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"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how ..."
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
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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Cited by 3 (2 self)
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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
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
Epistemic truth and excluded middle*
"... Abstract: Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemi ..."
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Abstract: Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemic conception of truth and the principle of excluded middle. In PART II I give a historical overview of different attitudes regarding the problem. In PART III I sketch a possible holistic solution. Part I The Problem §1. The epistemic conception of truth. The epistemic conception of truth can be formulated in many ways. But the basic idea is that truth is explained in terms of epistemic notions, like experience, argument, proof, knowledge, etc. One way of formulating this idea is by saying that truth and knowability coincide, i.e. for every statement S