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  1. Alice Ambrose (1935). Finitism in Mathematics (I). Mind 44 (174):186-203.
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  2. J. P. Bendegem (2012). A Defense of Strict Finitism. Constructivist Foundations 7 (2):141-149.
    Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism (which is now lacking) and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, (...)
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  3. Andrew Boucher, A Natural First-Order System of Arithmetic Which Proves Its Own Consistency.
    Herein is presented a natural first-order arithmetic system which can prove its own consistency, both in the weaker Godelian sense using traditional Godel numbering and, more importantly, in a more robust and direct sense; yet it is strong enough to prove many arithmetic theorems, including the Euclidean Algorithm, Quadratic Reciprocity, and Bertrand’s Postulate.
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  4. Michael Detlefsen (1998). Constructive Existence Claims. In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press 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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  5. Michael Detlefsen (1993). Hilbert's Formalism. 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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  6. Michael Detlefsen (1990). On an Alleged Refutation of Hilbert's Program Using Gödel's First Incompleteness Theorem. 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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  7. Michael Detlefsen (1979). On Interpreting Gödel's Second Theorem. 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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  8. Luca Incurvati (forthcoming). On the Concept of Finitism. Synthese:1-24.
    At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.
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  9. Marc A. Joseph (2001). Review of M. Marion, Wittgenstein, Finitism, and the Foundations of Mathematics. [REVIEW] Mind 110 (438):501-504.
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  10. 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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  11. David G. Stern (2001). Review of M. Marion, Wittgenstein, Finitism, and the Foundations of Mathematics. [REVIEW] Dialogue 40 (03):624-626.
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  12. W. W. Tait (2010). Gödel on Intuition and on Hilbert's Finitism. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic
    There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, G¨ odel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primarily to (...)
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  13. Adam Tamas Tuboly (2014). Unifying Historical Perspectives. [REVIEW] The Berlin Review of Books 2014.
    Greg Frost-Arnold’s book is a highly elegant edition and commentary of Carnap’s notes, claiming just as much as he is warranted on the basis of the manuscript and other relevant texts, and formulating his scholarly assumptions very carefully. Along the way he tries to unify the three historiographical strategies: narrative, argumentative and micro-historical.
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  14. Eric Winsberg (2000). Review of M. Marion, Wittgenstein, Finitism, and the Foundations of Mathematics. [REVIEW] Philosophy of Science 67 (3):533-.
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  15. Richard Zach (2003). The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program. 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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