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  1. Constructive aspects of Riemann’s permutation theorem for series.J. Berger, Douglas Bridges, Hannes Diener & Helmet Schwichtenberg - forthcoming - Logic Journal of the IGPL.
    The notions of permutable and weak-permutable convergence of a series|$\sum _{n=1}^{\infty }a_{n}$|of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD-|$\mathbb {N}$|implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem for (...)
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  • Bishop's Mathematics: a Philosophical Perspective.Laura Crosilla - forthcoming - In Handbook of Bishop's Mathematics. CUP.
    Errett Bishop's work in constructive mathematics is overwhelmingly regarded as a turning point for mathematics based on intuitionistic logic. It brought new life to this form of mathematics and prompted the development of new areas of research that witness today's depth and breadth of constructive mathematics. Surprisingly, notwithstanding the extensive mathematical progress since the publication in 1967 of Errett Bishop's Foundations of Constructive Analysis, there has been no corresponding advances in the philosophy of constructive mathematics Bishop style. The aim of (...)
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  • Intuitionistic fixed point logic.Ulrich Berger & Hideki Tsuiki - 2021 - Annals of Pure and Applied Logic 172 (3):102903.
    We study the system IFP of intuitionistic fixed point logic, an extension of intuitionistic first-order logic by strictly positive inductive and coinductive definitions. We define a realizability interpretation of IFP and use it to extract computational content from proofs about abstract structures specified by arbitrary classically true disjunction free formulas. The interpretation is shown to be sound with respect to a domain-theoretic denotational semantics and a corresponding lazy operational semantics of a functional language for extracted programs. We also show how (...)
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  • (1 other version)Logical pluralism and normativity.Stewart Shapiro & Teresa Kouri Kissel - 2020 - Inquiry: An Interdisciplinary Journal of Philosophy 63 (3-4):389-410.
    We are logical pluralists who hold that the right logic is dependent on the domain of investigation; different logics for different mathematical theories. The purpose of this article is to explore the ramifications for our pluralism concerning normativity. Is there any normative role for logic, once we give up its universality? We discuss Florian Steingerger’s “Frege and Carnap on the Normativity of Logic” as a source for possible types of normativity, and then turn to our own proposal, which postulates that (...)
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  • Choice Sequences and the Continuum.Casper Storm Hansen - 2020 - Erkenntnis 87 (2):517-534.
    According to L.E.J. Brouwer, there is room for non-definable real numbers within the intuitionistic ontology of mental constructions. That room is allegedly provided by freely proceeding choice sequences, i.e., sequences created by repeated free choices of elements by a creating subject in a potentially infinite process. Through an analysis of the constitution of choice sequences, this paper argues against Brouwer’s claim.
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  • Maddy On The Multiverse.Claudio Ternullo - 2019 - In Stefania Centrone, Deborah Kant & Deniz Sarikaya (eds.), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Springer Verlag. pp. 43-78.
    Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...)
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  • The entanglement of logic and set theory, constructively.Laura Crosilla - 2022 - Inquiry: An Interdisciplinary Journal of Philosophy 65 (6).
    ABSTRACT Theories of sets such as Zermelo Fraenkel set theory are usually presented as the combination of two distinct kinds of principles: logical and set-theoretic principles. The set-theoretic principles are imposed ‘on top’ of first-order logic. This is in agreement with a traditional view of logic as universally applicable and topic neutral. Such a view of logic has been rejected by the intuitionists, on the ground that quantification over infinite domains requires the use of intuitionistic rather than classical logic. In (...)
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  • Philosophy of Mathematical Practice — Motivations, Themes and Prospects†.Jessica Carter - 2019 - Philosophia Mathematica 27 (1):1-32.
    A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.
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  • The binary expansion and the intermediate value theorem in constructive reverse mathematics.Josef Berger, Hajime Ishihara, Takayuki Kihara & Takako Nemoto - 2019 - Archive for Mathematical Logic 58 (1-2):203-217.
    We introduce the notion of a convex tree. We show that the binary expansion for real numbers in the unit interval ) is equivalent to weak König lemma ) for trees having at most two nodes at each level, and we prove that the intermediate value theorem is equivalent to \ for convex trees, in the framework of constructive reverse mathematics.
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  • Convexity and unique minimum points.Josef Berger & Gregor Svindland - 2019 - Archive for Mathematical Logic 58 (1-2):27-34.
    We show constructively that every quasi-convex, uniformly continuous function \ with at most one minimum point has a minimum point, where C is a convex compact subset of a finite dimensional normed space. Applications include a result on strictly quasi-convex functions, a supporting hyperplane theorem, and a short proof of the constructive fundamental theorem of approximation theory.
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  • C. S. Peirce and Intersemiotic Translation.Joao Queiroz & Daniella Aguiar - 2015 - In Peter Pericles Trifonas (ed.), International Handbook of Semiotics. Dordrecht: Springer. pp. 201-215.
    Intersemiotic translation (IT) was defined by Roman Jakobson (The Translation Studies Reader, Routledge, London, p. 114, 2000) as “transmutation of signs”—“an interpretation of verbal signs by means of signs of nonverbal sign systems.” Despite its theoretical relevance, and in spite of the frequency in which it is practiced, the phenomenon remains virtually unexplored in terms of conceptual modeling, especially from a semiotic perspective. Our approach is based on two premises: (i) IT is fundamentally a semiotic operation process (semiosis) and (ii) (...)
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  • (1 other version)Logical pluralism and normativity.Teresa Kouri Kissel & Stewart Shapiro - 2017 - Inquiry: An Interdisciplinary Journal of Philosophy:1-22.
    We are logical pluralists who hold that the right logic is dependent on the domain of investigation; different logics for different mathematical theories. The purpose of this article is to explore the ramifications for our pluralism concerning normativity. Is there any normative role for logic, once we give up its universality? We discuss Florian Steingerger’s “Frege and Carnap on the Normativity of Logic” as a source for possible types of normativity, and then turn to our own proposal, which postulates that (...)
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  • 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.
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  • Hilbert’s Program.Richard Zach - 2014 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Stanford, CA: The Metaphysics Research Lab.
    In the early 1920s, the German mathematician David Hilbert (1862–1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification (...)
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  • (1 other version)An omniscience principle, the König Lemma and the Hahn-Banach theorem.Hajime Ishihara - 1990 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (3):237-240.
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  • (1 other version)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.
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  • (1 other version)A General Constructive Intermediate Value Theorem.Douglas S. Bridges - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (5):433-435.
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  • Pluralism in Mathematics: A New Position in Philosophy of Mathematics.Michèle Friend - 2013 - Dordrecht, Netherland: Springer.
    The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...)
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  • 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.
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  • 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.
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  • 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 (...)
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  • Constructive notions of strict convexity.Douglas S. Bridges - 1993 - Mathematical Logic Quarterly 39 (1):295-300.
    Two classically equivalent, but constructively inequivalent, strict convexity properties of a preference relation are discussed, and conditions given under which the stronger notion is a consequence of the weaker. The last part of the paper introduces uniformly rotund preferences, and shows that uniform rotundity implies strict convexity. The paper is written from a strictly constructive point of view, in which all proofs embody algorithms. MSC: 03F60, 90A06.
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  • 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.
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  • (1 other version)A Constructive Treatment of Open and Unopen Mapping Theorems.Douglas Bridges, William Julian & Ray Mines - 1989 - Mathematical Logic Quarterly 35 (1):29-43.
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  • (1 other version)A General Constructive Intermediate Value Theorem.Douglas S. Bridges - 1989 - Mathematical Logic Quarterly 35 (5):433-435.
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  • (1 other version)An omniscience principle, the König Lemma and the Hahn‐Banach theorem.Hajime Ishihara - 1990 - Mathematical Logic Quarterly 36 (3):237-240.
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  • (1 other version)Constructively Complete Finite Sets.Mark Mandelkern - 1988 - Mathematical Logic Quarterly 34 (2):97-103.
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  • Absolute Continuity and the Uniqueness of the Constructive Functional Calculus.Douglas Bridges & Hajime Ishihara - 1994 - 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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  • A Definitive Constructive Open Mapping Theorem?Douglas Bridges & Hajime Ishihara - 1998 - 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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  • 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.
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  • Intuitionism As A Kuhnian Revolution In Mathematics.Bruce Pourciau - 2000 - Studies in History and Philosophy of Science Part A 31 (2):297-329.
    In this paper it is argued, firstly, that Kuhnian revolutions in mathematics are logically possible, in the sense of not being inconsistent with the nature of mathematics; and, secondly, that Kuhnian revolutions are actually possible, in the sense that a Kuhnian paradigm for mathematics can be exhibited which would, if accepted by the mathematical community, produce a full Kuhnian revolution. These two arguments depend on first proving that a shift from a classical conception of mathematics to an intuitionist conception would (...)
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  • The applicability of mathematics as a scientific and a logical problem.Feng Ye - 2010 - Philosophia Mathematica 18 (2):144-165.
    This paper explores how to explain the applicability of classical mathematics to the physical world in a radically naturalistic and nominalistic philosophy of mathematics. The applicability claim is first formulated as an ordinary scientific assertion about natural regularity in a class of natural phenomena and then turned into a logical problem by some scientific simplification and abstraction. I argue that there are some genuine logical puzzles regarding applicability and no current philosophy of mathematics has resolved these puzzles. Then I introduce (...)
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  • Constructive mathematics.Douglas Bridges - 2008 - Stanford Encyclopedia of Philosophy.
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  • The axiom of choice.John L. Bell - 2008 - Stanford Encyclopedia of Philosophy.
    The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...)
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  • On the foundations of constructive mathematics – especially in relation to the theory of continuous functions.Frank Waaldijk - 2004 - 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 related to the definition in BISH of (...)
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  • Countable choice as a questionable uniformity principle.Peter M. Schuster - 2004 - Philosophia Mathematica 12 (2):106-134.
    Should weak forms of the axiom of choice really be accepted within constructive mathematics? A critical view of the Brouwer-Heyting-Kolmogorov interpretation, accompanied by the intention to include nondeterministic algorithms, leads us to subscribe to Richman's appeal for dropping countable choice. As an alternative interpretation of intuitionistic logic, we propose to renew dialogue semantics.
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  • Domains for computation in mathematics, physics and exact real arithmetic.Abbas Edalat - 1997 - Bulletin of Symbolic Logic 3 (4):401-452.
    We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and (...)
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  • 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.
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  • Logical pluralism.Jc Beall & Greg Restall - 2000 - Australasian Journal of Philosophy 78 (4):475 – 493.
    Consequence is at the heart of logic; an account of consequence, of what follows from what, offers a vital tool in the evaluation of arguments. Since philosophy itself proceeds by way of argument and inference, a clear view of what logical consequence amounts to is of central importance to the whole discipline. In this book JC Beall and Greg Restall present and defend what thay call logical pluralism, the view that there is more than one genuine deductive consequence relation, a (...)
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  • Characterising dominated weak-operator continuous functionals on subspaces of B.Douglas S. 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.
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  • Too simple solutions of hard problems.Peter M. Schuster - 2010 - Nordic Journal of Philosophical Logic 6 (2):138-146.
    Even after yet another grand conjecture has been proved or refuted, any omniscience principle that had trivially settled this question is just as little acceptable as before. The significance of the constructive enterprise is therefore not affected by any gain of knowledge. In particular, there is no need to adapt weak counterexamples to mathematical progress.
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  • Pluralism and Proofs.Greg Restall - 2014 - Erkenntnis 79 (S2):279-291.
    Beall and Restall’s Logical Pluralism (2006) characterises pluralism about logical consequence in terms of the different ways cases can be selected in the analysis of logical consequence as preservation of truth over a class of cases. This is not the only way to understand or to motivate pluralism about logical consequence. Here, I will examine pluralism about logical consequence in terms of different standards of proof. We will focus on sequent derivations for classical logic, imposing two different restrictions on classical (...)
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  • Computable Real‐Valued Functions on Recursive Open and Closed Subsets of Euclidean Space.Qing Zhou - 1996 - Mathematical Logic Quarterly 42 (1):379-409.
    In this paper we study intrinsic notions of “computability” for open and closed subsets of Euclidean space. Here we combine together the two concepts, computability on abstract metric spaces and computability for continuous functions, and delineate the basic properties of computable open and closed sets. The paper concludes with a comprehensive examination of the Effective Riemann Mapping Theorem and related questions.
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  • Open subspaces of locally compact metric spaces.Mark Mandelkern - 1993 - Mathematical Logic Quarterly 39 (1):213-216.
    Although classically every open subspace of a locally compact space is also locally compact, constructively this is not generally true. This paper provides a locally compact remetrization for an open set in a compact metric space and constructs a one-point compactification. MSC: 54D45, 03F60, 03F65.
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  • Gödel's functional interpretation and its use in current mathematics.Ulrich Kohlenbach - 2008 - Dialectica 62 (2):223–267.
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  • Why do mathematicians re-prove theorems?John W. Dawson Jr - 2006 - Philosophia Mathematica 14 (3):269-286.
    From ancient times to the present, the discovery and presentation of new proofs of previously established theorems has been a salient feature of mathematical practice. Why? What purposes are served by such endeavors? And how do mathematicians judge whether two proofs of the same theorem are essentially different? Consideration of such questions illuminates the roles that proofs play in the validation and communication of mathematical knowledge and raises issues that have yet to be resolved by mathematical logicians. The Appendix, in (...)
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  • Strict Finitism and the Logic of Mathematical Applications.Feng Ye - 2011 - Dordrecht, Netherland: Springer.
    This book intends to show that radical naturalism, nominalism and strict finitism account for the applications of classical mathematics in current scientific theories. The applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry. The fact that so much applied mathematics can be developed within such a weak, strictly finitistic system, is surprising in itself. It also shows that the applications of those classical theories to the (...)
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  • Locatedness and overt sublocales.Bas Spitters - 2010 - Annals of Pure and Applied Logic 162 (1):36-54.
    Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected.Bishop defines a metric space to be compact if it is complete and totally bounded. A subset of a totally bounded set is again totally bounded iff it is located. So a closed subset of a Bishop compact (...)
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  • Reverse-engineering Reverse Mathematics.Sam Sanders - 2013 - Annals of Pure and Applied Logic 164 (5):528-541.
    An important open problem in Reverse Mathematics is the reduction of the first-order strength of the base theory from IΣ1IΣ1 to IΔ0+expIΔ0+exp. The system ERNA, a version of Nonstandard Analysis based on the system IΔ0+expIΔ0+exp, provides a partial solution to this problem. Indeed, weak Königʼs lemma and many of its equivalent formulations from Reverse Mathematics can be ‘pushed down’ into ERNA, while preserving the equivalences, but at the price of replacing equality with ‘≈’, i.e. infinitesimal proximity . The logical principle (...)
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  • Finitary sequence spaces.Mark Mandelkern - 1993 - Mathematical Logic Quarterly 39 (1):416-430.
    This paper studies the metric structure of the space Hr of absolutely summable sequences of real numbers with at most r nonzero terms. Hr is complete, and is located and nowhere dense in the space of all absolutely summable sequences. Totally bounded and compact subspaces of Hr are characterized, and large classes of located, totally bounded, compact, and locally compact subspaces are constructed. The methods used are constructive in the strict sense. MSC: 03F65, 54E50.
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