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  1. Reason, Causation and Compatibility with the Phenomena.Basil Evangelidis - 2020 - Wilmington, Delaware, USA: Vernon Press.
    'Reason, Causation and Compatibility with the Phenomena' strives to give answers to the philosophical problem of the interplay between realism, explanation and experience. This book is a compilation of essays that recollect significant conceptions of rival terms such as determinism and freedom, reason and appearance, power and knowledge. This title discusses the progress made in epistemology and natural philosophy, especially the steps that led from the ancient theory of atomism to the modern quantum theory, and from mathematization to analytic philosophy. (...)
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  • Enciclopédia de Termos Lógico-Filosóficos.João Branquinho, Desidério Murcho & Nelson Gonçalves Gomes (eds.) - 2006 - São Paulo, SP, Brasil: Martins Fontes.
    Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...)
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  • 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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  • Truth, Reflection, and Hierarchies.Michael Glanzberg - 2005 - Synthese 142 (3):289 - 315.
    A common objection to hierarchical approaches to truth is that they fragment the concept of truth. This paper defends hierarchical approaches in general against the objection of fragmentation. It argues that the fragmentation required is familiar and unprob-lematic, via a comparison with mathematical proof. Furthermore, it offers an explanation of the source and nature of the fragmentation of truth. Fragmentation arises because the concept exhibits a kind of failure of closure under reflection. This paper offers a more precise characterization of (...)
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  • The Russellian Influence on Hilbert and His School.Paolo Mancosu - 2003 - Synthese 137 (1-2):59 - 101.
    The aim of the paper is to discuss the influence exercised by Russell's thought inGöttingen in the period leading to the formulation of Hilbert's program in theearly twenties. I show that after a period of intense foundational work, culminatingwith the departure from Göttingen of Zermelo and Grelling in 1910 we witnessa reemergence of interest in foundations of mathematics towards the end of 1914. Itis this second period of foundational work that is my specific interest. Through theuse of unpublished archival sources (...)
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  • Conditio Sine Qua Non? Zuordnung in the Early Epistemologies of Cassirer and Schlick.T. A. Ryckman - 1991 - Synthese 88 (1):57 - 95.
    In early major works, Cassirer and Schlick differently recast traditional doctrines of the concept and of the relation of concept to intuitive content along the lines of recent epistemological discussions within the exact sciences. In this, they attempted to refashion epistemology by incorporating as its basic principle the notion of functional coordination, the theoretical sciences' own methodological tool for dispensing with the imprecise and unreliable guide of intuitive evidence. Examining their respective reconstructions of the theory of knowledge provides an axis (...)
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  • Is Unsaying Polite?Berislav Žarnić - 2012 - In Majda Trobok, Nenad Miščević & Berislav Žarnić (eds.), Between Logic and Reality: Modeling Inference, Action and Understanding. Springer. pp. 201--224.
    This paper is divided in five sections. Section 11.1 sketches the history of the distinction between speech act with negative content and negated speech act, and gives a general dynamic interpretation for negated speech act. “Downdate semantics” for AGM contraction is introduced in Section 11.2. Relying on semantically interpreted contraction, Section 11.3 develops the dynamic semantics for constative and directive speech acts, and their external negations. The expressive completeness for the formal variants of natural language utterances, none of which is (...)
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  • Intensionality and the Gödel Theorems.David D. Auerbach - 1985 - Philosophical Studies 48 (3):337--51.
    Philosophers of language have drawn on metamathematical results in varied ways. Extensionalist philosophers have been particularly impressed with two, not unrelated, facts: the existence, due to Frege/Tarski, of a certain sort of semantics, and the seeming absence of intensional contexts from mathematical discourse. The philosophical import of these facts is at best murky. Extensionalists will emphasize the success and clarity of the model theoretic semantics; others will emphasize the relative poverty of the mathematical idiom; still others will question the aptness (...)
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  • Negation and Infinity.Kazimierz Trzęsicki - 2018 - Studies in Logic, Grammar and Rhetoric 54 (1):131-148.
    Infinity and negation are in various relations and interdependencies one to another. The analysis of negation and infinity aims to better understanding them. Semantical, syntactical, and pragmatic issues will be considered.
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  • Hilbert's 'Verunglückter Beweis', the First Epsilon Theorem, and Consistency Proofs.Richard Zach - 2004 - History and Philosophy of Logic 25 (2):79-94.
    In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert's epsilon-substitution method. There was, however, a second approach which was not reflected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's first epsilon theorem and a certain "general consistency result" due to Bernays. An analysis of the form of this so-called "failed proof" (...)
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  • Theological Underpinnings of the Modern Philosophy of Mathematics.Vladislav Shaposhnikov - 2016 - Studies in Logic, Grammar and Rhetoric 44 (1):147-168.
    The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.
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  • How Are Concepts of Infinity Acquired?Kazimierz Trzęsicki - 2015 - Studies in Logic, Grammar and Rhetoric 40 (1):179-217.
    Concepts of infinity have been subjects of dispute since antiquity. The main problems of this paper are: is the mind able to acquire a concept of infinity? and: how are concepts of infinity acquired? The aim of this paper is neither to say what the meanings of the word “infinity” are nor what infinity is and whether it exists. However, those questions will be mentioned, but only in necessary extent.
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  • Lorenzen's Proof of Consistency for Elementary Number Theory.Thierry Coquand & Stefan Neuwirth - 2020 - History and Philosophy of Logic 41 (3):281-290.
    We present a manuscript of Paul Lorenzen that provides a proof of consistency for elementary number theory as an application of the construction of the free countably complete pseudocomplemented semilattice over a preordered set. This manuscript rests in the Oskar-Becker-Nachlass at the Philosophisches Archiv of Universität Konstanz, file OB 5-3b-5. It has probably been written between March and May 1944. We also compare this proof to Gentzen's and Novikov's, and provide a translation of the manuscript.
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  • Philosophy of Mathematics and Computer Science.Kazimierz Trzęsicki - 2010 - Studies in Logic, Grammar and Rhetoric 22 (35).
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  • 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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  • 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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  • La constitución del programa de Hilbert.Max Fernández de Castro & Yolanda Torres Falcón - 2020 - Metatheoria – Revista de Filosofía E Historia de la Ciencia 10 (2):31--50.
    In the pages that follow, it is our intention to present a panoramic and schematic view of the evolution of the formalist program, which derives from recent studies of lecture notes that were unknown until very recently. Firstly, we analyze certain elements of the program. Secondly, we observe how, once the program was established in 1920, in the period up to 1931, different types of finitism with a common basis were tried out by Hilbert and Bernays, in an effort to (...)
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  • Objectivity Sans Intelligibility. Hermann Weyl's Symbolic Constructivism.Iulian D. Toader - 2011 - Dissertation, University of Notre Dame
    A new form of skepticism is described, which holds that objectivity and understanding are incompossible ideals of modern science. This is attributed to Weyl, hence its name: Weylean skepticism. Two general defeat strategies are then proposed, one of which is rejected as a failure.
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  • Pasch's Empiricism as Methodological Structuralism.Dirk Schlimm - 2020 - In Erich Reck & Georg Schiemer (eds.), The Pre-History of Mathematical Structuralism. New York: Oxford University Press. pp. 80-105.
  • Frege’s ‘On the Foundations of Geometry’ and Axiomatic Metatheory.Günther Eder - 2016 - Mind 125 (497):5-40.
    In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert’s methodology of proving the independence and consistency of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should be proved. Frege contends that independence proofs require the development of a ‘new science’ with its own basic truths. This paper aims to provide a reconstruction of this (...)
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  • Hilbert's Program Revisited.Panu Raatikainen - 2003 - Synthese 137 (1):157-177.
    After sketching the main lines of Hilbert's program, certain well-known and influential interpretations of the program are critically evaluated, and an alternative interpretation is presented. Finally, some recent developments in logic related to Hilbert's program are reviewed.
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  • Constructive Mathematics in Theory and Programming Practice.Douglas Bridges & Steeve Reeves - 1999 - 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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  • Are Mathematicians Better Described as Formalists or Pluralists?Andrea Pedeferri & Michele Friend - 2011 - Logic and Philosophy of Science 9 (1):173-180.
    In this paper we try to convert the mathematician who calls himself, or herself, “a formalist” to a position we call “meth-odological pluralism”. We show how the actual practice of mathe-matics fits methodological pluralism better than formalism while preserving the attractive aspects of formalism of freedom and crea-tivity. Methodological pluralism is part of a larger, more general, pluralism, which is currently being developed as a position in the philosophy of mathematics in its own right.1 Having said that, henceforth, in this (...)
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  • Hilbert and Set Theory.Burton Dreben & Akihiro Kanamori - 1997 - Synthese 110 (1):77-125.
  • Hilbert's Finitism: Historical, Philosophical, and Metamathematical Perspectives.Richard Zach - 2001 - Dissertation, University of California, Berkeley
    In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...)
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  • The Development of Mathematical Logic From Russell to Tarski, 1900-1935.Paolo Mancosu, Richard Zach & Calixto Badesa - 2009 - In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press.
    The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...)
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  • On the Syntax of Logic and Set Theory.Lucius T. Schoenbaum - 2010 - Review of Symbolic Logic 3 (4):568-599.
    We introduce an extension of the propositional calculus to include abstracts of predicates and quantifiers, employing a single rule along with a novel comprehension schema and a principle of extensionality, which are substituted for the Bernays postulates for quantifiers and the comprehension schemata of ZF and other set theories. We prove that it is consistent in any finite Boolean subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti, and others, and discuss the relationship of the system to other set-theoretic (...)
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  • On Turing’s Legacy in Mathematical Logic and the Foundations of Mathematics.Joan Bagaria - 2013 - Arbor 189 (764):a079.
  • Carnap’s Early Metatheory: Scope and Limits.Georg Schiemer, Richard Zach & Erich Reck - 2017 - Synthese 194 (1):33-65.
    In Untersuchungen zur allgemeinen Axiomatik and Abriss der Logistik, Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is much less confused and hopeless than it has often been made out to (...)
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  • The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program.Richard Zach - 2003 - Synthese 137 (1-2):211 - 259.
    After a brief flirtation with logicism around 1917, David Hilbertproposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays andWilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for everstronger and more comprehensive areas of mathematics, and finitisticproofs of consistency of these systems. Early advances in these areaswere made by Hilbert (and Bernays) in a series of lecture courses atthe (...)
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  • On Composition.Axel Arturo Barceló Aspeitia - unknown
    I can make no sense of a true, literal application of the notion of proper (nontemporal) components or parts to things that occupy no space. Others apparently can, and some construct elaborate theories of propositional structure… Perhaps I am blind to a possibility that other, cognitively better endowed philosophers see. From my unseeing perspective, though, it is more likely that the appearance of sight deceives.
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  • Poststructuralism and Deconstruction: A Mathematical History.Vladimir Tasic - 2012 - Cosmos and History 8 (1):177-198.
    Explaining his love of philosophy, Slavoj Žižek notes that he ‘secretly thinks reality exists so that we can speculate about it’. This article takes the view that links between mathematics and continental philosophy are part of reality, the reality of philosophy and its history, and hence require speculation. Examples from the work of Jacques Derrida and Henri Poincaré are discussed.
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  • ‘Metamathematics’ in Transition.Matthias Wille - 2011 - History and Philosophy of Logic 32 (4):333 - 358.
    In this paper, we trace the conceptual history of the term ?metamathematics? in the nineteenth century. It is well known that Hilbert introduced the term for his proof-theoretic enterprise in about 1922. But he was verifiably inspired by an earlier usage of the phrase in the 1870s. After outlining Hilbert's understanding of the term, we will explore the lines of inducement and elucidate the different meanings of ?metamathematics? in the final decades of the nineteenth century. Finally, we will investigate the (...)
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  • Harvard 1940–1941: Tarski, Carnap and Quine on a Finitistic Language of Mathematics for Science.Paolo Mancosu - 2005 - History and Philosophy of Logic 26 (4):327-357.
    Tarski, Carnap and Quine spent the academic year 1940?1941 together at Harvard. In their autobiographies, both Carnap and Quine highlight the importance of the conversations that took place among them during the year. These conversations centred around semantical issues related to the analytic/synthetic distinction and on the project of a finitist/nominalist construction of mathematics and science. Carnap's Nachlaß in Pittsburgh contains a set of detailed notes, amounting to more than 80 typescripted pages, taken by Carnap while these discussions were taking (...)
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  • Extensions of the Finitist Point of View.Matthias Schirn & Karl-Georg Niebergall - 2001 - History and Philosophy of Logic 22 (3):135-161.
    Hilbert developed his famous finitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939). The paper is in three sections. The first deals with Hilbert's introduction of a restricted ? -rule in his 1931 paper ?Die Grundlegung der elementaren Zahlenlehre?. The main question we discuss here is whether the finitist (meta-)mathematician would (...)
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  • Metaphors for Mathematics From Pasch to Hilbert.Dirk Schlimm - 2016 - Philosophia Mathematica 24 (3):308-329.
    How mathematicians conceive of the nature of mathematics is reflected in the metaphors they use to talk about it. In this paper I investigate a change in the use of metaphors in the late nineteenth and early twentieth centuries. In particular, I argue that the metaphor of mathematics as a tree was used systematically by Pasch and some of his contemporaries, while that of mathematics as a building was deliberately chosen by Hilbert to reflect a different view of mathematics. By (...)
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  • On the Concept of Finitism.Luca Incurvati - 2015 - Synthese 192 (8):2413-2436.
    At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.
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  • Constructive Mathematics.Douglas Bridges - 2008 - Stanford Encyclopedia of Philosophy.
  • Mathematical Instrumentalism, Gödel’s Theorem, and Inductive Evidence.Alexander Paseau - 2011 - Studies in History and Philosophy of Science Part A 42 (1):140-149.
    Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism (...)
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  • Hilbert's Axiomatic Method and Carnap's General Axiomatics.Michael Stöltzner - 2015 - Studies in History and Philosophy of Science Part A 53:12-22.
  • Logic and Games.Wilfrid Hodges - 2008 - Stanford Encyclopedia of Philosophy.