Mathematical Logic in the 20th Century

Author: Gerald E. Sacks
Publisher: World Scientific
ISBN: 9810247362
Category: Mathematics
Page: 693
View: 8110

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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 G”del, Kleene, Tarski, A Robinson, Kreisel, Cohen, Morley, Shelah, Hrushovski and Woodin.

Philosophy of Science, Logic and Mathematics in the Twentieth Century

Author: Stuart Shanker
Publisher: Psychology Press
ISBN: 9780415308816
Category: Philosophy
Page: 504
View: 912

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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

The Development of Modern Logic

Author: Leila Haaparanta
Publisher: Oxford University Press
ISBN: 9780199722723
Category: Philosophy
Page: 1008
View: 9315

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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.

Logic from Russell to Church

Author: Dov M. Gabbay,John Woods
Publisher: Elsevier
ISBN: 0080885470
Category: Mathematics
Page: 1068
View: 6825

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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

Classical Mathematical Logic

The Semantic Foundations of Logic
Author: Richard L. Epstein
Publisher: Princeton University Press
ISBN: 1400841550
Category: Mathematics
Page: 544
View: 4663

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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: 1071

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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.

Logic and the Modalities in the Twentieth Century

Author: Dov M. Gabbay,John Woods
Publisher: Elsevier
ISBN: 9780080463032
Category: Mathematics
Page: 732
View: 1616

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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.

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: 7171

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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

Mathematics of the 19th Century

Mathematical Logic, Algebra, Number Theory, Probability Theory
Author: Andreĭ Nikolaevich Kolmogorov,Adolʹf Pavlovich I͡Ushkevich
Publisher: Springer Science & Business Media
ISBN: 9783764364410
Category: Mathematics
Page: 308
View: 4594

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New Edition - New in Paperback - This is the second revised edition of the first volume of the outstanding collection of historical studies of mathematics in the nineteenth century compiled in three volumes by A. N. Kolmogorov and A. P. Yushkevich. This second edition was carefully revised by Abe Shenitzer, York University, Ontario, Canada. The historical period covered in this book extends from the early nineteenth century up to the end of the 1930s, as neither 1801 nor 1900 are, in themselves, turning points in the history of mathematics, although each date is notable fo a remarkable event: the first for the publication of Gauss' "Disquisitiones arithmeticae," the second for Hilbert's "Mathematical Problems." Beginning in the second quarter of the nineteenth century mathematics underwent a revolution as crucial and profound in its consequences for the general world outlook as the mathematical revolution in the beginning of the modern era. The main changes included a new statement of the problem of the existence of mathematical objects, particulary in the calculus, and soon thereafter the formation of non-standard structures in geometry, arithmetic and algebra. The primary objective of the work has been to treat the evolution of mathematics in the nineteenth century as a whole; the discussion is concentrated on the essential concepts, methods, and algorithms.

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: 8201

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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."

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
View: 4963

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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

Handbook of the History of Logic: Sets and extensions in the twentieth century

Author: Dov M. Gabbay,John Hayden Woods,Akihiro Kanamori
Publisher: Elsevier
ISBN: 0444516212
Category: Reference
Page: 865
View: 8699

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"Starting at the very beginning with Aristotle's founding contributions, logic has been graced by several periods in which the subject has flourished, attaining standards of rigour and conceptual sophistication underpinning a large and deserved reputation as a leading expression of human intellectual effort. It is widely recognized that the period from the mid-nineteenth century until the three-quarter mark of the century just past marked one of these golden ages, a period of explosive creativity and transforming insights. It has been said that ignorance of our history is a kind of amnesia, concerning which it is wise to note that amnesia is an illness. It would be a matter for regret, if we lost contact with another of logic's golden ages, one that greatly exceeds in reach that enjoyed by mathematical symbolic logic. This is the period between the eleventh and sixteenth centuries, loosely conceived of as the Middle Ages. The logic of this period does not have the expressive virtues afforded by the symbolic resources of uninterpreted calculi, but mediaeval logic rivals in range, originality and intellectual robustness a good deal of the modern record. The range of logic in this period is striking, extending from investigation of quantifiers and logic consequence to enquiries into logical truth; from theories of reference to accounts of identity; from work on the modalities to the stirrings of the logic of relations, from theories of meaning to analyses of the paradoxes, and more. While the scope of mediaeval logic is impressive, of greater importance is that nearly all of it can be read by the modern logician with at least some prospect of profit. The last thing that mediaeval logic is, is a museum piece." -- Publisher's website.

Infinitesimal Methods of Mathematical Analysis

Author: J S Pinto
Publisher: Elsevier
ISBN: 0857099507
Category: Mathematics
Page: 270
View: 7487

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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

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: 806

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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: 5698

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

Turing's Legacy

Developments from Turing's Ideas in Logic
Author: Rod Downey
Publisher: Cambridge University Press
ISBN: 1107043484
Category: Computers
Page: 539
View: 5461

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

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: 2787

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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.

Twentieth-Century Analytic Philosophy

Author: Avrum Stroll
Publisher: Columbia University Press
ISBN: 0231112211
Category: Philosophy
Page: 304
View: 4048

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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?

The Handy Math Answer Book

Author: Patricia Barnes-Svarney,Thomas E Svarney
Publisher: Visible Ink Press
ISBN: 1578593867
Category: Mathematics
Page: 528
View: 3892

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From modern-day challenges such as balancing a checkbook, following the stock market, buying a home, and figuring out credit card finance charges to appreciating historical developments by Pythagoras, Archimedes, Newton, and other mathematicians, this engaging resource addresses more than 1,000 questions related to mathematics. Organized into chapters that cluster similar topics in an easily accessible format, this reference provides clear and concise explanations about the fundamentals of algebra, calculus, geometry, trigonometry, and other branches of mathematics. It contains the latest mathematical discoveries, including newly uncovered historical documents and updates on how science continues to use math to make cutting-edge innovations in DNA sequencing, superstring theory, robotics, and computers. With fun math facts and illuminating figures, The Handy Math Answer Book explores the uses of math in everyday life and helps the mathematically challenged better understand and enjoy the magic of numbers.

The Rise of Modern Logic: from Leibniz to Frege

Author: Dov M. Gabbay,John Woods
Publisher: Elsevier
ISBN: 9780080532875
Category: Technology & Engineering
Page: 780
View: 8777

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With the publication of the present volume, the Handbook of the History of Logic turns its attention to the rise of modern logic. The period covered is 1685-1900, with this volume carving out the territory from Leibniz to Frege. What is striking about this period is the earliness and persistence of what could be called 'the mathematical turn in logic'. Virtually every working logician is aware that, after a centuries-long run, the logic that originated in antiquity came to be displaced by a new approach with a dominantly mathematical character. It is, however, a substantial error to suppose that the mathematization of logic was, in all essentials, Frege's accomplishment or, if not his alone, a development ensuing from the second half of the nineteenth century. The mathematical turn in logic, although given considerable torque by events of the nineteenth century, can with assurance be dated from the final quarter of the seventeenth century in the impressively prescient work of Leibniz. It is true that, in the three hundred year run-up to the Begriffsschrift, one does not see a smoothly continuous evolution of the mathematical turn, but the idea that logic is mathematics, albeit perhaps only the most general part of mathematics, is one that attracted some degree of support throughout the entire period in question. Still, as Alfred North Whitehead once noted, the relationship between mathematics and symbolic logic has been an "uneasy" one, as is the present-day association of mathematics with computing. Some of this unease has a philosophical texture. For example, those who equate mathematics and logic sometimes disagree about the directionality of the purported identity. Frege and Russell made themselves famous by insisting (though for different reasons) that logic was the senior partner. Indeed logicism is the view that mathematics can be re-expressed without relevant loss in a suitably framed symbolic logic. But for a number of thinkers who took an algebraic approach to logic, the dependency relation was reversed, with mathematics in some form emerging as the senior partner. This was the precursor of the modern view that, in its four main precincts (set theory, proof theory, model theory and recursion theory), logic is indeed a branch of pure mathematics. It would be a mistake to leave the impression that the mathematization of logic (or the logicization of mathematics) was the sole concern of the history of logic between 1665 and 1900. There are, in this long interval, aspects of the modern unfolding of logic that bear no stamp of the imperial designs of mathematicians, as the chapters on Kant and Hegcl make clear. Of the two, Hcgel's influence on logic is arguably the greater, serving as a spur to the unfolding of an idealist tradition in logic - a development that will be covered in a further volume, British Logic in the Nineteenth Century.