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Hilbert's program then and now

In Dale Jacquette (ed.), Philosophy of Logic. Amsterdam: North Holland. pp. 411–447 (2007)

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  1. Different Senses of Finitude: An Inquiry Into Hilbert’s Finitism.Sören Stenlund - 2012 - Synthese 185 (3):335-363.
    This article develops a critical investigation of the epistemological core of Hilbert's foundational project, the so-called the finitary attitude. The investigation proceeds by distinguishing different senses of 'number' and 'finitude' that have been used in the philosophical arguments. The usual notion of modern pure mathematics, i.e. the sense of number which is implicit in the notion of an arbitrary finite sequence and iteration is one sense of number and finitude. Another sense, of older origin, is connected with practices of counting (...)
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  • On Three Arguments Against Categorical Structuralism.Makmiller Pedroso - 2009 - Synthese 170 (1):21 - 31.
    Some mathematicians and philosophers contend that set theory plays a foundational role in mathematics. However, the development of category theory during the second half of the twentieth century has encouraged the view that this theory can provide a structuralist alternative to set-theoretical foundations. Against this tendency, criticisms have been made that category theory depends on set-theoretical notions and, because of this, category theory fails to show that set-theoretical foundations are dispensable. The goal of this paper is to show that these (...)
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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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  • On the Depth of Gödel’s Incompleteness Theorems.Yong Cheng - forthcoming - Philosophia Mathematica.
    ABSTRACT We use Gödel’s incompleteness theorems as a case study for investigating mathematical depth. We examine the philosophical question of what the depth of Gödel’s incompleteness theorems consists in. We focus on the methodological study of the depth of Gödel’s incompleteness theorems, and propose three criteria to account for the depth of the incompleteness theorems: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of Gödel’s incompleteness theorems.
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  • Gödel’s Incompleteness Theorem and the Anti-Mechanist Argument: Revisited.Yong Cheng - 2020 - Studia Semiotyczne 34 (1):159-182.
    This is a paper for a special issue of Semiotic Studies devoted to Stanislaw Krajewski’s paper. This paper gives some supplementary notes to Krajewski’s on the Anti-Mechanist Arguments based on Gödel’s incompleteness theorem. In Section 3, we give some additional explanations to Section 4–6 in Krajewski’s and classify some misunderstandings of Gödel’s incompleteness theorem related to AntiMechanist Arguments. In Section 4 and 5, we give a more detailed discussion of Gödel’s Disjunctive Thesis, Gödel’s Undemonstrability of Consistency Thesis and the definability (...)
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  • Beyond Logical Pluralism and Logical Monism.Pavel Arazim - 2020 - Logica Universalis 14 (2):151-174.
    Logical pluralism as a thesis that more than one logic is correct seems very plausible for two basic reasons. First, there are so many logical systems on the market today. And it is unclear how we should decide which of them gets the logical rules right. On the other hand, logical monism as the opposite thesis still seems plausible, as well, because of normativity of logic. An approach which would manage to bring a synthesis of both logical pluralism and logical (...)
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  • Proof Theory.Jeremy Avigad - unknown
    At the turn of the nineteenth century, mathematics exhibited a style of argumentation that was more explicitly computational than is common today. Over the course of the century, the introduction of abstract algebraic methods helped unify developments in analysis, number theory, geometry, and the theory of equations; and work by mathematicians like Dedekind, Cantor, and Hilbert towards the end of the century introduced set-theoretic language and infinitary methods that served to downplay or suppress computational content. This shift in emphasis away (...)
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  • Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic.Corey Edward Mulvihill - 2015 - Dissertation, University of Waterloo
    Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...)
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  • Anti-Foundational Categorical Structuralism.Darren McDonald - unknown
    The aim of this dissertation is to outline and defend the view here dubbed “anti-foundational categorical structuralism”. The program put forth is intended to provide an answer the question “what is mathematics?”. The answer here on offer adopts the structuralist view of mathematics, in that mathematics is taken to be “the science of structure” expressed in the language of category theory, which is argued to accurately capture the notion of a “structural property”. In characterizing mathematical theorems as both conditional and (...)
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  • In the Beginning Was the Verb: The Emergence and Evolution of Language Problem in the Light of the Big Bang Epistemological Paradigm.Edward G. Belaga - 2008 - Cognitive Philology 1 (1).
    The enigma of the Emergence of Natural Languages, coupled or not with the closely related problem of their Evolution is perceived today as one of the most important scientific problems. The purpose of the present study is actually to outline such a solution to our problem which is epistemologically consonant with the Big Bang solution of the problem of the Emergence of the Universe}. Such an outline, however, becomes articulable, understandable, and workable only in a drastically extended epistemic and scientific (...)
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  • Takeuti's Well-Ordering Proof: Finitistically Fine?Eamon Darnell & Aaron Thomas-Bolduc - 2018 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics The CSHPM 2017 Annual Meeting in Toronto, Ontario. Birkhäuser Basel.
    If it could be shown that one of Gentzen's consistency proofs for pure number theory could be shown to be finitistically acceptable, an important part of Hilbert's program would be vindicated. This paper focuses on whether the transfinite induction on ordinal notations needed for Gentzen's second proof can be finitistically justified. In particular, the focus is on Takeuti's purportedly finitistically acceptable proof of the well-ordering of ordinal notations in Cantor normal form. The paper begins with a historically informed discussion of (...)
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