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  1. What is Cantor's Continuum Problem?S. C. Kleene - 1948 - Journal of Symbolic Logic 13 (2):116-117.
  • Gleason's Theorem has a Constructive Proof.Fred Richman - 2000 - Journal of Philosophical Logic 29 (4):425-431.
    Gleason's theorem for 𝐑³ says that if f is a nonnegative function on the unit sphere with the property that f(x) + f(y) + f(z) is a fixed constant for each triple x, y, z of mutually orthogonal unit vectors, then f is a quadratic form. We examine the issues raised by discussions in this journal regarding the possibility of a constructive proof of Gleason's theorem in light of the recent publication of such a proof.
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  • Two Dogmas of Empiricism.Willard V. O. Quine - 1951 - Philosophical Review 60 (1):20–43.
    Modern empiricism has been conditioned in large part by two dogmas. One is a belief in some fundamental cleavage between truths which are analytic, or grounded in meanings independently of matters of fact, and truth which are synthetic, or grounded in fact. The other dogma is reductionism: the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience. Both dogmas, I shall argue, are ill founded. One effect of abandoning them is, as (...)
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  • Two Dogmas of Empiricism.John G. Kemeny - 1951 - Journal of Symbolic Logic 17 (4):281-283.
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  • Two Dogmas of Empiricism.W. V. O. Quine - 1951 - In Robert B. Talisse & Scott F. Aikin (eds.), The Pragmatism Reader: From Peirce Through the Present. Princeton University Press. pp. 202-220.
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  • Models and Reality.Hilary Putnam - 1980 - Journal of Symbolic Logic 45 (3):464-482.
  • Defending Abduction.Ilkka Niiniluoto - 1999 - Philosophy of Science 66 (3):451.
    Charles S. Peirce argued that, besides deduction and induction, there is a third mode of inference which he called " hypothesis " or " abduction." He characterized abduction as reasoning " from effect to cause," and as " the operation of adopting an explanatory hypothesis." Peirce ' s ideas about abduction, which are related also to historically earlier accounts of heuristic reasoning, have been seen as providing a logic of scientific discovery. Alternatively, abduction is interpreted as giving reasons for pursuing (...)
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  • Ontology and the Vicious Circle Principle.Stanley C. Martens - 1976 - Philosophical Review 85 (2):256.
  • Inference to the Best Explanation.Peter Lipton - 1991 - London and New York: Routledge/Taylor and Francis Group.
    How do we go about weighing evidence, testing hypotheses, and making inferences? The model of " inference to the best explanation " -- that we infer the hypothesis that would, if correct, provide the best explanation of the available evidence--offers a compelling account of inferences both in science and in ordinary life. Widely cited by epistemologists and philosophers of science, IBE has nonetheless remained little more than a slogan. Now this influential work has been thoroughly revised and updated, and features (...)
  • Inference to the Best Explanation.Peter Milne - 1993 - Philosophy and Phenomenological Research 53 (4):970-972.
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  • Inference to the Best Explanation.Peter Lipton - 2007 - Philosophy and Phenomenological Research 74 (2):421-423.
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  • Inference to the Best Explanation.Peter Lipton - 2004 - In Martin Curd & Stathis Psillos (eds.), The Routledge Companion to Philosophy of Science. Routledge. pp. 193.
    Science depends on judgments of the bearing of evidence on theory. Scientists must judge whether an observation or the result of an experiment supports, disconfirms, or is simply irrelevant to a given hypothesis. Similarly, scientists may judge that, given all the available evidence, a hypothesis ought to be accepted as correct or nearly so, rejected as false, or neither. Occasionally, these evidential judgments can be made on deductive grounds. If an experimental result strictly contradicts a hypothesis, then the truth of (...)
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  • Ontologies for Plane, Polygonal Mereotopology.Ian Pratt & Oliver Lemon - 1997 - Notre Dame Journal of Formal Logic 38 (2):225-245.
    Several authors have suggested that a more parsimonious and conceptually elegant treatment of everyday mereological and topological reasoning can be obtained by adopting a spatial ontology in which regions, not points, are the primitive entities. This paper challenges this suggestion for mereotopological reasoning in two-dimensional space. Our strategy is to define a mereotopological language together with a familiar, point-based interpretation. It is proposed that, to be practically useful, any alternative region-based spatial ontology must support the same sentences in our language (...)
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  • Real Analysis Without Classes.Geoffrey Hellman - 1994 - Philosophia Mathematica 2 (3):228-250.
    This paper explores strengths and limitations of both predicativism and nominalism, especially in connection with the problem of characterizing the continuum. Although the natural number structure can be recovered predicatively (despite appearances), no predicative system can characterize even the full predicative continuum which the classicist can recognize. It is shown, however, that the classical second-order theory of continua (third-order number theory) can be recovered nominalistically, by synthesizing mereology, plural quantification, and a modal-structured approach with essentially just the assumption that an (...)
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  • Quantum Mechanical Unbounded Operators and Constructive Mathematics – a Rejoinder to Bridges.Geoffrey Hellman - 1997 - Journal of Philosophical Logic 26 (2):121-127.
    As argued in Hellman (1993), the theorem of Pour-El and Richards (1983) can be seen by the classicist as limiting constructivist efforts to recover the mathematics for quantum mechanics. Although Bridges (1995) may be right that the constructivist would work with a different definition of 'closed operator', this does not affect my point that neither the classical unbounded operators standardly recognized in quantum mechanics nor their restrictions to constructive arguments are recognizable as objects by the constructivist. Constructive substitutes that may (...)
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  • Mathematics Without Numbers: Towards a Modal-Structural Interpretation.Geoffrey Hellman - 1989 - Oxford, England: Oxford University Press.
    Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...
  • Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem. [REVIEW]Geoffrey Hellman - 1993 - Journal of Philosophical Logic 22 (3):221 - 248.
  • Structuralism's Unpaid Epistemological Debts.Bob Hale - 1996 - Philosophia Mathematica 4 (2):124--47.
    One kind of structuralism holds that mathematics is about structures, conceived as a type of abstract entity. Another denies that it is about any distinctively mathematical entities at all—even abstract structures; rather it gives purely general information about what holds of any collection of entities conforming to the axioms of the theory. Of these, pure structuralism is most plausibly taken to enjoy significant advantages over platonism. But in what appears to be its most plausible—modalised—version, even restricted to elementary arithmetic, it (...)
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  • Mathematics Without Numbers: Towards a Modal-Structural Interpretation.Bob Hale & Geoffrey Hellman - 1992 - Philosophical Review 101 (4):919.
  • Theory and Evidence.Clark N. Glymour - 1980 - Princeton University Press.
  • Structuralism Without Structures.Hellman Geoffrey - 1996 - Philosophia Mathematica 4 (2):100-123.
    Recent technical developments in the logic of nominalism make it possible to improve and extend significantly the approach to mathematics developed in Mathematics without Numbers. After reviewing the intuitive ideas behind structuralism in general, the modal-structuralist approach as potentially class-free is contrasted broadly with other leading approaches. The machinery of nominalistic ordered pairing (Burgess-Hazen-Lewis) and plural quantification (Boolos) can then be utilized to extend the core systems of modal-structural arithmetic and analysis respectively to full, classical, polyadic third- and fourthorder number (...)
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  • What is Cantor's Continuum Problem?Kurt Gödel - 1947 - Journal of Symbolic Logic 13 (2):176--187.
  • Truth, Proof and Infinity a Theory of Constructions and Contructive Reasoning.P. Fletcher - 1998 - Dordrecht, Netherland: Springer.
    Constructive mathematics is based on the thesis that the meaning of a mathematical formula is given, not by its truth-conditions, but in terms of what constructions count as a proof of it. However, the meaning of the terms `construction' and `proof' has never been adequately explained. This monograph develops precise definitions of construction and proof, and describes the algorithmic substructure underlying intuitionistic logic. Interpretations of Heyting arithmetic and constructive analysis are given. The philosophical basis of constructivism is explored thoroughly in (...)
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  • Elements of Intuitionism.Nicolas D. Goodman - 1979 - Journal of Symbolic Logic 44 (2):276-277.
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  • Ontology and the Vicious Circle Principle.Charles S. Chihara - 1973 - Ithaca, NY, USA: Ithaca [N.Y.]Cornell University Press.
  • Ontology and the Vicious Circle Principle. [REVIEW]Mark Steiner - 1975 - Journal of Philosophy 72 (7):184-196.
  • Intellectual Autobiography.Rudolf Carnap & Paul Arthur Schilpp - 1971 - Journal of Symbolic Logic 36 (1):178-179.
  • Constructive Mathematics and Unbounded Operators — a Reply to Hellman.Douglas S. Bridges - 1995 - Journal of Philosophical Logic 24 (5):549 - 561.
    It is argued that Hellman's arguments purporting to demonstrate that constructive mathematics cannot cope with unbounded operators on a Hilbert space are seriously flawed, and that there is no evidence that his thesis is correct.
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  • Individuals and Points.Bowman L. Clark - 1985 - Notre Dame Journal of Formal Logic 26 (1):61-75.
  • A Calculus of Individuals Based on "Connection".Bowman L. Clarke - 1981 - Notre Dame Journal of Formal Logic 22 (3):204-218.
    Although Aristotle (Metaphysics, Book IV, Chapter 2) was perhaps the first person to consider the part-whole relationship to be a proper subject matter for philosophic inquiry, the Polish logician Stanislow Lesniewski [15] is generally given credit for the first formal treatment of the subject matter in his Mereology.1 Woodger [30] and Tarski [24] made use of a specific adaptation of Lesniewski's work as a basis for a formal theory of physical things and their parts. The term 'calculus of individuals' was (...)
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  • Discussion. Applied Constructive Mathematics: On Hellman's 'Mathematical Constructivism in Spacetime'.H. Billinge - 2000 - British Journal for the Philosophy of Science 51 (2):299-318.
    claims that constructive mathematics is inadequate for spacetime physics and hence that constructive mathematics cannot be considered as an alternative to classical mathematics. He also argues that the contructivist must be guilty of a form of a priorism unless she adopts a strong form of anti-realism for science. Here I want to dispute both claims. First, even if there are non-constructive results in physics this does not show that adequate constructive alternatives could not be formulated. Secondly, the constructivist adopts a (...)
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  • A Constructive Formulation of Gleason's Theorem.Helen Billinge - 1997 - Journal of Philosophical Logic 26 (6):661-670.
    In this paper I wish to show that we can give a statement of a restricted form of Gleason's Theorem that is classically equivalent to the standard formulation, but that avoids the counterexample that Hellman gives in "Gleason's Theorem is not Constructively Provable".
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  • Infinity: An Essay in Metaphysics.Jose Benardete - 1964 - Clarendon Press.
  • Intellectual Autobiography.Rudolf Carnap - 1963 - In Paul Arthur Schilpp (ed.), The Philosophy of Rudolf Carnap. LaSalle, Illinois: Open Court. pp. 3--84.
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  • Testing Scientific Theories.John Earman (ed.) - 1983 - Minneapolis: University of Minnesota Press.
    Rich with historical and cultural value, these works are published unaltered from the original University of Minnesota Press editions.
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  • Two Dogmas of Empiricism.W. Quine - 1951 - [Longmans, Green].
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  • Inference to the Best Explanation.Peter Lipton - 1991 - London and New York: Routledge.
    "How do we go about weighing evidence, testing hypotheses and making inferences? According to the model of 'inference to the Best explanation', we work out what to inter from the evidence by thinking about what would actually explain that evidence, and we take the ability of a hypothesis to explain the evidence as a sign that the hypothesis is correct. In inference to the Best Explanation, Peter Lipton gives this important and influential idea the development and assessment it deserves." "The (...)
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  • A Primer of Infinitesimal Analysis.John L. Bell - 1998 - Cambridge University Press.
    This is the first elementary book to employ the concept of infinitesimals.
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  • Inference to the Best Explanation.Peter Lipton - 1991 - London and New York: Routledge.
    How do we go about weighing evidence, testing hypotheses, and making inferences? According to the model of _Inference to the Best Explanation_, we work out what to infer from the evidence by thinking about what would actually explain that evidence, and we take the ability of a hypothesis to explain the evidence as a sign that the hypothesis is correct. In _Inference to the Best Explanation_, Peter Lipton gives this important and influential idea the development and assessment it deserves. The (...)
  • Philosophy of Natural Science.Carl Gustav Hempel - 1966 - Englewood Cliffs, N.J., Prentice-Hall.
  • Elements of Intuitionism.Michael Dummett - 1977 - Oxford University Press.
    This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics, for example Brouwer's proof of the Bar Theorem, valuation systems, and the completeness of intuitionistic first-order logic, have been completely revised.
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  • Varieties of Constructive Mathematics.D. S. Bridges - 1987 - Cambridge University Press.
    This is an introduction to, and survey of, the constructive approaches to pure mathematics. The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism. Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate. Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists. This account for non-specialists in these and other disciplines.
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  • Parts, Wholes, and Part-Whole Relations: The Prospects of Mereotopology.Achille C. Varzi - 1996 - Data and Knowledge Engineering 20:259–286.
    We can see mereology as a theory of parthood and topology as a theory of wholeness. How can these be combined to obtain a unified theory of parts and wholes? This paper examines various non-equivalent ways of pursuing this task, with specific reference to its relevance to spatio-temporal reasoning. In particular, three main strategies are compared: (i) mereology and topology as two independent (though mutually related) chapters; (ii) mereology as a general theory subsuming topology; (iii) topology as a general theory (...)
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  • Infinity, an essay in metaphysics. [REVIEW]R. BlanchĂ© - 1964 - Revue Philosophique de la France Et de l'Etranger 156:502-503.
     
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  • What is Cantor's Continuum Problem?Kurt Gödel - 1983 - In Paul Benacerraf & Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings (2nd Edition). Cambridge: Cambridge University Press. pp. 470-485.
  • Theory and Evidence.Clark Glymour - 1981 - Philosophy of Science 48 (3):498-500.
     
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  • Philosophy of Natural Science.Carl G. Hempel - 1967 - British Journal for the Philosophy of Science 18 (1):70-72.
     
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  • Theory and Evidence.Clark Glymour - 1982 - Erkenntnis 18 (1):105-130.
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  • Scientific Explanation.P. Kitcher & W. C. Salmon - 1992 - British Journal for the Philosophy of Science 43 (1):85-98.
     
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  • Theory and Evidence.Clark Glymour - 1980 - Ethics 93 (3):613-615.
     
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