Proofs and Refutations

Proofs and Refutations
Title Proofs and Refutations PDF eBook
Author Imre Lakatos
Publisher Cambridge University Press
Pages
Release 2015-10-08
Genre Science
ISBN 1316425339

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Imre Lakatos's Proofs and Refutations is an enduring classic, which has never lost its relevance. Taking the form of a dialogue between a teacher and some students, the book considers various solutions to mathematical problems and, in the process, raises important questions about the nature of mathematical discovery and methodology. Lakatos shows that mathematics grows through a process of improvement by attempts at proofs and critiques of these attempts, and his work continues to inspire mathematicians and philosophers aspiring to develop a philosophy of mathematics that accounts for both the static and the dynamic complexity of mathematical practice. With a specially commissioned Preface written by Paolo Mancosu, this book has been revived for a new generation of readers.

Proofs and Refutations

Proofs and Refutations
Title Proofs and Refutations PDF eBook
Author Imre Lakatos
Publisher Cambridge University Press
Pages 197
Release 2015-10-15
Genre Mathematics
ISBN 1107113466

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This influential book discusses the nature of mathematical discovery, development, methodology and practice, forming Imre Lakatos's theory of 'proofs and refutations'.

Proofs and Refutations

Proofs and Refutations
Title Proofs and Refutations PDF eBook
Author Imre Lakatos
Publisher
Pages
Release 1987
Genre
ISBN

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Proofs and Refutations

Proofs and Refutations
Title Proofs and Refutations PDF eBook
Author Imre Lakatos
Publisher Cambridge University Press
Pages 188
Release 1976-01-01
Genre Science
ISBN 9780521290388

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Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.

Proofs and Refutations

Proofs and Refutations
Title Proofs and Refutations PDF eBook
Author Imre Lakatos
Publisher
Pages
Release 1976
Genre Proof theory
ISBN

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In the Light of Logic

In the Light of Logic
Title In the Light of Logic PDF eBook
Author Solomon Feferman
Publisher Oxford University Press
Pages 353
Release 1998-11-19
Genre Philosophy
ISBN 0195359836

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In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantor elaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues that the freedom provided by Cantorian set theory was purchased at a heavy philosophical price, namely adherence to a form of mathematical platonism that is difficult to support. Beginning with a previously unpublished lecture for a general audience, Deciding the Undecidable, Feferman examines the famous list of twenty-three mathematical problems posed by David Hilbert, concentrating on three problems that have most to do with logic. Other chapters are devoted to the work and thought of Kurt Gödel, whose stunning results in the 1930s on the incompleteness of formal systems and the consistency of Cantors continuum hypothesis have been of utmost importance to all subsequent work in logic. Though Gödel has been identified as the leading defender of set-theoretical platonism, surprisingly even he at one point regarded it as unacceptable. In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part--if not all--of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, Is Cantor Necessary?, is answered with a resounding no. This volume of important and influential work by one of the leading figures in logic and the foundations of mathematics is essential reading for anyone interested in these subjects.

Encyclopedia of Educational Theory and Philosophy

Encyclopedia of Educational Theory and Philosophy
Title Encyclopedia of Educational Theory and Philosophy PDF eBook
Author D. C. Phillips
Publisher SAGE Publications
Pages 952
Release 2014-05-19
Genre Education
ISBN 1483364747

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Education is a field sometimes beset by theories-of-the-day and with easy panaceas that overpromise the degree to which they can alleviate pressing educational problems. The two-volume Encyclopedia of Educational Theory and Philosophy introduces readers to theories that have stood the test of time and those that have provided the historical foundation for the best of contemporary educational theory and practice. Drawing together a team of international scholars, this invaluable reference examines the global landscape of all the key theories and the theorists behind them and presents them in the context needed to understand their strengths and weaknesses. In addition to interpretations of long-established theories, this work offers essays on cutting-edge research and concise, to-the-point definitions of key concepts, ideas, schools, and figures. Features: Over 300 signed entries by trusted experts in the field are organized into two volumes and overseen by a distinguished General Editor and an international Editorial Board. Entries are followed by cross references and further reading suggestions. A Chronology of Theory within the field of education highlights developments over the centuries; a Reader’s Guide groups entries thematically, and a master Bibliography facilitates further study. The Reader’s Guide, detailed index, and cross references combine for strong search-and-browse capabilities in the electronic version. Available in a choice of print or electronic formats, Encyclopedia of Educational Theory and Philosophy is an ideal reference for anyone interested in the roots of contemporary educational theory.