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1 — 50 / 321
  1. added 2018-11-20
    Intuitionistic Mereology.Paolo Maffezioli & Achille C. Varzi - 2018 - Synthese:1-26.
    Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import.
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  2. added 2018-11-09
    On the Intuitionistic Background of Gentzen's 1935 and 1936 Consistency Proofs and Their Philosophical Aspects.Yuta Takahashi - 2018 - Annals of the Japan Association for Philosophy of Science 27:1-26.
    Gentzen's three consistency proofs for elementary number theory have a common aim that originates from Hilbert's Program, namely, the aim to justify the application of classical reasoning to quantified propositions in elementary number theory. In addition to this common aim, Gentzen gave a “finitist” interpretation to every number-theoretic proposition with his 1935 and 1936 consistency proofs. In the present paper, we investigate the relationship of this interpretation with intuitionism in terms of the debate between the Hilbert School and the Brouwer (...)
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  3. added 2018-10-08
    Constructivism for Philosophers (Be It a Remark on Realism).Ofer Gal - 2002 - Perspectives on Science 10 (4):523-549.
    : Bereft of the illusion of an epistemic vantage point external to science, what should be our commitment towards the categories, concepts and terms of that very science? Should we, despaired of the possibility to found these concepts on rock bottom, adopt empiricist skepticism? Or perhaps the inexistence of external foundations implies, rather, immunity for scientific ontology from epistemological criticism? Philosophy's "realism debate" died out without providing a satisfactory answer to the dilemma, which was taken over by the neighboring disciplines. (...)
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  4. added 2018-09-29
    Actual and Potential Infinity.Øystein Linnebo & Stewart Shapiro - 2017 - Noûs.
    The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.
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  5. added 2018-06-06
    Poincaré on the Foundation of Geometry in the Understanding.Jeremy Shipley - 2017 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. Springer. pp. 19-37.
    This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study (...)
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  6. added 2018-02-18
    Structuralism, Invariance, and Univalence†: Articles.Steve Awodey - 2014 - Philosophia Mathematica 22 (1):1-11.
    The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...)
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  7. added 2018-02-18
    A Proof System for Fork Algebras and its Applications to Reasoning in Logics Based on Intuitionism.M. Frias & E. Orlowska - 1995 - Logique Et Analyse 150:151-152.
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  8. added 2018-01-20
    Logic of Probability and Conjecture.Harry Crane - unknown
    I introduce a formalization of probability which takes the concept of 'evidence' as primitive. In parallel to the intuitionistic conception of truth, in which 'proof' is primitive and an assertion A is judged to be true just in case there is a proof witnessing it, here 'evidence' is primitive and A is judged to be probable just in case there is evidence supporting it. I formalize this outlook by representing propositions as types in Martin-Lof type theory (MLTT) and defining a (...)
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  9. added 2017-10-25
    Intuitionistic Logic and its Philosophy.Panu Raatikainen - 2013 - Al-Mukhatabat. A Trilingual Journal For Logic, Epistemology and Analytical Philosophy (6):114-127.
  10. added 2017-09-03
    The Justification of Identity Elimination in Martin-Löf’s Type Theory.Ansten Klev - forthcoming - Topoi:1-14.
    On the basis of Martin-Löf’s meaning explanations for his type theory a detailed justification is offered of the rule of identity elimination. Brief discussions are thereafter offered of how the univalence axiom fares with respect to these meaning explanations and of some recent work on identity in type theory by Ladyman and Presnell.
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  11. added 2017-02-27
    Hilbert’s Program.Richard Zach - 2003 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University.
    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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  12. added 2017-02-16
    Non-Realism, Nominalism and Strict Finitism the Sheer Complexity of It All.Jean Paul Van Bendegem - 2006 - Poznan Studies in the Philosophy of the Sciences and the Humanities 90:343-365.
  13. added 2017-02-15
    Hilbert's Finitism and the Notion of Infinity.Karl-Georg Niebergall & Matthias Schirn - 2003 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press.
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  14. added 2017-02-14
    Undecidable Theories and Reverse Mathematics.James H. Schmerl - 2005 - In Stephen Simpson (ed.), Reverse Mathematics 2001. pp. 21--349.
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  15. added 2017-02-14
    Free Sets and Reverse Mathematics.Carl G. Jockusch Jr - 2005 - In Stephen Simpson (ed.), Reverse Mathematics 2001. pp. 104.
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  16. added 2017-02-14
    Dirk van Dalen. Mystic, Geometer, and Intuitionist-The Life of LEJ Brouwer, Volume 1, The Dawning Revolution. Clarendon Press, Oxford, 1999. Pp. Xv+ 440. [REVIEW]Hiroshi Kaneko - 2002 - Annals of the Japan Association for Philosophy of Science 11 (1):51-56.
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  17. added 2017-02-14
    On Maximal Continuity Regulators for Constructive Functions.A. O. Slisenko - 1969 - In Studies in Constructive Mathematics and Mathematical Logic. New York: Consultants Bureau. pp. 82--84.
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  18. added 2017-02-13
    Topics in Reverse Mathematics.Mariagnese Giusto - 2003 - In Benedikt Löwe, Thoralf Räsch & Wolfgang Malzkorn (eds.), Foundations of the Formal Sciences Ii. Kluwer Academic Publishers. pp. 63--87.
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  19. added 2017-02-13
    Response Sequences Following “Wrongs” in a Concept Task.Jean L. Bresnahan & Martin M. Shapiro - 1973 - Bulletin of the Psychonomic Society 2 (4):193-195.
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  20. added 2017-02-12
    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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  21. added 2017-02-12
    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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  22. added 2017-02-12
    Constructive Well‐Orderings.Robin J. Grayson - 1982 - Mathematical Logic Quarterly 28 (33‐38):495-504.
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  23. added 2017-02-11
    Reverse Mathematics and Properties of Finite Character.Damir D. Dzhafarov & Carl Mummert - 2012 - Annals of Pure and Applied Logic 163 (9):1243-1251.
  24. added 2017-02-11
    Aligning the Weak König Lemma, the Uniform Continuity Theorem, and Brouwer’s Fan Theorem.Josef Berger - 2012 - Annals of Pure and Applied Logic 163 (8):981-985.
  25. added 2017-02-11
    The Dirac Delta Function in Two Settings of Reverse Mathematics.Sam Sanders & Keita Yokoyama - 2012 - Archive for Mathematical Logic 51 (1-2):99-121.
    The program of Reverse Mathematics (Simpson 2009) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property ${\int_\mathbb{R}f(x)\delta(x)\,dx=f(0)}$ of the Dirac delta function. We show that (...)
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  26. added 2017-02-11
    Reverse Mathematics and Grundy Colorings of Graphs.James H. Schmerl - 2010 - Mathematical Logic Quarterly 56 (5):541-548.
    The relationship of Grundy and chromatic numbers of graphs in the context of Reverse Mathematics is investi-gated.
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  27. added 2017-02-11
    Exact Calculation of Inverse Functions.Josef Berger - 2005 - Mathematical Logic Quarterly 51 (2):201-205.
    We represent continuous functions on compact intervals by sequences of functions defined on finite sets of rational numbers. We call this an exact representation. This enables us to calculate the values of the function arbitrarily exactly, without roundoff errors. As an application we develop a procedure to transfer an exact representation of an increasing function into an exact representation of the corresponding inverse function.
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  28. added 2017-02-10
    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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  29. added 2017-02-08
    The MRDP Theorem.Peter Smith - unknown
    Here is Hilbert is his famous address of 1900: The supply of problems in mathematics is inexhaustible, and as soon as one problem is solved numerous others come forth in its place. Permit me in the following, tentatively as it were, to mention particular definite problems, drawn from various branches of mathematics, from the discussion of which an advancement of science may be expected.
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  30. added 2017-02-08
    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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  31. added 2017-02-08
    Ideology and Analysis, A Rehabilitation of Metaphysical Ontology. Par Richard C. Hinners Bruges-New York, Desclée de Brouwer. 1966. 275 Pages. [REVIEW]J. A. Tremblay - 1967 - Dialogue 6 (1):117-118.
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  32. added 2017-02-07
    Dna Sequences From Below: A Nominalist Approach.Yu Lin & Peter Simons - unknown
    We define DNA sequence by a bottom-up approach, starting with a real sequence from an actual biological sample. By providing axioms for notions of string, substring and strand, we formally define a DNA sequence, and a DNA molecule as composed of two antiparallel strands. We note that a sequence is a kind of group in which each member stands a certain relation to every other. The spatial aspects of a DNA sequence are also described.
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  33. added 2017-02-07
    Intuitionism, Excluded Middle and Decidability: A Response to Weir on Dummett: A Response to Weir on Dummett.Alexander George - 1988 - Mind 97 (388):597-602.
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  34. added 2017-02-02
    An Excerpt and Fragments From His Cambridge Lectures on Intuitionism (1951).Lej Brouwer - unknown
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  35. added 2017-02-02
    Remarks on Finitism.William Tait - manuscript
    The background of these remarks is that in 1967, in ‘’Constructive reasoning” [27], I sketched an argument that finitist arithmetic coincides with primitive recursive arithmetic, P RA; and in 1981, in “Finitism” [28], I expanded on the argument. But some recent discussions and some of the more recent literature on the subject lead me to think that a few further remarks would be useful.
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  36. added 2017-02-02
    Finitism in Geometry.Jean-Paul Van Bendegem - 2008 - Stanford Encyclopedia of Philosophy.
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  37. added 2017-02-02
    Dummett's Case for Constructivist Logicism.Peter Sullivan - 2007 - In Randall E. Auxier & Lewis Edwin Hahn (eds.), The Philosophy of Michael Dummett. Open Court. pp. 753--85.
    Self‐evidently the standard work on the topic its whole title defines, Sir Michael Dummett’s Frege: Philosophy of Mathematics (FPM) is also the most profound and creative discussion in recent decades of the problems confronting the branch of philosophy mentioned after the colon. Chapters 14‐18 and 23‐24 of this book constitute a continuous and challenging diagnosis of these problems.1 They culminate in the proposal that these problems present an impasse that can be escaped only by adopting a constructivist understanding of mathematical (...)
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  38. added 2017-01-31
    Discussion. Applied Constructive Mathematics: On Hellman's 'Mathematical Constructivism in Spacetime'.H. Billinge - 2000 - British Journal for the Philosophy of Science 51 (2):299-318.
  39. added 2017-01-30
    Classical Mathematics: A Concise History of the Classical Era in Mathematics. Joseph Ehrenfried Hofmann.Carolyn Eisele - 1962 - Isis 53 (2):261-262.
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  40. added 2017-01-29
    A Logic Of Sequences.Norihiro Kamide - 2011 - Reports on Mathematical Logic:29-57.
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  41. added 2017-01-29
    Van Dalen, D. : "Brower's Cambridge Lectures on Intuitionism". [REVIEW]Joan Weiner - 1984 - British Journal for the Philosophy of Science 35:90.
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  42. added 2017-01-29
    Brouwer's Cambridge Lectures on Intuitionism.D. van Dalen - 1984 - British Journal for the Philosophy of Science 35 (1):90-94.
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  43. added 2017-01-29
    DUMMETT, MICHAEL: "Elements of Intuitionism". [REVIEW]Peter Eggenberger - 1980 - British Journal for the Philosophy of Science 31:299.
    Reviews Dummett's "Elements of intuitionism".
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  44. added 2017-01-28
    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.
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  45. added 2017-01-28
    Omniscience, Sequential Compactness, and the Anti-Specker Property.Douglas Bridges - 2011 - Logic Journal of the IGPL 19 (1):53-61.
    Working within Bishop-style constructive mathematics, we derive a number of results relating the nonconstructive LPO and sequential compactness property on the one hand, and the intuitionistically reasonable anti-Specker property on the other.
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  46. added 2017-01-28
    Indecomposability of Negative Dense Subsets of ℝ in Constructive Reverse Mathematics.Iris Loeb - 2009 - Logic Journal of the IGPL 17 (2):173-177.
    In 1970 Vesley proposed a substitute of Kripke's Scheme. In this paper it is shown that —over Bishop's constructive mathematics— the indecomposability of negative dense subsets of ℝ is equivalent to a weakening of Vesley's proposal. This result supports the idea that full Kripke's Scheme might not be necessary for most of intuitionistic mathematics. At the same time it contributes to the programme of Constructive Reverse Mathematics and gives a new answer to a 1997 question of Van Dalen.
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  47. added 2017-01-28
    Indecomposability of ℝ and ℝ \ {0} in Constructive Reverse Mathematics.Iris Loeb - 2008 - Logic Journal of the IGPL 16 (3):269-273.
    It is shown that—over Bishop's constructive mathematics—the indecomposability of ℝ is equivalent to the statement that all functions from a complete metric space into a metric space are sequentially nondiscontinuous. Furthermore we prove that the indecomposability of ℝ \ {0} is equivalent to the negation of the disjunctive version of Markov's Principle. These results contribute to the programme of Constructive Reverse Mathematics.
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  48. added 2017-01-28
    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.
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  49. added 2017-01-28
    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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  50. added 2017-01-28
    Weak Continuity Properties in Constructive Analysis.D. Bridges & L. Dediu - 1999 - Logic Journal of the IGPL 7 (3):277-281.
    Within Bishop's constructive mathematics we provide conditions that ensure weak continuity properties of mappings between metric and normed spaces.
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1 — 50 / 321