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  1. Was Sind und was Sollen Die Zahlen?Richard Dedekind - 1888 - Cambridge University Press.
    This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.
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  • How we learn mathematical language.Vann McGee - 1997 - Philosophical Review 106 (1):35-68.
    Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is א2 or whether there are (...)
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  • How We Learn Mathematical Language.Vann McGee - 1997 - Philosophical Review 106 (1):35-68.
    Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is א2 or whether there are (...)
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  • On Putnam and his models.Timothy Bays - 2001 - Journal of Philosophy 98 (7):331-350.
    It is not my claim that the ‘L¨ owenheim-Skolem paradox’ is an antinomy in formal logic; but I shall argue that it is an antinomy, or something close to it, in philosophy of language. Moreover, I shall argue that the resolution of the antinomy—the only resolution that I myself can see as making sense—has profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language.
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  • On Putnam and His Models.Timothy Bays - 2001 - Journal of Philosophy 98 (7):331.
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  • Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic.Steve Awodey & Erich H. Reck - 2002 - History and Philosophy of Logic 23 (1):1-30.
    This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
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  • XIII-Categoricity and Indefinite Extensibility.James Walmsley - 2002 - Proceedings of the Aristotelian Society 102 (3):217-235.
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  • On the concept of categoricity.Andrzej Grzegorczyk - 1962 - Studia Logica 13 (1):39 - 66.
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  • Mind and body.Hilary Putnam - 1981 - In Reason, truth, and history. New York: Cambridge University Press.
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  • Models and reality.Hilary Putnam - 1983 - In Realism and reason. New York: Cambridge University Press. pp. 1-25.
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  • Realism and Reason.Hilary Putnam - 1977 - Proceedings and Addresses of the American Philosophical Association 50 (6):483-498.
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  • Kreisel, the continuum hypothesis and second order set theory.Thomas Weston - 1976 - Journal of Philosophical Logic 5 (2):281 - 298.
    The major point of contention among the philosophers and mathematicians who have written about the independence results for the continuum hypothesis (CH) and related questions in set theory has been the question of whether these results give reason to doubt that the independent statements have definite truth values. This paper concerns the views of G. Kreisel, who gives arguments based on second order logic that the CH does have a truth value. The view defended here is that although Kreisel's conclusion (...)
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  • Logicism, Interpretability, and Knowledge of Arithmetic.Sean Walsh - 2014 - Review of Symbolic Logic 7 (1):84-119.
    A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...)
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  • Categoricity and indefinite extensibility.James Walmsley - 2002 - Proceedings of the Aristotelian Society 102 (3):217–235.
    Structure is central to the realist view of mathematical disciplines with intended interpretations and categoricity is a model-theoretic notion that captures the idea of the determination of structure by theory. By considering the cases of arithmetic and (pure) set theory, I investigate how categoricity results might offer support from within to the realist view. I argue, amongst other things, that second-order quantification is essential to the support the categoricity results provide. I also note how the findings on categoricity relate to (...)
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  • Internal Categoricity in Arithmetic and Set Theory.Jouko Väänänen & Tong Wang - 2015 - Notre Dame Journal of Formal Logic 56 (1):121-134.
    We show that the categoricity of second-order Peano axioms can be proved from the comprehension axioms. We also show that the categoricity of second-order Zermelo–Fraenkel axioms, given the order type of the ordinals, can be proved from the comprehension axioms. Thus these well-known categoricity results do not need the so-called “full” second-order logic, the Henkin second-order logic is enough. We also address the question of “consistency” of these axiom systems in the second-order sense, that is, the question of existence of (...)
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  • Second order logic or set theory?Jouko Väänänen - 2012 - Bulletin of Symbolic Logic 18 (1):91-121.
    We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each (...)
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  • Set Theory and Its Logic.Joseph S. Ullian & Willard Van Orman Quine - 1966 - Philosophical Review 75 (3):383.
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  • In Defense of Putnam’s Brains.Thomas Tymoczko - 1989 - Philosophical Studies 57 (3):281--97.
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  • Mathematics as a science of patterns. [REVIEW]Mark Steiner - 2000 - Philosophical Review 109 (1):115-118.
    For the past hundred years, mathematics, for its own reasons, has been shifting away from the study of “mathematical objects” and towards the study of “structures”. One would have expected philosophers to jump onto the bandwagon, as in many other cases, to proclaim that this shift is no accident, since mathematics is “essentially” about structures, not objects. In fact, structuralism has not been a very popular philosophy of mathematics, probably because of the hostility of Frege and other influential logicists, and (...)
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  • Completeness and categoricity: Frege, gödel and model theory.Stephen Read - 1997 - History and Philosophy of Logic 18 (2):79-93.
    Frege’s project has been characterized as an attempt to formulate a complete system of logic adequate to characterize mathematical theories such as arithmetic and set theory. As such, it was seen to fail by Gödel’s incompleteness theorem of 1931. It is argued, however, that this is to impose a later interpretation on the word ‘complete’ it is clear from Dedekind’s writings that at least as good as interpretation of completeness is categoricity. Whereas few interesting first-order mathematical theories are categorical or (...)
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  • Set Theory and Its Logic.J. C. Shepherdson & Willard Van Orman Quine - 1965 - Philosophical Quarterly 15 (61):371.
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  • Foundations without Foundationalism: A Case for Second-Order Logic.Gila Sher - 1994 - Philosophical Review 103 (1):150.
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  • Understanding the Infinite.Stewart Shapiro - 1996 - Philosophical Review 105 (2):256.
    Understanding the Infinite is a loosely connected series of essays on the nature of the infinite in mathematics. The chapters contain much detail, most of which is interesting, but the reader is not given many clues concerning what concepts and ideas are relevant for later developments in the book. There are, however, many technical cross-references, so the reader can expect to spend much time flipping backward and forward.
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  • Logical consequence, proof theory, and model theory.Stewart Shapiro - 2005 - In 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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  • Higher-Order Logic or Set Theory: A False Dilemma.S. Shapiro - 2012 - Philosophia Mathematica 20 (3):305-323.
    The purpose of this article is show that second-order logic, as understood through standard semantics, is intimately bound up with set theory, or some other general theory of interpretations, structures, or whatever. Contra Quine, this does not disqualify second-order logic from its role in foundational studies. To wax Quinean, why should there be a sharp border separating mathematics from logic, especially the logic of mathematics?
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  • Foundations Without Foundationalism: A Case for Second-Order Logic.Michael Potter - 1994 - Philosophical Quarterly 44 (174):127-129.
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  • Foundations without foundationalism: a case for second-order logic.Stewart Shapiro - 1991 - New York: 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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  • Models and reality.John R. Searle - 1990 - Behavioral and Brain Sciences 13 (2):399-399.
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  • Mathematics as a science of patterns: Ontology and reference.Michael Resnik - 1981 - Noûs 15 (4):529-550.
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  • Set Theory and its Logic: Revised Edition.Willard Van Orman Quine - 1963 - Harvard University Press.
    This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject.
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  • Set Theory and its Logic.Willard van Orman Quine - 1963 - Cambridge, MA, USA: Harvard University Press.
    This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject. The treatment of ordinal numbers has been strengthened and much simplified, especially in the theory of transfinite recursions, by adding an axiom and reworking the proofs. Infinite cardinals are treated anew in clearer and fuller terms than before. Improvements have been made all through the book; in various instances a proof has been shortened, a theorem strengthened, a space-saving (...)
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  • Realism and reason.Hilary Putnam (ed.) - 1983 - New York: Cambridge University Press.
    This is the third volume of Hilary Putnam's philosophical papers, published in paperback for the first time. The volume contains his major essays from 1975 to 1982, which reveal a large shift in emphasis in the 'realist'_position developed in his earlier work. While not renouncing those views, Professor Putnam has continued to explore their epistemological consequences and conceptual history. He now, crucially, sees theories of truth and of meaning that derive from a firm notion of reference as inadequate.
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  • Models and reality.Hilary Putnam - 1980 - Journal of Symbolic Logic 45 (3):464-482.
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  • Set Theory and its Philosophy: A Critical Introduction.Michael D. Potter - 2004 - Oxford, England: Oxford University Press.
    Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...)
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  • Mathematical determinacy and the transferability of aboutness.Stephen Pollard - 2007 - Synthese 159 (1):83-98.
    Competent speakers of natural languages can borrow reference from one another. You can arrange for your utterances of ‘Kirksville’ to refer to the same thing as my utterances of ‘Kirksville’. We can then talk about the same thing when we discuss Kirksville. In cases like this, you borrow “ aboutness ” from me by borrowing reference. Now suppose I wish to initiate a line of reasoning applicable to any prime number. I might signal my intention by saying, “Let p be (...)
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  • Open-endedness, schemas and ontological commitment.Nikolaj Jang Lee Linding Pedersen & Marcus Rossberg - 2010 - Noûs 44 (2):329-339.
    Second-order axiomatizations of certain important mathematical theories—such as arithmetic and real analysis—can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth-value. Second-order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth-values. The status of second-order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, Vann McGee has argued that one (...)
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  • The structuralist view of mathematical objects.Charles Parsons - 1990 - Synthese 84 (3):303 - 346.
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  • Recent Results in Set Theory.Andrzej Mostowski, Imre Lakatos, G. Kreisel, A. Robinson & A. Mostowski - 1972 - Journal of Symbolic Logic 37 (4):765-766.
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  • Zermelo's Axiom of Choice. Its Origins, Development, and Influence.Gregory H. Moore - 1984 - Journal of Symbolic Logic 49 (2):659-660.
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  • The model-theoretic argument against realism.G. H. Merrill - 1980 - Philosophy of Science 47 (1):69-81.
    In "Realism and Reason" Hilary Putnam has offered an apparently strong argument that the position of metaphysical realism provides an incoherent model of the relation of a correct scientific theory to the world. However, although Putnam's attack upon the notion of the "intended" interpretation of a scientific theory is sound, it is shown here that realism may be formulated in such a way that the realist need make no appeal to any "intended" interpretation of such a theory. Consequently, it can (...)
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  • WHAT CAN A CATEGORICITY THEOREM TELL US?Toby Meadows - 2013 - Review of Symbolic Logic (3):524-544.
    f The purpose of this paper is to investigate categoricity arguments conducted in second order logic and the philosophical conclusions that can be drawn from them. We provide a way of seeing this result, so to speak, through a first order lens divested of its second order garb. Our purpose is to draw into sharper relief exactly what is involved in this kind of categoricity proof and to highlight the fact that we should be reserved before drawing powerful philosophical conclusions (...)
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  • Informal Rigour and Completeness Proofs.Georg Kreisel - 1967 - In Imre Lakatos (ed.), Problems in the philosophy of mathematics. Amsterdam,: North-Holland Pub. Co.. pp. 138--157.
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  • A Semantic Conception of Truth?Vann McGee - 1993 - Philosophical Topics 21 (2):83-111.
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  • Set Theory and Its Logic. [REVIEW]Donald A. Martin - 1970 - Journal of Philosophy 67 (4):111-114.
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  • Multiple universes of sets and indeterminate truth values.Donald A. Martin - 2001 - Topoi 20 (1):5-16.
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  • Gödel's conceptual realism.Donald A. Martin - 2005 - Bulletin of Symbolic Logic 11 (2):207-224.
    Kurt Gödel is almost as famous—one might say “notorious”—for his extreme platonist views as he is famous for his mathematical theorems. Moreover his platonism is not a myth; it is well-documented in his writings. Here are two platonist declarations about set theory, the first from his paper about Bertrand Russell and the second from the revised version of his paper on the Continuum Hypotheses.Classes and concepts may, however, also be conceived as real objects, namely classes as “pluralities of things” or (...)
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  • The development of mathematical logic from Russell to Tarski, 1900-1935.Paolo Mancosu, Richard Zach & Calixto Badesa - 2011 - In Leila Haaparanta (ed.), The development of modern logic. New York: Oxford University Press.
    The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...)
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  • Structuralism reconsidered.Fraser MacBride - 2005 - In Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. pp. 563--589.
    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 (...)
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  • Putnam’s paradox.David Lewis - 1984 - Australasian Journal of Philosophy 62 (3):221 – 236.
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  • Understanding the infinite.Shaughan Lavine - 1994 - Cambridge: Harvard University Press.
    An engaging account of the origins of the modern mathematical theory of the infinite, his book is also a spirited defense against the attacks and misconceptions ...
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