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Stewart Shapiro [190]Stewart David Shapiro [1]
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Stewart Shapiro
Ohio State University
  1. Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - 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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  2. Institutionism, Pluralism, and Cognitive Command.Stewart Shapiro & William W. Taschek - 1996 - Journal of Philosophy 93 (2):74.
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  3.  86
    Foundations Without Foundationalism: A Case for Second-Order Logic.Stewart Shapiro - 1991 - Oxford University Press.
    The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...)
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  4.  89
    Vagueness in Context.Stewart Shapiro - 2006 - Oxford University Press.
    Stewart Shapiro's ambition in Vagueness in Context is to develop a comprehensive account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary according to their context: a person can be tall with respect to male accountants and not tall (even short) with respect to professional basketball players. The key feature of Shapiro's account is that the extensions of vague terms (...)
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  5. The Good, the Bad and the Ugly.Philip A. Ebert & Stewart Shapiro - 2009 - Synthese 170 (3):415-441.
    This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form (...)
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  6. 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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  7. Proof and Truth.Stewart Shapiro - 1998 - Journal of Philosophy 95 (10):493-521.
  8. Thinking About Mathematics: The Philosophy of Mathematics.Stewart Shapiro - 2000 - Oxford University Press.
    This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that mathematics is logic (logicism), (...)
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  9. Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2000 - Philosophical Quarterly 50 (198):120-123.
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  10.  14
    Neologicism, Frege's Constraint, and the Frege‐Heck Condition.Eric Snyder, Richard Samuels & Stewart Shapiro - forthcoming - Noûs.
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  11. Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-Mathematics.Stewart Shapiro - 2005 - Philosophia Mathematica 13 (1):61-77.
    There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...)
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  12.  61
    Actual and Potential Infinity.Øystein Linnebo & Stewart Shapiro - forthcoming - Noûs.
    The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.
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  13.  85
    The Oxford Handbook of Philosophy of Mathematics and Logic.Stewart Shapiro (ed.) - 2005 - Oxford University Press.
    Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in (...)
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  14.  55
    Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility.Stewart Shapiro - 2003 - British Journal for the Philosophy of Science 54 (1):59--91.
    The purpose of this paper is to assess the prospects for a neo-logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): PQ[Ext(P) = Ext(Q) [(BAD(P) & BAD(Q)) x(Px Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’. 1 Background: what and why? (...)
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  15. Conservativeness and Incompleteness.Stewart Shapiro - 1983 - Journal of Philosophy 80 (9):521-531.
  16.  48
    Frege Meets Dedekind: A Neologicist Treatment of Real Analysis.Stewart Shapiro - 2000 - Notre Dame Journal of Formal Logic 41 (4):335--364.
    This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a property (of (...)
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  17.  15
    Vagueness in Context. [REVIEW]Stewart Shapiro - 2008 - Philosophy and Phenomenological Research 76 (2):471-483.
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  18. The Classical Continuum Without Points.Geoffrey Hellman & Stewart Shapiro - 2013 - Review of Symbolic Logic 6 (3):488-512.
    We develop a point-free construction of the classical one- dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence (...)
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  19. The Objectivity of Mathematics.Stewart Shapiro - 2007 - Synthese 156 (2):337-381.
    The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.
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  20. Mathematics and Reality.Stewart Shapiro - 1983 - Philosophy of Science 50 (4):523-548.
    The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) survey of the major, traditional philosophies (...)
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  21. New V, ZF and Abstractiont.Stewart Shapiro & Alan Weir - 1999 - Philosophia Mathematica 7 (3):293-321.
    We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New V yields a system equivalent to (...)
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  22.  21
    Logical Consequence, Proof Theory, and Model Theory.Stewart Shapiro - 2005 - In The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. pp. 651--670.
    This chapter provides broad coverage of the notion of logical consequence, exploring its modal, semantic, and epistemic aspects. It develops the contrast between proof-theoretic notion of consequence, in terms of deduction, and a model-theoretic approach, in terms of truth-conditions. The main purpose is to relate the formal, technical work in logic to the philosophical concepts that underlie reasoning.
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  23.  59
    An “I” for an I: Singular Terms, Uniqueness, and Reference.Stewart Shapiro - 2012 - Review of Symbolic Logic 5 (3):380-415.
    There is an interesting logical/semantic issue with some mathematical languages and theories. In the language of (pure) complex analysis, the two square roots of i’ manage to pick out a unique object? This is perhaps the most prominent example of the phenomenon, but there are some others. The issue is related to matters concerning the use of definite descriptions and singular pronouns, such as donkey anaphora and the problem of indistinguishable participants. Taking a cue from some work in linguistics and (...)
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  24. Aristotelian Continua.Øystein Linnebo, Stewart Shapiro & Geoffrey Hellman - 2016 - Philosophia Mathematica 24 (2):214-246.
    In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, (...)
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  25.  79
    ‘Neo-Logicist‘ Logic is Not Epistemically Innocent.Stewart Shapiro & Alan Weir - 2000 - Philosophia Mathematica 8 (2):160--189.
    The neo-logicist argues tliat standard mathematics can be derived by purely logical means from abstraction principles—such as Hume's Principle— which are held to lie 'epistcmically innocent'. We show that the second-order axiom of comprehension applied to non-instantiated properties and the standard first-order existential instantiation and universal elimination principles are essential for the derivation of key results, specifically a theorem of infinity, but have not been shown to be epistemically innocent. We conclude that the epistemic innocence of mathematics has not been (...)
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  26. We Hold These Truths to Be Self-Evident: But What Do We Mean by That?Stewart Shapiro - 2009 - Review of Symbolic Logic 2 (1):175-207.
    At the beginning of Die Grundlagen der Arithmetik (§2) [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that (...)
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  27.  93
    Logical Consequence: Models and Modality.Stewart Shapiro - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press. pp. 131--156.
  28. Epistemology of Mathematics: What Are the Questions? What Count as Answers?Stewart Shapiro - 2011 - Philosophical Quarterly 61 (242):130-150.
    A paper in this journal by Fraser MacBride, ‘Can Ante Rem Structuralism Solve the Access Problem?’, raises important issues concerning the epistemological goals and burdens of contemporary philosophy of mathematics, and perhaps philosophy of science and other disciplines as well. I use a response to MacBride's paper as a framework for developing a broadly holistic framework for these issues, and I attempt to steer a middle course between reductive foundationalism and extreme naturalistic quietism. For this purpose the notion of entitlement (...)
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  29. Set Theory, Type Theory, and Absolute Generality.Salvatore Florio & Stewart Shapiro - 2014 - Mind 123 (489):157-174.
    In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are open-ended, or that neither (...)
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  30.  27
    Varieties of Logic.Stewart Shapiro - 2014 - Oxford University Press.
    Logical pluralism is the view that different logics are equally appropriate, or equally correct. Logical relativism is a pluralism according to which validity and logical consequence are relative to something. Stewart Shapiro explores various such views. He argues that the question of meaning shift is itself context-sensitive and interest-relative.
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  31. Modality and Ontology.Stewart Shapiro - 1993 - Mind 102 (407):455-481.
  32. So Truth is Safe From Paradox: Now What?Stewart Shapiro - 2010 - Philosophical Studies 147 (3):445-455.
    The article is part of a symposium on Hartry Field’s “Saving truth from paradox”. The book is one of the most significant intellectual achievements of the past decades, but it is not clear what, exactly, it accomplishes. I explore some alternatives, relating the developed view to the intuitive, pre-theoretic notion of truth.
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  33.  85
    Frege Meets Aristotle: Points as Abstracts.Stewart Shapiro & Geoffrey Hellman - 2015 - Philosophia Mathematica:nkv021.
    There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...)
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  34. The Guru, the Logician, and the Deflationist: Truth and Logical Consequence.Stewart Shapiro - 2003 - Noûs 37 (1):113–132.
    The purpose of this paper is to present a thought experiment and argument that spells trouble for “radical” deflationism concerning meaning and truth such as that advocated by the staunch nominalist Hartry Field. The thought experiment does not sit well with any view that limits a truth predicate to sentences understood by a given speaker or to sentences in (or translatable into) a given language, unless that language is universal. The scenario in question concerns sentences that are not understood but (...)
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  35.  93
    Mechanism, Truth, and Penrose's New Argument.Stewart Shapiro - 2003 - Journal of Philosophical Logic 32 (1):19-42.
    Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). If so, its premises are inconsistent. The usual (...)
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  36.  59
    Structure and Identity.Stewart Shapiro - 2006 - In Fraser MacBride (ed.), Identity and Modality. 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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  37. What is Mathematical Logic?John Corcoran & Stewart Shapiro - 1978 - Philosophia 8 (1):79-94.
    This review concludes that if the authors know what mathematical logic is they have not shared their knowledge with the readers. This highly praised book is replete with errors and incoherency.
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  38. Towards a Point-Free Account of the Continuous.Geoffrey Hellman & Stewart Shapiro - 2012 - Iyyun 61:263.
     
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  39.  14
    Vagueness in Context.Stewart Shapiro - 2006 - Oxford University Press UK.
    Stewart Shapiro's aim in Vagueness in Context is to develop both a philosophical and a formal, model-theoretic account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary with such contextual factors as the comparison class and paradigm cases. A person can be tall with respect to male accountants and not tall with respect to professional basketball players. The main feature (...)
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  40. Second-Order Languages and Mathematical Practice.Stewart Shapiro - 1985 - Journal of Symbolic Logic 50 (3):714-742.
  41.  84
    Sets and Abstracts – Discussion.Stewart Shapiro - 2004 - Philosophical Studies 122 (3):315-332.
  42. Simple Truth, Contradiction, and Consistency.Stewart Shapiro - 2004 - In G. Priest, J. C. Beall & B. Armour-Garb (eds.), The Law of Non-Contradiction. Oxford University Press.
  43.  83
    Reasoning with Slippery Predicates.Stewart Shapiro - 2008 - Studia Logica 90 (3):313-336.
    It is a commonplace that the extensions of most, perhaps all, vague predicates vary with such features as comparison class and paradigm and contrasting cases. My view proposes another, more pervasive contextual parameter. Vague predicates exhibit what I call open texture: in some circumstances, competent speakers can go either way in the borderline region. The shifting extension and anti-extensions of vague predicates are tracked by what David Lewis calls the “conversational score”, and are regulated by what Kit Fine calls penumbral (...)
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  44.  43
    ``Intuitionism, Pluralism, and Cognitive Command".Stewart Shapiro & William W. Taschek - 1996 - Journal of Philosophy 20 (2):74-88.
  45.  29
    The Status of Logic.Stewart Shapiro - 2000 - In Paul Boghossian & Christopher Peacocke (eds.), New Essays on the a Priori. Oxford University Press. pp. 333--338.
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  46.  3
    We Hold These Truths to Be Self-Evident: But What Do We Mean by That?: We Hold These Truths to Be Self-Evident.Stewart Shapiro - 2009 - Review of Symbolic Logic 2 (1):175-207.
    At the beginning of Die Grundlagen der Arithmetik [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both (...)
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  47. Where in the (World Wide) Web of Belief is the Law of Non-Contradiction?Jack Arnold & Stewart Shapiro - 2007 - Noûs 41 (2):276–297.
    It is sometimes said that there are two, competing versions of W. V. O. Quine’s unrelenting empiricism, perhaps divided according to temporal periods of his career. According to one, logic is exempt from, or lies outside the scope of, the attack on the analytic-synthetic distinction. This logic-friendly Quine holds that logical truths and, presumably, logical inferences are analytic in the traditional sense. Logical truths are knowable a priori, and, importantly, they are incorrigible, and so immune from revision. The other, radical (...)
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  48. All Things Indefinitely Extensible.Stewart Shapiro & Crispin Wright - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), ¸ Iterayo&Uzquiano:Ag. Clarendon Press. pp. 255--304.
     
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  49.  20
    Understanding the Infinite.Stewart Shapiro, Mary Ellen Smircich & Shaughan Lavine - 1996 - Philosophical Review 105 (2):256.
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  50.  74
    Principles of Reflection and Second-Order Logic.Stewart Shapiro - 1987 - Journal of Philosophical Logic 16 (3):309 - 333.
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