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Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
Hilbert’s “Verunglückter Beweis,” the first epsilon theorem and consistency proofs. History and Philosophy of Logic
"... Abstract. On the face of it, Hilbert’s Program was concerned with proving consistency of mathematical systems in a finitary way. This was to be accomplished by showing that that these systems are conservative over finitistically interpretable and obviously sound quantifierfree subsystems. One propo ..."
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Abstract. On the face of it, Hilbert’s Program was concerned with proving consistency of mathematical systems in a finitary way. This was to be accomplished by showing that that these systems are conservative over finitistically interpretable and obviously sound quantifierfree subsystems. One proposed method of giving such proofs is Hilbert’s epsilonsubstitution method. There was, however, a second approach which was not refelected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert’s first epsilon theorem and a certain “general consistency result. ” An analysis of this socalled “failed proof ” lends further support to an interpretation of Hilbert according to which he was expressly concerned with conservatitvity proofs, even though his publications only mention consistency as the main question. §1. Introduction. The aim of Hilbert’s program for consistency proofs in the 1920s is well known: to formalize mathematics, and to give finitistic consistency proofs of these systems and thus to put mathematics on a “secure foundation.” What is perhaps less well known is exactly how Hilbert thought this should be carried out. Over ten years before Gentzen developed sequent calculus formalizations
Incompleteness, Mechanism, And Optimism
, 1998
"... this paper would be "G odel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility". Adding "God and the Devil" would probably be redundant. Despite the breathtaking, whirlwind tour, I have the modest aim of forging connections between di#erent par ..."
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this paper would be "G odel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility". Adding "God and the Devil" would probably be redundant. Despite the breathtaking, whirlwind tour, I have the modest aim of forging connections between di#erent parts of this literature and clearing up some confusions, together with the less modest aim of not introducing any more confusions.
WE HOLD THESE TRUTHS TO BE SELFEVIDENT: BUT WHAT DO WE MEAN BY THAT?
"... Mathematicians at first distrusting the new ideas (Cantor made his first discoveries in 1873), then got used to them;... Waismann (1982, p. 102) Abstract. At the beginning of Die Grundlagen der Arithmetik (§2) [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where pro ..."
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Mathematicians at first distrusting the new ideas (Cantor made his first discoveries in 1873), then got used to them;... Waismann (1982, p. 102) Abstract. At the beginning of Die Grundlagen der Arithmetik (§2) [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of selfevidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both programs are undermined at a crucial point, namely when selfevidence is supported by holistic and even pragmatic considerations. At the beginning of Die Grundlagen der Arithmetik (§2) (1884), Gottlob Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”, noting that “Euclid gives proofs of many things which anyone would concede him without question”. Frege sets himself the task of providing proofs of such basic arithmetic propositions as