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What’s the Point of Complete Rigour?

Mind 125 (497):177-207 (2016)

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  1. The Philosophical Basis of Intuitionistic Logic.Michael Dummett - 1978 - In Truth and other enigmas. Cambridge: Harvard University Press. pp. 215--247.
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  • Review of William Lycan, "Real Conditionals". [REVIEW]Brian Weatherson - 2002 - Philosophical Review 111 (4):609-611.
    Over the last two decades, William Lycan’s work on the semantics of conditionals has been distinguished by his careful attention to the connection between syntax and semantics, and more generally by his impeccable methodology. Lycan takes compositionality seriously, so he requires that the meaning of compound expressions like ‘even if’ be a combination of the constituent expressions, here ‘even’ and ‘if’. After reading his work, it’s hard to take seriously work that does not share this methodology.
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  • The rationalist conception of logic.Steven J. Wagner - 1987 - Notre Dame Journal of Formal Logic 28 (1):3-35.
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  • Introduction to mathematical logic..Alonzo Church - 1944 - Princeton,: Princeton university press: London, H. Milford, Oxford university press. Edited by C. Truesdell.
    This book is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work.
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  • Ignorance: A Case for Scepticism.Peter K. Unger - 1975 - Oxford [Eng.]: Oxford University Press.
    In these challenging pages, Unger argues for the extreme skeptical view that, not only can nothing ever be known, but no one can ever have any reason at all for anything. A consequence of this is that we cannot ever have any emotions about anything: no one can ever be happy or sad about anything. Finally, in this reduction to absurdity of virtually all our supposed thought, he argues that no one can ever believe, or even say, that anything is (...)
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  • Studies in the way of words.Herbert Paul Grice - 1989 - Cambridge: Harvard University Press.
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  • The Seventeen Provers of the World.Freek Wiedijk - 2007 - Studia Logica 87 (2-3):369-374.
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  • A Critique of a Formalist-Mechanist Version of the Justification of Arguments in Mathematicians' Proof Practices.Yehuda Rav - 2007 - Philosophia Mathematica 15 (3):291-320.
    In a recent article, Azzouni has argued in favor of a version of formalism according to which ordinary mathematical proofs indicate mechanically checkable derivations. This is taken to account for the quasi-universal agreement among mathematicians on the validity of their proofs. Here, the author subjects these claims to a critical examination, recalls the technical details about formalization and mechanical checking of proofs, and illustrates the main argument with aanalysis of examples. In the author's view, much of mathematical reasoning presents genuine (...)
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  • A counterexample to modus ponens.Vann McGee - 1985 - Journal of Philosophy 82 (9):462-471.
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  • Believing the axioms. I.Penelope Maddy - 1988 - Journal of Symbolic Logic 53 (2):481-511.
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  • The derivation-indicator view of mathematical practice.Jody Azzouni - 2004 - Philosophia Mathematica 12 (2):81-106.
    The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.
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  • The Seventeen Provers of the World.Wiedijk Freek (ed.) - 2006 - Springer.
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  • (1 other version)Why Do We Prove Theorems?Yehuda Rav - 1999 - Philosophia Mathematica 7 (1):5-41.
    Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...)
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  • (1 other version)Ignorance : a case for scepticism.Peter Unger - 1975 - Revue Philosophique de la France Et de l'Etranger 166 (3):371-372.
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  • (1 other version)Ignorance: A Case for Scepticism.Peter Unger - 1975 - Philosophy 50 (194):489-490.
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  • 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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  • Intuitionism and Formalism.L. E. J. Brouwer - 1913 - Bulletin of the American Mathematical Society 20:81-96.
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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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  • Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl.Gottlob Frege - 1884 - Wittgenstein-Studien 3 (2):993-999.
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  • Formalism.Michael Detlefsen - 2005 - In Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic. Oxford and New York: Oxford University Press. pp. 236--317.
    A comprehensive historical overview of formalist ideas in the philosophy of mathematics.
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  • A renaissance of empiricism in the recent philosophy of mathematics.Imre Lakatos - 1976 - British Journal for the Philosophy of Science 27 (3):201-223.
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  • Philosophy of mathematics, selected readings.Paul Benacerraf & Hilary Putnam - 1966 - Revue Philosophique de la France Et de l'Etranger 156:501-502.
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  • (1 other version)Why Do We Prove Theorems?Yehuda Rav - 1998 - Philosophia Mathematica 6 (3):5-41.
    Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...)
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  • (2 other versions)Truth and Other Enigmas.Michael Dummett - 1978 - Philosophical Quarterly 31 (122):47-67.
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  • Mathematical Knowledge.Mark Steiner - 1977 - Mind 86 (343):467-469.
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  • Deduction, Computation, Experiment: Exploring the Effectiveness of Proof.Giovanna Corsi & Rossella Lupacchini (eds.) - 2008 - Berlin and Milano: Springer.
    The essays collected in this volume address such questions from different points of view and will interest students and scholars in several branches of scientific knowledge.
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  • What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians.Don Fallis - 2002 - Logique Et Analyse 45.
    Several philosophers have used the framework of means/ends reasoning to explain the methodological choices made by scientists and mathematicians (see, e.g., Goldman 1999, Levi 1962, Maddy 1997). In particular, they have tried to identify the epistemic objectives of scientists and mathematicians that will explain these choices. In this paper, the framework of means/ends reasoning is used to study an important methodological choice made by mathematicians. Namely, mathematicians will only use deductive proofs to establish the truth of mathematical claims. In this (...)
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  • Quine, analyticity and philosophy of mathematics.John P. Burgess - 2004 - Philosophical Quarterly 54 (214):38–55.
    Quine correctly argues that Carnap's distinction between internal and external questions rests on a distinction between analytic and synthetic, which Quine rejects. I argue that Quine needs something like Carnap's distinction to enable him to explain the obviousness of elementary mathematics, while at the same time continuing to maintain as he does that the ultimate ground for holding mathematics to be a body of truths lies in the contribution that mathematics makes to our overall scientific theory of the world. Quine's (...)
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  • Oxford Handbook of Philosophy of Mathematics and Logic.Stewart Shapiro (ed.) - 2005 - Oxford and New York: Oxford University Press.
    This Oxford Handbook covers the current state of the art in the philosophy of maths and logic in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 newly-commissioned chapters are by established experts in the field and contain both exposition and criticism as well as substantial development of their own positions. Select major positions are represented by two chapters - one supportive and one critical. The book includes a comprehensive (...)
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  • (1 other version)Real Conditionals.William G. Lycan - 2001 - Oxford, England: Oxford: Oxford University Press.
    This book contends that insufficient attention has been paid to the syntax of conditionals, as investigated by linguists.
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  • From Frege to Gödel.Jean Van Heijenoort (ed.) - 1967 - Cambridge,: Harvard University Press.
    The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for ...
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  • Real Conditionals.William G. Lycan - 2003 - Philosophical Quarterly 53 (210):134-137.
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  • (1 other version)From Frege to Gödel.Jean van Heijenoort - 1968 - Philosophy of Science 35 (1):72-72.
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  • Frege and the rigorization of analysis.William Demopoulos - 1994 - Journal of Philosophical Logic 23 (3):225 - 245.
    This paper has three goals: (i) to show that the foundational program begun in the Begriffsschroft, and carried forward in the Grundlagen, represented Frege's attempt to establish the autonomy of arithmetic from geometry and kinematics; the cogency and coherence of 'intuitive' reasoning were not in question. (ii) To place Frege's logicism in the context of the nineteenth century tradition in mathematical analysis, and, in particular, to show how the modern concept of a function made it possible for Frege to pursue (...)
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  • (2 other versions)Introduction to Mathematical Logic.Max Black - 1956 - Journal of Symbolic Logic 22 (3):286-289.
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  • Some Proposals for Reviving the Philosophy of Mathematics.Reuben Hersh - 1983 - Journal of Symbolic Logic 48 (3):871-872.
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  • Mathematical rigor--who needs it?Philip Kitcher - 1981 - Noûs 15 (4):469-493.
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  • The Epistemic Status of Probabilistic Proof.Don Fallis - 1997 - Journal of Philosophy 94 (4):165-186.
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  • (1 other version)The Non-Logical Basis of Metaphysics.D. E. Bradshaw - 1996 - Idealistic Studies 26 (1):1-16.
    Michael Dummett begins The Logical Basis of Metaphysics by noting that most of the work done in analytic philosophy seems disconcertingly remote from any concern with the “deep questions of great import for an understanding of the world” that the non-professional expects it to answer. In part, he says, this is because modern analytic philosophy is founded upon a more penetrating analysis of the general structure of our thoughts than was available to past ages, namely, the apparatus of modern logic, (...)
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  • Infinite Regrees and Foundations of Mathematics.Imre Lakatos - 1962 - Aristotelian Society Supplementary Volume 36:155--84.
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  • (1 other version)Mathematical knowledge.Mark Steiner - 1975 - Ithaca: Cornell University Press.
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