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

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  1.  31
    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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  2.  49
    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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  3.  13
    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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  4.  4
    Rudolf Taschner (2010). The Swap of Integral and Limit in Constructive Mathematics. Mathematical Logic Quarterly 56 (5):533-540.
    Integration within constructive, especially intuitionistic mathematics in the sense of L. E. J. Brouwer, slightly differs from formal integration theories: Some classical results, especially Lebesgue's dominated convergence theorem, have tobe substituted by appropriate alternatives. Although there exist sophisticated, but rather laborious proposals, e.g. by E. Bishop and D. S. Bridges , the reference to partitions and the Riemann-integral, also with regard to the results obtained by R. Henstock and J. Kurzweil , seems to give a better direction. Especially, (...)
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  5.  5
    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 (...)
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  6. Per Martin-Löf (1970). Notes on Constructive Mathematics. Stockholm,Almqvist & Wiksell.
     
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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.
     
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  8. A. O. Slisenko (ed.) (1969). Studies in Constructive Mathematics and Mathematical Logic. New York, Consultants Bureau.
     
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  9.  43
    Sarah Elizabeth Hoffman (1999). Mathematics as Make-Believe: A Constructive Empiricist Account. Dissertation, University of Alberta (Canada)
    Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of (...) are considered and rejected. Constructive empiricism cannot be realist about abstract objects; it must reject even the realism advocated by otherwise ontologically restrained and epistemologically empiricist indispensability theorists. Indispensability arguments rely on the kind of inference to the best explanation the rejection of which is definitive of constructive empiricism. On the other hand, formalist and logicist anti-realist positions are also shown to be untenable. It is argued that a constructive empiricist philosophy of mathematics must be fictionalist. Borrowing and developing elements from both Philip Kitcher's constructive naturalism and Kendall Walton's theory of fiction, the account of mathematics advanced treats mathematics as a collection of stories told about an ideal agent and mathematical objects as fictions. The account explains what true portions of mathematics are about and why mathematics is useful, even while it is a story about an ideal agent operating in an ideal world; it connects theory and practice in mathematics with human experience of the phenomenal world. At the same time, the make-believe and game-playing aspects of the theory show how we can make sense of mathematics as fiction, as stories, without either undermining that explanation or being forced to accept abstract mathematical objects into our ontology. All of this occurs within the framework that constructive empiricism itself provides the epistemological limitations it mandates, the semantic view of theories, and an emphasis on the pragmatic dimension of our theories, our explanations, and of our relation to the language we use. (shrink)
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  10.  19
    Hajime Ishihara (1992). Continuity Properties in Constructive Mathematics. Journal of Symbolic Logic 57 (2):557-565.
    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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  11.  17
    Maria Emilia Maietti & Giuseppe Rosolini (2013). Quotient Completion for the Foundation of Constructive Mathematics. Logica Universalis 7 (3):371-402.
    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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  12.  17
    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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  13.  27
    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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  14.  32
    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.
    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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  15.  20
    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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  16.  47
    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 (...)
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  17.  6
    Dirk van Dalen (1995). Why Constructive Mathematics? Vienna Circle Institute Yearbook 3:141-157.
    The situation in constructive mathematics in the nineties is so vastly different from that in the thirties, that it is worthwhile to pause a moment to survey the development in the intermediate years. In doing so, I follow the example of Heyting, who at certain intervals took stock of intuitionistic mathematics, which for a long time was the only variety of constructive mathematics. Heyting entered the foundational debate in 1930 at the occasion of the famous (...)
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  18.  1
    Maria Emilia Maietti (2009). A Minimalist Two-Level Foundation for Constructive Mathematics. Annals of Pure and Applied Logic 160 (3):319-354.
    We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin.One level is given by an intensional type theory, called Minimal type theory. This theory extends a previous version with collections.The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model.This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; (...)
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  19.  39
    Geoffrey Hellman (1997). Quantum Mechanical Unbounded Operators and Constructive Mathematics – a Rejoinder to Bridges. 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 (...)
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  20.  7
    Christian Espíndola (2016). Semantic Completeness of First-Order Theories in Constructive Reverse Mathematics. Notre Dame Journal of Formal Logic 57 (2):281-286.
    We introduce a general notion of semantic structure for first-order theories, covering a variety of constructions such as Tarski and Kripke semantics, and prove that, over Zermelo–Fraenkel set theory, the completeness of such semantics is equivalent to the Boolean prime ideal theorem. Using a result of McCarty, we conclude that the completeness of Kripke semantics is equivalent, over intuitionistic Zermelo–Fraenkel set theory, to the Law of Excluded Middle plus BPI. Along the way, we also prove the equivalence, over ZF, between (...)
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  21.  4
    Michael R. Koss (2015). Some Obstacles Facing a Semantic Foundation for Constructive Mathematics. Erkenntnis 80 (5):1055-1068.
    This paper discusses Michael Dummett’s attempt to base the use of intuitionistic logic in mathematics on a proof-conditional semantics. This project is shown to face significant obstacles resulting from the existence of variants of standard intuitionistic logic. In order to overcome these obstacles, Dummett and his followers must give an intuitionistically acceptable completeness proof for intuitionistic logic relative to the BHK interpretation of the logical constants, but there are reasons to doubt that such a proof is possible. The paper (...)
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  22.  10
    Douglas Bridges (2008). Constructive Mathematics. Stanford Encyclopedia of Philosophy.
  23.  3
    Hannes Diener & Iris Loeb (2009). Sequences of Real Functions on [0, 1] in Constructive Reverse Mathematics. Annals of Pure and Applied Logic 157 (1):50-61.
    We give an overview of the role of equicontinuity of sequences of real-valued functions on [0,1] and related notions in classical mathematics, intuitionistic mathematics, Bishop’s constructive mathematics, and Russian recursive mathematics. We then study the logical strength of theorems concerning these notions within the programme of Constructive Reverse Mathematics. It appears that many of these theorems, like a version of Ascoli’s Lemma, are equivalent to fan-theoretic principles.
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  24.  82
    W. W. Tait (1983). Against Intuitionism: Constructive Mathematics is Part of Classical Mathematics. [REVIEW] Journal of Philosophical Logic 12 (2):173 - 195.
  25.  7
    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.
  26.  55
    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.
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  27.  26
    Geoffrey Hellman (1993). Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem. [REVIEW] Journal of Philosophical Logic 22 (3):221 - 248.
  28.  10
    Hajime Ishihara (1991). Continuity and Nondiscontinuity in Constructive Mathematics. Journal of Symbolic Logic 56 (4):1349-1354.
    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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  29.  71
    James Robert Brown (2003). Science and Constructive Mathematics. Analysis 63 (1):48–51.
  30.  4
    R. J. Grayson (1982). Concepts of General Topology in Constructive Mathematics and in Sheaves, II. Annals of Mathematical Logic 23 (1):55-98.
  31. Hajime Ishihara (2006). Reverse Mathematics in Bishop’s Constructive Mathematics. Philosophia Scientiae:43-59.
  32.  12
    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.
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  33.  6
    J. R. Brown (2003). Science and Constructive Mathematics. Analysis 63 (1):48-51.
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  34.  10
    Gerhard Jäger (1991). Between Constructive Mathematics and PROLOG. Archive for Mathematical Logic 30 (5-6):297-310.
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  35.  26
    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 (...)
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  36.  6
    Matthew Hendtlass (2012). The Intermediate Value Theorem in Constructive Mathematics Without Choice. Annals of Pure and Applied Logic 163 (8):1050-1056.
  37.  21
    Michael Beeson (1978). Some Relations Between Classical and Constructive Mathematics. Journal of Symbolic Logic 43 (2):228-246.
  38.  4
    Jaap Van Oosten (2006). From Sets and Types to Topology and Analysis—Towards Practicable Foundations for Constructive Mathematics, Edited by Crosilla Laura and Schuster Peter, Oxford Logic Guides, Vol. 48. Clarendon Press, 2005, Xix+ 450 Pp. [REVIEW] Bulletin of Symbolic Logic 12 (4):611-612.
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  39. William A. Howard (1987). Review: Michael J. Beeson, Foundations of Constructive Mathematics. Metamathematical Studies. [REVIEW] Journal of Symbolic Logic 52 (1):278-279.
     
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  40.  11
    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.
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  41.  2
    R. J. Grayson (1982). A Correction to “Concepts of General Topology in Constructive Mathematics and in Sheaves”. Annals of Mathematical Logic 23 (1):99.
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  42.  2
    Iris Loeb (2009). Indecomposability of Negative Dense Subsets of ℝ in Constructive Reverse Mathematics. Logic Journal of the IGPL 17 (2):173-177.
    In 1970 Vesley proposed a substitute of Kripke's Scheme. In this paper it is shown that —over Bishop's constructive mathematics— the indecomposability of negative dense subsets of ℝ is equivalent to a weakening of Vesley's proposal. This result supports the idea that full Kripke's Scheme might not be necessary for most of intuitionistic mathematics. At the same time it contributes to the programme of Constructive Reverse Mathematics and gives a new answer to a 1997 question (...)
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  43.  8
    M. H. Löb (1956). Formal Systems of Constructive Mathematics. Journal of Symbolic Logic 21 (1):63-75.
  44.  4
    Wim Ruitenburg (1991). Inequality in Constructive Mathematics. Notre Dame Journal of Formal Logic 32 (4):533-553.
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  45.  6
    M. H. Lob (1956). Formal Systems of Constructive Mathematics. Journal of Symbolic Logic 21 (1):63 - 75.
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  46.  4
    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.
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  47.  2
    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 (4):1-2.
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  48.  2
    D. van Dalen (1991). Review: Douglas Bridges, Fred Richman, Varieties of Constructive Mathematics. [REVIEW] Journal of Symbolic Logic 56 (2):750-751.
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  49.  1
    Iris Loeb (2008). Indecomposability of ℝ and ℝ \ {0} in Constructive Reverse Mathematics. Logic Journal of the IGPL 16 (3):269-273.
    It is shown that—over Bishop's constructive mathematics—the indecomposability of ℝ is equivalent to the statement that all functions from a complete metric space into a metric space are sequentially nondiscontinuous. Furthermore we prove that the indecomposability of ℝ \ {0} is equivalent to the negation of the disjunctive version of Markov's Principle. These results contribute to the programme of Constructive Reverse Mathematics.
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  50.  1
    Maarten McKubre-Jordens (2012). Constructive Mathematics. In J. Feiser & B. Dowden (eds.), Internet Encyclopedia of Philosophy.
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