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Author : Rod Adams
Publisher : Docent Press
Page : 312 pages
File Size : 43,8 Mb
Release : 2011
Category : Mathematics
ISBN : 9780983700401
Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.
Author : Roman Murawski
Publisher : Springer Science & Business Media
Page : 395 pages
File Size : 50,9 Mb
Release : 2013-03-14
Category : Philosophy
ISBN : 9789401728669
Recursive Functions and Metamathematics deals with problems of the completeness and decidability of theories, using as its main tool the theory of recursive functions. This theory is first introduced and discussed. Then Gödel's incompleteness theorems are presented, together with generalizations, strengthenings, and the decidability theory. The book also considers the historical and philosophical context of these issues and their philosophical and methodological consequences. Recent results and trends have been included, such as undecidable sentences of mathematical content, reverse mathematics. All the main results are presented in detail. The book is self-contained and presupposes only some knowledge of elementary mathematical logic. There is an extensive bibliography. Readership: Scholars and advanced students of logic, mathematics, philosophy of science.
Author : Nigel Cutland
Publisher : Cambridge University Press
Page : 268 pages
File Size : 49,8 Mb
Release : 1980-06-19
Category : Computers
ISBN : 0521294657
What can computers do in principle? What are their inherent theoretical limitations? The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function - a function whose values can be calculated in an automatic way.
Author : Hartley Rogers
Publisher : Unknown
Page : 526 pages
File Size : 45,8 Mb
Release : 1967
Category : Mathematics
ISBN : UOM:39015013841039
Author : Herbert B. Enderton
Publisher : Academic Press
Page : 192 pages
File Size : 52,8 Mb
Release : 2010-12-30
Category : Computers
ISBN : 0123849594
Computability Theory: An Introduction to Recursion Theory provides a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. The text includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science. Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable way. Frequent historical information presented throughout More extensive motivation for each of the topics than other texts currently available Connects with topics not included in other textbooks, such as complexity theory
Author : Hans Hermes
Publisher : Springer
Page : 255 pages
File Size : 43,6 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662116869
The task of developing algorithms to solve problems has always been considered by mathematicians to be an especially interesting and im portant one. Normally an algorithm is applicable only to a narrowly limited group of problems. Such is for instance the Euclidean algorithm, which determines the greatest common divisor of two numbers, or the well-known procedure which is used to obtain the square root of a natural number in decimal notation. The more important these special algorithms are, all the more desirable it seems to have algorithms of a greater range of applicability at one's disposal. Throughout the centuries, attempts to provide algorithms applicable as widely as possible were rather unsuc cessful. It was only in the second half of the last century that the first appreciable advance took place. Namely, an important group of the inferences of the logic of predicates was given in the form of a calculus. (Here the Boolean algebra played an essential pioneer role. ) One could now perhaps have conjectured that all mathematical problems are solvable by algorithms. However, well-known, yet unsolved problems (problems like the word problem of group theory or Hilbert's tenth problem, which considers the question of solvability of Diophantine equations) were warnings to be careful. Nevertheless, the impulse had been given to search for the essence of algorithms. Leibniz already had inquired into this problem, but without success.
Author : R. G. Adams
Publisher : Unknown
Page : 0 pages
File Size : 46,8 Mb
Release : 1983
Category : Electronic
ISBN : OCLC:53517840
Author : Stephen Cole Kleene
Publisher : American Mathematical Soc.
Page : 110 pages
File Size : 53,5 Mb
Release : 1969
Category : Intuitionistic mathematics
ISBN : 9780821812891
This monograph carries out the program which the author formulated in earlier work, the formalization of the theory of recursive functions of type 0 and 1 and of the theory of realizability.
Author : Yuk Hui
Publisher : Rowman & Littlefield
Page : 288 pages
File Size : 41,5 Mb
Release : 2019-01-28
Category : Philosophy
ISBN : 9781786600547
This book is an investigation of algorithmic contingency and an elucidation of the contemporary situation that we are living in: the regular arrival of algorithmic catastrophes on a global scale. Through a historical analysis of philosophy, computation and media, this book proposes a renewed relation between nature and technics.
Author : Fabio Gadducci,Mirko Tavosanis
Publisher : Springer
Page : 321 pages
File Size : 43,5 Mb
Release : 2016-10-05
Category : Computers
ISBN : 9783319472867
This volume constitutes the refereed post-conference proceedings of the Third International Conference on the History and Philosophy of Computing, held in Pisa, Italy in October 2015. The 18 full papers included in this volume were carefully reviewed and selected from the 30 papers presented at the conference. They cover topics ranging from the world history of computing to the role of computing in the humanities and the arts.
Author : B. Jack Copeland,Carl J. Posy,Oron Shagrir
Publisher : MIT Press
Page : 373 pages
File Size : 45,6 Mb
Release : 2015-01-30
Category : Computers
ISBN : 9780262527484
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 : Borut Robič
Publisher : Springer Nature
Page : 422 pages
File Size : 48,8 Mb
Release : 2020-11-13
Category : Computers
ISBN : 9783662624210
This book offers an original and informative view of the development of fundamental concepts of computability theory. The treatment is put into historical context, emphasizing the motivation for ideas as well as their logical and formal development. In Part I the author introduces computability theory, with chapters on the foundational crisis of mathematics in the early twentieth century, and formalism. In Part II he explains classical computability theory, with chapters on the quest for formalization, the Turing Machine, and early successes such as defining incomputable problems, c.e. (computably enumerable) sets, and developing methods for proving incomputability. In Part III he explains relative computability, with chapters on computation with external help, degrees of unsolvability, the Turing hierarchy of unsolvability, the class of degrees of unsolvability, c.e. degrees and the priority method, and the arithmetical hierarchy. Finally, in the new Part IV the author revisits the computability (Church-Turing) thesis in greater detail. He offers a systematic and detailed account of its origins, evolution, and meaning, he describes more powerful, modern versions of the thesis, and he discusses recent speculative proposals for new computing paradigms such as hypercomputing. This is a gentle introduction from the origins of computability theory up to current research, and it will be of value as a textbook and guide for advanced undergraduate and graduate students and researchers in the domains of computability theory and theoretical computer science. This new edition is completely revised, with almost one hundred pages of new material. In particular the author applied more up-to-date, more consistent terminology, and he addressed some notational redundancies and minor errors. He developed a glossary relating to computability theory, expanded the bibliographic references with new entries, and added the new part described above and other new sections.
Author : Luca M. Possati
Publisher : Springer Nature
Page : 198 pages
File Size : 50,7 Mb
Release : 2022-06-24
Category : Philosophy
ISBN : 9783030636104
This book claims that continental philosophy gives us a new understanding of digital technology, and software in particular; its main thesis being that software is like a text, so it involves a hermeneutic process. A hermeneutic understanding of software allows us to explain those aspects of software that escape a strictly technical definition, such as the relationship with the user, the human being, and the social and cultural transformations that software produces. The starting point of the book is the fracture between living experience and the code. In the first chapter, the author argues that the code is the origin of the digital experience, while remaining hidden, invisible. The second chapter explores how the software can be seen as a text in Ricoeur’s sense. Before being an algorithm, code or problem solving, software is an act of interpretation. The third chapter connects software to the history of writing, following Kittler’s suggestions. The fourth chapter unifies the two parts of the book, the historical and the theoretical, from a Kantian perspective. The central thesis is that software is a form of reflective judgment, namely, digital reflective judgement.
Author : Oron Shagrir
Publisher : Oxford University Press
Page : 319 pages
File Size : 52,6 Mb
Release : 2022
Category : Computers
ISBN : 9780197552384
Computing systems are ubiquitous in contemporary life. Even the brain is thought to be a computing system of sorts. But what does it mean to say that a given organ or system "computes"? What is it about laptops, smartphones, and nervous systems that they are deemed to compute - and why does itseldom occur to us to describe stomachs, hurricanes, rocks, or chairs that way? These questions are key to laying the conceptual foundations of computational sciences, including computer science and engineering, and the cognitive and neural sciences.Oron Shagrir here provides an extended argument for the semantic view of computation, which states that semantic properties are involved in the nature of computing systems. The first part of the book provides general background. Although different in scope, these chapters have a common theme-namely,that the linkage between the mathematical theory of computability and the notion of physical computation is weak. The second part of the book reviews existing non-semantic accounts of physical computation. Shagrir analyze three influential accounts in greater depth and argues that none of theseaccounts is satisfactory, but each of them highlights certain key features of physical computation that he eventually adopts in his own semantic account of physical computation - a view that rests on a phenomenon known as simultaneous implementation (or "indeterminacy of computation"). Shagrircompletes the characterization of his account of computation and highlights the distinctive feature of computational explanations.
Author : Frederick C. Hennie
Publisher : Addison Wesley Publishing Company
Page : 392 pages
File Size : 46,7 Mb
Release : 1977
Category : Algorithms
ISBN : UCSD:31822004868626