Constructivity and Computability in Historical and Philosophical Perspective

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Author: Jacques Dubucs,Michel Bourdeau

Publisher: Springer

ISBN: 9401792178

Category: Philosophy

Page: 214

View: 6516

Ranging from Alan Turing’s seminal 1936 paper to the latest work on Kolmogorov complexity and linear logic, this comprehensive new work clarifies the relationship between computability on the one hand and constructivity on the other. The authors argue that even though constructivists have largely shed Brouwer’s solipsistic attitude to logic, there remain points of disagreement to this day. Focusing on the growing pains computability experienced as it was forced to address the demands of rapidly expanding applications, the content maps the developments following Turing’s ground-breaking linkage of computation and the machine, the resulting birth of complexity theory, the innovations of Kolmogorov complexity and resolving the dissonances between proof theoretical semantics and canonical proof feasibility. Finally, it explores one of the most fundamental questions concerning the interface between constructivity and computability: whether the theory of recursive functions is needed for a rigorous development of constructive mathematics. This volume contributes to the unity of science by overcoming disunities rather than offering an overarching framework. It posits that computability’s adoption of a classical, ontological point of view kept these imperatives separated. In studying the relationship between the two, it is a vital step forward in overcoming the disagreements and misunderstandings which stand in the way of a unifying view of logic.

Mathematical Intuitionism

Introduction to Proof Theory

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Author: Al'bert Grigor'evi_ Dragalin

Publisher: American Mathematical Soc.

ISBN: 0821845209

Category: Mathematics

Page: 228

View: 1095

In the area of mathematical logic, a great deal of attention is now being devoted to the study of nonclassical logics. This book intends to present the most important methods of proof theory in intuitionistic logic and to acquaint the reader with the principal axiomatic theories based on intuitionistic logic.

Lectures on Soft Computing and Fuzzy Logic

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Author: Antonio Di Nola,Giangiacomo Gerla

Publisher: Springer Science & Business Media

ISBN: 9783790813968

Category: Computers

Page: 337

View: 9777

The present volume collects selected papers arising from lectures delivered by the authors at the School on Fuzzy Logic and Soft Computing held during the years 1996/97/98/99 and sponsored by the Salerno University. The authors contributing to this volume agreed with editors to write down, to enlarge and, in many cases, to rethink their original lectures, in order to offer to readership, a more compact presentation of the proposed topics. The aim of the volume is to offer a picture, as a job in progress, of the effort that is coming in founding and developing soft computing's techniques. The volume contains papers aimed to report on recent results containing genuinely logical aspects of fuzzy logic. The topics treated in this area cover algebraic aspects of Lukasiewicz Logic, Fuzzy Logic as the logic of continuous t-norms, Intuitionistic Fuzzy Logic. Aspects of fuzzy logic based on similar ity relation are presented in connection with the problem of flexible querying in deductive database. Departing from fuzzy logic, some papers present re sults in Probability Logic treating computational aspects, results based on indishernability relation and a non commutative version of generalized effect algebras. Several strict applications of soft computing are presented in the book. Indeed we find applications ranging among pattern recognition, image and signal processing, evolutionary agents, fuzzy cellular networks, classi fication in fuzzy environments. The volume is then intended to serve as a reference work for foundational logico-algebraic aspect of Soft Computing and for concrete applications of soft computing technologies.

Mechanism, Mentalism and Metamathematics

An Essay on Finitism

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Author: J. Webb

Publisher: Springer Science & Business Media

ISBN: 940157653X

Category: Philosophy

Page: 285

View: 8504

This book grew out of a graduate student paper [261] in which I set down some criticisms of J. R. Lucas' attempt to refute mechanism by means of G6del's theorem. I had made several such abortive attempts myself and had become familiar with their pitfalls, and especially with the double edged nature of incompleteness arguments. My original idea was to model the refutation of mechanism on the almost universally accepted G6delian refutation of Hilbert's formalism, but I kept getting stuck on questions of mathematical philosophy which I found myself having to beg. A thorough study of the foundational works of Hilbert and Bernays finally convinced me that I had all too naively and uncritically bought this refutation of formalism. I did indeed discover points of surprisingly close contact between formalism and mechanism, but also that it was possible to under mine certain strong arguments against these positions precisely by invok ing G6del's and related work. I also began to realize that the Church Turing thesis itself is the principal bastion protecting mechanism, and that G6del's work was perhaps the best thing that ever happened to both mechanism and formalism. I pushed these lines of argument in my dis sertation with the patient help of my readers, Raymond Nelson and Howard Stein. I would especially like to thank the latter for many valuable criticisms of my dissertation as well as some helpful suggestions for reor ganizing it in the direction of the present book.

Combinators, λ-Terms and Proof Theory

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Author: S. Stenlund

Publisher: Springer Science & Business Media

ISBN: 940102913X

Category: Philosophy

Page: 177

View: 4003

The aim of this monograph is to present some of the basic ideas and results in pure combinatory logic and their applications to some topics in proof theory, and also to present some work of my own. Some of the material in chapter 1 and 3 has already appeared in my notes Introduction to Combinatory Logic. It appears here in revised form since the presen tation in my notes is inaccurate in several respects. I would like to express my gratitude to Stig Kanger for his invalu able advice and encouragement and also for his assistance in a wide variety of matters concerned with my study in Uppsala. I am also in debted to Per Martin-USf for many valuable and instructive conversa tions. As will be seen in chapter 4 and 5, I also owe much to the work of Dag Prawitz and W. W. Tait. My thanks also to Craig McKay who read the manuscript and made valuable suggestions. I want, however, to emphasize that the shortcomings that no doubt can be found, are my sole responsibility. Uppsala, February 1972.

Proof Theory and Automated Deduction

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Author: Jean Goubault-Larrecq,I. Mackie

Publisher: Springer Science & Business Media

ISBN: 9781402003684

Category: Computers

Page: 444

View: 2923

Proof Theory and Automated Deduction is written for final-year undergraduate and first-year post-graduate students. It should also serve as a valuable reference for researchers in logic and computer science. It covers basic notions in logic, with a particular stress on proof theory, as opposed to, for example, model theory or set theory; and shows how they are applied in computer science, and especially the particular field of automated deduction, i.e. the automated search for proofs of mathematical propositions. We have chosen to give an in-depth analysis of the basic notions, instead of giving a mere sufficient analysis of basic and less basic notions. We often derive the same theorem by different methods, showing how different mathematical tools can be used to get at the very nature of the objects at hand, and how these tools relate to each other. Instead of presenting a linear collection of results, we have tried to show that all results and methods are tightly interwoven. We believe that understanding how to travel along this web of relations between concepts is more important than just learning the basic theorems and techniques by rote. Audience: The book is a valuable reference for researchers in logic and computer science.

Logic Colloquium 2004

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Author: Alessandro Andretta,Keith Kearnes,Domenico Zambella

Publisher: Cambridge University Press

ISBN: 0521884241

Category: Computers

Page: 220

View: 508

A collection of surveys, tutorials, and research papers from the 2004 Logic Colloquium.

The Journal of Symbolic Logic

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Author: Alonzo Church

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: N.A

View: 4903

Includes lists of members.