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  1. The Methodological Roles of Tolerance and Conventionalism in the Philosophy of Mathematics: Reconsidering Carnap's Logic of Science.Emerson P. Doyle - 2014 - Dissertation, University of Western Ontario
    This dissertation makes two primary contributions. The first three chapters develop an interpretation of Carnap's Meta-Philosophical Program which places stress upon his methodological analysis of the sciences over and above the Principle of Tolerance. Most importantly, I suggest, is that Carnap sees philosophy as contiguous with science—as a part of the scientific enterprise—so utilizing the very same methods and subject to the same limitations. I argue that the methodological reforms he suggests for philosophy amount to philosophy as the explication of (...)
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  • Godel's Interpretation of Intuitionism.William Tait - 2006 - Philosophia Mathematica 14 (2):208-228.
    Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will (1) criticize this foundation, (2) propose a quite different one, and (3) note that essentially the latter foundation also underlies the Curry-Howard type theory, and hence Heyting's intuitionistic conception of logic. Thus the Dialectica interpretation (in (...)
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  • Reading Gentzen's Three Consistency Proofs Uniformly.Ryota Akiyoshi & Yuta Takahashi - 2013 - Journal of the Japan Association for Philosophy of Science 41 (1):1-22.
  • Gödel's Reformulation of Gentzen's First Consistency Proof for Arithmetic: The No-Counterexample Interpretation.W. W. Tait - 2005 - Bulletin of Symbolic Logic 11 (2):225-238.
    The last section of “Lecture at Zilsel’s” [9, §4] contains an interesting but quite condensed discussion of Gentzen’s first version of his consistency proof for P A [8], reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen’s result (in game-theoretic terms), fill in the details (with some corrections) of Godel's reformulation, and discuss the relation between the two proofs.
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  • On Reflection Principles.Peter Koellner - 2009 - Annals of Pure and Applied Logic 157 (2-3):206-219.
    Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a series of limitative results concerning this approach. These results collectively show that general reflection principles are either weak ) or inconsistent. The philosophical significance of these (...)
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