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  1. Toshiyasu Arai (2003). Avigad Jeremy. Update Procedures and the 1-Consistency of Arithmetic. Mathematical Logic Quarterly, Vol. 48 (2002), Pp. 3–13. [REVIEW] Bulletin of Symbolic Logic 9 (1):45-47.
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  2. 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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  3. Josef Berger & Hajime Ishihara (2005). Brouwer's Fan Theorem and Unique Existence in Constructive Analysis. Mathematical Logic Quarterly 51 (4):360-364.
    Many existence propositions in constructive analysis are implied by the lesser limited principle of omniscience LLPO; sometimes one can even show equivalence. It was discovered recently that some existence propositions are equivalent to Bouwer's fan theorem FAN if one additionally assumes that there exists at most one object with the desired property. We are providing a list of conditions being equivalent to FAN, such as a unique version of weak König's lemma. This illuminates the relation between FAN and LLPO. Furthermore, (...)
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  4. Patricia A. Blanchette (1990). Logicism Reconsidered. Dissertation, Stanford University
    This thesis is an examination of Frege's logicism, and of a number of objections which are widely viewed as refutations of the logicist thesis. In the view offered here, logicism is designed to provide answers to two questions: that of the nature of arithmetical truth, and that of the source of arithmetical knowledge. ;The first objection dealt with here is the view that logicism is not an epistemologically significant thesis, due to the fact that the epistemological status of logic itself (...)
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  5. Ross T. Brady (2012). The Consistency of Arithmetic, Based on a Logic of Meaning Containment. Logique Et Analyse 55 (219).
  6. D. S. Bridges (2004). Constructive Complements of Unions of Two Closed Sets. Mathematical Logic Quarterly 50 (3):293.
    It is well known that in Bishop-style constructive mathematics, the closure of the union of two subsets of ℝ is ‘not’ the union of their closures. The dual situation, involving the complement of the closure of the union, is investigated constructively, using completeness of the ambient space in order to avoid any application of Markov's Principle.
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  7. 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. 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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  9. 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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  10. John P. Burgess (2012). Frege’s Theorem by Richard G. Heck, Jr. Journal of Philosophy 109 (12):728-732.
  11. John P. Burgess (2012). Richard G. Heck, Jr.: Frege’s Theorem. [REVIEW] Journal of Philosophy 109 (12):728-733.
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  12. J. Posy Carl (1998). Brouwer Versus Hilbert: 1907–1928. Science in Context 11 (2):291.
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  13. Abel Casanave (2004). Chateaubriand's Logicism. Manuscrito 27 (1):13-20.
    In his doctoral dissertation, O. Chateaubriand favored Dedekind’s analysis of the notion of number; whereas in Logical Forms, he favors a fregean approach to the topic. My aim in this paper is to examine the kind of logicism he defends. Three aspects will be considered: the concept of analysis; the universality of arithmetical properties and their definability; the irreducibility of arithmetical objects.
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  14. Brian Coffey (1952). The Foundations of Arithmetic. Modern Schoolman 29 (2):157-157.
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  15. Thierry Coquand (2004). About Brouwer's Fan Theorem. Revue Internationale de Philosophie 230:483-489.
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  16. John Corcoran (2008). Subregular Tetrahedra. Bulletin of Symbolic Logic 14:411-2.
    This largely expository lecture deals with aspects of traditional solid geometry suitable for applications in logic courses. Polygons are plane or two-dimensional; the simplest are triangles. Polyhedra [or polyhedrons] are solid or three-dimensional; the simplest are tetrahedra [or triangular pyramids, made of four triangles]. -/- A regular polygon has equal sides and equal angles. A polyhedron having congruent faces and congruent [polyhedral] angles is not called regular, as some might expect; rather they are said to be subregular—a word coined for (...)
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  17. James X. Corgan (1968). Fossils and Evolution General Palaeontology A. Brouwer R. H. Kaye. BioScience 18 (3):250-250.
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  18. Z. Damnjanovic (2012). Truth Through Proof: A Formalist Foundation for Mathematics * by Alan Weir. Analysis 72 (2):415-418.
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  19. Oswald Demuth (1969). Lebesgue Integral in Constructive Analysis. In A. O. Slisenko (ed.), Studies in Constructive Mathematics and Mathematical Logic. New York, Consultants Bureau 9--14.
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  20. Charles A. Ellwood (1916). The Social Problem: A Constructive Analysis. Journal of Philosophy, Psychology and Scientific Methods 13 (3):81-82.
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  21. Solomon Feferman, Challenges to Predicative Foundations of Arithmetic.
    This is a sequel to our article “Predicative foundations of arithmetic” (1995), referred to in the following as [PFA]; here we review and clarify what was accomplished in [PFA], present some improvements and extensions, and respond to several challenges. The classic challenge to a program of the sort exemplified by [PFA] was issued by Charles Parsons in a 1983 paper, subsequently revised and expanded as Parsons (1992). Another critique is due to Daniel Isaacson (1987). Most recently, Alexander George and Daniel (...)
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  22. A. C. Fox (1924). Number System of Arithmetic and Algebra. [REVIEW] Australasian Journal of Philosophy 2:71.
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  23. David Watson Galloway (1991). Finitism: An Essay on Hilbert's Programme. Dissertation, Massachusetts Institute of Technology
    In this thesis, I discuss the philosophical foundations of Hilbert's Consistency Programme of the 1920's, in the light of the incompleteness theorems of Godel. ;I begin by locating the Consistency Programme within Hilbert's broader foundational project. I show that Hilbert's main aim was to establish that classical mathematics, and in particular classical analysis, is a conservative extension of finitary mathematics. Accepting the standard identification of finitary mathematics with primitive recursive arithmetic, and classical analysis with second order arithmetic, I report upon (...)
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  24. I. Grattan-Guinness (2003). Mathematics in and Behind Russell's Logicism and its Reception'. In Nicholas Griffin (ed.), Bulletin of Symbolic Logic. Cambridge University Press 51.
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  25. Matthew Hendtlass (2012). The Intermediate Value Theorem in Constructive Mathematics Without Choice. Annals of Pure and Applied Logic 163 (8):1050-1056.
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  26. Norihiro Kamide (2011). A Logic Of Sequences. Reports on Mathematical Logic:29-57.
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  27. U. Kohlenbach (2002). On Weak Markov's Principle. Mathematical Logic Quarterly 48 (S1):59-65.
    We show that the so-called weak Markov's principle which states that every pseudo-positive real number is positive is underivable in [MATHEMATICAL SCRIPT CAPITAL T]ω ≔ E-HAω + AC. Since [MATHEMATICAL SCRIPT CAPITAL T]ω allows one to formalize Bishop's constructive mathematics, this makes it unlikely that WMP can be proved within the framework of Bishop-style mathematics . The underivability even holds if the ine.ective schema of full comprehension for negated formulas is added, which allows one to derive the law of excluded (...)
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  28. J. Largeault (1992). Brève note sur l'intuitionnisme de Brouwer. Revue Philosophique de la France Et de l'Etranger 182 (3):317 - 324.
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  29. V. A. Lifshits (1969). On Constructive Groups. In A. O. Slisenko (ed.), Studies in Constructive Mathematics and Mathematical Logic. New York, Consultants Bureau 32--35.
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  30. Rod McBeth (1980). Fundamental Sequences for Exponential Polynomials. Mathematical Logic Quarterly 26 (7‐9):115-122.
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  31. Charles Mccarty (2013). Brouwer's Weak Counterexamples and Testability: Further Remarks. Review of Symbolic Logic 6 (3):513-523.
    Straightforwardly and strictly intuitionistic inferences show that the BrouwerKolmogorov (BHK) interpretation, in the presence of a formulation of the recognition principle, entails the validity of the Law of Testability: that the form s original weak counterexample reasoning was fallacious. The results of the present article extend and refine those of McCarty, C. (2012). Antirealism and Constructivism: Brouwer’s Weak Counterexamples. The Review of Symbolic Logic. First View. Cambridge University Press.
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  32. Joan Rand Moschovakis (2010). Unavoidable Sequences in Constructive Analysis. Mathematical Logic Quarterly 56 (2):205-215.
    Five recursively axiomatizable theories extending Kleene's intuitionistic theory FIM of numbers and numbertheoretic sequences are introduced and shown to be consistent, by a modified relative realizability interpretation which verifies that every sequence classically defined by a Π11 formula is unavoidable and that no sequence can fail to be classically Δ11. The analytical form of Markov's Principle fails under the interpretation. The notion of strongly inadmissible rule of inference is introduced, with examples.
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  33. Satyadev Nandakumar (2009). A Characterization of Constructive Dimension. Mlq 55 (2):185-200.
    In the context of Kolmogorov's algorithmic approach to the foundations of probability, Martin-Löf defined the concept of an individual random sequence using the concept of a constructive measure 1 set. Alternate characterizations use constructive martingales and measures of impossibility. We prove a direct conversion of a constructive martingale into a measure of impossibility and vice versa such that their success sets, for a suitably defined class of computable probability measures, are equal. The direct conversion is then generalized to give a (...)
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  34. Andrzej Orlicki (1987). Constructive and Locally Constructive Endofunctors on the Category of Enumerated Sets. Mathematical Logic Quarterly 33 (4):371-384.
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  35. J. M. P. (1965). The Foundations of Intuitionistic Mathematics. [REVIEW] Review of Metaphysics 19 (1):154-155.
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  36. Diederick Raven (2001). Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer. Volume 1: The Dawning Revolution. [REVIEW] British Journal for the History of Science 34 (1):97-124.
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  37. Edoardo Rivello (2015). Cofinally Invariant Sequences and Revision. Studia Logica 103 (3):599-622.
    Revision sequences are a kind of transfinite sequences which were introduced by Herzberger and Gupta in 1982 as the main mathematical tool for developing their respective revision theories of truth. We generalise revision sequences to the notion of cofinally invariant sequences, showing that several known facts about Herzberger’s and Gupta’s theories also hold for this more abstract kind of sequences and providing new and more informative proofs of the old results.
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  38. Murray Rosenblatt & Errett Bishop (1985). Errett Bishop Reflections on Him and His Research. Monograph Collection (Matt - Pseudo).
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  39. Matthias Schirn (2010). Consistency, Models, and Soundness. Axiomathes 20 (2-3):153-207.
    This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. (...)
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  40. Yaroslav Sergeyev (2007). Blinking Fractals and Their Quantitative Analysis Using Infinite and Infinitesimal Numbers. Chaos, Solitons and Fractals 33 (1):50-75.
    The paper considers a new type of objects – blinking fractals – that are not covered by traditional theories studying dynamics of self-similarity processes. It is shown that the new approach allows one to give various quantitative characteristics of the newly introduced and traditional fractals using infinite and infinitesimal numbers proposed recently. In this connection, the problem of the mathematical modelling of continuity is discussed in detail. A strong advantage of the introduced computational paradigm consists of its well-marked numerical character (...)
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  41. Harvey Siegel (2010). Penelope MaddySecond Philosophy: A Naturalistic Method. British Journal for the Philosophy of Science 61 (4):897-903.
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  42. Jakob G. Simonsen (2006). On Local Non‐Compactness in Recursive Mathematics. Mathematical Logic Quarterly 52 (4):323-330.
    A metric space is said to be locally non-compact if every neighborhood contains a sequence that is eventually bounded away from every element of the space, hence contains no accumulation point. We show within recursive mathematics that a nonvoid complete metric space is locally non-compact iff it is without isolated points.The result has an interesting consequence in computable analysis: If a complete metric space has a computable witness that it is without isolated points, then every neighborhood contains a computable sequence (...)
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  43. Mark Steiner (1991). Hilbert's Program: An Essay on Mathematical Instrumentalism by Michael Detlefsen. Journal of Philosophy 88 (6):331-336.
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  44. N. Tennant (2000). Penelope Maddy, Naturalism in Mathematics. Philosophia Mathematica 8 (3):316-338.
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  45. Silvio Valentini (2005). The Problem of the Formalization of Constructive Topology. Archive for Mathematical Logic 44 (1):115-129.
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  46. Mark van Atten (2004). Hesseling Dennis E.. Gnomes in the Fog. The Reception of Brouwer's Intuitionism in the 1920s. Science Networks. Historical Studies, Vol. 28. Birkhäuser, Boston, 2003, Xxiii+ 447 Pp. [REVIEW] Bulletin of Symbolic Logic 10 (3):423-427.
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  47. 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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  48. Alan Weir (1998). Dummett on Impredicativity. Grazer Philosophische Studien 55:65-101.
    Gödel and others held that impredicative specification is illegitimate in a constructivist framework but legitimate elsewhere. Michael Dummett argues to the contrary that impredicativity, though not necessarily illicit, needs justification regardless of whether one assumes the context is realist or constructivist. In this paper I defend the Gödelian position arguing that Dummett seeks a reduction of impredicativity to predicativity which is neither possible nor necessary. The argument is illustrated by considering first highly predicative versions of the equinumerosity axiom for cardinal (...)
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  49. Feng Ye (2000). Strict Constructivism and the Philosophy of Mathematics. Dissertation, Princeton University
    The dissertation studies the mathematical strength of strict constructivism, a finitistic fragment of Bishop's constructivism, and explores its implications in the philosophy of mathematics. ;It consists of two chapters and four appendixes. Chapter 1 presents strict constructivism, shows that it is within the spirit of finitism, and explains how to represent sets, functions and elementary calculus in strict constructivism. Appendix A proves that the essentials of Bishop and Bridges' book Constructive Analysis can be developed within strict constructivism. Appendix B further (...)
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Logicism in Mathematics
  1. H. Leitgeb A. Hieke (ed.) (2009). Reduction – Abstraction – Analysis. Ludwig Wittgenstein Society.
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