77 found
Order:
Disambiguations
Douglas Bridges [42]Douglas S. Bridges [35]
  1.  6
    Constructive Analysis.Errett Bishop & Douglas Bridges - 1987 - Journal of Symbolic Logic 52 (4):1047-1048.
    Direct download  
     
    Export citation  
     
    Bookmark   55 citations  
  2.  15
    Constructive Mathematics.Douglas Bridges - 2008 - Stanford Encyclopedia of Philosophy.
  3.  3
    Apartness Spaces as a Framework for Constructive Topology.Douglas Bridges & Luminiţa Vîţă - 2003 - 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.
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   12 citations  
  4.  4
    Apartness Spaces as a Framework for Constructive Topology.Douglas Bridges & Luminia Vî - 2003 - 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.
    Direct download  
     
    Export citation  
     
    Bookmark   10 citations  
  5. Uniformly Convex Banach Spaces Are Reflexive—Constructively.Douglas S. Bridges, Hajime Ishihara & Maarten McKubre‐Jordens - 2013 - 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.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  6.  16
    Constructive Notions of Equicontinuity.Douglas S. Bridges - 2009 - 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.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  7.  16
    Strong Continuity Implies Uniform Sequential Continuity.Douglas Bridges, Hajime Ishihara, Peter Schuster & Luminiţa Vîţa - 2005 - Archive for Mathematical Logic 44 (7):887-895.
  8.  10
    The Pseudocompactness of [0.1] Is Equivalent to the Uniform Continuity Theorem.Douglas Bridges & Hannes Diener - 2007 - 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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  9.  37
    Constructive Mathematics in Theory and Programming Practice.Douglas Bridges & Steeve Reeves - 1999 - 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.
    Direct download (15 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  10.  12
    The Anti-Specker Property, a Heine–Borel Property, and Uniform Continuity.Josef Berger & Douglas Bridges - 2008 - 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.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  11.  10
    Product a-Frames and Proximity.Douglas S. Bridges - 2008 - 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.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  12. Corrigendum to "a Proof-Technique in Uniform Space Theory".Douglas Bridges & Luminiţa Vîţă - 2004 - Journal of Symbolic Logic 69 (1):328-328.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark  
  13.  24
    Constructive Mathematics and Unbounded Operators — a Reply to Hellman.Douglas S. Bridges - 1995 - 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.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  14.  34
    Apartness, Topology, and Uniformity: A Constructive View.Douglas Bridges, Peter Schuster & Luminiţa Vîţă - 2002 - 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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  15.  9
    A Bizarre Property Equivalent To The -Fan Theorem.Josef Berger & Douglas Bridges - 2006 - 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.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  16.  15
    Glueing Continuous Functions Constructively.Douglas S. Bridges & Iris Loeb - 2010 - 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.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  17.  17
    The Fan Theorem and Unique Existence of Maxima.Josef Berger, Douglas Bridges & Peter Schuster - 2006 - 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.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  18.  4
    Constructive Mathematics in Theory and Programming Practice.Douglas Bridges & Steeve Reeves - 1998 - Philosophia Mathematica 6 (3):65-104.
    The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics. 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.
    Direct download  
     
    Export citation  
     
    Bookmark   4 citations  
  19.  16
    Compactness Notions for an Apartness Space.Douglas S. Bridges - 2012 - Archive for Mathematical Logic 51 (5-6):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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  20.  8
    A Constructive Treatment of Urysohn's Lemma in an Apartness Space.Douglas Bridges & Hannes Diener - 2006 - 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 (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  21.  10
    Double Sequences, Almost Cauchyness and BD-N.Josef Berger, Douglas Bridges & Erik Palmgren - 2012 - Logic Journal of the IGPL 20 (1):349-354.
    It is shown that, relative to Bishop-style constructive mathematics, the boundedness principle BD-N is equivalent both to a general result about the convergence of double sequences and to a particular one about Cauchyness in a semi-metric space.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  22.  33
    A Proof-Technique in Uniform Space Theory.Douglas Bridges & Luminiţa Vîţă - 2003 - 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.
    Direct download (9 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  23.  11
    How to Construct a Product of a‐Frames.Douglas S. Bridges - 2012 - 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.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  24.  9
    Reflections on Function Spaces.Douglas S. Bridges - 2012 - Annals of Pure and Applied Logic 163 (2):101-110.
  25.  58
    Can Constructive Mathematics Be Applied in Physics?Douglas S. Bridges - 1999 - 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.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  26.  14
    Characterising Near Continuity Constructively.Douglas Bridges & Luminiţa Vîţă - 2001 - Mathematical Logic Quarterly 47 (4):535-538.
    The relation between near continuity and sequential continuity for mappings between metric spaces is explored constructively. It is also shown that the classical implications “near continuity implies sequential continuity” and “near continuity implies apart continuity” are essentially nonconstructive.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  27.  11
    Continuous Isomorphisms From R Onto a Complete Abelian Group.Douglas Bridges & Matthew Hendtlass - 2010 - Journal of Symbolic Logic 75 (3):930-944.
    This paper provides a Bishop-style constructive analysis of the contrapositive of the statement that a continuous homomorphism of R onto a compact abelian group is periodic. It is shown that, subject to a weak locatedness hypothesis, if G is a complete (metric) abelian group that is the range of a continuous isomorphism from R, then G is noncompact. A special case occurs when G satisfies a certain local path-connectedness condition at 0. A number of results about one-one and injective mappings (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  28.  24
    Constructive Truth in Practice.Douglas Bridges - 1998 - In H. G. Dales & Gianluigi Oliveri (eds.), Truth in Mathematics. Oxford University Press, Usa. pp. 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.
    Direct download  
     
    Export citation  
     
    Bookmark   1 citation  
  29.  8
    Uniform Continuity Properties of Preference Relations.Douglas S. Bridges - 2008 - 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.
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  30.  15
    On Weak Operator Compactness of the Unit Ball ofL.Douglas S. Bridges - 1978 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (31-36):493-494.
  31.  5
    A Weak Constructive Sequential Compactness Property And The Fan Theorem.Douglas Bridges - 2005 - Logic Journal of the IGPL 13 (2):151-158.
    A weak constructive sequential compactness property of metric spaces is introduced. It is proved that for complete, totally bounded metric spaces this property is equivalent to Brouwer's fan theorem for detachable bars. Our results form a part of constructive reverse mathematics.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  32.  27
    A Constructive Treatment of Open and Unopen Mapping Theorems.Douglas Bridges, William Julian & Ray Mines - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (1):29-43.
  33.  17
    A Criterion for Compactness in Metric Spaces?Douglas S. Bridges - 1979 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (7-12):97-98.
  34.  11
    A Constructive Version of the Spectral Mapping Theorem.Douglas Bridges & Robin Havea - 2001 - Mathematical Logic Quarterly 47 (3):299-304.
    The spectral mapping theorem in a unital Banach algebra is examined for its constructive content.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  35.  10
    Sequentially Continuous Linear Mappings in Constructive Analysis.Douglas Bridges & Ray Mines - 1998 - Journal of Symbolic Logic 63 (2):579-583.
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  36.  8
    Bounded Variation Implies Regulated: A Constructive Proof.Douglas Bridges & Ayan Mahalanobis - 2001 - Journal of Symbolic Logic 66 (4):1695-1700.
    It is shown constructively that a strongly extensional function of bounded variation on an interval is regulated, in a sequential sense that is classically equivalent to the usual one.
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  37.  6
    On Weak Operator Compactness of the Unit Ball of L(H).Douglas S. Bridges - 1978 - Mathematical Logic Quarterly 24 (31‐36):493-494.
  38.  11
    Sequential, Pointwise, and Uniform Continuity: A Constructive Note.Douglas S. Bridges - 1993 - 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.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  39.  8
    Complements of Intersections in Constructive Mathematics.Douglas S. Bridges & Hajime Ishihara - 1994 - 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 (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  40.  3
    Weak-Operator Continuity and the Existence of Adjoints.Douglas Bridges & Luminita Dediu - 1999 - 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.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  41.  17
    A Note on Morse's Lambda-Notation in Set Theory.Douglas S. Bridges - 1978 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (8):113-114.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  42.  16
    On the Constructive Convergence of Series of Independent Functions.Douglas S. Bridges - 1979 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (3-6):93-96.
  43.  15
    A Constructive Treatment of Open and Unopen Mapping Theorems.Douglas Bridges, William Julian & Ray Mines - 1989 - Mathematical Logic Quarterly 35 (1):29-43.
  44.  15
    Square Roots and Powers in Constructive Banach Algebra Theory.Douglas S. Bridges & Robin S. Havea - 2012 - In S. Barry Cooper (ed.), How the World Computes. pp. 68--77.
    Direct download  
     
    Export citation  
     
    Bookmark  
  45.  13
    A General Constructive Intermediate Value Theorem.Douglas S. Bridges - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (5):433-435.
  46.  5
    A Criterion for Compactness in Metric Spaces?Douglas S. Bridges - 1979 - Mathematical Logic Quarterly 25 (7‐12):97-98.
  47.  17
    Geometric Intuition and Elementary Constructive Analysis.Douglas S. Bridges - 1979 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (33):521-523.
  48.  12
    The Anti-Specker Property, Positivity, and Total Boundedness.Douglas Bridges & Hannes Diener - 2010 - 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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  49.  9
    The Continuum Hypothesis Implies Excluded Middle.Douglas S. Bridges - 2016 - In Peter Schuster & Dieter Probst (eds.), Concepts of Proof in Mathematics, Philosophy, and Computer Science. De Gruyter. pp. 111-114.
    Direct download  
     
    Export citation  
     
    Bookmark  
  50.  7
    Characterising Dominated Weak-Operator Continuous Functionals on Subspaces of B.Douglas Bridges - 2013 - 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.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
1 — 50 / 77