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Information processing, computation, and cognition
 JOURNAL OF BIOLOGICAL PHYSICS
"... Computation and information processing are among the most fundamental notions in cognitive science. They are also among the most imprecisely discussed. Many cognitive scientists take it for granted that cognition involves computation, information processing, or both – although others disagree veheme ..."
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Computation and information processing are among the most fundamental notions in cognitive science. They are also among the most imprecisely discussed. Many cognitive scientists take it for granted that cognition involves computation, information processing, or both – although others disagree vehemently. Yet different cognitive scientists use ‘computation ’ and ‘information processing ’ to mean different things, sometimes without realizing that they do. In addition, computation and information processing are surrounded by several myths; first and foremost, that they are the same thing. In this paper, we address this unsatisfactory state of affairs by presenting a general and theoryneutral account of computation and information processing. We also apply our framework by analyzing the relations between computation and information processing on one hand and classicism and connectionism/computational neuroscience on the other. We defend the relevance to cognitive science of both computation, at least in a generic sense, and information processing, in three important senses of the term. Our account advances several foundational debates in cognitive science by untangling some of their conceptual knots in a theoryneutral way. By leveling the playing field, we pave the way for the future resolution of the debates ’ empirical aspects.
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
Mathematical Definability
"... Introduction One might fairly say that the mathematical analysis of definability began in 1931, with the appearance of Godel's Incompleteness Theorem [Godel, 1931]. Godel showed that for sufficiently strong formal systems T , there exist undecidable statements ' such that there is no proof of ' or ..."
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Introduction One might fairly say that the mathematical analysis of definability began in 1931, with the appearance of Godel's Incompleteness Theorem [Godel, 1931]. Godel showed that for sufficiently strong formal systems T , there exist undecidable statements ' such that there is no proof of ' or of :' within T . This theorem pointed to an intrinsic incompleteness within the formal notion of proof. The method of computation by algorithm is more general than that of verification by formal proof. It too was shown to be incomplete, but it took some time to develop the technical apparatus needed to state this incompleteness correctly. Kleene [1987], in his biographical memoir of Godel, recalls this development and we summarize some of his remarks. Kleene describes the intuitive notion of an algorithm as follows. An algorithm is a procedure described in advance such that, whenev
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."
Church’s Thesis
"... In this project we will learn about both primitive recursive and general recursive functions. We will also learn about Turing computable functions, and will discuss why the class of general recursive functions coincides with the class of Turing computable functions. We will introduce the effectively ..."
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In this project we will learn about both primitive recursive and general recursive functions. We will also learn about Turing computable functions, and will discuss why the class of general recursive functions coincides with the class of Turing computable functions. We will introduce the effectively calculable functions, and the ideas behind Alonzo Church’s (1903–1995) proposal to identify the
Extensions, Automorphisms, and Definability
 CONTEMPORARY MATHEMATICS
"... This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to ..."
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This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to know the dynamical properties of a c.e. set We , not merely whether an element x is enumerated in We but when, relative to its appearance in other c.e. sets. We present here
Gödel on Intuition and on Hilbert’s finitism
"... There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the con ..."
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There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small doublereversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of firstorder number theory, P A; but starting in the Dialectica paper
Z(y)
, 2007
"... Abstract. This article demonstrates the invalidity of Theorem VI in Gödel’s monograph of 1931, by showing that (15) xBκ(17Gen r) − → Bewκ[Sb(r17 Z(x))], (16) xBκ(17Gen r) − → Bewκ[Neg(Sb(r17 Z(x))], (derived by means of definition (8.1) Q(x, y) ≡ xBκ[Sb(y19)] respectively ..."
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Abstract. This article demonstrates the invalidity of Theorem VI in Gödel’s monograph of 1931, by showing that (15) xBκ(17Gen r) − → Bewκ[Sb(r17 Z(x))], (16) xBκ(17Gen r) − → Bewκ[Neg(Sb(r17 Z(x))], (derived by means of definition (8.1) Q(x, y) ≡ xBκ[Sb(y19)] respectively
OBSERVATIONS CONCERNING GÖDEL’S 1931
, 2003
"... Abstract. This article demonstrates the invalidity of the socalled Gödel’s first incompleteness theorem, Theorem VI in Gödel’s 1931 article, showing that propositions (15) and (16), derived from the definition 8.1, in its proof, are false in PA. Introduction. Developed as a consequence of the crisi ..."
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Abstract. This article demonstrates the invalidity of the socalled Gödel’s first incompleteness theorem, Theorem VI in Gödel’s 1931 article, showing that propositions (15) and (16), derived from the definition 8.1, in its proof, are false in PA. Introduction. Developed as a consequence of the crisis of the foundation of mathematics due to the discovery of the antinomies, Hilbert’s formalism planned as criterions of adequacy for the axiomatic systems, the achievement of their coherence and completeness [8][9]. The result of incompleteness for any system embodying the arithmetic of the positive integers, obtained by Gödel in 1931, grafted on Hilbert’s