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  1. Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle.Maria Carla Galavotti (ed.) - 2004 - Dordrecht: Springer Verlag.
    The Institute Vienna Circle held a conference in Vienna in 2003, Cambridge and Vienna a?
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  • Gentzen’s “cut rule” and quantum measurement in terms of Hilbert arithmetic. Metaphor and understanding modeled formally.Vasil Penchev - 2022 - Logic and Philosophy of Mathematics eJournal 14 (14):1-37.
    Hilbert arithmetic in a wide sense, including Hilbert arithmetic in a narrow sense consisting by two dual and anti-isometric Peano arithmetics, on the one hand, and the qubit Hilbert space (originating for the standard separable complex Hilbert space of quantum mechanics), on the other hand, allows for an arithmetic version of Gentzen’s cut elimination and quantum measurement to be described uniformy as two processes occurring accordingly in those two branches. A philosophical reflection also justifying that unity by quantum neo-Pythagoreanism links (...)
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  • Constructivity and Computability in Historical and Philosophical Perspective.Jacques Dubucs & Michel Bourdeau (eds.) - 2014 - Dordrecht, Netherland: Springer.
    Ranging from Alan Turing’s seminal 1936 paper to the latest work on Kolmogorov complexity and linear logic, this comprehensive new work clarifies the relationship between computability on the one hand and constructivity on the other. The authors argue that even though constructivists have largely shed Brouwer’s solipsistic attitude to logic, there remain points of disagreement to this day. Focusing on the growing pains computability experienced as it was forced to address the demands of rapidly expanding applications, the content maps the (...)
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  • Both Classical & Quantum Information; Both Bit & Qubit: Both Physical & Transcendental Time.Vasil Penchev - 2021 - Philosophy of Science eJournal (Elsevier: SSRN) 14 (22):1-24.
    Information can be considered as the most fundamental, philosophical, physical and mathematical concept originating from the totality by means of physical and mathematical transcendentalism (the counterpart of philosophical transcendentalism). Classical and quantum information, particularly by their units, bit and qubit, correspond and unify the finite and infinite. As classical information is relevant to finite series and sets, as quantum information, to infinite ones. A fundamental joint relativity of the finite and infinite, of the external and internal is to be investigated. (...)
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  • Nothing but Gold: Complexities in terms of Non-difference and Identity. Part 2. Contrasting Equivalence, Equality, Identity, and Non-difference.Alberto Anrò - 2021 - Journal of Indian Philosophy 49 (3):387-420.
    The present paper is a continuation of a previous one by the same title, the content of which faced the issue concerning the relations of coreference and qualification in compliance with the Navya-Nyāya theoretical framework, although prompted by the Advaita-Vedānta enquiry regarding non-difference. In a complementary manner, by means of a formal analysis of equivalence, equality, and identity, this section closes the loop by assessing the extent to which non-difference, the main issue here, cannot be reduced to any of the (...)
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  • Mary Shepherd on the role of proofs in our knowledge of first principles.M. Folescu - 2022 - Noûs 56 (2):473-493.
    This paper examines the role of reason in Shepherd's account of acquiring knowledge of the external world via first principles. Reason is important, but does not have a foundational role. Certain principles enable us to draw the required inferences for acquiring knowledge of the external world. These principles are basic, foundational and, more importantly, self‐evident and thus justified in other ways than by demonstration. Justificatory demonstrations of these principles are neither required, nor possible. By drawing on textual and contextual evidence, (...)
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  • The Relationship of Arithmetic As Two Twin Peano Arithmetic(s) and Set Theory: A New Glance From the Theory of Information.Vasil Penchev - 2020 - Metaphilosophy eJournal (Elseviers: SSRN) 12 (10):1-33.
    The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s (...)
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  • Georg Cantor’s Ordinals, Absolute Infinity & Transparent Proof of the Well-Ordering Theorem.Hermann G. W. Burchard - 2019 - Philosophy Study 9 (8).
    Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add Zermelo's choice function. (...)
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  • Immanent Reasoning or Equality in Action: A Plaidoyer for the Play Level.Nicolas Clerbout, Ansten Klev, Zoe McConaughey & Shahid Rahman - 2018 - Cham, Switzerland: Springer Verlag.
    This monograph proposes a new way of implementing interaction in logic. It also provides an elementary introduction to Constructive Type Theory. The authors equally emphasize basic ideas and finer technical details. In addition, many worked out exercises and examples will help readers to better understand the concepts under discussion. One of the chief ideas animating this study is that the dialogical understanding of definitional equality and its execution provide both a simple and a direct way of implementing the CTT approach (...)
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  • Treatise on intuitionistic type theory.Johan Georg Granström - 2011 - New York: Springer.
    Prolegomena It is fitting to begin this book on intuitionistic type theory by putting the subject matter into perspective. The purpose of this chapter is to ...
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  • The prehistory of the subsystems of second-order arithmetic.Walter Dean & Sean Walsh - 2017 - Review of Symbolic Logic 10 (2):357-396.
    This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincar\'e to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak K\"onig's (...)
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  • Logic, Mathematics, Philosophy, Vintage Enthusiasms: Essays in Honour of John L. Bell.David DeVidi, Michael Hallett & Peter Clark (eds.) - 2011 - Dordrecht, Netherland: Springer.
    The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...)
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  • (1 other version)Hintikka’s Take on the Axiom of Choice and the Constructivist Challenge.Radmila Jovanović - 2013 - Revista de Humanidades de Valparaíso 2:135-150.
    In the present paper we confront Martin- Löf’s analysis of the axiom of choice with J. Hintikka’s standing on this axiom. Hintikka claims that his game theoretical semantics for Independence Friendly Logic justifies Zermelo’s axiom of choice in a first-order way perfectly acceptable for the constructivists. In fact, Martin- Löf’s results lead to the following considerations:Hintikka preferred version of the axiom of choice is indeed acceptable for the constructivists and its meaning does not involve higher order logic.However, the version acceptable (...)
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  • A Structured Argumentation Framework for Modeling Debates in the Formal Sciences.Marcos Cramer & Jérémie Dauphin - 2020 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 51 (2):219-241.
    Scientific research in the formal sciences comes in multiple degrees of formality: fully formal work; rigorous proofs that practitioners know to be formalizable in principle; and informal work like rough proof sketches and considerations about the advantages and disadvantages of various formal systems. This informal work includes informal and semi-formal debates between formal scientists, e.g. about the acceptability of foundational principles and proposed axiomatizations. In this paper, we propose to use the methodology of structured argumentation theory to produce a formal (...)
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  • Consistency, Models, and Soundness.Matthias Schirn - 2010 - Axiomathes 20 (2):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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  • The Axiom of Choice and the Road Paved by Sierpiński.Valérie Lynn Therrien - 2020 - Hopos: The Journal of the International Society for the History of Philosophy of Science 10 (2):504-523.
    From 1908 to 1916, articles supporting the axiom of choice were scant. The situation changed in 1916, when Wacław Sierpiński published a series of articles reviving the debate. The posterity of the axiom of choice as we know it would be unimaginable without Sierpiński’s efforts.
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  • A Theory of Infinitary Relations Extending Zermelo’s Theory of Infinitary Propositions.R. Gregory Taylor - 2016 - Studia Logica 104 (2):277-304.
    An idea attributable to Russell serves to extend Zermelo’s theory of systems of infinitely long propositions to infinitary relations. Specifically, relations over a given domain \ of individuals will now be identified with propositions over an auxiliary domain \ subsuming \. Three applications of the resulting theory of infinitary relations are presented. First, it is used to reconstruct Zermelo’s original theory of urelements and sets in a manner that achieves most, if not all, of his early aims. Second, the new (...)
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  • Andrew Aberdein and Ian J. Dove (eds): The Argument of Mathematics (Logic, Epistemology and the Unity of Science, Vol. 30): Springer, Dordrecht, The Netherlands, 2013, x + 393 pp. [REVIEW]David Hitchcock - 2014 - Argumentation 28 (2):245-258.
    Post-war argumentation theorists have tended to regard argumentation as one thing and mathematical proof as another. Perelman (1958, 1969), for example, defined the word ‘argumentation’ stipulatively as a contrast term to ‘demonstration’: whereas mathematical reasoning as theorized by modern formal logic, he writes, is a matter of deducing theorems from axioms in accordance with stipulated rules of transformation, argumentation aims at gaining the adherence of minds (Perelman 1969, pp. 1–2). Toulmin (1958) contrasted his “jurisprudential model” of argument, according to which (...)
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  • From the axiom of choice to choice sequences.H. Jervell - 1996 - Nordic Journal of Philosophical Logic 1 (1):95-98.
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  • On Carnap: Reflections of a metaphysical student. [REVIEW]Abner Shimony - 1992 - Synthese 93 (1-2):261 - 274.
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  • “Mathematics is the Logic of the Infinite”: Zermelo’s Project of Infinitary Logic.Jerzy Pogonowski - 2021 - Studies in Logic, Grammar and Rhetoric 66 (3):673-708.
    In this paper I discuss Ernst Zermelo’s ideas concerning the possibility of developing a system of infinitary logic that, in his opinion, should be suitable for mathematical inferences. The presentation of Zermelo’s ideas is accompanied with some remarks concerning the development of infinitary logic. I also stress the fact that the second axiomatization of set theory provided by Zermelo in 1930 involved the use of extremal axioms of a very specific sort.1.
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  • Representation of Functions and Total Antisymmetric Relations in Monadic Third Order Logic.M. Randall Holmes - 2019 - Journal of Philosophical Logic 48 (2):263-278.
    We analyze the representation of binary relations in general, and in particular of functions and of total antisymmetric relations, in monadic third order logic, that is, the simple typed theory of sets with three types. We show that there is no general representation of functions or of total antisymmetric relations in this theory. We present partial representations of functions and of total antisymmetric relations which work for large classes of these relations, and show that there is an adequate representation of (...)
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  • ¿Es necesario el Axioma de Zermelo para comprender la teoría de la medida?Carmen Martínez-Adame - 2013 - Metatheoria – Revista de Filosofía E Historia de la Ciencia 3:37--64.
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  • Scientific phenomena and patterns in data.Pascal Ströing - 2018 - Dissertation, Lmu München
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  • An inferential community: Poincaré’s mathematicians.Michel Dufour & John Woods - 2011 - In Frank Zenker (ed.), Proceedings of the 9th International Conference of the Ontario Society for the Study of Argumentation (OSSA), May 18-21, 2011. pp. 156-166.
    Inferential communities are communities using specific substantial argumentative schemes. The religious or scientific communities are examples. I discuss the status of the mathematical community as it appears through the position held by the French mathematician Henri Poincaré during his famous ar-guments with Russell, Hilbert, Peano and Cantor. The paper focuses on the status of complete induction and how logic and psychology shape the community of mathematicians and the teaching of mathematics.
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