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  1. T. Achourioti & M. van Lambalgen (forthcoming). A Formalisation of Kant's Transcendental Logic. Review of Symbolic Logic.
    Although Kant envisaged a prominent role for logic in the argumentative structure of his Critique of pure reason, logicians and philosophers have generally judged Kant's logic negatively. What Kant called `general' or `formal' logic has been dismissed as a fairly arbitrary subsystem of first order logic, and what he called `transcendental logic' is considered to be not a logic at all: no syntax, no semantics, no definition of validity. Against this, we argue that Kant's `transcendental logic' is a logic in (...)
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  2. Alice Ambrose (1933). A Controversy in the Logic of Mathematics. Philosophical Review 42 (6):594-611.
  3. Michael A. Arbib (1990). A Piagetian Perspective on Mathematical Construction. Synthese 84 (1):43 - 58.
    In this paper, we offer a Piagetian perspective on the construction of the logico-mathematical schemas which embody our knowledge of logic and mathematics. Logico-mathematical entities are tied to the subject's activities, yet are so constructed by reflective abstraction that they result from sensorimotor experience only via the construction of intermediate schemas of increasing abstraction. The axiom set does not exhaust the cognitive structure (schema network) which the mathematician thus acquires. We thus view truth not as something to be defined within (...)
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  4. Jeremy Avigad & Jeffrey Helzner (2002). Transfer Principles in Nonstandard Intuitionistic Arithmetic. Archive for Mathematical Logic 41 (6):581-602.
    Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these rules destroy (...)
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  5. H. Billinge (2000). Applied Constructive Mathematics: On Hellman's 'Mathematical Constructivism in Spacetime'. 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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  6. Helen Billinge (2003). Did Bishop Have a Philosophy of Mathematics? Philosophia Mathematica 11 (2):176-194.
    When Bishop published Foundations of Constructive Analysis he showed that it was possible to do ordinary analysis within a constructive framework. Bishop's reasons for doing his mathematics constructively are explicitly philosophical. In this paper, I will expound, examine, and amplify his philosophical arguments for constructivism in mathematics. In the end, however, I argue that Bishop's philosophical comments cannot be rounded out into an adequate philosophy of constructive mathematics.
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  7. Anthony Birch (2007). Waismann's Critique of Wittgenstein. Analysis and Metaphysics 6 (2007):263-272.
    Friedrich Waismann, a little-known mathematician and onetime student of Wittgenstein's, provides answers to problems that vexed Wittgenstein in his attempt to explicate the foundations of mathematics through an analysis of its practice. Waismann argues in favor of mathematical intuition and the reality of infinity with a Wittgensteinian twist. Waismann's arguments lead toward an approach to the foundation of mathematics that takes into consideration the language and practice of experts.
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  8. Michel J. Blais (1989). A Pragmatic Analysis of Mathematical Realism and Intuitionism. Philosophia Mathematica (1):61-85.
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  9. D. S. Bridges (1987). Varieties of Constructive Mathematics. 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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  10. Douglas S. Bridges (1995). Constructive Mathematics and Unbounded Operators — a Reply to Hellman. 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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  11. L. E. J. Brouwer (1981). Brouwer's Cambridge Lectures on Intuitionism. Cambridge University Press.
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  12. James Robert Brown (2003). Science and Constructive Mathematics. Analysis 63 (1):48–51.
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  13. P. Cariani (2012). Infinity and the Observer: Radical Constructivism and the Foundations of Mathematics. Constructivist Foundations 7 (2):116-125.
    Problem: There is currently a great deal of mysticism, uncritical hype, and blind adulation of imaginary mathematical and physical entities in popular culture. We seek to explore what a radical constructivist perspective on mathematical entities might entail, and to draw out the implications of this perspective for how we think about the nature of mathematical entities. Method: Conceptual analysis. Results: If we want to avoid the introduction of entities that are ill-defined and inaccessible to verification, then formal systems need to (...)
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  14. Laura Crosilla & Peter Schuster (eds.) (2005). From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics. Oxford University Press.
    This edited collection bridges the foundations and practice of constructive mathematics and focuses on the contrast between the theoretical developments, which have been most useful for computer science (ie: constructive set and type theories), and more specific efforts on constructive analysis, algebra and topology. Aimed at academic logician, mathematicians, philosophers and computer scientists with contributions from leading researchers, it is up to date, highly topical and broad in scope.
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  15. E. B. Davies (2005). A Defence of Mathematical Pluralism. Philosophia Mathematica 13 (3):252-276.
    We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context.
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  16. Michael de (2013). Empirical Negation. Acta Analytica 28 (1):49-69.
    An extension of intuitionism to empirical discourse, a project most seriously taken up by Dummett and Tennant, requires an empirical negation whose strength lies somewhere between classical negation (‘It is unwarranted that. . . ’) and intuitionistic negation (‘It is refutable that. . . ’). I put forward one plausible candidate that compares favorably to some others that have been propounded in the literature. A tableau calculus is presented and shown to be strongly complete.
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  17. David Dedivi (2004). Choice Principles and Constructive Logics. Philosophia Mathematica 12 (3):222-243.
    to constructive systems is significant for contemporary metaphysics. However, many are surprised by these results, having learned that the Axiom of Choice (AC) is constructively valid. Indeed, even among specialists there were, until recently, reasons for puzzlement-rival versions of Intuitionistic Type Theory, one where (AC) is valid, another where it implies classical logic. This paper accessibly explains the situation, puts the issues in a broader setting by considering other choice principles, and draws philosophical morals for the understanding of quantification, choice (...)
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  18. M. Detlefsen (1998). Walter van Stigt. Brouwer's Intuitionism. Amsterdam: North-Holland Publishing Co., 1990. Pp. Xxvi + 530. ISBN 0-444-88384-3 (Cloth). [REVIEW] Philosophia Mathematica 6 (2):235-241.
  19. Michael Detlefsen (1995). Wright on the Non-Mechanizability of Intuitionist Reasoning. Philosophia Mathematica 3 (1):103-119.
    Crispin Wright joins the ranks of those who have sought to refute mechanist theories of mind by invoking Gödel's Incompleteness Theorems. His predecessors include Gödel himself, J. R. Lucas and, most recently, Roger Penrose. The aim of this essay is to show that, like his predecessors, Wright, too, fails to make his case, and that, indeed, he fails to do so even when judged by standards of success which he himself lays down.
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  20. Michael Dummett (1998). Truth From the Constructive Standpoint. Theoria 64 (2-3):122-138.
  21. Michael A. E. Dummett (2000). Elements of Intuitionism. Oxford University Press.
    This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an informal but thorough introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics has been completely revised for this second edition. Brouwer's proof of the Bar Theorem has been reworked, the account of valuation systems simplified, and the treatment of generalized Beth Trees and the completeness of intuitionistic first-order logic rewritten. Readers (...)
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  22. Michael A. E. Dummett (1974). Intuitionistic Mathematics and Logic. Mathematical Institute.
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  23. William J. Edgar (1973). Is Intuitionism the Epistemically Serious Foundation for Mathematics? Philosophia Mathematica (2):113-133.
  24. Solomon Feferman (2008). Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on Finitism, Constructivity and Hilbert's Program. Dialectica 62 (2: Table of Contents"/> Select):179–203.
    This is a survey of Gödel's perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert's program, using his published and unpublished articles and lectures as well as the correspondence between Bernays and Gödel on these matters. There is also an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end.
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  25. Eduardo L. Fermé & Sven Ove Hansson (1999). Selective Revision. Studia Logica 63 (3):331-342.
    We introduce a constructive model of selective belief revision in which it is possible to accept only a part of the input information. A selective revision operator ο is defined by the equality K ο α = K * f(α), where * is an AGM revision operator and f a function, typically with the property ⊢ α → f(α). Axiomatic characterizations are provided for three variants of selective revision.
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  26. Peter Fletcher (2002). A Constructivist Perspective on Physics. Philosophia Mathematica 10 (1):26-42.
    This paper examines the problem of extending the programme of mathematical constructivism to applied mathematics. I am not concerned with the question of whether conventional mathematical physics makes essential use of the principle of excluded middle, but rather with the more fundamental question of whether the concept of physical infinity is constructively intelligible. I consider two kinds of physical infinity: a countably infinite constellation of stars and the infinitely divisible space-time continuum. I argue (contrary to Hellman) that these do not. (...)
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  27. Miriam Franchella (2008). Mark Van Atten. Brouwer Meets Husserl: On the Phenomenology of Choice Sequences. Philosophia Mathematica 16 (2):276-281.
  28. Yvon Gauthier, Constructive Truth and Certainty in Logic and Mathematics.
    The theme « Truth and Certainty » is reminiscent of Hegel’s dialectic of prominent in the Phänomenologie des Geistes, but I want to treat it from a different angle in the perspective of the constructivist stance in the foundations of logic and mathematics. Although constructivism stands in opposition to mathematical realism, it is not to be considered as an idealist alternative in the philosophy of mathematics. It is true that Brouwer’s intuitionism, as a variety of constructivism, (...)
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  29. D. A. Gillies (1980). Brouwer's Philosophy of Mathematics: Review of L. E. J. Brouwer (A. Heyting and H. Freudenthal Eds.), Collected Works. [REVIEW] Erkenntnis 15 (1):105 - 126.
  30. Johan Georg Granström (2011). Treatise on Intuitionistic Type Theory. Springer.
    Prolegomena It is fitting to begin this book on intuitionistic type theory by putting the subject matter into perspective. The purpose of this chapter is to ...
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  31. R. G. Heck (2013). Sir Michael Anthony Eardley Dummett, 1925-2011. Philosophia Mathematica 21 (1):1-8.
    A remembrance of Dummett's work on philosophy of mathematcis.
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  32. Gerhard Heinzmann & Giuseppina Ronzitti (eds.) (2006). Constructivism: Mathematics, Logic, Philosophy and Linguistics.
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  33. Geoffrey Hellman (2006). Pluralism and the Foundations of Mathematics. In ¸ Itekellersetal:Sp.
    A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose problematic relationship is discussed. The first case illustrates the hypothesis that a sufficiently rich subject matter may require a multiplicity of approaches. The second case, while in some respects special to mathematics, raises issues of (...)
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  34. Geoffrey Hellman (1998). Mathematical Constructivism in Spacetime. British Journal for the Philosophy of Science 49 (3):425-450.
    To what extent can constructive mathematics based on intuitionistc logic recover the mathematics needed for spacetime physics? Certain aspects of this important question are examined, both technical and philosophical. On the technical side, order, connectivity, and extremization properties of the continuum are reviewed, and attention is called to certain striking results concerning causal structure in General Relativity Theory, in particular the singularity theorems of Hawking and Penrose. As they stand, these results appear to elude constructivization. On the philosophical side, it (...)
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  35. A. Heyting (1971). Intuitionism. Amsterdam,North-Holland Pub. Co..
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  36. A. Heyting (1955). G. F. C. Griss and His Negationless Intuitionistic Mathematics. Synthese 9 (1):91 - 96.
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  37. Arend Heyting (1974). Intuitionistic Views on the Nature of Mathematics. Synthese 27 (1-2):79 - 91.
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  38. B. Kerkhove & J. P. Bendegem (2012). The Many Faces of Mathematical Constructivism. Constructivist Foundations 7 (2):97-103.
    Context: As one of the major approaches within the philosophy of mathematics, constructivism is to be contrasted with realist approaches such as Platonism in that it takes human mental activity as the basis of mathematical content. Problem: Mathematical constructivism is mostly identified as one of the so-called foundationalist accounts internal to mathematics. Other perspectives are possible, however. Results: The notion of “meaning finitism” is exploited to tie together internal and external directions within mathematical constructivism. The various contributions to this issue (...)
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  39. Charles F. Kielkopf (1995). ‘Surveyablity’ Should Not Be Formalized. Philosophia Mathematica 3 (2):175-178.
    There is a review of how Mark Addis has made a case that it would require great effort for scant philosophical profit to formalize a notion of surveyability as a metamathematical predicate demarcating strict finitistic mathematics. It is then suggested how the notion of surveyability is useful in informal philosophizing about mathematics.
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  40. Charles F. Kielkopf (1970). Strict Finitism. The Hague,Mouton.
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  41. Stephen Cole Kleene (1965). The Foundations of Intuitionistic Mathematics. Amsterdam, North-Holland Pub. Co..
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  42. Hugh Lehman (1983). Intuitionism and Platonism on Infinite Totalities. Idealistic Studies 13 (3):190-198.
  43. M. Lievers (2004). Critical Studies / Book Reviews. [REVIEW] Philosophia Mathematica 12 (2):176-186.
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  44. Sten Lindström & Erik Palmgren (2009). Introduction: The Three Foundational Programmes. In Sten Lindström, Erik Palmgren, Krister Segerberg & Viggo Stoltenberg-Hansen (eds.), Logicism, Intuitionism and Formalism: What has become of them? Springer.
  45. Sten Lindström, Erik Palmgren, Krister Segerberg & Viggo Stoltenberg-Hansen (eds.) (2009). Logicism, Intuitionism, and Formalism - What has Become of Them? Springer.
    These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics.A special section is concerned with constructive ...
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  46. I. Loeb (2012). Questioning Constructive Reverse Mathematics. Constructivist Foundations 7 (2):131-140.
    Context: It is often suggested that the methodology of the programme of Constructive Reverse Mathematics (CRM) can be sufficiently clarified by a thorough understanding of Brouwer’s intuitionism, Bishop’s constructive mathematics, and classical Reverse Mathematics. In this paper, the correctness of this suggestion is questioned. Method: We consider the notion of a mathematical programme in order to compare these schools of mathematics in respect of their methodologies. Results: Brouwer’s intuitionism, Bishop’s constructive mathematics, and classical Reverse Mathematics are historical influences upon the (...)
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  47. Ofra Magidor (2012). Strict Finitism and the Happy Sorites. Journal of Philosophical Logic 41 (2):471-491.
    Call an argument a ‘happy sorites’ if it is a sorites argument with true premises and a false conclusion. It is a striking fact that although most philosophers working on the sorites paradox find it at prima facie highly compelling that the premises of the sorites paradox are true and its conclusion false, few (if any) of the standard theories on the issue ultimately allow for happy sorites arguments. There is one philosophical view, however, that appears to allow for at (...)
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  48. Ofra Magidor (2007). Strict Finitism Refuted? Proceedings of the Aristotelian Society 107 (1pt3):403-411.
    In his paper ‘Wang’s Paradox’, Michael Dummett provides an argument for why strict finitism in mathematics is internally inconsistent and therefore an untenable position. Dummett’s argument proceeds by making two claims: (1) Strict finitism is committed to the claim that there are sets of natural numbers which are closed under the successor operation but nonetheless have an upper bound; (2) Such a commitment is inconsistent, even by finitistic standards. -/- In this paper I claim that Dummett’s argument fails. I question (...)
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  49. K. Mainzer (1972). Mathematischer Konstruktivismus Im Lichte-Kantischer Philosophie. Philosophia Mathematica (1):3-26.
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  50. Per Martin-Löf (1970). Notes on Constructive Mathematics. Stockholm,Almqvist & Wiksell.
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  51. Charles McCarty (2008). Intuitionism and Logical Syntax. Philosophia Mathematica 16 (1):56-77.
    , Rudolf Carnap became a chief proponent of the doctrine that the statements of intuitionism carry nonstandard intuitionistic meanings. This doctrine is linked to Carnap's ‘Principle of Tolerance’ and claims he made on behalf of his notion of pure syntax. From premises independent of intuitionism, we argue that the doctrine, the Principle, and the attendant claims are mistaken, especially Carnap's repeated insistence that, in defining languages, logicians are free of commitment to mathematical statements intuitionists would reject. I am grateful to (...)
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  52. Grigori Mints (2006). Notes on Constructive Negation. Synthese 148 (3):701 - 717.
    We put together several observations on constructive negation. First, Russell anticipated intuitionistic logic by clearly distinguishing propositional principles implying the law of the excluded middle from remaining valid principles. He stated what was later called Peirce’s law. This is important in connection with the method used later by Heyting for developing his axiomatization of intuitionistic logic. Second, a work by Dragalin and his students provides easy embeddings of classical arithmetic and analysis into intuitionistic negationless systems. In the last section, we (...)
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  53. I. Moerdijk (1998). Sets, Topoi and Intuitionism. Philosophia Mathematica 6 (2):169-177.
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  54. Charles Parsons (1990). On Constructive Interpretation of Predicative Mathematics. Garland Pub..
  55. Michael Potter (1998). Classical Arithmetic as Part of Intuitionistic Arithmetic. Grazer Philosophische Studien 55:127-41.
    Argues that classical arithmetic can be viewed as a proper part of intuitionistic arithmetic. Suggests that this largely neutralizes Dummett's argument for intuitionism in the case of arithmetic.
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  56. B. Pourciau (2000). Intuitionism as a (Failed) Kuhnian Revolution in Mathematics. Studies in History and Philosophy of Science Part A 31 (2):297-329.
    In this paper it is argued, firstly, that Kuhnian revolutions in mathematics are logically possible, in the sense of not being inconsistent with the nature of mathematics; and, secondly, that Kuhnian revolutions are actually possible, in the sense that a Kuhnian paradigm for mathematics can be exhibited which would, if accepted by the mathematical community, produce a full Kuhnian revolution. These two arguments depend on first proving that a shift from a classical conception of mathematics to an intuitionist conception would (...)
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  57. A. Quale (2012). On the Role of Constructivism in Mathematical Epistemology. Constructivist Foundations 7 (2):104-111.
    Context: the position of pure and applied mathematics in the epistemic conflict between realism and relativism. Problem: To investigate the change in the status of mathematical knowledge over historical time: specifically, the shift from a realist epistemology to a relativist epistemology. Method: Two examples are discussed: geometry and number theory. It is demonstrated how the initially realist epistemic framework – with mathematics situated in a platonic ideal reality from where it governs our physical world – became untenable, with the advent (...)
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  58. F. Richman (1998). Review of R. Hersh, What is Mathematics, Really?. Philosophia Mathematica 6 (2):245-255.
  59. Fred Richman (2000). Review of P. Fletcher, Truth, Proof and Infinity: A Theory of Constructive Reasoning. Philosophia Mathematica 8 (2):214-220.
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  60. Fred Richman (1994). Review of A. S. Troelstra and D. Van Dalen, Constructivism in Mathematics: An Introduction. [REVIEW] Philosophia Mathematica 2 (1).
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  61. Fred Richman (ed.) (1981). Constructive Mathematics: Proceedings of the New Mexico State University Conference Held at Las Cruces, New Mexico, August 11-15, 1980. [REVIEW] Springer-Verlag.
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  62. Giuseppina Ronzitti (2004). On Some Difficulties Concerning the Definition of an Intuitionistic Concept of Countable Set. In Libor Behounek (ed.), Logica Yearbook 2003.
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  63. Dirk Schlimm (2005). Against Against Intuitionism. Synthese 147 (1):171 - 188.
    The main ideas behind Brouwer’s philosophy of Intuitionism are presented. Then some critical remarks against Intuitionism made by William Tait in “Against Intuitionism” [Journal of Philosophical Logic, 12, 173–195] are answered.
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  64. Andrzej Sendlewski (1995). Axiomatic Extensions of the Constructive Logic with Strong Negation and the Disjunction Property. Studia Logica 55 (3):377 - 388.
    We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B V, thenA×B is a homomorphic image of some well-connected algebra ofV.We prove:• each varietyV of Nelson algebras with PQWC lies in the fibre –1(W) for some varietyW of Heyting algebras having PQWC, • for any varietyW of Heyting algebras with PQWC the least and the (...)
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  65. Stewart Shapiro & William W. Taschek (1996). ``Intuitionism, Pluralism, and Cognitive Command". Journal of Philosophy 20 (2):74-88.
  66. Sanford Shieh (1998). Undecidability in Anti-Realism. Philosophia Mathematica 6 (3):324-333.
    In this paper I attempt to clarify a relatively little-studied aspect of Michael Dummett's argument for intuitionism: its use of the notion of ‘undecidable’ sentence. I give a new analysis of this concept in epistemic terms, with which I resolve some puzzles and questions about how it works in the anti-realist critique of classical logic.
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  67. A. O. Slisenko (ed.) (1969). Studies in Constructive Mathematics and Mathematical Logic. New York, Consultants Bureau.
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  68. Ivahn Smadja (2010). How Discrete Patterns Emerge From Algorithmic Fine-Tuning: A Visual Plea for Kroneckerian Finitism. Topoi 29 (1):61-75.
    This paper sets out to adduce visual evidence for Kroneckerian finitism by making perspicuous some of the insights that buttress Kronecker’s conception of arithmetization as a process aiming at disclosing the arithmetical essence enshrined in analytical formulas, by spotting discrete patterns through algorithmic fine-tuning. In the light of a fairly tractable case study, it is argued that Kronecker’s main tenet in philosophy of mathematics is not so much an ontological as a methodological one, inasmuch as highly demanding requirements regarding mathematical (...)
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  69. B. G. Sundholm (1998). Inference, Consequence, Implication: A Constructivist's Perspective. Philosophia Mathematica 6 (2):178-194.
    An implication is a proposition, a consequence is a relation between propositions, and an inference is act of passage from certain premise-judgements to another conclusion-judgement: a proposition is true, a consequence holds, whereas an inference is valid. The paper examines interrelations, differences, refinements and linguistic renderings of these notions, as well as their history. The truth of propositions, respectively the holding of consequences, are treated constructively in terms of verification-objects. The validity of an inference is elucidated in terms of the (...)
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  70. Nobu-Yuki Suzuki (2003). Halldén-Completeness in Super-Intuitionistic Predicate Logics. Studia Logica 73 (1):113 - 130.
    One criterion of constructive logics is the disjunction, property (DP). The Halldén-completeness is a weak DP, and is related to the relevance principle and variable separation. This concept is well-understood in the case of propositional logics. We extend this notion to predicate logics. Then three counterparts naturally arise. We discuss relationships between these properties and meet-irreducibility in the lattice of logics.
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  71. W. W. Tait (1983). Against Intuitionism: Constructive Mathematics is Part of Classical Mathematics. Journal of Philosophical Logic 12 (2):173 - 195.
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  72. William Tait (2006). Godel's Interpretation of Intuitionism. Philosophia Mathematica 14 (2):208-228.
    Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will (1) criticize this foundation, (2) propose a quite different one, and (3) note that essentially the latter foundation also underlies the Curry-Howard type theory, and hence Heyting's intuitionistic conception of logic. Thus the Dialectica interpretation (in (...)
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  73. R. Tieszen (1998). Perspectives on Intuitionism. Philosophia Mathematica 6 (2):129-130.
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  74. Richard Tieszen (2000). The Philosophical Background of Weyl's Mathematical Constructivism. Philosophia Mathematica 8 (3):274-301.
    Weyl's inclination toward constructivism in the foundations of mathematics runs through his entire career, starting with Das Kontinuum. Why was Weyl inclined toward constructivism? I argue that Weyl's general views on foundations were shaped by a type of transcendental idealism in which it is held that mathematical knowledge must be founded on intuition. Kant and Fichte had an impact on Weyl but HusserFs transcendental idealism was even more influential. I discuss Weyl's views on vicious circularity, existence claims, meaning, the continuum (...)
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  75. A. S. Troelstra (1998). Concepts and Axioms. Philosophia Mathematica 6 (2):195-208.
    The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic logic in connection with a ‘theory of meaning’.
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  76. A. S. Troelstra (1988). Constructivism in Mathematics: An Introduction. Sole Distributors for the U.S.A. And Canada, Elsevier Science Pub. Co..
    Provability, Computability and Reflection.
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  77. A. S. Troelstra (1977). Choice Sequences: A Chapter of Intuitionistic Mathematics. Clarendon Press.
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  78. A. S. Troelstra (1975). Axioms for Intuitionistic Mathematics Incompatible with Classical Logic. Mathematisch Instituut.
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  79. A. S. Troelstra (1973). Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. New York,Springer.
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  80. Eric P. Tsui-James (1998). Dummett, Brouwer and the Metaphysics of Mathematics. Grazer Philosophische Studien 55:143-168.
    Although Brouwer is well known for his Intuitionistic philosophy of mathematics, a constructivist philosophy which calls for restricted use of certain logical principles, there is much less awareness of the well-developed metaphysical basis which underlies those restrictions. In the first half of this paper I outline a basic interpretation of Brouwer's metaphysics, and then in the second half consider the compatibility of that metaphysics with Dummett's argument for a principled non-metaphysical approach to intuitionism. I conclude that once the variously misleading (...)
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  81. John Tucker (1969). An Outline of a New Programme for the Foundations of Mathematics. Philosophia Mathematica (1-2):28-37.
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  82. Mark van Atten (2003). Brouwer, as Never Read by Husserl. Synthese 137 (1-2):3-19.
    Even though Husserl and Brouwer have never discussed each other's work, ideas from Husserl have been used to justify Brouwer's intuitionistic logic. I claim that a Husserlian reading of Brouwer can also serve to justify the existence of choice sequences as objects of pure mathematics. An outline of such a reading is given, and some objections are discussed.
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  83. Mark van Atten, Dirk van Dalen & And Richard Tieszen (2002). Brouwer and Weyl: The Phenomenology and Mathematics of the Intuitive Continuumt. Philosophia Mathematica 10 (2):203-226.
    Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception of (...)
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  84. Markus Sebastiaan Paul Rogier van Atten (2007). Brouwer Meets Husserl: On the Phenomenology of Choice Sequences. Springer.
    Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? Mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice sequence. But (...)
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  85. D. van Dalen (1998). From a Brouwerian Point of View. Philosophia Mathematica 6 (2):209-226.
    We discuss a number of topics that are central in Brouwer's intuitionism. A complete treatment is beyond the scope of the paper, the reader may find it a useful introduction to Brouwer's papers.
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  86. Frank Waaldijk (2005). On the Foundations of Constructive Mathematics – Especially in Relation to the Theory of Continuous Functions. Foundations of Science 10 (3).
    We discuss the foundations of constructive mathematics, including recursive mathematics and intuitionism, in relation to classical mathematics. There are connections with the foundations of physics, due to the way in which the different branches of mathematics reflect reality. Many different axioms and their interrelationship are discussed. We show that there is a fundamental problem in BISH (Bishop’s school of constructive mathematics) with regard to its current definition of ‘continuous function’. This problem is closely related to the definition in BISH of (...)
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  87. Kai F. Wehmeier (1996). Classical and Intuitionistic Models of Arithmetic. Notre Dame Journal of Formal Logic 37 (3):452-461.
    Given a classical theory T, a Kripke model K for the language L of T is called T-normal or locally PA just in case the classical L-structure attached to each node of K is a classical model of T. Van Dalen, Mulder, Krabbe, and Visser showed that Kripke models of Heyting Arithmetic (HA) over finite frames are locally PA, and that Kripke models of HA over frames ordered like the natural numbers contain infinitely many PA-nodes. We show that Kripke models (...)
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  88. Dag Westerståhl (2004). Perspectives on the Dispute Between Intuitionistic and Classical Mathematics. In Christer Svennerlind (ed.), Ursus Philosophicus. Essays dedicated to Björn Haglund on his sixtieth birthday. Philosophical Communications.
    It is not unreasonable to think that the dispute between classical and intuitionistic mathematics might be unresolvable or 'faultless', in the sense of there being no objective way to settle it. If so, we would have a pretty case of relativism. In this note I argue, however, that there is in fact not even disagreement in any interesting sense, let alone a faultless one, in spite of appearances and claims to the contrary. A position I call classical pluralism is sketched, (...)
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  89. Crispin Wright (1995). Intuitionists Are Not (Turing) Machines. Philosophia Mathematica 3 (1):86-102.
    Lucas and Penrose have contended that, by displaying how any characterisation of arithmetical proof programmable into a machine allows of diagonalisation, generating a humanly recognisable proof which eludes that characterisation, Gödel's incompleteness theorem rules out any purely mechanical model of the human intellect. The main criticisms of this argument have been that the proof generated by diagonalisation (i) will not be humanly recognisable unless humans can grasp the specification of the object-system (Benacerraf); and (ii) counts as a proof only on (...)
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  90. Feng Ye (2000). Toward a Constructive Theory of Unbounded Linear Operators. Journal of Symbolic Logic 65 (1):357-370.
    We show that the following results in the classical theory of unbounded linear operators on Hilbert spaces can be proved within the framework of Bishop's constructive mathematics: the Kato-Rellich theorem, the spectral theorem, Stone's theorem, and the self-adjointness of the most common quantum mechanical operators, including the Hamiltonians of electro-magnetic fields with some general forms of potentials.
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