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Author: Adam Olszewski
Publisher: Walter de Gruyter
ISBN: 3110325462
Category : Philosophy
Languages : en
Pages : 551
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Book Description
Church's Thesis (CT) was first published by Alonzo Church in 1935. CT is a proposition that identifies two notions: an intuitive notion of a effectively computable function defined in natural numbers with the notion of a recursive function. Despite of the many efforts of prominent scientists, Church's Thesis has never been falsified. There exists a vast literature concerning the thesis. The aim of the book is to provide one volume summary of the state of research on Church's Thesis. These include the following: different formulations of CT, CT and intuitionism, CT and intensional mathematics, CT and physics, the epistemic status of CT, CT and philosophy of mind, provability of CT and CT and functional programming.
Author: Adam Olszewski
Publisher: Walter de Gruyter
ISBN: 3110325462
Category : Philosophy
Languages : en
Pages : 551
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Book Description
Church's Thesis (CT) was first published by Alonzo Church in 1935. CT is a proposition that identifies two notions: an intuitive notion of a effectively computable function defined in natural numbers with the notion of a recursive function. Despite of the many efforts of prominent scientists, Church's Thesis has never been falsified. There exists a vast literature concerning the thesis. The aim of the book is to provide one volume summary of the state of research on Church's Thesis. These include the following: different formulations of CT, CT and intuitionism, CT and intensional mathematics, CT and physics, the epistemic status of CT, CT and philosophy of mind, provability of CT and CT and functional programming.
Author: P. Dybjer
Publisher: Springer Science & Business Media
ISBN: 9400744358
Category : Philosophy
Languages : en
Pages : 399
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Book Description
This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice (ZFC). This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics?
Author: Gualtiero Piccinini
Publisher: Oxford University Press, USA
ISBN: 0199658854
Category : Computers
Languages : en
Pages : 324
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Book Description
Computation permeates our world, but a satisfactory philosophical theory of what it is has been lacking. Gualtiero Piccinini presents a mechanistic account of what makes a physical system a computing system. He argues that computation does not entail representation or information-processing, although information-processing entails computation.
Author: Ruth Hagengruber
Publisher: Routledge
ISBN: 1317317556
Category : History
Languages : en
Pages : 303
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Book Description
Over the last four decades computers and the internet have become an intrinsic part of all our lives, but this speed of development has left related philosophical enquiry behind. Featuring the work of computer scientists and philosophers, these essays provide an overview of an exciting new area of philosophy that is still taking shape.
Author: Wilfried Sieg
Publisher: Oxford University Press
ISBN: 0195372220
Category : Computers
Languages : en
Pages : 452
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Book Description
David Hilbert was one of the great mathematicians who expounded the centrality of their subject in human thought. In this collection of essays, Wilfried Sieg frames Hilbert's foundational work, from 1890 to 1939, in a comprehensive way and integrates it with modern proof theoretic investigations.
Author: B. Jack Copeland
Publisher: MIT Press
ISBN: 0262527480
Category : Computers
Languages : en
Pages : 373
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Book Description
Computer scientists, mathematicians, and philosophers discuss the conceptual foundations of the notion of computability as well as recent theoretical developments. In the 1930s a series of seminal works published by Alan Turing, Kurt Gödel, Alonzo Church, and others established the theoretical basis for computability. This work, advancing precise characterizations of effective, algorithmic computability, was the culmination of intensive investigations into the foundations of mathematics. In the decades since, the theory of computability has moved to the center of discussions in philosophy, computer science, and cognitive science. In this volume, distinguished computer scientists, mathematicians, logicians, and philosophers consider the conceptual foundations of computability in light of our modern understanding. Some chapters focus on the pioneering work by Turing, Gödel, and Church, including the Church-Turing thesis and Gödel's response to Church's and Turing's proposals. Other chapters cover more recent technical developments, including computability over the reals, Gödel's influence on mathematical logic and on recursion theory and the impact of work by Turing and Emil Post on our theoretical understanding of online and interactive computing; and others relate computability and complexity to issues in the philosophy of mind, the philosophy of science, and the philosophy of mathematics. Contributors Scott Aaronson, Dorit Aharonov, B. Jack Copeland, Martin Davis, Solomon Feferman, Saul Kripke, Carl J. Posy, Hilary Putnam, Oron Shagrir, Stewart Shapiro, Wilfried Sieg, Robert I. Soare, Umesh V. Vazirani
Author: Gualtiero Piccinini
Publisher: Oxford University Press
ISBN: 0192636049
Category : Science
Languages : en
Pages : 413
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Book Description
In Neurocognitive Mechanisms Gualtiero Piccinini presents the most systematic, rigorous, and comprehensive philosophical defence to date of the computational theory of cognition. His view posits that cognition involves neural computation within multilevel neurocognitive mechanisms, and includes novel ideas about ontology, functions, neural representation, neural computation, and consciousness. He begins by defending an ontologically egalitarian account of composition and realization, according to which all levels are equally real. He then explicates multiple realizability and mechanisms within this ontologically egalitarian framework, defends a goal-contribution account of teleological functions, and defends a mechanistic version of functionalism. This provides the foundation for a mechanistic account of computation, which in turn clarifies the ways in which the computational theory of cognition is a multilevel mechanistic theory supported by contemporary cognitive neuroscience. Piccinini argues that cognition is computational at least in a generic sense. He defends the computational theory of cognition from standard objections, yet also rebuts putative a priori arguments. He contends that the typical vehicles of neural computations are representations, and that, contrary to the received view, the representations posited by the computational theory of cognition are observable and manipulatable in the laboratory. He also contends that neural computations are neither digital nor analog; instead, neural computations are sui generis. He concludes by investigating the relation between computation and consciousness, suggesting that consciousness may be a functional phenomenon without being computational in nature. This book will be of interest to philosophers of cognitive science as well as neuroscientists.
Author: Leon Horsten
Publisher: Oxford University Press
ISBN: 0191077690
Category : Mathematics
Languages : en
Pages : 272
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Book Description
The logician Kurt Gödel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is not equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the mathematical mind is more powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency if rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Gödel's disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field.
Author: Liesbeth De Mol
Publisher: Springer
ISBN: 331997226X
Category : Philosophy
Languages : en
Pages : 284
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Book Description
This book presents a historical and philosophical analysis of programming systems, intended as large computational systems like, for instance, operating systems, programmed to control processes. The introduction to the volume emphasizes the contemporary need of providing a foundational analysis of such systems, rooted in a broader historical and philosophical discussion. The different chapters are grouped around three major themes. The first concerns the early history of large systems developed against the background of issues related to the growing semantic gap between hardware and code. The second revisits the fundamental issue of complexity of large systems, dealt with by the use of formal methods and the development of `grand designs’ like Unix. Finally, a third part considers several issues related to programming systems in the real world, including chapters on aesthetical, ethical and political issues. This book will interest researchers from a diversity of backgrounds. It will appeal to historians, philosophers, as well as logicians and computer scientists who want to engage with topics relevant to the history and philosophy of programming and more specifically the role of programming systems in the foundations of computing.
Author: Peter Smith
Publisher: Cambridge University Press
ISBN: 0521857848
Category : Mathematics
Languages : en
Pages : 376
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Book Description
Peter Smith examines Gödel's Theorems, how they were established and why they matter.
