Mathematical Logic in the 20th Century

Author: Gerald E Sacks
Publisher: World Scientific
ISBN: 9814490199
Category: Mathematics
Page: 708
View: 5791

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This invaluable book is a collection of 31 important — both in ideas and results — papers published by mathematical logicians in the 20th Century. The papers have been selected by Professor Gerald E Sacks. Some of the authors are Gödel, Kleene, Tarski, A Robinson, Kreisel, Cohen, Morley, Shelah, Hrushovski and Woodin. Contents:The Independence of the Continuum Hypothesis (P J Cohen)Higher Set Theory and Mathematical Practice (H M Friedman)Measurable Cardinals and Analytic Games (D A Martin)Categoricity in Power (M Morley)Hyperanalytic Predicates (Y N Moschovakis)Stable Theories (S Shelah)On the Singular Cardinals Problem (J Silver)On Degrees of Recursive Unsolvability (C Spector)Denumerable Models of Complete Theories (R L Vaught)Supercompact Cardinals, Sets of Reals, and Weakly Homogeneous Trees (W H Woodin)and other papers Readership: Researchers and graduate students in logic. Keywords:JFM 65.0185.02;Mathematical LogicReviews:“… overall we must be grateful for getting all this wonderful stuff available in one convenient place.” Zentralblatt MATH “This volume makes an indispensable addition to the numerous lecture notes and monographs on mathematical logic lying on our bookshelves.”Acta Scientiarum Mathematicarum

Philosophy of Science, Logic and Mathematics in the Twentieth Century

Author: Stuart Shanker
Publisher: Psychology Press
ISBN: 9780415308816
Category: Philosophy
Page: 504
View: 4416

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Contents - Introduction. 1. Philosophy of logic 2. Philosophy of mathematics in the 20th century. 3. Frege 4. Wittgenstein's Tractatus 5. Logical postivism 6. The philosophy of physics 7. The philosophy of science 8. Chance, cause and conduct; probability

Logic from Russell to Church

Author: Dov M. Gabbay,John Woods
Publisher: Elsevier
ISBN: 0080885470
Category: Mathematics
Page: 1068
View: 5024

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This volume is number five in the 11-volume Handbook of the History of Logic. It covers the first 50 years of the development of mathematical logic in the 20th century, and concentrates on the achievements of the great names of the period--Russell, Post, Gödel, Tarski, Church, and the like. This was the period in which mathematical logic gave mature expression to its four main parts: set theory, model theory, proof theory and recursion theory. Collectively, this work ranks as one of the greatest achievements of our intellectual history. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, and artificial intelligence, for whom the historical background of his or her work is a salient consideration. • The entire range of modal logic is covered • Serves as a singular contribution to the intellectual history of the 20th century • Contains the latest scholarly discoveries and interpretative insights

Logic and the Modalities in the Twentieth Century

Author: Dov M. Gabbay,John Woods
Publisher: Elsevier
ISBN: 9780080463032
Category: Mathematics
Page: 732
View: 7172

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Logic and the Modalities in the Twentieth Century is an indispensable research tool for anyone interested in the development of logic, including researchers, graduate and senior undergraduate students in logic, history of logic, mathematics, history of mathematics, computer science and artificial intelligence, linguistics, cognitive science, argumentation theory, philosophy, and the history of ideas. This volume is number seven in the eleven volume Handbook of the History of Logic. It concentrates on the development of modal logic in the 20th century, one of the most important undertakings in logic’s long history. Written by the leading researchers and scholars in the field, the volume explores the logics of necessity and possibility, knowledge and belief, obligation and permission, time, tense and change, relevance, and more. Both this volume and the Handbook as a whole are definitive reference tools for students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, artificial intelligence, for whom the historical background of his or her work is a salient consideration. · Detailed and comprehensive chapters covering the entire range of modal logic. · Contains the latest scholarly discoveries and interpretative insights that answer many questions in the field of logic.

The Development of Modern Logic

Author: Leila Haaparanta
Publisher: Oxford University Press
ISBN: 9780199722723
Category: Philosophy
Page: 1008
View: 9001

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This edited volume presents a comprehensive history of modern logic from the Middle Ages through the end of the twentieth century. In addition to a history of symbolic logic, the contributors also examine developments in the philosophy of logic and philosophical logic in modern times. The book begins with chapters on late medieval developments and logic and philosophy of logic from Humanism to Kant. The following chapters focus on the emergence of symbolic logic with special emphasis on the relations between logic and mathematics, on the one hand, and on logic and philosophy, on the other. This discussion is completed by a chapter on the themes of judgment and inference from 1837-1936. The volume contains a section on the development of mathematical logic from 1900-1935, followed by a section on main trends in mathematical logic after the 1930s. The volume goes on to discuss modal logic from Kant till the late twentieth century, and logic and semantics in the twentieth century; the philosophy of alternative logics; the philosophical aspects of inductive logic; the relations between logic and linguistics in the twentieth century; the relationship between logic and artificial intelligence; and ends with a presentation of the main schools of Indian logic. The Development of Modern Logic includes many prominent philosophers from around the world who work in the philosophy and history of mathematics and logic, who not only survey developments in a given period or area but also seek to make new contributions to contemporary research in the field. It is the first volume to discuss the field with this breadth of coverage and depth, and will appeal to scholars and students of logic and its philosophy.

The Threat of Logical Mathematism

A Study on the Critique of Mathematical Logic in Germany at the Turn of the 20th Century
Author: Jarmo Pulkkinen
Publisher: Peter Lang Pub Incorporated
ISBN: 9783631474099
Category: Mathematics
Page: 187
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A survey of the critique of mathematical logic in Germany discussing the role and significance of logic and the relationships between logic, mathematics, linguistics, and psychology, as advanced by contemporary German philosophers. Covers the development of German logic from 1830-1920, and gives detailed accounts of the arguments of three individual critics--Fritz Mauthner, Heinrich Rickert, and Theodor Ziehen. Lacks an index. Annotation copyright by Book News, Inc., Portland, OR

Sets and Extensions in the Twentieth Century

Author: N.A
Publisher: Elsevier
ISBN: 0080930662
Category: Mathematics
Page: 880
View: 1036

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Set theory is an autonomous and sophisticated field of mathematics that is extremely successful at analyzing mathematical propositions and gauging their consistency strength. It is as a field of mathematics that both proceeds with its own internal questions and is capable of contextualizing over a broad range, which makes set theory an intriguing and highly distinctive subject. This handbook covers the rich history of scientific turning points in set theory, providing fresh insights and points of view. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in mathematics, the history of philosophy, and any discipline such as computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration Serves as a singular contribution to the intellectual history of the 20th century Contains the latest scholarly discoveries and interpretative insights

Classical Mathematical Logic

The Semantic Foundations of Logic
Author: Richard L. Epstein
Publisher: Princeton University Press
ISBN: 1400841550
Category: Mathematics
Page: 544
View: 1933

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In Classical Mathematical Logic, Richard L. Epstein relates the systems of mathematical logic to their original motivations to formalize reasoning in mathematics. The book also shows how mathematical logic can be used to formalize particular systems of mathematics. It sets out the formalization not only of arithmetic, but also of group theory, field theory, and linear orderings. These lead to the formalization of the real numbers and Euclidean plane geometry. The scope and limitations of modern logic are made clear in these formalizations. The book provides detailed explanations of all proofs and the insights behind the proofs, as well as detailed and nontrivial examples and problems. The book has more than 550 exercises. It can be used in advanced undergraduate or graduate courses and for self-study and reference. Classical Mathematical Logic presents a unified treatment of material that until now has been available only by consulting many different books and research articles, written with various notation systems and axiomatizations.

Representation and Productive Ambiguity in Mathematics and the Sciences

Author: Emily R. Grosholz
Publisher: Clarendon Press
ISBN: 0191538515
Category: Mathematics
Page: 330
View: 4397

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Emily Grosholz offers an original investigation of demonstration in mathematics and science, examining how it works and why it is persuasive. Focusing on geometrical demonstration, she shows the roles that representation and ambiguity play in mathematical discovery. She presents a wide range of case studies in mechanics, topology, algebra, logic, and chemistry, from ancient Greece to the present day, but focusing particularly on the seventeenth and twentieth centuries. She argues that reductive methods are effective not because they diminish but because they multiply and juxtapose modes of representation. Such problem-solving is, she argues, best understood in terms of Leibnizian 'analysis' - the search for conditions of intelligibility. Discovery and justification are then two aspects of one rational way of proceeding, which produces the mathematician's formal experience. Grosholz defends the importance of iconic, as well as symbolic and indexical, signs in mathematical representation, and argues that pragmatic, as well as syntactic and semantic, considerations are indispensable for mathematical reasoning. By taking a close look at the way results are presented on the page in mathematical (and biological, chemical, and mechanical) texts, she shows that when two or more traditions combine in the service of problem solving, notations and diagrams are sublty altered, multiplied, and juxtaposed, and surrounded by prose in natural language which explains the novel combination. Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous. Grosholtz's arguments, which invoke Descartes, Locke, Hume, and Kant, will be of considerable interest to philosophers and historians of mathematics and science, and also have far-reaching consequences for epistemology and philosophy of language.

Mathematics and the Historian's Craft

The Kenneth O. May Lectures
Author: Michael Kinyon,Glen van Brummelen
Publisher: Springer Science & Business Media
ISBN: 0387282726
Category: Mathematics
Page: 358
View: 2483

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The Kenneth May Lectures have never before been published in book form Important contributions to the history of mathematics by well-known historians of science Should appeal to a wide audience due to its subject area and accessibility

A Mathematics Sampler

Topics for the Liberal Arts
Author: William P. Berlinghoff,Kerry E. Grant,Dale Skrien
Publisher: Rowman & Littlefield
ISBN: 9780742502024
Category: Mathematics
Page: 602
View: 5443

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Now in its fifth edition, A Mathematics Sampler presents mathematics as both science and art, focusing on the historical role of mathematics in our culture. It uses selected topics from modern mathematics including computers, perfect numbers, and four-dimensional geometry to exemplify the distinctive features of mathematics as an intellectual endeavor, a problem-solving tool, and a way of thinking about the rapidly changing world in which we live. A Mathematics Sampler also includes unique LINK sections throughout the book, each of which connects mathematical concepts with areas of interest throughout the humanities. The original course on which this text is based was cited as an innovative approach to liberal arts mathematics in Lynne Cheney's report, "50 HOURS: A Core Curriculum for College Students," published by the National Endowment for the Humanities."

Infinitesimal Methods of Mathematical Analysis

Author: J S Pinto
Publisher: Elsevier
ISBN: 0857099507
Category: Mathematics
Page: 270
View: 9437

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This modern introduction to infinitesimal methods is a translation of the book Métodos Infinitesimais de Análise Matemática by José Sousa Pinto of the University of Aveiro, Portugal and is aimed at final year or graduate level students with a background in calculus. Surveying modern reformulations of the infinitesimal concept with a thoroughly comprehensive exposition of important and influential hyperreal numbers, the book includes previously unpublished material on the development of hyperfinite theory of Schwartz distributions and its application to generalised Fourier transforms and harmonic analysis. This translation by Roy Hoskins was also greatly assisted by the comments and constructive criticism of Professor Victor Neves, of the University of Aveiro. Surveys modern reformulations of the infinitesimal concept with a comprehensive exposition of important and influential hyperreal numbers Includes material on the development of hyperfinite theory of Schwartz distributions and its application to generalised Fourier transforms and harmonic analysis

Principia Mathematica.

Author: Alfred North Whitehead,Bertrand Russell
Publisher: N.A
Category: Logic, Symbolic and mathematical
Page: 167
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Mathematics of the 19th Century

Mathematical Logic Algebra Number Theory Probability Theory
Publisher: Birkhäuser
ISBN: 303485112X
Category: Mathematics
Page: 308
View: 2586

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This multi-authored effort, Mathematics of the nineteenth century (to be fol lowed by Mathematics of the twentieth century), is a sequel to the History of mathematics fram antiquity to the early nineteenth century, published in three 1 volumes from 1970 to 1972. For reasons explained below, our discussion of twentieth-century mathematics ends with the 1930s. Our general objectives are identical with those stated in the preface to the three-volume edition, i. e. , we consider the development of mathematics not simply as the process of perfecting concepts and techniques for studying real-world spatial forms and quantitative relationships but as a social process as weIl. Mathematical structures, once established, are capable of a certain degree of autonomous development. In the final analysis, however, such immanent mathematical evolution is conditioned by practical activity and is either self-directed or, as is most often the case, is determined by the needs of society. Proceeding from this premise, we intend, first, to unravel the forces that shape mathe matical progress. We examine the interaction of mathematics with the social structure, technology, the natural sciences, and philosophy. Throughan anal ysis of mathematical history proper, we hope to delineate the relationships among the various mathematical disciplines and to evaluate mathematical achievements in the light of the current state and future prospects of the science. The difficulties confronting us considerably exceeded those encountered in preparing the three-volume edition.

Computer Science Logic

14th International Workshop, CSL 2000 Annual Conference of the EACSL Fischbachau, Germany, August 21-26, 2000 Proceedings
Author: Germany) Workshop on Computer Science Logic 2000 (Fischbachau,Peter Clote,European Association for Computer Science Logic. Conference
Publisher: Springer Science & Business Media
ISBN: 3540678956
Category: Computers
Page: 541
View: 9847

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This book constitutes the refereed proceedings of the 13th International Workshop on Computer Science Logic, CSL 2000, held in Fischbachau, Germany as the 8th Annual Conference of the EACSL in August 2000. The 28 revised full papers presented together with eight invited papers were carefully reviewed and selected by the program committee. Among the topics covered are automated deduction, theorem proving, categorical logic, term rewriting, finite model theory, higher order logic, lambda and combinatory calculi, computational complexity, logic programing, constraints, linear logic, modal logic, temporal logic, model checking, formal specification, formal verification, program transformation, etc.

Turing's Legacy

Developments from Turing's Ideas in Logic
Author: Rod Downey
Publisher: Cambridge University Press
ISBN: 1107043484
Category: Computers
Page: 539
View: 7299

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A collection of essays celebrating the influence of Alan Turing's work in logic, computer science and related areas.

Logic, Epistemology, and the Unity of Science

Author: Shahid Rahman,John Symons,Dov M. Gabbay,jean paul van bendegem
Publisher: Springer Science & Business Media
ISBN: 1402028083
Category: Philosophy
Page: 626
View: 7427

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The first volume in this new series explores, through extensive co-operation, new ways of achieving the integration of science in all its diversity. The book offers essays from important and influential philosophers in contemporary philosophy, discussing a range of topics from philosophy of science to epistemology, philosophy of logic and game theoretical approaches. It will be of interest to philosophers, computer scientists and all others interested in the scientific rationality.

Mathematical Ideas and Sociological Theory

Current State and Prospects
Author: Thomas J. Fararo
Publisher: Taylor & Francis
ISBN: 9780677166353
Page: 185
View: 8268

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First Published in 1984. Routledge is an imprint of Taylor & Francis, an informa company.

Mathematical Logic

Author: R.O. Gandy,C.E.M. Yates
Publisher: Elsevier
ISBN: 9780080535920
Category: Computers
Page: 306
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Mathematical Logic is a collection of the works of one of the leading figures in 20th-century science. This collection of A.M. Turing's works is intended to include all his mature scientific writing, including a substantial quantity of unpublished material. His work in pure mathematics and mathematical logic extended considerably further; the work of his last years, on morphogenesis in plants, is also of the greatest originality and of permanent importance. This book is divided into three parts. The first part focuses on computability and ordinal logics and covers Turing's work between 1937 and 1938. The second part covers type theory; it provides a general introduction to Turing's work on type theory and covers his published and unpublished works between 1941 and 1948. Finally, the third part focuses on enigmas, mysteries, and loose ends. This concluding section of the book discusses Turing's Treatise on the Enigma, with excerpts from the Enigma Paper. It also delves into Turing's papers on programming and on minimum cost sequential analysis, featuring an excerpt from the unpublished manuscript. This book will be of interest to mathematicians, logicians, and computer scientists.

Twentieth-Century Analytic Philosophy

Author: Avrum Stroll
Publisher: Columbia University Press
ISBN: 0231112211
Category: Philosophy
Page: 304
View: 9321

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Avrum Stroll investigates the "family resemblances" between that impressive breed of thinkers known as analytic philosophers. In so doing, he grapples with the point and purpose of doing philosophy: What is philosophy? What are its tasks? What kind of information, illumination, and understanding is it supposed to provide if it is not one of the natural sciences?