Why Prove it Again? Alternative Proofs in Mathematical Practice
This monograph considers several well-known mathematical theorems and asks the question, “Why prove it again?” while examining alternative proofs. It explores the different rationales mathematicians may have for pursuing and presenting n
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Why Prove it Again? Alternative Proofs in Mathematical Practice
John W. Dawson, Jr.
Why Prove it Again? Alternative Proofs in Mathematical Practice
with the assistance of Bruce S. Babcock and with a chapter by Steven H. Weintraub
John W. Dawson, Jr. Penn State York York, PA, USA
ISBN 978-3-319-17367-2 ISBN 978-3-319-17368-9 (eBook) DOI 10.1007/978-3-319-17368-9 Library of Congress Control Number: 2015936605 Mathematics Subject Classification (2010): 00A35, 00A30, 01A05, 03A05, 03F99 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)
To Solomon Feferman, friend and mentor, who suggested I write this book
Preface
This book is an elaboration of themes that I previously explored in my paper “Why do mathematicians re-prove theorems?” (Dawson 2006). It addresses two basic questions concerning mathematical practice: 1. What rationales are there for presenting new proofs of previously established mathematical results? and 2. How do mathematicians judge whether two proofs of a given result are essentially different? The discovery and presentation of new proofs of results already proven by other means has been a salient feature of mathematical practice since ancient times.1 Yet historians and philosophers of mathematics have paid surprisingly little attention to that phenomenon, and mathematical logicians have so far made little progress in developing formal criteria for distinguishing different proofs from one another, or for recognizing when proofs are substantially the same. A number of books and papers have compared alternative proofs of particular theorems (see the references in succeeding chapters), but no extended general study of the roles of alternative proofs in mathematical practice seems hitherto to have been undertaken. Cons
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