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107
"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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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.
Traditional logic, modern logic and natural language
"... DRAFT June 2009. The paper is for a Festschrift and this draft has removed a number of personal references. 1 The questions... Wikipedia [38] defines: traditional logic is ‘a loose name for the way of doing logic that began with Aristotle, and that was dominant until the advent of modern predicate l ..."
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DRAFT June 2009. The paper is for a Festschrift and this draft has removed a number of personal references. 1 The questions... Wikipedia [38] defines: traditional logic is ‘a loose name for the way of doing logic that began with Aristotle, and that was dominant until the advent of modern predicate logic in the late nineteenth century’. It is of great interest to place the transitions between traditional and modern logic. In this paper I will say where I think the main differences lie. In my last section I will comment on... the relationship between some traditional argument forms and natural language argument. The strength of traditional logic is sometimes measured in terms of the valid inference patterns that it recognises. Among other patterns: (1) “Some P R all Q ” implies “All Q are Red by some P ”. and the pattern behind some inferences that De Morgan studied: (2) “All horses are animals. So, all horse tails are animal tails.”. This is not a new measure; it was widely used in the mid 20th century
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...
The Discovery Of My Completeness Proofs
 Bulletin of Symbolic Logic
, 1996
"... This paper deals with aspects of my doctoral dissertation 1 ..."
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This paper deals with aspects of my doctoral dissertation 1
The history and concept of computability
 in Handbook of Computability Theory
, 1999
"... We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in th ..."
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We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in their present roles, and how
Computability and Incomputability
"... The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930’s working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal cr ..."
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The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930’s working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal credit. This is incorrect. It was Turing alone who achieved the characterization, in the opinion of Gödel. We also explore Turing’s oracle machine and its analogous properties in analysis. Keywords: Turing amachine, computability, ChurchTuring Thesis, Kurt Gödel, Alan Turing, Turing omachine, computable approximations,
Alan Turing and the Mathematical Objection
 Minds and Machines 13(1
, 2003
"... Abstract. This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet accord ..."
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Abstract. This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the “mathematical objection ” to his view that machines can think. Logicomathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human mathematicians presumably do. Key words: artificial intelligence, ChurchTuring thesis, computability, effective procedure, incompleteness, machine, mathematical objection, ordinal logics, Turing, undecidability The ‘skin of an onion ’ analogy is also helpful. In considering the functions of the mind or the brain we find certain operations which we can express in purely mechanical terms. This we say does not correspond to the real mind: it is a sort of skin which we must strip off if we are to find the real mind. But then in what remains, we find a further skin to be stripped off, and so on. Proceeding in this way, do we ever come to the ‘real ’ mind, or do we eventually come to the skin which has nothing in it? In the latter case, the whole mind is mechanical (Turing, 1950, p. 454–455). 1.
Canonical typing and Πconversion in the Barendregt Cube
, 1996
"... In this article, we extend the Barendregt Cube with \Piconversion (which is the analogue of betaconversion, on product type level) and study its properties. We use this extension to separate the problem of whether a term is typable from the problem of what is the type of a term. ..."
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Cited by 4 (3 self)
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In this article, we extend the Barendregt Cube with \Piconversion (which is the analogue of betaconversion, on product type level) and study its properties. We use this extension to separate the problem of whether a term is typable from the problem of what is the type of a term.
Canonical typing and Πconversion
, 1997
"... In usual type theory, if a function f is of type oe ! oe and an argument a is of type oe, then the type of fa is immediately given to be oe and no mention is made of the fact that what has happened is a form of ficonversion. A similar observation holds for the generalized Cartesian product typ ..."
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In usual type theory, if a function f is of type oe ! oe and an argument a is of type oe, then the type of fa is immediately given to be oe and no mention is made of the fact that what has happened is a form of ficonversion. A similar observation holds for the generalized Cartesian product types, \Pi x:oe : . In fact, many versions of type theory assume that fi holds of both types and terms, yet only a few attempt to study the theory where terms and types are really treated equally and where ficonversion is used for both. A unified treatment however, of types and terms is becoming indispensible especially in the approaches which try to generalise many systems under a unique one. For example, [Barendregt 91] provides the Barendregt cube and the Pure Type Systems (PTSs) which are a generalisation of many type theories. Yet even such a generalisation does not use ficonversion for both types and terms. This is unattractive, in a calculus where types have the same syntax as terms (such as the calculi of the cube or the PTSs). For example, in those systems, even though compatibility holds for the typing of abstraction, it does not hold for the typing of application. That is, even though M : N ) y:P :M : \Pi y:P :N holds, the following does not hold: Based on this observation, we present a calculus in which the conversion rules apply to types as well as terms. Abstraction and application, moreover, range over both types and terms. We extend the calculus with a canonical type operator in order to associate types to terms. The type of fa will then be Fa, where F is the type of f and the statement \Gamma ` t : oe from usual type theory is split in two statements in our system: \Gamma ` t and (\Gamma; t) = oe. Such a splitting enables us to discuss the two questio...