From Dedekind to Gödel

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Publisher : Springer Science & Business Media
ISBN 13 : 9401584788
Total Pages : 585 pages
Book Rating : 4.4/5 (15 download)

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Book Synopsis From Dedekind to Gödel by : Jaakko Hintikka

Download or read book From Dedekind to Gödel written by Jaakko Hintikka and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 585 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discussions of the foundations of mathematics and their history are frequently restricted to logical issues in a narrow sense, or else to traditional problems of analytic philosophy. From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics illustrates the much greater variety of the actual developments in the foundations during the period covered. The viewpoints that serve this purpose included the foundational ideas of working mathematicians, such as Kronecker, Dedekind, Borel and the early Hilbert, and the development of notions like model and modelling, arbitrary function, completeness, and non-Archimedean structures. The philosophers discussed include not only the household names in logic, but also Husserl, Wittgenstein and Ramsey. Needless to say, such logically-oriented thinkers as Frege, Russell and Gödel are not entirely neglected, either. Audience: Everybody interested in the philosophy and/or history of mathematics will find this book interesting, giving frequently novel insights.

From Frege to Gödel

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Author :
Publisher : Harvard University Press
ISBN 13 : 9780674324497
Total Pages : 684 pages
Book Rating : 4.3/5 (244 download)

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Book Synopsis From Frege to Gödel by : Jean van Heijenoort

Download or read book From Frege to Gödel written by Jean van Heijenoort and published by Harvard University Press. This book was released on 1967 with total page 684 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gathered together here are the fundamental texts of the great classical period in modern logic. A complete translation of Gottlob Frege’s Begriffsschrift—which opened a great epoch in the history of logic by fully presenting propositional calculus and quantification theory—begins the volume, which concludes with papers by Herbrand and by Gödel.

An Introduction to Gödel's Theorems

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Publisher : Cambridge University Press
ISBN 13 : 1139465937
Total Pages : 376 pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis An Introduction to Gödel's Theorems by : Peter Smith

Download or read book An Introduction to Gödel's Theorems written by Peter Smith and published by Cambridge University Press. This book was released on 2007-07-26 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt: In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.

Frege, Dedekind, and Peano on the Foundations of Arithmetic (Routledge Revivals)

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Publisher : Routledge
ISBN 13 : 113672107X
Total Pages : 115 pages
Book Rating : 4.1/5 (367 download)

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Book Synopsis Frege, Dedekind, and Peano on the Foundations of Arithmetic (Routledge Revivals) by : Donald Gillies

Download or read book Frege, Dedekind, and Peano on the Foundations of Arithmetic (Routledge Revivals) written by Donald Gillies and published by Routledge. This book was released on 2013-01-11 with total page 115 pages. Available in PDF, EPUB and Kindle. Book excerpt: First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic.

Contingent Computation

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Publisher : Rowman & Littlefield
ISBN 13 : 1786606097
Total Pages : 249 pages
Book Rating : 4.7/5 (866 download)

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Book Synopsis Contingent Computation by : M. Beatrice Fazi

Download or read book Contingent Computation written by M. Beatrice Fazi and published by Rowman & Littlefield. This book was released on 2018-11-15 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: In Contingent Computation, M. Beatrice Fazi offers a new theoretical perspective through which we can engage philosophically with computing. The book proves that aesthetics is a viable mode of investigating contemporary computational systems. It does so by advancing an original conception of computational aesthetics that does not just concern art made by or with computers, but rather the modes of being and becoming of computational processes. Contingent Computation mobilises the philosophies of Gilles Deleuze and Alfred North Whitehead in order to address aesthetics as an ontological study of the generative potential of reality. Through a novel philosophical reading of Gödel’s incompleteness theorems and of Turing’s notion of incomputability, Fazi finds this potential at the formal heart of computational systems, and argues that computation is a process of determining indeterminacy. This indeterminacy, which is central to computational systems, does not contradict their functionality. Instead, it drives their very operation, albeit in a manner that might not always fit with the instrumental, representational and cognitivist purposes that we have assigned to computing.

From Frege to Gödel

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Author :
Publisher : Harvard University Press
ISBN 13 : 0674257243
Total Pages : 684 pages
Book Rating : 4.6/5 (742 download)

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Book Synopsis From Frege to Gödel by : Jean van Heijenoort

Download or read book From Frege to Gödel written by Jean van Heijenoort and published by Harvard University Press. This book was released on 2002-01-15 with total page 684 pages. Available in PDF, EPUB and Kindle. Book excerpt: The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for the first time. Modern logic, heralded by Leibniz, may be said to have been initiated by Boole, De Morgan, and Jevons, but it was the publication in 1879 of Gottlob Frege’s Begriffsschrift that opened a great epoch in the history of logic by presenting, in full-fledged form, the propositional calculus and quantification theory. Frege’s book, translated in its entirety, begins the present volume. The emergence of two new fields, set theory and foundations of mathematics, on the borders of logic, mathematics, and philosophy, is depicted by the texts that follow. Peano and Dedekind illustrate the trend that led to Principia Mathematica. Burali-Forti, Cantor, Russell, Richard, and König mark the appearance of the modern paradoxes. Hilbert, Russell, and Zermelo show various ways of overcoming these paradoxes and initiate, respectively, proof theory, the theory of types, and axiomatic set theory. Skolem generalizes Löwenheim’s theorem, and he and Fraenkel amend Zermelo’s axiomatization of set theory, while von Neumann offers a somewhat different system. The controversy between Hubert and Brouwer during the twenties is presented in papers of theirs and in others by Weyl, Bernays, Ackermann, and Kolmogorov. The volume concludes with papers by Herbrand and by Gödel, including the latter’s famous incompleteness paper. Of the forty-five contributions here collected all but five are presented in extenso. Those not originally written in English have been translated with exemplary care and exactness; the translators are themselves mathematical logicians as well as skilled interpreters of sometimes obscure texts. Each paper is introduced by a note that sets it in perspective, explains its importance, and points out difficulties in interpretation. Editorial comments and footnotes are interpolated where needed, and an extensive bibliography is included.

Reflections on Kurt Gödel

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Publisher : MIT Press
ISBN 13 : 9780262730877
Total Pages : 366 pages
Book Rating : 4.7/5 (38 download)

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Book Synopsis Reflections on Kurt Gödel by : Hao Wang

Download or read book Reflections on Kurt Gödel written by Hao Wang and published by MIT Press. This book was released on 1990-03-14 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: Newton/Descartes. Einstein/Gödel. The seventeenth century had its scientific and philosophical geniuses. Why shouldn't ours have them as well? Kurt Gödel was indisputably one of the greatest thinkers of our time, and in this first extended treatment of his life and work, Hao Wang, who was in close contact with Gödel in his last years, brings out the full subtlety of Gödel's ideas and their connection with grand themes in the history of mathematics and philosophy. The subjects he covers include the completeness of elementary logic, the limits of formalization, the problem of evidence, the concept of set, the philosophy of mathematics, time, and relativity theory, metaphysics and religion, as well as general ideas on philosophy as a worldview. Wang, whose reflections on his colleague also serve to clarify his own philosophical thoughts, distinguishes his ideas from those of Gödel's and on points of agreement develops Gödel's views further. The book provides a generous array of information on and interpretation of the two main phases of Gödel's career - the years between 1924 and 1939 at the University of Vienna, which were marked by intense mathematical creativity, and the period from 1940 to his death in 1978, during which he was affiliated with the Institute for Advanced Studies in Princeton, a time in which Gödel's interests steadily shifted from questions of logic to metaphysics. And it also examines Gödel's relations with the Vienna Circle, his philosophical differences with Carnap and Wittgenstein, the intimate and mutually fruitful friendship with Einstein, and the periodic bouts of depression for which Gödel was hospitalized a number of times over the course of his life. A Bradford Book.

Philosophy and Logic In Search of the Polish Tradition

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Publisher : Springer Science & Business Media
ISBN 13 : 9781402017216
Total Pages : 332 pages
Book Rating : 4.0/5 (172 download)

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Book Synopsis Philosophy and Logic In Search of the Polish Tradition by : Jaakko Hintikka

Download or read book Philosophy and Logic In Search of the Polish Tradition written by Jaakko Hintikka and published by Springer Science & Business Media. This book was released on 2003-11-30 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains papers on truth, logic, semantics, and history of logic and philosophy. These papers are dedicated to Jan Wolenski to honor his 60th birthday. Jan Wolenski is professor of philosophy at the Department of Philosophy of the Jagiellonian University in Cracow, Poland. He is likely to be the most well-known Polish philosopher of this time, best known for his work on the history of the philosophy and logic of the Lvov-Warsaw School.

Against the Current

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Author :
Publisher : Walter de Gruyter
ISBN 13 : 3110322005
Total Pages : 471 pages
Book Rating : 4.1/5 (13 download)

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Book Synopsis Against the Current by : Guillermo E. Rosado Haddock

Download or read book Against the Current written by Guillermo E. Rosado Haddock and published by Walter de Gruyter. This book was released on 2013-05-02 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present collection of seventeen papers, most of them already published in international philosophical journals, deals both with issues in the philosophy of logic, the philosophy of mathematics, the philosophy of language and epistemology. The first part contains critical assessments and somewhat deviant renderings of the work of two seminal philosophers, Frege and Husserl, as well as of the young Carnap and Kripke. The second part contains analyses of central issues in the philosophy of logic, the philosophy of mathematics and semantics, including arguments on behalf of Platonism in the philosophy of mathematics, a defense of second-order logic, a new definition of analyticity, a sketch of a semantics for mathematical statements and a critique of Kripke’s possible world semantics for modal logic.

Wittgenstein and the Creativity of Language

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Publisher : Springer
ISBN 13 : 1137472545
Total Pages : 519 pages
Book Rating : 4.1/5 (374 download)

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Book Synopsis Wittgenstein and the Creativity of Language by : Sebastian Sunday Grève

Download or read book Wittgenstein and the Creativity of Language written by Sebastian Sunday Grève and published by Springer. This book was released on 2016-01-26 with total page 519 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is the first to focus on a particular complex of questions that have troubled Wittgenstein scholarship since its very beginnings. The authors re-examine Wittgenstein’s fundamental insights into the workings of human linguistic behaviour, its creative extensions and its philosophical capabilities, as well as his creative use of language. It offers insight into a variety of topics including painting, politics, literature, poetry, literary theory, mathematics, philosophy of language, aesthetics and philosophical methodology.

Kurt Gödel

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Publisher : Cambridge University Press
ISBN 13 : 1139487752
Total Pages : 384 pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis Kurt Gödel by : Solomon Feferman

Download or read book Kurt Gödel written by Solomon Feferman and published by Cambridge University Press. This book was released on 2010-04-19 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: Kurt Gödel (1906–1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating Gödel's centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of Gödel's writings are among the topics covered. Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible sets.

The History of Continua

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Publisher : Oxford University Press
ISBN 13 : 0192537490
Total Pages : 320 pages
Book Rating : 4.1/5 (925 download)

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Book Synopsis The History of Continua by : Stewart Shapiro

Download or read book The History of Continua written by Stewart Shapiro and published by Oxford University Press. This book was released on 2020-12-01 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical and philosophical thought about continuity has changed considerably over the ages. Aristotle insisted that continuous substances are not composed of points, and that they can only be divided into parts potentially. There is something viscous about the continuous. It is a unified whole. This is in stark contrast with the prevailing contemporary account, which takes a continuum to be composed of an uncountably infinite set of points. This vlume presents a collective study of key ideas and debates within this history. The opening chapters focus on the ancient world, covering the pre-Socratics, Plato, Aristotle, and Alexander. The treatment of the medieval period focuses on a (relatively) recently discovered manuscript, by Bradwardine, and its relation to medieval views before, during, and after Bradwardine's time. In the so-called early modern period, mathematicians developed the calculus and, with that, the rise of infinitesimal techniques, thus transforming the notion of continuity. The main figures treated here include Galileo, Cavalieri, Leibniz, and Kant. In the early party of the nineteenth century, Bolzano was one of the first important mathematicians and philosophers to insist that continua are composed of points, and he made a heroic attempt to come to grips with the underlying issues concerning the infinite. The two figures most responsible for the contemporary orthodoxy regarding continuity are Cantor and Dedekind. Each is treated in an article, investigating their precursors and influences in both mathematics and philosophy. A new chapter then provides a lucid analysis of the work of the mathematician Paul Du Bois-Reymond, to argue for a constructive account of continuity, in opposition to the dominant Dedekind-Cantor account. This leads to consideration of the contributions of Weyl, Brouwer, and Peirce, who once dubbed the notion of continuity "the master-key which . . . unlocks the arcana of philosophy". And we see that later in the twentieth century Whitehead presented a point-free, or gunky, account of continuity, showing how to recover points as a kind of "extensive abstraction". The final four chapters each focus on a more or less contemporary take on continuity that is outside the Dedekind-Cantor hegemony: a predicative approach, accounts that do not take continua to be composed of points, constructive approaches, and non-Archimedean accounts that make essential use of infinitesimals.

Language, Truth and Logic in Mathematics

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Publisher : Springer Science & Business Media
ISBN 13 : 9401720452
Total Pages : 339 pages
Book Rating : 4.4/5 (17 download)

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Book Synopsis Language, Truth and Logic in Mathematics by : Jaakko Hintikka

Download or read book Language, Truth and Logic in Mathematics written by Jaakko Hintikka and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: One can distinguish, roughly speaking, two different approaches to the philosophy of mathematics. On the one hand, some philosophers (and some mathematicians) take the nature and the results of mathematicians' activities as given, and go on to ask what philosophical morals one might perhaps find in their story. On the other hand, some philosophers, logicians and mathematicians have tried or are trying to subject the very concepts which mathematicians are using in their work to critical scrutiny. In practice this usually means scrutinizing the logical and linguistic tools mathematicians wield. Such scrutiny can scarcely help relying on philosophical ideas and principles. In other words it can scarcely help being literally a study of language, truth and logic in mathematics, albeit not necessarily in the spirit of AJ. Ayer. As its title indicates, the essays included in the present volume represent the latter approach. In most of them one of the fundamental concepts in the foundations of mathematics and logic is subjected to a scrutiny from a largely novel point of view. Typically, it turns out that the concept in question is in need of a revision or reconsideration or at least can be given a new twist. The results of such a re-examination are not primarily critical, however, but typically open up new constructive possibilities. The consequences of such deconstructions and reconstructions are often quite sweeping, and are explored in the same paper or in others.

Philosophy in an Age of Science

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Publisher : Harvard University Press
ISBN 13 : 0674050134
Total Pages : 672 pages
Book Rating : 4.6/5 (74 download)

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Book Synopsis Philosophy in an Age of Science by : Hilary Putnam

Download or read book Philosophy in an Age of Science written by Hilary Putnam and published by Harvard University Press. This book was released on 2012-04-17 with total page 672 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hilary Putnam's unceasing self-criticism has led to the frequent changes of mind he is famous for, but his thinking is also marked by considerable continuity. A simultaneous interest in science and ethicsÑunusual in the current climate of contentionÑhas long characterized his thought. In Philosophy in an Age of Science, Putnam collects his papers for publicationÑhis first volume in almost two decades. Mario De Caro and David Macarthur's introduction identifies central themes to help the reader negotiate between Putnam past and Putnam present: his critique of logical positivism; his enduring aspiration to be realist about rational normativity; his anti-essentialism about a range of central philosophical notions; his reconciliation of the scientific worldview and the humanistic tradition; and his movement from reductive scientific naturalism to liberal naturalism. Putnam returns here to some of his first enthusiasms in philosophy, such as logic, mathematics, and quantum mechanics. The reader is given a glimpse, too, of ideas currently in development on the subject of perception. Putnam's work, contributing to a broad range of philosophical inquiry, has been said to represent a Òhistory of recent philosophy in outline.Ó Here it also delineates a possible future.

Mathematical Knowledge, Objects and Applications

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Publisher : Springer Nature
ISBN 13 : 3031216555
Total Pages : 404 pages
Book Rating : 4.0/5 (312 download)

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Book Synopsis Mathematical Knowledge, Objects and Applications by : Carl Posy

Download or read book Mathematical Knowledge, Objects and Applications written by Carl Posy and published by Springer Nature. This book was released on 2023-05-05 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a survey of a number of the major issues in the philosophy of mathematics, such as ontological questions regarding the nature of mathematical objects, epistemic questions about the acquisition of mathematical knowledge, and the intriguing riddle of the applicability of mathematics to the physical world. Some of these issues go back to the nascent years of mathematics itself, others are just beginning to draw the attention of scholars. In addressing these questions, some of the papers in this volume wrestle with them directly, while others use the writings of philosophers such as Hume and Wittgenstein to approach their problems by way of interpretation and critique. The contributors include prominent philosophers of science and mathematics as well as promising younger scholars. The volume seeks to share the concerns of philosophers of mathematics with a wider audience and will be of interest to historians, mathematicians and philosophers alike.

New Makers of Modern Culture

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Publisher : Routledge
ISBN 13 : 113409454X
Total Pages : 905 pages
Book Rating : 4.1/5 (34 download)

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Book Synopsis New Makers of Modern Culture by : Wintle Justin

Download or read book New Makers of Modern Culture written by Wintle Justin and published by Routledge. This book was released on 2013-05-13 with total page 905 pages. Available in PDF, EPUB and Kindle. Book excerpt: New Makers of Modern Culture is the successor to the classic reference works Makers of Modern Culture and Makers of Nineteenth-Century Culture, published by Routledge in the early 1980s. The set was extremely successful and continues to be used to this day, due to the high quality of the writing, the distinguished contributors, and the cultural sensitivity shown in the selection of those individuals included. New Makers of Modern Culture takes into full account the rise and fall of reputation and influence over the last twenty-five years and the epochal changes that have occurred: the demise of Marxism and the collapse of the Soviet Union; the rise and fall of postmodernism; the eruption of Islamic fundamentalism; the triumph of the Internet. Containing over eight hundred essay-style entries, and covering the period from 1850 to the present, New Makers includes artists, writers, dramatists, architects, philosophers, anthropologists, scientists, sociologists, major political figures, composers, film-makers and many other culturally significant individuals and is thoroughly international in its purview. Next to Karl Marx is Bob Marley, next to John Ruskin is Salmon Rushdie, alongside Darwin is Luigi Dallapiccola, Deng Xiaoping runs shoulders with Jacques Derrida, Julia Kristeva with Kropotkin. Once again, Wintle has enlisted the services of many distinguished writers and leading academics, such as Sam Beer, Bernard Crick, Edward Seidensticker and Paul Preston. In a few cases, for example Michael Holroyd and Philip Larkin, contributors are themselves the subject of entries. With its global reach, New Makers of Modern Culture provides a multi-voiced witness of the contemporary thinking world. The entries carry short bibliographies and there is thorough cross-referencing. There is an index of names and key terms.

New Makers of Modern Culture

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Publisher : Routledge
ISBN 13 : 1136768823
Total Pages : 1812 pages
Book Rating : 4.1/5 (367 download)

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Book Synopsis New Makers of Modern Culture by : Justin Wintle

Download or read book New Makers of Modern Culture written by Justin Wintle and published by Routledge. This book was released on 2016-04-22 with total page 1812 pages. Available in PDF, EPUB and Kindle. Book excerpt: New Makers of Modern Culture will be widely acquired by both higher education and public libraries. Bibliographies are attached to entries and there is thorough cross- referencing.