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  1. Why Logical Pluralism?Colin R. Caret - forthcoming - Synthese:1-22.
    This paper scrutinizes the debate over logical pluralism. I hope to make this debate more tractable by addressing the question of motivating data: what would count as strong evidence in favor of logical pluralism? Any research program should be able to answer this question, but when faced with this task, many logical pluralists fall back on brute intuitions. This sets logical pluralism on a weak foundation and makes it seem as if nothing pressing is at stake in the debate. The (...)
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  2. The Entanglement of Logic and Set Theory, Constructively.Laura Crosilla - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    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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  3. Choice Sequences and the Continuum.Casper Storm Hansen - forthcoming - Erkenntnis:1-18.
    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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  4. Intuitionistic Mereology.Paolo Maffezioli & Achille C. Varzi - forthcoming - 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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  5. Constructive Thinking.B. Thayer-Bacon & C. Thayer-Bacon - forthcoming - Philosophy.
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  6. A Dilemma for Mathematical Constructivism.Samuel Kahn - 2021 - Axiomathes 31 (1):63-72.
    In this paper I argue that constructivism in mathematics faces a dilemma. In particular, I maintain that constructivism is unable to explain (i) the application of mathematics to nature and (ii) the intersubjectivity of mathematics unless (iii) it is conjoined with two theses that reduce it to a form of mathematical Platonism. The paper is divided into five sections. In the first section of the paper, I explain the difference between mathematical constructivism and mathematical Platonism and I outline my argument. (...)
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  7. Ludwig Wittgenstein, Dictating Philosophy To Francis Skinner: The Wittgenstein-Skinner Manuscripts. Transcribed and Edited, with an Introduction, Introductory Chapters and Notes by Arthur Gibson.Arthur Gibson & Niamh O'Mahony (eds.) - 2020, December 1 - Berlin, Germany: Springer.
  8. Sense, Reference, and Computation.Bruno Bentzen - 2020 - Perspectiva Filosófica 47 (2):179-203.
    In this paper, I revisit Frege's theory of sense and reference in the constructive setting of the meaning explanations of type theory, extending and sharpening a program–value analysis of sense and reference proposed by Martin-Löf building on previous work of Dummett. I propose a computational identity criterion for senses and argue that it validates what I see as the most plausible interpretation of Frege's equipollence principle for both sentences and singular terms. Before doing so, I examine Frege's implementation of his (...)
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  9. Logical Revision by Counterexamples: A Case Study of the Paraconsistent Counterexample to Ex Contradictione Quodlibet.Seungrak Choi - 2019 - In Proceedings of the 14th and 15th Asian Logic Conferences. pp. 141-167.
    It is often said that a correct logical system should have no counterexample to its logical rules and the system must be revised if its rules have a counterexample. If a logical system (or theory) has a counterexample to its logical rules, do we have to revise the system? In this paper, focussing on the role of counterexamples to logical rules, we deal with the question. -/- We investigate two mutually exclusive theories of arithmetic - intuitionistic and paraconsistent theories. The (...)
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  10. Indeterminism in Physics, Classical Chaos and Bohmian Mechanics.\\ Are Real Numbers Really Real?Nicolas Gisin - 2019 - Erkenntnis:1-13.
    It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is (...)
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  11. Considerações de Brouwer sobre espaço e infinitude: O idealismo de Brouwer Diante do Problema Apresentado por Dummett Quanto à Possibilidade Teórica de uma Infinitude Espacial.Paulo Júnio de Oliveira - 2019 - Kinesis 11:94-108.
    Resumo Neste artigo, será discutida a noção de “infinitude cardinal” – a qual seria predicada de um “conjunto” – e a noção de “infinitude ordinal” – a qual seria predicada de um “processo”. A partir dessa distinção conceitual, será abordado o principal problema desse artigo, i.e., o problema da possibilidade teórica de uma infinitude de estrelas tratado por Dummett em sua obra Elements of Intuitionism. O filósofo inglês sugere que, mesmo diante dessa possibilidade teórica, deveria ser possível predicar apenas infinitude (...)
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  12. The Justification of Identity Elimination in Martin-Löf’s Type Theory.Ansten Klev - 2019 - Topoi 38 (3):577-590.
    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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  13. Eta-Rules in Martin-Löf Type Theory.Ansten Klev - 2019 - Bulletin of Symbolic Logic 25 (3):333-359.
    The eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Löf type theory. They are, however, justified by the meaning explanations, and a higher-order eta rule is part of that type theory. The main aim of this paper is to clarify this somewhat puzzling situation. It will be argued that lower-order eta rules do not, whereas the (...)
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  14. Rumfitt on the Logic of Set Theory.Øystein Linnebo - 2019 - Inquiry: An Interdisciplinary Journal of Philosophy 62 (7):826-841.
    ABSTRACTAccording to a famous argument by Dummett, the concept of set is indefinitely extensible, and the logic appropriate for reasoning about the instances of any such concept is intuitionistic, not classical. But Dummett's argument is widely regarded as obscure. This note explains how the final chapter of Rumfitt's important new book advances our understanding of Dummett's argument, but it also points out some problems and unanswered questions. Finally, Rumfitt's reconstruction of Dummett's argument is contrasted with my own preferred alternative.
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  15. Actual and Potential Infinity.Øystein Linnebo & Stewart Shapiro - 2019 - Noûs 53 (1):160-191.
    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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  16. Inference Rules and the Meaning of the Logical Constants.Hermógenes Oliveira - 2019 - Dissertation, Eberhard Karls Universität Tübingen
    The dissertation provides an analysis and elaboration of Michael Dummett's proof-theoretic notions of validity. Dummett's notions of validity are contrasted with standard proof-theoretic notions and formally evaluated with respect to their adequacy to propositional intuitionistic logic.
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  17. Constructive Mathematics and Equality.Bruno Bentzen - 2018 - Dissertation, Sun Yat-Sen University
    The aim of the present thesis is twofold. First we propose a constructive solution to Frege's puzzle using an approach based on homotopy type theory, a newly proposed foundation of mathematics that possesses a higher-dimensional treatment of equality. We claim that, from the viewpoint of constructivism, Frege's solution is unable to explain the so-called ‘cognitive significance' of equality statements, since, as we shall argue, not only statements of the form 'a = b', but also 'a = a' may contribute to (...)
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  18. Wittgenstein on Cantor's Proof.Chrysoula Gitsoulis - 2018 - In Gabriele M. Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics: Contributions of the 41st International Wittgenstein Symposium. pp. 67-69.
    Cantor’s proof that the reals are uncountable forms a central pillar in the edifices of higher order recursion theory and set theory. It also has important applications in model theory, and in the foundations of topology and analysis. Due partly to these factors, and to the simplicity and elegance of the proof, it has come to be accepted as part of the ABC’s of mathematics. But even if as an Archimedean point it supports tomes of mathematical theory, there is a (...)
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  19. 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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  20. 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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  21. Brouwer's Conception of Truth.Casper Storm Hansen - 2016 - Philosophia Mathematica 24 (3):379-400.
    In this paper it is argued that the understanding of Brouwer as replacing truth conditions with assertability or proof conditions, in particular as codified in the so-called Brouwer-Heyting-Kolmogorov Interpretation, is misleading and conflates a weak and a strong notion of truth that have to be kept apart to understand Brouwer properly: truth-as-anticipation and truth- in-content. These notions are explained, exegetical documentation provided, and semi-formal recursive definitions are given.
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  22. Differential Calculus Based on the Double Contradiction.Kazuhiko Kotani - 2016 - Open Journal of Philosophy 6 (4):420-427.
    The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...)
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  23. Brouwer’s Certainties: Mysticism, Mathematics, and the Ego: Dirk van Dalen: L. E. J. Brouwer: Topologist, Intuitionist, Philosopher—How Mathematics is Rooted in Life. London, Heidelberg, Dordrecht: Springer, 2013, Xii+875pp, 97 Illus., £24.95 HB.Jeremy Gray - 2015 - Metascience 24 (1):127-134.
    The lives of few mathematicians offer the drama that is presented by the life of L. E. J. Brouwer, correctly identified on the cover of this book as a topologist, intuitionist, and philosopher, and before we go any further, it will be worth indicating why.It is not just that Brouwer would rank high among mathematicians for his work in topology alone: he set standards for rigour and created a theory of dimension for topological spaces, and his fixed-point theorem is of (...)
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  24. L. E. J. Brouwer and Karl Popper: Two Perspectives on Mathematics.Alexander John Naraniecki - 2015 - Cosmos and History 11 (1):239-255.
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  25. Constructive Realism in Mathematics.Ilkka Niiniluoto - 2015 - In Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics. De Gruyter. pp. 339-354.
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  26. 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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  27. Frege Meets Brouwer.Stewart Shapiro & Øystein Linnebo - 2015 - Review of Symbolic Logic 8 (3):540-552.
    We show that, by choosing definitions carefully, a version of Frege's theorem can be proved in intuitionistic logic.
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  28. Mysticism and Mathematics: Brouwer, Gödel, and the Common Core Thesis.Robert Tragesser, Mark van Atten & Mark Atten - 2015 - In Robert Tragesser, Mark van Atten & Mark Atten (eds.), Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer. Springer Verlag.
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  29. Gödel and Brouwer: Two Rivalling Brothers.Mark van Atten & Mark Atten - 2015 - In Mark van Atten & Mark Atten (eds.), Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer. Springer Verlag.
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  30. On A.A. Markov’s Attitude Towards Brouwer’s Intuitionism.Ioannis M. Vandoulakis - 2015 - Philosophia Scientae 19:143-158.
    The paper examines Andrei A. Markov’s critical attitude towards L.E.J. Brouwer’s intuitionism, as is expressed in his endnotes to the Russian translation of Heyting’s Intuitionism, published in Moscow in 1965. It is argued that Markov’s algorithmic approach was shaped under the impact of the mathematical style and values prevailing in the Petersburg mathematical school, which is characterized by the proclaimed primacy of applications and the search for rigor and effective solutions.
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  31. Structuralism, Invariance, and Univalence.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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  32. Reverse Mathematics and Isbell's Zig-Zag Theorem.Takashi Sato - 2014 - Mathematical Logic Quarterly 60 (4-5):348-353.
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  33. Why Did Weyl Think That Formalism's Victory Against Intuitionism Entails a Defeat of Pure Phenomenology?Iulian D. Toader - 2014 - History and Philosophy of Logic 35 (2):198-208.
    This paper argues that Weyl took formalism to prevail over intuitionism with respect to supporting scientific objectivity, rather than grounding classical mathematics, and that he thought this was enough for rejecting pure phenomenology as well.
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  34. Brouwer’s Fan Theorem as an Axiom and as a Contrast to Kleene’s Alternative.Wim Veldman - 2014 - Archive for Mathematical Logic 53 (5-6):621-693.
    The paper is a contribution to intuitionistic reverse mathematics. We introduce a formal system called Basic Intuitionistic MathematicsBIM, and then search for statements that are, over BIM, equivalent to Brouwer’s Fan Theorem or to its positive denial, Kleene’s Alternative to the Fan Theorem. The Fan Theorem is true under the intended intuitionistic interpretation and Kleene’s Alternative is true in the model of BIM consisting of the Turing-computable functions. The task of finding equivalents of Kleene’s Alternative is, intuitionistically, a nontrivial extension (...)
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  35. Review of M. Van Atten, P. Boldini, M. Bourdeau, and G. Heinzmann (Eds.), _One Hundred Years of Intuitionism (1907–2007): The Cerisy Conference. [REVIEW]J. L. Bell - 2013 - Philosophia Mathematica 21 (3):392-399.
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  36. A First Constructive Look at the Comparison of Projections.D. S. Bridges & L. S. Vita - 2013 - Logic Journal of the IGPL 21 (1):14-27.
  37. Empirical Negation.Michael De - 2013 - Acta Analytica 28 (1):49-69.
    An extension of intuitionism to empirical discourse, a project most seriously taken up by Dummett and Tennant, requires an empirical negation whose strength lies somewhere between classical negation (‘It is unwarranted that. . . ’) and intuitionistic negation (‘It is refutable that. . . ’). I put forward one plausible candidate that compares favorably to some others that have been propounded in the literature. A tableau calculus is presented and shown to be strongly complete.
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  38. Sir Michael Anthony Eardley Dummett, 1925-2011.R. G. Heck - 2013 - Philosophia Mathematica 21 (1):1-8.
    A remembrance of Dummett's work on philosophy of mathematcis.
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  39. Modal-Epistemic Arithmetic and the Problem of Quantifying In.Jan Heylen - 2013 - Synthese 190 (1):89-111.
    The subject of this article is Modal-Epistemic Arithmetic (MEA), a theory introduced by Horsten to interpret Epistemic Arithmetic (EA), which in turn was introduced by Shapiro to interpret Heyting Arithmetic. I will show how to interpret MEA in EA such that one can prove that the interpretation of EA is MEA is faithful. Moreover, I will show that one can get rid of a particular Platonist assumption. Then I will discuss models for MEA in light of the problems of logical (...)
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  40. Lipschitz Functions in Constructive Reverse Mathematics.I. Loeb - 2013 - Logic Journal of the IGPL 21 (1):28-43.
  41. Antirealism and Constructivism: Brouwer’s Weak Counterexamples: Antirealism and Constructivism: Brouwer’s Weak Counterexamples.Charles Mccarty - 2013 - Review of Symbolic Logic 6 (1):147-159.
    Strictly intuitionistic inferences are employed to demonstrate that three conditions—the existence of Brouwerian weak counterexamples to _Test_, the recognition condition, and the _BHK_ interpretation of the logical signs—are together inconsistent. Therefore, if the logical signs in mathematical statements governed by the recognition condition are constructive in that they satisfy the clauses of the _BHK_, then every relevant instance of the classical principle _Test_ is true intuitionistically, and the antirealistic critique of conventional logic, once thought to yield such weak counterexamples, is (...)
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  42. Brouwer’s Weak Counterexamples and Testability: Further Remarks: 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 Brouwer– Heyting–Kolmogorov interpretation, in the presence of a formulation of the recognition principle, entails the validity of the Law of Testability: that the form ¬ f V ¬¬ f is valid. Therefore, the BHK and recognition, as described here, are inconsistent with the axioms both of intuitionistic mathematics and of Markovian constructivism. This finding also implies that, if the BHK and recognition are suitably formulated, then Brouwer’s original weak counterexample reasoning was fallacious. (...)
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  43. Intuitionistic Logic and its Philosophy.Panu Raatikainen - 2013 - Al-Mukhatabat. A Trilingual Journal For Logic, Epistemology and Analytical Philosophy (6):114-127.
  44. Norman Sieroka. Umgebungen: Symbolischer Konstruktivismus im Anschluss an Hermann Weyl und Fritz Medicus. Zurich: Chronos, 2010. Pp. 416. €43.00. [REVIEW]Thomas Ryckman - 2013 - Hopos: The Journal of the International Society for the History of Philosophy of Science 3 (1):164-168.
  45. 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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  46. Numbers and Everything.Gonçalo Santos - 2013 - Philosophia Mathematica 21 (3):297-308.
    I begin by drawing a parallel between the intuitionistic understanding of quantification over all natural numbers and the generality relativist understanding of quantification over absolutely everything. I then argue that adoption of an intuitionistic reading of relativism not only provides an immediate reply to the absolutist's charge of incoherence but it also throws a new light on the debates surrounding absolute generality.
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  47. L.E.J. Brouwer: Topologist, Intuitionist, Philosopher: How Mathematics is Rooted in Life.D. van Dalen - 2013 - Springer.
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  48. A Solution to the Surprise Exam Paradox in Constructive Mathematics.Mohammad Ardeshir & Rasoul Ramezanian - 2012 - Review of Symbolic Logic 5 (4):679-686.
    We represent the well-known surprise exam paradox in constructive and computable mathematics and offer solutions. One solution is based on Brouwer’s continuity principle in constructive mathematics, and the other involves type 2 Turing computability in classical mathematics. We also discuss the backward induction paradox for extensive form games in constructive logic.
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  49. On the Constructive Notion of Closure Maps.Mohammad Ardeshir & Rasoul Ramezanian - 2012 - Mathematical Logic Quarterly 58 (4-5):348-355.
    Let A be a subset of the constructive real line. What are the necessary and sufficient conditions for the set A such that A is continuously separated from other reals, i.e., there exists a continuous function f with f−1 = A? In this paper, we study the notions of closed sets and closure maps in constructive reverse mathematics.
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  50. 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.
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