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  1. 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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  2. How to Nominalize Formalism.J. 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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  3. The Instrumentalist and Formalist Elements of Berkeley's Philosophy of Mathematics.Robert J. Baum - 1972 - Studies in History and Philosophy of Science Part A 3 (2):119-134.
    The main thesis of this paper is that, Contrary to general belief, George berkeley did in fact express a coherent philosophy of mathematics in his major published works. He treated arithmetic and geometry separately and differently, And this paper focuses on his philosophy of arithmetic, Which is shown to be strikingly similar to the 19th and 20th century philosophies of mathematics known as 'formalism' and 'instrumentalism'. A major portion of the paper is devoted to showing how this philosophy of mathematics (...)
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  4. Waismann's Critique of Wittgenstein.Anthony Birch - 2007 - Analysis and Metaphysics 6 (2007):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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  5. Alan Weir. Truth Through Proof: A Formalist Foundation for Mathematics. Oxford: Clarendon Press, 2010. ISBN 978-0-19-954149-2. Pp. Xiv+281. [REVIEW]J. 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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  6. 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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  7. Outlines of a Formalist Philosophy of Mathematics.Haskell B. Curry - 1951 - Amsterdam: North-Holland Pub. Co..
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  8. To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism.Haskell B. Curry, J. Roger Hindley & J. P. Seldin (eds.) - 1980 - Academic Press.
  9. Truth Through Proof: A Formalist Foundation for Mathematics * by Alan Weir.Z. Damnjanovic - 2012 - Analysis 72 (2):415-418.
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  10. Formalism.Michael Detlefsen - 2005 - In Stewart Shapiro (ed.), The 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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  11. 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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  12. 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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  13. Hilbert's Formalism.Michael Detlefsen - 1993 - Revue Internationale de Philosophie 47 (186):285-304.
    Various parallels between Kant's critical program and Hilbert's formalistic program for the philosophy of mathematics are considered.
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  14. On an Alleged Refutation of Hilbert's Program Using Gödel's First Incompleteness Theorem.Michael Detlefsen - 1990 - Journal of Philosophical Logic 19 (4):343 - 377.
    It is argued that an instrumentalist notion of proof such as that represented in Hilbert's viewpoint is not obligated to satisfy the conservation condition that is generally regarded as a constraint on Hilbert's Program. A more reasonable soundness condition is then considered and shown not to be counter-exemplified by Godel's First Theorem. Finally, attention is given to the question of what a theory is; whether it should be seen as a "list" or corpus of beliefs, or as a method for (...)
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  15. Hilbert's Program: An Essay on Mathematical Instrumentalism.Michael Detlefsen - 1986 - Reidel.
    An Essay on Mathematical Instrumentalism M. Detlefsen. THE PHILOSOPHICAL FUNDAMENTALS OF HILBERT'S PROGRAM 1. INTRODUCTION In this chapter I shall attempt to set out Hilbert's Program in a way that is more revealing than ...
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  16. On Interpreting Gödel's Second Theorem.Michael Detlefsen - 1979 - Journal of Philosophical Logic 8 (1):297 - 313.
    In this paper I have considered various attempts to attribute significance to Gödel's second incompleteness theorem (G2 for short). Two of these attempts (Beth-Cohen and the position maintaining that G2 shows the failure of Hilbert's Program), I have argued, are false. Two others (an argument suggested by Beth, Cohen and ??? and Resnik's Interpretation), I argue, are groundless.
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  17. Consistency Problem and “Unexpected Hanging Paradox” (An Answering to P=NP Problem).Farzad Didehvar - unknown
    Abstract The Theory of Computation in its existed form is based on Church –Turing Thesis. Throughout this paper, we show that the Turing computation model of this theory leads us to a contradiction. In brief, by applying a well-known paradox (Unexpected hanging paradox) we show a contradiction in the Theory when we consider the Turing model as our Computation model.
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  18. Formalism and its Limits. Investigations Into the Recent Philosophy of Mathematics.Anita Dilger - 1987 - Philosophy and History 20 (2):145-146.
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  19. Is Intuitionism the Epistemically Serious Foundation for Mathematics?William J. Edgar - 1973 - Philosophia Mathematica (2):113-133.
  20. 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.
  21. Hilbert's Program Modi Ed.Solomon Feferman - unknown
    The background to the development of proof theory since 1960 is contained in the article (MATHEMATICS, FOUNDATIONS OF), Vol. 5, pp. 208- 209. Brie y, Hilbert's program (H.P.), inaugurated in the 1920s, aimed to secure the foundations of mathematics by giving nitary consistency proofs of formal systems such as for number theory, analysis and set theory, in which informal mathematics can be represented directly. These systems are based on classical logic and implicitly or explicitly depend on the assumption of \completed (...)
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  22. 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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  23. 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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  24. 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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  25. 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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  26. Finitism: An Essay on Hilbert's Programme.David Watson Galloway - 1991 - Dissertation, Massachusetts Institute of Technology
    In this thesis, I discuss the philosophical foundations of Hilbert's Consistency Programme of the 1920's, in the light of the incompleteness theorems of Godel. ;I begin by locating the Consistency Programme within Hilbert's broader foundational project. I show that Hilbert's main aim was to establish that classical mathematics, and in particular classical analysis, is a conservative extension of finitary mathematics. Accepting the standard identification of finitary mathematics with primitive recursive arithmetic, and classical analysis with second order arithmetic, I report upon (...)
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  27. Hilbert Et la Notion D’Existence En Mathématiques.Yvon Gauthier - 2005 - Dialogue 44 (2):399-402.
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  28. Hilbert's Philosophy of Mathematics.Marcus Giaquinto - 1983 - British Journal for the Philosophy of Science 34 (2):119-132.
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  29. Study of Analytic Number Theory: Riemann’s Hypothesis and Prime Number Theorem with Addendum on Integer Partitions.Lukasz Andrzej Glinka - 2013 - Cambridge International Science Publishing.
    This monograph explores several classical issues of modern mathematics, and discusses both the historical and research aspects. The brief historical part is focused on Riemann’s zeta function and few related problems. The research part starts from direct formulation of simple proofs of both the prime number theorem and Riemann’s hypothesis, two intriguing problems of modern mathematics, which applies the concept of Mertens’s function and is based on Apostol’s and Littlewood’s criterions of equivalence. The second research problem discussed in this book (...)
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  30. Towards a Theory of Mathematical Research Programmes (I).Michael Hallett - 1979 - British Journal for the Philosophy of Science 30 (1):1-25.
  31. Axiomatic Thinking.David Hilbert - 1970 - Philosophia Mathematica (1-2):1-12.
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  32. 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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  33. 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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  34. 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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  35. 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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  36. 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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  37. 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.
  38. Logicism, Intuitionism, and Formalism - What has Become of Them?Sten Lindström, Erik Palmgren, Krister Segerberg & Viggo Stoltenberg-Hansen (eds.) - 2009 - 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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  39. Hilbert program of formalism as a working philosophical direction for consideration of the bases of mathematics.N. V. Mikhailova - 2015 - Liberal Arts in Russia 4 (6):534-545.
    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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  40. 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.
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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. 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.
  43. 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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  44. 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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  45. 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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  46. 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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  47. Why Do We Prove Theorems?Y. 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. 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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  49. Review of J. Azzouni, Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences[REVIEW]Michael D. Resnik - 1995 - Philosophia Mathematica 3 (3).
  50. Review of R. Hersh, What is Mathematics, Really?.F. Richman - 1998 - Philosophia Mathematica 6 (2):245-255.
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