A Forgotten Theory of Proofs ? (original) (raw)
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The study of the foundations of mathematics is in a crisis: the failure of logicism has invited a satisfactory alternative. This should be a better proof theory; Gentzen's natural deduction came to provide this but had a limited success. The discussion on this practically faded out. Solomon Feferman and his coworkers propose tentatively a synthesis that relies on achievements from diverse directions. Presenting sympathetically ideas of Hilbert, Brouwer, Gentzen, Weyl, and Bishop, they hope to change in the standard textbooks of logic. This requires a more problem-oriented presentatlon of the background material that should make the situation more accessible to the ordinary philosophy faculty. 'Ihk invites supplementary discussions of achievements that they are thus far sadly overlooked, especially those of Popper, Robinson, and Lakatos.
From mathematical axioms to mathematical rules of proof: recent developments in proof analysis
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019
A short text in the hand of David Hilbert, discovered in Göttingen a century after it was written, shows that Hilbert had considered adding a 24th problem to his famous list of mathematical problems of the year 1900. The problem he had in mind was to find criteria for the simplicity of proofs and to develop a general theory of methods of proof in mathematics. In this paper, it is discussed to what extent proof theory has achieved the second of these aims. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.
Proof Theory in Philosophy of Mathematics
Philosophy Compass, 2010
A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.
The Fundamental Problem of General Proof Theory
Studia Logica, 2018
I see the question what it is that makes an inference valid and thereby gives a proof its epistemic power as the most fundamental problem of general proof theory. It has been surprisingly neglected in logic and philosophy of mathematics with two exceptions: Gentzen's remarks about what justifies the rules of his system of natural deduction and proposals in the intuitionistic tradition about what a proof is. They are reviewed in the paper and I discuss to what extent they succeed in answering what a proof is. Gentzen's ideas are shown to give rise to a new notion of valid argument. At the end of the paper I summarize and briefly discuss an approach to the problem that I have proposed earlier.
Structures of Proofs, unpublished paper, August 2003.
2003
This unpublished paper discusses and illustrates what we call the hierarchical structure of proofs, as well as the linear structure of proofs. It also discusses what we have come to call the formal-rhetorical part of a proof and the problem-solving part of a proof. Lastly, it discusses the idea of "logic in action".
A Natural Interpretation of Classical Proofs
2006
of accuracy. On the other hand, the constructivist claims that to know that a particular sequence of rational numbers converges is to know how to approximate the corresponding real number to an arbitrarily given degree of accuracy. Hence the constructivist rejects the law of excluded middle. Using the law of excluded middle one can also argue that every sequence of rational numbers that does not converge must diverge. However, one can not from the knowledge that a particular sequence of rational numbers does not converge construct the corresponding witness. The example just given illustrates the direct nature of constructive existence, as opposed to the indirect nature of classical existence, and shows why the law of excluded middle is not accepted as a law of constructive logic. Finally, I would like to thank my supervisor, Per Martin-Löf, for posing the problem of investigating how the double-negation interpretation operates on derivations and not only on formulas as well as for his continued guidance of my work. Without him, this thesis would never have come into existence. Jens Brage Reynolds (1972), and Plotkin (1975) for the foundations of CPS translations and Reynolds (1993) for the early history of continuations. To my knowledge there is no interpretation of classical logic in constructive logic that makes full use of the syntactic-semantic method of constructive type theory. With this thesis I hope to fill this gap. The subject of interpretations of classical logic in constructive logic began with the double negation interpretation of classical logic in minimal logic due to Kolmogorov (1925). The double negation interpretation was then followed by the interpretation of Peano arithmetic in Heyting arithmetic due to Gödel (1933) and the interpretation of classical logic in intuitionistic logic due to Kuroda (1951). Yet it was not until Griffin (1990) showed how to extend the formulae-as-types correspondence to classical logic that significant growth took place. His solution was to include operations on the flow of control, similar to call/cc of Scheme, into the notion of computation given by a simply typed call-by-value λ-calculus. After that Parigot (1992) introduced his λµ-calculus to realize classical proofs as programs. The λµ-calculus extended the simply typed λ-calculus with operators that can be used to model operations on the flow of control. The development then took the form of CPS translations of different λµ-calculi into different λ-calculi. See Ong (1996) and Ong and Stewart (1997) for call-by-value respectively call-byname CPS translations of Parigot's λµ-calculus into the simply typed λ-calculus. See Selinger (2001, p. 24) for an informal description of the semantics of the λµcalculus.
Analytic Methods for the Logic of Proofs
Journal of Logic and Computation, 2008
The logic of proofs (LP) was proposed as Gödel's missed link between Intuitionistic and S4-proofs, but so far the tableau-based methods proposed for LP have not explored this closeness with S4 and contain rules whose analycity is not immediately evident. We study possible formulations of analytic tableau proof methods for LP that preserve the subformula property. Two sound and complete tableau decision methods of increasing degree of analycity are proposed, KELP and preKELP. The latter is particularly inspired on S4-proofs. The crucial role of proof constants in the structure of LP-proofs methods is analysed. In particular, a method for the abduction of proof constant specifications in strongly analytic preKELP proofs is presented; abduction heuristics and the complexity of the method are discussed.