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Douglas S. Bridges [34]Douglas Bridges [33]
  1. Errett Bishop & Douglas Bridges (1987). Constructive Analysis. Journal of Symbolic Logic 52 (4):1047-1048.
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  2. Douglas Bridges & Luminiţa Vîţă (2004). Corrigendum to "a Proof-Technique in Uniform Space Theory". Journal of Symbolic Logic 69 (1):328-328.
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  3.  8
    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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  4. Douglas Bridges & Luminiţa Vîţă (2003). Apartness Spaces as a Framework for Constructive Topology. Annals of Pure and Applied Logic 119 (1-3):61-83.
    An axiomatic development of the theory of apartness and nearness of a point and a set is introduced as a framework for constructive topology. Various notions of continuity of mappings between apartness spaces are compared; the constructive independence of one of the axioms from the others is demonstrated; and the product apartness structure is defined and analysed.
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  5.  9
    Douglas Bridges, Constructive Mathematics. Stanford Encyclopedia of Philosophy.
  6.  8
    Douglas Bridges, Hajime Ishihara, Peter Schuster & Luminiţa Vîţa (2005). Strong Continuity Implies Uniform Sequential Continuity. Archive for Mathematical Logic 44 (7):887-895.
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  7.  3
    Douglas S. Bridges (2008). Product a-Frames and Proximity. Mathematical Logic Quarterly 54 (1):12-26.
    Continuing the study of apartness in lattices, begun in [8], this paper deals with axioms for a product a-frame and with their consequences. This leads to a reasonable notion of proximity in an a-frame, abstracted from its counterpart in the theory of set-set apartness.
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  8.  2
    Douglas S. Bridges (2016). The Continuum Hypothesis Implies Excluded Middle. In Peter Schuster & Dieter Probst (eds.), Concepts of Proof in Mathematics, Philosophy, and Computer Science. De Gruyter 111-114.
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  9.  16
    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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  10.  8
    Josef Berger & Douglas Bridges (2008). The Anti-Specker Property, a Heine–Borel Property, and Uniform Continuity. Archive for Mathematical Logic 46 (7-8):583-592.
    Working within Bishop’s constructive framework, we examine the connection between a weak version of the Heine–Borel property, a property antithetical to that in Specker’s theorem in recursive analysis, and the uniform continuity theorem for integer-valued functions. The paper is a contribution to the ongoing programme of constructive reverse mathematics.
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  11.  7
    Douglas Bridges & Hannes Diener (2007). The Pseudocompactness of [0.1] Is Equivalent to the Uniform Continuity Theorem. Journal of Symbolic Logic 72 (4):1379 - 1384.
    We prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into R is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.
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  12.  9
    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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  13.  48
    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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  14.  2
    Douglas S. Bridges (2016). Apartness Spaces and Uniform Neighbourhood Structures. Annals of Pure and Applied Logic 167 (9):850-864.
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  15.  14
    Douglas S. Bridges & Robin S. Havea (2012). Square Roots and Powers in Constructive Banach Algebra Theory. In S. Barry Cooper (ed.), How the World Computes. 68--77.
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  16.  9
    Douglas Bridges & Luminiţa Vîţă (2003). A Proof-Technique in Uniform Space Theory. Journal of Symbolic Logic 68 (3):795-802.
    In the constructive theory of uniform spaces there occurs a technique of proof in which the application of a weak form of the law of excluded middle is circumvented by purely analytic means. The essence of this proof-technique is extracted and then applied in several different situations.
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  17.  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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  18.  15
    Douglas Bridges, Peter Schuste & Luminiţa Vîţă (2002). Apartness, Topology, and Uniformity: A Constructive View. Mathematical Logic Quarterly 48 (4):16-28.
    The theory of apartness spaces, and their relation to topological spaces (in the point–set case) and uniform spaces (in the set–set case), is sketched. New notions of local decomposability and regularity are investigated, and the latter is used to produce an example of a classically metrisable apartness on R that cannot be induced constructively by a uniform structure.
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  19.  3
    Douglas S. Bridges (2008). Uniform Continuity Properties of Preference Relations. Notre Dame Journal of Formal Logic 49 (1):97-106.
    The anti-Specker property, a constructive version of sequential compactness, is used to prove constructively that a pointwise continuous, order-dense preference relation on a compact metric space is uniformly sequentially continuous. It is then shown that Ishihara's principle BD-ℕ implies that a uniformly sequentially continuous, order-dense preference relation on a separable metric space is uniformly continuous. Converses of these two theorems are also proved.
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  20.  12
    Josef Berger, Douglas Bridges & Peter Schuster (2006). The Fan Theorem and Unique Existence of Maxima. Journal of Symbolic Logic 71 (2):713 - 720.
    The existence and uniqueness of a maximum point for a continuous real—valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem.
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  21.  8
    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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  22.  2
    Douglas Bridges & Hannes Diener (2006). A Constructive Treatment of Urysohn's Lemma in an Apartness Space. Mathematical Logic Quarterly 52 (5):464-469.
    This paper is dedicated to Prof. Dr. Günter Asser, whose work in founding this journal and maintaining it over many difficult years has been a major contribution to the activities of the mathematical logic community.At first sight it appears highly unlikely that Urysohn's Lemma has any significant constructive content. However, working in the context of an apartness space and using functions whose values are a generalisation of the reals, rather than real numbers, enables us to produce a significant constructive version (...)
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  23.  10
    Douglas Bridges (1998). Constructive Truth in Practice. In H. G. Dales & Gianluigi Oliveri (eds.), Truth in Mathematics. Oxford University Press, Usa 53--69.
    In this chapter, which has evolved over the last ten years to what I hope will be its perfect Platonic form, I shall first discuss those features of constructive mathematics that distinguish it from its traditional, or classical, counterpart, and then illustrate the practice of that distinction in aspects of complex analysis whose classical treatment ought to be familiar to a beginning graduate student of pure mathematics.
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  24.  1
    Josef Berger & Douglas Bridges (2006). A Bizarre Property Equivalent To The -Fan Theorem. Logic Journal of the IGPL 14 (6):867-871.
    It is shown, with intuitionistic logic, that if every locally constant function from to has a property akin to constancy, then the fan theorem for -bars holds, and conversely.
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  25.  4
    Douglas Bridges & Ray Mines (1998). Sequentially Continuous Linear Mappings in Constructive Analysis. Journal of Symbolic Logic 63 (2):579-583.
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  26.  4
    Douglas S. Bridges (1979). A Criterion for Compactness in Metric Spaces? Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (7-12):97-98.
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  27.  4
    Douglas S. Bridges (1978). A Note on Morse's Lambda-Notation in Set Theory. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (8):113-114.
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  28.  4
    Douglas S. Bridges (1979). On the Constructive Convergence of Series of Independent Functions. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (3-6):93-96.
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  29.  4
    Douglas S. Bridges (1978). On Weak Operator Compactness of the Unit Ball ofL. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (31-36):493-494.
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  30.  12
    Douglas S. Bridges, Hajime Ishihara & Maarten McKubre‐Jordens (2013). Uniformly Convex Banach Spaces Are Reflexive—Constructively. Mathematical Logic Quarterly 59 (4-5):352-356.
    We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive.
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  31.  3
    Douglas Bridges, Peter Schuster & Luminiţa Vîţă (2002). Apartness, Topology, and Uniformity: A Constructive View. Mathematical Logic Quarterly 48 (S1):16-28.
    The theory of apartness spaces, and their relation to topological spaces and uniform spaces , is sketched. New notions of local decomposability and regularity are investigated, and the latter is used to produce an example of a classically metrisable apartness on R that cannot be induced constructively by a uniform structure.
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  32.  11
    Douglas S. Bridges (2012). Compactness Notions for an Apartness Space. Archive for Mathematical Logic 51 (5):517-534.
    Two new notions of compactness, each classically equivalent to the standard classical one of sequential compactness, for apartness spaces are examined within Bishop-style constructive mathematics.
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  33.  4
    Douglas Bridges, William Julian & Ray Mines (1989). A Constructive Treatment of Open and Unopen Mapping Theorems. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (1):29-43.
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  34.  4
    Douglas S. Bridges (1989). A General Constructive Intermediate Value Theorem. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (5):433-435.
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  35.  10
    Douglas Bridges, Fred Richman & Peter Schuster (1999). Linear Independence Without Choice. Annals of Pure and Applied Logic 101 (1):95-102.
    The notions of linear and metric independence are investigated in relation to the property: if U is a set of n+1 independent vectors, and X is a set of n independent vectors, then adjoining some vector in U to X results in a set of n+1 independent vectors. It is shown that this property holds in any normed linear space. A related property – that finite-dimensional subspaces are proximinal – is established for strictly convex normed spaces over the real or (...)
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  36.  1
    Douglas S. Bridges (1978). On Weak Operator Compactness of the Unit Ball of L(H). Mathematical Logic Quarterly 24 (31‐36):493-494.
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  37.  1
    Douglas Bridges & Luminita Dediu (1999). Weak-Operator Continuity and the Existence of Adjoints. Mathematical Logic Quarterly 45 (2):203-206.
    It is shown, within constructive mathematics, that the unit ball B1 of the set of bounded operators on a Hilbert space H is weak-operator totally bounded. This result is then used to prove that the weak-operator continuity of the mapping T → AT on B1 is equivalent to the existence of the adjoint of A.
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  38.  8
    Douglas Bridges & Hajime Ishihara (1994). Absolute Continuity and the Uniqueness of the Constructive Functional Calculus. Mathematical Logic Quarterly 40 (4):519-527.
    The constructive functional calculus for a sequence of commuting selfadjoint operators on a separable Hilbert space is shown to be independent of the orthonormal basis used in its construction. The proof requires a constructive criterion for the absolute continuity of two positive measures in terms of test functions.
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  39.  2
    Douglas S. Bridges (1993). Sequential, Pointwise, and Uniform Continuity: A Constructive Note. Mathematical Logic Quarterly 39 (1):55-61.
    The main result of this paper is a weak constructive version of the uniform continuity theorem for pointwise continuous, real-valued functions on a convex subset of a normed linear space. Recursive examples are given to show that the hypotheses of this theorem are necessary. The remainder of the paper discusses conditions which ensure that a sequentially continuous function is continuous. MSC: 03F60, 26E40, 46S30.
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  40.  7
    Douglas S. Bridges (2012). Reflections on Function Spaces. Annals of Pure and Applied Logic 163 (2):101-110.
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  41.  8
    Douglas S. Bridges (2006). Church's Thesis and Bishop's Constructivism. In A. Olszewski, J. Wole'nski & R. Janusz (eds.), Church's Thesis After Seventy Years. Ontos Verlag 1--58.
  42.  3
    Douglas S. Bridges (2013). Characterising Dominated Weak-Operator Continuous Functionals on Subspaces Of. Annals of Pure and Applied Logic 164 (4):416-420.
    A characterisation of a type of weak-operator continuous linear functional on certain linear subsets of B, where H is a Hilbert space, is derived within Bishop-style constructive mathematics.
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  43.  3
    Douglas S. Bridges (2007). Constructing Local Optima on a Compact Interval. Archive for Mathematical Logic 46 (2):149-154.
    The existence of either a maximum or a minimum for a uniformly continuous mapping f of a compact interval into ${\mathbb{R}}$ is established constructively under the hypotheses that f′ is sequentially continuous and f has at most one critical point.
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  44.  2
    Douglas S. Bridges (1979). A Criterion for Compactness in Metric Spaces? Mathematical Logic Quarterly 25 (7‐12):97-98.
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  45.  5
    Douglas S. Bridges (1978). A Note on Morse's Lambda‐Notation in Set Theory. Mathematical Logic Quarterly 24 (8):113-114.
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  46.  5
    Douglas S. Bridges (1979). Geometric Intuition and Elementary Constructive Analysis. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (33):521-523.
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  47.  5
    Douglas Bridges & Matthew Hendtlass (2010). Continuous Homomorphisms of R Onto a Compact Group. Mathematical Logic Quarterly 56 (2):191-197.
    It is shown within Bishop's constructive mathematics that, under one extra, classically automatic, hypothesis, a continuous homomorphism from R onto a compact metric abelian group is periodic, but that the existence of the minimum value of the period is not derivable.
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  48.  5
    Douglas S. Bridges (2012). How to Construct a Product of a‐Frames. Mathematical Logic Quarterly 58 (4‐5):281-293.
    It is shown how, under certain circumstances and within Bishop-style constructive mathematics, one can construct a product of two a-frames.
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  49.  5
    Douglas Bridges & Hajime Ishihara (1998). A Definitive Constructive Open Mapping Theorem? Mathematical Logic Quarterly 44 (4):545-552.
    It is proved, within Bishop's constructive mathematics , that, in the context of a Hilbert space, the Open Mapping Theorem is equivalent to a principle that holds in intuitionistic mathematics and recursive constructive mathematics but is unlikely to be provable within BISH.
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  50.  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 principle and (...)
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