The purpose of this article is to examine aspects of the development of the concept and theory of computability through the theory of recursive functions. Following a brief introduction, Section 2 is devoted to the presuppositions of computability. It focuses on certain concepts, beliefs and theorems necessary for a general property of computability to be formulated and developed into a mathematical theory. The following two sections concern situations in which the presuppositions were realized and the theory of computability was developed. (...) It is suggested in Section 3 that a central item was the problem of generalizing Gödel's incompleteness theorem. It is shown that this involved both the characterization of recursiveness and the attempt to clarify and formulate the notion of an effective process as it relates to the syntax of deductive systems. Section 4 concerns the decision problems which grew from the Hilbert program. Section 5 is devoted to the development of an informal' technique in the theory of computability often called ?argument by Church's thesis? (shrink)

Cognitive science uses the notion of computational information processing to explain cognitive information processing. Some philosophers have argued that anything can be described as doing computational information processing; if so, it is a vacuous notion for explanatory purposes.An attempt is made to explicate the notions of cognitive information processing and computational information processing and to specify the relationship between them. It is demonstrated that the resulting notion of computational information processing can only be realized in a restrictive class of dynamical (...) systems called physical notational systems (after Goodman's theory of notationality), and that the systems generally appealed to by cognitive science-physical symbol systems-are indeed such systems. Furthermore, it turns out that other alternative conceptions of computational information processing, Fodor's (1975) Language of Thought and Cummins' (1989) Interpretational Semantics appeal to substantially the same restrictive class of systems. (shrink)

The idea that human thought requires the execution of mental algorithms provides a foundation for research programs in cognitive science, which are largely based upon the computational conception of language and mentality. Consideration is given to recent work by Penrose, Searle, and Cleland, who supply various grounds for disputing computationalism. These grounds in turn qualify as reasons for preferring a non-computational, semiotic approach, which can account for them as predictable manifestations of a more adquate conception. Thinking does not ordinarily require (...) the execution of mental algorithms, which appears to be at best no more than one rather special kind of thinking. (shrink)

The idea that human thought requires the execution of mental algorithms provides a foundation for research programs in cognitive science, which are largely based upon the computational conception of language and mentality. Consideration is given to recent work by Penrose, Searle, and Cleland, who supply various grounds for disputing computationalism. These grounds in turn qualify as reasons for preferring a non-computational, semiotic approach, which can account for them as predictable manifestations of a more adquate conception. Thinking does not ordinarily require (...) the execution of mental algorithms, which appears to be at best no more than one rather special kind of thinking. (shrink)

This paper argues for a noncognitiveist computationalism in the philosophy of mind. It further argues that both humans and computers have intentionality, that is, their mental states are semantical -- they are about things in their worlds.

The following paper presents a characterization of three distinctions fundamental to computationalism, viz., the distinction between analog and digital machines, representation and nonrepresentation-using systems, and direct and indirect perceptual processes. Each distinction is shown to rest on nothing more than the methodological principles which justify the explanatory framework of the special sciences.

Looks at accounts by Locke, Fodor, Dretske, and Millikan concerning the nature of mental representation, and discusses connectionism and representation.

The Church-Turing thesis makes a bold claim about the theoretical limits to computation. It is based upon independent analyses of the general notion of an effective procedure proposed by Alan Turing and Alonzo Church in the 1930''s. As originally construed, the thesis applied only to the number theoretic functions; it amounted to the claim that there were no number theoretic functions which couldn''t be computed by a Turing machine but could be computed by means of some other kind of effective (...) procedure. Since that time, however, other interpretations of the thesis have appeared in the literature. In this paper I identify three domains of application which have been claimed for the thesis: (1) the number theoretic functions; (2) all functions; (3) mental and/or physical phenomena. Subsequently, I provide an analysis of our intuitive concept of a procedure which, unlike Turing''s, is based upon ordinary, everyday procedures such as recipes, directions and methods; I call them mundane procedures. I argue that mundane procedures can be said to be effective in the same sense in which Turing machine procedures can be said to be effective. I also argue that mundane procedures differ from Turing machine procedures in a fundamental way, viz., the former, but not the latter, generate causal processes. I apply my analysis to all three of the above mentioned interpretations of the Church-Turing thesis, arguing that the thesis is (i) clearly false under interpretation (3), (ii) false in at least some possible worlds (perhaps even in the actual world) under interpretation (2), and (iii) very much open to question under interpretation (1). (shrink)

Gathered together here are the fundamental texts of the great classical period in modern logic. A complete translation of Gottlob Frege's Begriffsschrift--which opened a great epoch in the history of logic by fully presenting propositional calculus and quantification theory--begins the volume, which concludes with papers by Herbrand and by Gödel.

Determinism is a perennial topic of philosophical discussion. Very little acquaintance with the philosophical literature is needed to reveal the Tower of ...

This systematic investigation of computation and mental phenomena by a noted psychologist and computer scientist argues that cognition is a form of computation, that the semantic contents of mental states are encoded in the same general way as computer representations are encoded. It is a rich and sustained investigation of the assumptions underlying the directions cognitive science research is taking. 1 The Explanatory Vocabulary of Cognition 2 The Explanatory Role of Representations 3 The Relevance of Computation 4 The Psychological Reality (...) of Programs: Strong Equivalence 5 Constraining Functional Architecture 6 The Bridge from Physical to Symbolic: Transduction 7 Functional Architecture and Analogue Processes 8 Mental Imagery and Functional Architecture 9 Epilogue: What is Cognitive Science the Science of? (shrink)

Winner of the Wolf Prize for his contribution to our understanding of the universe, Penrose takes on the question of whether artificial intelligence will ever ...

Presenting a look at the human mind's capacity while criticizing artificial intelligence, the author makes suggestions about classical and quantum physics and ..

"A valuable collection both for original source material as well as historical formulations of current problems."-- The Review of Metaphysics "Much more than a mere collection of papers . . . a valuable addition to the literature."-- Mathematics of Computation An anthology of fundamental papers on undecidability and unsolvability by major figures in the field, this classic reference opens with Godel's landmark 1931 paper demonstrating that systems of logic cannot admit proofs of all true assertions of arithmetic. Subsequent papers by (...) Godel, Church, Turing, and Post single out the class of recursive functions as computable by finite algorithms. Additional papers by Church, Turing, and Post cover unsolvable problems from the theory of abstract computing machines, mathematical logic, and algebra, and material by Kleene and Post includes initiation of the classification theory of unsolvable problems. Suitable for graduate and undergraduate courses. 1965 ed. (shrink)

Wilfred Sieg. Mechanical Procedures and Mathematical Experience.