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  1. Jeremy Avigad & Richard Zach, The Epsilon Calculus. Stanford Encyclopedia of Philosophy.
    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..
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  2. J. Azzouni (2005). How to Nominalize Formalism. 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. Robert J. Baum (1972). The Instrumentalist and Formalist Elements of Berkeley's Philosophy of Mathematics. 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. Anthony Birch (2007). Waismann's Critique of Wittgenstein. 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. J. P. Burgess (2011). Alan Weir. Truth Through Proof: A Formalist Foundation for Mathematics. Oxford: Clarendon Press, 2010. ISBN 978-0-19-954149-2. Pp. Xiv+281. [REVIEW] Philosophia Mathematica 19 (2):213-219.
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  6. Haskell B. Curry (1951/1970). Outlines of a Formalist Philosophy of Mathematics. Amsterdam,North-Holland Pub. Co..
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  7. Haskell B. Curry, J. Roger Hindley & J. P. Seldin (eds.) (1980). To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism. Academic Press.
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  8. Anita Dilger (1987). Formalism and its Limits. Investigations Into the Recent Philosophy of Mathematics. Philosophy and History 20 (2):145-146.
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  9. William J. Edgar (1973). Is Intuitionism the Epistemically Serious Foundation for Mathematics? Philosophia Mathematica (2):113-133.
  10. S. Feferman (2012). 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] Philosophia Mathematica 20 (3):387-400.
  11. Solomon Feferman (2008). Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on Finitism, Constructivity and Hilbert's Program. Dialectica 62 (2: Table of Contents"/> Select):179–203.
  12. José Ferreirós (2009). Hilbert, Logicism, and Mathematical Existence. 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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  13. Curtis Franks (2009). The Autonomy of Mathematical Knowledge: Hilbert's Program Revisited. Cambridge University Press.
    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.
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  14. Joseph S. Fulda (2009). Rendering Conditionals in Mathematical Discourse with Conditional Elements. 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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  15. Marcus Giaquinto (1983). Hilbert's Philosophy of Mathematics. British Journal for the Philosophy of Science 34 (2):119-132.
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  16. David Hilbert (1970). Axiomatic Thinking. Philosophia Mathematica (1-2):1-12.
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  17. Thomas Hofweber (2000). Proof-Theoretic Reduction as a Philosopher's Tool. 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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  18. Luca Incurvati (2009). Does Truth Equal Provability in the Maximal Theory? 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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  19. R. Jones (2012). Craig Smorynski. Adventures in Formalism. London: College Publications, 2012. Isbn 978-1-84890-060-8. Pp. XII + 606. Philosophia Mathematica 20 (3):401-403.
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  20. Sten Lindström & Erik Palmgren (2009). Introduction: The Three Foundational Programmes. In Sten Lindström, Erik Palmgren, Krister Segerberg & Viggo Stoltenberg-Hansen (eds.), Logicism, Intuitionism and Formalism: What has become of them? Springer.
  21. Sten Lindström, Erik Palmgren, Krister Segerberg & Viggo Stoltenberg-Hansen (eds.) (2009). Logicism, Intuitionism, and Formalism - What has Become of Them? Springer.
    These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics.A special section is concerned with constructive ...
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  22. Lydia Patton (2014). Hilbert's Objectivity. 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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  23. Volker Peckhaus (2003). The Pragmatism of Hilbert's Programme. 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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  24. Y. Rav (1999). Why Do We Prove Theorems? 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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  25. Yehuda Rav (2007). A Critique of a Formalist-Mechanist Version of the Justification of Arguments in Mathematicians' Proof Practices. 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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  26. Michael D. Resnik (1995). Review of J. Azzouni, Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences. [REVIEW] Philosophia Mathematica 3 (3).
  27. F. Richman (1998). Review of R. Hersh, What is Mathematics, Really?. Philosophia Mathematica 6 (2):245-255.
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  28. M. A. Rozov (1989). The Mode of Existence of Mathematical Objects. Philosophia Mathematica (2):105-111.
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  29. Richard Tieszen (1993). Review of J. O'Neill, Worlds Without Content: Against Formalism. [REVIEW] Husserl Studies 10 (3).
  30. Iulian D. Toader (2014). Why Did Weyl Think That Formalism's Victory Against Intuitionism Entails a Defeat of Pure Phenomenology? History and Philosophy of Logic 35 (2):198-208.
    It has been contended that it is unjustified to believe, as Weyl did, that formalism's victory against intuitionism entails a defeat of the phenomenological approach to mathematics. The reason for this contention, recently put forth by Paolo Mancosu and Thomas Ryckman, is that, unlike intuitionistic Anschauung, phenomenological intuition could ground classical mathematics. I argue that this indicates a misinterpretation of Weyl's view, for he did not take formalism to prevail over intuitionism with respect to grounding classical mathematics. I also point (...)
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  31. Steven J. Wagner (1996). Review of J. O'Neill, Worlds Without Content: Against Formalism. [REVIEW] Philosophia Mathematica 4 (3).
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  32. Judson C. Webb (1997). Hilbert's Formalism and Arithmetization of Mathematics. Synthese 110 (1):1-14.
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  33. Alan Weir, Formalism in the Philosophy of Mathematics.
    The guiding idea behind formalism is that mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess. This idea has some intuitive plausibility: consider the tyro toiling at multiplication tables or the student using a standard algorithm for differentiating or integrating a function. It also corresponds to some aspects of the practice of (...)
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  34. Alan Weir, A Neo-Formalist Approach to Mathematical Truth.
    I outline a variant on the formalist approach to mathematics which rejects textbook formalism's highly counterintuitive denial that mathematical theorems express truths while still avoiding ontological commitment to a realm of abstract objects. The key idea is to distinguish the sense of a sentence from its explanatory truth conditions. I then look at various problems with the neo-formalist approach, in particular at the status of the notion of proof in a formal calculus and at problems which Gödelian results seem to (...)
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  35. Alan Weir (2010). Truth Through Proof: A Formalist Foundation for Mathematics. OUP Oxford.
    Truth Through Proof defends an anti-platonist philosophy of mathematics derived from game formalism. Classic formalists claimed implausibly that mathematical utterances are truth-valueless moves in a game. 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. This framework allows for sentences whose truth-conditions are not representational, which are made true or false by conditions residing (...)
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  36. Richard Zach (2005). Critical Study of Michael Potter’s Reason’s Nearest Kin. Notre Dame Journal of Formal Logic 46:503-513.
    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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