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  1. T. Achourioti & M. van Lambalgen (2011). A Formalization of Kant's Transcendental Logic. Review of Symbolic Logic 4 (2):254-289.
    Although Kant (1998) envisaged a prominent role for logic in the argumentative structure of his Critique of Pure Reason, logicians and philosophers have generally judged Kantgeneralformaltranscendental logics is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first-order logic. The main technical application of the formalism developed here is a formal proof that Kants logic is after all a distinguished subsystem of first-order logic, namely what (...)
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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. Mohammad Ardeshir & Rasoul Ramezanian (2012). A Solution to the Surprise Exam Paradox in Constructive Mathematics. Review of Symbolic Logic 5 (4):679-686.
    We represent the well-known surprise exam paradox in constructive and computable mathematics and offer solutions. One solution is based on Brouwer’s continuity principle in constructive mathematics, and the other involves type 2 Turing computability in classical mathematics. We also discuss the backward induction paradox for extensive form games in constructive logic.
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  5. Mohammad Ardeshir & Rasoul Ramezanian (2012). On the Constructive Notion of Closure Maps. Mathematical Logic Quarterly 58 (4‐5):348-355.
    Let A be a subset of the constructive real line. What are the necessary and sufficient conditions for the set A such that A is continuously separated from other reals, i.e., there exists a continuous function f with f−1 = A? In this paper, we study the notions of closed sets and closure maps in constructive reverse mathematics.
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  6. 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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  7. Steve Awodey (2013). Structuralism, Invariance, and Univalence. Philosophia Mathematica 22 (1):nkt030.
    The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...)
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  8. Michael Beeson (1978). Some Relations Between Classical and Constructive Mathematics. Journal of Symbolic Logic 43 (2):228-246.
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  9. Michael J. Beeson (1977). Principles of Continuous Choice and Continuity of Functions in Formal Systems for Constructive Mathematics. Annals of Mathematical Logic 12 (3):249-322.
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  10. J. L. Bell (2013). Review of M. Van Atten, P. Boldini, M. Bourdeau, and G. Heinzmann (Eds.), _One Hundred Years of Intuitionism (1907–2007): The Cerisy Conference. [REVIEW] Philosophia Mathematica 21 (3):392-399.
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  11. S. Berardi (2004). A Generalization of Conservativity Theorem for Classical Versus Intuitionistic Arithmetic. Mathematical Logic Quarterly 50 (1):41.
    A basic result in intuitionism is Π02-conservativity. Take any proof p in classical arithmetic of some Π02-statement , with P decidable). Then we may effectively turn p in some intuitionistic proof of the same statement. In a previous paper [1], we generalized this result: any classical proof p of an arithmetical statement ∀x.∃y.P, with P of degree k, may be effectively turned into some proof of the same statement, using Excluded Middle only over degree k formulas. When k = 0, (...)
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  12. 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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  13. 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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  14. 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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  15. Michel J. Blais (1989). A Pragmatic Analysis of Mathematical Realism and Intuitionism. Philosophia Mathematica (1):61-85.
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  16. 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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  17. D. Bridges & R. Havea (2001). A Constructive Version of the Spectral Mapping Theorem. Mathematical Logic Quarterly 47 (3):299-304.
    The spectral mapping theorem in a unital Banach algebra is examined for its constructive content.
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  18. Douglas Bridges, Constructive Mathematics. Stanford Encyclopedia of Philosophy.
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  19. Douglas S. Bridges (2009). Constructive Notions of Equicontinuity. Archive for Mathematical Logic 48 (5):437-448.
    In the informal setting of Bishop-style constructive reverse mathematics we discuss the connection between the antithesis of Specker’s theorem, Ishihara’s principle BD-N, and various types of equicontinuity. In particular, we prove that the implication from pointwise equicontinuity to uniform sequential equicontinuity is equivalent to the antithesis of Specker’s theorem; and that, for a family of functions on a separable metric space, the implication from uniform sequential equicontinuity to uniform equicontinuity is equivalent to BD-N.
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  20. Douglas S. Bridges (1999). Can Constructive Mathematics Be Applied in Physics? Journal of Philosophical Logic 28 (5):439-453.
    The nature of modern constructive mathematics, and its applications, actual and potential, to classical and quantum physics, are discussed.
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  21. 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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  22. Douglas S. Bridges & Hannes Diener (2010). The Anti-Specker Property, Positivity, and Total Boundedness. Mathematical Logic Quarterly 56 (4):434-441.
    Working within Bishop-style constructive mathematics, we examine some of the consequences of the anti-Specker property, known to be equivalent to a version of Brouwer's fan theorem. The work is a contribution to constructive reverse mathematics.
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  23. Douglas S. Bridges & Hajime Ishihara (1994). Complements of Intersections in Constructive Mathematics. Mathematical Logic Quarterly 40 (1):35-43.
    We examine, from a constructive perspective, the relation between the complements of S, T, and S ∩ T in X, where X is either a metric space or a normed linear space. The fundamental question addressed is: If x is distinct from each element of S ∩ T, if s ϵ S, and if t ϵ T, is x distinct from s or from t? Although the classical answer to this question is trivially affirmative, constructive answers involve Markov's principle and (...)
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  24. Douglas S. Bridges & Iris Loeb (2010). Glueing Continuous Functions Constructively. Archive for Mathematical Logic 49 (5):603-616.
    The glueing of (sequentially, pointwise, or uniformly) continuous functions that coincide on the intersection of their closed domains is examined in the light of Bishop-style constructive analysis. This requires us to pay attention to the way that the two domains intersect.
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  25. Douglas Bridges & Ayan Mahalanobis (2000). Sequential Continuity of Functions in Constructive Analysis. Mathematical Logic Quarterly 46 (1):139-143.
    It is shown that in any model of constructive mathematics in which a certain omniscience principle is false, for strongly extensional functions on an interval the distinction between sequentially continuous and regulated disappears. It follows, without the use of Markov's Principle, that any recursive function of bounded variation on a bounded closed interval is recursively sequentially continuous.
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  26. Douglas Bridges & Steeve Reeves (1999). Constructive Mathematics in Theory and Programming Practice. Philosophia Mathematica 7 (1):65-104.
    The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). it gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on Martin-L6f 's theory of types as a formal system for BISH.
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  27. L. E. J. Brouwer (1981). Brouwer's Cambridge Lectures on Intuitionism. Cambridge University Press.
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  28. James Robert Brown (2003). Science and Constructive Mathematics. Analysis 63 (1):48–51.
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  29. 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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  30. Francesco Ciraulo (2008). A Constructive Semantics for Non-Deducibility. Mathematical Logic Quarterly 54 (1):35-48.
    This paper provides a constructive topological semantics for non-deducibility of a first order intuitionistic formula. Formal topology theory, in particular the recently introduced notion of a binary positivity predicate, and co-induction are two needful tools.
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  31. L. Schuster Crosilla & Jaap van Oosten (2006). REVIEWS-From Sets and Types to Topology and Analysis--Towards Practicable Foundations for Constructive Mathematics. Bulletin of Symbolic Logic 12 (4):611-612.
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  32. 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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  33. Michael E. Cuffaro (2012). Kant's Views on Non-Euclidean Geometry. Proceedings of the Canadian Society for History and Philosophy of Mathematics 25:42-54.
    Kant's arguments for the synthetic a priori status of geometry are generally taken to have been refuted by the development of non-Euclidean geometries. Recently, however, some philosophers have argued that, on the contrary, the development of non-Euclidean geometry has confirmed Kant's views, for since a demonstration of the consistency of non-Euclidean geometry depends on a demonstration of its equi-consistency with Euclidean geometry, one need only show that the axioms of Euclidean geometry have 'intuitive content' in order to show that both (...)
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  34. Dirk Dalen (1978). Brouwer: The Genesis of His Intuitionism. Dialectica 32 (3‐4):291-303.
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  35. Dirk Van Dalen (1995). Hermann Weyl's Intuitionistic Mathematics. Bulletin of Symbolic Logic 1 (2):145 - 169.
  36. 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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  37. 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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  38. 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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  39. 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.
  40. Michael Detlefsen (1995). Review of J. Folina, Poincare and the Philosophy of Mathematics. [REVIEW] Philosophia Mathematica 3 (2):208-218.
  41. 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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  42. Hannes Diener (2012). Reclassifying the Antithesis of Specker's Theorem. Archive for Mathematical Logic 51 (7-8):687-693.
    It is shown that a principle, which can be seen as a constructivised version of sequential compactness, is equivalent to a form of Brouwer’s fan theorem. The complexity of the latter depends on the geometry of the spaces involved in the former.
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  43. Ag Dragalin (1973). Constructive Mathematics and Models of Enturnonistic Theories. In Patrick Suppes (ed.), Logic, Methodology and Philosophy of Science. New York,American Elsevier Pub. Co.. 111.
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  44. Michael Dummett (1998). Truth From the Constructive Standpoint. Theoria 64 (2-3):122-138.
  45. 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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  46. Michael A. E. Dummett (1974). Intuitionistic Mathematics and Logic. Mathematical Institute.
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  47. William J. Edgar (1973). Is Intuitionism the Epistemically Serious Foundation for Mathematics? Philosophia Mathematica (2):113-133.
  48. Miklós Erdélyi‐Szabó (1997). Decidability in the Constructive Theory of Reals as an Ordered ℚ‐Vectorspace. Mathematical Logic Quarterly 43 (3):343-354.
    We show that various fragments of the intuitionistic/constructive theory of the reals are decidable.
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  49. I͡Uriĭ Leonidovich Ershov (2000). Constructive Models. Consultants Bureau.
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  50. Solomon Feferman, Relationships Between Constructive, Predicative and Classical Systems of Analysis.
    Both the constructive and predicative approaches to mathematics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unrestricted use of the law of excluded middle on the one hand, and of apparently circular \impredicative" de nitions on the other. But the positive redevelopment of mathematics along constructive, resp. predicative grounds did not emerge as really viable alternatives to (...)
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