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  1. Numerical Infinities and Infinitesimals: Methodology, Applications, and Repercussions on Two Hilbert Problems.Yaroslav Sergeyev - 2017 - EMS Surveys in Mathematical Sciences 4 (2):219–320.
    In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...)
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  2. Genetic Counseling in Historical Perspective: Understanding Our Hereditary Past and Forecasting Our Genomic Future. [REVIEW]618 622 - 2013 - Studies in History and Philosophy of Science Part A 44 (4):Devon-Stillwell.
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  3. Subregular Tetrahedra.John Corcoran - 2008 - Bulletin of Symbolic Logic 14 (3):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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  4. A Logic Of Sequences.Norihiro Kamide - 2011 - Reports on Mathematical Logic:29-57.
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  5. Logicism Reconsidered.Patricia A. Blanchette - 1990 - 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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  6. Finitism: An Essay on Hilbert's Programme.David Watson Galloway - 1991 - 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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  7. Strict Constructivism and the Philosophy of Mathematics.Feng Ye - 2000 - 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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  8. Facets of Infinity: A Theory of Finitistic Truth.Zlatan Damnjanovic - 1992 - Dissertation, Princeton University
    The thesis critically examines the question of the philosophical coherence of finitism, the view which seeks to interpret mathematics without postulating an actual infinity of mathematical objects. It is argued that a widely accepted characterization of finitism, most recently expounded by Tait, is inadequate, and a new characterization based on the notion of elementary abstraction is proposed. It is further argued that the notion of elementary abstraction better explains the bearing of Godel's incompleteness theorems on the issue of the coherence (...)
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  9. The Social Problem: A Constructive Analysis.Charles A. Ellwood - 1916 - Journal of Philosophy, Psychology and Scientific Methods 13 (3):81-82.
  10. Errett Bishop Reflections on Him and His Research.Murray Rosenblatt & Errett Bishop - 1985
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  11. The Foundations of Intuitionistic Mathematics. [REVIEW]J. M. P. - 1965 - Review of Metaphysics 19 (1):154-155.
  12. Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer. Volume 1: The Dawning Revolution. [REVIEW]Diederick Raven - 2001 - British Journal for the History of Science 34 (1):97-124.
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  13. Frege’s Theorem by Richard G. Heck, Jr. [REVIEW]John P. Burgess - 2012 - Journal of Philosophy 109 (12):728-732.
  14. Hilbert's Program: An Essay on Mathematical Instrumentalism by Michael Detlefsen. [REVIEW]Mark Steiner - 1991 - Journal of Philosophy 88 (6):331-336.
  15. Cofinally Invariant Sequences and Revision.Edoardo Rivello - 2015 - 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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  16. Brouwer Versus Hilbert: 1907–1928.J. Posy Carl - 1998 - Science in Context 11 (2):291.
  17. Avigad Jeremy. Update Procedures and the 1-Consistency of Arithmetic. Mathematical Logic Quarterly, Vol. 48 (2002), Pp. 3–13. [REVIEW]Toshiyasu Arai - 2003 - Bulletin of Symbolic Logic 9 (1):45-47.
  18. 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]Mark van Atten - 2004 - Bulletin of Symbolic Logic 10 (3):423-427.
  19. Penelope Maddy, Naturalism in Mathematics.N. Tennant - 2000 - Philosophia Mathematica 8 (3):316-338.
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  20. 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]Jaap Van Oosten - 2006 - Bulletin of Symbolic Logic 12 (4):611-612.
  21. Mathematics in and Behind Russell's Logicism and its Reception'.I. Grattan-Guinness - 2003 - In Nicholas Griffin (ed.), Bulletin of Symbolic Logic. Cambridge University Press. pp. 51.
  22. Lebesgue Integral in Constructive Analysis.Oswald Demuth - 1969 - In A. O. Slisenko (ed.), Studies in Constructive Mathematics and Mathematical Logic. New York: Consultants Bureau. pp. 9--14.
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  23. On Constructive Groups.V. A. Lifshits - 1969 - In A. O. Slisenko (ed.), Studies in Constructive Mathematics and Mathematical Logic. New York: Consultants Bureau. pp. 32--35.
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  24. About Brouwer's Fan Theorem.Thierry Coquand - 2004 - Revue Internationale de Philosophie 230:483-489.
  25. Arithmetic of Infinity.Yaroslav D. Sergeyev - 2013 - E-book.
    This book presents a new type of arithmetic that allows one to execute arithmetical operations with infinite numbers in the same manner as we are used to do with finite ones. The problem of infinity is considered in a coherent way different from (but not contradicting to) the famous theory founded by Georg Cantor. Surprisingly, the introduced arithmetical operations result in being very simple and are obtained as immediate extensions of the usual addition, multiplication, and division of finite numbers to (...)
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  26. A Characterization of Constructive Dimension.Satyadev Nandakumar - 2009 - Mathematical Logic Quarterly 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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  27. Brouwer's Weak Counterexamples and Testability: Further Remarks.Charles Mccarty - 2013 - 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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  28. Lawless Mind.John Martin Fischer - 1991 - Philosophical Books 32 (4):240-241.
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  29. Unavoidable Sequences in Constructive Analysis.Joan Rand Moschovakis - 2010 - 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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  30. 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.
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  31. 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 (...)
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  32. On Local Non‐Compactness in Recursive Mathematics.Jakob G. Simonsen - 2006 - 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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  33. Constructive Complements of Unions of Two Closed Sets.Douglas S. Bridges - 2004 - 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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  34. On Weak Markov's Principle.Ulrich Kohlenbach - 2002 - 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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  35. Constructive and Locally Constructive Endofunctors on the Category of Enumerated Sets.Andrzej Orlicki - 1987 - Mathematical Logic Quarterly 33 (4):371-384.
  36. Fundamental Sequences for Exponential Polynomials.Rod McBeth - 1980 - Mathematical Logic Quarterly 26 (7‐9):115-122.
  37. 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.
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  38. Brouwer's Fan Theorem and Unique Existence in Constructive Analysis.Josef Berger & Hajime Ishihara - 2005 - 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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  39. The Problem of the Formalization of Constructive Topology.Silvio Valentini - 2004 - Archive for Mathematical Logic 44 (1):115-129.
  40. Penelope Maddy * Second Philosophy: A Naturalistic Method.Harvey Siegel - 2010 - British Journal for the Philosophy of Science 61 (4):897-903.
  41. Consistency, Models, and Soundness.Matthias Schirn - 2010 - 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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  42. The Consistency of Arithmetic, Based on a Logic of Meaning Containment.Ross T. Brady - 2012 - Logique Et Analyse 55 (219).
  43. The Intermediate Value Theorem in Constructive Mathematics Without Choice.Matthew Hendtlass - 2012 - Annals of Pure and Applied Logic 163 (8):1050-1056.
  44. Richard G. Heck, Jr.: Frege’s Theorem. [REVIEW]John P. Burgess - 2012 - Journal of Philosophy 109 (12):728-733.
  45. Brève note sur l'intuitionnisme de Brouwer.J. Largeault - 1992 - Revue Philosophique de la France Et de l'Etranger 182 (3):317 - 324.
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  46. Dummett on Impredicativity.Alan Weir - 1998 - 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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  47. Chateaubriand's Logicism.Abel Casanave - 2004 - 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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  48. The Foundations of Arithmetic.Brian Coffey - 1952 - Modern Schoolman 29 (2):157-157.
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  49. Truth Through Proof: A Formalist Foundation for Mathematics * by Alan Weir.Z. Damnjanovic - 2012 - Analysis 72 (2):415-418.
  50. What Finitism Could Not Be (Lo Que El Finitismo No Podría Ser).Matthias Schirn & Karl-Georg Niebergall - 2003 - Critica 35 (103):43 - 68.
    In his paper "Finitism" (1981), W.W. Tait maintains that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentially primitive recursive. In this paper, we attempt to show that his thesis "The finitist functions are precisely the primitive recursive functions" is disputable and that another, likewise defended by (...)
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1 — 50 / 809