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  1. (1 other version)Metamathematics of First-Order Arithmetic.Petr Hajék & Pavel Pudlák - 1994 - Studia Logica 53 (3):465-466.
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  • Subsystems of Second Order Arithmetic.Stephen G. Simpson - 1999 - Studia Logica 77 (1):129-129.
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  • (2 other versions)The Foundations of Arithmetic. A Logico-Mathematical Enquiry into the Concept of Number.Max Black - 1951 - Journal of Symbolic Logic 16 (1):67-67.
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  • Finitude and Hume’s Principle.Richard G. Heck - 1997 - Journal of Philosophical Logic 26 (6):589-617.
    The paper formulates and proves a strengthening of ‘Frege’s Theorem’, which states that axioms for second-order arithmetic are derivable in second-order logic from Hume’s Principle, which itself says that the number of Fs is the same as the number ofGs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. ‘Finite Hume’s Principle’ also suffices for (...)
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  • Frege’s Conception of Numbers as Objects.Crispin Wright - 1983 - Critical Philosophy 1 (1):97.
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  • Frege and the Paradox of Analysis.Michael Dummett - 1991 - In Frege and Other Philosophers. Oxford, England: Oxford University Press UK.
    This chapter is about Frege's understanding of the aim of philosophical analyses of familiar concepts, illustrated by his procedure in his Grunndlagen der Arithmetik and by Edmund Husserl's criticism of that book.
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  • Frege’s Philosophy of Mathematics. [REVIEW]Sanford Shieh - 1997 - Philosophical Review 106 (2):275.
    The days when Frege was more footnoted than read are now long gone; still, until very recently he has been read rather selectively. No doubt many had an inkling that there’s more to Frege than the sense/reference distinction; but few, one suspects, thought that his philosophy of mathematics was as fertile and intriguing as the present collection demonstrates. Perhaps, as Paul Benacerraf’s essay in this collection suggests, logical positivism should be held partly responsible for the neglect of this aspect of (...)
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  • Mathematics in Philosophy, Selected Essays.Stewart Shapiro - 1983 - Journal of Symbolic Logic 53 (1):320.
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  • The Philosophy of Philosophy.Timothy Williamson - 2007 - Malden, MA: Wiley-Blackwell.
    The second volume in the _Blackwell Brown Lectures in Philosophy_, this volume offers an original and provocative take on the nature and methodology of philosophy. Based on public lectures at Brown University, given by the pre-eminent philosopher, Timothy Williamson Rejects the ideology of the 'linguistic turn', the most distinctive trend of 20th century philosophy Explains the method of philosophy as a development from non-philosophical ways of thinking Suggests new ways of understanding what contemporary and past philosophers are doing.
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  • (1 other version)Philosophy of Mathematics.Stewart Shapiro - 2003 - In Peter Clark & Katherine Hawley (eds.), Philosophy of science today. New York: Oxford University Press.
    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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  • The Logicism of Frege, Dedekind, and Russell.William Demopoulos & Peter Clark - 2005 - In Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic. Oxford and New York: Oxford University Press. pp. 129--165.
    The common thread running through the logicism of Frege, Dedekind, and Russell is their opposition to the Kantian thesis that our knowledge of arithmetic rests on spatio-temporal intuition. Our critical exposition of the view proceeds by tracing its answers to three fundamental questions: What is the basis for our knowledge of the infinity of the numbers? How is arithmetic applicable to reality? Why is reasoning by induction justified?
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  • On formal and informal provability.Hannes Leitgeb - 2009 - In Ø. Linnebo O. Bueno (ed.), New Waves in Philosophy of Mathematics. Palgrave-Macmillan. pp. 263--299.
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  • The Tarskian Turn: Deflationism and Axiomatic Truth.Leon Horsten - 2011 - MIT Press.
    The work of mathematician and logician Alfred Tarski (1901--1983) marks the transition from substantial to deflationary views about truth.
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  • Cardinality, Counting, and Equinumerosity.Richard G. Heck - 2000 - Notre Dame Journal of Formal Logic 41 (3):187-209.
    Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that Frege (...)
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  • The Structure of Appearance.Nelson Goodman - 1951 - Cambridge, MA, USA: Harvard University Press.
    With this third edition of Nelson Goodman's The Structure of Appear ance, we are pleased to make available once more one of the most in fluential and important works in the philosophy of our times. Professor Geoffrey Hellman's introduction gives a sustained analysis and appreciation of the major themes and the thrust of the book, as well as an account of the ways in which many of Goodman's problems and projects have been picked up and developed by others. Hellman also (...)
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  • A mathematical introduction to logic.Herbert Bruce Enderton - 1972 - New York,: Academic Press.
    A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional (...)
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  • The foundations of arithmetic: a logico-mathematical enquiry into the concept of number.Gottlob Frege - 1968 - Evanston, Ill.: Northwestern University Press. Edited by J. L. Austin.
    § i. After deserting for a time the old Euclidean standards of rigour, mathematics is now returning to them, and even making efforts to go beyond them. ...
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  • The reason's proper study: essays towards a neo-Fregean philosophy of mathematics.Crispin Wright & Bob Hale - 2001 - Oxford: Clarendon Press. Edited by Crispin Wright.
    Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as the (...)
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  • Mathematics as a science of patterns.Michael David Resnik - 1997 - New York ;: Oxford University Press.
    This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...)
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  • Accuracy and actuality.G. Hellman - 1978 - Erkenntnis 12 (2):209 - 228.
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  • (1 other version)Conceptual truth.Timothy Williamson - 2006 - Aristotelian Society Supplementary Volume 80 (1):1–41.
    The paper criticizes epistemological conceptions of analytic or conceptual truth, on which assent to such truths is a necessary condition of understanding them. The critique involves no Quinean scepticism about meaning. Rather, even granted that a paradigmatic candidate for analyticity is synonymy with a logical truth, both the former and the latter can be intelligibly doubted by linguistically competent deviant logicians, who, although mistaken, still constitute counterexamples to the claim that assent is necessary for understanding. There are no analytic or (...)
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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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  • (1 other version)Fixing Frege.John P. Burgess - 2005 - Princeton University Press.
    The great logician Gottlob Frege attempted to provide a purely logical foundation for mathematics. His system collapsed when Bertrand Russell discovered a contradiction in it. Thereafter, mathematicians and logicians, beginning with Russell himself, turned in other directions to look for a framework for modern abstract mathematics. Over the past couple of decades, however, logicians and philosophers have discovered that much more is salvageable from the rubble of Frege's system than had previously been assumed. A variety of repaired systems have been (...)
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  • A Mathematical Introduction to Logic.Herbert Enderton - 2001 - Bulletin of Symbolic Logic 9 (3):406-407.
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  • Mathematical Thought and its Objects.Charles Parsons - 2007 - New York: Cambridge University Press.
    Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the concept of intuition (...)
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  • Arithmetization of Metamathematics in a General Setting.Solomon Feferman - 1960 - Journal of Symbolic Logic 31 (2):269-270.
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  • On the proof of Frege's theorem.George Boolos - 1996 - In Adam Morton & Stephen P. Stich (eds.), Benacerraf and His Critics. Blackwell. pp. 143--59.
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  • Frege's theorem.Richard G. Heck - 2011 - New York: Clarendon Press.
    The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues.
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  • Subsystems of Second Order Arithmetic.Stephen George Simpson - 1998 - Springer Verlag.
    Stephen George Simpson. with definition 1.2.3 and the discussion following it. For example, taking 90(n) to be the formula n §E Y, we have an instance of comprehension, VYEIXVn(n€X<—>n¢Y), asserting that for any given set Y there exists a ...
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  • Frege: The Last Logicist.Paul Benacerraf - 1981 - Midwest Studies in Philosophy 6 (1):17-36.
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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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  • Abstract objects.Bob Hale - 1987 - New York, NY, USA: Blackwell.
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  • Ontological reduction and the world of numbers.W. V. Quine - 1964 - Journal of Philosophy 61 (7):209-216.
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  • (1 other version)Three Grades of Modal Involvement.W. V. Quine - 1976 - In Willard Van Orman Quine (ed.), The ways of paradox, and other essays. Cambridge: Harvard University Press. pp. 158-176.
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  • (1 other version)Metamathematics of First-Order Arithmetic.P. Hájek & P. Pudlák - 2000 - Studia Logica 64 (3):429-430.
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  • Abstract Objects.John P. Burgess - 1992 - Philosophical Review 101 (2):414.
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  • (1 other version)The Foundations of Arithmetic. A Logico-Mathematical Enquiry into the Concept of Number. [REVIEW]E. N. - 1951 - Journal of Philosophy 48 (10):342.
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  • On definitional equivalence and related topics.J. Corcoran - 1980 - History and Philosophy of Logic 1:231.
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  • Frege’s Conception of Logic.Patricia Blanchette - 2012 - Oxford, England: Oup Usa.
    In Frege's Conception of Logic Patricia A. Blanchette explores the relationship between Gottlob Frege's understanding of conceptual analysis and his understanding of logic.
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  • (1 other version)Where do the natural numbers come from?Harold T. Hodes - 1990 - Synthese 84 (3):347-407.
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  • (1 other version)Reals by Abstraction.Bob Hale - 2000 - Philosophia Mathematica 8 (2):100--123.
    On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to (...)
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  • John P. Burgess, Fixing Frege. [REVIEW]Pierre Swiggers - 2006 - Tijdschrift Voor Filosofie 68 (3):665-665.
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  • (2 other versions)A Mathematical Introduction to Logic.J. R. Shoenfield - 1973 - Journal of Symbolic Logic 38 (2):340-341.
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  • (1 other version)Is Hume's Principle Analytic?Crispin Wright - 1999 - Notre Dame Journal of Formal Logic 40 (1):6-30.
    One recent `neologicist' claim is that what has come to be known as "Frege's Theorem"–the result that Hume's Principle, plus second-order logic, suffices for a proof of the Dedekind-Peano postulate–reinstates Frege's contention that arithmetic is analytic. This claim naturally depends upon the analyticity of Hume's Principle itself. The present paper reviews five misgivings that developed in various of George Boolos's writings. It observes that each of them really concerns not `analyticity' but either the truth of Hume's Principle or our entitlement (...)
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  • (1 other version)Frege’s Theorem: An Introduction.Richard G. Heck - 1999 - The Harvard Review of Philosophy 7 (1):56-73.
    A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.
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  • On the harmless impredicativity of N=('Hume's Principle').Crispin Wright - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today: Papers From a Conference Held in Munich From June 28 to July 4,1993. Oxford, England: Clarendon Press. pp. 339--68.
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  • Quasi finitely axiomatizable totally categorical theories.Gisela Ahlbrandt & Martin Ziegler - 1986 - Annals of Pure and Applied Logic 30 (1):63-82.
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  • Frege's theory of numbers.Charles Parsons - 1964 - In Max Black (ed.), Philosophy in America. Ithaca: Routledge. pp. 180-203.
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  • Frege's philosophy of mathematics.William Demopoulos (ed.) - 1995 - Cambridge: Harvard University Press.
    Widespread interest in Frege's general philosophical writings is, relatively speaking, a fairly recent phenomenon. But it is only very recently that his philosophy of mathematics has begun to attract the attention it now enjoys. This interest has been elicited by the discovery of the remarkable mathematical properties of Frege's contextual definition of number and of the unique character of his proposals for a theory of the real numbers. This collection of essays addresses three main developments in recent work on Frege's (...)
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  • Mathematics in philosophy: selected essays.Charles Parsons - 1983 - Ithaca, N.Y.: Cornell University Press.
    This important book by a major American philosopher brings together eleven essays treating problems in logic and the philosophy of mathematics.
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