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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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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
The ChurchTuring thesis: Consensus and opposition
 Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006
, 2006
"... Many years ago, I wrote [7]: It is truly remarkable (Gödel...speaks of a kind of miracle) that it has proved possible to give a precise mathematical characterization of the class of processes that can be carried out by purely machanical means. It is in fact the possibility of such a characterization ..."
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Many years ago, I wrote [7]: It is truly remarkable (Gödel...speaks of a kind of miracle) that it has proved possible to give a precise mathematical characterization of the class of processes that can be carried out by purely machanical means. It is in fact the possibility of such a characterization that underlies the ubiquitous applicability of digital computers. In addition it has made it possible to prove the algorithmic unsolvability of important problems, has provided a key tool in mathematical logic, has made available an array of fundamental models in theoretical computer science, and has been the basis of a rich new branch of mathemtics. A few years later I wrote [8]: Thesubject...isAlanTuring’sdiscoveryoftheuniversal(orallpurpose) digitalcomputerasamathematicalabstraction....Wewill tryto show how this very abstract work helped to lead Turing and John von Neumann to the modern concept of the electronic computer.
Logic and artificial intelligence
 The Stanford Encyclopedia of Philosophy. Fall 2003. http://plato.stanford.edu/archives/fall2003/entries/logicai
"... www.rthomaso.eecs.umich.edu ..."
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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Cited by 5 (1 self)
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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
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
"... ..."
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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Cited by 4 (2 self)
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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.
from Turing to Dijkstra”
"... Turing’s involvement with computer building was popularized in the 1970s and later. Most notable are the books by Brian Randell (1973), Andrew Hodges (1983), and Martin Davis (2000). A central question is whether John von Neumann was influenced by Turing’s 1936 paper when he helped build the EDVAC m ..."
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Turing’s involvement with computer building was popularized in the 1970s and later. Most notable are the books by Brian Randell (1973), Andrew Hodges (1983), and Martin Davis (2000). A central question is whether John von Neumann was influenced by Turing’s 1936 paper when he helped build the EDVAC machine, even though he never cited Turing’s work. This question remains unsettled up till this day. As remarked by Charles Petzold, one standard history barely mentions Turing, while the other, written by a logician, makes Turing a key player. Contrast these observations then with the fact that Turing’s 1936 paper was cited and heavily discussed in 1959 among computer programmers. In 1966, the first Turing award was given to a programmer, not a computer builder, as were several subsequent Turing awards. An historical investigation of Turing’s influence on computing, presented here, shows that Turing’s 1936 notion of universality became increasingly relevant among programmers during the 1950s. The central thesis of this paper states that Turing’s influence was felt more in programming after his death than in computer building during the 1940s. 1