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167 results

Page 1 of 9

  • Naturalism, Realism, and Normativity

  • HILARY PUTNAM
  • Journal:
    Journal of the American Philosophical Association / Volume 1 / Issue 2 / Summer 2015
    Published online by Cambridge University Press:
    19 June 2015, pp. 312-328

  • This essay describes three commitments that have become central to the author's philosophical outlook, namely, to liberal naturalism, to metaphysical realism, and to the epistemic and ontological objectivity of normative judgments. Liberal naturalism is contrasted with familiar scientistic versions of naturalism and their project of forcing explanations in every field into models derived from one or another particular science. The form of metaphysical realism that the author endorses rejects every form of verificationism, including the author's one-time ‘internal realism’, and insists that our claims about the world are true or false and not just epistemically successful or unsuccessful and that the terms they contain typically refer to real entities. ‘Representationalism is no sin’. The central part of the essay is an account of truth based on a detailed analysis of Tarski's theory of truth and of the insights we can get from it as well as of the respects in which Tarski is misleading. (This part goes beyond what the author has previously published on the subject.) The account of the objectivity of the normative in this essay draws on insights from Dewey as well as Scanlon.

Philosophy of Mathematics
  • Selected Readings
  • 2nd edition
  • Edited by Paul Benacerraf, Hilary Putnam
  • Published online:
    05 June 2012
    Print publication:
    27 January 1984
  • The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.

Reason, Truth and History
  • Hilary Putnam
  • Published online:
    05 June 2012
    Print publication:
    31 December 1981
  • Hilary Putnam deals in this book with some of the most fundamental persistent problems in philosophy: the nature of truth, knowledge and rationality. His aim is to break down the fixed categories of thought which have always appeared to define and constrain the permissible solutions to these problems.

Kurt Gödel and the Foundations of Mathematics
  • Horizons of Truth
  • Edited by Matthias Baaz, Christos H. Papadimitriou, Hilary W. Putnam, Dana S. Scott, Charles L. Harper, Jr
  • Published online:
    07 September 2011
    Print publication:
    06 June 2011
  • This volume commemorates the life, work and foundational views of Kurt Gödel (1906–78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.

  • Preface

  • Edited by Matthias Baaz, Technische Universität Wien, Austria, Christos H. Papadimitriou, University of California, Berkeley, Hilary W. Putnam, Harvard University, Massachusetts, Dana S. Scott, Carnegie Mellon University, Pennsylvania, Charles L. Harper, Jr
  • Book:
    Kurt Gödel and the Foundations of Mathematics
    Published online:
    07 September 2011
    Print publication:
    06 June 2011, pp xv-xvi

  • Summary

    Kurt Gödel and the Foundations of Mathematics: Horizons of Truth is the culmination of a creative research initiative coorganized by the Kurt Gödel Society, Vienna; the Institute for Experimental Physics; the Kurt Gödel Research Center; the Institute Vienna Circle; the Vienna University of Technology; the Austrian Academy of Sciences; and the Anton Zeilinger Group at the University of Vienna, where the Gödel centenary celebratory symposium “Horizons of Truth: Logics, Foundations of Mathematics, and the Quest for Understanding the Nature of Knowledge” was held from April 27 to April 29, 2006.

    More than twenty invited world-renowned researchers in the fields of mathematics, logic, computer science, physics, philosophy, theology, and the history of science attended the symposium, giving the participants the remarkable opportunity to present their ideas about Gödel's work and its influence on various areas of intellectual endeavor. These fascinating interdisciplinary lectures provided new insights into Gödel's life and work and their implications for future generations of researchers.

    The interaction among international scholars who only rarely, if ever, have the opportunity to hold discussions in the same room – and some of whom almost never write articles – has produced a book that contains chapters expanded and developed to take advantage of the rich intellectual exchange that took place in Vienna. Written by some of the most renowned figures of the scientific and academic world, the resulting volume is an opus of current research and thinking that is built on the work and inspiration of Gödel.

  • 15 - The Gödel Theorem and Human Nature

  • Edited by Matthias Baaz, Technische Universität Wien, Austria, Christos H. Papadimitriou, University of California, Berkeley, Hilary W. Putnam, Harvard University, Massachusetts, Dana S. Scott, Carnegie Mellon University, Pennsylvania, Charles L. Harper, Jr
  • Book:
    Kurt Gödel and the Foundations of Mathematics
    Published online:
    07 September 2011
    Print publication:
    06 June 2011, pp 325-338

  • Summary

    In the Encyclopedia of Philosophy (Edwards, 1967, 348–49), the article by Jean van Heijenoort titled “Gödel's Theorem” begins with the following terse paragraphs:

    By Gödel's theorem the following statement is generally meant:

    In any formal system adequate for number theory there exists an undecidable formula, that is, a formula that is not provable and whose negation is not provable. (This statement is occasionally referred to as Gödel's first theorem.)

    A corollary to the theorem is that the consistency of a formal system adequate for number theory cannot be proved within the system. (Sometimes it is this corollary that is referred to as Gödel's theorem; it is also referred to as Gödel's second theorem.)

    These statements are somewhat vaguely formulated generalizations of results published in 1931 by Kurt Gödel, then in Vienna.

    Despite its forbidding technicality, the Gödel (1931) theorem has never stopped generating enormous interest. Much of that interest has been piqued because with the proof of the Gödel theorem, the human mind has succeeded in proving that – at least in any fixed, consistent system with a fixed, finite set of axioms that are at least minimally adequate for number theory and in a system that has the usual logic as the instrument with which deductions are to be made from those axioms – there has to be a mathematical statement the human mind cannot prove.

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