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  1. 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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  • 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 The analysis of unpublished material presented (...)
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  • Completeness Before Post: Bernays, Hilbert, and the Development of Propositional Logic.Richard Zach - 1999 - Bulletin of Symbolic Logic 5 (3):331-366.
    Some of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917-1923. The aim of this paper is to describe these results, focussing (...)
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  • Dedekind and Cassirer on Mathematical Concept Formation†.Audrey Yap - 2014 - Philosophia Mathematica 25 (3):369-389.
    Dedekind's major work on the foundations of arithmetic employs several techniques that have left him open to charges of psychologism, and through this, to worries about the objectivity of the natural-number concept he defines. While I accept that Dedekind takes the foundation for arithmetic to lie in certain mental powers, I will also argue that, given an appropriate philosophical background, this need not make numbers into subjective mental objects. Even though Dedekind himself did not provide that background, one can nevertheless (...)
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  • Taking Stock: Hale, Heck, and Wright on Neo-Logicism and Higher-Order Logic.Crispin Wright - 2021 - Philosophia Mathematica 29 (3): 392--416.
    ABSTRACT Four philosophical concerns about higher-order logic in general and the specific demands placed on it by the neo-logicist project are distinguished. The paper critically reviews recent responses to these concerns by, respectively, the late Bob Hale, Richard Kimberly Heck, and myself. It is argued that these score some successes. The main aim of the paper, however, is to argue that the most serious objection to the applications of higher-order logic required by the neo-logicist project has not been properly understood. (...)
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  • Ontic Vagueness and Metaphysical Indeterminacy.J. Robert G. Williams - 2008 - Philosophy Compass 3 (4):763-788.
    Might it be that world itself, independently of what we know about it or how we represent it, is metaphysically indeterminate? This article tackles in turn a series of questions: In what sorts of cases might we posit metaphysical indeterminacy? What is it for a given case of indefiniteness to be 'metaphysical'? How does the phenomenon relate to 'ontic vagueness', the existence of 'vague objects', 'de re indeterminacy' and the like? How might the logic work? Are there reasons for postulating (...)
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  • Transfinite Cardinals in Paraconsistent Set Theory.Zach Weber - 2012 - Review of Symbolic Logic 5 (2):269-293.
    This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.
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  • Naturalism, Fallibilism, and the a Priori.Lisa Warenski - 2009 - Philosophical Studies 142 (3):403-426.
    This paper argues that a priori justification is, in principle, compatible with naturalism—if the a priori is understood in a way that is free of the inessential properties that, historically, have been associated with the concept. I argue that empirical indefeasibility is essential to the primary notion of the a priori ; however, the indefeasibility requirement should be interpreted in such a way that we can be fallibilist about apriori-justified claims. This fallibilist notion of the a priori accords with the (...)
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  • What is a Higher Level Set?Dimitris Tsementzis - 2016 - Philosophia Mathematica:nkw032.
    Structuralist foundations of mathematics aim for an ‘invariant’ conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. I argue in favor of the former over the latter. First, I explain why to choose between them we need to ask the question of what is the correct ‘categorified’ version of a set. Second, I argue in favor of groupoids over categories as ‘categorified’ sets by introducing a pre-formal understanding of groupoids as (...)
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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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  • Cantor's Abstractionism and Hume's Principle.Claudio Ternullo & Luca Zanetti - 2021 - History and Philosophy of Logic 43 (3):284-300.
    Richard Kimberly Heck and Paolo Mancosu have claimed that the possibility of non-Cantorian assignments of cardinalities to infinite concepts shows that Hume's Principle (HP) is not implicit in the concept of cardinal number. Neologicism would therefore be threatened by the ‘good company' HP is kept by such alternative assignments. In his review of Mancosu's book, Bob Hale argues, however, that ‘getting different numerosities for different countable infinite collections depends on taking the groups in a certain order – but it is (...)
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  • Zermelo's Cantorian Theory of Systems of Infinitely Long Propositions.R. Gregory Taylor - 2002 - Bulletin of Symbolic Logic 8 (4):478-515.
    In papers published between 1930 and 1935. Zermelo outlines a foundational program, with infinitary logic at its heart, that is intended to (1) secure axiomatic set theory as a foundation for arithmetic and analysis and (2) show that all mathematical propositions are decidable. Zermelo's theory of systems of infinitely long propositions may be termed "Cantorian" in that a logical distinction between open and closed domains plays a signal role. Well-foundedness and strong inaccessibility are used to systematically integrate highly transfinite concepts (...)
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  • Godel's Unpublished Papers on Foundations of Mathematics.W. W. Tatt - 2001 - Philosophia Mathematica 9 (1):87-126.
  • Godel's Interpretation of Intuitionism.William Tait - 2006 - Philosophia Mathematica 14 (2):208-228.
    Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will (1) criticize this foundation, (2) propose a quite different one, and (3) note that essentially the latter foundation also underlies the Curry-Howard type theory, and hence Heyting's intuitionistic conception of logic. Thus the Dialectica interpretation (in (...)
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  • On the Origin of Symbolic Mathematics and Its Significance for Wittgenstein’s Thought.Sören Stenlund - 2015 - Nordic Wittgenstein Review 4 (1):7-92.
    The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also (...)
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  • Le « Wittgenstein intermédiaire » et les mathématiques modernes.Sören Stenlund & Anne-Marie Boisvert - 2012 - Philosophiques 39 (1):125-161.
    Dans cet article, j’essaie de montrer que le dépassement et le rejet du dogmatisme sont un aspect décisif du changement dans la pensée de Wittgenstein qui a eu lieu au début des années 30, quand il commence à mettre en valeur l’autonomie de la grammaire du langage et à parler d’images grammaticales et de jeux de langage en tant qu’objets de comparaison. En examinant certains traits fondamentaux de ce changement, je mettrai en évidence l’impulsion et les idées décisives que Wittgenstein (...)
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  • The Modernity of Dedekind’s Anticipations Contained in What Are Numbers and What Are They Good For?J. Soliveres Tur & J. Climent Vidal - 2018 - Archive for History of Exact Sciences 72 (2):99-141.
    We show that Dedekind, in his proof of the principle of definition by mathematical recursion, used implicitly both the concept of an inductive cone from an inductive system of sets and that of the inductive limit of an inductive system of sets. Moreover, we show that in Dedekind’s work on the foundations of mathematics one can also find specific occurrences of various profound mathematical ideas in the fields of universal algebra, category theory, the theory of primitive recursive mappings, and set (...)
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  • The Ways of Hilbert's Axiomatics: Structural and Formal.Wilfried Sieg - 2014 - Perspectives on Science 22 (1):133-157.
    Hilbert gave lectures on the foundations of mathematics throughout his career. Notes for many of them have been preserved and are treasures of information; they allow us to reconstruct the path from Hilbert's logicist position, deeply influenced by Dedekind and presented in lectures starting around 1890, to the program of finitist proof theory in the early 1920s. The development toward proof theory begins, in some sense, in 1917 when Hilbert gave his talk Axiomatisches Denken in Zürich. This talk is rooted (...)
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  • Hilbert's Programs: 1917–1922.Wilfried Sieg - 1999 - Bulletin of Symbolic Logic 5 (1):1-44.
    Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has (...)
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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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  • 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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  • A Justification for the Quantificational Hume Principle.Chris Scambler - 2021 - Erkenntnis 86 (5):1293-1308.
    In recent work Bruno Whittle has presented a new challenge to the Cantorian idea that there are different infinite cardinalities. Most challenges of this kind have tended to focus on the status of the axioms of standard set theory; Whittle’s is different in that he focuses on the connection between standard set theory and intuitive concepts related to cardinality. Specifically, Whittle argues we are not in a position to know a principle I call the Quantificational Hume Principle, which connects the (...)
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  • Which Arithmetization for Which Logicism? Russell on Relations and Quantities in The Principles of Mathematics.Sébastien Gandon - 2008 - History and Philosophy of Logic 29 (1):1-30.
    This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible 'by logical principles from logical principles' does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV-V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: Russell's logicism (...)
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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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  • Zermelo's Analysis of 'General Proposition'.R. Gregory Taylor - 2009 - History and Philosophy of Logic 30 (2):141-155.
    On Zermelo's view, any mathematical theory presupposes a non-empty domain, the elements of which enjoy equal status; furthermore, mathematical axioms must be chosen from among those propositions that reflect the equal status of domain elements. As for which propositions manage to do this, Zermelo's answer is, those that are ?symmetric?, meaning ?invariant under domain permutations?. We argue that symmetry constitutes Zermelo's conceptual analysis of ?general proposition?. Further, although others are commonly associated with the extension of Klein's Erlanger Programme to logic, (...)
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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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  • Yablo’s Paradox in Second-Order Languages: Consistency and Unsatisfiability.Lavinia María Picollo - 2013 - Studia Logica 101 (3):601-617.
    Stephen Yablo [23,24] introduces a new informal paradox, constituted by an infinite list of semi-formalized sentences. It has been shown that, formalized in a first-order language, Yablo’s piece of reasoning is invalid, for it is impossible to derive falsum from the sequence, due mainly to the Compactness Theorem. This result casts doubts on the paradoxical character of the list of sentences. After identifying two usual senses in which an expression or set of expressions is said to be paradoxical, since second-order (...)
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  • Open-Endedness, Schemas and Ontological Commitment.Nikolaj Jang Lee Linding Pedersen & Marcus Rossberg - 2010 - Noûs 44 (2):329-339.
    Second-order axiomatizations of certain important mathematical theories—such as arithmetic and real analysis—can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth-value. Second-order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth-values. The status of second-order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, Vann McGee has argued that one (...)
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  • The Pragmatism of Hilbert's Programme.Volker Peckhaus - 2003 - Synthese 137 (1-2):141 - 156.
    It is shown that David Hilbert's formalistic approach to axiomaticis accompanied by a certain pragmatism that is compatible with aphilosophical, or, so to say, external foundation of mathematics.Hilbert's foundational programme can thus be seen as areconciliation of Pragmatism and Apriorism. This interpretation iselaborated by discussing two recent positions in the philosophy ofmathematics which are or can be related to Hilbert's axiomaticalprogramme and his formalism. In a first step it is argued that thepragmatism of Hilbert's axiomatic contradicts the opinion thatHilbert style (...)
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  • Friedman on Implicit Definition: In Search of the Hilbertian Heritage in Philosophy of Science.Woosuk Park - 2012 - Erkenntnis 76 (3):427-442.
    Michael Friedman’s project both historically and systematically testifies to the importance of the relativized a priori. The importance of implicit definitions clearly emerges from Schlick’s General Theory of Knowledge . The main aim of this paper is to show the relationship between both and the relativized a priori through a detailed discussion of Friedman’s work. Succeeding with this will amount to a contribution to recent scholarship showing the importance of Hilbert for Logical Empiricism.
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  • Epistemic Modality and Hyperintensionality in Mathematics.Hasen Khudairi - 2017 - Dissertation, University of St Andrews
    This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; and (...)
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  • What Are Sets and What Are They For?Alex Oliver & Timothy Smiley - 2006 - Philosophical Perspectives 20 (1):123–155.
  • The Pursuit of Rigor: Hilbert's Axiomatic Method and the Objectivity of Mathematics.Yoshinori Ogawa - 2004 - Annals of the Japan Association for Philosophy of Science 12 (2):89-108.
  • Peirce’s Topical Theory of Continuity.Matthew E. Moore - 2015 - Synthese 192 (4):1-17.
    In the last decade of his life C.S. Peirce began to formulate a purely geometrical theory of continuity to supersede the collection-theoretic theory he began to elaborate around the middle of the 1890s. I argue that Peirce never succeeded in fully formulating the later theory, and that while that there are powerful motivations to adopt that theory within Peirce’s system, it has little to recommend it from an external perspective.
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  • Mathematical Roots of Phenomenology: Husserl and the Concept of Number.Mirja Hartimo - 2006 - History and Philosophy of Logic 27 (4):319-337.
    The paper examines the roots of Husserlian phenomenology in Weierstrass's approach to analysis. After elaborating on Weierstrass's programme of arithmetization of analysis, the paper examines Husserl's Philosophy of Arithmetic as an attempt to provide foundations to analysis. The Philosophy of Arithmetic consists of two parts; the first discusses authentic arithmetic and the second symbolic arithmetic. Husserl's novelty is to use Brentanian descriptive analysis to clarify the fundamental concepts of arithmetic in the first part. In the second part, he founds the (...)
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  • Una reevaluación del convencionalismo geométrico de Poincaré.Pablo Melogno - 2018 - Dianoia 63 (81):37-59.
    Resumen: Janet Folina ha propuesto una interpretación del convencionalismo de Poincaré contraria a la que ofrecen Michael Friedman y Robert DiSalle. Ambos afirman que la propuesta de Poincaré queda refutada por la relati-vidad general pues supone una noción restrictiva de los principios a priori. Folina sostiene que el convencionalismo de Poincaré no es contradictorio con la relatividad general porque permite una noción relativizada de los princi-pios a priori. Intento mostrar que la estrategia de Folina es ineficaz porque Poincaré no puede (...)
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  • Wittgenstein and Brouwer.Mathieu Marion - 2003 - Synthese 137 (1-2):103 - 127.
    In this paper, I present a summary of the philosophical relationship betweenWittgenstein and Brouwer, taking as my point of departure Brouwer's lecture onMarch 10, 1928 in Vienna. I argue that Wittgenstein having at that stage not doneserious philosophical work for years, if one is to understand the impact of thatlecture on him, it is better to compare its content with the remarks on logics andmathematics in the Tractactus. I thus show that Wittgenstein's position, in theTractactus, was already quite close to (...)
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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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  • 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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  • Grundlagen, Section 64: Frege's Discussion of Definitions by Abstraction in Historical Context.Paolo Mancosu - 2015 - History and Philosophy of Logic 36 (1):62-89.
    I offer in this paper a contextual analysis of Frege's Grundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege's discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlömilch, and the tradition of textbooks in geometry for secondary schools . In addition, I put Frege's considerations in the context of a widespread debate in Germany (...)
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  • Between Russell and Hilbert: Behmann on the Foundations of Mathematics.Paolo Mancosu - 1999 - Bulletin of Symbolic Logic 5 (3):303-330.
    After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Göttingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to (...)
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  • El Enfoque Epistemológico de David Hilbert: El a Priori Del Conocimiento y El Papel de la Lógica En la Fundamentación de la Ciencia.Rodrigo Lopez-Orellana - 2019 - Principia: An International Journal of Epistemology 23 (2):279-308.
    This paper explores the main philosophical approaches of David Hilbert’s theory of proof. Specifically, it is focuses on his ideas regarding logic, the concept of proof, the axiomatic, the concept of truth, metamathematics, the a priori knowledge and the general nature of scientific knowledge. The aim is to show and characterize his epistemological approach on the foundation of knowledge, where logic appears as a guarantee of that foundation. Hilbert supposes that the propositional apriorism, proposed by him to support mathematics, sustains (...)
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  • Practical Reasoning and the Witnessably Rigorous Proof.Eric Livingston - 2020 - Synthese 199 (1-2):2277-2291.
    This paper introduces an anthropological approach to the foundations of mathematics. Traditionally, the philosophy of mathematics has focused on the nature and origins of mathematical truth. Mathematicians, however, treat mathematical arguments as determining mathematical truth: if an argument is found to describe a witnessably rigorous proof of a theorem, that theorem is considered—until the need for further examination arises—to be true. The anthropological question is how mathematicians, as a practical matter and as a matter of mathematical practice, make such determinations. (...)
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  • 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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  • Metaontological Minimalism.Øystein Linnebo - 2012 - Philosophy Compass 7 (2):139-151.
    Can there be objects that are ‘thin’ in the sense that very little is required for their existence? A number of philosophers have thought so. For instance, many Fregeans believe it suffices for the existence of directions that there be lines standing in the relation of parallelism; other philosophers believe it suffices for a mathematical theory to have a model that the theory be coherent. This article explains the appeal of thin objects, discusses the three most important strategies for articulating (...)
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  • Dummett on Indefinite Extensibility.Øystein Linnebo - 2018 - Philosophical Issues 28 (1):196-220.
    Dummett’s notion of indefinite extensibility is influential but obscure. The notion figures centrally in an alternative Dummettian argument for intuitionistic logic and anti-realism, distinct from his more famous, meaning-theoretic arguments to the same effect. Drawing on ideas from Dummett, a precise analysis of indefinite extensibility is proposed. This analysis is used to reconstruct the poorly understood alternative argument. The plausibility of the resulting argument is assessed.
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  • Aristotelian Continua.Øystein Linnebo, Stewart Shapiro & Geoffrey Hellman - 2016 - Philosophia Mathematica 24 (2):214-246.
    In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, (...)
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  • Axiomatics Without Foundations. On the Model-Theoretical Viewpoint In Modern Axiomatics.Johannes Lenhard - 2005 - Philosophia Scientiae 9 (2: Aperçus philosophiques en log):97-107.
    Two conflicting interpretations of modern axiomatics will be considered. The logico-analytical interpretation goes back to Pasch, while the model-theoretical approach stems from Hilbert. This perspective takes up the distinction between logic as calculus ratiocinator versus lingua characterica that Heijenoort and Hintikka placed emphasis on. It is argued that the Heijenoort-Hintikka distinction can be carried over from logic to mathematical axiomatics. In particular, the model-theoretical viewpoint is deeply connected to a philosophy of mathematics that is not committed to a foundational perspective, (...)
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  • How to Be a Structuralist All the Way Down.Elaine Landry - 2011 - Synthese 179 (3):435 - 454.
    This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the "algebraic" approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a "foundation", (...)
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  • Dedekind's Logicism.Ansten Mørch Klev - 2015 - Philosophia Mathematica:nkv027.
    A detailed argument is provided for the thesis that Dedekind was a logicist about arithmetic. The rules of inference employed in Dedekind's construction of arithmetic are, by his lights, all purely logical in character, and the definitions are all explicit; even the definition of the natural numbers as the abstract type of simply infinite systems can be seen to be explicit. The primitive concepts of the construction are logical in their being intrinsically tied to the functioning of the understanding.
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