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This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.
This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.
I first became interested in Husserl and Heidegger as long ago as 1980, when as an undergraduate at the Freie Universitat Berlin I studied the books by Professor Ernst Tugendhat. Tugendhat's at tempt to bring together analytical and continental philosophy has never ceased to fascinate me, and even though in more recent years other influences have perhaps been stronger, I should like to look upon the present study as still being indebted to Tugendhat's initial incentive. It was my good fortune that for personal reasons I had to con tinue my academic training from 1981 onwards in Finland. Even though Finland is a stronghold of analytical philosophy, it also has a tradition of combining continental and Anglosaxon philosophical thought. Since I had already admired this line of work in Tugendhat, it is hardly surprising that once in Finland I soon became impressed by Professor Jaakko Hintikka's studies on Husserl and intentionality, and by Professor Georg Henrik von Wright's analytical hermeneu tics. While the latter influence has-at least in part-led to a book on the history of hermeneutics, the former influence has led to the present work. My indebtedness to Professor Hintikka is enormous. Not only is the research reported here based on his suggestions, but Hintikka has also commented extensively on different versions of the manuscript, helped me to make important contacts, found a publisher for me, and-last but not least-was a never failing source of encouragement.
Intended for both the student and the professional, this work addresses the art and science of decision-making. It presents a very practical approach to decision-making that has a sound theoretical foundation, known as the analytic hierarchy process.
Hailed as “lucid and magisterial” by The Observer, this book is universally acclaimed as the outstanding one-volume work on the subject of Western philosophy. Considered to be one of the most important philosophical works of all time, the History of Western Philosophy is a dazzlingly unique exploration of the ideologies of significant philosophers throughout the ages—from Plato and Aristotle through to Spinoza, Kant and the twentieth century. Written by a man who changed the history of philosophy himself, this is an account that has never been rivaled since its first publication over sixty years ago. Since its first publication in 1945, Lord Russell’s A History of Western Philosophy is still unparalleled in its comprehensiveness, its clarity, its erudition, its grace, and its wit. In seventy-six chapters he traces philosophy from the rise of Greek civilization to the emergence of logical analysis in the twentieth century. Among the philosophers considered are: Pythagoras, Heraclitus, Parmenides, Empedocles, Anaxagoras, the Atomists, Protagoras, Socrates, Plato, Aristotle, the Cynics, the Sceptics, the Epicureans, the Stoics, Plotinus, Ambrose, Jerome, Augustine, Benedict, Gregory the Great, John the Scot, Aquinas, Duns Scotus, William of Occam, Machiavelli, Erasmus, More, Bacon, Hobbes, Descartes, Spinoza, Leibniz, Locke, Berkeley, Hume, Rousseau, Kant, Hegel, Schopenhauer, Nietzsche, the Utilitarians, Marx, Bergson, James, Dewey, and lastly the philosophers with whom Lord Russell himself is most closely associated—Cantor, Frege, and Whitehead, coauthor with Russell of the monumental Principia Mathematica.
This book provides a comprehensive examination of the use of MPEG-2, MPEG-4, MPEG-7, MPEG-21, and MPEG-A standards, providing a detailed reference to their application. In this book, the authors address five leading MPEG standards: MPEG-2, MPEG-4, MPEG-7, MPEG-21, and MPEG-A, focusing not only on the standards themselves, but specifically upon their application (e.g. for broadcasting media, personalised advertising and news, multimedia collaboration, digital rights management, resource adaptation, digital home systems, and so on); including MPEG cross-breed applications. In the evolving digital multimedia landscape, this book provides comprehensive coverage of the key MPEG standards used for generation and storage, distribution and dissemination, and delivery of multimedia data to various platforms within a wide variety of application domains. It considers how these MPEG standards may be used, the context of their use, and how supporting and complementary technologies and the standards interact and add value to each other. Key Features: Integrates the application of five popular MPEG standards (MPEG-2, MPEG-4, MPEG-7, MPEG-21, and MPEG-A) into one single volume, including MPEG cross-breed applications Up-to-date coverage of the field based on the latest versions of the five MPEG standards Opening chapter provides overviews of each of the five MPEG standards Contributions from leading MPEG experts worldwide Includes an accompanying website with supporting material (www.wiley.com/go/angelides_mpeg) This book provides an invaluable reference for researchers, practitioners, CTOs, design engineers, and developers. Postgraduate students taking MSc, MRes, MPhil and PhD courses in computer science and engineering, IT consultants, and system developers in the telecoms, broadcasting and publishing sectors will also find this book of interest.
The Cambridge History of Western Music Theory is the first comprehensive history of Western music theory to be published in the English language. A collaborative project by leading music theorists and historians, the volume traces the rich panorama of music-theoretical thought from the Ancient Greeks to the present day. Recognizing the variety and complexity of music theory as an historical subject, the volume has been organized within a flexible framework. Some chapters are defined chronologically within a restricted historical domain, whilst others are defined conceptually and span longer historical periods. Together the thirty-one chapters present a synthetic overview of the fascinating and complex subject that is historical music theory. Richly enhanced with illustrations, graphics, examples and cross-citations as well as being thoroughly indexed and supplemented by comprehensive bibliographies of the most important primary and secondary literature, this book will be an invaluable resource for students and scholars alike.

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