Hilbert's Program Then and Now

Zach, Richard

doi:10.1016/b978-044451541-4/50014-2

Cited by 44 publications

(28 citation statements)

References 75 publications

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“…Following e.g. [144] we call this kind of modified Hilbert's programme 'generalised Hilbert's programme'. 19 Another kind of modification of Hilbert's programme, proposed by Kreisel and then especially brought forward by Feferman, is usually referred to as 'relativised Hilbert's programme' [55].…”

Section: Generalised and Relativised Hilbert's Programmesmentioning

confidence: 99%

“…Regrettably, we here do no justice to the complexity of Hilbert's thought (as well as of his collaborators') nor take into due account the deep and rich literature on the subject. For an excellent recent survey on Hilbert's programme, also in the light of the latest contributions in proof theory, we refer to Zach's[144]; see also[91] and[123,6], the latter especially with regard to the historical progression of the programme and the developments of metamathematics and proof theory 3. We wish to highlight the purely syntactic character of the notion of consistency, and recall that, as clarified in[123,6], Hilbert and Bernays had a fundamental role in…”

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confidence: 99%

“…See e.g [144]. for a concise summary.10 In[70] Hilbert calls number theory the 'purest and simplest offspring of the human mind'.…”

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confidence: 99%

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Finite Methods in Mathematical Practice

Schuster¹,

Crosilla²

2014

Formalism and Beyond

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In the present contribution we look at the legacy of Hilbert's programme in some recent developments in mathematics. Hilbert's ideas have seen new life in generalised and relativised forms by the hands of proof theorists and have been a source of motivation for the so-called reverse mathematics programme initiated by H. Friedman and S. Simpson. More recently Hilbert's programme has inspired T. Coquand and H. Lombardi to undertake a new approach to constructive algebra in which strong emphasis is laid on the use of finite methods. The main aim is to eliminate the ideal objects and in so doing obtain more elementary and informative proofs. We survey some work in commutative algebra -mainly about and around the Zariski spectrum and the Krull dimension of a commutative ring -which witnesses the feasibility of such a revised Hilbert's programme.

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Section: Generalised and Relativised Hilbert's Programmesmentioning

confidence: 99%

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confidence: 99%

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Finite Methods in Mathematical Practice

Schuster¹,

Crosilla²

2014

Formalism and Beyond

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“…His investigation was motivated by an outstanding problem in the foundations of mathematics called decision problem [Hod02] which was posed by David Hilbert. He worked on a formalistic approach on the foundations of mathematics also known as Hilbert's program [Zac03]. The decision problem is to show that there exists a definite method to decide with a finite number of elementary steps whether a mathematical proposition is provable or not.…”

Section: Turing Machines and Computational Complexity Classesmentioning

confidence: 99%

Evolving blackbox quantum algorithms using genetic programming

Stadelhofer

Banzhaf

Suter

2008

AIEDAM

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Although it is known that quantum computers can solve certain computational problems exponentially faster than classical computers, only a small number of quantum algorithms have been developed so far. Designing such algorithms is complicated by the rather nonintuitive character of quantum physics. In this paper we present a genetic programming system that uses some new techniques to develop and improve quantum algorithms. We have used this system to develop two formerly unknown quantum algorithms. We also address a potential deficiency of the quantum decision tree model used to prove lower bounds on the query complexity of the parity problem.

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“…14 14 What Hilbert meant by finitism has been a subject of extensive discussion in the literature; the trouble is that he made no specific effort to delineate it and instead relied on informal explanations and examples. A strong case has been made by Zach (1998, 2001, 2003) among others that Hilbert's conception of finitism definitely goes beyond PRA to include Ackermann's function and other functions obtained by multiply nested recursions, together with certain forms of transfinite recursion (in view of Hilbert 1926). …”

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Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert's program*

Feferman¹

2008

Dialectica

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The correspondence between Paul Bernays and Kurt Gödel is one of the most extensive in the two volumes of Gödel's collected works devoted to his letters of (primarily) scientific, philosophical and historical interest. It ranges from 1930 to 1975 and deals with a rich body of logical and philosophical issues, including the incompleteness theorems, finitism, constructivity, set theory, the philosophy of mathematics, and postKantian philosophy, and contains Gödel's thoughts on many topics that are not expressed elsewhere. In addition, it testifies to their life-long warm personal relationship. I have given a detailed synopsis of the Bernays Gödel correspondence, with explanatory background, in my introductory note to it in Vol. IV of Gödel's Collected Works, pp. 41-79. References to individual items by Gödel follow the system of these volumes, which are either of the form Gödel 19xx or of the form *Gödel 19xx with possible further addition of a letter in the case of multiple publications within a given year; the former are from CW I or CW II, while the latter are from CW III. Thus, for example, Gödel 1931 is the famous incompleteness paper, while Gödel 1931c is a review that Gödel wrote of an article by Hilbert, both in CW I; Gödel °1933o is notes for a lecture, "The present situation in the foundations of mathematics," to be found in CW III. Pagination is by reference to these volumes, e.g. Gödel 1931, CW I, p. 181, or simply, CW I, p. 181. In the case of correspondence, reference is by letter number and/or date within a given body of correspondence, as e.g. (Gödel to Bernays) letter #56, or equivalently 2 Dec. 1965, under Bernays in CW IV. When an item in question was originally written in German, my quotation from it is taken from the facing English translation. Finally, reference will be made to various of the introductory notes written by the editors and colleagues that accompany most of the pieces or bodies of correspondence. incompleteness theorem: no consistent formal axiomatic extension of a system that contains a sufficient amount of arithmetic is complete. Both of these deserved Hilbert's approbation, but not a word passed from him in public or in writing at the time. In fact, there are no communications between Hilbert and Gödel and they never met. Perhaps the second incompleteness theorem on the unprovability of consistency of a system took Hilbert by surprise. We don't know exactly what he made of it, but we can appreciate that it might have been quite disturbing, for he had invested a great deal of thought and emotion in his finitary consistency program which became problematic as a result. There is just one comment, of a dismissive character, that he made about it four years later; I will return to that in the following.

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Hilbert's Program Then and Now

Cited by 44 publications

References 75 publications

Finite Methods in Mathematical Practice

Finite Methods in Mathematical Practice

Evolving blackbox quantum algorithms using genetic programming

Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert's program*

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