perspective both the identification of isomorphic systems and the reduction (...)
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  • Epistemological Challenges to Mathematical Platonism.Øystein Linnebo - 2006 - Philosophical Studies 129 (3):545-574.
    Since Benacerraf’s “Mathematical Truth” a number of epistemological challenges have been launched against mathematical platonism. I first argue that these challenges fail because they unduely assimilate mathematics to empirical science. Then I develop an improved challenge which is immune to this criticism. Very roughly, what I demand is an account of how people’s mathematical beliefs are responsive to the truth of these beliefs. Finally I argue that if we employ a semantic truth-predicate rather than just a deflationary one, there surprisingly (...)
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  • Weaseling Away the Indispensability Argument.J. Melia - 2000 - Mind 109 (435):455-480.
    According to the indispensability argument, the fact that we quantify over numbers, sets and functions in our best scientific theories gives us reason for believing that such objects exist. I examine a strategy to dispense with such quantification by simply replacing any given platonistic theory by the set of sentences in the nominalist vocabulary it logically entails. I argue that, as a strategy, this response fails: for there is no guarantee that the nominalist world that go beyond the set of (...)
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  • Einige Bemerkungen Zur Axiomatischen Begründung der Mengenlehre.Thoralf Skolem - 1922 - In ¸ Iteskolem:Swl. pp. 137--52.
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  • Über Grenzzahlen Und Mengenbereiche: Neue Untersuchungen Über Die Grundlagen der Mengenlehre.Ernst Zermelo - 1930 - Fundamenta Mathematicæ 16:29--47.
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  • Reasoning with Arbitrary Objects.John Macnamara - 1988 - Journal of Symbolic Logic 53 (1):305.
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  • On What There Is.Charles A. Baylis - 1954 - Journal of Symbolic Logic 19 (3):222-223.
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  • Logical Structuralism and Benacerraf’s Problem.Audrey Yap - 2009 - Synthese 171 (1):157-173.
    There are two general questions which many views in the philosophy of mathematics can be seen as addressing: what are mathematical objects, and how do we have knowledge of them? Naturally, the answers given to these questions are linked, since whatever account we give of how we have knowledge of mathematical objects surely has to take into account what sorts of things we claim they are; conversely, whatever account we give of the nature of mathematical objects must be accompanied by (...)
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  • Predicativity and Structuralism in Dedekind’s Construction of the Reals.Audrey Yap - 2009 - Erkenntnis 71 (2):157-173.
    It is a commonly held view that Dedekind's construction of the real numbers is impredicative. This naturally raises the question of whether this impredicativity is justified by some kind of Platonism about sets. But when we look more closely at Dedekind's philosophical views, his ontology does not look Platonist at all. So how is his construction justified? There are two aspects of the solution: one is to look more closely at his methodological views, and in particular, the places in which (...)
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  • Structure in Mathematics and Logic: A Categorical Perspective.S. Awodey - 1996 - Philosophia Mathematica 4 (3):209-237.
    A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with the (...)
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  • Indispensability Arguments and Instrumental Nominalism.Richard Pettigrew - 2012 - Review of Symbolic Logic 5 (4):687-709.
    In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that abstract mathematical objects exist. I wish to defend a particular objection to such arguments that has become increasingly popular recently. It is called instrumental nominalism. I consider the recent versions of this view and conclude that it has yet to be given an adequate formulation. I provide such a formulation and show that it can be used to answer the indispensability arguments. -/- There (...)
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  • Cohesive Toposes and Cantor's 'Lauter Einsen'.F. W. Lawvere - 1994 - Philosophia Mathematica 2 (1):5-15.
    For 20th century mathematicians, the role of Cantor's sets has been that of the ideally featureless canvases on which all needed algebraic and geometrical structures can be painted. (Certain passages in Cantor's writings refer to this role.) Clearly, the resulting contradication, 'the points of such sets are distinc yet indistinguishable', should not lead to inconsistency. Indeed, the productive nature of this dialectic is made explicit by a method fruitful in other parts of mathematics (see 'Adjointness in Foundations', Dialectia 1969). This (...)
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  • On What There's Not.Joseph Melia - 1995 - Analysis 55 (4):223 - 229.
    (1) The average Mum has 2.4 children. (2) The number of Argle’s fingers equals the number of Bargle’s toes. (3) There are two possible ways in which Joe could win this chess game. In the right contexts, and outside the philosophy room, all the above sentences may be completely uncontroversial. For instance, if we know that Joe could win either by exchanging queens and entering an endgame, or by initiating a kingside attack then, if ignorant of Quine’s work on ontology, (...)
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  • Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2002 - Philosophy and Phenomenological Research 65 (2):467-475.
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  • Mathematics without Numbers. Towards a Modal-Structural Interpretation.Geoffrey Hellman - 1991 - Tijdschrift Voor Filosofie 53 (4):726-727.
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  • Reasoning with Arbitrary Objects.Kit Fine - 1986 - Revue Philosophique de la France Et de l'Etranger 176 (3):402-403.
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  • 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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  • Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2000 - Philosophical Quarterly 50 (198):120-123.
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  • How to Defeat Opposition to Moore.Ernest Sosa - 1999 - Noûs 33 (s13):141-153.
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  • Functional Semantics of Algebraic Theories.F. William Lawvere - 1974 - Journal of Symbolic Logic 39 (2):340-341.
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