Search results for 'Constructive mathematics' (try it on Scholar)

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  1. Douglas S. Bridges (1999). Can Constructive Mathematics Be Applied in Physics? Journal of Philosophical Logic 28 (5):439-453.score: 90.0
    The nature of modern constructive mathematics, and its applications, actual and potential, to classical and quantum physics, are discussed.
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  2. D. S. Bridges (1987). Varieties of Constructive Mathematics. Cambridge University Press.score: 90.0
    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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  3. Laura Crosilla & Peter Schuster (eds.) (2005). From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics. Oxford University Press.score: 90.0
    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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  4. Douglas S. Bridges & Hajime Ishihara (1994). Complements of Intersections in Constructive Mathematics. Mathematical Logic Quarterly 40 (1):35-43.score: 76.0
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  5. Rudolf Taschner (2010). The Swap of Integral and Limit in Constructive Mathematics. Mathematical Logic Quarterly 56 (5):533-540.score: 76.0
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  6. Per Martin-Löf (1970). Notes on Constructive Mathematics. Stockholm,Almqvist & Wiksell.score: 75.0
     
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  7. 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.score: 75.0
  8. A. O. Slisenko (ed.) (1969). Studies in Constructive Mathematics and Mathematical Logic. New York, Consultants Bureau.score: 75.0
  9. Hajime Ishihara (1992). Continuity Properties in Constructive Mathematics. Journal of Symbolic Logic 57 (2):557-565.score: 63.0
    The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We deal with principles which are equivalent to the statements "every mapping is sequentially nondiscontinuous", "every sequentially nondiscontinuous mapping is sequentially continuous", and "every sequentially continuous mapping is continuous". As corollaries, we show that every mapping of a complete separable space is continuous in constructive recursive mathematics (the Kreisel-Lacombe-Schoenfield-Tsejtin theorem) and in intuitionism.
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  10. Frank Waaldijk (2005). On the Foundations of Constructive Mathematics – Especially in Relation to the Theory of Continuous Functions. Foundations of Science 10 (3):249-324.score: 60.0
    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 (...)
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  11. H. Billinge (2000). Applied Constructive Mathematics: On Hellman's 'Mathematical Constructivism in Spacetime'. British Journal for the Philosophy of Science 51 (2):299-318.score: 60.0
    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 (...)
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  12. Douglas Bridges & Steeve Reeves (1999). Constructive Mathematics in Theory and Programming Practice. Philosophia Mathematica 7 (1):65-104.score: 60.0
    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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  13. Maria Emilia Maietti & Giuseppe Rosolini (2013). Quotient Completion for the Foundation of Constructive Mathematics. Logica Universalis 7 (3):371-402.score: 60.0
    We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere hyperdoctrine for which we describe a notion of quotient completion. That notion includes the exact completion on a category with weak finite limits as an instance as well as examples from type theory that fall apart from this.
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  14. Mohammad Ardeshir & Rasoul Ramezanian (2012). A Solution to the Surprise Exam Paradox in Constructive Mathematics. Review of Symbolic Logic 5 (4):679-686.score: 60.0
    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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  15. Douglas S. Bridges (1995). Constructive Mathematics and Unbounded Operators — a Reply to Hellman. Journal of Philosophical Logic 24 (5):549 - 561.score: 60.0
    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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  16. Geoffrey Hellman (1997). Quantum Mechanical Unbounded Operators and Constructive Mathematics – a Rejoinder to Bridges. Journal of Philosophical Logic 26 (2):121-127.score: 57.0
    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 (...)
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  17. Richard L. Tieszen (2005). Phenomenology, Logic, and the Philosophy of Mathematics. Cambridge University Press.score: 51.0
    Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this book is divided into three parts. Part I, Reason, Science, and Mathematics contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay oN phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some (...)
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  18. Stewart Shapiro (ed.) (1985). Intentional Mathematics. Sole Distributors for the U.S.A. And Canada, Elsevier Science Pub. Co..score: 51.0
    Among the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.
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  19. I. Loeb (2012). Questioning Constructive Reverse Mathematics. Constructivist Foundations 7 (2):131-140.score: 51.0
    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 (...)
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  20. Yvon Gauthier, Constructive Truth and Certainty in Logic and Mathematics.score: 51.0
    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 (...)
     
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  21. Charles S. Chihara (1990). Constructibility and Mathematical Existence. Oxford University Press.score: 48.0
    Chihara here develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. He utilizes this system in the analysis of the nature of mathematics, and discusses many recent works in the philosophy of mathematics from the viewpoint of the constructibility theory developed. This innovative analysis will appeal to mathematicians and philosophers of logic, mathematics, and science.
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  22. R. J. Grayson (1982). Concepts of General Topology in Constructive Mathematics and in Sheaves, II. Annals of Mathematical Logic 23 (1):55-98.score: 46.0
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  23. Anne S. Troelstra (1977). Proof Theory and Constructive Mathematics. In Jon Barwise & H. Jerome Keisler (eds.), Handbook of Mathematical Logic. North-Holland Pub. Co.. 973--1052.score: 46.0
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  24. 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.score: 46.0
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  25. Douglas S. Bridges (2009). Constructive Notions of Equicontinuity. Archive for Mathematical Logic 48 (5):437-448.score: 46.0
    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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  26. R. J. Grayson (1982). A Correction to “Concepts of General Topology in Constructive Mathematics and in Sheaves”. Annals of Mathematical Logic 23 (1):99.score: 46.0
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  27. Gerhard Jäger (1991). Between Constructive Mathematics and PROLOG. Archive for Mathematical Logic 30 (5-6):297-310.score: 46.0
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  28. W. W. Tait (1983). Against Intuitionism: Constructive Mathematics is Part of Classical Mathematics. [REVIEW] Journal of Philosophical Logic 12 (2):173 - 195.score: 45.0
  29. William W. Tait (2006). Gödel's Correspondence on Proof Theory and Constructive Mathematics Kurt Gödel. Collected Works. Volume IV: Selected Correspondence A–G; Volume V: Selected Correspondence H–Z. Solomon Feferman, John W. Dawson, Warren Goldfarb, Charles Parsons, and Wilfried Sieg, Eds. Oxford: Oxford University Press, 2002. Pp. Xi+ 662; Xxiii+ 664. ISBN 0-19-850073-4; 0-19-850075-0. [REVIEW] Philosophia Mathematica 14 (1):76-111.score: 45.0
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  30. James Robert Brown (2003). Science and Constructive Mathematics. Analysis 63 (1):48–51.score: 45.0
  31. Geoffrey Hellman (1993). Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem. [REVIEW] Journal of Philosophical Logic 22 (3):221 - 248.score: 45.0
  32. M. R. Koss (2012). Giovanni Sommaruga, Ed. Foundational Theories of Classical and Constructive Mathematics. Dordrecht: Springer, 2011. Isbn 978-94-007-0430-5. Pp. XI + 314. [REVIEW] Philosophia Mathematica 20 (2):267-271.score: 45.0
  33. Michael Beeson (1978). Some Relations Between Classical and Constructive Mathematics. Journal of Symbolic Logic 43 (2):228-246.score: 45.0
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  34. M. H. Löb (1956). Formal Systems of Constructive Mathematics. Journal of Symbolic Logic 21 (1):63-75.score: 45.0
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  35. Hajime Ishihara (1991). Continuity and Nondiscontinuity in Constructive Mathematics. Journal of Symbolic Logic 56 (4):1349-1354.score: 45.0
    The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that "every mapping is strongly extensional", "every sequentially nondiscontinuous mapping is sequentially continuous", and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of the Kreisel-Lacombe-Shoenfield-Tsĕitin theorem.
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  36. Douglas Bridges, Constructive Mathematics. Stanford Encyclopedia of Philosophy.score: 45.0
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  37. M. H. Lob (1956). Formal Systems of Constructive Mathematics. Journal of Symbolic Logic 21 (1):63 - 75.score: 45.0
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  38. B. G. Sundholm, Tractarian Expressions and Their Use in Constructive Mathematics.score: 45.0
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  39. D. van Dalen (1991). Review: Douglas Bridges, Fred Richman, Varieties of Constructive Mathematics. [REVIEW] Journal of Symbolic Logic 56 (2):750-751.score: 45.0
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  40. Wim Ruitenburg (1991). Inequality in Constructive Mathematics. Notre Dame Journal of Formal Logic 32 (4):533-553.score: 45.0
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  41. Julian C. Cole (2012). Giovanni Sommaruga (Ed.), Foundational Theories of Classical and Constructive Mathematics, Springer, The Western Ontario Series in Philosophy of Science, Vol. 76, 2011, Pp. Xi+314. ISBN 978-94-007-0430-5 (Hardcover) US $139.00. [REVIEW] Studia Logica 100 (5):1047-1050.score: 45.0
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  42. J. van Oosten (2006). From Sets and Types to Topology and Analysis Towards Practicable Foundations for Constructive Mathematics (Book Review). Bulletin of Symbolic Logic 12:1-2.score: 45.0
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  43. Roy T. Cook (2012). G. Sommaruga (Editor), Foundational Theories of Classical and Constructive Mathematics. Bulletin of Symbolic Logic 18 (1):128.score: 45.0
     
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  44. 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.score: 45.0
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  45. Ag Dragalin (1973). Constructive Mathematics and Models of Enturnonistic Theories. In. In Patrick Suppes (ed.), Logic, Methodology and Philosophy of Science. New York,American Elsevier Pub. Co.. 111.score: 45.0
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  46. Matthew Hendtlass (2012). The Intermediate Value Theorem in Constructive Mathematics Without Choice. Annals of Pure and Applied Logic 163 (8):1050-1056.score: 45.0
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  47. William A. Howard (1987). Review: Michael J. Beeson, Foundations of Constructive Mathematics. Metamathematical Studies. [REVIEW] Journal of Symbolic Logic 52 (1):278-279.score: 45.0
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  48. Hajime Ishihara (2006). Reverse Mathematics in Bishop's Constructive Mathematics. Philosophia Scientiae:43-59.score: 45.0
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  49. Boris A. Kushner (1999). Hellman Geoffrey. Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem. Journal of Philosophical Logic, Vol. 22 (1993), Pp. 221–248. [REVIEW] Journal of Symbolic Logic 64 (1):397-398.score: 45.0
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  50. Boris A. Kushner (1999). Review: Geoffrey Hellman, Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem; Douglas S. Bridges, Constructive Mathematics and Unbounded Operators -- A Reply to Hellman; Geoffrey Hellman, Quantum Mechanical Unbounded Operators and Constructive Mathematics -- A Rejoinder to Bridges. [REVIEW] Journal of Symbolic Logic 64 (1):397-398.score: 45.0
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