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  1. Self-Graphing Equations.Samuel Alexander - manuscript
    Can you find an xy-equation that, when graphed, writes itself on the plane? This idea became internet-famous when a Wikipedia article on Tupper’s self-referential formula went viral in 2012. Under scrutiny, the question has two flaws: it is meaningless (it depends on fonts) and it is trivial. We fix these flaws by formalizing the problem.
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  2. Leibniz's and Kant's Philosophical Ideas Vs. Hilbert's Program.Roman Murawski - unknown - Poznan Studies in the Philosophy of the Sciences and the Humanities 98:29-39.
  3. Frege, Hankel, and Formalism in the Foundations.Richard Lawrence - 2021 - Journal for the History of Analytical Philosophy 9 (11).
    Frege says, at the end of a discussion of formalism in the Foundations of Arithmetic, that his own foundational program “could be called formal” but is “completely different” from the view he has just criticized. This essay examines Frege’s relationship to Hermann Hankel, his main formalist interlocutor in the Foundations, in order to make sense of these claims. The investigation reveals a surprising result: Frege’s foundational program actually has quite a lot in common with Hankel’s. This undercuts Frege’s claim that (...)
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  4. An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs.Paolo Mancosu, Sergio Galvan & Richard Zach - 2021 - Oxford: Oxford University Press.
    An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic, natural deduction and the normalization theorems, the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these (...)
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  5. 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 - Berlin, Germany: Springer.
    This book publishes the previously unpublished Wittgenstein-Skinner Archive held in Trinity College Cambridge Wren Library. The principal Editor is Arthur Gibson, joined by the Editor Niamh O'Mahony in the editing project. The manuscripts were transcribed by Arthur Gibson, checked and edited by Niamh O'Mahony and Arthur Gibson, with additional assistance from Kelsey Gibson. The Chapters that reproduce the Archive, including the Preface, and Part I (chapters 1 and 2) are authored by Arthur Gibson. Arthur Gibson and Niamh O'Mahony jointly edited (...)
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  6. All Science as Rigorous Science: The Principle of Constructive Mathematizability of Any Theory.Vasil Penchev - 2020 - Logic and Philosophy of Mathematics eJournal 12 (12):1-15.
    A principle, according to which any scientific theory can be mathematized, is investigated. Social science, liberal arts, history, and philosophy are meant first of all. That kind of theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather (...)
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  7. 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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  8. Why Logical Pluralism?Colin R. Caret - 2019 - Synthese 198 (Suppl 20):4947-4968.
    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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  9. 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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  10. A Case Study of Misconceptions Students in the Learning of Mathematics; The Concept Limit Function in High School.Widodo Winarso & Toheri Toheri - 2017 - Jurnal Riset Pendidikan Matematika 4 (1): 120-127.
    This study aims to find out how high the level and trends of student misconceptions experienced by high school students in Indonesia. The subject of research that is a class XI student of Natural Science (IPA) SMA Negeri 1 Anjatan with the subject matter limit function. Forms of research used in this study is a qualitative research, with a strategy that is descriptive qualitative research. The data analysis focused on the results of the students' answers on the test essay subject (...)
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  11. Laplacian Growth Without Surface Tension in Filtration Combustion: Analytical Pole Solution.Oleg Kupervasser - 2016 - Complexity 21 (5):31-42.
    Filtration combustion is described by Laplacian growth without surface tension. These equations have elegant analytical solutions that replace the complex integro-differential motion equations by simple differential equations of pole motion in a complex plane. The main problem with such a solution is the existence of finite time singularities. To prevent such singularities, nonzero surface tension is usually used. However, nonzero surface tension does not exist in filtration combustion, and this destroys the analytical solutions. However, a more elegant approach exists for (...)
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  12. A New Reading and Comparative Interpretation of Gödel’s Completeness (1930) and Incompleteness (1931) Theorems.Vasil Penchev - 2016 - Логико-Философские Штудии 13 (2):187-188.
    Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation. Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity. That is nothing else than a new reading at issue and comparative interpretation of Gödel’s papers (1930; 1931) meant here. Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of infinity. The most (...)
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  13. Hilbert Program of Formalism as a Working Philosophical Direction for Consideration of the Bases of Mathematics.N. V. Mikhailova - 2015 - Liberal Arts in Russiaроссийский Гуманитарный Журналrossijskij Gumanitarnyj Žurnalrossijskij Gumanitaryj Zhurnalrossiiskii Gumanitarnyi Zhurnal 4 (6):534.
    In the article, philosophical and methodological analysis of the program of Hilbert’s formalism as a really working direction for consideration of the bases of modern mathematics is presented. For the professional mathematicians methodological advantages of the program of formalism advanced by David Hilbert, consist primarily in the fact that the highest possible level of theoretical rigor of modern mathematical theories was practically represented there. To resolve the fundamental difficulties of the problem of bases of mathematics, according to Hilbert, the theory (...)
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  14. Fourteen Arguments in Favour of a Formalist Philosophy of Real Mathematics.Karlis Podnieks - 2015 - Baltic Journal of Modern Computing 3 (1):1-15.
    The formalist philosophy of mathematics (in its purest, most extreme version) is widely regarded as a “discredited position”. This pure and extreme version of formalism is called by some authors “game formalism”, because it is alleged to represent mathematics as a meaningless game with strings of symbols. Nevertheless, I would like to draw attention to some arguments in favour of game formalism as an appropriate philosophy of real mathematics. For the most part, these arguments have not yet been used or (...)
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  15. Hilbert's Objectivity.Lydia Patton - 2014 - Historia Mathematica 41 (2):188-203.
    Detlefsen (1986) reads Hilbert's program as a sophisticated defense of instrumentalism, but Feferman (1998) has it that Hilbert's program leaves significant ontological questions unanswered. One such question is of the reference of individual number terms. Hilbert's use of admittedly "meaningless" signs for numbers and formulae appears to impair his ability to establish the reference of mathematical terms and the content of mathematical propositions (Weyl (1949); Kitcher (1976)). The paper traces the history and context of Hilbert's reasoning about signs, which illuminates (...)
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  16. 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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  17. Alan Weir , Truth Through Proof: A Formalist Foundation for Mathematics . Reviewed By.Julian C. Cole - 2012 - Philosophy in Review 32 (6):529-532.
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  18. Truth Through Proof: A Formalist Foundation for Mathematics * by Alan Weir.Z. Damnjanovic - 2012 - Analysis 72 (2):415-418.
  19. Curtis Franks. The Autonomy of Mathematical Knowledge: Hilbert's Program Revisted. Cambridge: Cambridge University Press, 2009. Isbn 978-0-521-51437-8. Pp. XIII+213. [REVIEW]S. Feferman - 2012 - Philosophia Mathematica 20 (3):387-400.
  20. Hilbert Versus Hindman.Jeffry L. Hirst - 2012 - Archive for Mathematical Logic 51 (1-2):123-125.
    We show that a statement HIL, which is motivated by a lemma of Hilbert and close in formulation to Hindman’s theorem, is actually much weaker than Hindman’s theorem. In particular, HIL is finitistically reducible in the sense of Hilbert’s program, while Hindman’s theorem is not.
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  21. Craig Smorynski. Adventures in Formalism. London: College Publications, 2012. Isbn 978-1-84890-060-8. Pp. XII + 606.R. Jones - 2012 - Philosophia Mathematica 20 (3):401-403.
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  22. 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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  23. Alan Weir. Truth Through Proof: A Formalist Foundation for Mathematics. Oxford: Clarendon Press, 2010. ISBN 978-0-19-954149-2. Pp. Xiv&Plus;281: Critical Studies/Book Reviews. [REVIEW]John P. Burgess - 2011 - Philosophia Mathematica 19 (2):213-219.
    Alan Weir’s new book is, like Darwin’s Origin of Species, ‘one long argument’. The author has devised a new kind of have-it-both-ways philosophy of mathematics, supposed to allow him to say out of one side of his mouth that the integer 1,000,000 exists and even that the cardinal ℵω exists, while saying out of the other side of his mouth that no numbers exist at all, and the whole book is devoted to an exposition and defense of this new view. (...)
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  24. Неразрешимост на първата теорема за непълнотата. Гьоделова и Хилбертова математика.Vasil Penchev - 2010 - Philosophical Alternatives 19 (5):104-119.
    Can the so-ca\led first incompleteness theorem refer to itself? Many or maybe even all the paradoxes in mathematics are connected with some kind of self-reference. Gбdel built his proof on the ground of self-reference: а statement which claims its unprovabllity. So, he demonstrated that undecidaЬle propositions exist in any enough rich axiomatics (i.e. such one which contains Peano arithmetic in some sense). What about the decidabllity of the very first incompleteness theorem? We can display that it fulfills its conditions. That's (...)
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  25. Truth Through Proof: A Formalist Foundation for Mathematics.Alan Weir - 2010 - Oxford, England: Oxford University Press.
    Truth Through Proof defends an anti-platonist philosophy of mathematics derived from game formalism. Alan Weir aims to develop a more satisfactory successor to game formalism utilising a widely accepted, broadly neo-Fregean framework, in which the proposition expressed by an utterance is a function of both sense and background circumstance.
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  26. Hilbert, Logicism, and Mathematical Existence.José Ferreirós - 2009 - Synthese 170 (1):33 - 70.
    David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new (...)
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  27. The Autonomy of Mathematical Knowledge: Hilbert's Program Revisited.Curtis Franks - 2009 - Cambridge University Press.
    Most scholars think of David Hilbert's program as the most demanding and ideologically motivated attempt to provide a foundation for mathematics, and because they see technical obstacles in the way of realizing the program's goals, they regard it as a failure. Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles. He weaves together an original historical account, philosophical analysis, and his own development (...)
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  28. Rendering Conditionals in Mathematical Discourse with Conditional Elements.Joseph S. Fulda - 2009 - Journal of Pragmatics 41 (7):1435-1439.
    In "Material Implications" (1992), mathematical discourse was said to be different from ordinary discourse, with the discussion centering around conditionals. This paper shows how.
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  29. Does Truth Equal Provability in the Maximal Theory?Luca Incurvati - 2009 - Analysis 69 (2):233-239.
    According to the received view, formalism – interpreted as the thesis that mathematical truth does not outrun the consequences of our maximal mathematical theory – has been refuted by Goedel's theorem. In support of this claim, proponents of the received view usually invoke an informal argument for the truth of the Goedel sentence, an argument which is supposed to reconstruct our reasoning in seeing its truth. Against this, Field has argued in a series of papers that the principles involved in (...)
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  30. Introduction: The Three Foundational Programmes.Sten Lindström & Erik Palmgren - 2009 - In Sten Lindström, Erik Palmgren, Krister Segerberg & Viggo Stoltenberg-Hansen (eds.), Logicism, Intuitionism and Formalism: What has become of them? Springer.
  31. The Epsilon Calculus.Jeremy Avigad & Richard Zach - 2008 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University.
    The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term εx A denotes some x satisfying A(x), if there is one. In Hilbert's Program, the epsilon terms play the role of ideal elements; the aim of Hilbert's finitistic consistency proofs is to give a procedure which removes such terms (...)
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  32. Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on Finitism, Constructivity and Hilbert's Program.Solomon Feferman - 2008 - Dialectica 62 (2):179-203.
    This is a survey of Gödel's perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert's program, using his published and unpublished articles and lectures as well as the correspondence between Bernays and Gödel on these matters. There is also an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end.
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  33. Logicism, Intuitionism, and Formalism - What has Become of Them?Sten Lindstr©œm, Erik Palmgren, Krister Segerberg & Viggo Stoltenberg-Hansen (eds.) - 2008 - Berlin, Germany: Springer.
    The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in (...)
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  34. Waismann's Critique of Wittgenstein.Anthony Birch - 2007 - Analysis and Metaphysics 6:263-272.
    Friedrich Waismann, a little-known mathematician and onetime student of Wittgenstein's, provides answers to problems that vexed Wittgenstein in his attempt to explicate the foundations of mathematics through an analysis of its practice. Waismann argues in favor of mathematical intuition and the reality of infinity with a Wittgensteinian twist. Waismann's arguments lead toward an approach to the foundation of mathematics that takes into consideration the language and practice of experts.
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  35. A Critique of a Formalist-Mechanist Version of the Justification of Arguments in Mathematicians' Proof Practices.Yehuda Rav - 2007 - Philosophia Mathematica 15 (3):291-320.
    In a recent article, Azzouni has argued in favor of a version of formalism according to which ordinary mathematical proofs indicate mechanically checkable derivations. This is taken to account for the quasi-universal agreement among mathematicians on the validity of their proofs. Here, the author subjects these claims to a critical examination, recalls the technical details about formalization and mechanical checking of proofs, and illustrates the main argument with aanalysis of examples. In the author's view, much of mathematical reasoning presents genuine (...)
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  36. Axiomatics and Problematics as Two Modes of Formalisation: Deleuze's Epistemology of Mathematics'.Daniel W. Smith - 2006 - In Simon B. Duffy (ed.), Virtual Mathematics: The Logic of Difference. Clinamen. pp. 145--168.
  37. How to Nominalize Formalism &Dagger.Jody Azzouni - 2005 - Philosophia Mathematica 13 (2):135-159.
    Formalism shares with nominalism a distaste for _abstracta_. But an honest exposition of the former position risks introducing _abstracta_ as the stuff of syntax. This article describes the dangers, and offers a new escape route from platonism for the formalist. It is explained how the needed role of derivations in mathematical practice can be explained, not by a commitment to the derivations themselves, but by the commitment of the mathematician to a practice which is in accord with a theory of (...)
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  38. Formalism.Michael Detlefsen - 2005 - In Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. pp. 236--317.
    A comprehensive historical overview of formalist ideas in the philosophy of mathematics.
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  39. Hilbert Et la Notion D’Existence En Mathématiques.Yvon Gauthier - 2005 - Dialogue 44 (2):399-402.
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  40. Critical Study of Michael Potter’s Reason’s Nearest Kin. [REVIEW]Richard Zach - 2005 - Notre Dame Journal of Formal Logic 46 (4):503-513.
    Critical study of Michael Potter, Reason's Nearest Kin. Philosophies of Arithmetic from Kant to Carnap. Oxford University Press, Oxford, 2000. x + 305 pages.
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  41. On the Meaning of Hilbert's Consistency Problem (Paris, 1900).Enrico Moriconi - 2003 - Synthese 137 (1-2):129 - 139.
    The theory that ``consistency implies existence'' was put forward by Hilbert on various occasions around the start of the last century, and it was strongly and explicitly emphasized in his correspondence with Frege. Since (Gödel's) completeness theorem, abstractly speaking, forms the basis of this theory, it has become common practice to assume that Hilbert took for granted the semantic completeness of second order logic. In this paper I maintain that this widely held view is untrue to the facts, and that (...)
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  42. 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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  43. 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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  44. 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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  45. What Does Gödel's Second Theorem Say?Michael Detlefsen - 2001 - Philosophia Mathematica 9 (1):37-71.
    We consider a seemingly popular justification (we call it the Re-flexivity Defense) for the third derivability condition of the Hilbert-Bernays-Löb generalization of Godel's Second Incompleteness Theorem (G2). We argue that (i) in certain settings (rouglily, those where the representing theory of an arithmetization is allowed to be a proper subtheory of the represented theory), use of the Reflexivity Defense to justify the tliird condition induces a fourth condition, and that (ii) the justification of this fourth condition faces serious obstacles. We (...)
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  46. Proof-Theoretic Reduction as a Philosopher's Tool.Thomas Hofweber - 2000 - Erkenntnis 53 (1-2):127-146.
    Hilbert’s program in the philosophy of mathematics comes in two parts. One part is a technical part. To carry out this part of the program one has to prove a certain technical result. The other part of the program is a philosophical part. It is concerned with philosophical questions that are the real aim of the program. To carry out this part one, basically, has to show why the technical part answers the philosophical questions one wanted to have answered. Hilbert (...)
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  47. Why Do We Prove Theorems?Yehuda Rav - 1999 - Philosophia Mathematica 7 (1):5-41.
    Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...)
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  48. Constructive Existence Claims.Michael Detlefsen - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press. pp. 1998--307.
    It is a commonplace of constructivist thought that a claim that an object of a certain kind exists is to be backed by an explicit display or exhibition of an object that is manifestly of that kind. Let us refer to this requirement as the exhibition condition. The main objective of this essay is to examine this requirement and to arrive at a better understanding of its epistemic character and the role that it plays in the two main constructivist philosophies (...)
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  49. Review of R. Hersh, What is Mathematics, Really?.F. Richman - 1998 - Philosophia Mathematica 6 (2):245-255.
  50. Numbers and Functions in Hilbert's Finitism.Richard Zach - 1998 - Taiwanese Journal for History and Philosophy of Science 10:33-60.
    David Hilbert's finitistic standpoint is a conception of elementary number theory designed to answer the intuitionist doubts regarding the security and certainty of mathematics. Hilbert was unfortunately not exact in delineating what that viewpoint was, and Hilbert himself changed his usage of the term through the 1920s and 30s. The purpose of this paper is to outline what the main problems are in understanding Hilbert and Bernays on this issue, based on some publications by them which have so far received (...)
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