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Computability and recursion
 BULL. SYMBOLIC LOGIC
, 1996
"... We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they b ..."
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Cited by 38 (1 self)
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We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than
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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Cited by 10 (0 self)
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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
"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 sho ..."
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Cited by 9 (3 self)
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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.
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
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HIERARCHICAL REPRESENTATION OF LEGAL KNOWLEDGE WITH METAPROGRAMMING IN LOGIC
 J. LOGIC PROGRAMMING
, 1994
"... We present an application of metaprogramming in logic that, unlike most metaprogramming applications, is not primarily concerned with controlling the execution of logic programs. Metalevel computation is used to define theories from schemata that were either given explicitly or obtained by abstracti ..."
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Cited by 6 (2 self)
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We present an application of metaprogramming in logic that, unlike most metaprogramming applications, is not primarily concerned with controlling the execution of logic programs. Metalevel computation is used to define theories from schemata that were either given explicitly or obtained by abstraction from other theories. Our main application is a representation of legal knowledge in a metalogic programming language. We argue that legal knowledge is multilayered and therefore a single level representation language lacks the needed expressiveness. We show that legal rules can be partitioned into primary, secondary, tertiary, quaternary, and higher level rules. Our classification enables us to define a multilevel model of legal knowledge and a onetoone correspondence with levels of metaprogramming in logic. We show that this framework has a potential for capturing important legal interpretation principles such as analogia legis, lex specialis Zegi generuli derogut, etc. We have a running example from commercial law that utilizes rules up to the tertiary level, emphasizing unulogiu legis. The example is expressed in a multilevel metalogic programming language that provides a naming convention and employs reflection between levels.
The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program
, 2001
"... . After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the ..."
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. After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Gttingen between 1917 and 1923, and notably in Ackermann 's dissertation of 1924. The main innovation was the invention of the ecalculus, on which Hilbert's axiom systems were based, and the development of the esubstitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the enotation and provides an analysis of Ackermann's consisten...
Brouwer and Fraenkel on Intuitionism
"... In the present paper the story is told of the brief and far from tranquil encounter of L.E.J. Brouwer and A. Fraenkel. The relationship which started in perfect harmony, ended in irritation and reproaches.1 The mutual appreciation at the outset is beyond question. All the more deplorable is the sud ..."
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In the present paper the story is told of the brief and far from tranquil encounter of L.E.J. Brouwer and A. Fraenkel. The relationship which started in perfect harmony, ended in irritation and reproaches.1 The mutual appreciation at the outset is beyond question. All the more deplorable is the sudden outbreak of an emotional disagreement in 1927. Looking at the Brouwer–Fraenkel episode, one should keep in mind that at that time the socalled Grundlagenstreit2 was in full swing. An emotional man like Brouwer, who easily suffered under stress, was already on edge when Fraenkel’s book Zehn Vorlesungen über die Grundlegung der Mengenlehre, [Fraenkel 1927] was about to appear. With the Grundlagenstreit reaching (in print!) a level of personal abuse unusual in the quiet circles of pure mathematics, Brouwer was rather sensitive, where the expositions of his ideas were concerned. So when he thought that he detected instances of misconception and misrepresentation in the case of his intuitionism, he felt slighted. We will mainly look at Brouwer’s reactions. since
www.elsevier.com/locate/dsw A methodology for conducting interdisciplinary social research
"... The Vienna Circle idea that science is an interdisciplinary enterprise leads to the question of how the knowledge of the natural sciences can be used to further understanding in the social sciences. Analysis of the practice of social research shows there is no easy answer to this question. Ideologie ..."
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The Vienna Circle idea that science is an interdisciplinary enterprise leads to the question of how the knowledge of the natural sciences can be used to further understanding in the social sciences. Analysis of the practice of social research shows there is no easy answer to this question. Ideologies colour the use of exact knowledge in social research methods; even in the natural sciences ideological misunderstandings seem inevitable in research practice. The concept of strangi®cation is introduced to describe this situation and to give a framework for a methodology to handle this problem of scienti®c regression. It is applied to the problem how the mathematical theory of complexity can be used in
BERNAYS AND SET THEORY
"... We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. ..."
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We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles.